Mock Modular Mathieu Moonshine Modules

Cheng et al. Mathematical Sciences (2015) 2:13 DOI 10.1186/s40687-015-0034-9

RESEARCH Open Access Mock modular Mathieu moonshine modules

Miranda C N Cheng1*, Xi Dong2, John F R Duncan3, Sarah Harrison2, Shamit Kachru2 and Timm Wrase2

*Correspondence: [email protected] Abstract 1 Institute of Physics We construct super vertex operator algebras which lead to modules for moonshine and Korteweg‑de Vries Institute for Mathematics, relations connecting the four smaller sporadic simple Mathieu groups with distin- University of Amsterdam, guished mock modular forms. Starting with an orbifold of a free fermion theory, any Amsterdam, The Netherlands subgroup of Co0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimen- Full list of author information N = 4 is available at the end of the sional representation commutes with a certain superconformal algebra. Simi- article larly, any subgroup of Co0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N = 2 superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the N = 4 (resp. N = 2) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensionalZ -graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.

1 Introduction The investigation of moonshine connecting modular objects, sporadic groups, and 2d conformal field theories has been revitalized in recent years by the discovery of several new classes of examples. While monstrous moonshine [3, 15, 21, 30, 36, 37, 44] remains the best understood and prototypical case, a new class of umbral moonshines tying mock modular forms to automorphism groups of Niemeier lattices has recently been uncovered [13, 14] (cf. also [6]). The best studied example, and the first to be discovered, involves the group M24 and was discovered through the study of the elliptic genus of K3 [35]. The twining functions have been constructed in 4[ , 31, 38, 39] and were proved to be the graded characters of an infinite-dimensional M24-module in [43]. Steps towards a better and deeper understanding of this mock modular moonshine can be found in [7, 8, 18, 40–42, 46, 56, 57, 59–61], and particularly in [9], where the importance of K3 surface geometry for all cases of umbral moonshine is elucidated. Possible connections to space- time physics in string theory have been discussed in [5, 47–49, 63]. Evidence for a deep connection between monstrous and umbral moonshine has appeared in [25, 55].

© 2015 Cheng et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Cheng et al. Mathematical Sciences (2015) 2:13 Page 2 of 89

In none of these cases, however, has a connection to an underlying conformal field theorya (whose Hilbert space furnishes the underlying module) been established. The goal of this paper is to provide first examples of mock modular moonshine for sporadic simple groups G, where the underlying G-module can be explicitly constructed in the state space of a simple and soluble conformal field theory. Our starting point is the Conway module sketched in [36], studied in detail in [24], and revisited recently in [27]. The original construction was in terms of a supersymmetric theory of bosons on the E8 root lattice, but this has the drawback of obscuring the true Z symmetries of the model. In [24], a different formulation of the same theory, as a 2 orbifold of the theory of 24 free chiral fermions, was introduced. A priori, the theory N 1 has a Spin(24) symmetry. However, one can also view this theory as an = supercon- N 1 formal field theory. The choice of = structure breaks the Spin(24) symmetry to a N 1 subgroup. In [24] it was shown that the subgroup preserving the natural choice of = structure is precisely the Conway group Co0, a double cover of the sporadic group Co1. In [24] it was shown that this action can be used to attach a normalized principal modulus (i.e. normalized Hauptmodul) for a genus zero group to every element of Co0. In this paper, we show that generalizations of the basic strategy of [24, 27] can be used to construct a wide variety of new examples of mock modular moonshine. Instead of N 1 choosing an = superconformal structure, we choose larger extended chiral algebras A. The subgroup of Spin(24) that commutes with a given choice can be determined by simple geometric considerations; in the cases of interest to us, it will be a subgroup that preserves point-wise a 2-plane or a 3-plane in the 24 dimensional representation of Co0, or equivalently, a subgroup that acts trivially on two or three of the free fermions in some basis. In the rest of the paper, we will use 24 to denote the unique non-trivial 24-dimensional representation of Co0, and use the term n-plane to refer to a n-dimensional subspace in 24. It is natural to ask about the role in moonshine, or geometry, of n-planes in 24 for other values of n. One of the inspirations for our analysis here is the recent result of Gab- erdiel–Hohenegger–Volpato [40] which indicates the importance of 4-planes in 24 for non-linear K3 sigma models. The relationship between their results and the Co0-module considered here is studied in [25], where connections to umbral moonshine for various higher n are also established. We refer the reader to Sect. 9, or the recent articles [1, 10], n 1 for a discussion of the interesting case that = . A N 4 N 2 In this work we will focus on the cases where is an = or = superconformal algebra, though other possibilities exist. In the first case, we demonstrate that any subgroup of Co0 that preserves a 3-plane in the 24-dimensional representation can stabilise N 4 an = structure. The groups that arise are discussed in e.g. Chapter 10 of the book by Conway and Sloane [16]. They include in particular the Mathieu groups M22 and M11. In N 2 the second case, where the group need only fix a 2-plane in order to preserve an = superconformal algebra, there are again many possibilities (again, see [16]), including the larger Mathieu groups M23 and M12. Note that the larger the superconformal algebra we wish to preserve, the smaller the global symmetry group is. Corresponding to cer- N 0, 1, 2, 4 tain specific choices of the = algebras, we have the global symmetry groups Spin(24) Co M M ⊃ 0 ⊃ 23 ⊃ 22. Cheng et al. Mathematical Sciences (2015) 2:13 Page 3 of 89

Co N 4 We should stress again that there are other 0 subgroups that preserve = resp. N 2 24 = superconformal algebras arising from 3-planes resp. 2-planes in . Some examples are: the group U4(3) for the former case, and the McLaughlin (McL) and Hig- man–Sims (HS) sporadic groups, and also U6(2) for the latter case. However, only for the Mathieu groups do the twined partition functions of the module display uniformly a special property, which we regard as an essential feature of the moonshine phenomena. Namely, all the mock modular forms obtained via twining by elements of the Mathieu groups are encoded in Jacobi forms that are constant in the elliptic variable, in the limit as the modular variable tends to any cusp other than the infinite one. This property also holds for the Jacobi forms of the Mathieu moonshine mentioned above, and may be regarded as a counter-part to the genus zero property of monstrous moonshine, as we explain in more detail in Sect. 8. The importance of this property is its predictive power: it allows us to write down trace functions for the actions of Mathieu group elements with little more information than a certain fixed multiplier system, and the levels of the functions we expect to find. A priori these are just guesses, but the constructions we present here verify their validity. This may be compared to the predictive power of the genus zero property of monstrous R moonshine: if Ŵquantum gravity is discussed in [29]. The organization of the paper is as follows. We begin in Sect. 2 with a review of the module discussed in [27]. In Sect. 3, we describe methods for endowing this module N 4 N 2 with = and = structure. In Sect. 4, we discuss what this does to the manifest symmetry group of the model, reducing the symmetry from Co0 to a variety of other possible groups which preserve a 3-plane (respectively 2-plane) in the 24 of Co0. In Sects. 4 and 5, we discuss the action of these Co0-subgroups on the modules and compute the corresponding twining functions. We identify M23, M22, McL, HS, U6(2) and U4(3) as some of the most interesting Co0-subgroups preserving some extended superconformal algebra. In Sects. 6 and 7, we discuss in some detail the decomposition of the graded partition function of our chiral conformal field theory N 4 N 2 into characters of irreducible representations of the = and = superconformal algebras. In Sect. 8, we discuss the special property we require from a moonshine twining function, and show how this property singles out the Mathieu groups Cheng et al. Mathematical Sciences (2015) 2:13 Page 4 of 89

in our setup. We close with a discussion in Sect. 9. The appendices contain a number of tables: character tables for the various groups we discuss, tables of coefficients of the vector-valued mock modular forms that arise as our twining functions, and tables describing the decompositions of our modules into irreducible representations of the various groups.

2 The free field theory The chiral 2d conformal field theory that will play a starring role in this paper has two i different constructions. The first is described in [37] and starts with 8 free bosons X compactified on the 8-dimensional torus given by the E8 root lattice, together with their i Z Fermi superpartners ψ . One then orbifolds by the 2 symmetry (Xi, ψi) ( Xi, ψi). (2.1) → − − Note that, in more mathematical terms, a chiral 2d conformal field theory can be understood to mean a super vertex operator algebra (usually assumed to be simple, and of CFT-type in the sense of [23]), together with a (simple) canonically-twisted module. These two spaces are referred to as the Neveu–Schwarz (NS) and Ramond (R) sectors of the theory, respectively. The compactification of free bosons on the torus defined by a lattice L manifests, in the NS sector, as the usual lattice vertex algebra construction, and their Fermi superpartners are then realized by a Clifford module, or free fermion, super vertex algebra, where the underlying orthogonal space comes equipped with an isometric embedding of L. In the cases under consideration, there is a unique simple canonically-twisted module up to equivalence (cf. e.g. [24]), and hence a unique choice of R sector. The orbifold procedure is described in detail in the language of vertex algebra in [24]. In what follows, a field of dimension d is a vertex operator or intertwining operator L(0)v dv attached to a vector v in the NS or R sector, respectively, with = . A current is a field of dimension 1. A field is called primary if its corresponding vector is a highest weight for the Virasoro (Lie) algebra. A ground state is an L(0)-eigenvector of minimal eigenvalue. E N 1 The 8 construction just described has manifest = supersymmetry, in the sense that the Neveu–Schwarz and Ramond algebras act naturally on the NS and R sectors, respectively. After orbifolding we obtain a c 12 theory with no primary fields of 1 = dimension 2. The partition functions (i.e., graded dimensions) of this free field theory can easily be computed. For example, the NS sector partition function is given by 4 4 4 1 E4(τ)θ3(τ,0) θ4(τ,0) θ2(τ,0) Z E (τ) 16 16 (2.2) NS, 8 = 2 η(τ)12 + θ (τ,0)4 + θ (τ,0)4 2 4

1/2 1/2 3/2 q− 0 276q 2048q 11202q , (2.3) = + + + + +··· where E4 is the weight 4 Eisenstein series, being the theta series of the E8 lattice, 1/24 n η(τ) q n∞ 1(1 q ) is the Dedekind eta function, and θi are the Jacobi theta func- = = − q ǫ(τ) tions recorded in “Appendix A”. We have also set = and we use the shorthand e(x) e2πix notation = throughout this paper. Cheng et al. Mathematical Sciences (2015) 2:13 Page 5 of 89

One recognizes representations of the Co1 sporadic group appearing in the q-series (2.3): apart from 276, which is the minimal dimension of a faithful irreducible representation (cf. [31]), one can also observe 2048 1 276 1771, (2.4) = + + 11202 1 276 299 1771 8855, = + + + + (2.5) ··· Co0 2.Co1 In fact, this theory has a =∼ symmetry, which we call non-manifest since the action of Co0 is not obvious from the given description. Note that we sometimes use n or Z Z Z n to denote /n depending on the context. A better realization, for our purposes, was discussed in detail in [28] (cf. also [40]). The E8 orbifold theory is equivalent to a theory of 24 free chiral fermions 1, 2, ..., 24, also Z α α orbifolded by the 2 symmetry →− . This gives an alternative description of the Conway module above. The partition function from this “free fermion” point of view is more naturally written as

1 4 θ 12(τ,0) Z (τ) i . (2.6) NS,fermion = 2 η12(τ) i 2 =

This is equal to (2.2) according to non-trivial identities satisfied by theta functions. Note θ (τ,0) 0 that 1 = . N 1 The free fermion theory has a manifest Spin(24) symmetry, but not a manifest = N 1 supersymmetry. However, one can construct an = supercurrent as follows. There 212 4096 is a unique (up to scale) NS ground state, but there are = linearly independent Ramond sector ground states, which may be obtained by acting on a given fixed R sector ground state with the fermion zero modes i(0). It will be convenient to label the result- s F12 F / / ing 4096 Ramond sector ground states by vectors ˜ 2 , where ˜ 2 1 2, 1 2 . 3∈ = {− } We therefore have 4096 spin fields of dimension2 which implement the flow from the NS to the R sector. Denoting these fields asW s, one can try to find a linear combination

W csWs = (2.7) s F12 ∈˜ 2

N 1 which will serve as an = supercurrent (i.e., field whose modes generate actions of the Neveu–Schwarz and Ramond super Lie algebras). As demonstrated in [28], and as we will review in the next section, there exists a set of values cs such that the operator product expansion of W and the stress tensor T close properly, defining actions of the Neveu–Schwarz and Ramond algebras. Any choice of W breaks the Spin(24) symmetry, since the Ramond sector ground states split into two 2048-dimensional irreducible representation of Spin(24). It is proven in N 1 [28] that the subgroup of Spin(24) that stabilizes a suitably chosen = supercurrent is exactly the Conway group Co0. In brief, the method of [28] is to identify a certain ele- 12 mentary abelian subgroup of order 2 in Spin(24) (which should be regarded as a copy of the extended Golay code in Spin(24)). The action of this subgroup on the Ramond sector ground states singles out a particular choice of W, with the property that it is not Cheng et al. Mathematical Sciences (2015) 2:13 Page 6 of 89

annihilated by the zero mode of any dimension 1 field in the theory. It follows from this (cf. Proposition 4.8 of [28]) that the subgroup of Spin(24) that stabilizes W is a reductive algebraic group of dimension 0, and hence finite. On the other hand, one can show (cf. Proposition 4.7 of [28]) that this group contains Co0, by virtue of the choice of subgroup 12 2 . We obtain that the full stabilizer of W in Spin(24) is Co0 by verifying (cf. Proposition 4.9 of [28]) that Co0 is a maximal subgroup, subject to being finite. N 1 In the rest of this paper, we extend this idea as follows. Instead of choosing an = N 1 supercurrent and viewing the theory as an = super conformal field theory, we N 4 choose various other super extensions of the Virasoro algebra. We will argue that = N 2 and = superconformal presentations of the theory are in one to one correspondence with choices of subgroups of Co0 which fix a 3-plane (respectively, 2-plane) in the 24 dimensional representation. This leads us naturally to theories with various interesting symmetry groups, whose twining functions are easily computed in terms of the partition function (or elliptic genus) of the free fermion conformal field theory. These functions in turn are expressed nicely in terms of mock modular forms, and thus we establish mock modular moonshine relations for subgroups of Co0 via this family of modules.

3 The superconformal algebras We first discuss the largest superconformal algebra (SCA) we will consider, which gives N 4 rise to smaller global symmetry groups. We will construct an = SCA in the free fermion orbifold theory. Our strategy is to first construct the SU(2) fields, and act with N 1 N 4 them on an = supercurrent to generate the full = SCA. We consequently N 2 N 4 obtain actions of the = SCA by virtue of its embeddings in the = SCA. In this process, we break the Co0 symmetry group down to a proper subgroup as we will discuss N 4 N 2 in Sect. 4. We refer the reader to [32, 50, 52] for background on the = and = superconformal algebras. We start with 24 real free fermions 1, 2, ..., 24 . Picking out the first three fermions, we obtain the currents Ji: Ji iǫijk j k , i, j, k 1, 2, 3 . (3.1) =− ∈{ } They form an affine SU(2) algebra with level 2 as may be seen from their operator product expansion (OPE), 1 i Ji(z)Jj(0) δij ǫ J (0) . (3.2) ∼ z2 + z ijk k

N 1 Ji The next step is to pick an = supercurrent and act with on it. As we reviewed in N 1 Sect. 2, an = supercurrent exists in this model and may be written as a linear combination of spin fields. Moreover, it may be chosen so that its stabilizer in Spin(24) is precisely Co0. We will present a very general version of the construction now, and then N 4 extend it to find the = SCA. N 1 To write the = supercurrent explicitly, we first group the 24 real fermions into 12 complex ones and bosonize them: Cheng et al. Mathematical Sciences (2015) 2:13 Page 7 of 89

1/2 iHa ψa 2− ( 2a 1 i 2a) e , ≡ − + =∼ 1/2 iHa ψa 2− ( 2a 1 i 2a) e− , a 1, 2, ..., 12. (3.3) ¯ ≡ − − =∼ = H (H , ..., H ) N 1 In terms of the bosonic fields = 1 12 , an = supercurrent W may be written as

is H W wse · cs(p), = (3.4) s F12 ∈˜ 2

where each component of s (s1, s2, ..., s12) takes the values 1/2, and the coefficients C = ± ws belong to . We have introduced cocycle operators cs(p) to ensure that the fields with integer spins (i.e., corresponding to even parity vectors) commute with all other operators, and the fields with half integral spins (corresponding to odd parity vectors) anticommute amongst themselves. The cocycle operators depend on the zero-mode operators p which are characterized by the commutation relation

ik H ik H p, e · ke · , (3.5) = k (k , ..., k ) where = 1 12 is an arbitrary 12-tuple of complex numbers. The associativity and closure of the OPE of the “dressed” vertex operators ik H Vk e · ck(p) (3.6) = requires that

ck(p k′)ck (p) e(k, k′)ck k (p), (3.7) + ′ = + ′ where the ǫ(k, k′) satisfy the 2-cocycle condition

e(k, k′) e(k k′, k′′) e(k′, k′′)ǫ(k, k′ k′′). (3.8) + = + Moreover, in order for Vk to have the desired (anti)commutation relation, the condition k k k2k 2 e(k, k′) ( 1) · ′+ ′ e(k′, k) (3.9) = − should be imposed. An explicit description of the cocycle for a general vertex operator ik H e · may be chosen as iπk M p ck(p) e · · (3.10) = M 12 12 according to [33]. In our case, is a × matrix that has the block form 00 0 0 M 00 4 10 0 0 M 1 M 0 , M , 4 4 4  1100 (3.11) =  1 1 M  = 4 4 4 11 10      − −  1 4 4 where 4 is the × matrix with all entries set to 1. Generically, the OPE of W with itself is 1 T(0) wŴαβγ δ w W (z)W (0) ww ¯ α β γ δ(0), ∼ ¯ z3 + 4z + 96z (3.12) Cheng et al. Mathematical Sciences (2015) 2:13 Page 8 of 89

where we have defined

ws w sc s(s), (3.13) ¯ = − −

αβγ δ α β γ δ Ŵ Ŵ Ŵ Ŵ Ŵ c s(s s) 2s α 1 2s β 2s γ 2s δ , ss ss ′ 2 2 1 ′ = ′ − − − ⌈ ⌉+ ··· − ⌈ 2 ⌉ − ⌈ ⌉+ ··· − ⌈ 2 ⌉ (3.14)

αβγ δ for α<β<γ <δ. The other components ofŴ are defined by the requirement that N 1 it is totally antisymmetrized. For W to be an = supercurrent, the last terms in (3.12) must vanish, wŴαβγ δ w 0, α, β, γ , δ 1, 2, ..., 12 , (3.15) ¯ = ∀ ∈{ } and the first two terms must have the correct normalization,

ww w swscs( s) 8. ¯ = − − = (3.16) s F12 ∈˜ 2

(ws) N 1 From now on we presume to be chosen a solution such that W is an = supercurrent stabilized by Co0, as described in Sect. 2. Ji N 1 We may now act with the SU(2) currents on our = supercurrent W. In order to do this, write the SU(2) currents in (3.1) in bosonized form, 1 iH1 iH1 iH2 iπp1 iH2 iπp1 J1 e e− e e e− e− , (3.17) =−2 − +

i iH1 iH1 iH2 iπp1 iH2 iπp1 J2 e e− e e e− e− , (3.18) = 2 + +

J3 i∂H1. (3.19) = iπp Here we have included the cocycles e± 1. We now extract the singular terms of the OPEs, i Ji(z)W (0) Wi(0), (3.20) ∼−2z

where Wi are slightly modified combinations of spin fields,

is H W1 2s2wRse · cs(p), (3.21) =− s

is H W2 i 4s1s2wRse · cs(p), (3.22) = s

is H W3 i 2s1wse · cs(p), (3.23) = s Rs ( s , s , s , ..., s ) and where ≡ − 1 − 2 3 12 . Cheng et al. Mathematical Sciences (2015) 2:13 Page 9 of 89

Wi N 1 We claim that all three defined above are valid = supercurrents. This is because we may obtain, for instance, W3 from W by rotating the 1–2 plane by π, and N 1 the conditions (3.15) and (3.16), for being an = supercurrent, are invariant under SO(24) rotations. We may obtain W2 and W3 similarly. This shows that each of theW i is N 1 an = supercurrent. Furthermore, we can check using the identity (3.15) that the OPEs of the Wi are given by 8 2 2 1 Wi(z)Wj(0) δij T(0) 2iǫijk Jk (0) ∂Jk (0) , (3.24) ∼ z3 + z + z2 + z

2 ∂ W (z)Wi(0) 2i Ji(0), (3.25) ∼− z2 + z

i Ji(z)Wj(0) (δijW ǫ W ). (3.26) ∼ 2z + ijk k

This shows that W, Wi, and Ji, together with the stress tensor T, defined as

1 1 T α∂ α ∂Ha∂Ha, (3.27) =−2 α =−2 a N 4 c 12 N 1 form an = SCA with central charge = . We may recombine the four = Wi N 4 supercurrents W, into the more conventional = supercurrents 1/2 1/2 W ± 2− (W iW ) , W ± 2− i(W iW ), (3.28) 1 ≡ ± 3 2 ≡± 1 ± 2 2 2 which transform according to the representation + ¯ of SU(2). In terms of these super- N 4 c 12 currents we obtain the standard (small) = SCA with central charge = , characterized by the following set of OPEs: 6 2 1 T(z)T(0) T(0) ∂T(0), (3.29) ∼ z4 + z2 + z

3 1 T(z)W ±(0) W ±(0) ∂W ±(0), (3.30) a ∼ 2z2 a + z a

1 1 T(z)Ji(0) Ji(0) ∂Ji(0), (3.31) ∼ z2 + z

8 2 i 2 1 W +(z)W −(0) δab T(0) 2σ Ji(0) ∂Ji(0) , (3.32) a b ∼ z3 + z − ab z2 + z

W +(z)W +(0) W −(z)W −(0) 0, (3.33) a b ∼ a b ∼

1 i Ji(z)W +(0) σ W +(0), (3.34) a ∼−2z ab b

1 i Ji(z)W −(0) σ ∗W −(0), a ∼ 2z ab b (3.35) Cheng et al. Mathematical Sciences (2015) 2:13 Page 10 of 89

1 i Ji(z)Jj(0) δij ǫ J (0). (3.36) ∼ z2 + z ijk k

i Here σ are the Pauli matrices. Now we can generalize our formula for the partition function (2.6) to include a grading by the U(1) charge under the Cartan generator of the SU(2). The U(1) charge operator J0 is, by definition, twice the zero-mode of the J3 current. From this and the definition J i ψ ψ J ψ 3 =− 1 2 = 1 ¯ 1, we see that under 0 the complex fermion 1 has charge 2 while the other 11 complex fermions are neutral. Therefore, the U(1)-graded NS sector partition function becomes

4 11 1 θi(τ,2z)θi(τ,0) ZNS(τ, z) . (3.37) = 2 η(τ)12 i 2 =

In the above discussion, we have chosen the first three fermions out of a total of 24 N 1 to generate a set of SU(2) currents. Together with an = supercurrent they gen- N 4 erate a full = SCA. It is clear that we are free to choose any three fermions for this purpose. In fact, we could choose an arbitrary three-dimensional subspace of the N 4 24-dimensional vector space spanned by the fermions, and obtain an = SCA. For N 1 a given = supercurrent, not all choices of 3-plane are equivalent, as we will see in Sect. 4. Observe that we could instead have chosen to single out only two real fermions, and construct a U(1) current algebra instead of an SU(2) current algebra. Completely analo- N 2 gous manipulations then show that each such choice provides an = superconformal Co N 2 algebra. As a result we can equip the 0 theory with = structure in such a way that the global symmetry group is broken to subgroups G of Co0 which stabilize 2-planes in 24. N 1 To summarize the results of this section, we have shown how to construct an = supercurrent for the chiral conformal field theory described, in the previous section, as an orbifold of 24 free fermions. We have also shown how choices of 2- and 3-planes N 2 in the space spanned by the generating fermions give rise to actions of the = and N 4 = superconformal algebras (respectively) on the theory. As reviewed in Sect. 2, a N 1 Co suitable choice of = structure reduces the global symmetry of the theory to 0. In the next section we will discuss the finite simple groups that appear when we impose the N 2 N 4 richer, = and = superconformal structures.

4 Global symmetries N 1 N 4 Co Enhancing the = structure of the theory to = breaks the 0 symmetry. We now show that for a specific choice of 3-plane in24 , resulting in a specific copy of N 4 Co M the = SCA, the stabilising subgroup of 0 is the sporadic group 22. Similarly, N 2 for a specific choice of 2-plane, resulting in a specific copy of the = SCA, the stabilising subgroup of Co0 is the sporadic group M23. This amounts to a proof that the model described in Sect. 2 results in an infinite-dimensional M22 (resp. M23)-module underlying the mock modular forms described in Sect. 6 (resp. Sect. 7) arising from its Cheng et al. Mathematical Sciences (2015) 2:13 Page 11 of 89

N 4 N 2 interpretation as an = (resp. = ) module. More generally, we establish the modules for 3- (2-)plane-fixing subgroups of the largest Mathieu group M24 by fixing a spe- N 4 N 2 cific copy of = ( = ) SCA. N 0 Recall that the theory regarded as an = theory (i.e., with no extension of the Virasoro action) has a Spin(24) symmetry resulting from the SO(24) rotations on the N 1 24-dimensional space, and a suitable choice of = supercurrent breaks the Spin(24) group down to its subgroup Co0. The groupCo 0 is the automorphism group of the Leech lattice Leech, and various interesting subgroups of Co0 can be identified as stabilizers of suitably chosen lattice vectors in Leech. To study the automorphism group of the module when fixing more structure—more supersymmetries in this case—it will therefore be useful to describe the enhanced supersymmetries in terms of Leech lattice vectors. In Chapter 10 of [16] it is shown that if we choose an appropriate tetrahedron in the 16 (2, 2, 2, 2, 3, 3) Leech lattice, whose edges have lengths squared × in the normali- sation described below, the subgroup of Co0 that leaves all vertices of the tetrahedron M eγ γ 1, 2, ..., 24 invariant is 22. To be more precise, let , for ∈{ }, be an orthonormal R24 G P( 1, ..., 24 ). basis of , and choose a copy of the extended binary Golay code in { } Then we may realize Leech as the lattice generated by the vectors 2 γ C eγ for G 24 ∈ C together with 4e1 γ 1 eγ. (One can show that all 24 vectors of the form ∈ − + = 4e 24 e T α γ 1 γ are in Leech .) Define the tetrahedron α,β to be that whose four verti- − + = 24 24 { } ces are O 0, Xα 4eα γ 1 eγ, Xβ 4eβ γ 1 eγ and Pαβ 4eα 4eβ, for any = = + = = + = = + α, β 1, 2, ..., 24 with α β. For every such T α,β , the subgroup fixing every vertex ∈{ } �= { } M 27 32 5 7 11 443, 520 is a copy of 22, a sporadic simple group of order · · · · = and the subgroup of M24 fixinge α and eβ. α, β M From the above discussion, it is clear that given { }, a copy of 22 stabilises the real 24 N span of eα, eβ and γ 1 eγ. Given a suitable choice of the 1 superconformal alge- = = R3 24 bra, this copy of in then determines, up to rotations, the three fermions, denoted 1,2,3, from which the SU(2) current algebra was built in Sect. 3. By definition then, a M N 4 copy of 22 leaves the = superconformal algebra invariant. G N 2 A natural question is: what is the symmetry group that fixes a given choice of = superconformal structure? Given the above description of the M22 action, we can choose R2 R3 24 R2 the generated by eα and γ 1 eγ and use the two free fermions lying in the ⊂ = N 2 N 4 to construct the = sub-algebra of the = SCA. Specifically, the U(1) action is R2 rotation of the . From the above discussion, it is not hard to see that there is a copy 24 N of M23 fixinge α and γ 1 eγ and hence stabilising the 2 structure. Recall that M23 = = 27 32 5 7 11 23 10, 200, 960 is a sporadic simple group of order · · · · · = . In terms of the Leech lattice, it corresponds to the fact that the stabiliser of the triangle in Leech whose 24 edges have lengths squared 16 (6, 3, 2), with vertices chosen to be O, Xα and 2 γ 1 eγ × = M Co , is a copy of 23 inside the copy of 0 stabilising Leech. This furnishes a proof that the theory described in Sect. 2 leads to modules for M22 and M23 which explicitly realize the mock modular forms to be defined in Sects. 6 and 7. We should mention that by stabilizing different choices of geometric structure, other than the tetrahedron and triangle just discussed, leading to M22 and M23, respectively, we can determine other global symmetry groups G. Indeed, our method constructs a N 4 N 2 G < Co G-module with = ( = ) superconformal symmetry for any subgroup 0 Cheng et al. Mathematical Sciences (2015) 2:13 Page 12 of 89

which fixes a 3-plane (2-plane) in24 . Since, as we will see in Sect. 6 (Sect. 7), such modules furnish assignments of mock modular forms to the elements of their global symmetry groups, it is an interesting question to classify the global symmetry groups G < Co0 that can arise. We conclude this section with a discussion of some of these possibilities. Certainly a full classification is beyond the scope of this work, so we restrict our attention (mostly) to sporadic simple examples. Indeed, the Conway group Co0 is a rich source of sporadic simple groups, for no less than 12 of the 26 sporadic simple groups are involved in Co0 (cf. [17]), in the sense that they may be obtained by taking quotients of subgroups of Co0. Of these 12, all but 3 are actually realised as subgroups, and 6 of these 9 sporadic simple groups appear as subgroups of Co0 fixing (at least) a 2-plane in24 . These six 2-plane fixing groups are the smaller Mathieu groups, M23, M22, M12 and M11, the Higman–Sims group HS, and the McLaughlin group McL. Some 2-planes they fix are described explicitly in Chapter 10 of [16]. N 4 = modules From the character tables (cf. [17]) of the six sporadic 2-plane fixing subgroups ofCo 0 it is clear that M22 and M11 are the only examples that fix a 3-plane. Even though M11 is not a subgroup of M22, it turns out that the mock modular forms attached to M11 by our N 4 = construction (and the analysis of Sect. 6) are a proper subset of those attached to M22, since the conjugacy classes of Co0 appearing in a 3-plane-fixing subgroup M11 are a proper subset of those appearing in a subgroup M22. For this reason we focus on M22 N 4 when discussing mock modular forms attached to sporadic simple groups via the = construction in this work. If we expand our attention to simple, not necessarily sporadic subgroups of Co0, then there is one example which is larger than M22 (which has order 443,520). Namely, the group U4(3), with order 3,265,920, can arise as the stabilizer of a suitably chosen 3-plane in the 24 of Co0 [16]. TheU 4(3) characters are presented in Appendix Table 20, the coefficients in the associated (twined) vector valued mock modular forms in Appendix Tables 3 and 4, and the decomposition of the module into irreducible representations of the group in Appendix Tables 26 and 27. As we shall see in Sect. 8, the Jacobi forms attached to M22 (and therefore also those M N 4 attached to 11) by the = construction are distinguished in that they satisfy a natural analogue of the genus zero condition of monstrous moonshine. By contrast, this property does not hold for all the Jacobi forms arising from U4(3). This is the main rea- M N 4 son for our focus on 22 in the context of = supersymmetry (Appendix Tables 1, 2, 5, 6, 15). N 2 = modules We have focused on the example of M23, with order 10,200,960, in this section. Since M22 and M11 are subgroups of M23 we do not consider them further in the context of N 2 Co , = structures. Of the remaining sporadic simple 2-plane-fixing subgroups of 0 the largest is the McLaughlin group McL, which is actually considerably larger than M23, having order 898,128,000. Its characters are presented in Appendix Table 21, the coefficients of the (twined) mock modular forms in Appendix Tables 7 and 8, and the decomposition of the module into irreducible representations of the group in Appendix Tables 33, 34, 35, 36, 37, 38. Cheng et al. Mathematical Sciences (2015) 2:13 Page 13 of 89

The next largest example, also larger than M23, is the Higman–Sims group HS, with order 44,352,000. Its characters are presented in Appendix Table 22, the coefficients of the (twined) mock modular forms in Appendix Tables 9 and 10, and the decomposition of the module into irreducible representations of the group in Appendix Tables 39, 40, 41, 42, 43, 44. If we expand our attention to simple groups fixing a 2-plane in24 then there is one example larger than McL. Namely, the group U6(2), of order 9,196,830,720, fixes any triangle in 24 whose three sides are vectors of minimal length in the Leech lattice. The characters of U6(2) are given in Appendix Tables 17, 18, 19, the coefficients of the (twined) mock modular forms in Appendix Tables 11, 12, 13, 14, and the decomposition of the module into irreducible representations of the group in Appendix Tables 45, 46, 47, 48, 49, 50, 51, 52, 53, 54. N 4 In direct analogy with the case of = structure, it will develop in Sect. 8 that the Jacobi forms attached to M12 and M23 satisfy a natural analogue of the genus zero condition of monstrous moonshine, and, contrastingly, this property fails in general for the modular forms arising from the other, non-Mathieu, 2-plane-fixing simple groups mentioned above. For these reasons, and since M12 is relatively small, we focus on M23 in our N 2 discussion of = supersymmetry.

5 Twining the module In the last sections, we have described how to equip the orbifolded free fermion the- N 4 N 2 ory with = and = superconformal structures. In this section we will use the Ramond sector of our theory to attach two variable formal power series—the g-twined g Co graded R sector partition function, cf. (5.10)—to each element ∈ 0 that preserves at least a 2-plane in 24. Let us denote the Ramond sector by V, and let us choose a U(1) charge operator J0. N 4 This will be twice the Cartan generator of theSU (2) in the = case, or the single N 2 U(1) generator in the case of = SCA. Then it is natural to define the Ramond-sector U(1)-graded partition function, or elliptic genus,

F L0 c/24 J0 Z(τ, z) T rV ( 1) q − y (5.1) = −

4 1 1 i 1 11 ( 1) + θi(τ,2z)θ (τ,0) (5.2) = 2 η12(τ) − i i 2 =

4 4 4 1 E4(τ)θ (τ, z) θi(τ, z) 1 8 , (5.3) = 2 η12(τ) + θ (τ,0) i 2 i =

J y e(z) where we have introduced a chemical potential for the 0 charges and set = z C ( 1)F Id for ∈ . Also, we define − as an operator on V by requiring that it act as on Id the untwisted free fermion contribution to V, and as − on the twisted fermion contribution. N 2 As is expected for 2d conformal field theories with ≥ supersymmetry, the m c elliptic genus (5.1) transforms as a Jacobi form of weight 0, index = 6 and level 1. Cheng et al. Mathematical Sciences (2015) 2:13 Page 14 of 89

c 12 Z ( , µ) Z , µ Z Explicitly, and since = in our case, this means that |2 = for all ∈ , Z γ Z γ SL (Z) and |0,2 = for all ∈ 2 , where the elliptic and modular slash operators are defined by

2 (φ m( , µ))(τ, z) e(m( τ 2 z))φ(τ, z τ µ), | = + + + cz2 k aτ b z (5.4) (φ k,mγ )(τ, z) e( m cτ d )(cτ d)− φ cτ +d , cτ d , | = − + + + + ab , µ Z γ SL (Z) respectively, for ∈ and = cd ∈ 2 . (A Jacobi form of level N is only φ γ φ γ Ŵ (N) required to satisfy |k,m = for in the congruence subgroup 0 (cf. (6.24)).) As we have seen in Sect. 2, the two different ways of writing this function, 5.2( ) and (5.3), are intuitively connected more closely with the free fermion and E8 root lattice descriptions of the theory, respectively. Of course, the U(1)-graded NS sector partition function (3.37) is related to the above, graded Ramond sector partition function by a spectral flow transformation

1/2 2 τ 1 ZNS(τ, z) q y− Z(τ, z + ). (5.5) = − 2

There is a natural way in which one can twine the above function under certain subgroups of Co0. From the previous discussions, we see that the representation 24 plays a central role in the way various subgroups of Co0 act on the model. Let’s denote by ℓg,k ℓ k 1, ...12 g Co and ¯g,k, for = , the 12 complex conjugate pairs of eigenvalues of ∈ 0 when acting on 24. This information is conveniently encoded in the so-called Frame shape of g, given by

mn �g Ln ,1 L1 < L2 < L3 , and mn Z, mn 0, = n ≤ ··· ∈ �=

Lnmn 24 ℓ , ℓ satisfying n = , through the fact that the 12 pairs { g,k ¯g,k } are precisely the 24 roots solving the equation (xLn 1)mn 0. n − = N 2 As discussed in Sects. 3 and 4, in order to preserve at least = superconformal symmetry and hence be able to twine the graded R-sector partition function (5.1), the subgroup G must leave at least a 2-dimensional subspace in 24 pointwise invariant. In the graded partition function this corresponds to leaving the factor θi(τ,2z) in (5.2) invariant. As a result, for every conjugacy class [g] of such a group G we can choose ℓg ℓg 1 ,1 = ¯ ,1 = . It is easy to see that when acting on the untwisted free fermions of the theory, contributing the terms involving θi with i 3, 4 in (5.2), the group element g sim- 11 = ply replaces θi (τ,0) with

12 θi(τ, ρg,k ) (5.6) k 2 =

e(ρ ) ℓ where g,k = g,k. Cheng et al. Mathematical Sciences (2015) 2:13 Page 15 of 89

When trying to do the same for the contribution from the twisted fermions, contributing the term involving θ2 in (5.2), however, we see that the above simple consideration θ (τ, ρ) θ (τ, ρ 1), suffers from an ambiguity. This can be seen from the fact that 2 =− 2 + and hence the answer cannot be determined simply by looking at the g-eigenvalues on 24. This of course is a reflection of the fact that the global symmetry group, with no superconformal structure imposed, is Spin(24), which is a 2-fold cover of SO(24). As a result, to specify the twining of the twisted fermion contribution, we also need to know 12 the action of G on the faithful 2 -dimensional representation of Spin(24) spanned by Ramond sector ground states in the free fermion theory (cf. Sect. 2), henceforth denoted 4096 4096 1 276 1771 24 2024 , which decomposes as = + + + + in terms of the irreducible representations of Co0. Note that, according to the orbifold construction, just “half” of the Ramond sector ground states in the free fermion theory will contribute to the Ramond sector V of the orbifold theory under consideration. In terms of the Co0 action, the two “halves” 24 2024 Co 1 276 1771 are + , where 0 acts faithfully, and + + , where the action factors Co Co /2 through 1 = 0 . In practice, both choices give rise to equivalent theories (i.e., iso- morphic super vertex operator algebras, cf. [24, 27]), but they are inequivalent as Co0 24 2024 -modules. For us, the ground states represented by + lie in the R sector, V, and 1 1 276 1771 Co N 1 the in + + represents the 0-invariant = supercurrent in the NS sector of our orbifold theory. The above discussion serves to remind us that there is, really, a vanishing term

1 1 11 0 θ1(τ,2z)θ (τ,0) (5.7) = 2 η12(τ) 1

g Co in (5.2), which, for certain ∈ 0, will make a non-vanishing contribution to the g e g-twined version of (5.1). It vanishes when = is the identity because the Ramond sector ground states in the free fermion theory come in pairs with opposite eigenvalues for F ( 1) ψa ψa, − . Moreover, exchanging the pair corresponds to complex conjugation ↔ ¯ a 1, ..., 12 for = , of the complex fermions. Recall that one of the complex fermions, denoted ψ1 in (3.19), was used to construct the U(1) charge operator J0, and we are inter- ested in the graded partition function where we introduce a chemical potential z for this ψ ψ operator. Because exchanging 1 ↔ ¯ 1 also induces a flip ofU (1) charges, captured by z z ↔− , the contribution of the first complex fermion does not vanish, corresponding to the fact that the identity

θ1(τ, z) θ1(τ, z 2) θ1(τ, z) (5.8) = + =− − θ (τ, z) z Z only forces 1 to vanish at ∈ . Consequently, the g-twining of (5.7) makes a non- ρ Z k 2, ..., 12 zero contribution to the g-twining of (5.2) if and only if g,k �∈ for all = . In other words, it is non-zero only when the cyclic group generated by g fixes nothing but a 2-plane. By inspection we find that, among the groups we consider, such group elements 23AB M 6AB, 12AB, 12DE, 18AB U (2) must be in the conjugacy classes ⊂ 23, ⊂ 6 , 15AB, 30AB 20AB ⊂ McL, or ⊂HS. The pairs of these conjugacy classes corresponding to the letters A and B (or D and E) are mutually inverse, and so their respective traces, Cheng et al. Mathematical Sciences (2015) 2:13 Page 16 of 89

on any representation, are related by complex conjugation. In terms of our construction, ψ ψ choosing one over the other is the same as choosing what one labels 1 and ¯ 1, and the same as choosing an orientation on the 2-plane fixed by the group element in24 . As a result, from (5.8) we see that the θ1 term in the partition functions twined by these conjugate A (D) and B (E) classes come with an opposite sign. ρ 0, 1/2 Let us work with the principal branch of the logarithm, and choose g,k ∈[ ] in (5.6). Then, by direct computation—we must compute directly, for the choice of labels for mutually inverse conjugacy classes is not natural—we find that the signs in (5.10) are

23A M23, 20A ⊂ HS, lǫg,1 1 for g in ⊂ (5.9) =  15A 30A McL,  12A ∪ 12D ⊂6B 18B U (2), ∪ ∪ ∪ ⊂ 6  ǫg 1 23B M 20B HS and ,1 =− for the inverse classes, ⊂ 23, ⊂ ,&c. g G Putting these different contributions together, we conclude that for every[ ]⊂ where G is a subgroup of Co0 preserving (at least) a 2-plane in 24, the corresponding g-twined U(1)-graded R sector partition function reads

F L0 c/24 J0 Zg (τ, z) T rV g( 1) q − y (5.10) = −

4 12 1 1 i 1 ( 1) + ǫg,i θi(τ,2z) θi(τ, ρg,k ), (5.11) = 2 η(τ)12 − i 1 k 2 = =

where

T r4096g 1, 1 when 12 cos(πρ ) 0 212 12 cos(πρ ) k 1 g,k ǫg,2 k 1 g,k ∈ {− } = �= (5.12) = = 12 0 when k 1 cos(πρg,k ) 0 = =

ǫg ǫg 1, ,3 = ,4 = (5.13) and where the ǫg,1 are as determined in the preceding paragraph. In this section we have introduced the g-twined U(1)-graded Ramond sector partition Zg g Co function, or g-twined elliptic genus of our theory, , for any ∈ 0 fixing a 2-plane in 24. We have also derived an explicit formula (5.10) for Zg, in terms of the Frame shapes g T r4096g g G and values . This Frame shape and trace value data is collected, for ∈ , G Co for various ⊂ 0, in “Appendix B”. In Sects. 6 and 7 we will see how the above twining leads to the mock modular forms playing the role of the McKay–Thompson series in these new examples of mock modular moonshine.

6 The N = 4 decompositions From the discussion in Sect. 3 it is clear that the orbifold theory discussed in Sect. 2 N 4 can be equipped with = superconformal structure. In this section we will study the N 4 decomposition of the Ramond sector V into irreducible representations of the = Cheng et al. Mathematical Sciences (2015) 2:13 Page 17 of 89

SCA and see how the decomposition leads to mock modular forms relevant for the M22 moonshine which we will discuss in Sect. 8. N 4 Recall (cf. [32]) that the = superconformal algebra contains subalgebras isomor- phic to the affineSU (2) and Virasoro Lie algebras. In a unitary representation the former m 1 m > 1 of these acts with level − , for some integer , and the latter with central charge c 6(m 1). = − N 4 The unitary irreducible highest weight representationsv m =h,j are labeled by the 1 3 ; eigenvalues of L0 and 2 J0 acting on the highest weight state, which we denote by h and j, respectively. Cf. [33, 34]. The superconformal algebra has two types of highest weight Ramond sector representations: the massless (or BPS) representations with c m 1 1 m 1 h 24 4− and j 0, 2 , ..., 2− , and the massive (or non-BPS) representations = = m 1 ∈{1 m 1 } h > − j , 1, ..., − with 4 and ∈{2 2 }. Their graded characters, defined as N 3 3 4 J0 J0 L0 c/24 chm =h,j (τ, z) trvN 4 ( 1) y q − , (6.1) ; = m h=,j − ; are given by

N 4 1 chm =h,j (τ, z) (�1,1(τ, z))− µm j(τ, z) (6.2) ; = ;

and

2 N c j 4 1 h 24 m chm =h,j (τ, z) (�1,1(τ, z))− q − − θm,2j(τ, z) θm, 2j(τ, z) (6.3) ; = − − in the massless and massive cases, respectively, [34]. In the above formulas, the function µm j(τ, z) ; is defined by setting k 2j k 2j 1 k 1 2j 1 2j mk2 2mk (yq )− (yq )− + (yq ) + µm j(τ, z) ( 1) + q y + +···+ , (6.4) ; = − 1 yqk k Z ∈ −

and 1,1 is a meromorphic Jacobi form (cf. Sect. 8 of [19] for more on meromorphic Jacobi forms) of weight 1 and index 1 given by 3 θ1(τ,2z) η(τ) y 1 2 2 � (τ, z) i + (y y− )q . (6.5) 1,1 =− (θ (τ, z))2 = y 1 − − +··· 1 − Finally, we have used the theta functions

k k2/4m k θm r(τ, z) e( ) q y , , = 2 (6.6) k r (mod 2m) =

2m Z> r m Z defined for all ∈ 0 and − ∈ , and satisfying θm,r(τ, z) θm,r 2m(τ, z) e(m)θm, r(τ, z). = + = − −

θm (θm r) r m Z/2mZ Note that the vector-valued theta function = , , − ∈ , is a vector-valued Jacobi form of weight 1/2 and index m satisfying Cheng et al. Mathematical Sciences (2015) 2:13 Page 18 of 89

1 i m 2 S 1 z θm(τ, z) e( τ z ) θ .θm( τ , τ ) = 2m τ − − Tθ .θm(τ 1, z) = + (6.7) θm(τ, z 1) e(m(τ 2z 1))θm(τ, z τ), = + = + + +

Sθ Tθ 2m 2m where the and matrices are × matrices with entries

rr r r r2 (Sθ ) e( ′ )e( − + ′ ), (Tθ ) e( )δ . (6.8) r,r′ = 2m 2 r,r′ = − 4m r,r′

m Z N 2 We will take ∈ for the rest of this section. When we consider = decompositions in the next section, we will use the theta function with half-integral indices. From the above discussion, it is clear that the graded partition function of a module c 6(m 1) N 4 for the = − = SCA admits the following decomposition ZN 4,m r2 N 4 = cr′ (n 4m ) ch =m 1 r (τ, z). = − m 4− n, 2 n 0,0 r m 1 ; + (6.9) r≥0 when ≤ ≤ n>−0 �=

Furthermore, from the identity

r (r n 1)2 r n 1 − + µm r ( 1) (r 1)µm 0 ( 1) − nq− 4m (θm,r n 1 θm, (r n 1)) ; 2 = − + ; + − − + − − − + n 1 =

we arrive at

ZN 4,m 1 (m) = (�1,1(τ, z))− c0 µm 0(τ, z) Fr (τ) θm,r(τ, z) , (6.10) =  ; +  r Z/2mZ ∈�   where

2 (m) ∞ r2 n r F (τ) cr n q − 4m ,1 r m 1, (6.11) r = − 4m ≤ ≤ − n 0 =

m 1 − r r2 c0 ( 1) (r 1) c′ , (6.12) = − + r − 4m r 0 =

2 m 1 r′ r r′ 2 r −r ( 1) − (r′ 1 r) cr′ 4m , n 0 c n r ′= − + − ′ − = r 4m r2 (6.13) − =  c�′ n , n > 0. � � � �  r − 4m � �  (m) (m) F (Fr ) r Z/2mZ The rest of the components of = , ∈ , are defined by setting

(m) (m) (m) Fr (τ) F r (τ) Fr 2m(τ). (6.14) =− − = + Cheng et al. Mathematical Sciences (2015) 2:13 Page 19 of 89

(m) (m) Recall that µm 0(τ, z) f0 (τ, z) f0 (τ, z), a specialisation of the Appell–Lerch ; =− + − sum

mk2 2mk ( ) q y f m (τ, z) (6.15) u = 1 yqk e( u) k Z ∈ − − Z studied in [64], has the following relation to the modular group SL2( ): let the (non- µm 0(τ, z) holomorphic) completion of ; be 1 z z 1 µm 0(τ, τ, ) µm 0(τ, ) e 8 ˆ ; ¯ = ; − − √2m i ∞ 1/2 θm,r(τ, z) (τ ′ τ)− Sm,r( τ ) dτ ′ . (6.16) × + −¯′ r Z/2mZ τ ∈ −¯

Z ⋉Z2 Then µm 0 transforms like a Jacobi form of weight 1 and index m for SL2( ) . Here ˆ ; Sm (Sm r) SL (Z) = , is the vector-valued cusp form for 2 whose components are given by the unary theta functions

k k2/4m 1 ∂ Sm,r(τ) e kq θm,r(τ, z) z 0. = 2 = 2πi ∂z | = k r (mod 2m) =

Sm r(τ) 2m Z For later use, note that the theta series , is defined for all ∈ and r m Z/2mZ − ∈ . Z(m) The way in which the functions and µm 0 transform under the Jacobi group shows ˆ ; (m) that the non-holomorphic function r Z/2mZ Fr (τ) θm,r(τ, z) transforms as a Jacobi ∈ ˆ SL (Z) ⋉Z2 form of weight 1 and index m under 2 , where i (m) (m) 1 1 ∞ 1/2 Fr (τ) Fr (τ) c0e 8 (τ ′ τ)− Sm,r( τ ′) dτ ′. ˆ = + − √2m τ + −¯ −¯ (m) (m) F (Fr ) r Z/2mZ In other words, = , ∈ is a vector-valued mock modular form with a vector-valued shadow c0 Sm, whose r-th component is given by Sm,r(τ), with the multi- Z plier for SL2( ) given by the inverse of the multiplier system of Sm (cf. (6.8)). Now we are ready to apply the above discussion to the U(1)-graded Ramond sector partition function of the theory, discussed in Sect. 5. Recall that in this case we have c 12 m 3 N 4 = , so = in (6.2) and (6.3). The = decomposition of (5.1) gives Z z = N =4 + N =4 + N =4 + N =4 + N =4 +··· (τ, ) 21 ch ; 1 ch ; 1 560 ch ; 3 1 8470 ch ; 5 1 70576 ch ; 7 1 3 2 ,0 3 2 ,1 3 2 , 2 3 2 , 2 3 2 , 2 + N =4 + N =4 + N =4 +··· 210 ch ; 3 4444 ch ; 5 42560 ch ; 7 (6.17) 3 2 ,1 3 2 ,1 3 2 ,1

1 (�1,1(τ, z))− 24 µ3 0(τ, z) hr(τ)θ3,r(τ, z) (6.18) =  ; +  r Z/6Z ∈�   Cheng et al. Mathematical Sciences (2015) 2:13 Page 20 of 89

1 α β 2 ... �− q y α β /12 > 3 where stand for terms with expansion 1,1 with − . More Fou- rier coefficients of the functionsh r(τ) are recorded in “Appendix C”, where h hg for 2 = g 1A. c (n r ) [ ]= Note that all the graded multiplicities r′ − 12 appear to be non-negative. Of N 4 course, this is guaranteed by the fact that V is a module for the = SCA as shown in Sect. 3. In particular, the Fourier coefficients ofh r(τ) appear to be all non-negative apart 1/12 from that of the polar term 2q− in h1. − Z Z From the above discussion we see that h (hr), for r /6 , is a weight 1/2 vector- Z = ∈ valued mock modular form for SL2( ) with 6 components (but just 2 linearly independent components, since h0 h3 0, h 1 h1, and h 2 h2), with shadow given by = = − =− − =− 24 S3, and multiplier system inverse to that of S3. This is to be contrasted with the elliptic genus of a generic non-chiral super conformal c 6 field theory. For example, the sigma model of a K3 surface has = , and the elliptic genus is given by

FL FR J0 L0 c/24 L0 c/24 EG(τ, z K3) TrHRR ( 1) + y q − q ˜ −˜ ; = − ¯ 20 chN 4 2 chN 4 90 chN 4 462 chN 4 1540 chN 4 2 1=,0 2 1=, 1 2 5=, 1 2 9=, 1 2 13=, 1 = ; 4 − ; 4 2 + ; 4 2 + ; 4 2 + ; 4 2 +··· (6.19)

1 (�1,1(τ, z))− 24 µ2 0(τ, z) (θ2,1(τ, z) θ2, 1(τ, z)) = ; + − − (6.20) 1/8 7/8 15/8 23/8 ( 2q− 90q 462q 1540q ) , × − + + + +··· 1 α β 2 ... �− q y α β /8 > 3 where stand for terms with expansion 1,1 with − . In this case, the chN 4 coefficient multiplying the massless character 2 =1 , 1 is negative, arising from the Witten ; 4 2 N 4 index of the right-moving massless multiplets paired with the representation v 1= 1 of the 2 4 , 2 N 4 ; left-moving = SCA. In Sect. 3 we have shown that the theory under consideration, as a module for the N 4 = SCA, admits a faithful action via automorphisms by a group G, as long as G is a 0 g G subgroup of Co fixing at least a 3-plane. For any such ∈ , the g-twined graded partition function Zg (τ, z) is given by (5.10), and from the fact that the action of g commutes N 4 Zg (τ, z) with the = SCA, we expect to admit a decomposition

1 Zg (τ, z) (�1,1(τ, z))− (Tr24g)µ3 0(τ, z) hg,r(τ)θ3,r(τ, z) . (6.21) =  ; +  r Z/6Z ∈�   Moreover, the coefficients of

r2/12 ∞ n r2/12 hg,r(τ) arq− (Tr G g) q − (6.22) = + Vr,n n 1 =

must be characters of the G-module

∞ V G V G = r,n (6.23) r 1,2 n 1 = = Cheng et al. Mathematical Sciences (2015) 2:13 Page 21 of 89

arising from the orbifold theory discussed in Sect. 2. N 4 Indeed, the multiplicities of the = multiplets in the decomposition (6.17) are sug- b h 1/2 j 0 gestive of the following group theoretic interpretation : the 21 = , = massless representations transform as the 21-dimensional irreducible representation of M22, and h 3/2 j 1/2 χ χ similarly, the 560 = , = massive representations transform as 10 + 11 (see 280 280 M “Appendix B”), or “ + ”, under 22, etc. We have explicitly computed the first 30 or so coefficients of each q-series hg,r(τ) for all conjugacy classes [g] of G, for G M22 and G U4(3). These can be found in the = = G tables in “Appendix C”. Subsequently, we compute the first 30 or so G-modules Vr,n in terms of their decompositions into irreducible representations. They can be found in the tables in “Appendix D”. Finally we would like to discuss the mock modular property of the functions hg (hg r). SL (Z) = , Recall that the Hecke congruence subgroups of 2 are defined as ab Ŵ0(N) SL2(Z) c=0 mod N . (6.24) = cd ∈ |

We expect Zg to be a weak Jacobi form of weight zero and index 2 (possibly with multi- 2 Ŵ (og ) ⋉Z g g G plier) for the group 0 , where o is the order of the group element ∈ . This can be verified explicitly from the expression (5.10). Repeating the similar arguments as above, we conclude that each vector-valued function hg is a vector-valued mock modular form of weight 1/2 with shadow (Tr24g)S3 for the congruence subgroup Ŵ0(og ). Note (Tr24g) 0 g M that �= for all ∈ 22 which are the cases of our main interest. For these cases the multiplier of hg is again given by the inverse of the multiplier system of S3, now restricted to Ŵ0(og ). In this section we have analyzed the decomposition of the Ramond sector of our orbi- N 4 fold theory into irreducible modules for the = SCA, and we have demonstrated N 4 that the generating functions of irreducible = SCA module multiplicities furnish a vector-valued mock modular form. We have also demonstrated that these multiplicities are dimensions of modules for subgroups G < Co0 that point-wise fix a 3-plane in 24, and we have analyzed the modularity of the resulting, g-twined multiplicity generat- g G ing functions, for ∈ . We have verified that each suchg -twining results in a vector- valued mock modular form with a specified shadow function. In the next section we will N 2 present directly analogous considerations for = superconformal structures arising from 2-planes in 24.

7 The N = 2 decompositions As discussed in Sect. 4, the theory presented in Sect. 2 can be regarded as a module for N 2 N 4 G < Co an = SCA as well as for an = SCA. Moreover, for every subgroup 0 N 2 fixing a 2-plane there is an = SCA commuting with the action of G on the theory. As a result, and as we will now demonstrate, the decomposition of the partition func- N 2 tion (5.10) twined by elements of G into = characters leads to sets of vector-valued mock modular forms, now of weight 1 / 2 and index 3 / 2, which are the graded characters of an infinite-dimensional G-module inherited from the Co0-module structure on V (cf. Sect. 5). Cheng et al. Mathematical Sciences (2015) 2:13 Page 22 of 89

hg (hg j) To see what these (vector-valued) mock modular forms ˜ = ˜ , are, let us start N 2 by recalling the characters of the irreducible representations of the = SCA. For the SCA with central charge c 3(2ℓ 1) 3c, the unitary irreducible high- N 2 = + = ˆ est weight representations vℓ h=,Q are labeled by the two quantum numbers h and Q ; which are the eigenvalues of L0 and J0, respectively, when acting on the highest weight state [22, 51]. Just as in the N 4 case, there are two types of Ramond sector high- = c c est weight representations: the massless (or BPS) representations with h 24 8ˆ and c c c c = = Q 2ˆ 1, 2ˆ 2, ..., 2ˆ 1, 2ˆ , and the massive (or non-BPS) representations with ∈ {−c + − +c c − } c c c h > ˆ Q ˆ 1, ˆ 2, ..., ˆ 2, ˆ 1, ˆ Q 0 8 and ∈ {− 2 + − 2 + 2 − 2 − 2 }, �= . From now on we will con- ℓ c centrate on the case when is half-integral, and hence ˆ and c are even. The graded characters, defined as

N 3 3 2 J0 J0 L0 c/24 chℓ h=,Q(τ, z) trvN 2 ( 1) y˜ q − , (7.1) ; = ℓ h=,Q − ; are given by

2 N c j 2 ℓ 1 h 24 4ℓ chℓ h=,Q(τ, z) e( 2 )(�1, 1 (τ, z))− q − − θℓ,j(τ, z), ; = − 2 (7.2) j sgn(Q)(Q 1/2), = | |−

for the massive representations, and

N 1 2 1 2 ℓ Q / 1 Q 2 (ℓ) chℓ c=/24,Q(τ, z) e + 2+ (�1, 1 (τ, z))− y + fu (τ, z u), ; = − 2 + (7.3) 1 (1 2Q)τ u + , = 2 + 4ℓ c N 2 c for the massless representations (with Q 2ˆ). The characterch ℓ c=/24,Q(τ, z) for Q 2ˆ �= ; = is given in (7.5). In the above formula, we have used the Appell–Lerch sum (6.15) and defined 3 η(τ) 1 1/2 1/2 2 �1, 1 i q (y y− ) O(q ). 2 =− θ (τ, z) = y1/2 y 1/2 + − + − 1 − − Note that the above characters transform according to the rule

N 2 N 2 chℓ c=/24,Q(τ, z) chℓ c=/24, Q(τ, z) ; = ; − −

under charge conjugation. From the relation between the massless and massive characters Q 1 N 2 N 2 n | |− k N 2 N 2 chℓ c=/24,Q chℓ c=/24, Q q− ( 1) chℓ,n= c/24,Q k chℓ,n= c/24,k Q ; + ; − = − + − + + − k 0 = Q N 2 2( 1) chℓ c=/24,0, n > 0, Q ℓ, (7.4) + − ; | |≤ c ˆ 1 2 − N 2 n N 2 k N 2 N 2 ch = c q− ch = c ( 1) ch = c ch = c ℓ c/24, 2ˆ =  ℓ n c/24, 2ˆ + − ℓ,n c/24, 2ˆ k + ℓ,n c/24,k 2ˆ  ; ; + k 1 � + − + − �  �=  c  ˆ N 2  ( 1) 2 chℓ c=/24,0, (7.5) + − ; Cheng et al. Mathematical Sciences (2015) 2:13 Page 23 of 89

as well as the charge conjugation symmetry of the theory, we expect the U(1)-graded Ramond sector partition function of a theory that is invariant under charge conjugation to admit a decomposition

N 2, N 2 N 2 Z ℓ C c z C n ℓ z error = 0′ chℓ =,0(τ, ) ℓ′ 4 ch =c n 1 (τ, )< 24 ℓ 24 ,ℓ 2 = ; + n 0 − ; + + − ≥ j2 N 2 N 2 Cj′ n ch =c 1 (τ, z) ch =c (τ, z) 4ℓ ℓ n,j ℓ n, j 1 + − ; 24 + + 2 + 24 2 n 0,j 1 , 3 ,...,ℓ 1 ; + − + ≥ ∈{ 2 2 − }

(7.6)

ℓ 1 (ℓ) e 2 (�1, 1 )− C0 µℓ 0(τ, z) Fj (τ)θℓ,j(τ, z) (7.7) = 2  ˜ ; + ˜  − j ℓ Z/2ℓZ � � − �∈   N 2 c 3(2ℓ 1) when the = SCA has even central charge, = + . In the last equation, we have defined

1 1/2 (ℓ) 1 τ µℓ 0 e( ) y fu (τ, u z), u , ˜ ; = 4 + = 2 + 4ℓ

and

2 (ℓ) (ℓ) (ℓ) j2 j n 4ℓ Fj (τ) F j (τ) Fj 2ℓ(τ) Cj n 4ℓ q − , (7.8) ˜ = ˜− = ˜ + = − n 0 ≥

j 1 j2 C0 C′ 2 ( 1) + 2 C′ , (7.9) = 0 + − j − 4ℓ j 1 , 3 ,...,ℓ ∈{ 2 2 }

ℓ j (j k)2 − k j2 k 0( 1) Cj′ k +4ℓ n 0 Cj n = − 2 + − = (7.10) 4ℓ  j − = �C′ n � � n > 0. � �  j − 4ℓ � �  N 4 Similar to the case of massless = characters, through its relation to the Appell–Lerch sum, µℓ 0 admits a completion which transforms as a weight one, half-integral index Jac- ˜ ; obi form under the Jacobi group. More precisely, defineµ ℓ 0 by replacing µm 0 with µℓ 0 ˜ ; ; ˜ ; and the integer m with the half-integral ℓ in (6.16). Thenµ ℓ 0 transforms like a Jacobi form ˜ ; Z ⋉Z2 of weight 1 and index ℓ under the group SL2( ) . Following the same computation as F (ℓ) ( F (ℓ)) j 1/2 Z/2ℓZ in the previous section, we hence conclude that ˜ = ˜j , where − ∈ , is C Sℓ C (Sℓ j(τ)) a vector-valued mock modular form with a vector-valued shadow 0 = 0 , . Now we are ready to apply the above discussion to the U(1)-graded R sector partition N 2 function of the orbifold theory of Sect. 2. The = decomposition gives Z , z 23 chN 2 chN 2 770 chN 2 chN 2 (τ ) 3 =1 ,0 3 =1 ,2 3 =3 ,1 3 =3 , 1 = 2 ; 2 + 2 ; 2 + 2 ; 2 + 2 ; 2 − 13915 chN 2 chN 2 3 =5 ,1 3 =5 , 1 + 2 ; 2 + 2 ; 2 − +··· 231 chN 2 5796 chN 2 (7.11) 3 =3 ,2 3 =5 ,2 ... + 2 ; 2 + 2 ; 2 + Cheng et al. Mathematical Sciences (2015) 2:13 Page 24 of 89

1 23 47 3 1 24 24 24 e �− 1 24 µ 3 q− 770 q 13915 q c ... θ 3 1 θ 3 1 4 1, 2 0 2 , 2 2 , 2 = − 2 ˜ ; + − + + + + − 3 5 13 8 8 8 (q− 231 q 5796q )θ3 , 3 (7.12) + + + +··· 2 2 1 α β 2 ... �− q y α β /6 > 2 where denote the terms with expansion 1, 1 with . Again, we − 2 − N 2 observe that all the multiplicities of the representations with characters ch 3 = appear 2 h,Q N 2 ; to be non-negative, consistent with our construction of V as an = SCA module. In general, from the previous sections we have seen that the graded partition function twined by any element g of a subgroup G of Co0 should admit a decomposition into N 2 = characters. We write

3 1 Zg (τ, z) e( 4 )�− 1 (Tr24g) µ 3 0(τ, z) hg,j(τ)θ 3 ,j . (7.13) = 1, 2  ˜ 2 + ˜ 2  − ; j 1/2, 1/2,3/2 ∈{ �− }   Moreover, from the discussion in Sect. 2 we have seen that

hg,1/2(τ) hg, 1/2( τ), (7.14) ˜ = ˜ − −¯

and the coefficients of these functions

j2/6 ∞ n j2/6 hg,j(τ) ajq− (TrV G g) q − (7.15) ˜ = + ˜r,n n 1 =

are given by characters of a G-module

∞ V G V G ˜ = ˜j,n (7.16) j 1/2,1/2,3/2 n 1 =− =

which descends from the orbifold theory in Sect. 2. In particular, for any n, the G-mod- G G ule V 1/2,n is the dual of V1/2,n. From the above discussion, we conclude that hg (hg,j) ˜− ˜ ˜ = ˜ is a vector-valued mock modular form for Ŵ0(og ) with shadow (Tr24g)S3/2. Recall that (Tr24g) 0 g M �= for all ∈ 22 which are the cases of our main interest. For these case the multiplier of h˜ g is given by the inverse of the multiplier system of S3/2, restricted to Ŵ0(og ). To summarize, we have analyzed the decomposition of the Ramond sector of our orbi- N 2 fold theory into irreducible modules for the = SCA in this section, and we have N 2 demonstrated that the generating functions of irreducible = SCA module multiplicities also furnish a vector-valued mock modular form. We have demonstrated that these multiplicities are dimensions of modules for subgroups G < Co0 that point-wise fix a 2-plane in24 , and we have observed that the resulting, g-twined multiplicity gen- g G erating functions, for ∈ , are vector-valued mock modular forms with a certain extra 1/2 symmetry, relating the components labelled by ± by complex conjugation.

8 Mathieu moonshine In the previous sections we have seen that the orbifold theory described in Sect. 2 leads to infinite-dimensional G-modules underlying a set of vector-valued mock modular Cheng et al. Mathematical Sciences (2015) 2:13 Page 25 of 89

N 4 N 2 Co forms from its = ( = ) structures for any subgroup G of 0 fixing at least a 3-plane (2-plane) in 24. In this section we will discuss a natural property of the vector- valued mock modular forms that distinguishes subgroups of M24 from other 3-plane (2-plane) fixing subgroups. These considerations lead to mock modular Mathieu moonshine involving distinguished vector-valued mock modular forms of weight 1 / 2. Recall the celebrated genus zero condition of monstrous moonshine, which states that the monstrous McKay–Thompson series are Hauptmoduls with only a polar term 1 q− at the cusp represented by i , and no poles at any other cusps. To be more pre- ∞n Tg (τ) q tr ♮ g cise, denote by n 1 Vn the graded character of the moonshine module ♮ ♮ = ≥− V n 1 Vn of Frenkel–Lepowsky–Meurman [37]. ThenT g (τ) is a function invari- = ≥− Ŵ < SL (R) ant under the action of a particular g 2 (specified in [15]), such that

(i) qTg (τ) O(1) as τ i , and = → ∞ (8.1) (ii) Tg (γ τ) O(1) as τ i whenever γ SL2(Z) and γ Ŵg . = → ∞ ∈ ∞ �∈ ∞

g M Similarly, in Mathieu moonshine [35], it follows from the results of [7] that if ∈ 24 and Zg denotes the g-twined K3 elliptic genus then

1 (i) Zg (τ, z) cg y y− as τ i , and = + + → ∞ (8.2) (ii) Zg γ (τ, z) cg γ as τ i whenever γ SL (Z) and γ Ŵg , |0,1 = , → ∞ ∈ 2 ∞ �∈ ∞

cg , cg γ C τ i Zg γ (cf. (5.4)) for some , ∈ . In other words, the → ∞ limit of |0,1 is a z-independent constant whenever γ is not in the invariance group Ŵg. (Note that Ŵg is always a SL (Z) g M subgroup of 2 for ∈ 24). A natural question is therefore: among the subgroups of Co0 fixing 2- or 3-planes for which we have constructed a module in this work, for which of these do the associated modular objects satisfy a condition analogous to the preceding cases of moonshine described above? We will see presently that the Mathieu groups M23, M22, M12 and M11 are distinguished in our setting, in that the graded characters of their respective modules yield weight zero weak Jacobi forms satisfying the conditions

2 2 (i) Zg (τ, z) cg y y− as τ i , and = + + → ∞ (8.3) (ii) Zg γ (τ, z) cg γ as τ i whenever γ SL (Z) and γ Ŵg , |0,2 = , → ∞ ∈ 2 ∞ �∈ ∞

cg , cg γ C for some , ∈ . On the other hand, all the other groups mentioned in Sect. 4 contain elements g for which the condition (ii) in (8.3) is not satisfied. Thus the conditions (8.3) single out the Mathieu groups as the sporadic simple subgroups of Co0 with this moonshine property. The constructions we have presented in this paper provide con- crete realizations of the underlying mock modular Mathieu moonshine modules. Note that the conditions (8.3) impose restrictions on the degrees of the poles (if any) of the mock modular forms hg, h˜ g (cf. Sects. 6, 7) at all cusps. In fact, for the case that M M N 4 G is a copy of 22 or 11 preserving = supersymmetry, the corresponding mock modular forms hg only have poles at the infinite cusp. This property also holds for the M N 4 mock modular forms attached to 24 via = decomposition of the twined K3 elliptic genera of Mathieu moonshine, satisfying (8.2), as was demonstrated in [7]. In more Cheng et al. Mathematical Sciences (2015) 2:13 Page 26 of 89

physical terms, (8.2) and (8.3) can be interpreted as the condition that the elliptic genus of any cyclic orbifold of the theory receives no contributions from U(1)-charged ground states in twisted sectors. Zg i To investigate the behaviour of at cusps other than ∞, we first note that for any positve integer N,

12 2 4 10 ρℓ i 1 Zg (τ, z) e N ǫi N ( 1) + �i N (τ, z), |0,2 N 1 = 2 , − , ℓ 1 i 1 = =

where 12 2 2 θi(τ,2z) ρk N �i N (τ, z) e τ θi(τ, ρ Nρ τ), (8.4) , = η12(τ) 2 k + k k 2 = ǫi,N = ǫi 2 N and when | , and ǫ1,N ǫ1 =− ǫ2,N ǫ3 1 (8.5) =− =−  ǫ3,N ǫ2 =−  ǫ4,N ǫ4 1 =− =−  otherwise. The above expressions can be derived from the transformation laws of Jacobi Z theta functions under SL2( ). τ i Near the infinite cusp, → ∞, the different contributions have the following leading behaviour: 12 1 1 ρk fN,1(�g )/2 1 1/og �1,N (τ, z) e ( ρ ) Nρ q (y y− ) 1 O(q ) = 2 + 2 − k ⌊ k ⌋− 2 − [ + ] � k 1 � �= 12 ρk fN,1(�g )/2 1 1/og �2,N (τ, z) e ρ Nρ q (y y− ) 1 O(q ) = − k ⌊ k ⌋− 2 + [ + ] �k 1 � �= 12 1 fN,2(�g )/2 1/2 2 2 �3,N (τ, z) e ρ Nρ q 1 q (y y− ) = − k ⌊ 2 + k ⌋ + + � k 1 � �= � � 1 Nρ 12 1 1 ⌊ 2 + k ⌋ 1 Nρ Nρ Nρ n 1 e( ρ ) q⌊ 2 + k ⌋+ 2 − k 1 e(ρ )q 2 + k − ×  + − k + k  k 2 � � n 1 � � �= �=  1 O(q1/og )  ×[ + ] 12 1 1 fN,2(�g )/2 1/2 2 2 �4,N (τ, z) e ( ρ ) Nρ q 1 q (y y− ) = − 2 + k ⌊ 2 + k ⌋ − + � k 1 � �= � � 1 Nρ 12 1 1 ⌊ 2 + k ⌋ 1 Nρ Nρ Nρ n 1 e( ρ ) q⌊ 2 + k ⌋+ 2 − k 1 e(ρ ) q 2 + k − ×  − − k − k  k 2 � � n 1 � � �= �=  1 O(q1/og ) (8.6) ×[ + ]

with 12 2 fN (�g ) 1 (Nρ 1/2) Nρ (1 Nρ 2Nρ ) , ,1 =− + k − +⌊ k ⌋ +⌊ k ⌋− k (8.7) k 1 = Cheng et al. Mathematical Sciences (2015) 2:13 Page 27 of 89

12 2 2 1 1 fN,2(�g ) 1 N ρ Nρk (2Nρk Nρk ) , (8.8) =− + k −⌊2 + ⌋ −⌊2 + ⌋ k 1 =

where x denotes the largest integer that is not greater than x. ⌊ ⌋ 10 γ Comparing with the second condition in (8.3), we see that it is satisfied for = N 1 if and only if the g satisfies

fN,1(�g )>0, (8.9)

together with

fN,2(�g ) 1 fN,2(�g )> 1, Nρk Nρk 0 (8.10) − 2 + 2 +⌊ ⌋− ≥ 1 12 1 for the case ǫ3,N ǫ4,N ǫ(2 k 1 2 N ρk ) 1, and = ⌊ ⌋ = fN,2(�g ) 0 (8.11) ≥ 1 12 1 for the case ǫ3,N ǫ4,N ǫ(2 k 1 2 Nρk ) 1. = ⌊ ⌋ =− Now we are left to check explicitly the condition (ii) in (8.3) for the various 2- and 3-plane preserving subgroups discussed in Sect. 4. First, recall that, for n a positive integer, each cusp of Ŵ0(n) is represented by a rational number of the form u / v where u and v are coprime u /v v v positive integers, v a divisor of n, and u / v is equivalent ′ ′ if and only if = ′, and u u mod (v, n/v) = ′ . (note that the infinite cusp is also represented by 1 / n.) Via direct g Co 24 computation using the data of the eigenvalues of ∈ 0 acting on , we note that among all the groups we have considered in Sect. 4, the groups U4(3), U6(2) and McL all contain a 9 3 conjugacy classe with Frame shape �g 3 /1 , and HS has a conjugacy class with Frame 5 1 = shape �g 5 /1 . One can explicitly check that f1,1(�g ) 0 for these classes and hence = 10 = 0 1 Zg (τ, z) q y the corresponding twining |0,2 11 has a non-vanishing coefficient for as τ i τ 1 i → ∞. (Note that the cusp at = is not equivalent to the cusp at ∞ in these cases.) This excludes the groups U4(3), U6(2), McL and HS as candidates for moonshine satisfying (8.3). For the subgroups of M24, a simple analysis of the cusp representatives of Ŵg Ŵ(og ) 10 = γ N < n shows that it is sufficient to verify (8.3) for = N 1 for all N|n and . For these cases, we explicitly verify that (8.9)–(8.11) are satisfied and hence the moonshine condition (8.3) is met. We therefore conclude that we have established mock modular moonshine for all but the largest of the sporadic simple Mathieu groups, together with explicit constructions of the corresponding modules. Moreoever, the corresponding twined graded characters are mock modular forms arising from Jacobi forms satisfying the distinguishing conditions (8.3), which we may recognise as furnishing a natural analogue of the powerful principal modulus property (a.k.a. genus zero property) of monstrous moonshine. The reader will note that many of the numbers which occur as dimensions of irreducible representations of M23 also occur as dimensions of irreducible representations for M24. Indeed, looking at the Tables in “Appendix C”, one is tempted to guess that there is an alternative construction, or hidden symmetry in our model, which yields an M24 -module with the same graded dimensions. In fact, the procedure we have explained for computing twinings can be carried out for any element of M24, regarded as a subgroup Cheng et al. Mathematical Sciences (2015) 2:13 Page 28 of 89

of Co0, for any such element fixes a 2-space in24 . However, there is no 2-space that is fixed by every element of a given copy of M24, and explicit computations reveal that any M24-module structure on the module we have constructed would have to involve virtual representations. This indicates that there is no direct extension to M24 of the Mathieu moonshine modules we have considered here, despite the prominent role M24 plays in incorporating the different groups of moonshine in the current setting. Nevertheless, there is a certain modification of our method for which M24 is now known to play a leading role. We refer the reader to the next, and final section for a description of this.

9 Discussion In this paper we have demonstrated that, starting with the free field Co0 module of [24], G Co one can construct explicit examples of modules for various subgroups ⊂ 0 which underlie certain mock modular forms. In particular, subgroups which preserve a 3-plane 24 N 4 N 2 (respectively 2-plane) in the give rise to = ( = ) modules with G symmetry. This gives completely explicit examples of mock modular moonshine for the smaller Mathieu groups, where the modules are known, and where the twining functions are distinguished in a manner directly similar to Mathieu moonshine. Other examples, including modules for the sporadic groups McL and HS, are also described. N 2 N 4 There are several future directions. We considered here the = and = extended chiral algebras, and the subgroups of Co0 that they preserve. Other extended chiral algebras may also yield interesting results. For instance, supersymmetric sigma models with target a Spin(7) manifold give rise to an extended chiral algebra [58], whose N 1 representations were studied in [45]. It is an extension of the = superconformal N 2 algebra where instead of adding a U(1) current (which extends the theory to an = superconformal theory), one chooses an additional Ising factor. Conjectural characters for the unitary, irreducible representations of this algebra were worked out in [2], where it was shown that there is a suggestive relation between the decomposition of the elliptic genus of a Spin(7) manifold into these characters and irreducible representations of finite groups. This connection was made precise in [12], where it was shown that the c 12 N 1 same = theory can be viewed as an SCFT with extended = symmetry, and thus yields theories with global symmetry groups M24, Co2, and Co3. The partition function twined by these symmetries, when decomposed into characters of the Spin(7) algebra, gives rise to two-component vector-valued mock modular forms encoding infinite- dimensional modules for the corresponding sporadic groups. The motivation that led, eventually, to the present study was actually to find connec- c 12 tions between geometrical target manifolds associated to = conformal field theories, and sporadic groups. The elliptic genera of Calabi-Yau fourfolds were computed in [54], for instance; their structure is reminiscent of some of the modules we have seen here, and we intend to further explore and describe some of these connections in a future publication. Likewise, hyperkähler fourfolds, as well as the Spin(7) manifolds mentioned above, provide a wide class of geometries where an analogue of the connections between M24 and K3 may be sought. Last but not least, there are suggestive connections between the trace functions in moonshine modules, and certain special properties of the underlying conformal field theory. Both the CFT appearing in monstrous moonshine and the Co0 module that Cheng et al. Mathematical Sciences (2015) 2:13 Page 29 of 89

played a starring role in this paper appear to play special roles also in AdS3 quantum gravity, where they are candidates for CFT duals to pure (super)gravity [62]. The genus zero property of the twining functions in monstrous moonshine can be reformulated as a condition that these class functions should be expressed as Rademacher sums based on a fixed polar part [7, 29]; this latter description then applies uniformly to monstrous moonshine and Mathieu moonshine. In this paper we demonstrate that a similar criterion also applies to our mock modular Mathieu moonshines. In particular, we have shown that the Jacobi forms relevant for Mathieu moonshine display a specific asymptotic behaviour near non-infinite cusps, which, in physical terms, can be interpreted as a condition on orbifolds of the theory. Pursuing a deeper understanding of this property constitutes an enticing direction for the future.

10 Endnote aSee however the recent work [26] which constructs the super vertex operator algebra X E3 underlying the = 8 case of umbral moonshine. b N 4 The observation that the decomposition into = characters of (a multiple of) the function Z(τ, z) returns positive integers that are suggestive of representations of the Mathieu group M22 was first communicated privately by Jeff Harvey to J.D. in 2010.

Authors’ contributions Each author contributing equally to all aspects of the production of this article. All authors read and approved the final manuscript. Author details 1 Institute of Physics and Korteweg‑de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Neth- erlands. 2 Department of Physics and SLAC, Stanford University, Stanford, CA 94305, USA. 3 Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA. Acknowledgments We are grateful to Jeff Harvey, Sander Mack-Crane and Daniel Whalen for conversations about related subjects. We are grateful to Daniel Baumann, and the anonymous referees, for very helpful comments on earlier drafts. We note that the appearance of dimensions of M22 representations in an N = 4 decomposition of the partition function of the model studied in this paper was described in correspondence from J. Harvey to J. Duncan in 2010. The expression (5.2) for Z(τ, z) first came to our attention during the course of conversations with S. Mack-Crane. We thank the Simons Center for Geometry and Physics, and in particular the organizers of the workshop on “Mock Modular Forms, Moonshine, and String Theory,” for hospitality when this work was initiated. S.K. is grateful to the Aspen Center for Physics for providing the rockies during the completion of this work. X.D., S.H. and S.K. are supported by the U.S. National Science Foundation Grant PHY-0756174, the Department of Energy under contract DE-AC02-76SF00515, and the John Templeton Founda- tion. J.D. is supported in part by the Simons Foundation (#316779) and by the U.S. National Science Foundation (DMS 1203162). T.W. is supported by a Research Fellowship (Grant number WR 166/1-1) of the German Research Foundation (DFG). Miranda C N Cheng is on leave from CNRS, Paris. Compliance with ethical guidelines Competing interests The authors declare that they have no competing interests.

Appendix A: Jacobi theta functions θi(τ, z) q e(τ) y e(z) We define the Jacobi theta functions as follows for = and = :

1/8 1/2 ∞ n n 1 n 1 θ1(τ, z) iq y (1 q )(1 yq )(1 y− q − ), (9.1) =− − − − n 1 = 1/8 1/2 ∞ n n 1 n 1 θ (τ, z) q y (1 q )(1 yq )(1 y− q − ), 2 = − + + n 1 = (9.2) Cheng et al. Mathematical Sciences (2015) 2:13 Page 30 of 89

∞ n n 1/2 1 n 1/2 θ (τ, z) (1 q )(1 yq − )(1 y− q − ), 3 = − + + (9.3) n 1 =

∞ n n 1/2 1 n 1/2 θ4(τ, z) (1 q )(1 yq − )(1 y− q − ). (9.4) = − − − n 1 =

Z ⋉Z2 They transform in the following way under the groupSL 2( ) .

1 1 z θ (τ, z) i α− (τ, z)θ , e( 1/8)θ1(τ 1, z) 1 = 1 −τ τ = − + µ 1 2 ( 1) + e ( τ 2 z) θ (τ, z τ µ), = − 2 + 1 + + 1 1 z θ (τ, z) α− (τ, z)θ , e( 1/8)θ (τ 1, z) 2 = 4 −τ τ = − 2 + ( 1)µe 1 ( 2τ 2 z) θ (τ, z τ µ), = − 2 + 2 + + (9.5) 1 1 z θ (τ, z) α− (τ, z)θ , θ (τ 1, z) 3 = 3 −τ τ = 4 + e 1 ( 2τ 2 z) θ (τ, z τ µ), = 2 + 3 + + 1 1 z θ (τ, z) α− (τ, z)θ , θ (τ 1, z) 4 = 2 −τ τ = 3 + ( 1) e 1 ( 2τ 2 z) θ (τ, z τ µ). = − 2 + 4 + + 2 α(τ, z) √ iτe( z ) , µ Z Here = − 2τ , and ∈ . The weight four Eisenstein series E4 can be written in terms of the Jacobi theta functions as

1 8 8 8 E4(τ) θ (τ,0) θ (τ,0) θ (τ,0) . (9.6) = 2 2 + 3 + 4

Appendix B: Character tables B1: Frame shapes and spinor representations See Tables 1, 2, 3, 4, 5, 6.

Table 1 Frame shapes and spinor characters for M22

[g] 1A 2A 3A 4A 4B 5A 6A 7A 7B 8A 11A 11B

24 8 8 6 6 4 4 4 4 4 4 4 4 2 2 3 2 3 3 3 3 2 2 2 2 2 2 g 1 1 2 1 3 1 2 4 1 2 4 1 5 1 2 3 6 1 7 1 7 1 2.4.8 1 11 1 11 Tr4096g 2,048 0 64 0 0 0 0 8 8 0 4 4

Table 2 Frame shapes and spinor characters for U4(3)

[g] 1A 2A 3A 3BCD 4AB 5A 6A 6BC 7AB 8A 9ABCD 12A

24 8 8 9 6 6 4 2 4 4 4 153.64 2 2 3 2 3 3 2 2 3 3 2 2 g 1 1 2 3 1 3 1 2 4 1 5 1 2 3 6 1 7 1 2.4.8 1 9 1.2 3.12 13 24 32 42 Rg 2,048 0 8 64 0 0 72 0 8 0 4 0 − Cheng et al. Mathematical Sciences (2015) 2:13 Page 31 of 89

Table 3 Frame shapes and spinor characters for M23

[g] 1A 2A 3A 4A 5A 6A 7AB 8A 11AB 14AB 15AB 23AB

24 8 8 6 6 4 2 4 4 4 2 2 3 2 3 3 2 2 2 2 g 1 1 2 1 3 1 2 4 1 5 1 2 3 6 1 7 1 2.4.8 1 11 1.2.7.14 1.3.5.15 1.23 Tr4096g 2,048 0 64 0 0 0 8 0 4 0 4 2

Table 4 Frame shapes and spinor characters for McL

[g] 1A 2A 3A 3B 4A 5A 5B 6A 6B 7AB

24 8 8 9 6 6 4 2 4 55 4 4 153.64 2 2 3 2 3 3 g 1 1 2 3 1 3 1 2 4 1 5 1 2 3 6 1 7 13 1 24 Tr4096g 2,048 0 8 64 0 4 0 72 0 8 − − [g] 8A 9AB 10A 11AB 12A 14AB 15AB 30AB

2 2 3 3 3 2 2 2 2 2 2 2 2.3.5.30 g 1 2.4.8 1 9 1 5.10 1 11 1.2 3.12 1.2.7.14 1 15 32 22 42 3.5 6.10 Tr4096g 0 4 20 4 0 0 2 2

Table 5 Frame shapes and spinor characters for HS

[g] 1A 2A 2B 3A 4A 4BC 5A 5BC 6A 6B

24 8 8 12 6 6 6 4 4 2 4 55 4 4 3 3 2 2 3 2 g 1 1 2 2 1 3 2 4 1 2 4 1 5 2 6 1 2 3 6 14 1 Tr4096g 2,048 0 0 64 0 0 4 0 0 0 − [g] 7A 8ABC 10A 10B 11AB 12A 15A 20AB

3 3 2 2 3 2 2 2 2 2 2 2 1.2.10.20 g 1 7 1 2.4.8 1 5.10 2 10 1 11 1 4.6 12 1.3.5.15 22 32 4.5 Tr4096g 8 0 20 0 4 0 4 0

Table 6 Frame shapes and spinor characters for U6(2)

[g] 1A 2A 2B 2C 3A 3B 3C 4A 4B 4CDE 4F 4G

24 16 8 8 12 6 6 9 6 6 8 8 8 4 2 4 4 4 4 2 4 g 1 2 1 2 2 1 3 3 1 3 1 4 4 1 2 4 2 4 1 2 4 18 13 28 24 Tr4069g 2048 0 0 0 64 –8 64 256 0 0 0 0 [g] 5A 6AB 6C 6D 6E 6F 6G 6H 7A 8A 8BCD

4 4 6 2464 153.64 2 2 2 2142.65 2 2 2 2 3 3 3 3 1484 2 2 g 1 5 1.6 1 2 3 6 1 2 3 6 2 6 1 7 1 2.4.8 2233 1232 24 34 2242 Tr4096g 0 0 0 72 0 0 0 0 8 32 0 [g] 9ABC 10A 11AB 12AB 12C 12DE 12FGH 12I 15A 18AB

3 3 2 3 2 2 3 3 2 2 2 2 3 3 2 2 2 g 1 9 1 2.10 1 11 2.3 12 1 3 4 12 1 12 1.2 3.12 2.4.6.12 1.3.5.15 1.2.18 32 52 1.4.63 2262 2.3.4.6 42 6.9 Tr4096g 4 0 4 4 16 12 0 0 4 0 Cheng et al. Mathematical Sciences (2015) 2:13 Page 32 of 89 11 11 1 1 b 11A 11B 11B 11A 1A 1 − 1 0 0 0 1 0 0 11B b 11 11 1 1 b 11B 11A 11A 11B 1A 1 − 1 0 0 0 1 0 0 11A b − 1 − 1 4A 8A 8A 8A 8A 1 − 1 1 − 1 0 0 − 1 0 0 1 8A 7 7 b 7B 7A 7A 1A 7B 1 0 − 1 1 0 0 0 0 0 0 7B b 7 7 b 7A 7B 7B 1A 7A 1 0 − 1 1 0 0 0 0 0 0 7A b 3A 2A 6A 6A 6A 1 − 1 0 0 1 0 1 − 1 1 1 1 − 2 6A 5A 5A 1A 5A 5A 1 1 0 0 0 − 1 − 1 0 1 0 0 0 5A 2A 4B 4B 4B 4B 1 1 1 1 − 1 − 1 2 − 2 − 1 0 0 1 4B 2A 4A 4A 4A 4A 1 1 1 1 3 3 − 2 − 2 − 1 0 0 1 4A )/ 2 p √ i + 3A 1A 3A 3A 3A 1 3 0 0 1 0 1 3 − 3 1 1 − 2 3A ( − 1 = p 14 . b . 22 13 , 1A 2A 2A 2A 2A 1 5 − 3 − 3 7 3 10 2 7 − 8 − 8 1 2A M 12 , 11 , 10 , , 9 , 1A 1A 1A 1A 1A 1 21 45 45 55 99 154 210 231 280 280 385 1A 8 , Character table of Character ] ] ] ] ] 2 3 5 7 11 1 2 3 4 5 6 7 8 9 10 11 12 B2: Irreducible representations B2: Irreducible 7 See Tables 7 Table [ g ] χ χ χ χ χ χ χ χ χ χ χ χ [ g [ g [ g [ g [ g Cheng et al. Mathematical Sciences (2015) 2:13 Page 33 of 89 0 0 6A 4A 12A 12A 1 − 2 0 0 1 1 − 1 1 − 1 − 1 1 0 0 0 12A 0 0 0 0 3 3 b 3 3 b + + 9C 3A 9C 9D 1 0 − 1 2 0 − 1 0 0 0 0 0 − 1 0 − 1 9D 1 1 1 1 1 1 3 3 b 3 3 b + + 9D 3A 9D 9C 1 0 − 1 2 0 − 1 0 0 0 0 0 − 1 0 − 1 9C 1 1 1 1 1 1 3 3 b 3 3 b + + 9A 3A 9A 9B 1 0 2 − 1 0 − 1 0 0 1 1 0 0 0 − 1 0 − 1 9B 1 1 1 1 3 3 b 3 3 b + + 9B 3A 9B 9A 1 0 2 − 1 0 − 1 0 0 1 1 0 0 0 − 1 0 − 1 9A 1 1 1 1 4A 8A 8A 8A 1 − 1 − 1 − 1 0 0 1 0 0 0 0 0 1 1 0 0 − 1 0 8A 0 0 7 7 7B 7A 7A 1A 1 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 1 0 7B b b 7 7 7A 7B 7B 1A 1 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 1 0 7A b b 3C 2A 6C 6C 1 − 1 3 0 1 0 0 − 1 1 1 − 2 − 2 − 1 2 − 2 2 0 0 0 0 6C 3B 2A 6B 6B 1 − 1 0 3 1 0 0 − 1 − 2 − 2 1 1 2 − 1 − 2 2 0 0 0 0 6B 3A 2A 6A 6A 1 2 0 0 1 − 3 3 5 − 2 − 2 − 2 − 2 − 1 − 1 1 2 0 0 0 0 6A 5A 5A 1A 5A 1 1 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 − 1 1 5A 2A 4B 4B 4B 1 1 − 1 − 1 2 0 1 − 2 0 0 0 0 − 1 − 1 0 0 0 0 1 0 4B 2A 4A 4A 4A 1 1 3 3 − 2 4 5 − 2 0 0 0 0 − 1 − 1 4 0 0 0 − 3 0 4A )/ 2 p √ i 3D 1A 3D 3D 1 3 − 1 − 1 0 5 0 3 1 1 1 1 0 0 − 3 2 1 1 0 − 4 3D + ( − 1 3C 1A 3C 3C 1 3 − 1 8 9 − 4 0 3 1 1 10 10 − 9 18 6 2 − 8 − 8 0 − 4 3C = p b . 3B 1A 3B 3B 1 3 8 − 1 9 − 4 0 3 10 10 1 1 18 − 9 6 2 − 8 − 8 0 − 4 3B ( 3 ) 4 U 3A 1A 3A 3A 1 − 6 8 8 9 5 27 21 10 10 10 10 − 9 − 9 − 39 − 34 − 8 − 8 0 32 3A 1A 2A 2A 2A 1 5 3 3 10 12 − 3 2 − 8 − 8 − 8 − 8 11 11 4 − 16 0 0 9 0 2A Character table of Character 1A 1A 1A 1A 1 21 35 35 90 140 189 210 280 280 280 280 315 315 420 560 640 640 729 896 1A ] ] ] ] 2 3 5 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 8 Table [ g ] χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ [ g [ g [ g [ g Cheng et al. Mathematical Sciences (2015) 2:13 Page 34 of 89 23 23 23B 23B 23B 23A 23A 23A 1A 1 − 1 0 1 0 0 0 − 1 − 1 1 1 1 1 − 1 − 1 b b 23 23 23A 23A 23A 23B 23B 23B 1A 1 − 1 0 1 0 0 0 − 1 − 1 1 1 1 1 b − 1 − 1 b 15 15 15B 15B 5A 3A 15A 15A 15B 1 − 1 0 1 1 0 0 0 − 1 0 0 0 0 b 1 1 b 15 15 15A 15A 5A 3A 15B 15B 15A 1 − 1 0 1 1 0 0 0 − 1 0 0 0 0 b 1 1 b 7 7 b b 7 7 14B 7B 14A 14A 2A 14B 14B 1 − 1 1 0 0 0 − 1 0 0 − 1 1 − − 0 0 b b 7 7 b b 7 7 14A 7A 14B 14B 2A 14A 14A 1 − 1 1 0 0 0 − 1 0 0 − 1 1 − − 0 0 b b 11 11 11B 11A 11B 11B 11A 1A 11B 1 0 − 1 0 0 0 0 0 0 1 0 1 0 0 1 b b 11 11 11A 11B 11A 11A 11B 1A 11A 1 0 − 1 0 0 0 0 0 0 1 0 1 0 0 1 b b 8A 4A 8A 8A 8A 8A 8A 1 0 0 − 1 − 1 − 1 − 1 0 0 0 0 1 0 − 1 0 0 − 1 7 7 7 7 7B 7B 7A 7A 1A 7B 7B 1 1 − 1 0 0 0 1 0 0 0 0 − 1 1 b b b b 7 7 7 7 7A 7A 7B 7B 1A 7A 7A 1 1 − 1 0 0 0 1 0 0 0 0 − 1 1 b b b b 6A 3A 2A 6A 6A 6A 6A 1 0 0 0 1 − 2 1 1 1 1 1 0 0 0 0 0 − 1 )/ 2 5A 5A 5A 1A 5A 5A 5A 1 2 0 0 0 1 1 1 − 2 0 0 1 1 0 0 0 − 1 p √ i + 4A 2A 4A 4A 4A 4A 4A 1 2 1 1 2 − 1 − 1 − 1 1 − 2 − 2 0 0 2 2 − 1 0 ( − 1 = p b . 3A 3A 1A 3A 3A 3A 3A 1 4 0 0 5 6 − 3 − 3 1 5 5 − 4 − 4 0 0 0 − 1 23 M 2A 1A 2A 2A 2A 2A 2A 1 6 − 3 − 3 22 7 7 7 13 − 14 − 14 0 0 − 18 − 18 27 8 1A 1A 1A 1A 1A 1A 1A 1 22 45 45 230 231 231 231 253 770 770 896 896 990 990 1035 2024 Character table of Character ] ] ] ] ] ] 2 3 5 7 11 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 9 Table [ g ] χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ [ g [ g [ g [ g [ g [ g Cheng et al. Mathematical Sciences (2015) 2:13 Page 35 of 89 15 15 30B 15B 10A 6A 30A 30A 1 − 2 2 1 0 − 1 1 0 − 1 0 − 1 0 0 0 0 0 0 0 b b 15 15 30A 15A 10A 6A 30B 30B 1 − 2 2 1 0 − 1 1 0 − 1 0 − 1 0 0 0 0 0 0 0 b b 15 15 15B 15B 5A 3A 15A 15A 1 0 0 − 1 0 1 1 0 − 1 0 1 0 2 2 0 0 0 0 b b 15 15 15A 15A 5A 3A 15B 15B 1 0 0 − 1 0 1 1 0 − 1 0 1 0 2 2 0 0 0 0 b b 7 7 b 7 7 14B 7B 14A 14A 2A 14B 1 − 1 0 0 0 0 0 1 − 1 − 1 1 0 0 0 0 0 b b − − b 7 7 b 7 7 14A 7A 14B 14B 2A 14A 1 − 1 0 0 0 0 0 1 − 1 − 1 1 0 0 0 0 0 b b − b − 12A 6A 4A 12A 12A 12A 1 − 1 − 1 1 1 1 − 1 0 0 1 0 0 0 0 0 0 1 1 0 0 11 11 11B 11A 11B 11B 11A 1A 1 0 0 − 1 0 0 1 0 0 1 0 − 1 0 0 0 0 0 0 b b 11 11 11A 11B 11A 11A 11B 1A 1 0 0 − 1 0 0 1 0 0 1 0 − 1 0 0 0 0 0 0 b b 10A 5A 10A 2A 10A 10A 1 1 2 − 2 1 1 0 0 0 − 1 1 0 2 3 − 1 0 0 0 0 0 9B 9A 3A 9A 9B 9A 1 1 0 0 − 1 − 1 − 1 − 1 − 2 1 1 0 0 0 0 1 0 0 0 0 9A 9B 3A 9B 9A 9B 1 1 0 0 − 1 − 1 − 1 − 1 − 2 1 1 0 0 0 0 1 0 0 0 0 8A 4A 8A 8A 8A 8A 1 0 − 1 0 0 0 0 0 0 0 0 0 0 1 0 − 1 − 1 − 1 0 0 7 7 7 7 b b 7B 7B 7A 7A 1A 7B 1 1 0 0 0 0 0 0 0 − 1 − 1 − 1 − 1 0 0 0 − − b − − b 7 7 7 7 b b 7A 7A 7B 7B 1A 7A 1 1 0 0 0 0 0 0 0 − 1 − 1 − 1 − 1 0 0 0 3 − b − − b − 3 b + 6B 3B 2A 6B 6B 6B 1 0 − 2 1 1 1 0 0 1 − 2 2 − 1 0 0 1 0 0 − 2 − 2 3 1 = a 6A 3A 2A 6A 6A 6A 1 3 7 1 7 7 0 0 − 5 4 − 4 5 − 6 0 4 0 0 − 5 − 5 0 , )/ 2 p 5B 5B 5B 1A 5B 5B 1 2 1 2 0 0 1 1 0 0 0 0 2 − 2 − 1 − 1 − 1 0 0 0 √ i + 5A 5A 5A 1A 5A 5A 1 − 3 6 2 − 5 − 5 − 4 − 4 0 − 5 − 5 0 2 3 19 − 6 − 6 0 0 0 ( − 1 = 4A 2A 4A 4A 4A 4A 1 2 − 1 4 − 2 − 2 0 0 2 0 0 4 0 3 0 3 3 − 2 − 2 − 3 p b 3B 3B 1A 3B 3B 3B 1 4 6 9 5 5 − 4 − 4 13 10 10 − 9 0 0 9 0 0 6 6 − 5 3A 3A 1A 3A 3A 3A 1 − 5 15 9 − 13 − 13 32 32 − 5 − 44 − 44 45 54 0 36 0 0 15 15 40 2A 1A 2A 2A 2A 2A 1 6 7 28 − 14 − 14 0 0 70 64 − 64 20 − 48 63 − 56 − 45 − 45 10 10 105 Character table of McL . Character 1A 1A 1A 1A 1A 1A 1 22 231 252 770 770 896 896 1750 3520 3520 4500 4752 5103 5544 8019 8019 8250 8250 9625 ] ] ] ] ] 2 3 5 7 11 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 10 Table [ g ] χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ [ g [ g [ g [ g [ g Cheng et al. Mathematical Sciences (2015) 2:13 Page 36 of 89 15 15 b b 30B 0 0 − − 15 15 b b 30A 0 0 − − 15 15 15B 0 0 b b 15 15 15A 0 0 b b 14B 0 0 0 0 14A 0 0 0 0 12A − 1 − 1 0 0 11B 0 0 0 0 11A 0 0 0 0 10A − 1 − 1 0 0 9B 0 0 a a 9A a 0 0 a 8A 0 0 1 1 7B 0 0 0 0 7A 0 0 0 0 6B 0 0 0 0 6A 0 0 3 3 5B 1 1 0 0 5A 6 6 − 5 − 5 4A 0 0 − 1 − 1 3B − 8 − 8 0 0 3A − 80 − 80 27 27 2A 0 0 − 21 − 21 continued 1A 9856 9856 10395 10395 21 22 23 24 10 Table [ g ] χ χ χ χ Cheng et al. Mathematical Sciences (2015) 2:13 Page 37 of 89 5 5 a 0 0 20B 0 0 1 0 0 0 0 10A 20B 4A 20B 20A 1 − 1 0 − 2 0 0 0 0 1 1 a 5 5 0 0 a 20A 0 0 1 0 0 0 0 10A 20A 4A 20A 20B 1 − 1 0 − 2 0 0 0 0 1 1 a 1 1 15A 1 − 1 0 − 1 0 − 1 − 1 0 15A 5B 3A 15A 15A 1 − 1 0 1 1 1 − 1 1 0 0 0 0 0 12A 0 0 0 0 − 1 − 1 1 − 1 6B 4A 12A 12A 12A 1 0 − 1 1 − 1 − 1 0 0 0 1 − 1 11 11 b 11B 0 0 0 0 1 0 0 0 11A 11B 11B 11A 1A 1 0 0 0 0 0 − 1 0 0 0 0 b 11 11 b 11A 0 0 0 0 1 0 0 0 11B 11A 11A 11B 1A 1 0 0 0 0 0 − 1 0 0 0 0 b 10B − 1 1 1 0 − 2 1 0 1 1 0 5B 10B 2B 10B 10B 1 − 2 1 0 0 0 1 1 − 1 0 0 10A 0 0 0 2 − 1 0 0 0 0 1 5A 10A 2A 10A 10A 1 1 − 2 0 0 0 0 2 1 − 1 1 8C 1 0 0 0 0 0 0 1 − 1 0 4C 8C 8C 8C 8C 1 0 − 1 0 − 2 2 1 − 1 − 1 0 0 8B 1 0 0 0 0 0 0 1 − 1 0 4C 8B 8B 8B 8B 1 0 − 1 0 2 − 2 1 − 1 − 1 0 0 8A 1 0 0 0 0 0 − 2 1 1 0 4B 8A 8A 8A 8A 1 0 1 0 0 0 − 1 − 1 1 − 2 0 7A − 1 0 0 − 1 0 1 0 0 0 0 7A 7A 7A 1A 7A 1 1 0 0 0 0 0 0 0 0 0 6B − 2 0 0 2 0 0 − 1 − 1 − 1 1 3A 2A 6B 6B 6B 1 0 1 1 1 1 0 − 2 0 1 1 6A 0 − 2 − 2 0 0 − 2 1 − 1 1 1 3A 2B 6A 6A 6A 1 − 2 1 1 − 1 − 1 2 0 0 − 1 1 5C 0 1 1 1 1 − 2 0 0 0 0 5C 5C 1A 5C 5C 1 2 2 − 1 − 1 − 1 0 1 − 2 0 0 5B − 5 1 1 − 4 6 − 7 0 5 5 0 5B 5B 1A 5B 5B 1 2 − 3 4 4 4 5 1 3 0 0 p √ i = 5A 0 − 4 − 4 6 11 8 0 0 0 − 5 5A 5A 1A 5A 5A 1 − 3 2 4 4 4 0 6 − 7 − 5 − 5 p a , 4C 1 0 0 0 − 2 0 2 − 3 1 − 2 2A 4C 4C 4C 4C 1 2 1 − 2 2 2 3 − 1 1 − 2 − 2 )/ 2 p √ i 4B 1 0 0 0 − 2 0 6 5 − 3 − 2 2A 4B 4B 4B 4B 1 2 5 6 − 2 − 2 − 1 − 1 5 2 − 2 + ( − 1 = 4A − 15 0 0 0 6 0 − 10 5 − 35 − 10 2A 4A 4A 4A 4A 1 − 6 5 − 2 − 10 − 10 15 15 21 − 14 − 10 p b 3A 6 − 4 − 4 − 6 0 4 − 5 − 1 − 1 5 3A 1A 3A 3A 3A 1 4 5 1 1 1 4 6 0 5 5 2B 9 16 16 0 18 16 10 − 19 1 10 1A 2B 2B 2B 2B 1 − 2 1 10 − 10 − 10 11 − 9 9 − 10 10 2A 25 0 0 32 − 6 0 − 10 5 5 − 14 1A 2A 2A 2A 2A 1 6 13 10 10 10 15 7 21 34 − 14 Character table of HS . Character 1A 825 896 896 1056 1386 1408 1750 1925 1925 770 1A 1A 1A 1A 1A 1 22 77 154 154 154 175 231 693 770 770 ] ] ] ] ] 2 3 5 7 11 13 14 15 16 17 18 19 20 21 12 1 2 3 4 5 6 7 8 9 10 11 11 Table [ g ] χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ [ g [ g [ g [ g [ g Cheng et al. Mathematical Sciences (2015) 2:13 Page 38 of 89 20B − 1 0 0 20A − 1 0 0 15A 0 0 1 12A 0 1 0 11B 1 0 − 1 11A 1 0 − 1 10B 0 0 − 1 10A − 1 0 0 8C 0 0 0 8B 0 0 0 8A 0 0 0 7A 0 − 1 1 6B 0 1 0 6A 0 − 1 2 5C 0 0 0 5B 0 0 − 5 5A − 5 0 0 4C 0 2 0 4B − 8 2 0 4A 24 10 0 3A 0 5 − 4 2B 0 − 10 − 16 2A 24 − 50 0 continued 1A 2520 2750 3200 22 23 24 11 Table [ g ] χ χ χ Cheng et al. Mathematical Sciences (2015) 2:13 Page 39 of 89

Table 12 Character table of U6(2) - Part I

[g] 1A 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 5A g2 1A 1A 1A 1A 3A 3B 3C 2A 2A 2B 2B 2B 2B 2B 5A [ ] g3 1A 2A 2B 2C 1A 1A 1A 4A 4B 4C 4D 4E 4F 4G 5A [ ] g5 1A 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 1A [ ] g7 1A 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 5A [ ] g11 1A 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 5A [ ] χ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 χ2 22 10 6 2 4 5 4 6 2 2 2 2 2 2 2 − − − − − χ3 231 39 7 9 6 15 6 23 7 1 1 1 1 1 1 − − − − − − χ4 252 60 28 12 9 9 9 12 4 4 4 4 4 4 2 − χ5 385 95 17 7 25 7 2 1 9 5 5 5 7 3 0 − − − − − χ6 440 120 24 8 35 8 1 24 8 0 0 0 8 0 0 − χ7 560 80 16 16 20 20 2 16 16 0 0 0 0 0 0 − − − χ8 616 24 40 8 14 5 13 8 8 8 8 8 0 0 1 − − − χ9 770 30 14 10 5 13 5 34 6 2 2 2 2 2 0 − − − − − − − − χ10 770 30 14 10 5 13 5 34 6 2 2 2 2 2 0 − − − − − − − − χ11 1155 195 35 19 30 6 3 13 3 11 5 5 3 3 0 − − − − χ12 1155 195 35 19 30 6 3 13 3 5 11 5 3 3 0 − − − − χ13 1155 195 35 19 30 6 3 13 3 5 5 11 3 3 0 − − − − χ14 1386 246 58 30 36 9 9 6 2 2 2 2 6 6 1 − − − − − − − χ15 1540 260 4 28 55 1 1 12 12 4 4 4 4 4 0 − − − − χ16 3080 440 40 8 65 2 7 40 8 0 0 0 8 0 0 − − − − − χ17 3080 440 40 8 65 2 7 40 8 0 0 0 8 0 0 − − − − − χ18 3520 320 64 0 10 44 10 64 0 0 0 0 0 0 0 − − χ19 4620 180 44 36 15 57 12 28 20 4 4 4 4 4 0 − − − − − − χ20 4928 320 64 64 4 68 5 0 0 0 0 0 0 0 2 − − χ21 5544 24 56 24 9 36 9 72 24 0 0 0 8 0 1 − − − − χ22 6160 400 48 16 40 50 4 48 16 0 0 0 0 0 0 − − − χ23 6160 400 48 16 40 50 4 48 16 0 0 0 0 0 0 − − − χ24 6930 690 98 42 45 36 9 18 6 6 6 6 10 2 0 − − − − χ25 8064 384 128 0 36 36 18 0 0 0 0 0 0 0 1 − − − χ26 9240 360 88 8 30 33 3 8 8 24 8 8 0 0 0 − − − − − − − − χ27 9240 360 88 8 30 33 3 8 8 8 24 8 0 0 0 − − − − − − − − χ28 9240 360 88 8 30 33 3 8 8 8 8 24 0 0 0 − − − − − − − − χ29 10395 315 21 45 0 27 0 75 13 1 1 1 3 1 0 − − − − − − − χ30 10395 315 21 45 0 27 0 75 13 1 1 1 3 1 0 − − − − − − − χ31 10395 315 21 45 0 27 0 21 19 15 1 1 3 1 0 − − − − − − χ32 10395 315 21 45 0 27 0 21 19 1 15 1 3 1 0 − − − − − − χ33 10395 315 21 45 0 27 0 21 19 1 1 15 3 1 0 − − − − − − χ34 11264 1024 0 0 104 32 4 0 0 0 0 0 0 0 1 − − χ35 13860 420 100 36 45 9 18 84 20 4 4 4 4 4 0 − − − χ36 14784 1344 64 0 114 12 6 64 0 0 0 0 0 0 1 − − − − χ37 18711 1161 57 63 81 0 0 9 15 3 3 3 1 5 1 − − − − − χ38 18711 1431 87 9 81 0 0 9 9 9 9 9 1 1 1 − − − − − − − − χ39 20790 810 6 18 0 54 0 6 14 6 6 6 6 2 0 − − − − − χ40 20790 810 6 18 0 54 0 6 14 6 6 6 6 2 0 − − − − − χ41 24640 960 64 64 20 92 11 0 0 0 0 0 0 0 0 − − − − − χ42 25515 405 117 27 0 0 0 27 21 3 3 3 3 3 0 − − − χ43 25515 405 117 27 0 0 0 27 21 3 3 3 3 3 0 − − − χ44 32768 0 0 0 64 64 8 0 0 0 0 0 0 0 2 − − − χ45 37422 270 30 54 81 0 0 18 6 6 6 6 2 6 2 − − − − − − − χ46 40095 1215 81 9 0 0 0 81 9 3 3 3 9 3 0 − − − − − Cheng et al. Mathematical Sciences (2015) 2:13 Page 40 of 89

√ √ Table 13 Character table of U6(2) - Part II. aq = q + 6i 3, c−3 = (1 − 3i 3)/2

[g] 6A 6B 6C 6D 6E 6F 6G 6H 7A 8A 8B 8C 8D 9A 9B 9C g2 3B 3B 3A 3B 3A 3C 3C 3C 7A 4B 4C 4D 4E 9B 9A 9C [ ] g3 2A 2A 2A 2B 2B 2A 2B 2C 7A 8A 8B 8C 8D 3B 3B 3B [ ] g5 6B 6A 6C 6D 6E 6F 6G 6H 7A 8A 8B 8C 8D 9B 9A 9C [ ] g7 6A 6B 6C 6D 6E 6F 6G 6H 1A 8A 8B 8C 8D 9A 9B 9C [ ] g11 6B 6A 6C 6D 6E 6F 6G 6H 7A 8A 8B 8C 8D 9B 9A 9C [ ] χ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 χ2 1 1 4 3 0 2 0 2 1 2 0 0 0 1 1 1 − − − − χ3 3 3 6 7 2 0 2 0 0 3 1 1 1 0 0 0 − − − − − χ4 3 3 9 1 1 3 1 3 0 0 0 0 0 0 0 0 − − χ5 5 5 1 1 5 2 2 2 0 1 1 1 1 1 1 1 − − − − − − − χ6 12 12 3 0 3 3 3 1 1 0 0 0 0 2 2 1 − − − χ7 8 8 4 4 4 2 2 2 0 0 0 0 0 1 1 2 − − − − − − − − χ8 3 3 6 5 2 3 1 1 0 0 0 0 0 2 2 2 − − − − − − − χ9 a 3 a 3 3 7 1 3 1 1 0 2 0 0 0 1 1 1 − − − − − − − χ10 a 3 a 3 3 7 1 3 1 1 0 2 0 0 0 1 1 1 − − − − − − − χ11 6 6 6 2 2 3 1 1 0 1 3 1 1 0 0 0 − − − − − χ12 6 6 6 2 2 3 1 1 0 1 1 3 1 0 0 0 − − − − − χ13 6 6 6 2 2 3 1 1 0 1 1 1 3 0 0 0 − − − − − χ14 3 3 12 1 4 3 1 3 0 2 0 0 0 0 0 0 − − − − − − χ15 17 17 1 1 5 1 1 1 0 0 0 0 0 1 1 1 − − − − χ16 a 8 a 8 1 2 1 1 1 1 0 0 0 0 0 c 3 c 3 1 − − − − ¯− − χ17 a 8 a 8 1 2 1 1 1 1 0 0 0 0 0 c 3 c 3 1 − − − ¯− − − χ18 4 4 14 4 2 4 2 0 1 0 0 0 0 1 1 1 − − − − χ19 9 9 9 7 5 0 4 0 0 0 0 0 0 0 0 0 − − χ20 4 4 4 4 4 5 1 1 0 0 0 0 0 1 1 2 − − − − − χ21 12 12 9 4 1 3 1 3 0 0 0 0 0 0 0 0 − − χ22 a4 a4 8 6 0 2 0 2 0 0 0 0 0 c 3 c 3 1 − − − − − −¯− χ23 a4 a4 8 6 0 2 0 2 0 0 0 0 0 c 3 c 3 1 − − − −¯− − − χ24 12 12 3 4 5 3 1 3 0 2 0 0 0 0 0 0 − − − − − − − χ25 12 12 12 4 4 0 2 0 0 0 0 0 0 0 0 0 − − − − χ26 9 9 6 1 2 3 1 1 0 0 0 0 0 0 0 0 − − − χ27 9 9 6 1 2 3 1 1 0 0 0 0 0 0 0 0 − − − χ28 9 9 6 1 2 3 1 1 0 0 0 0 0 0 0 0 − − − χ29 9 9 0 3 0 0 0 0 0 1 1 1 1 0 0 0 − − − χ30 9 9 0 3 0 0 0 0 0 1 1 1 1 0 0 0 − − − χ31 9 9 0 3 0 0 0 0 0 1 3 1 1 0 0 0 − − − − χ32 9 9 0 3 0 0 0 0 0 1 1 3 1 0 0 0 − − − − χ33 9 9 0 3 0 0 0 0 0 1 1 1 3 0 0 0 − − − − χ34 16 16 8 0 0 4 0 0 1 0 0 0 0 1 1 1 − − − − χ35 3 3 3 1 5 0 2 0 0 0 0 0 0 0 0 0 − − − − χ36 12 12 6 4 2 0 2 0 0 0 0 0 0 0 0 0 − − − − − χ37 0 0 9 0 3 0 0 0 0 1 1 1 1 0 0 0 − − χ38 0 0 9 0 3 0 0 0 0 1 1 1 1 0 0 0 − − − − − χ39 18i√3 18i√3 0 6 0 0 0 0 0 2 0 0 0 0 0 0 − − χ40 18i√3 18i√3 0 6 0 0 0 0 0 2 0 0 0 0 0 0 − − χ41 12 12 12 4 4 3 1 1 0 0 0 0 0 2 2 1 − − − χ42 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 − − − − χ43 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 − − − − χ44 0 0 0 0 0 0 0 0 1 0 0 0 0 2 2 1 − Cheng et al. Mathematical Sciences (2015) 2:13 Page 41 of 89

Table 13 continued

χ45 0 0 9 0 3 0 0 0 0 2 0 0 0 0 0 0 − − χ46 0 0 0 0 0 0 0 0 1 3 1 1 1 0 0 0 −

√ 2 6 3 1 √ 2 Table 14 Character√ table of U6( ) - Part III. aq = q + i , bp = (− + i p)/ , n 3 dm = m + in

[g] 10A 11A 11B 12A 12B 12C 12D 12E 12F 12G 12H 12I 15A 18A 18B g2 5A 11B 11A 6B 6A 6C 6B 6A 6D 6D 6D 6E 15A 9B 9A [ ] g3 10A 11A 11B 4A 4A 4A 4B 4B 4C 4D 4E 4F 5A 6A 6B [ ] g5 2A 11A 11B 12B 12A 12C 12E 12D 12F 12G 12H 12I 3A 18B 18A [ ] g7 10A 11B 11A 12A 12B 12C 12D 12E 12F 12G 12H 12I 15A 18A 18B [ ] g11 10A 1A 1A 12B 12A 12C 12E 12D 12F 12G 12H 12I 15A 18B 18A [ ] χ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 χ2 0 0 0 3 3 0 1 1 1 1 1 2 1 1 1 − − − − − − − − − χ3 1 0 0 5 5 2 1 1 1 1 1 2 1 0 0 − − − − χ4 0 1 1 3 3 3 1 1 1 1 1 1 1 0 0 − − − − − − χ5 0 0 0 1 1 1 3 3 1 1 1 1 0 1 1 − − − − − − χ6 0 0 0 6 6 3 2 2 0 0 0 1 0 0 0 − χ7 0 1 1 2 2 4 2 2 0 0 0 0 0 1 1 − − − − χ8 1 0 0 1 1 2 1 1 1 1 1 0 1 0 0 − 3 − 3 − 1 −1 − − − 1 1 χ9 0 0 0 d d 1 d d 1 1 1 1 0 d d −2 ¯ −2 0− 0 − 0 0− −3 −3 1 1 1 1 χ10 0 0 0 d−2 d−2 1 d0 d0− 1 1 1 1 0 d0− d0 ¯− − − χ11 0 0 0 4 4 2 0 0 2 2 2 0 0 0 0 − − − − χ12 0 0 0 4 4 2 0 0 2 2 2 0 0 0 0 − − − − χ13 0 0 0 4 4 2 0 0 2 2 2 0 0 0 0 − − − − χ14 1 0 0 3 3 0 1 1 1 1 1 0 1 0 0 − − − χ15 0 0 0 3 3 3 3 3 1 1 1 1 0 1 1 − 3 − 3 − 1 − 1 − 1 2 − 1 2 χ16 0 0 0 d d 1 d d 0 0 0 1 0 / / −5 ¯ −5 1− ¯1− d−1/2 d¯ −1/2 −3 −3 1 1 −1 2 −1 2 χ17 0 0 0 d d 1 d d 0 0 0 1 0 / / ¯ −5 −5 ¯1− 1− d−1/2 d−1/2 − − ¯− − χ18 0 0 0 8 8 2 0 0 0 0 0 0 0 1 1 − − − χ19 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 − χ20 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 − − χ21 1 0 0 0 0 3 0 0 0 0 0 1 1 0 0 − − 1 1 1/2 1/2 χ22 0 0 0 6b3 6b3 0 d d 0 0 0 0 0 ¯ 1− ¯1− d−1/2 d−1/2 − − − ¯− 1 1 1/2 1/2 χ23 0 0 0 6b3 6b3 0 d d 0 0 0 0 0 ¯ ¯1− 1− d−1/2 d−1/2 − − ¯− − χ24 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 − χ25 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 − − χ26 0 0 0 1 1 2 1 1 3 1 1 0 0 0 0 − − χ27 0 0 0 1 1 2 1 1 1 3 1 0 0 0 0 − − χ28 0 0 0 1 1 2 1 1 1 1 3 0 0 0 0 − 2 2 − χ29 0 0 0 a 3 a 3 0 d− d− 1 1 1 0 0 0 0 −¯− − − 1 ¯ 1 − − − −2 −2 χ30 0 0 0 a 3 a 3 0 d−1 d−1 1 1 1 0 0 0 0 − − −¯− ¯− − − − − χ31 0 0 0 3 3 0 1 1 3 1 1 0 0 0 0 − − − − χ32 0 0 0 3 3 0 1 1 1 3 1 0 0 0 0 − − − − χ33 0 0 0 3 3 0 1 1 1 1 3 0 0 0 0 − − − − χ34 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 − − χ35 0 0 0 3 3 3 1 1 1 1 1 1 0 0 0 − − − χ36 1 0 0 8 8 2 0 0 0 0 0 0 1 0 0 − χ37 1 0 0 0 0 3 0 0 0 0 0 1 1 0 0 − − − Cheng et al. Mathematical Sciences (2015) 2:13 Page 42 of 89

Table 14 continued

χ38 1 0 0 0 0 3 0 0 0 0 0 1 1 0 0 − 1 − χ39 0 0 0 6b3 6 0 d 1 0 0 0 0 0 0 0 b¯3 1 d¯ 1 6 −1 −1 χ40 0 0 0 6b3 b3 0 d 1 d 1 0 0 0 0 0 0 0 ¯ ¯− − χ41 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

χ42 0 b11 b11 0 0 0 0 0 0 0 0 0 0 0 0 − −¯ χ43 0 b11 b11 0 0 0 0 0 0 0 0 0 0 0 0 −¯ − χ44 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 − − χ45 0 0 0 0 0 3 0 0 0 0 0 1 1 0 0 − χ46 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Appendix C: Coefficient tables See Tables 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28.

Table 15 The twined series for M22. The table shows the Fourier coefficients multiplying −D/12 q in the q-expansion of the function hg,1(τ) −D[g] 1A 2A 3A 4AB 5A 6A 7AB 8A 11AB

1 2 2 2 2 2 2 2 2 2 − − − − − − − − − − 11 560 16 2 0 0 2 0 0 1 − − 23 8470 54 8 2 0 0 0 2 0 − − 35 70576 144 16 0 4 0 2 0 0 − − 47 435820 332 4 4 0 4 0 0 0 − 59 2187328 704 50 0 8 2 4 0 0 − − − − 71 9493330 1394 58 6 0 2 0 2 0 − − 83 36792560 2640 38 0 0 6 0 0 2 − 95 130399766 4822 172 6 14 4 0 2 2 − − − 107 429229920 8480 174 0 0 2 0 0 2 − − − 119 1327987562 14506 104 6 22 8 6 2 0 − − 131 3895785632 24288 502 0 8 6 4 0 2 − − − − 143 10912966810 39770 466 10 0 2 10 2 2 − − 155 29351354032 63888 316 0 28 12 0 0 0 − − 167 76141761850 101018 1232 14 0 8 0 2 0 − − − 179 191223891936 157344 1098 0 56 6 0 0 0 − − 191 466389602756 241764 710 12 14 18 0 0 0 − − 203 1107626293840 366960 2876 0 0 12 10 0 0 − − − 215 2567229121428 550676 2472 12 62 8 6 4 4 − − 227 5818567673360 817840 1658 0 0 26 20 0 2 − − 239 12917927405852 1203068 6148 20 112 20 0 0 4 − − 251 28135083779792 1753840 5174 0 48 10 4 0 0 − − − − 263 60194978116000 2535488 3382 24 0 34 0 4 2 − − − 275 126660856741328 3637232 12634 0 112 26 0 0 3 − − − − − 287 262393981258310 5179526 10430 22 0 14 20 2 0 − Cheng et al. Mathematical Sciences (2015) 2:13 Page 43 of 89 0 0 0 4 0 0 0 − 5 0 − 1 3 0 − 1 − 1 1 0 0 0 − 3 0 0 1 0 1 1 11AB 3 16 0 0 0 − 12 0 0 − 2 8 0 0 0 − 6 0 0 1 4 0 0 0 − 2 0 0 1 8A (τ) ,2 g h 0 − 10 0 − 20 18 9 0 0 − 5 0 − 11 10 3 0 0 − 4 0 − 4 5 1 0 0 − 1 0 1 7AB − 3 4 2 0 − 3 − 6 − 1 1 4 2 − 1 − 2 0 3 1 0 − 2 − 2 − 1 0 3 1 − 1 − 1 1 6A 196 0 − 84 0 − 51 104 0 − 42 0 − 28 48 0 − 17 0 − 14 22 0 − 6 0 − 6 7 0 − 1 0 1 5A in the q -expansion of the function / 12 D − q − 697 608 496 − 420 − 352 280 244 − 202 − 166 144 112 − 88 − 72 52 44 − 36 − 27 24 16 − 12 − 8 4 4 − 2 1 4AB − 15099 4228 6146 − 7480 2077 2946 − 3545 981 1340 − 1582 431 558 − 656 183 217 − 248 70 70 − 81 24 19 − 19 7 3 1 3A − 2097 1792 − 496 420 − 1056 864 − 244 202 − 494 416 − 112 88 − 216 168 − 44 36 − 83 64 − 16 12 − 24 16 − 4 2 1 2A . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . 22 M 219021850730991 105323108387200 49851590654096 23199658816580 10602363945184 4751675601024 2085157528300 894352929498 374172530930 152335803872 60186506768 23000871960 8469129448 2990338680 1006567156 320626372 95731405 26455264 6649200 1481964 281512 42560 4444 210 1 1A The twined series for The [ g ] 284 272 260 248 236 224 212 200 188 176 164 152 140 128 116 104 92 80 68 56 44 32 20 8 − 4 16 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 44 of 89 0 0 12A 0 − 2 − 2 0 − 2 0 − 2 0 0 0 0 0 0 0 − 2 0 − 2 0 0 0 0 0 2 14 − 32 9ABCD 32 − 2 11 − 1 1 1 − 2 4 1 − 7 2 3 − 7 5 4 − 11 13 3 − 22 10 9 − 22 17 (τ) ,1 0 − 4 8A 0 − 2 − 2 0 2 0 0 0 − 2 0 − 2 0 2 0 2 0 − 2 0 0 0 4 0 0 g h 0 7AB 0 − 2 20 0 0 2 0 − 4 0 0 0 0 6 4 − 10 0 0 0 0 10 6 − 20 0 − 4 − 34 6BC − 26 − 2 14 2 0 0 − 4 − 2 2 6 4 − 2 − 8 − 6 2 12 8 − 6 − 18 − 12 8 26 20 − 10 − 58 6A − 26 − 2 38 2 0 0 2 − 2 − 4 6 10 − 2 − 14 − 6 14 12 − 4 − 6 − 6 − 12 − 4 26 38 − 10 in the q -expansion of the function 0 5A − 112 − 2 0 0 0 − 4 0 8 0 0 − 14 0 22 − 8 0 − 28 0 56 − 14 0 − 62 0 112 − 48 / 12 D − q − 24 4AB 0 − 2 22 0 − 2 0 4 0 − 6 0 6 0 − 6 0 10 0 − 14 0 12 0 − 12 0 20 0 3382 3BCD − 12634 − 2 10430 2 − 8 16 4 − 50 58 38 − 172 174 104 − 502 466 316 − 1232 1098 710 − 2876 2472 1658 − 6148 5174 − 5067686 3A − 7287046 − 2 − 10348570 − 34 − 116 − 272 − 662 − 1454 − 2732 − 5254 − 9802 − 16782 − 28930 − 49066 − 79058 − 127484 − 203264 − 313578 − 482842 − 736772 − 1098876 − 1634074 − 2412226 − 3502486 . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . 2535488 2A − 3637232 − 2 5179526 − 16 54 − 144 332 − 704 1394 − 2640 4822 − 8480 14506 − 24288 39770 − 63888 101018 − 157344 241764 − 366960 550676 − 817840 1203068 − 1753840 ( 3 ) 4 U The twined series for The 60194978116000 1A 126660856741328 − 2 262393981258310 560 8470 70576 435820 2187328 9493330 36792560 130399766 429229920 1327987562 3895785632 10912966810 29351354032 76141761850 191223891936 466389602756 1107626293840 2567229121428 5818567673360 12917927405852 28135083779792 [ g ] 263 17 Table 275 − 1 287 11 23 35 47 59 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239 251 − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 45 of 89 0 − 1 − 2 1 − 1 2 0 1 2 0 1 − 1 0 0 0 1 0 1 1 1 1 − 1 1 2 12A − 2 − 16 − 16 25 − 6 − 17 24 − 4 − 10 11 − 9 − 5 9 1 − 5 7 − 2 − 3 3 − 2 − 1 1 0 − 30 1 34 9ABCD (τ) ,2 g 0 0 0 − 12 0 0 − 2 8 0 0 0 − 6 0 0 1 4 0 0 0 − 2 0 0 3 1 16 8A h 0 − 20 18 9 0 0 − 5 0 − 11 10 3 0 0 − 4 0 − 4 5 1 0 0 − 1 0 0 1 − 10 7AB 2 0 − 3 − 6 − 1 1 4 2 − 1 − 2 0 3 1 0 − 2 − 2 − 1 0 3 1 − 1 − 1 − 3 1 4 6BC − 994 840 − 705 582 − 487 403 − 326 272 − 223 178 − 144 117 − 89 72 − 56 40 − 31 24 − 15 13 − 7 5 − 1401 1 1186 6A in the q -expansion of the function / 12 D − 84 0 − 51 104 0 − 42 0 − 28 48 0 − 17 0 − 14 22 0 − 6 0 − 6 7 0 − 1 0 196 1 0 5A − q 496 − 420 − 352 280 244 − 202 − 166 144 112 − 88 − 72 52 44 − 36 − 27 24 16 − 12 − 8 4 4 − 2 − 697 1 608 4AB 6146 − 7480 2077 2946 − 3545 981 1340 − 1582 431 558 − 656 183 217 − 248 70 70 − 81 24 19 − 19 7 3 − 15099 1 4228 3BCD 4635278 3186440 2189455 1485462 993025 662499 436490 280208 181097 114066 69904 42861 25759 14440 8440 4552 2313 1176 577 197 97 21 9475383 1 6654706 3A − 496 420 − 1056 864 − 244 202 − 494 416 − 112 88 − 216 168 − 44 36 − 83 64 − 16 12 − 24 16 − 4 2 − 2097 1 1792 2A . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . ( 3 ) 4 U The twined series for The 49851590654096 23199658816580 10602363945184 4751675601024 2085157528300 894352929498 374172530930 152335803872 60186506768 23000871960 8469129448 2990338680 1006567156 320626372 95731405 26455264 6649200 1481964 281512 42560 4444 210 219021850730991 1 105323108387200 1A [ g ] 260 248 236 224 212 200 188 176 164 152 140 128 116 104 92 80 68 56 44 32 20 8 284 − 4 272 18 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 46 of 89 23 23 b 23B 0 − 1 0 0 0 − 1 0 0 0 0 1 0 0 0 0 0 0 − 1 0 1 0 1 0 b 2 )/ p √ i 23 23 + b 23A 0 − 1 0 0 0 − 1 0 0 0 0 1 0 0 0 0 0 0 − 1 0 1 0 1 0 b − 1 ( = p b . 15AB 0 − 1 1 0 1 − 1 0 0 0 0 − 1 0 0 0 1 − 2 0 0 1 1 − 1 0 0 1 1 (τ) 2 1 , g ˜ h 14AB 1 − 1 0 0 0 0 0 0 0 − 1 0 0 0 0 − 1 0 1 0 0 1 0 2 0 − 2 0 11AB 0 − 1 0 0 1 0 − 1 − 1 0 − 1 2 0 0 0 0 3 0 − 2 − 1 0 − 4 3 0 0 0 8A 1 − 1 − 1 0 0 − 1 − 1 2 0 2 2 − 3 − 1 − 2 − 4 4 1 2 6 − 5 0 − 4 − 7 7 1 in the q -expansion of the function 7AB − 1 − 1 0 0 0 − 2 4 2 0 − 5 0 0 − 6 10 3 0 − 9 0 0 − 11 22 8 0 − 22 0 / 24 D − q 6A − 2 − 1 1 1 − 1 3 − 3 3 − 4 4 − 6 6 − 5 9 − 11 10 − 11 15 − 17 16 − 19 22 − 26 29 − 29 5A 0 − 1 − 5 0 4 2 0 0 − 18 15 11 0 0 − 45 34 22 0 0 − 107 79 56 0 0 − 218 154 4A − 1 − 1 5 − 2 4 − 13 − 9 24 14 − 38 − 20 63 35 − 108 − 60 170 87 − 250 − 124 371 190 − 554 − 283 799 395 3A 10 − 1 21 5 31 59 85 135 200 300 414 610 835 1165 1581 2158 2865 3855 5051 6656 8649 11250 14430 18581 23631 . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . 2A 43 − 1 − 119 − 14 308 − 693 1407 − 2772 5306 − 9710 17136 − 29589 50155 − 83160 135148 − 216482 342259 − 533610 821296 − 1250717 1886234 − 2816798 4167709 − 6116665 8909383 23 M 1A 13915 − 1 132825 770 915124 5069867 24053215 101268540 387746282 1372935090 4552039296 14265412315 42568680715 121665949240 334658246604 889413095662 2291482148835 5739333227670 14008317423968 33388385201699 77853744768906 177881794535250 398808419854845 878461575586727 1903241478167799 The twined series for The [ g ] 47 19 Table − 1 71 23 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 47 of 89 0 0 0 0 0 0 − 1 0 − 1 0 0 0 0 0 1 0 − 1 1 0 1 0 1 1 1 23AB 0 − 4 − 1 2 0 0 − 2 1 − 1 0 0 1 2 2 0 0 − 2 0 0 0 0 − 2 1 1 0 15AB 0 (τ) 2 3 , g 0 0 0 − 1 − 1 − 1 0 0 − 1 0 1 1 1 0 0 1 0 0 − 1 − 1 0 0 1 1 14AB 0 ˜ h − 2 − 1 0 4 0 0 0 1 1 − 3 0 − 1 − 1 0 2 0 0 0 1 0 − 1 0 1 − 5 11AB 0 − 7 7 4 2 3 − 5 − 3 − 1 − 2 4 2 1 2 − 3 − 1 − 1 − 1 1 1 1 0 − 1 1 − 5 8A − 3 0 − 16 0 3 19 − 9 0 0 − 9 0 1 9 − 3 0 0 − 3 0 0 3 − 1 0 0 1 − 15 7AB 0 in the q -expansion of the function 46 23 40 − 32 − 13 − 28 24 10 16 − 18 − 7 − 10 10 4 8 − 4 − 3 − 6 4 1 2 − 2 1 − 32 6A − 50 / 24 D − q 68 92 − 136 0 0 37 46 − 64 0 0 16 17 − 25 0 0 7 6 − 9 0 0 1 1 1 0 5A 0 305 617 − 201 − 424 146 295 − 101 − 195 59 122 − 36 − 78 25 50 − 15 − 29 7 15 − 3 − 7 1 4 − 1 1 − 901 4A 34110 26646 − 10281 15912 12120 − 4645 6948 5256 − 1926 2880 2070 − 759 1062 762 − 260 360 228 − 75 90 60 − 15 18 6 1 − 21840 3A 5388145 − 3660335 2466359 − 1647128 1088842 − 711985 460571 − 294483 185563 − 115046 70308 − 42286 24809 − 14174 7945 − 4333 2255 − 1113 525 − 239 97 − 28 7 1 − 7867397 2A . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . 23 M The twined series for The 676433083819185 305326880332593 135349351964727 58848028309224 25058448433770 10432762525295 4238788584987 1676994065901 644418434331 239806592730 86116649220 29717775186 9804660777 3073155810 907876585 250188435 63447087 14525511 2922381 494385 65505 5796 231 1 1473468429847035 1A [ g ] 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 − 9 567 20 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 48 of 89 ) ) ) ) 6 ) 1 ) 3 ) 4 2 5 3 ) 2 ) 2 ) 3 ) 3 ) 3 ) 5 ) 4 6 ) B 2 ) 1 ) 1 ) − − − − − − − − − − − − − − − 30 − − − 15 )/ 2 15 15 15 15 15 15 15 15 15 15 15 15 15 15 A , 15 15 15 15 15 b b b b b p 15 15 15 15 3 9 − 1 30 2 b 2 b b b b b ( b ( b ( b ( 2 b ( 2 b ( 3 b ( ( 4 ( 5 b ( 4 ( 4 ( 5 b ( 6 b ( 7 b ( 8 b ( ( 9 b ( 10 b √ i + ) ) ) ) ) 1 ) 1 ) ) ) 1 ) B 15 15 15 15 15 2 1 ) 15 b − − − 3 2 b 2 b b 3 b 5 b 15 − − 15 ( − 1 15 15 15 15 15 15 15 15 − − − − − − A , 15 15 b b b 15 = 15 15 3 b − 1 0 2 0 2 b b − b − b − b − − 2 b − b ( b ( 2 b ( b ( 1 ( 4 ( 2 ( 1 ( 4 ( 1 ( ( 2 15 p b . (τ) 2 1 − 1 1 0 0 0 0 0 0 − 1 0 0 0 0 − 1 0 1 0 0 1 0 2 0 − 2 0 14AB 0 , g ˜ h − 1 − 1 2 − 2 2 − 3 3 − 4 4 − 5 6 − 7 9 − 9 11 − 12 14 − 16 17 − 20 22 − 25 28 − 31 12A 1 − 1 0 0 1 0 − 1 − 1 0 − 1 2 0 0 0 0 3 0 − 2 − 1 0 − 4 3 0 0 0 11AB 0 − 1 − 2 1 − 2 2 − 3 3 − 4 5 − 4 6 − 5 10 − 12 8 − 11 15 − 19 18 − 16 22 − 26 30 − 32 10A 1 − 1 1 0 − 2 2 1 − 3 2 0 − 3 4 1 − 5 3 1 − 6 6 2 − 7 6 0 − 9 8 3 9AB − 1 − 1 1 − 1 0 − 1 − 1 2 0 2 2 − 3 − 1 − 2 − 4 4 1 2 6 − 5 0 − 4 − 7 7 1 8A 0 in the q -expansion of the function − 1 − 1 0 0 − 2 4 2 0 − 5 0 0 − 6 10 3 0 − 9 0 0 − 11 22 8 0 − 22 0 7AB 0 / 24 D − q − 1 − 2 1 − 1 3 − 3 3 − 4 4 − 6 6 − 5 9 − 11 10 − 11 15 − 17 16 − 19 22 − 26 29 − 29 6B 1 − 1 7 22 32 60 81 141 194 304 411 612 829 1179 1567 2167 2860 3864 5032 6679 8624 11272 14413 18602 23599 6A 7 − 1 0 − 5 4 2 0 0 − 18 15 11 0 0 − 45 34 22 0 0 − 107 79 56 0 0 − 218 154 5B 0 − 1 − 10 − 25 − 26 − 58 − 85 − 135 − 218 − 285 − 404 − 610 − 835 − 1210 − 1546 − 2138 − 2865 − 3855 − 5157 − 6576 − 8594 − 11250 − 14430 − 18798 − 23476 5A − 5 − 1 − 1 5 4 − 13 − 9 24 14 − 38 − 20 63 35 − 108 − 60 170 87 − 250 − 124 371 190 − 554 − 283 799 395 4A − 2 − 1 10 21 31 59 85 135 200 300 414 610 835 1165 1581 2158 2865 3855 5051 6656 8649 11250 14430 18581 23631 3B 5 − 1 − 17 − 42 − 68 − 112 − 167 − 279 − 394 − 600 − 837 − 1208 − 1667 − 2345 − 3153 − 4313 − 5748 − 7692 − 10096 − 13333 − 17280 − 22500 − 28887 − 37138 − 47253 3A − 13 − 1 43 − 119 308 − 693 1407 − 2772 5306 − 9710 17136 − 29589 50155 − 83160 135148 − 216482 342259 − 533610 821296 − 1250717 1886234 − 2816798 4167709 − 6116665 8909383 2A − 14 . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The McL . twined series for The − 1 13915 132825 915124 5069867 24053215 101268540 387746282 1372935090 4552039296 14265412315 42568680715 121665949240 334658246604 889413095662 2291482148835 5739333227670 14008317423968 33388385201699 77853744768906 177881794535250 398808419854845 878461575586727 1903241478167799 1A 770 [ g ] 21 Table − 1 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 23 − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 49 of 89 9 8 6 5 6 6 4 4 2 22 3 20 2 16 2 17 2 16 0 14 2 12 1 10 30AB 10 − 1 0 0 3 0 0 0 0 0 0 − 1 0 0 0 0 − 3 0 0 0 0 0 0 1 0 15AB 0 0 − 1 0 1 1 1 0 0 1 1 0 0 0 0 − 1 0 − 1 0 0 − 1 0 − 1 1 − 1 14AB 0 (τ) 2 3 − 13 8 − 9 6 − 5 5 − 3 4 − 2 29 3 − 25 − 3 23 2 − 18 − 2 20 1 − 16 − 1 16 1 − 14 12A 12 , g ˜ h 1 1 − 3 0 − 1 − 1 0 2 0 − 5 0 0 0 − 2 1 − 1 0 0 − 1 4 0 0 1 0 11AB 0 18 − 16 18 − 16 9 − 9 10 − 8 5 − 62 − 3 50 5 − 50 − 4 44 2 − 38 − 3 32 2 − 30 1 31 10A − 23 9 0 0 0 0 0 − 5 0 0 15 3 0 0 0 0 0 0 0 0 0 0 − 10 1 0 9AB 0 − 1 − 2 4 2 1 2 − 3 − 1 − 1 − 5 − 1 − 3 1 − 7 1 7 1 4 0 2 − 1 3 1 − 5 8A − 3 0 − 9 0 1 9 − 3 0 0 − 3 − 15 0 0 0 0 3 − 16 − 1 0 0 3 0 19 1 − 9 7AB 0 in the q -expansion of the function / 24 D 10 16 − 18 − 7 − 10 10 4 8 − 4 − 32 − 3 − 50 − 6 46 4 23 1 40 2 − 32 − 2 − 13 1 − 28 6B 24 − q 1999 1408 1071 710 551 361 289 164 122 21637 63 17155 57 13231 22 10442 22 7876 5 6124 7 4532 1 3530 6A 2580 − 64 0 0 16 17 − 25 0 0 7 0 6 0 − 9 68 0 92 0 − 136 1 0 1 0 1 37 5B 46 5301 3806 2880 2070 1536 1077 735 510 360 237 43710 161 34110 81 26718 60 20652 30 15774 21 12120 6 9270 1 6987 5A − 195 59 122 − 36 − 78 25 50 − 15 − 29 7 − 901 15 305 − 3 617 − 7 − 201 1 − 424 4 146 − 1 295 1 − 101 4A 5256 − 1926 2880 2070 − 759 1062 762 − 260 360 228 − 21840 − 75 34110 90 26646 60 − 10281 − 15 15912 18 12120 6 − 4645 1 6948 3B 13140 3879 7200 5175 1518 2655 1905 505 900 570 43725 159 85275 225 66615 150 20562 30 39780 45 30300 15 9260 1 17370 3A − 294483 185563 − 115046 70308 − 42286 24809 − 14174 7945 − 4333 2255 − 7867397 − 1113 5388145 525 − 3660335 − 239 2466359 97 − 1647128 − 28 1088842 7 − 711985 1 460571 2A . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The McL . twined series for The 1676994065901 644418434331 239806592730 86116649220 29717775186 9804660777 3073155810 907876585 250188435 63447087 1473468429847035 14525511 676433083819185 2922381 305326880332593 494385 135349351964727 65505 58848028309224 5796 25058448433770 231 10432762525295 1 4238788584987 1A [ g ] 375 351 327 303 279 255 231 207 183 159 567 135 543 111 519 87 495 63 471 39 447 15 423 − 9 399 22 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 50 of 89 B 1 ) 1 ) 1 ) 1 ) 1 ) 20 + − + − + A , 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 a a a a a 0 0 − 1 0 0 0 0 0 0 a a a a a a a a a 20 2 a 2 ( ( a ( ( ( 0 0 0 1 − 1 0 1 1 − 1 0 1 0 0 0 − 1 0 − 2 0 0 1 1 − 1 0 0 1 15A p √ i 1 − 1 − 1 1 1 4 5 7 − 1 2 − 3 − 3 2 0 − 2 − 5 2 − 7 5 3 − 6 5 0 − 8 1 12A = p a . (τ) − 1 0 0 0 0 0 0 0 − 1 0 1 − 1 0 − 1 2 0 3 0 − 2 − 1 0 − 4 3 0 0 11AB 2 1 , g ˜ h 0 0 0 − 1 2 0 2 − 2 − 1 0 0 0 2 − 1 − 1 − 1 2 0 0 1 − 1 − 4 0 0 2 10B − 3 − 5 1 1 2 6 − 12 30 − 1 − 2 − 2 3 − 4 5 − 4 10 8 − 11 15 − 19 18 − 16 22 − 26 − 32 10A − 1 − 1 0 − 1 − 1 − 3 − 4 7 − 1 1 0 2 0 2 2 − 2 4 1 2 6 − 5 0 − 4 − 7 1 8ABC 4 − 6 0 0 − 2 0 3 − 22 − 1 − 1 0 2 0 − 5 0 10 0 − 9 0 0 − 11 22 8 0 0 7A − 3 − 5 1 1 3 6 − 11 29 − 1 − 2 − 1 3 − 4 4 − 6 9 10 − 11 15 − 17 16 − 19 22 − 26 − 29 6B in the q -expansion of the function − 3 − 9 1 3 1 6 − 7 27 − 1 0 − 3 3 − 4 4 − 4 9 12 − 13 13 − 15 18 − 21 22 − 24 − 33 6A / 24 D − q 0 0 0 − 5 2 0 34 − 218 − 1 0 4 0 − 18 15 11 − 45 22 0 0 − 107 79 56 0 0 154 5BC − 85 − 835 − 5 − 25 − 58 − 610 − 1546 − 18798 − 1 − 10 − 26 − 135 − 218 − 285 − 404 − 1210 − 2138 − 2865 − 3855 − 5157 − 6576 − 8594 − 11250 − 14430 − 23476 5A − 9 35 − 2 5 − 13 63 − 60 799 − 1 − 1 4 24 14 − 38 − 20 − 108 170 87 − 250 − 124 371 190 − 554 − 283 395 4BC − 185 − 1645 − 10 − 35 − 125 − 1145 − 3244 − 36305 − 1 − 25 − 60 − 240 − 394 − 630 − 860 − 2420 − 4126 − 5665 − 7930 − 10260 − 12909 − 17146 − 23010 − 29195 − 46925 4A 85 835 5 21 59 610 1581 18581 − 1 10 31 135 200 300 414 1165 2158 2865 3855 5051 6656 8649 11250 14430 23631 3A − 1425 − 50085 10 129 667 29715 − 135268 6118263 − 1 − 45 − 300 2820 − 5278 9634 − 17176 82944 216822 − 342085 533110 − 821544 1251459 − 1885854 2815690 − 4168275 − 8908593 2B 1407 50155 − 14 − 119 − 693 − 29589 135148 − 6116665 − 1 43 308 − 2772 5306 − 9710 17136 − 83160 − 216482 342259 − 533610 821296 − 1250717 1886234 − 2816798 4167709 8909383 2A . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The HS . twined series for The 24053215 42568680715 770 132825 5069867 14265412315 334658246604 878461575586727 − 1 13915 915124 101268540 387746282 1372935090 4552039296 121665949240 889413095662 2291482148835 5739333227670 14008317423968 33388385201699 77853744768906 177881794535250 398808419854845 1903241478167799 1A [ g ] 143 287 23 71 119 263 335 551 23 Table − 1 47 95 167 191 215 239 311 359 383 407 431 455 479 503 527 575 − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 51 of 89 0 0 0 0 0 2 − 1 − 1 0 0 0 0 1 − 1 0 − 1 0 0 0 1 1 0 1 − 1 20AB 0 1 0 − 0 0 0 1 − 4 2 2 − 1 0 0 2 − 2 0 0 0 0 0 0 − 2 − 2 1 1 1 15A − 2 − 4 0 0 0 − 1 0 0 0 5 2 0 0 0 1 0 0 0 1 − 1 0 0 0 1 0 12A − 5 1 1 0 − 3 0 − 2 − 1 − 1 − 1 0 2 0 0 0 4 0 1 0 0 − 1 0 0 1 0 11AB (τ) 2 3 , g ˜ h 0 0 0 0 0 0 4 1 − 1 4 0 0 0 − 1 − 2 0 − 1 0 0 0 1 − 3 1 1 − 2 10B − 62 18 − 16 50 18 − 16 − 50 9 − 9 44 10 − 8 − 38 5 − 3 32 5 − 4 − 30 2 − 3 31 2 1 − 23 10A − 5 − 1 − 2 − 3 4 2 − 7 1 2 7 − 3 − 1 4 − 1 − 1 2 1 1 3 1 0 − 5 − 1 1 − 3 8ABC − 15 0 − 9 0 0 1 0 9 − 3 − 16 0 0 0 − 3 0 3 0 3 19 − 1 0 − 9 0 1 0 7A − 32 10 16 − 50 − 18 − 7 46 − 10 10 23 4 8 40 − 4 − 3 − 32 − 6 4 − 13 1 2 − 28 − 2 1 24 6B in the q -expansion of the function 90 − 36 0 0 0 19 0 0 0 − 59 − 10 0 0 0 9 0 0 0 45 − 5 0 0 0 1 0 6A / 24 D − q 0 − 64 0 0 0 16 68 17 − 25 92 0 0 − 136 7 6 0 − 9 0 0 0 1 37 1 1 46 5BC 43710 3806 2880 34110 2070 1536 26718 1077 735 20652 510 360 15774 237 161 12120 81 60 9270 30 21 6987 6 1 5301 5A − 901 59 122 305 − 36 − 78 617 25 50 − 201 − 15 − 29 − 424 7 15 146 − 3 − 7 295 1 4 − 101 − 1 1 − 195 4BC 86395 7835 5850 68625 4140 2930 53817 2169 1554 41015 1025 675 31320 471 319 24450 189 105 18775 65 36 13851 15 1 10269 4A − 21840 − 1926 2880 34110 2070 − 759 26646 1062 762 − 10281 − 260 360 15912 228 − 75 12120 90 60 − 4645 − 15 18 6948 6 1 5256 3A 7865595 − 185445 115290 − 5387535 − 70380 42130 3661569 − 24759 14274 − 2466761 − 7975 4275 1646280 − 2241 1143 − 1088550 − 531 225 712575 − 95 36 − 460773 − 9 1 294093 2B − 7867397 185563 − 115046 5388145 70308 − 42286 − 3660335 24809 − 14174 2466359 7945 − 4333 − 1647128 2255 − 1113 1088842 525 − 239 − 711985 97 − 28 460571 7 1 − 294483 2A . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The HS . twined series for The 1473468429847035 644418434331 239806592730 676433083819185 86116649220 29717775186 305326880332593 9804660777 3073155810 135349351964727 907876585 250188435 58848028309224 63447087 14525511 25058448433770 2922381 494385 10432762525295 65505 5796 4238788584987 231 1 1676994065901 1A [ g ] 567 351 327 543 303 279 519 255 231 495 207 183 471 159 135 447 111 87 423 63 39 399 15 − 9 375 24 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 52 of 89 3 3 3 3 3 3 3 3 3 3 a a 3 3 3 3 3 3 3 a p 3 3 3 3 11766 a 3 4284 5658 a 2496 a 3366 a 9318 a 7194 1092 a 1938 a 1428 a √ 408 a 606 a 306 a 90 a 768 a 156 a i 204 3 30 a 78 a 18 a 12 a 36 a B 6 a = 6 p A , a − 3– − 10 − 18 − 26 − 43 − 69 − 100 − 148 − 213 − 298 − 419 − 583 − 793 − 1073 − 1444 − 1914 − 2532 − 3327 − 4328 − 5614 − 7237 − 9268 − 11827 − 1 − 5 6 - part I. 0 − 5 4 2 0 0 − 18 15 11 0 0 − 45 34 22 0 0 − 107 79 56 0 0 − 218 154 − 1 0 5A (τ) 2 1 , g ˜ h − 2 5 4 − 13 − 9 24 14 − 38 − 20 63 35 − 108 − 60 170 87 − 250 − 124 371 190 − 554 − 283 799 395 − 1 − 1 4G 2 − 7 − 4 11 7 − 20 − 14 42 24 − 69 − 37 104 52 − 162 − 85 254 136 − 381 − 190 546 269 − 785 − 393 − 1 3 4F − 2 5 4 − 13 − 9 24 14 − 38 − 20 63 35 − 108 − 60 170 87 − 250 − 124 371 190 − 554 − 283 799 395 − 1 − 1 4CDE − 6 − 15 − 28 − 69 − 97 − 108 − 190 − 334 − 440 − 541 − 805 − 1264 − 1652 − 1978 − 2789 − 4090 − 5192 − 6269 − 8478 − 11782 − 14739 − 17753 − 23265 − 1 − 13 4B in the q -expansion of the function / 24 D − 34 505 1412 3627 8559 19068 40154 81250 158736 300987 555307 1001016 1767388 3062830 5217427 8751462 14473168 23625763 38101658 60762514 95895325 149872439 232094167 − 1 139 4A q 5 21 31 59 85 135 200 300 414 610 835 1165 1581 2158 2865 3855 5051 6656 8649 11250 14430 18581 23631 − 1 10 3C − 13 − 42 − 68 − 112 − 167 − 279 − 394 − 600 − 837 − 1208 − 1667 − 2345 − 3153 − 4313 − 5748 − 7692 − 10096 − 13333 − 17280 − 22500 − 28887 − 37138 − 47253 − 1 − 17 3B 5 21 31 59 85 135 200 300 414 610 835 1165 1581 2158 2865 3855 5051 6656 8649 11250 14430 18581 23631 − 1 10 3A 10 129 − 300 667 − 1425 2820 − 5278 9634 − 17176 29715 − 50085 82944 − 135268 216822 − 342085 533110 − 821544 1251459 − 1885854 2815690 − 4168275 6118263 − 8908593 − 1 − 45 2C − 14 − 119 308 − 693 1407 − 2772 5306 − 9710 17136 − 29589 50155 − 83160 135148 − 216482 342259 − 533610 821296 − 1250717 1886234 − 2816798 4167709 − 6116665 8909383 − 1 43 2B . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . ( 2 ) 6 U − 30 − 199 180 − 917 1055 − 3300 4490 − 10894 15456 − 32005 46795 − 87784 128780 − 225074 330755 − 548970 801024 − 1277277 1851562 − 2861710 4109885 − 6190873 8814743 − 1 − 5 2A The twined series for The 770 132825 915124 5069867 24053215 101268540 387746282 1372935090 4552039296 14265412315 42568680715 121665949240 334658246604 889413095662 2291482148835 5739333227670 14008317423968 33388385201699 77853744768906 177881794535250 398808419854845 878461575586727 1903241478167799 − 1 13915 1A [ g ] 23 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 25 Table − 1 47 − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 53 of 89 B 3 3 3 3 3 3 3 3 3 3 a a a a a a a a a a 18 3 3 3 3 3 a a a a a 3 3 3 A , p 1 0 1 + − 1 2 − 1 + 2 − 2 + − 1 2 2 + − 2 + − 1 3 + − 3 + − 3 4 − 4 + 2 − 1 2 5 + − 4 + − 1 + − 1 √ − a − a − a 18 i = p a 15A 0 0 1 − 1 0 0 0 0 − 1 0 0 0 1 − 2 0 0 1 1 − 1 0 0 1 1 1 − 1 12I − 1 0 − 1 − 1 1 1 − 2 0 0 0 − 1 − 1 1 0 − 1 − 1 1 0 − 1 0 2 1 − 3 − 1 − 1 - part II. (τ) 2 1 , g 12FGH 1 − 1 − 2 2 − 3 3 − 4 4 − 5 6 − 7 9 − 9 11 − 12 14 − 16 17 − 20 22 − 25 28 − 31 2 − 1 ˜ h 3 3 3 3 a 3 3 E 3 3 3 3 3 3 3 a a a 3 3 a a 3 a a a a a a 3 3 3 a a 12 a 2 a 3 3 a a a , a a D 3 3 a − 1 0 − 1 + 3 + − 1 2 3 − 1 3 2 + − 5 2 5 + − 1 2 2 + 7 − 5 8 5 + 4 − 8 2 11 + 5 − 8 10 4 + 11 − 6 12 11 + − 18 9 19 + 11 − 12 15 − 1 a 12 12C 1 − 2 1 − 1 3 -3 3 -4 4 − 6 6 − 5 9 − 11 10 − 11 15 − 17 16 − 19 22 − 26 29 − 29 − 1 3 3 3 3 3 3 3 3 3 3 a 3 3 3 3 3 3 a 3 3 3 3 3 11811 a 4320 a 5625 a 2526 a 3333 a 9285 a 7221 a 3 1080 a 1434 1923 a 414 3 3 585 a 210 a 303 a 99 a 792 a 150 a B 42 a 69 a 27 a 15 a 6 a 3 a 12 a 12 A , in the q -expansion of the function − 1 − 5 + − 19 + − 45 + − 103 + − 213 + − 297 + − 576 + − 797 + − 1067 + − 1448 + − 1911 + − 2540 + − 3314 + − 4342 + − 7244 + − 9259 + − 11846 + − 8 + − 27 + − 66 + − 146 + − 425 + − 5603 + − 2 + 12 / 24 D − q 11AB − 1 0 1 − 1 0 2 0 0 0 3 0 − 2 − 1 0 − 4 0 0 0 0 0 − 1 − 1 0 3 0 10A − 1 0 0 0 0 1 0 1 0 − 4 0 0 − 1 3 2 0 2 − 2 1 − 2 0 1 0 0 0 9ABC − 1 1 − 2 1 2 − 3 4 − 5 3 1 − 6 6 2 − 7 6 − 9 8 3 0 2 − 3 0 1 0 − 1 8BCD − 1 1 0 − 1 0 2 − 3 − 2 − 4 4 1 2 6 − 5 0 − 7 7 1 − 1 − 1 2 2 − 1 − 4 0 8A − 1 − 1 0 − 5 − 2 − 4 3 8 − 16 14 − 9 18 − 12 11 − 18 − 35 35 − 25 1 3 4 6 − 5 18 2 7A − 1 − 1 0 4 0 0 0 10 3 0 − 9 0 0 − 11 22 0 − 22 0 0 − 2 2 − 5 − 6 8 0 6H − 1 0 − 3 − 3 − 4 − 4 6 9 − 7 12 − 13 13 − 15 18 − 21 − 24 27 − 33 3 1 3 4 − 9 22 1 6G − 1 − 2 − 1 − 3 − 4 − 6 6 9 − 11 10 − 11 15 − 17 16 − 19 − 26 29 − 29 1 3 3 4 − 5 22 1 . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . ( 2 ) 6F − 1 4 − 3 5 8 0 2 − 19 17 − 8 − 1 − 3 21 − 30 31 8 − 13 35 − 1 1 − 9 − 4 7 − 22 − 3 6 U 6E − 1 − 2 − 1 − 3 − 4 − 6 6 9 − 11 10 − 11 15 − 17 16 − 19 − 26 29 − 29 1 3 3 4 − 5 22 1 6D − 1 7 32 81 194 411 612 1179 1567 2167 2860 3864 5032 6679 8624 14413 18602 23599 22 60 141 304 829 11272 7 The twined series for The 6C − 1 − 14 − 33 − 91 − 208 − 426 − 598 − 1147 − 1603 − 2138 − 2887 − 3825 − 5085 − 6624 − 8687 − 14482 − 18523 − 23689 − 19 − 53 − 129 − 292 − 845 − 11206 − 3 [ g ] 26 Table − 1 47 95 143 191 239 263 311 335 359 383 407 431 455 479 527 551 575 71 119 167 215 287 503 23 − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 54 of 89 3530 2580 1999 1408 1071 710 551 361 289 164 122 63 21637 57 17155 22 13231 22 10442 5 7876 7 6124 1 4532 6D 6948 5256 3898 2880 2070 1497 1062 762 524 360 228 149 43616 90 34110 60 26646 33 20631 18 15912 6 12120 1 9227 6C 3474 2628 1963 1440 1035 738 531 381 269 180 114 71 21761 45 17055 30 13323 18 10350 9 7956 3 6060 1 4592 6AB - part I (τ) 2 3 , g 37 46 − 64 0 0 16 17 − 25 0 0 7 6 0 − 9 0 0 68 0 92 1 − 136 1 0 1 0 5A ˜ h − 101 − 195 59 122 − 36 − 78 25 50 − 15 − 29 7 15 − 901 − 3 305 − 7 617 1 − 201 4 − 424 − 1 146 1 295 4G 91 189 − 61 − 126 44 82 − 23 − 46 9 27 − 9 − 17 891 5 − 311 9 − 631 1 215 − 4 432 − 1 − 142 1 − 289 4F − 101 − 195 59 122 − 36 − 78 25 50 − 15 − 29 7 15 − 901 − 3 305 − 7 617 1 − 201 4 − 424 − 1 146 1 295 4CDE 6875 5037 3947 2986 2052 1426 1097 802 505 323 239 167 42747 93 34465 49 27217 33 20407 20 15448 7 12298 1 9535 4B in the q -expansion of the function / 24 D − q 7377627 4376077 2555051 1465578 824596 453842 243945 127490 64553 31395 14719 6487 200804347 2685 129280929 993 82458145 353 52070711 84 32531448 23 20093306 1 12257551 4A 4645 − 6948 5256 − 1926 2880 2070 − 759 1062 762 − 260 360 228 − 75 − 21840 90 34110 60 26646 − 15 − 10281 18 15912 6 12120 1 3C 9260 17370 13140 3879 7200 5175 1518 2655 1905 505 900 570 159 43725 225 85275 150 66615 30 20562 45 39780 15 30300 1 3B − 4645 6948 5256 − 1926 2880 2070 − 759 1062 762 − 260 360 228 − 75 − 21840 90 34110 60 26646 − 15 − 10281 18 15912 6 12120 1 3A 712575 − 460773 294093 − 185445 115290 − 70380 42130 − 24759 14274 − 7975 4275 − 2241 1143 7865595 − 531 − 5387535 225 3661569 − 95 − 2466761 36 1646280 − 9 − 1088550 1 2C − 711985 460571 − 294483 185563 − 115046 70308 − 42286 24809 − 14174 7945 − 4333 2255 − 1113 − 7867397 525 5388145 − 239 − 3660335 97 2466359 − 28 − 1647128 7 1088842 1 2B . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . ( 2 ) 6 U − 675025 488475 − 273555 201115 − 103590 78660 − 36270 29097 − 11166 10025 − 2925 3183 − 505 − 7692805 909 5524785 − 15 − 3553935 225 2548791 36 − 1583640 39 1137450 1 2A The twined series for The 10432762525295 4238788584987 1676994065901 644418434331 239806592730 86116649220 29717775186 9804660777 3073155810 907876585 250188435 63447087 14525511 1473468429847035 2922381 676433083819185 494385 305326880332593 65505 135349351964727 5796 58848028309224 231 25058448433770 1 1A [ g ] 423 399 375 351 327 303 279 255 231 207 183 159 135 567 111 543 87 519 63 495 39 471 15 447 − 9 27 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 55 of 89 0 18AB 0 1 2 0 0 0 0 0 0 0 0 0 0 − 1 − 1 0 0 − 1 0 0 0 0 0 1 15A − 2 1 0 1 0 − 2 2 0 − 1 0 − 4 0 0 0 0 − 2 0 0 2 2 1 0 0 − 1 1 12I 4 1 − 1 2 − 4 2 0 1 − 1 0 2 2 − 2 1 0 0 0 0 2 − 2 1 2 0 2 0 - part II (τ) 2 3 , g ˜ h 12FGH − 14 1 16 − 1 − 16 1 20 − 2 − 18 2 23 − 3 − 25 3 29 − 2 4 − 3 5 − 5 6 − 9 8 − 13 12 12DE 14 1 − 14 1 16 − 1 − 20 0 28 − 2 − 23 3 25 − 1 − 33 2 − 4 7 − 5 5 − 8 9 − 8 5 − 12 12C − 60 1 19 2 92 − 6 − 168 5 119 0 34 − 6 − 138 1 52 16 − 24 20 2 − 18 5 34 − 72 50 16 12AB 3414 1 4654 5 6152 3 7788 20 10400 30 13357 39 16917 79 21907 130 156 275 383 513 764 1069 1368 1985 2644 in the q -expansion of the function / 24 D − 11AB 0 1 0 0 4 − 1 0 0 − 1 1 − 2 0 0 0 − 5 0 2 0 − 1 − 1 0 − 3 q 1 1 0 10A 5 1 0 − 1 0 1 0 0 − 4 0 0 − 1 0 0 0 3 0 0 − 1 − 3 0 0 0 0 0 9ABC 0 1 − 10 0 0 0 0 0 0 0 0 0 0 3 15 0 0 − 5 0 0 0 0 0 9 0 8BCD − 5 1 3 − 1 2 0 4 1 7 1 − 7 1 − 3 − 1 − 5 − 1 − 1 − 3 2 1 2 4 − 2 − 1 − 3 8A 27 1 − 25 3 30 0 − 36 1 55 − 3 − 51 9 45 − 5 − 69 3 − 9 9 − 6 9 − 14 24 − 18 15 − 27 7A − 9 1 19 0 3 0 0 − 1 − 16 3 0 0 0 0 − 15 − 3 0 0 − 3 9 1 0 − 9 0 0 . The table shows the Fourier coefficients multiplying coefficients the Fourier table shows The . 6H 0 1 45 0 0 0 0 − 5 − 59 0 0 0 0 9 90 0 0 − 10 0 0 19 0 0 − 36 0 ( 2 ) 6 U 6G − 28 1 − 13 − 2 − 32 2 40 1 23 4 46 − 6 − 50 − 3 − 32 − 4 8 4 10 − 10 − 7 − 18 16 10 24 6F 0 1 − 43 0 0 0 0 3 69 0 0 0 0 − 7 − 94 0 0 14 0 0 − 21 0 0 28 0 6E − 28 1 − 13 − 2 − 32 2 40 1 23 4 46 − 6 − 50 − 3 − 32 − 4 8 4 10 − 10 − 7 − 18 16 10 24 The twined series for The [ g ] 28 Table 399 − 9 423 15 447 39 471 63 495 87 519 111 543 135 567 159 183 207 231 − D 255 279 303 327 351 375 Cheng et al. Mathematical Sciences (2015) 2:13 Page 56 of 89 22 M 12 11213482124 5050837768 2228498796 961480364 404852680 165992492 66095608 25478420 9473140 3381724 1152800 372564 113216 31928 8240 1900 380 60 8 0 0 χ of n χ 11 8155231968 3673355264 1620714052 699266588 294432862 120725462 48067156 18531236 6888684 2459952 838076 271160 82218 23284 5966 1394 268 48 4 1 0 χ 10 8155231968 3673355264 1620714052 699266588 294432862 120725462 48067156 18531236 6888684 2459952 838076 271160 82218 23284 5966 1394 268 48 4 1 0 χ 9 6728109656 3030488952 1337108322 576882244 242915594 99592828 39659114 15285968 5684522 2028652 691920 223388 68016 19112 4966 1132 232 32 6 0 0 χ into irreducible representations representations into irreducible (τ) ,1 g h 8 6116448648 2754999510 1215547168 524442118 220829728 90540888 36052396 13897114 5167368 1844428 628868 203204 61752 17410 4508 1028 208 34 4 0 0 χ in the function 7 4485422600 2020314748 891413422 384582826 161947152 66393106 26440708 10189794 3790274 1352058 461482 148822 45402 12708 3334 738 160 22 4 0 0 χ / 12 D − q 6 2883475316 1298781032 573046530 247235148 104106714 42682664 16996718 6551148 2436238 869408 296536 95744 29148 8192 2130 480 100 16 2 0 0 χ 5 1601947284 721533716 318366892 137347678 57840438 23710442 9443990 3638646 1354036 482650 164946 53078 16254 4516 1204 258 60 6 2 0 0 χ 4 1310656938 590364314 260469430 112383780 47318486 19403004 7724632 2978518 1106932 395460 134624 43616 13194 3754 952 228 42 8 0 0 0 χ 3 1310656938 590364314 260469430 112383780 47318486 19403004 7724632 2978518 1106932 395460 134624 43616 13194 3754 952 228 42 8 0 0 0 χ 2 611659466 275489850 121561770 52439410 22086048 9052216 3606408 1388930 517270 184100 63080 20228 6220 1710 472 92 26 2 0 0 0 χ 1 29128906 13116978 5789798 2496320 1052210 430782 171904 66016 24720 8706 3040 950 300 78 26 2 2 0 0 0 − 2 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the Fourier the decomposition table shows The 239 227 215 203 191 179 167 155 143 131 119 107 95 83 71 59 47 35 23 11 − D − 1 Appendix D: Decomposition tables D: Decomposition Appendix 29 , 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 . See Tables 29 Table Cheng et al. Mathematical Sciences (2015) 2:13 Page 57 of 89 22 M 12 43273949924 20138593100 9203440774 4124718268 1810032796 776347956 324802452 132236044 52245218 19965996 7351704 2595774 873748 278332 83094 22964 5778 1284 242 38 4 0 0 χ of n χ 11 31471963978 14646249036 6693411661 2999795266 1316387235 564616772 236220080 96171541 37996546 14520766 5346659 1887842 635466 202408 60440 16702 4196 936 179 26 3 0 0 χ 10 31471963978 14646249036 6693411661 2999795266 1316387235 564616772 236220080 96171541 37996546 14520766 5346659 1887842 635466 202408 60440 16702 4196 936 179 26 3 0 0 χ into irreducible representations representations into irreducible 9 25964369548 12083156304 5522064385 2474830808 1086019814 465808750 194881422 79341684 31347100 11979584 4411059 1557452 524228 167022 49855 13770 3468 770 147 24 1 0 0 χ (τ) ,2 g h 8 23603973264 10984686398 5020058920 2249846600 987290154 423462681 177165164 72128538 28497414 10890628 4009962 1415886 476604 151798 45338 12528 3138 706 136 18 2 1 0 χ in the function / 12 D 7 17309580294 8055436866 3681376388 1649887462 724012941 310539240 129921032 52894358 20898099 7986426 2940645 1038325 349513 111316 33238 9192 2307 516 96 15 2 0 0 χ − q 6 11127587228 5178495256 2366599091 1060641874 465436954 199632354 83520644 34003540 13434478 5134124 1890431 667490 224684 71562 21367 5906 1486 332 61 10 1 0 0 χ 5 6181993066 2876941610 1314777297 589245588 258575987 110906886 46400388 18890818 7463621 2852302 1050214 370835 124829 39754 11871 3286 821 184 35 5 1 0 0 χ 4 5057994232 2353861448 1075726834 482109963 211562274 90741956 37963923 15456158 6106599 2333682 859281 303407 102132 32528 9711 2686 675 149 28 5 1 0 0 χ 3 5057994232 2353861448 1075726834 482109963 211562274 90741956 37963923 15456158 6106599 2333682 859281 303407 102132 32528 9711 2686 675 149 28 5 1 0 0 χ 2 2360397820 1098468036 502005985 224984946 98728774 42346314 17716592 7212750 2849794 1089096 400933 141611 47678 15162 4535 1260 310 70 15 1 1 0 0 χ 1 The table shows the decomposition of the Fourier coefficients multiplying coefficients of the Fourier the decomposition table shows The 112400070 52307782 23905058 10713686 4701307 2016488 843662 343436 135731 51870 19067 6752 2277 716 215 62 15 2 2 0 0 0 1 χ 260 248 236 224 212 200 188 176 164 152 140 128 116 104 92 80 68 56 44 32 20 8 − D − 4 30 Table Cheng et al. Mathematical Sciences (2015) 2:13 Page 58 of 89 - ( 3 ) 4 U of 10 n 22496001167 10859139417 5160722937 2412136251 1107491427 498851309 220092869 94962320 39982901 16394948 6526850 2516637 935210 334074 113706 36834 11125 3164 801 189 34 7 0 0 0 χ χ 9 22496001167 10859139417 5160722937 2412136251 1107491427 498851309 220092869 94962320 39982901 16394948 6526850 2516637 935210 334074 113706 36834 11125 3164 801 189 34 7 0 0 0 χ 8 16872013154 8144312112 3870548166 1809082032 830621184 374129064 165070996 71217412 29987754 12294420 4895338 1886748 701512 250260 85316 27532 8350 2344 606 132 26 4 0 0 0 χ into irreducible representations (τ) ,1 g h 7 15184775484 7329886448 3483475608 1628176080 747550924 336717232 148559990 64096288 26987278 11065144 4405146 1698156 631070 225288 76674 24784 7492 2112 532 120 22 4 0 0 0 χ in the function 6 11248072954 5429518660 2580396830 1206043980 553762164 249414368 110054184 47475942 19994842 8195334 3264774 1257446 468176 166678 57060 18306 5622 1548 422 82 22 2 0 0 0 χ / 12 D − q 5 7230903886 3490390608 1658826474 775306952 355990060 160334544 70749080 30518832 12853814 5267788 2098802 808100 300968 107060 36676 11732 3618 984 270 52 14 0 0 0 0 χ 4 2812006070 1357371466 645093274 301506846 138437866 62351670 27512230 11868186 4998134 2048450 815984 314208 116944 41622 14228 4554 1398 380 102 20 4 0 0 0 0 χ 3 2812006070 1357371466 645093274 301506846 138437866 62351670 27512230 11868186 4998134 2048450 815984 314208 116944 41622 14228 4554 1398 380 102 20 4 0 0 0 0 χ 2 1687237678 814425664 387072550 180905952 83070266 37411832 16511002 7121124 3000480 1229276 490188 188592 70446 24972 8640 2748 860 232 72 12 6 0 0 0 0 χ 1 80345934 38777988 18432618 8612692 3955942 1780628 786382 338664 142936 58388 23350 8908 3368 1156 420 124 38 8 4 0 0 0 0 0 − 2 χ The table shows the decomposition of the the decomposition multiplying table shows coefficients The Fourier 31 287 275 263 251 239 227 215 203 191 179 167 155 143 131 119 107 95 83 71 59 47 35 23 11 Table − D − 1 part I Cheng et al. Mathematical Sciences (2015) 2:13 Page 59 of 89 - ( 3 ) 4 U of n 20 0 0 2 16 116 596 2586 10064 35728 117656 364172 1068560 2993494 8051776 20888326 52460200 127950562 303871648 704309074 1596305804 3544000144 7718796324 16514369292 34749166956 71987317446 χ χ 19 0 0 2 16 100 480 2130 8192 29148 95744 296536 869408 2436238 6551148 16996718 42682664 104106714 247235148 573046530 1298781032 2883475316 6280138936 13436398852 28272484272 58570125574 χ 18 0 0 2 14 86 432 1864 7216 25568 84136 260274 763500 2138648 5751956 14921266 37473264 91395836 217054970 503083826 1140227136 2531441198 5513444680 11796004996 24820871652 51419567602 χ into irreducible representations (τ) ,1 g h 17 0 0 2 14 86 432 1864 7216 25568 84136 260274 763500 2138648 5751956 14921266 37473264 91395836 217054970 503083826 1140227136 2531441198 5513444680 11796004996 24820871652 51419567602 χ in the function 16 0 1 1 16 74 391 1626 6379 22345 73819 227677 668609 1871144 5034454 13055603 32792777 79970261 189931510 440195866 997717729 2215005306 4824304831 10321492731 21718346527 44992097964 χ / 12 D − q 15 0 0 2 10 62 290 1242 4756 16856 55278 171022 501258 1404072 3775230 9793632 24593074 59982140 142445378 330156976 748280636 1661276556 3618212136 7741166340 16288726840 33744168882 χ 14 0 0 2 6 46 206 934 3532 12636 41346 128270 375590 1053068 2830546 7345188 18442654 44986624 106828846 247617812 561199172 1245957224 2713634904 5805874860 12216494216 25308126986 χ 13 0 0 2 6 46 206 934 3532 12636 41346 128270 375590 1053068 2830546 7345188 18442654 44986624 106828846 247617812 561199172 1245957224 2713634904 5805874860 12216494216 25308126986 χ 12 0 0 0 7 34 189 801 3164 11125 36834 113706 334074 935210 2516637 6526850 16394948 39982901 94962320 220092869 498851309 1107491427 2412136251 5160722937 10859139417 22496001167 χ 11 0 0 0 7 34 189 801 3164 11125 36834 113706 334074 935210 2516637 6526850 16394948 39982901 94962320 220092869 498851309 1107491427 2412136251 5160722937 10859139417 22496001167 χ The table shows the decomposition of the the decomposition multiplying table shows coefficients The Fourier 32 Table − D − 1 11 23 35 47 59 part II 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239 251 263 275 287 Cheng et al. Mathematical Sciences (2015) 2:13 Page 60 of 89 - ( 3 ) 4 U of n 10 χ 0 0 1 3 26 129 573 2276 8226 27505 86349 256447 726201 1972157 5160351 13060772 χ 32080056 76677496 178770283 407382190 908985651 1989002072 4273978516 9029770416 18777609455 9 0 0 1 3 26 129 573 2276 8226 27505 86349 256447 726201 1972157 5160351 13060772 χ 32080056 76677496 178770283 407382190 908985651 1989002072 4273978516 9029770416 18777609455 8 0 1 0 3 21 104 428 1720 6189 20668 64812 192443 544792 1479426 3870656 9796194 χ 24061078 57509747 134079840 305540174 681744339 1491758880 3205494608 6772343354 14083228602 into irreducible representations (τ) ,2 g h 7 0 0 1 4 16 92 396 1558 5574 18616 58374 173260 490425 1331594 3483848 8817060 21655492 χ 51759604 120673444 274988136 613572808 1342587586 2884951508 6095118216 12674919317 in the function 6 0 0 1 1 14 64 285 1142 4114 13740 43187 128231 363072 986096 2580201 6530326 16040080 χ 38338768 89385017 203691226 454492898 994500692 2136989558 4514885396 9388804084 / 12 D − q 5 0 0 1 2 8 42 185 742 2648 8852 27785 82484 233470 634020 1658877 4198392 10311866 χ 24646996 57462835 130945830 292176084 639325160 1373783498 2902432966 6035669290 4 0 0 0 1 4 18 74 288 1038 3458 10820 32105 90862 246650 645246 1632930 4010504 χ 9585456 22347452 50924466 113625730 248629024 534252608 1128728962 2347212278 3 0 0 0 1 4 18 74 288 1038 3458 10820 32105 90862 246650 645246 1632930 4010504 χ 9585456 22347452 50924466 113625730 248629024 534252608 1128728962 2347212278 2 0 0 0 0 2 8 37 170 606 2040 6453 19200 54343 147802 386845 979182 2405530 χ 5750076 13406477 30552132 68171414 149171154 320543266 677225340 1408309053 1 1 0 0 0 1 0 3 10 30 98 315 932 2585 7068 18479 46678 114644 273958 χ 638587 1455274 3246731 7103972 15265066 32250408 67064204 The table shows the decomposition of the the decomposition multiplying table shows coefficients The Fourier 33 Table − D − 4 part I 8 20 32 44 56 68 80 92 104 116 128 140 152 164 176 188 200 212 224 236 248 260 272 284 Cheng et al. Mathematical Sciences (2015) 2:13 Page 61 of 89 - ( 3 ) 4 U of n 20 χ 60088352145 28895264904 13676730320 6364807716 2908753703 1303622708 572065334 245367882 102655962 41794712 16513050 6310848 2323911 820618 276276 88062 26305 7278 1842 410 81 14 1 0 0 χ 19 48888805889 23509622436 11127587228 5178495256 2366599091 1060641874 465436954 199632354 83520644 34003540 13434478 5134124 1890431 667490 224684 71562 21367 5906 1486 332 61 10 1 0 0 χ 18 42920200803 20639439888 9769068796 4546274088 2077669720 931151309 408612706 175259316 73323427 29851826 11794091 4507162 1659543 585938 197214 62812 18749 5178 1299 289 54 8 1 0 0 χ into irreducible representations (τ) ,1 g h 17 42920200803 20639439888 9769068796 4546274088 2077669720 931151309 408612706 175259316 73323427 29851826 11794091 4507162 1659543 585938 197214 62812 18749 5178 1299 289 54 8 1 0 0 χ in the function 16 37555131344 18059479228 8547913920 3977974852 1817950926 814750620 357531400 153348868 64156020 26119004 10318996 3943264 1451756 512502 172448 54888 16366 4512 1124 248 46 6 0 0 0 χ / 12 D − q 15 28166327222 13544593866 6410924444 2983474090 1363458009 611059410 268146424 115010066 48115930 19588686 7738814 2957154 1088680 384273 129268 41142 12251 3372 842 182 31 5 0 0 0 χ 14 21124777990 10158468546 4808209568 2237616450 1022601213 458299758 201113152 86259890 36088522 14692426 5804744 2218282 816744 288351 97044 30902 9223 2544 636 142 27 3 0 0 0 χ 13 21124777990 10158468546 4808209568 2237616450 1022601213 458299758 201113152 86259890 36088522 14692426 5804744 2218282 816744 288351 97044 30902 9223 2544 636 142 27 3 0 0 0 χ 12 18777609455 9029770416 4273978516 1989002072 908985651 407382190 178770283 76677496 32080056 13060772 5160351 1972157 726201 256447 86349 27505 8226 2276 573 129 26 3 1 0 0 χ 11 18777609455 9029770416 4273978516 1989002072 908985651 407382190 178770283 76677496 32080056 13060772 5160351 1972157 726201 256447 86349 27505 8226 2276 573 129 26 3 1 0 0 χ The table shows the decomposition of the the decomposition multiplying table shows coefficients The Fourier 34 284 272 260 248 236 224 212 200 188 176 164 152 140 128 116 104 92 80 68 56 44 32 20 8 Table − D − 4 part II Cheng et al. Mathematical Sciences (2015) 2:13 Page 62 of 89 - 23 M of n χ 9 0 0 0 4 24 122 604 2500 9646 34002 112980 353668 1056018 3017118 8300712 22057824 56834066 142342008 347432516 828078940 1930905618 4411737282 9891102068 21787212032 47203452886 χ 8 0 0 0 2 21 113 547 2283 8789 31062 103119 322951 964080 2754882 7578641 20140092 51891287 129965340 317219382 756074191 1762997173 4028112844 9030998389 19892682446 43098788160 χ 7 0 0 0 2 21 113 547 2283 8789 31062 103119 322951 964080 2754882 7578641 20140092 51891287 129965340 317219382 756074191 1762997173 4028112844 9030998389 19892682446 43098788160 χ into irreducible representations (τ) 2 1 , g ˜ h 6 0 0 1 3 23 115 552 2289 8800 31076 103141 322980 964123 2754938 7578723 20140197 51891433 129965529 317219639 756074520 1762997610 4028113401 9030999117 19892683368 43098789349 χ in the function / 24 D − q 5 0 0 1 3 24 110 555 2266 8792 30883 102784 321420 960228 2742534 7546650 20051805 51668700 129399884 315850951 752794475 1755376093 4010659821 8991927075 19806533653 42912264213 χ 4 0 0 0 1 4 23 104 450 1705 6066 20061 62965 187731 536800 1476142 3923759 10108150 25318792 61794663 147289246 343437978 784701896 1759278672 3875207902 8395853450 χ 3 0 0 0 1 4 23 104 450 1705 6066 20061 62965 187731 536800 1476142 3923759 10108150 25318792 61794663 147289246 343437978 784701896 1759278672 3875207902 8395853450 χ 2 0 0 0 0 4 10 57 217 851 2945 9865 30717 91938 262221 722082 1917740 4942779 12376668 30213098 72004775 167908464 383624438 860103679 1894529513 4104664777 χ 1 − 1 0 0 0 1 0 3 11 40 132 455 1393 4196 11898 32865 87129 224778 562443 1373557 3272651 7632762 17436751 39096804 86113417 186578208 χ The table shows the decomposition of the the decomposition table shows The multiplying coefficients Fourier 35 part I Table − D − 1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 Cheng et al. Mathematical Sciences (2015) 2:13 Page 63 of 89 - 23 M of n χ 17 377627303791 174297913365 79128667059 35293998406 15447177148 6624675836 2779430485 1138754982 454660159 176470194 66400838 24139819 8446311 2830347 903231 272375 76950 20084 4777 1003 182 26 3 0 0 χ 16 193104947248 89129562686 40463558496 18048043408 7899140768 3387607678 1421306694 582313360 232499596 90238598 33956132 12343520 4319566 1447084 462028 139204 39394 10246 2454 508 96 12 2 0 0 χ into irreducible representations (τ) 15 2 1 184708934751 85254464108 38704205367 17263391744 7555669131 3240340814 1359497348 557004065 222385345 86318725 32477570 11808192 4130944 1384656 441658 133306 37596 9848 2324 496 87 14 1 0 0 χ , g ˜ h 14 in the function 184708934751 85254464108 38704205367 17263391744 7555669131 3240340814 1359497348 557004065 222385345 86318725 32477570 11808192 4130944 1384656 441658 133306 37596 9848 2324 496 87 14 1 0 0 χ / 24 D − q 13 167170968143 77159558254 35029285569 15624224127 6838273397 2932664346 1230418627 504113505 201271974 78121437 29394631 10686484 3738996 1252987 399819 120586 34052 8892 2111 444 80 11 1 0 0 χ 12 167170968143 77159558254 35029285569 15624224127 6838273397 2932664346 1230418627 504113505 201271974 78121437 29394631 10686484 3738996 1252987 399819 120586 34052 8892 2111 444 80 11 1 0 0 χ 11 143662505417 66309028068 30103271271 13427082844 5876631765 2520265222 1057386979 433225520 172966448 67136828 25260382 9184195 3212975 1076965 343525 103695 29245 7661 1811 389 68 11 1 0 0 χ 10 143662505416 66309028068 30103271271 13427082844 5876631765 2520265222 1057386979 433225520 172966448 67136828 25260382 9184195 3212975 1076965 343525 103695 29245 7661 1811 389 68 11 1 1 0 χ The table shows the decomposition of the the decomposition table shows The multiplying coefficients Fourier 36 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part II Table − D − 1 Cheng et al. Mathematical Sciences (2015) 2:13 Page 64 of 89 - 23 M of n χ 9 0 0 0 2 12 76 354 1584 6186 22554 76160 243296 736834 2136178 5947048 15983498 41590710 105130940 258745612 621494632 1459516642 3356890620 7572573738 16776640436 36544317696 χ 8 0 0 0 2 10 65 327 1439 5650 20585 69539 222072 672862 1950253 5430053 14593319 37974566 95988148 236247364 567449375 1332604789 3064982875 6914095094 15317791616 33366565723 χ 7 0 0 0 2 10 65 327 1439 5650 20585 69539 222072 672862 1950253 5430053 14593319 37974566 95988148 236247364 567449375 1332604789 3064982875 6914095094 15317791616 33366565723 χ into irreducible representations (τ) 2 3 , g ˜ h 6 0 1 0 1 12 71 324 1451 5666 20571 69575 222128 672826 1950361 5430193 14593220 37974823 95988502 236247135 567449989 1332605575 3064982355 6914096414 15317793334 33366564639 χ in the function / 24 D − q 5 0 0 1 2 10 72 317 1456 5614 20527 69200 221297 669671 1942293 5406043 14531106 37808739 95575487 235220486 564999523 1326827403 3051727864 6884144714 15251512500 33222076673 χ 4 0 0 0 0 2 12 66 278 1108 3996 13574 43224 131140 379810 1058004 2842550 7398130 18698262 46023364 110540364 259601046 597070540 1346907692 2983976756 6499993004 χ 3 0 0 0 0 2 12 66 278 1108 3996 13574 43224 131140 379810 1058004 2842550 7398130 18698262 46023364 110540364 259601046 597070540 1346907692 2983976756 6499993004 χ 2 0 0 1 0 2 8 29 148 536 1969 6612 21229 63977 185925 516995 1390158 3616159 9142805 22498248 54045257 126911807 291907776 658478668 1458848820 3177751973 χ 1 1 0 0 0 1 0 1 8 25 90 301 979 2892 8477 23485 63229 164314 415733 1022444 2456955 5768329 13269156 29930003 66312927 144441012 χ The table shows the decomposition of the the decomposition table shows The multiplying coefficients Fourier 37 part I Table − D − 9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567 Cheng et al. Mathematical Sciences (2015) 2:13 Page 65 of 89 - 23 M of n χ 17 292354823899 134212929307 60580719175 26855037539 11676191108 4971917548 2069990475 841030812 332736007 127861367 47580312 17086836 5896262 1945438 609707 180159 49625 12594 2879 582 97 13 1 0 0 χ 16 149499558772 68631657584 30978745714 13732710656 5970765562 2542467146 1058511762 430076524 170146608 65385226 24329868 8738192 3014774 995038 311662 92194 25342 6460 1462 302 48 8 0 0 0 χ into irreducible representations (τ) 15 2 3 142999706146 65647584650 29631903464 13135596047 5711193875 2431907356 1012501148 411370026 162753713 62539370 23273934 8357121 2884379 951374 298347 88055 24308 6143 1418 280 49 6 1 0 0 χ , g ˜ h 14 in the function 142999706146 65647584650 29631903464 13135596047 5711193875 2431907356 1012501148 411370026 162753713 62539370 23273934 8357121 2884379 951374 298347 88055 24308 6143 1418 280 49 6 1 0 0 χ / 24 D − q 13 129421909124 59414411523 26818346389 11888393029 5168908580 2201005302 916360448 372313294 147298443 56602448 21063317 7563999 2610275 861169 269912 79749 21967 5568 1278 254 42 6 0 0 0 χ 12 129421909124 59414411523 26818346389 11888393029 5168908580 2201005302 916360448 372313294 147298443 56602448 21063317 7563999 2610275 861169 269912 79749 21967 5568 1278 254 42 6 0 0 0 χ 11 111221993159 51059233415 23047036703 10216574441 4442040173 1891483817 787500731 319954666 126586391 48641673 18102016 6500043 2243393 739984 232064 68482 18920 4783 1099 220 41 4 1 0 0 χ 10 111221993159 51059233415 23047036703 10216574441 4442040173 1891483817 787500731 319954666 126586391 48641673 18102016 6500043 2243393 739984 232064 68482 18920 4783 1099 220 41 4 1 0 0 χ The table shows the decomposition of the the decomposition table shows The multiplying coefficients Fourier 38 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part II Table − D − 9 Cheng et al. Mathematical Sciences (2015) 2:13 Page 66 of 89 8 0 0 0 0 1 5 24 101 386 1370 4541 14231 42467 121375 333865 887304 2286050 5725731 13975118 33309272 77669274 177460315 397863480 876380147 1898731972 χ of McL - part I n χ 7 0 0 0 0 1 5 24 101 386 1370 4541 14231 42467 121375 333865 887304 2286050 5725731 13975118 33309272 77669274 177460315 397863480 876380147 1898731972 χ into irreducible representations representations into irreducible (τ) 2 1 , 6 g 0 0 0 1 1 7 22 92 336 1191 3909 12259 36503 104379 286918 762670 1964545 4920869 12009751 28625800 66746652 152506321 341912958 753141920 1631720453 χ ˜ h in the function / 24 D 5 − 0 1 0 1 2 7 23 93 336 1192 3911 12260 36504 104381 286919 762672 1964549 4920870 12009753 28625805 66746654 152506325 341912961 753141922 1631720458 χ q 4 0 0 0 0 1 1 8 29 113 381 1292 3990 11987 34082 94007 249432 643218 1610016 3931104 9367446 21845882 49908836 111902115 246477854 534024793 χ 3 0 0 0 0 1 2 8 27 103 357 1181 3673 10971 31296 86127 228754 589479 1476138 3603184 8587440 20024583 45751192 102575149 225940987 489518783 χ 2 0 0 0 0 1 1 3 5 13 37 122 357 1065 2983 8247 21796 56231 140568 343337 817799 1907498 4357044 9769828 21517663 46622397 χ 1 The table shows the decomposition of the Fourier coefficients multiplying coefficients of the Fourier the decomposition table shows The − 1 0 0 0 0 0 0 1 0 2 6 18 49 135 379 995 2564 6386 15622 37167 86756 198014 444173 977997 2119404 χ

− D − 1 Table 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 Cheng et al. Mathematical Sciences (2015) 2:13 Page 67 of 89 McL - of n 16 χ 16993217734 7843413933 3560784919 1588233526 695120690 298112039 125073387 51244665 20459305 7941451 2987879 1086405 380029 127410 40626 12269 3459 909 214 46 8 2 0 0 0 χ 15 11748391635 5422610680 2461774474 1098039598 480575987 206102830 86469919 35428696 14144455 5490489 2065592 751132 262694 88095 28072 8486 2385 628 146 32 5 1 0 0 0 χ into irreducible representations (τ) 14 2 1 10813885859 4991249366 2265963373 1010687552 442353736 189704760 79593979 32608993 13020310 5053141 1901666 691150 241943 81007 25887 7783 2212 572 138 27 5 1 0 0 0 χ , g ˜ h 13 in the function 10070049592 4647951685 2110092165 941176611 411922132 176659468 74116978 30367390 12123772 4706100 1770483 643803 225153 75500 24056 7271 2039 536 123 27 4 0 0 0 0 χ / 24 D − q 12 9536046829 4401459954 1998200428 891261397 390081042 167289297 70187986 28756211 11481470 4456239 1676845 609553 213315 71465 22816 6874 1946 507 120 25 5 1 0 0 0 χ 11 7459290309 3442931988 1563028664 697170138 305126393 130859914 54901023 22494840 8980403 3486207 1311412 476982 166766 55963 17816 5400 1513 402 92 22 3 1 0 0 0 χ 10 7459319124 3442912979 1563042221 697161444 305132577 130856090 54903746 22493230 8981556 3485567 1311878 476743 166945 55882 17881 5375 1535 396 99 21 5 1 1 0 0 χ 9 3708476391 1711669416 777084367 346597295 151701090 65055014 27296592 11182141 4465544 1732646 652314 236919 83033 27745 8900 2657 765 193 49 8 3 0 0 0 0 χ The table shows the decomposition of the the decomposition table shows multiplying The coefficients Fourier 40 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part II Table − D − 1 Cheng et al. Mathematical Sciences (2015) 2:13 Page 68 of 89 McL - of n χ 24 22028253520 10167382771 4615836219 2058818686 901084142 386440376 162132948 66427799 26521644 10294273 3873312 1408230 492679 165131 52683 15900 4489 1175 279 60 11 2 1 0 0 χ 23 22028253515 10167382764 4615836213 2058818682 901084137 386440375 162132945 66427796 26521644 10294270 3873309 1408229 492677 165130 52683 15899 4487 1175 279 59 11 1 0 0 0 χ into irreducible representations (τ) 22 2 1 20886051640 9640181689 4376498280 1952063468 854361992 366401965 153726332 62983039 25146550 9760341 3672503 1335140 467137 156541 49950 15064 4252 1110 263 55 10 1 0 0 0 χ , g ˜ h 21 in the function 20396558046 9414224038 4273934999 1906304694 834342808 357811165 150125530 61505504 24558072 9531019 3586785 1303640 456320 152796 48824 14687 4167 1076 261 53 10 1 0 0 0 χ / 24 D − q 20 20396558046 9414224038 4273934999 1906304694 834342808 357811165 150125530 61505504 24558072 9531019 3586785 1303640 456320 152796 48824 14687 4167 1076 261 53 10 1 0 0 0 χ 19 17482746319 8069346991 3663364558 1633981014 715147222 306697753 128677387 52720052 21049092 8169871 3074116 1117568 391030 131025 41815 12607 3561 928 221 46 8 1 0 0 0 χ 18 17482746319 8069346991 3663364558 1633981014 715147222 306697753 128677387 52720052 21049092 8169871 3074116 1117568 391030 131025 41815 12607 3561 928 221 46 8 1 0 0 0 χ 17 16993217734 7843413933 3560784919 1588233526 695120690 298112039 125073387 51244665 20459305 7941451 2987879 1086405 380029 127410 40626 12269 3459 909 214 46 8 2 0 0 0 χ The table shows the decomposition of the the decomposition table shows multiplying The coefficients Fourier 41 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part III Table − D − 1 Cheng et al. Mathematical Sciences (2015) 2:13 Page 69 of 89 8 0 0 0 0 0 2 14 62 247 905 3060 9775 29645 85899 239219 642880 1672987 4228700 10408024 24998994 58708484 135028617 304603271 674830133 1469977138 χ of McL - part I n χ 7 0 0 0 0 0 2 14 62 247 905 3060 9775 29645 85899 239219 642880 1672987 4228700 10408024 24998994 58708484 135028617 304603271 674830133 1469977138 χ into irreducible representations representations into irreducible (τ) 2 3 , 6 g 0 0 0 0 1 3 12 55 219 777 2644 8404 25493 73822 205653 552420 1437895 3633969 8944652 21483285 50453393 116039405 261770033 579930706 1263264431 χ ˜ h in the function / 24 D 5 − 0 0 0 0 1 3 12 55 219 777 2644 8404 25493 73822 205653 552420 1437895 3633969 8944652 21483285 50453393 116039405 261770033 579930706 1263264431 χ q 4 0 0 1 0 1 1 5 23 72 259 865 2785 8308 24238 67254 180937 470406 1189748 2926781 7031919 16510850 37978515 85667592 189800269 413425592 χ 3 0 1 0 1 1 4 6 24 72 245 804 2556 7659 22218 61728 165837 431408 1090489 2683334 6445575 15135916 34812830 78530602 173981507 378977806 χ 2 0 0 0 0 0 0 0 3 5 22 73 251 716 2128 5867 15803 41051 103927 255433 614030 1441297 3315763 7478672 16570437 36091901 χ 1 The table shows the decomposition of the Fourier coefficients multiplying coefficients of the Fourier the decomposition table shows The 1 0 0 0 1 0 1 1 2 2 6 18 36 105 276 730 1881 4756 11625 27977 65536 150811 339981 753426 1640491 χ

Table − D − 9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567 Cheng et al. Mathematical Sciences (2015) 2:13 Page 70 of 89 McL - of n χ 16 0 0 0 0 4 25 130 562 2235 8093 27451 87506 265369 768802 2141239 5753503 14973440 37845752 93150396 223734839 525430528 1208473427 2726136749 6039574829 13155976526 χ 15 0 0 1 1 6 21 95 398 1566 5607 19023 60533 183543 531575 1480572 3977741 10352451 26165020 64401011 154680790 363262412 835486965 1884740065 4175506851 9095496261 χ into irreducible representations (τ) 14 2 3 0 0 0 1 2 19 81 365 1414 5173 17445 55752 168786 489419 1362371 3661782 9527924 24084770 59275991 142379305 334361281 769034337 1734806216 3843378364 8371967563 χ , g ˜ h 13 in the function 0 0 0 0 2 14 78 334 1328 4792 16274 51846 157282 455558 1268952 3409380 8873332 22426902 55200610 132583074 311367160 716131132 1615490472 3579004576 7796138458 χ / 24 D − q 12 0 0 0 1 2 15 75 320 1251 4559 15394 49139 148894 431517 1201486 3228953 8402301 21238355 52272318 125553959 294852626 678158814 1529814670 3389217725 7382694796 χ 11 0 0 0 0 2 10 56 244 986 3536 12064 38380 116510 337390 940034 2525280 6573020 16612158 40889694 98208856 230643438 530465432 1196662060 2651110530 5774922228 χ 10 0 0 0 0 1 13 54 254 975 3566 12026 38471 116393 337637 939698 2525910 6572140 16613698 40887536 98212448 230638380 530473486 1196650730 2651128016 5774897738 χ 9 0 0 0 0 1 7 24 129 482 1776 5969 19154 57809 167931 467083 1255917 3267135 8260107 20326853 48828286 114662453 263731843 594922426 1318037615 2871035001 χ The table shows the decomposition of the the decomposition table shows multiplying The coefficients Fourier 43 part II Table − D − 9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567 Cheng et al. Mathematical Sciences (2015) 2:13 Page 71 of 89 McL - of n χ 24 17054036789 7829083804 3533877823 1566542113 681112299 290027769 120749919 49059813 19409807 7458445 2775597 996685 343976 113467 35576 10503 2898 733 168 34 6 1 0 0 0 χ 23 17054036789 7829083804 3533877823 1566542113 681112299 290027769 120749919 49059813 19409807 7458445 2775597 996685 343976 113467 35576 10503 2898 733 168 34 6 1 0 0 0 χ into irreducible representations (τ) 22 2 3 16169749982 7423133737 3350637733 1485315445 645794430 274989746 114488565 46516151 18403192 7071840 2631602 945024 326136 107589 33718 9968 2743 695 161 31 5 1 0 0 0 χ , g ˜ h 21 in the function 15790750779 7249168236 3272097750 1450509683 630654439 268547565 111803325 45427164 17971134 6906560 2569637 923068 318375 105135 32895 9753 2671 685 153 32 5 1 0 0 0 χ / 24 D − q 20 15790750779 7249168236 3272097750 1450509683 630654439 268547565 111803325 45427164 17971134 6906560 2569637 923068 318375 105135 32895 9753 2671 685 153 32 5 1 0 0 0 χ 19 13534944287 6213562155 2804662530 1243288847 540564243 230181494 95832723 38936669 15404421 5919487 2202779 791067 272959 90070 28225 8336 2298 583 132 27 5 0 0 0 0 χ 18 13534944287 6213562155 2804662530 1243288847 540564243 230181494 95832723 38936669 15404421 5919487 2202779 791067 272959 90070 28225 8336 2298 583 132 27 5 0 0 0 0 χ 17 13155976526 6039574829 2726136749 1208473427 525430528 223734839 93150396 37845752 14973440 5753503 2141239 768802 265369 87506 27451 8093 2235 562 130 25 4 0 0 0 0 χ The table shows the decomposition of the the decomposition table shows multiplying The coefficients Fourier 44 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part III Table − D − 9 Cheng et al. Mathematical Sciences (2015) 2:13 Page 72 of 89 - part- HS 8 9912751919 4575295928 2077143971 926456288 405495839 173892768 72963281 29890200 11936173 4631476 1743556 633338 221911 74177 23777 7112 2039 515 131 24 6 0 0 0 0 χ of n χ 7 7509616830 3466163569 1573573454 701874719 307184590 131743166 55271192 22646724 9040892 3509785 1320213 480211 167874 56348 17929 5435 1521 407 92 21 3 1 0 0 0 χ into irreducible representations representations into irreducible 6 6608519893 3050184616 1384771352 617631664 270334509 115925926 48643912 19925689 7958171 (τ) 3087208 1162651 422054 148048 49392 15887 4720 1372 339 90 14 5 0 0 0 0 χ 2 1 , g ˜ h 5 6608519893 3050184616 1384771352 617631664 270334509 115925926 48643912 19925689 7958171 3087208 1162651 422054 148048 49392 15887 4720 1372 339 90 14 5 0 0 0 0 in the function χ / 24 D − q 4 6608457916 3050227006 1384742351 617651188 270321371 115934578 48638189 19929384 7955779 3088701 1161707 422628 147695 49595 15766 4787 1334 357 80 19 2 1 0 0 0 χ 3 3304250864 1525098819 692381433 308818831 135165363 57964332 24321125 9963420 3978753 1543852 581198 211115 73979 24733 7929 2372 681 174 44 8 3 0 0 0 0 χ 2 944081846 435736187 197828069 88231069 38620903 16560006 6949871 2846173 1137213 440909 166233 60245 21207 7047 2292 672 203 50 16 2 2 0 0 0 0 χ 1 42910275 19807924 8990981 4011289 1754960 753088 315659 129518 51591 20108 7516 2759 948 330 99 33 7 4 0 0 0 0 0 0 − 1 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the decomposition the Fourier shows table The 45 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 Table − D − 1 I Cheng et al. Mathematical Sciences (2015) 2:13 Page 73 of 89 - part- HS 16 of n 45315309735 20915725781 9495455515 4235268766 1853668197 794956158 333534637 136648460 54560442 21175541 7968570 2896427 1013725 339513 108441 32642 9247 2396 575 117 22 2 0 0 0 χ χ 15 38449273233 17746732420 8056712554 3593587213 1572792424 674519766 282991729 115949075 46290660 17969139 6760018 2458353 859693 288353 91864 27789 7803 2061 478 106 17 3 0 0 0 χ into irreducible representations representations into irreducible 14 38449273233 17746732420 8056712554 3593587213 1572792424 674519766 282991729 115949075 46290660 17969139 6760018 2458353 859693 288353 91864 27789 7803 2061 478 106 17 3 0 0 0 χ (τ) 2 1 , g ˜ h 13 35402559005 16340430775 7418312272 3308813035 1448172795 621063719 260571613 106758449 42624415 16544170 6225096 2263145 791857 265365 84686 25544 7219 1887 449 95 18 3 1 0 0 χ in the function / 24 D − q 12 33042340266 15251101928 6923735648 3088240863 1351617768 579666271 243195757 99644149 39780943 15442404 5809369 2112736 738797 247836 78950 23897 6709 1777 412 94 15 4 0 0 0 χ 11 33042340266 15251101926 6923735648 3088240865 1351617767 579666270 243195756 99644150 39780944 15442403 5809370 2112736 738796 247836 78950 23898 6708 1776 413 94 15 3 0 1 0 χ 10 33042457875 15251021256 6923790672 3088203718 1351642677 579649779 243206606 99637115 39785466 15439550 5811159 2111639 739461 247447 79177 23769 6780 1739 432 85 19 2 1 0 0 χ 9 29738144692 13725964622 6231379962 2779404150 1216464055 521694040 218879656 89677291 35804287 13897177 5228981 1901066 665120 222914 71118 21454 6059 1585 375 79 14 3 0 0 0 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the decomposition the Fourier shows table The 46 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 − 1 Table − D II Cheng et al. Mathematical Sciences (2015) 2:13 Page 74 of 89 - part- HS 24 of 0 0 1 9 67 361 1743 7290 28003 99000 328520 1029079 3071601 8777736 24146350 64169990 165332469 414090303 1010705622 2408967373 n 5617163685 12834168544 28774079262 63381031865 137319059194 χ χ 23 0 0 1 9 56 317 1487 6289 24028 85159 282195 884610 2639284 7544029 20749740 55147824 142079975 355863082 868568813 2070216248 4827235467 11029385777 24727692056 54468122422 118008497342 χ into irreducible representations representations into irreducible 22 0 0 1 7 53 286 1371 5744 22047 77977 258692 810439 2418829 6912578 19015093 50534149 130198910 326096807 795929685 1897063405 4423514093 10106911327 22659582306 49912570342 108138748123 χ (τ) 2 1 , g ˜ h 21 0 0 1 6 40 221 1046 4396 16834 59591 197583 619159 1847627 5280632 14525165 38602966 99456838 249102903 608000215 1449148385 3379069491 7720563319 17309394714 38127670972 82605970008 χ in the function / 24 D − q 20 0 0 1 4 42 215 1054 4374 16866 59519 197695 618937 1847957 5280034 14526071 38601412 99459154 249099125 608005819 1449139547 3379082403 7720543541 17309423366 38127628080 82606031374 χ 19 0 0 0 6 35 202 941 4009 15273 54213 179524 563011 1679391 4800963 13203972 35094660 90413520 226459775 552723198 1317413812 3071871642 7018707863 15735791983 34661549588 75096290859 χ 18 0 0 0 5 27 164 756 3230 12281 43635 144411 453032 1351104 3862871 10623310 28236528 72743550 182203997 444703911 1059955647 2471536933 5647056684 12660561531 27887702980 60420314767 χ 17 0 0 0 4 26 162 740 3180 12073 42967 142122 445985 1329897 3802615 10457085 27795624 71606337 179357824 437754013 1043395661 2432916045 5558825615 12462733462 27451966739 59476232959 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the decomposition the Fourier shows table The 47 Table − D − 1 III 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 Cheng et al. Mathematical Sciences (2015) 2:13 Page 75 of 89 - part- HS 8 7674283501 3523111975 1590230348 704954744 306494073 130517592 54334528 22079147 8733350 3357136 1248628 448887 154628 51207 15972 4768 1298 349 73 21 3 1 0 1 0 χ of n χ 7 5813889540 2668997408 1204737801 534044327 232200431 98871502 41165993 16724285 6617566 2542349 946482 339698 117337 38662 12169 3567 1000 251 61 12 3 0 1 0 0 χ into irreducible representations representations into irreducible 6 5116172310 2348751902 1060145461 469974829 204325655 87013768 36221486 14720250 5821517 (τ) 2238459 832131 299362 102985 34168 10609 3191 850 232 45 13 1 1 0 0 0 χ 2 3 , g ˜ h 5 5116172310 2348751902 1060145461 469974829 204325655 87013768 36221486 14720250 5821517 2238459 832131 299362 102985 34168 10609 3191 850 232 45 13 1 1 0 0 0 in the function χ / 24 D − q 4 5116227145 2348714622 1060170981 469957788 204337166 87006255 36226467 14717082 5823587 2237183 832940 298883 103287 34000 10710 3138 882 217 54 10 3 0 0 0 0 χ 3 2558094287 1174370946 530077046 234984819 102164805 43505965 18111488 7359765 2911159 1119026 416242 149641 51536 17078 5333 1585 436 118 25 6 2 0 1 0 0 χ 2 730874824 335540316 151446502 67140916 29188063 12431424 5173804 2103267 831414 319869 118790 42824 14653 4906 1504 455 117 36 4 2 0 0 0 0 0 χ 1 33223874 15250485 6885123 3051197 1327268 564823 235400 95506 37900 14492 5448 1937 680 224 76 19 8 2 1 0 1 0 0 0 1 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the decomposition the Fourier shows table The 48 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 Table − D − 9 I Cheng et al. Mathematical Sciences (2015) 2:13 Page 76 of 89 - part- HS 16 of 35082549764 16105571907 7269672253 3222614325 1401136588 596634205 248396323 100925428 39927180 15344148 5709189 2050680 707419 233542 73107 21661 5937 1520 344 71 10 3 0 0 0 χ n χ 15 29767082573 13665284242 6168239640 2734316595 1188857894 506225002 210766813 85629401 33880203 13017514 4845163 1739300 600589 197917 62148 18296 5071 1265 299 54 10 1 0 0 0 χ into irreducible representations representations into irreducible 14 29767082573 13665284242 6168239640 2734316595 1188857894 506225002 210766813 85629401 33880203 13017514 4845163 1739300 600589 197917 62148 18296 5071 1265 299 54 10 1 0 0 0 χ (τ) 2 3 , g ˜ h 13 27408265179 12582461650 5679443225 2517659061 1094643301 466117227 194061566 78846635 31194087 11986913 4460701 1601901 552755 182391 57171 16879 4655 1183 268 56 9 1 0 0 0 χ in the function / 24 D − q 12 25581089850 11743601341 5300833219 2349801630 1021675875 435036721 181127903 73587635 29116039 11186751 4163910 1494707 516126 170086 53426 15709 4367 1089 254 48 10 0 0 0 0 χ 11 25581089850 11743601341 5300833219 2349801630 1021675875 435036721 181127903 73587635 29116039 11186751 4163910 1494707 516126 170086 53426 15709 4367 1089 254 48 10 0 0 0 0 χ 10 25580985954 11743672359 5300784874 2349834128 1021654112 435051059 181118490 73593695 29112145 11189191 4162388 1495628 515565 170410 53238 15813 4310 1118 239 54 7 1 0 0 0 χ 9 23022947110 10569264743 4770733874 2114832583 919501026 391537803 163012176 66230865 26203117 10068963 3747019 1345538 464346 153185 48025 14181 3908 990 227 46 7 1 0 0 0 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the decomposition the Fourier shows table The 49 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 Table − D − 9 II Cheng et al. Mathematical Sciences (2015) 2:13 Page 77 of 89 - part- HS 24 of 0 0 0 5 34 213 1043 4587 18023 65550 221641 707534 2143915 6213691 17301389 46495887 120993550 305831383 752720667 1807974775 4245880168 9765479321 22029335806 48804725354 106310810073 χ n χ 23 0 0 1 4 32 181 904 3932 15532 56267 190608 607872 1842742 5339398 14869371 39955933 103981276 262820443 646874829 1553720168 3648816574 8392189392 18931489848 41941519745 91360913541 χ into irreducible representations representations into irreducible 22 0 0 0 4 28 168 822 3612 14204 51610 174568 557162 1688384 4893216 13625020 36615272 95282852 240841718 592768390 1423778908 3343632878 7690311850 17348106814 38433714858 83719772698 χ (τ) 2 3 , g ˜ h 21 0 0 0 2 20 125 628 2750 10850 39398 133362 425535 1289797 3737680 10408176 27969538 72786002 183975195 452810502 1087606345 2554167059 5874538248 13252033063 29359075739 63952619706 χ in the function / 24 D − q 20 0 0 0 4 20 131 622 2770 10828 39464 133278 425729 1289537 3738214 10407432 27970908 72784072 183978541 452805738 1087614155 2554155967 5874555816 13252008175 29359113857 63952566006 χ 19 0 0 0 2 19 111 580 2489 9884 35784 121313 386726 1172777 3397533 9462567 25425987 66170567 167247845 411649683 988727552 2321978347 5340477195 12047321308 26690041661 58138785609 χ 18 0 0 1 2 19 93 468 2007 7976 28783 97662 311150 943669 2733527 7613625 20456746 53239679 134562341 331202866 795500581 1868200964 4296791328 9692937895 21474037147 46776819698 χ 17 0 0 1 2 19 89 466 1973 7859 28328 96152 306248 929020 2690703 7494835 20136861 52408277 132459084 326029062 783069157 1839012867 4229650434 9541491409 21138496831 46045944874 χ The table shows the decomposition of the Fourier coefficients multiplying coefficients of the decomposition the Fourier shows table The 50 Table − D − 9 III 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567 Cheng et al. Mathematical Sciences (2015) 2:13 Page 78 of 89 - ( 2 ) 6 U 10 of 0 1 0 1 2 5 9 27 61 180 487 1411 3914 10878 29109 76503 195033 486240 1181610 2810645 6541292 14931866 33447965 73643797 159488167 χ n χ 9 0 0 0 1 1 4 7 22 56 172 475 1394 3892 10843 29067 76443 194953 486133 1181472 2810456 6541054 14931553 33447564 73643280 159487514 χ 8 0 0 0 0 1 0 3 10 32 103 330 998 2936 8287 22680 60008 154261 385653 940408 2239707 5220237 11923091 26726129 58860381 127512423 χ into irreducible representations (τ) 2 1 , g ˜ h 7 0 0 0 0 0 2 3 13 32 112 312 971 2723 7741 20811 55156 140844 352289 856699 2040543 4750527 10850420 24309087 53536679 115951616 χ in the function 6 0 0 0 0 0 2 3 11 33 99 280 807 2265 6244 16751 43863 111842 278362 676374 1607611 3741059 8536684 19121267 42092946 91155986 χ / 24 D − q 5 0 0 0 0 0 0 1 3 18 54 196 592 1803 5080 14074 37214 96109 240240 586899 1397738 3260343 7446731 16697919 36775178 79680703 χ 4 0 0 0 0 0 1 2 8 17 57 156 461 1279 3556 9534 25066 63872 159223 386854 920162 2141266 4887625 10947884 24103804 52199413 χ 3 0 0 0 0 1 2 5 10 28 67 187 477 1318 3451 9188 23558 59837 147655 358129 848112 1971916 4492407 10058179 22125220 47903632 χ 2 0 0 0 0 1 1 3 4 8 15 36 73 179 404 1029 2473 6120 14678 35220 82414 190679 431992 964995 2117328 4579167 χ 1 − 1 0 0 0 0 0 0 1 0 2 3 9 12 33 66 160 334 795 1763 4088 9121 20500 45009 98354 210968 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 51 part I Table − D − 1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575 Cheng et al. Mathematical Sciences (2015) 2:13 Page 79 of 89 - ( 2 ) 6 U 19 956216022 441360427 200393822 89385449 39131147 16782848 7045333 2886846 1154153 448042 169159 61506 21717 7273 2383 715 219 55 17 3 1 0 0 0 0 χ of n χ 18 728720690 336394392 152753296 68152090 29842923 12806125 5378796 2206704 883312 343936 130238 47724 16983 5815 1946 625 204 63 22 8 3 1 1 0 0 χ 17 637564041 294300936 133631622 59615100 26101618 11198333 4702278 1928238 771388 300016 113440 41450 14694 4992 1653 519 164 49 16 5 2 1 0 0 0 χ into irreducible representations (τ) 2 1 , g ˜ h 16 637563387 294300418 133631223 59614785 26101380 11198147 4702139 1928130 771309 299955 113397 41417 14671 4975 1642 510 159 45 14 3 1 0 0 0 0 χ in the function / 24 D − q 15 318652800 147060137 66762700 29770742 13029827 5584747 2343230 958742 382864 148102 55773 20097 7052 2302 744 207 61 11 4 0 0 0 0 0 0 χ 14 286811937 132364423 60096749 26797674 11731080 5027797 2110561 863390 345208 133476 50419 18136 6424 2090 693 189 62 12 5 0 1 0 0 0 0 χ 13 238965109 110289605 50061479 22326103 9767990 4187951 1755665 718884 286479 111048 41585 15071 5207 1732 532 157 39 10 1 0 0 0 0 0 0 χ 12 238965109 110289605 50061479 22326103 9767990 4187951 1755665 718884 286479 111048 41585 15071 5207 1732 532 157 39 10 1 0 0 0 0 0 0 χ 11 238965109 110289605 50061479 22326103 9767990 4187951 1755665 718884 286479 111048 41585 15071 5207 1732 532 157 39 10 1 0 0 0 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 52 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part II − 1 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 80 of 89 - ( 2 ) 6 U 28 of 1912148601 882558528 400669808 178705755 78215160 33540710 14072625 5764588 2301754 892976 336080 122027 42728 14266 4564 1360 388 97 24 4 1 0 0 0 0 χ n χ 27 1912148601 882558528 400669808 178705755 78215160 33540710 14072625 5764588 2301754 892976 336080 122027 42728 14266 4564 1360 388 97 24 4 1 0 0 0 0 χ 26 1912148601 882558528 400669808 178705755 78215160 33540710 14072625 5764588 2301754 892976 336080 122027 42728 14266 4564 1360 388 97 24 4 1 0 0 0 0 χ into irreducible representations (τ) 2 1 , g ˜ h 25 1668815367 770250507 349688430 155968449 68265693 29274597 12283559 5031900 2009529 779662 293554 106605 37369 12481 4006 1194 345 86 22 4 1 0 0 0 0 χ in the function / 24 D − q 24 1434203044 661990039 300539689 134058491 58675953 25167016 10559896 4327763 1728219 671268 252683 92034 32232 10856 3473 1064 302 83 20 5 1 0 0 0 0 χ 23 1274986283 588519370 267204168 119196928 52179508 22383781 9395420 3851728 1539418 598368 225714 82356 29006 9816 3191 991 296 85 24 7 2 1 0 0 0 χ 22 1274985629 588518852 267203769 119196613 52179270 22383595 9395281 3851620 1539339 598307 225671 82323 28983 9799 3180 982 291 81 22 5 1 0 0 0 0 χ 21 1147598073 529752466 240528777 107312059 46979165 20159114 8462435 3471701 1387781 540363 203895 74730 26332 9022 2932 942 281 89 24 8 2 1 0 0 0 χ 20 1019815583 470720019 213691022 95319311 41714567 17892382 7505177 3076064 1227450 476877 179162 65313 22754 7688 2420 750 204 59 12 4 0 1 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 53 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part III − 1 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 81 of 89 - ( 2 ) 6 U 37 of 3872097331 1787223481 811353013 361897629 158383477 67927686 28495813 11676421 4660418 1809476 680273 247540 86401 29034 9196 2800 769 208 44 11 1 0 0 0 0 χ n χ 36 3059214800 1411960695 640979164 285875651 125107300 53644469 22502085 9215765 3677682 1426142 535979 194405 67811 22579 7143 2112 581 141 30 4 0 0 0 0 0 χ 35 2868612560 1324106543 601164912 268167991 117386017 50353789 21132786 8662775 3461147 1345063 506964 184876 64970 21956 7089 2189 641 181 47 12 3 1 0 0 0 χ into irreducible representations (τ) 2 1 , g ˜ h 34 2331033239 1075912674 488448796 217864013 95353051 40892977 17157026 7029317 2806561 1089307 409888 149008 52139 17469 5576 1679 476 124 30 6 1 0 0 0 0 χ in the function / 24 D − q 33 2151118874 992844595 450733710 201030242 87984244 37727937 15828769 6483169 2588431 1003898 377745 137052 47964 15981 5106 1512 430 105 26 4 1 0 0 0 0 χ 32 2151118874 992844595 450733710 201030242 87984244 37727937 15828769 6483169 2588431 1003898 377745 137052 47964 15981 5106 1512 430 105 26 4 1 0 0 0 0 χ 31 2151118874 992844595 450733710 201030242 87984244 37727937 15828769 6483169 2588431 1003898 377745 137052 47964 15981 5106 1512 430 105 26 4 1 0 0 0 0 χ 30 2151521277 993104358 450899874 201135486 88050195 37768797 15853785 6498279 2597424 1009164 380778 138765 48909 16488 5373 1647 495 135 39 9 3 0 0 0 0 χ 29 2151522265 993105129 450900478 201135952 88050558 37769072 15853998 6498438 2597544 1009253 380846 138812 48944 16513 5391 1659 504 140 43 11 4 1 1 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 54 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part IV Table − D − 1 Cheng et al. Mathematical Sciences (2015) 2:13 Page 82 of 89 - ( 2 ) 6 U 46 of 8297132508 3829570352 1738525374 775414671 339359186 145527486 61050043 25008800 9982360 3873101 1456375 528970 184763 61760 19609 5869 1631 415 91 17 2 0 0 0 0 n χ χ 45 7744222864 3574428126 1622719280 723786642 316772980 135851564 56994271 23351239 9321948 3618306 1360994 494841 172971 57985 18452 5575 1557 409 94 21 3 0 0 0 0 χ 44 6781185810 3129929312 1420941137 633786948 277390339 118961732 49911204 20449076 8164490 3168960 1192384 433503 151675 50831 16220 4892 1384 362 87 18 3 1 0 0 0 χ into irreducible representations (τ) 2 1 , g ˜ h 43 5280310504 2437220934 1106464669 493536047 216007314 92643752 38869434 15927891 6359343 2469363 929109 338162 118296 39773 12681 3864 1088 295 69 17 3 1 0 0 0 χ in the function / 24 D − q 42 5280310504 2437220934 1106464669 493536047 216007314 92643752 38869434 15927891 6359343 2469363 929109 338162 118296 39773 12681 3864 1088 295 69 17 3 1 0 0 0 χ 41 5099152355 2353547126 1068489749 476572073 208588414 89450970 37532560 15375595 6140008 2382438 896888 325787 114151 38155 12230 3657 1049 265 67 12 3 0 0 0 0 χ 40 4302423193 1985829461 901543092 402117862 175998083 75478392 31668560 12974774 5180729 2010778 756753 275091 96303 32259 10311 3105 881 228 55 11 2 0 0 0 0 χ 39 4302423193 1985829461 901543092 402117862 175998083 75478392 31668560 12974774 5180729 2010778 756753 275091 96303 32259 10311 3105 881 228 55 11 2 0 0 0 0 χ 38 3872125240 1787204411 811366088 361888871 158389396 67923794 28498396 11674769 4661495 1808802 680702 247286 86560 28943 9251 2771 786 199 49 9 2 0 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 55 575 551 527 503 479 455 431 407 383 359 335 311 287 263 239 215 191 167 143 119 95 71 47 23 part V Table − D − 1 Cheng et al. Mathematical Sciences (2015) 2:13 Page 83 of 89 - ( 2 ) 6 U 10 of 123491729 56711601 25615443 11363145 4947675 2109987 881263 359275 143220 55450 21026 7697 2782 964 342 113 43 14 5 2 1 0 0 0 0 χ n χ 9 123491729 56711601 25615443 11363145 4947675 2109987 881263 359275 143220 55450 21026 7697 2782 964 342 113 43 14 5 2 1 0 0 0 0 χ 8 98720482 45326175 20462399 9073094 3946210 1681362 700427 284997 112936 43501 16267 5889 2037 693 223 66 20 7 0 0 0 0 0 0 0 χ into irreducible representations (τ) 2 3 , g ˜ h 7 89782368 41222644 18617246 8255052 3593518 1530976 639120 259946 103514 39848 15072 5434 1952 648 228 66 24 6 2 0 0 0 0 0 0 χ in the function 6 70582578 32418588 14643912 6498220 2829836 1207687 504591 206063 82214 31983 12149 4502 1639 585 213 79 30 13 5 2 1 1 0 0 0 χ / 24 D − q 5 61684063 28320987 12782104 5667740 2463620 1049704 436768 177721 70199 27087 10035 3642 1242 423 128 42 10 4 1 0 0 0 0 0 0 χ 4 40418996 18562586 8384938 3719775 1619959 691001 288683 117781 47011 18206 6947 2556 928 334 124 40 19 8 4 1 1 0 1 0 0 χ 3 37091514 17046286 7702776 3422339 1491474 638341 267074 109822 43965 17366 6655 2574 946 379 140 62 24 14 5 4 1 1 0 1 0 χ 2 3545538 1632596 738943 329562 144100 62208 26176 10973 4447 1817 716 299 107 54 16 8 2 2 0 0 0 0 0 0 0 χ 1 164118 75899 34864 15666 7070 3111 1400 608 287 122 64 30 17 10 5 2 2 1 1 0 1 0 0 0 1 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 56 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part I Table − D − 9 Cheng et al. Mathematical Sciences (2015) 2:13 Page 84 of 89 - ( 2 ) 6 U 19 740284029 339876627 153417371 68021751 29576649 12599677 5246147 2133713 844320 325236 121084 43783 15102 5088 1596 497 139 44 9 4 1 0 0 0 0 χ of n χ 18 564184563 259049862 116954636 51864617 22560343 9614460 4006776 1631017 646742 249613 93406 33922 11856 4034 1310 416 127 40 10 4 0 0 0 0 0 χ 17 493602894 226631616 102311008 45366836 19730666 8406902 3502378 1425020 564586 217714 81284 29442 10248 3458 1106 350 100 30 8 2 0 0 0 0 0 χ into irreducible representations (τ) 2 3 , g ˜ h 16 493602894 226631616 102311008 45366836 19730666 8406902 3502378 1425020 564586 217714 81284 29442 10248 3458 1106 350 100 30 8 2 0 0 0 0 0 χ in the function / 24 D − q 15 246680793 113244297 51105811 22654755 9845654 4192520 1743695 708547 279626 107493 39741 14297 4842 1607 475 143 32 9 0 0 0 0 0 0 0 χ 14 222026326 101933701 46000805 20394871 8863180 3775558 1570043 638563 251918 97039 35856 12975 4370 1484 436 136 30 11 0 0 0 0 0 0 0 χ 13 185003006 84919350 38326695 16985215 7383374 3142048 1307537 530506 209675 80286 29814 10611 3638 1172 363 97 26 5 1 0 0 0 0 0 0 χ 12 185003006 84919350 38326695 16985215 7383374 3142048 1307537 530506 209675 80286 29814 10611 3638 1172 363 97 26 5 1 0 0 0 0 0 0 χ 11 185003006 84919350 38326695 16985215 7383374 3142048 1307537 530506 209675 80286 29814 10611 3638 1172 363 97 26 5 1 0 0 0 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 57 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part II − 9 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 85 of 89 - ( 2 ) 6 U 28 of 1480349568 679593708 306745221 135979088 59118234 25174005 10479406 4257972 1683993 647221 240627 86454 29759 9834 3057 909 243 63 13 3 0 0 0 0 0 χ n χ 27 1480349568 679593708 306745221 135979088 59118234 25174005 10479406 4257972 1683993 647221 240627 86454 29759 9834 3057 909 243 63 13 3 0 0 0 0 0 χ 26 1480349568 679593708 306745221 135979088 59118234 25174005 10479406 4257972 1683993 647221 240627 86454 29759 9834 3057 909 243 63 13 3 0 0 0 0 0 χ into irreducible representations (τ) 2 3 , g ˜ h 25 1291964951 593117732 267714806 118679661 51598009 21972919 9147020 3717138 1470206 565195 210189 75588 26013 8627 2689 801 219 59 11 4 1 0 0 0 0 χ in the function / 24 D − q 24 1110356978 509745480 230098050 102004268 44354258 18887958 7865522 3196124 1265124 486308 181184 65100 22548 7442 2352 698 196 48 12 2 0 0 0 0 0 χ 23 987096186 453185010 204577702 90701880 39444133 16801605 6998318 2845570 1126986 433851 161869 58400 20286 6779 2165 662 193 55 15 4 1 0 0 0 0 χ 22 987096186 453185010 204577702 90701880 39444133 16801605 6998318 2845570 1126986 433851 161869 58400 20286 6779 2165 662 193 55 15 4 1 0 0 0 0 χ 21 888501021 407928442 184167767 81656325 35518958 15130943 6305860 2564491 1017030 391657 146652 52957 18566 6225 2051 630 204 60 20 7 3 1 1 0 0 χ 20 789548632 362451055 163610220 72522016 31535156 13425927 5591173 2270753 899003 345074 128664 46068 15977 5228 1672 477 142 32 9 2 1 0 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 58 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part III Table − D − 9 Cheng et al. Mathematical Sciences (2015) 2:13 Page 86 of 89 - ( 2 ) 6 U of 37 n 2997763988 1376174113 621183917 275355244 119725056 50975995 21225130 8621691 3411830 1310274 487925 174934 60491 19864 6268 1822 517 123 32 5 2 0 0 0 0 χ χ 36 2368380807 1087225658 490719721 217516112 94560774 40258951 16756387 6805582 2690675 1033033 383784 137526 47238 15497 4796 1390 367 89 16 3 0 0 0 0 0 χ 35 2220884907 1019607264 460266342 204055733 88736216 37794342 15741552 6399104 2534016 975053 363656 131014 45506 15140 4830 1471 424 118 35 8 2 1 0 0 0 χ into irreducible representations (τ) 2 3 , g ˜ h 34 1804658930 828475182 373954834 165771974 72075158 30690911 12777678 5191571 2053906 789280 293697 105478 36393 12011 3763 1113 306 77 17 4 1 0 0 0 0 χ in the function / 24 D − q 33 1665348005 764516151 345069777 152965722 66500652 28316673 11786521 4788741 1893498 727612 270372 97104 33372 11019 3411 1011 267 69 13 3 0 0 0 0 0 χ 32 1665348005 764516151 345069777 152965722 66500652 28316673 11786521 4788741 1893498 727612 270372 97104 33372 11019 3411 1011 267 69 13 3 0 0 0 0 0 χ 31 1665348005 764516151 345069777 152965722 66500652 28316673 11786521 4788741 1893498 727612 270372 97104 33372 11019 3411 1011 267 69 13 3 0 0 0 0 0 χ 30 1665696713 764740650 345212969 153056164 66557162 28351578 11807813 4801562 1901107 732055 272918 98541 34164 11444 3633 1124 322 95 24 8 2 1 0 0 0 χ 29 1665696713 764740650 345212969 153056164 66557162 28351578 11807813 4801562 1901107 732055 272918 98541 34164 11444 3633 1124 322 95 24 8 2 1 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 59 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part IV − 9 Table − D Cheng et al. Mathematical Sciences (2015) 2:13 Page 87 of 89 - ( 2 ) 6 U 46 of 6423520020 2948823657 1330999539 589999521 256509711 109216491 45465426 18468738 7304754 2805657 1043352 374214 128899 42381 13215 3861 1044 255 54 9 0 0 0 0 0 n χ χ 45 5995502658 2752364303 1242355809 550717850 239444613 101955129 42447872 17244669 6822630 2621064 975422 350044 120825 39779 12474 3662 1012 250 58 9 2 0 0 0 0 χ 44 5249918560 2410108430 1087869446 482245140 209673562 89282411 37171558 15102657 5975070 2296034 854421 306831 105878 34934 10951 3233 891 225 51 11 2 0 0 0 0 χ into irreducible representations (τ) 2 3 , g ˜ h 43 4087990850 1876700029 847121958 375524639 163281960 69528362 28950941 11762638 4655102 1788772 666188 239200 82728 27278 8608 2536 715 179 45 8 2 0 0 0 0 χ in the function / 24 D − q 42 4087990850 1876700029 847121958 375524639 163281960 69528362 28950941 11762638 4655102 1788772 666188 239200 82728 27278 8608 2536 715 179 45 8 2 0 0 0 0 χ 41 3947676846 1812299925 818019198 362631248 157661257 67138492 27949916 11357471 4492367 1726958 642249 230873 79550 26322 8205 2457 662 175 38 9 1 1 0 0 0 χ 40 3330883649 1529134899 690217270 305972205 133032283 56648802 23584843 9583033 3791267 1457089 542191 194787 67201 22198 6951 2060 566 145 32 7 1 0 0 0 0 χ 39 3330883649 1529134899 690217270 305972205 133032283 56648802 23584843 9583033 3791267 1457089 542191 194787 67201 22198 6951 2060 566 145 32 7 1 0 0 0 0 χ 38 2997739281 1376190840 621172387 275362888 119719865 50979357 21222871 8623112 3410897 1310841 487555 175147 60355 19937 6220 1846 503 130 28 6 1 0 0 0 0 χ The table shows the decomposition of the the decomposition multiplying coefficients table shows The Fourier 60 567 543 519 495 471 447 423 399 375 351 327 303 279 255 231 207 183 159 135 111 87 63 39 15 part V Table − D − 9 Cheng et al. Mathematical Sciences (2015) 2:13 Page 88 of 89

Received: 1 July 2014 Accepted: 15 June 2015

References 1. Benjamin, N., Harrison, S.M., Kachru, S., Paquette, N.M., Whalen, D.: On the elliptic genera of manifolds of Spin(7) holonomy. arXiv:1412.2804 [hep-th] 2. Benjamin, N, Harrison, S.M., Kachru, S., Paquette, N.M., Whalen, D.: On the elliptic genera of manifolds of Spin(7) holonomy. arXiv:1412.2804 3. Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405 (1992) 4. Cheng, M.C.N.: K3 surfaces, N 4 Dyons, and the Mathieu Group M24. Commun. Num. Theor. Phys. 4, 623 (2010). arXiv:1005.5415 [hep-th] = 5. Cheng, M.C.N., Dong, X., Duncan, J., Harvey, J., Kachru, S., Wrase, T.: Mathieu moonshine and N 2 string compactifi- cations. JHEP 1309, 030 (2013). arXiv:1306.4981 [hep-th] = 6. Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Weight one Jacobi forms and umbral moonshine (to appear) 7. Cheng, M.C.N., Duncan, J.F.R.: On Rademacher sums, the largest Mathieu group, and the Holographic modularity of moonshine. Commun. Num. Theor. Phys. 6, 697 (2012). arXiv:1110.3859 [math.RT] 8. Cheng, M.C.N., Duncan, J.F.R.: On the discrete groups of Mathieu moonshine. arXiv:1212.0906 [math.NT] 9. Cheng, M.C.N., Harrison, S.: Umbral moonshine and K3 surfaces. arXiv:1406.0619 [hep-th] 10. Cheng, M.C.N., Harrison, S.M., Kachru, S., Whalen, D.: Exceptional algebra and Sporadic Groups at c 12. arXiv:1503.07219 = 11. Cheng, M.C.N., Duncan, J.F.R.: Rademacher sums and rademacher series. arXiv:1210.3066 [math.NT] 12. Cheng, M.C.N., Harrison, S.M., Kachru, S., Whalen, D.: Exceptional algebra and Sporadic Groups at c 12. arXiv:1503.07219 [hep-th] = 13. Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Umbral moonshine. arXiv:1204.2779 [math.RT] 14. Cheng, M.C.N., Duncan, J.F.R. Harvey, J.A.: Umbral moonshine and the Niemeier lattices. Research in the Mathemati- cal Sciences 1:3. arXiv:1307.5793 [math.RT] (2014) 15. Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308 (1979) 16. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Springer, New York (1999) 17. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With comput. assist. from J. G. Thackray., Clarendon Press, Oxford (1985) 18. Creutzig, T., Hoehn, G.: Mathieu moonshine and the geometry of K3 surfaces. arXiv:1309.2671 [math.QA] 19. Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes, wall crossing, and mock modular forms. arXiv:1208.4074 [hep-th] 20. Dijkgraaf, R., Maldacena, J.M., Moore, G.W., Verlinde, E.P.: A black hole farey tail. hep-th/0005003 21. Dixon, L.J., Ginsparg, P.H., Harvey, J.A.: Beauty and the beast: superconformal symmetry in a monster module. Comm. Math. Phys. 119, 221 (1988) 22. Dobrev, V.K.: Characters of the unitarizable highest weight modules over the N = 2 superconformal algebras. Phys. Lett. B. 186, 43 (1987) 23. Dong, C., Li, H., Mason, G., Montague, P.S.: The radical of a vertex operator algebra. In: The monster and Lie algebras (Columbus, OH, 1996), Ohio State Univ. Math. Res. Inst. Publ., 7 (1998) 24. Duncan, J.F.: Super-moonshine for Conway’s largest sporadic group. Duke Math. J. 139(2), 255–315 (2007). arXiv:math/0502267 25. Duncan, J.F.R., Mack-Crane, S.: Derived equivalences of K3 surfaces and twined elliptic genera (to appear) 26. Duncan, J.F.R., Harvey, J.A.: The umbral mooonshine module for the unique unimodular Niemeier Root System. arXiv:1412.8191 27. Duncan, J.F.R., Mack-Crane, S.: The moonshine module for Conway’s Group. arXiv:1409.3829 28. Duncan, J.F.R., Griffin, M.J., Ono, K.: Proof of the umbral moonshine conjecture. arXiv:1503.01472 29. Duncan, J.F., Frenkel, I.B.: Rademacher sums, moonshine and gravity. Commun. Num. Theor. Phys. 5, 849 (2011). arXiv:0907.4529 [math.RT]] 30. Duncan, J.F.R., Griffin, M.J., Ono, K.: Moonshine. arXiv:1411.6571 31. Eguchi, T., Hikami, K.: Note on twisted elliptic genus of K3 surface. Phys. Lett. B 694, 446 (2011). arXiv:1008.4924 [hep-th] 32. Eguchi, T., Taormina, A.: Unitary representations of the N = 4 superconformal algebra. Phys. Lett. B. 196, 1 (1987) 33. Eguchi, T., Taormina, A.: On the unitary representations of N = 2 and N = 4 superconformal algebras. Phys. Lett. B. 210, 125 (1988) 34. Eguchi, T., Taormina, A.: Character formulas for the N = 4 superconformal algebra. Phys. Lett. B. 200, 315 (1988) 35. Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 Surface and the Mathieu group M24. Exper. Math. 20, 91 (2011). arXiv:1004.0956 [hep-th] 36. Frenkel, I., Lepowsky, J., Meurman, A.: A moonshine module for the monster, in vertex operators in mathematics and physics (Berkeley, CA 1983), vol. 3, Math. Sci. Res. Inst. Publ., Springer (1985) 37. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster, vol. 134, Pure and Applied Math- ematics, Elsevier Science (1989) 38. Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu twining characters for K3. JHEP 1009, 058 (2010). arXiv:1006.0221 [hep-th] 39. Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu moonshine in the elliptic genus of K3. JHEP 1010, 062 (2010). arXiv:1008.3778 [hep-th] 40. Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Symmetries of K3 sigma models. Commun. Num. Theor. Phys. 6, 1 (2012). arXiv:1106.4315 [hep-th] Cheng et al. Mathematical Sciences (2015) 2:13 Page 89 of 89

41. Gaberdiel, M.R., Persson, D., Ronellenfitsch, H., Volpato, R.: Generalised Mathieu moonshine. Commun. Num. Theor. Phys. 7, 145 (2013). arXiv:1211.7074 [hep-th] 42. Gaberdiel, M.R., Persson, D., Volpato, R.: Generalised moonshine and holomorphic orbifolds. arXiv:1302.5425 [hep-th] 43. Gannon, T.: Much ado about Mathieu. arXiv:1211.5531 [math.RT] 44. Gannon, T.: Moonshine beyond the monster: the bridge connecting algebras, modular forms, and physics. Cam- bridge University Press, Cambridge Monographs on Mathematical Physics (2006) 45. Gepner, D., Noyvert, B.: Unitary representations of SW(3/2,2) superconformal algebra. Nucl. Phys. B. 610, 545 (2001). arXiv:hep-th/0101116 46. Govindarajan, S.: Unravelling Mathieu moonshine. Nucl. Phys. B 864, 823 (2012). arXiv:1106.5715 [hep-th] 47. Harrison, S., Kachru, S., Paquette, N.M.: Twining genera of (0,4) supersymmetric Sigma models on K3. JHEP 1404, 048 (2014). arXiv: 1309.0510 [hep-th] 48. Harvey, J.A., Murthy, S.: Moonshine in fivebrane spacetimes. JHEP 1401, 146 (2014). arXiv:1307.7717 [hep-th] 49. Hohenegger, S., Stieberger, S.: BPS saturated string amplitudes: K3 elliptic genus and Igusa Cusp form. Nucl. Phys. B 856, 413 (2012). arXiv:1108.0323 [hep-th] 50. Kac, V.: Vertex algebras for beginners. University Lecture Series, American Mathematical Society 10 (1988) 51. Kiritsis, E.: Character formulae and the structure of the representations of the N = 1, N = 2 superconformal algebras. Int. J. Mod. Phys. A. 3, 1871 (1988) 52. Kostelecky, V.A., Lechtenfeld, O., Lerche, W., Samuel, S., Watamura, S.: Conformal techniques, bosonization and tree level string amplitudes. Nucl. Phys. B. 288, 173 (1987) 53. Manschot, J., Moore, G.W.: A modern farey tail. Commun. Num. Theor. Phys. 4, 103 (2010). arXiv:0712.0573 [hep-th] 54. Neumann, C.D.D.: The elliptic genus of Calabi-Yau 3-folds and 4-folds: product formulae and generalized Kac–Moody algebras. J. Geom. Phys. 29, 5 (1999). arXiv:hep-th/9607029 55. Ono, K., Rolen, L., Trebat-Leder, S.: Classical and umbral moonshine: connections and p-adic properties. J. Ramanujan Math. Soc. (to appear) (arXiv e-prints (2014)) 56. Persson, D., Volpato, R.: Second quantized Mathieu moonshine. arXiv:1312.0622 [hep-th] 57. Raum, M.: M24-twisted product expansions are Siegel modular forms. arXiv:1208.3453 [math.NT] 58. Shatashvili, S.L., Vafa, C.: Superstrings and manifold of exceptional holonomy. Selecta Math. 1, 347 (1995). arXiv:hep-th/9407025 59. Taormina, A., Wendland, K.: The overarching finite symmetry group of Kummer surfaces in the Mathieu groupM 24. JHEP 1308, 125 (2013). arXiv:1107.3834 [hep-th] 60. Taormina, A., Wendland, K.: The symmetries of the tetrahedral kummer surface in the Mathieu group M24. arXiv:1008.0954 [hep-th] 61. Taormina, A., Wendland, K.: Symmetry-surfing the moduli space of Kummer K3s. arXiv:1303.2931 [hep-th] 62. Witten, E.: Three-dimensional gravity revisited. arXiv:0706.3359 [hep-th] 63. Wrase, T.: Mathieu moonshine in four dimensional N = ∞ theories. JHEP 1404, 069 (2014). arXiv:1402.2973 [hep-th] 64. Zwegers, S.: Mock theta functions. arXiv:0807.4834 [math.NT]