In "A proof of the Thompson Moonshine Conjecture" [GM16], the McKay-Thompson series are written as linear combinations of eta quotients to provide a way to compute their behaviour at cusps. Here we give these linear combinations. The data are to be interpreted as follows.
The file EtaWt[k]_N.m contains a basis of the space M k(N) of modular forms of weight k for the group Γ0(N) consisting of eta quotients (this was computed using MAGMA scripts by Rouse and Webb, see here). Here, an eta quotient
∏ d|N η(dτ)rd
is represented by a list of pairs [<d,rd> : d|N]. The file EtaBases.m contains all the data from all the files EtaWt[k]_N.m as a single list.
The file EtaCoeffs[g].m, where [g] represents a conjugacy class of the Thompson group, contains a list of rational numbers such that the McKay-Thompson series of [g] is the linear combination of the eta quotients in the file EtaWt[k]_N.m , where N is the level N[g] specified in table A.5 of [GM16] with these rational numbers as coefficients, divided by a certain eta quotient g specified by the level (see the MAGMA file etahalf.m). The file EtaCoeffsAll.m is again a compilation of all the EtaCoeffs-files.
The file EtaCoeffs1A.m contains the list
[ 76, 0, 3968, 3592, 2, 912 ].The corresponding g from is given by the list
[<1,2>,<2,-1>,<4,18>].So consulting the file
EtaWt10_4.m we find that the McKay-Thompson series of the class 1A is given by
76η(τ)30η(2τ)-19η(4τ)-10 +3968η(τ)6η(2τ)-19η(4τ)14 +3592η(τ)14η(2τ)-19η(4τ)6 +2η(τ)38η(2τ)-19 +912η(τ)22η(2τ)-19η(4τ)-2.
| EtaCoeffs | EtaBases |