# Computing Thompson Series for the Monster Group

I am trying to do some experimentation with the values of Thompson series, but I have having a hard time finding a table that has these Thompson series with as many terms as I'd like. The tables I've seen only give me the required Head character values up to the $q^{10}$ term.

Can someone point me to a resource that either has more terms, or (even better), a resource that might help me compute them myself in sage?

Thanks a lot for any help.

• The periodicity should follow from asymptotic formulas for the coefficients of these modular functions analogous to those given by Hardy and Ramanujan for the partition function. There are also various elementary approaches; the simplest example is a proof that the $X_0(2)$ Hauptmodul $$\left(\frac{\eta(q)}{\eta(q^2)}\right)^{24} = q^{-1} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4 + - \cdots$$ has alternating signs by rearranging the product $\eta(q)/\eta(q^2)$ as $(1-q)(1-q^3)(1-q^5)(1-q^7)\cdots$ which clearly becomes nonnegative on substituting $-q$ for $q$. – Noam D. Elkies Aug 25 '13 at 4:53