$q$-expansion of Modular forms

Question

I am trying to compute the $q$-expansion of $g\theta_2$ and $g\theta_4$, the $q$-expansion of modular forms of weight $3/2$ and level $128$ and trivial character and character $\chi_8$ respectively.

We have $$g=q\prod_{n=1}^{\infty}\big(1-q^{8n}\big)\big(1-q^{16n}\big),\quad \theta_t=\sum_{-\infty}^{\infty}q^{tn^2}.$$

I would like to know the following:

• Are there formulas to find the coefficient of $q^d$ for any $d$?
• Is there a computer program to find it? (I should think so.)

I appreciate any help, or references to resources where I can find the answer. Thanks!

EDIT: I know the first few terms of $g\theta_2$: $$g\theta_2=q+2q^3+q^9-2q^{11}-4q^{17}-2q^{19}-3q^{25}+4q^{33}-4q^{35}+\dots$$

Answer 1

As stated in the comments, it often helps to work out the initial terms and then consult the OEIS. In your case,

1. A034950 for $g\theta_2$, odd-indexed coefficients,
2. A080966 for $g\theta_4$, coefficients with index $\equiv1\pmod{4}$

are relevant. Among other things, you can find

1. recipes for computation e.g. with Pari/GP, usually by expanding products and quotients of the Dedekind eta function,
2. via the crossrefs: additive decompositions in terms of representation counts that are related to Tunnell's criterion for congruent numbers.

I suppose you know the latter because the notation used in your question matches the notation in Tunnell's 1983 paper.