This question is a follow-up to Monstrous Moonshine for Thompson group $Th$? and is based on various comments to that question, in particular S. Carnahan's mention of the connection to known Moonshine for the Thompson group via the fact that $$j(\tau/3)= q^{-1/9}+248 q^{2/9} +4124 q^{5/9} + \cdots $$ is the character of the 3C twisted module of $V^\natural$. This connection arises from the fact that $j^{1/3}$ is the Borcherds lift of a weakly holomorphic weight $1/2$ modular form for $\Gamma_0(4)$
$$b_{1A}(\tau) \equiv b(\tau)- 4 \theta(\tau) = \sum_{n=-3}^\infty c_{1A}(n) q^n$$
where $b(\tau)$ was defined in the question cited earlier and
$$\theta(\tau) = \sum_{n=-\infty}^{+\infty} q^{n^2} $$.
This means that there is a Borcherds product
$$ j(\tau)^{1/3}= \frac{E_4}{\eta^8} =\prod_{n=1}^\infty (1-q^n)^{c_{1A}(n^2)} $$
The fact that the coefficients $c_{1A}(n^2)$ are related to sums of dimensions of irreducible representations of the Thompson group
is thus explained by the previous known example of Moonshine for the Thompson group, but this does not (at least naively) explain
why the coefficients $c_{1A}(m)$ for $m$ that are not squares are also given by sums of dimensions of Thompson irreps with small
multiplicities. Thus a perhaps more refined version of the earlier question is whether there is some additional Moonshine
phenomenon going on for $b_{1A}(\tau)$ that goes beyond that connected with $j^{1/3}$. One way to test this further is to
compute McKay-Thompson series by replacing the dimensions of irreps by characters. The low order terms for some simple examples lead
to (using ATLAS notation)
$$ b_{1A}(\tau)= q^{-3}-248 q + (27000-248)q^4 -85995 q^5+1707264 q^8 - (4096000+248)q^9+44330496 q^{12} + \cdots $$ $$ b_{2A}(\tau)= q^{-3}+8q+128 q^4 +21 q^5-768 q^8+8 q^9 + 3584 q^{12} + \cdots $$ $$ b_{3A}(\tau) = q^{-3} -14 q -41 q^4-78 q^9+168 q^{12} + \cdots $$ $$ b_{3B}(\tau) = q^{-3}-5 q +22 q^4+27 q^5-54 q^8+3 q^9+6 q^{12} + \cdots $$
The dimensions of Thompson irreps were given in the earlier question and I repeat them here:
$$1, \color{blue}{248}, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$
My questions are
How do I determine whether the expansions given above agree with the $q$ expansions of weakly holomorphic weight $1/2$ modular forms on some congruence subgroup like $\Gamma_0(8)$ or $\Gamma_0(12)$? I'm not sure what the rules of the game are here, so am not sure exactly which congruence subgroup one should look at.
Is there a way to determine the above McKay-Thompson like series in terms of those for $j^{1/3}$ when twisted by these elements of the Thompson group?
