I tried asking this on math exchange, but no luck, so thought I'd try here.
Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the congruence subgroup defined by $\Gamma^0(N)=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}:ad-bc=1 \ ,b\equiv 0 \pmod{N}\right\}$
My question is: What is the size of $\Gamma^0(N)\backslash M_2(m,\mathbb{Z})$?
A few thoughts: the $N=1$ case is easy: you get $\sigma_1(m)$, the sum of divisors. This is easy to see both directly, and by using the identification with the Hecke ring. Of course, $\sigma_1(m)$ are the coefficients of the weight 2 Eisenstein series for $SL_2(\mathbb{Z})$, and for $N>1$ I expect some linear combination of the coefficients from Eisenstein series for smaller groups, depending on $N$ and $m$. However, all my attempts at calculations get bogged down in mess fairly quickly, and I can't find the result I'm looking for anywhere. Any help, whether you know the answer or a reference where it might be found would be very much appreciated!
