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This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup $$ \pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:\;a,b,c,d\in\...

This is a follow up on another MO question. Question. For $n\geq2$, the following is always an integer. Is it not? $$\frac{(2^n-2)(2^{n-1}-2)\cdots(2^3-2)(2^2-2)}{n!}.$$