Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Question

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a simple google search. Certainly, there should be some explicit expression, no?

Answer 1

I found the following in some notes of mine.

The index of $\Gamma_0(N)$ in $\operatorname{SL}_2(\mathbb Z)$ is $\psi(N) = N \prod_{p \mid N} (1+1/p)$ and the index of $\Gamma_1(N)$ in $\Gamma_0(N)$ is $\varphi(N) = N \prod_{p \mid N} (1-1/p)$, so $$ [\operatorname{SL}_2(\mathbb Z):\Gamma_1(N)] = N^2 \prod_{p \mid N} (1 - 1/p^2) . $$

A discussion of this can be found on pages 22-24 of Kilford's book.