How many cusps does the congruence subgroup $\Gamma(N)$ have? 


Thanks;



The number of Cusps 
This can be found in most books on modular forms, there's a lot of detailed information in Chapter 3 of Diamond-Shurman. It's
$$ \frac 1 2 N^2 \prod\_{p \mid N}\left(1 - \frac 1 {p^2}\right)$$
if $N \geq 3$ and $3$ if $N=2$. The factor $N^2 \prod\_{p \mid N}\left(1 - \frac 1 {p^2}\right)$ arises as the number of elements of order $N$ in $(\mathbf Z/N)^2$.


