Let $\Gamma$ be a congruence subgroup of $\operatorname {SL}\_2 (\mathbb Z)$. Let $N$ be the smallest positive integer such that $\begin{pmatrix} 1 & N \\0 &1 \end{pmatrix}\in \Gamma$. 


Is necessarily $\Gamma(N)\subset \Gamma$? 



Congruence subgroups and translations
No. $\Gamma$ can be $\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname {SL}\_2 (\mathbb Z):\text{$M\mid c$ and $N\mid b$}\right\}$, which contains $\Gamma(N)$ if and only if $M\mid N$.


