How $\Gamma(N)$ is contained in $\Gamma_1(N)$?

Question

The following is from the book Modular Forms by W Stein:

By the very same book "a congruence subgroup is a subgroup of $SL_2(\mathbb{Z})$ that contains $\Gamma(N)$ for some $N$". So $\Gamma(N)$ must be of the form: So how $\Gamma(N)$ is contained in $\Gamma_1(N)$ when $a \equiv d \equiv 1$? (it must be $a \equiv d \equiv 0$)

Answer 1

The identity element of $SL_2(\Bbb Z/n\Bbb Z)$ is the identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Note that the zero matrix is not even an element of $SL_2(\Bbb Z/n\Bbb Z)$, because it has determinant zero.