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

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 $$a \equiv d \equiv 0$$)

• What you wrote as the zero matrix should be the identity matrix. – RghtHndSd Nov 5 '18 at 2:20
• @RghtHndSd, isn't Γ(N) the kernel of $SL_2(\mathbb{Z})$ to $SL_2(\mathbb{Z/NZ})$ map so it id in mod N? – user231343 Nov 5 '18 at 2:24

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.