Are the principal congruence subgroups of SL(2,Z) normally generated by a single element? Let $N\ge 3$, then would I be correct in saying that the principal congruence subgroup $\Gamma(N)$ (defined to be the 2x2 matrices in $SL(2,\mathbb{Z}$) congruent to the identity mod $N$) is the smallest normal subgroup of $SL(2,\mathbb{Z}$) containing the matrix $[[1,N],[0,1]]$?
Edit: I'm beginning to feel like this is false. If it's false, is there a way to describe the generators which are not parabolic (ie, conjugate to $[[1,N],[0,1]]$)?

If $X(N)$ has genus $g$, then $\pi_1(X(N))^{ab} = \mathbb Z^{2g}$, and $\pi_1(Y(N))^{ab} = \mathbb Z^{2g + s -1}$, where $s$ is the number of cusps. Since $\pi_1(Y(N)) = \Gamma(N)$ for $N\geq 3$, it must be that $\Gamma(N)^{ab} = \mathbb Z^{2g+s-1}$. In particular, $\Gamma(N)$ cannot be generated by less than $2g+s-1$ elements. 
