For positive integers $n$ and $L$, denote by $SL_n(Z,L)$ the level $L$ congruence subgroup of $SL_n(Z)$, i.e. the kernel of the homomorphism $SL_n(Z)\rightarrow SL_n(Z/LZ)$.
For $n$ at least $3$, it is known that $SL_n(Z,L)$ is normally generated (as a subgroup of $SL_n(Z)$) by Lth powers of elementary matrices. Indeed, this is essentially equivalent to the congruence subgroup problem for $SL_n(Z)$.
However, this fails for $SL_2(Z,L)$ since $SL_2(Z)$ does not have the congruence subgroup property.
Question : Is there a nice generating set for $SL_2(Z,L)\ ?$ I'm sure this is in the literature somewhere, but I have not been able to find it.