TITLE: Generators for congruence subgroups of SL_2
QUESTION [22 upvotes]: 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.

REPLY [5 votes]: I was at a workshop on noncongruence modular forms recently (a noncongruence subgroup of $SL_2(\mathbb{Z})$ is a subgroup of finite index not containing any $\Gamma(N)$ ) when this question came up. I believe that the consensus was that although one can, in principle, compute a generating set for $\Gamma(N)$, the algorithm is not terribly effective for $N$ large (where by large I mean greater than $13$ or so). The algorithm involves computing the Farey symbol associated to $\Gamma(N)$ and getting coset representatives. The difficulty is that the index of $\Gamma(N)$ increases very rapidly, making the calculation of the Farey symbol very lengthy. 
Ling Long and Chris Kurth have written a paper about the algorithm (it references the paper of Kulkarni that algori mentioned), which is available here.