For positive integers n and L, denote by SL_n(Z,L) the level L congruence subgroup of SL_n(Z), ie the kernel of the homomorphism SL_n(Z)-->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.