Coset representatives of principal congruence subgroups $\Gamma_l$ of $SL(n,\mathbb{Z})$ Consider the level $l$ principal congruence subgroup $\Gamma_l$ of the special linear group $SL(n,\mathbb{Z})$ defined as the kernel of the natural map $\phi : SL(n,\mathbb{Z}) \rightarrow SL(n,\mathbb{Z}/l\mathbb{Z})$.
Then the cosets of $\Gamma_l$ partition $SL(n,\mathbb{Z})$.
My questions are:


*

*What is the index of $\Gamma_l$?

*How to explicitly construct the cosets of $\Gamma_l$?
p.s. This question was asked at Mathoverflow, but put on hold for some reason. So I think I may ask it again here.
 A: Hints.


*

*Show that the canonical map $SL(n, \mathbf{Z}) \to SL(n, \mathbf{Z}/l\mathbf{Z})$ is surjective. 

*Using the first isomorphism theorem, the index of $\Gamma_l$ (traditionally called $\Gamma(l)$) is the order of the finite group $SL(n, \mathbf{Z}/l\mathbf{Z})$. 

*We compute the order of the group $GL(n, \mathbf{Z}/l\mathbf{Z})$ and realise the group $SL(n, \mathbf{Z}/l\mathbf{Z})$ as the kernel of the determinant map $\det: GL(n, \mathbf{Z}/l\mathbf{Z}) \to (\mathbf{Z}/l\mathbf{Z})^\ast$ (which is surjective). 

*The Chinese remainder theorem lifts to the isomorphism of groups 
$$GL(n, \mathbf{Z}/l\mathbf{Z}) \overset{\sim}{\to} \prod_{p^\alpha \parallel l} GL(n, \mathbf{Z}/p^\alpha\mathbf{Z}).$$

*For primes $p$, the order of the group $GL(n, \mathbf{Z}/p\mathbf{Z})$ is quite easy to compute. Study the canonical surjection 
$$GL(n, \mathbf{Z}/p^{\alpha+1}\mathbf{Z}) \to GL(n, \mathbf{Z}/p^\alpha\mathbf{Z})$$
to find the order of $GL(n, \mathbf{Z}/p^{\alpha+1}\mathbf{Z})$. (Here, show first that the kernel is a $p$-group using binomial theorem.) 


Here is an in-principle answer for coset representatives: the coset representatives for $\Gamma(l)$ in $SL(n, \mathbf{Z})$ is just given by lifts of elements in $SL(n, \mathbf{Z}/l\mathbf{Z})$. This answer is not as "satisfactory" or as "computable" in the case $n = 2$, where there is a rather explicit recipe.  

For n = 2, one can construct coset representatives for the principal congruence subgroup starting with ideas in this answer of mine. 

