Coset representatives of principal congruence subgroups $\Gamma_l$ of $SL(n,\mathbb{Z})$

Question

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:

1. What is the index of $\Gamma_l$?

2. 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.

Answer 1

Hints.

1. Show that the canonical map $SL(n, \mathbf{Z}) \to SL(n, \mathbf{Z}/l\mathbf{Z})$ is surjective.
2. Using the first isomorphism theorem, the index of $\Gamma_l$ (traditionally called $\Gamma(l)$) is the order of the finite group $SL(n, \mathbf{Z}/l\mathbf{Z})$.
3. 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).
4. The Chinese remainder theorem lifts to the isomorphism of groups $$GL(n, \mathbf{Z}/l\mathbf{Z}) \overset{\sim}{\to} \prod_{p^\alpha \parallel l} GL(n, \mathbf{Z}/p^\alpha\mathbf{Z}).$$
5. 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.