You will have to decompose

$$ Endo_{SL(3, \mathbb{Z})} ( Ind_{\Gamma_0(N)}^{SL(3, \mathbb{Z})} 1 ) .$$

This will give you the analog of the Atkin-Lehner theory. However, I have some doubts that this exists in the literature. This question of mine gives you the resaon why it has not been done for $d,g,h = 0 \bmod N$:

Parabolic induction GL(n,Zp)

I am happy, if somebody proves me wrong though.

Edit: I forgot to mention, that the case for $N$ square free is in general possible, since you can rely on the representation theory of reductive groups over residue fields.