When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for small values of $n$ or $p$).

For $\ell > 1$ we have the map $SL_n(\mathbb{Z}/p^\ell \mathbb{Z}) \to SL_n(\mathbb{Z}/p\mathbb{Z})$ so we still get a representation of this size. I was wondering if this is still the smallest possible dimension (again maybe with a finite number of exceptions for small $p$ and $n$), and if not are there known lower bounds for how small such a representation can be?

I found some calculations of characters for small values of $n$ that suggest this could be true, but otherwise couldn't find much. Is there anywhere in the literature that addresses this sort of question? I'm also interested in similar results for $Sp_{2n}(\mathbb{Z}/p^\ell \mathbb{Z})$.