The Computation of Fourier transforms on $SL_2(mathbb{Z}/p^nmathbb{Z}) and related numerical experiments.

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $SL_2({mathbb Z} /p^n {mathbb Z})$ for odd primes $p$ and $ngeq 2$. For $n=2$, the construction is a complete set of irreducible complex representations, while for $nu003e2$, all but a handful are obtained. We also produce an algorithm for the computation of a Fourier transform for a function on $SL_2({mathbb Z} /p^2 {mathbb Z})$. With this in hand we explore the spectrum of a collection of Cayley graphs on these groups, extending analogous computations for Cayley graphs on $SL_2({mathbb Z}/p {mathbb Z})$ and suggesting conjectures for the expansion properties of such graphs.