  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 $n\geq 2$. For $n=2$, the construction is a complete set of
irreducible complex representations, while for $n>2$, 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.
