In this paper we extend a conjecture of A. Ash and W. Sinnott
relating niveau 1 Galois representations to the $\mod p$ cohomology of
congruence subgroups of ${\rm SL}\sb n(\mathbb {Z})$ to include Galois
representations of higher niveau. We then present computational
evidence for our conjecture in the case $n=3$ in the form of
three-dimensional Galois representations which appear to correspond to
cohomology eigenclasses as predicted by the conjecture. Our examples
include Galois representations with nontrivial weight and level, as
well as irreducible three-dimensional representations that are in no
obvious way related to lower-dimensional representations. In addition,
we prove that certain symmetric square representations are actually
attached to cohomology eigenclasses predicted by the conjecture.