Conjecturally, any "algebraic'' automorphic representation on
$\GL(n)$ should have an $n$-dimensional Galois representation
attached. Many examples of algebraic automorphic representations come
from the cohomology over $\bold C$ of congruence subgroups of
$\GL(n,\bold Z)$. On the other hand, the first author has
conjectured that for any Hecke eigenclass in the mod $p$ cohomology of
a congruence subgroup of $\GL(n,\Z)$ there should be an attached
$n$-dimensional Galois representation.
¶ By computer, we found Hecke eigenclasses in the mod $p$ cohomology of
certain congruence subgroups of $\SL(3,\bold Z)$. In a range of
examples, we then found a Galois representation (uniquely determined
up to isomorphism by our data) that seemed to be attached to the
Hecke eigenclass.