Experimental indications of three-dimensional Galois representations from the cohomology of SL(3, Z)

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 Cof congruence subgroups of GL(n;Z). On the other hand, the first author has conjectured that for any Hecke eigenclass in the mod pcohomology 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 pcohomology of certain congruence subgroups of SL(3;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.