Galois representations, Hecke operators, and the mod-p cohomology of GL(3,Z) with twisted coefficients

We compute the degree 3 homology of GL(3, Z) with coefficients in the modul e of homogeneous polynomials in three variables of degree g over F-p, for g less than or equal to 200 and p less than or equal to 541. The homology ha s a "boundary part" and a "quasicuspidal" part which we determine. By conjecture a Hecke eigenclass in the homology has an attached Galois rep resentation into GL(3, (F) over bar(p)). The conjecture is proved for the b oundary part and explored experimentally for the quasicuspidal part.