Reducible Galois representations and the homology of GL(3,Z)

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $\rho$ is squarefree, we prove that $\rho$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $\Gamma$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, Pollack, and Sinnott.