Mod 2 homology for GL(4)GL(4) and Galois representations

We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ   of SL(4,Z)SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ   with F2F2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in [AGM11]. Our method of computation was justified in [AGM12].