Cohomology with Sym^g coefficients for congruence subgroups of SL₄(Z) and Galois representations

We extend the computations in [2–6] to find the cohomology in degree five of a congruence subgroup Γ of SL₄(Z) with coefficients in Sym^g(K⁴), twisted by a nebentype character η, along with the action of the Hecke algebra. This is the top cuspidal degree. In this paper we take K=F, a finite field of large characteristic, as a proxy for C. For each Hecke eigenclass found, we produce the unique Galois representation that appears to be attached to it. The computations require modifications to our previous algorithms to accommodate the fact that the coefficients are not one-dimensional. Types of attached Galois representations arise that were not found in our previous papers, and we must modify the Galois Finder accordingly.