Cohomology with twisted one-dimensional coefficients for congruence subgroups of SL₄(Z) and Galois representations

We extend the computations in [2–4] to find the cohomology in degree five of a congruence subgroup of SL₄(Z) with coefficients in a field twisted by a nebentype character, along with the action of the Hecke algebra on the cohomology. This is the top cuspidal degree. For each Hecke eigenclass we find, we produce the unique Galois representation that appears to be attached to it. The computations require serious modifications to our previous algorithms. Nontrivial coefficients add a layer of complication to our data structures. New possibilities must be taken into account in the Galois Finder, the code that finds the Galois representations. We have improved the Galois Finder to report when the attached Galois representation is uniquely determined by our data.