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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Cohomology of arithmetic groups
- Galois representations
- Modular symbols
- Steinberg module
- Voronoi complex