In this paper, we examine the conjecture of Avner Ash, Darrin Doud, David Pollack, and Warren Sinnott relating Galois representations to the mod p cohomology of congruence subgroups of the general linear group of n dimensions over the integers. We present computational evidence for this conjecture (the ADPS Conjecture) for the case n = 3 by finding Galois representations which appear to correspond to cohomology eigenclasses predicted by the ADPS Conjecture for the prime p, level N, and quadratic nebentype. The examples include representations which appear to be attached to cohomology eigenclasses which arise from D8, S3, A5, and S5 extensions. Other examples include representations which are reducible as sums of characters, representations which are symmetric squares of two-dimensional representations, and representations which arise from modular forms, as predicted by Jean-Pierre Serre for n = 2.

We provide algorithms and documention to compute the cohomology of congruence subgroups of the special linear group over the integers when n=3 using the well-rounded retract and the Voronoi decomposition. We define the Sharbly complex and how one acts on a k-sharbly by the Hecke operators. Since the norm of a sharbly is not preserved by the Hecke operators we also examine the reduction techniques described by Gunnells and present our implementation of said techniques for n=3.