Computations directly on the cuspidal cohomology of congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$

Abstract

Ash, Grayson, and Green [J. Number Theory 19 (1984), pp. 412-436] compute the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of $\mathsf{SL}(3, \mathbb{Z})$. This subspace contains the cuspidal cohomology, which is of primary interest. We extend their work, introducing a method that allows for computing the action of Hecke operators directly on the cuspidal cohomology. Using this method, we obtain data for prime level less than 3500, finding seven additional levels at which nonzero cuspidal classes appear and calculating local factors for five of these levels.