Computing Hecke eigenvalues below the cohomological dimension

Abstract

Let $\funnyGamma$ be a torsion-free finite-index subgroup of $\SL_{n} (\Z )$ or $\GL_{n} (\Z )$, and let $\nu $ be the cohomological dimension of $\funnyGamma $. We present an algorithm to compute the eigenvalues of the Hecke operators on $H^{\nu -1} (\funnyGamma ;\Z )$, for n= 2, 3, and 4. In addition, we describe a modification of the modular symbol algorithm of Ash and Rudolph for computing Hecke eigenvalues on $H^{\nu } (\funnyGamma ;\Z )$.