COMPUTING HECKE EIGENVALUES BELOW THE COHOMOLOGICAL DIMENSION

PAUL E. GUNNELLS

Abstract. Let Γ be a torsion-free finite-index subgroup of SL_{n}(ℤ) or GL_{n}(ℤ), and let ν be the cohomological dimension of Γ. We present an algorithm to compute the eigenvalues of the Hecke operators on H^{ν−1}(Γ; 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^{ν}(Γ; ℤ).