  We show that, in a restricted range, the divisor function of integers in
residue classes modulo a prime follows a Gaussian distribution, and a similar
result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore,
we obtain the joint distribution of these arithmetic functions in two related
residue classes. These results follow from asymptotic evaluations of the
relevant moments, and depend crucially on results on the independence of
monodromy groups related to products of Kloosterman sums.
