  Given a self-dual cuspidal automorphic representation for GL(2) over a number
field, we establish the existence of an infinite number of Hecke eigenvalues
that are greater than an explicit positive constant, and an infinite number of
Hecke eigenvalues that are less than an explicit negative constant. This
provides an answer to a question of Serre. We also consider analogous problems
for cuspidal automorphic representations that are not self-dual.

Number Theory