Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)

We consider a variant of the strong multiplicity one theorem. Let π 1 and π 2 be two unitary cuspidal automorphic representations for GL ( 2 ) that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which | a v ( π 1 ) | > | a v ( π 2 ) | , where a v ( π i ) is the trace of the Langlands conjugacy class of π i at v . One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which | a v ( π 1 ) | ≠ | a v ( π 2 ) | .