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

Abstract

We consider a variant of the strong multiplicity one theorem. Let $π_{1}$ and $π_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1}) \right\rvert > \left\lvert a_{v}(π_{2}) \right\rvert$, where $a_{v}(π_{i})$ is the trace of 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 $\left\lvert a_{v}(π_{1})\right\rvert \neq \left\lvert a_{v}(π_{2}) \right\rvert$.