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

We consider a variant of the strong multiplicity one theorem. Let ${\pi }_1$ and ${\pi }_2$ be two unitary cuspidal automorphic representations for $\mathrm{G}\mathrm{L}(2)$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\mid a_v({\pi }_1)\mid >\mid a_v({\pi }_2)\mid$, where $a_v({\pi }_i)$ is the trace of Langlands conjugacy class of ${\pi }_i$ 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 $\mid a_v({\pi }_1)\mid \mathrm{\ne }\mid a_v({\pi }_2)\mid$.