  We consider a variant of the strong multiplicity one theorem. Let $\pi_{1}$
and $\pi_{2}$ 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 $\left\lvert
a_{v}(\pi_{1}) \right\rvert > \left\lvert a_{v}(\pi_{2}) \right\rvert$, 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 $\left\lvert
a_{v}(\pi_{1})\right\rvert \neq \left\lvert a_{v}(\pi_{2}) \right\rvert$.
