A constraint for twist equivalence of cusp forms on GL$(n)$

Abstract

This note answers, and generalizes, a question of Kaisa Matomäki. We show that given two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL(n)$ over a number field $F$ of respective conductors $N_1$, $N_2$, every character $\chi$ such that $\pi_1\otimes\chi\simeq\pi_2$ of conductor $Q$, satisfies the bound: $Q^n\mid N_1N_2$. If at every finite place $v$, $\pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1,N_2].$