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

Abstract

This note answers, and generalizes, a question of Kaisa Matomaki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ over a number field $F$ of respective conductor $N_1,$ $N_2,$ 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 $[N_1, N_2].$