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

This Note answers, and generalizes, a question of Kaisa Matomäki. We show that give two cuspidal automorphic representations $π_1$ and $π_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $χ$ such that $π_1\otimesχ\simeqπ_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $π_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$

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