Non-Vanishing of Cubic Twists of $G{L}_n(\mathbb{Q})$ $L$-functions

Let $\pi$ be an irreducible, cuspidal automorphic representation of $G{L}_n({\mathbb{A}}_{\mathbb{Q}})$ ($n\ge 3$), which is tempered only for $n=3$. Let $s$ be a complex number such that $\mathrm{\Re }(s)\mathrm{\notin }[1\mathrm{/}n,1-1\mathrm{/}n]$ if $n\mathrm{\ne }4$; $\mathrm{\Re }(s)\mathrm{\notin }[1\mathrm{/}5,4\mathrm{/}5]$ if $n=4$, then we show that there are infinitely many primitive cubic Dirichlet characters $\chi$ such that $L(s,\pi \times \chi )\mathrm{\ne }0$. Similar results were previously known only for primitive Dirichlet characters without any restriction on the order and quadratic Dirichlet characters.