The Artin reciprocity says that if $$ \chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C $$ is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a Hecke character.
So if $\rho$ is a $\ell$-adic Galois representation that is attached to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A_{\mathbb Q})$ (more precisely, a strictly compatible system of $\ell$-adic Galois representations which is $\mathbb Q$-rational compatible), then the $L$-function $$ L(s,\rho \otimes \chi) $$ is automorphic.
In particular, if $\rho=\operatorname{Sym}^n(\rho_f)$ is a symmetric power of $\rho_f$ a $\ell$-adic Galois representation attached to a non-CM newform $f$, then by the recent work of Newton and Thorne, we have nice analytic properties (analytic continuation and functional equations) of $$ L(s,\operatorname{Sym}^n(\rho_f) \otimes \chi). $$
I wonder whether I understand correctly, or if there are some gaps I didn't catch, because at some points I just used the facts that I've read as sentences in some textbooks or papers. In that case, I would appreciate it if you point out those points.
Thank you so much.
