Selberg's orthogonality conjecture and symmetric power L-functions

Let pi be a cuspidal representation of GL(2)(A(Q)) defined by a non-CM holomorphic newform of weight w >= 2, and let K/Q be a totally real Galois extension with Galois group G. In this article, under Selberg's orthogonality conjecture, we show that for any irreducible character chi of G, the twisted symmetric power L-function L(s, Sym(m) pi x chi) is a primitive function in the Selberg class, and it is automorphic subject to further the solvability of K/Q. The key new idea is to apply the work of Barnet-Lamb, Geraghty, Harris, and Taylor on the potential automorphy of Sym(m) pi. (C) 2021 Elsevier Inc. All rights reserved.