Selberg's orthogonality conjecture and symmetric power L-functions

Let π 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 χ of G, the twisted symmetric power L-function L ( s , Sym m π × χ ) 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 π .