Let π be a cuspidal representation of GL2(AQ) 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,Symmπ×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 Symmπ.