Generation of cyclotomic Hecke fields by L-values of cusp forms on GL(2) with certain Zₚ twist

Let F be a number field, f an algebraic automorphic newform on GL(2) over F, p an odd prime does not divide the class number of F and the level of f. We prove that f is determined by its L-values twisted by Galois characters φ of certain Zₚ-extension of F. Furthermore, if F is totally real or CM, then under some mild assumption on f, the compositum of the Hecke field of f and the cyclotomic field Q(φ) is generated by the algebraic L-values of f twisted by Galois characters φ of certain Zₚ-extension of F.