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

Abstract

Let F be a number field, f an algebraic automorphic newform on $\mathrm{GL}\left(2\right)$ 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 $Z_p$-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 $Q\left(\varphi \right)$ is generated by the algebraic L-values of f twisted by Galois characters ϕ of certain $Z_p$-extension of F.