Bounds for modular L-functions in the level aspect

Abstract

Let f be a primitive (holomorphic or Maaß) cusp form of level q and non-trivial nebentypus. Then for Re s = 1 2 the associated L-function satisfies L ( f , s ) ≪ q 1 4 − 1 1889 , where the implied constant depends polynomially on s and the Archimedean parameters of f (weight or Laplacian eigenvalue).