Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

Abstract

We prove a result on the distribution of Hecke eigenvalues, µ F (p r ) (for r = 1, 2 or 3) of a non Saito-Kurokawa lift F of degree 2. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an F, which is the best possible in terms of orders of magnitude. k subspace of S 2 k orthogonal to S 2, * k . Our main result Theorem 1 implies (via Theorem 2) in particular that the Ramanujan-Petersson conjecture for eigenforms in S 2,⊥ k is optimal, in a sense described below.