Sign changes of Fourier coefficients of Maass eigenforms

Let f be a Maass eigenform that is a new form of level N on Γ0(N), with Laplace eigenvalue 1/4 + ν f 2 . Then, all its Fourier coefficients {λ f (n)} n=1 ∞ are real, and we may normalize so that λ f (1) = 1. It is proved, in this paper, that the first sign change in the sequence {λ f (n)} n=1 ∞ occurs at some n satisfying n ≪ {(3+|ν f |2)N}1/2−δ and (n, N) = 1. This generalizes the previous results for holomorphic Hecke eigenforms.