On the first sign change of Fourier coefficients of cusp forms

Abstract

Let f be a nonzero cusp form of even integral weight k⩾2 on the Hecke congruence subgroup Γ0(N) with N squarefree. Suppose that the normalized Fourier coefficients λf(n) of f are real. We prove that the first sign change of λf(n) occurs in the range n≪(kN)2+ε. This improves upon the earlier result of Choie and Kohnen [2].