On the first sign change of Hecke eigenvalues of newforms

Let f be a non-zero cusp form of even integral weight k ≥ 2 and level N with real Fourier coefficients a(n) (n ∈ N). Using a classical theorem of Landau together with the facts that the Hecke L-function of f is entire and the Rankin-Selberg zeta function of f has a pole at its abscissa of convergence with residue essentially equal to the square of the Petersson norm of f , one easily shows that there are infinitely many n ∈ N such that a(n) > 0 as well as infinitely many n with a(n) 1, Siegel’s argument does not apply and the result above is not known. Now suppose that N is squarefree and f is a newform which is a normalized Hecke eigenform (so a(1) = 1 and a(n) is the n-th Hecke eigenvalue of f ). We shall prove