Let f and g be two Hecke-Maass cusp forms of weight zero for SL 2 (Z) with Laplacian eigenvalues 1 4 +u 2 and 1 4 +v 2 , respectively. Then both have real Fourier coefficients say, λ f (n) and λ g (n), and we may normalize f and g so that λ f (1) = 1 = λ g (1). In this article, we first prove that the sequence {λ f (n)λ g (n)} n∈N has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of f and g.
This article deals with various kinds of quantitative results about the comparison between the normalized Hecke eigenvalues of two distinct Siegel cuspidal Hecke eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts. We also prove some simultaneous sign change results for their eigenvalues.
We prove some non-vanishing results of Hilbert Poincaré series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincaré series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level. To prove later results, we generalize a method of E. Kowalski et. al.
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