Abstract
Let f and g be two Hecke–Maass cusp forms of weight zero for \(SL_2({\mathbb {Z}})\) with Laplacian eigenvalues \(\frac{1}{4}+u^2\) and \(\frac{1}{4}+v^2\), respectively. Then both have real Fourier coefficients say, \(\lambda _f(n)\) and \(\lambda _g(n)\), and we may normalize f and g so that \(\lambda _f(1)=1=\lambda _g(1)\). In this article, we first prove that the sequence \(\{\lambda _f(n)\lambda _g(n)\}_{n \in {\mathbb {N}}}\) 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.
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The authors would like to thank the referee for numerous useful comments and substantial corrections.
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Kumari, M., Sengupta, J. The first simultaneous sign change for Fourier coefficients of Hecke–Maass forms. Ramanujan J 55, 205–218 (2021). https://doi.org/10.1007/s11139-020-00268-9
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DOI: https://doi.org/10.1007/s11139-020-00268-9