Abstract
Let π be a self-dual irreducible cuspidal automorphic representation of GL2(\({\mathbb{A}_\mathbb{Q}}\)) with trivial central character. Its Hecke eigenvalue ⁁π (n) is a real multiplicative function in n. We show that λπ (n) < 0 for some \(n \ll Q_\pi ^{2/5}\), where Qπ denotes (a special value of) the analytic conductor. The value \({2 \over 5}\) is the first explicit exponent for Hecke-Maass newforms.
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Acknowledgements
The first author was supported by General Research Fund of the Research Grants Council of Hong Kong (Grant Nos. 17313616 and 17305617). The third author was supported by National Natural Science Foundation of China (Grant No. 11871193) and the Program for Young Scholar of Henan Province (Grant No. 2019GGJS026). The fourth author was supported by National Natural Science Foundation of China (Grant No. 11871344). The authors are grateful to the referees for careful reading and thank Professor Matomäki for her C++ code.
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Lau, YK., Ng, M.H., Tang, H. et al. On the first negative Hecke eigenvalue of an automorphic representation of GL2(\({\mathbb{A}_\mathbb{Q}}\)). Sci. China Math. 64, 2381–2394 (2021). https://doi.org/10.1007/s11425-020-1789-5
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DOI: https://doi.org/10.1007/s11425-020-1789-5