Signs of the Second Coefficients of Hecke Polynomials

2024/07/15 by Erick Ross, Hui Xue, Ross, Erick +1
Mathematics · #Mathematical functions and polynomials #Analytic Number Theory Research #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2407.10951

Abstract

Let T_m(N, k, χ) be the m-th Hecke operator of level N, weight k ≥ 2, and nebentypus χ, where N is coprime to m. We first show that for any given m ≥ 1, the second coefficient of the characteristic polynomial of T_m(N, k, χ) is nonvanishing for all but finitely many triples (N,k,χ). Furthermore, for χ trivial and any fixed m, we determine the sign of the second coefficient for all but finitely many pairs (N,k). Finally, for χ trivial and m=3,4, we compute the sign of the second coefficient for all pairs (N,k).

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