Papers › Signs of the Second Coefficients of Hecke Polynomials
Signs of the Second Coefficients of Hecke Polynomials
Erick Ross, Hui Xue
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Let Tₘ(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 Tₘ(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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