Papers › Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace
Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace
William Cason, Akash Jim, Charlie Medlock, Erick Ross, Trevor Vilardi, Hui Xue
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For m ≥1, let N ≥1 be coprime to m, k ≥2, and χ be a Dirichlet character modulo N with χ(-1)=(-1)ᵏ. Then let Tₘⁿᵉʷ(N,k,χ) denote the restriction of the m-th Hecke operator to the space Sₖⁿᵉʷ(Γ₀(N), χ). We demonstrate that for fixed m and trivial character χ, the second coefficient of the characteristic polynomial of Tₘⁿᵉʷ(N,k) vanishes for only finitely many pairs (N,k), and we further determine the sign. To demonstrate our method, for m=2,4, we also compute all pairs (N,k) for which the second coefficient vanishes. In the general character case, we also show that excluding an infinite family where Sₖⁿᵉʷ(Γ₀(N), χ) is trivial, the second coefficient of the characteristic polynomial of Tₘⁿᵉʷ(N,k,χ) vanishes for only finitely many triples (N,k,χ).
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