Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910835256852480 |
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| author | Cason, William Jim, Akash Medlock, Charlie Ross, Erick Vilardi, Trevor Xue, Hui |
| author_facet | Cason, William Jim, Akash Medlock, Charlie Ross, Erick Vilardi, Trevor Xue, Hui |
| contents | For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $χ$ be a Dirichlet character modulo $N$ with $χ(-1)=(-1)^k$. Then let $T_m^{\text{new}}(N,k,χ)$ denote the restriction of the $m$-th Hecke operator to the space $S_k^{\text{new}}(Γ_0(N), χ)$. We demonstrate that for fixed $m$ and trivial character $χ$, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k)$ 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 $S_k^{\text{new}}(Γ_0(N), χ)$ is trivial, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k,χ)$ vanishes for only finitely many triples $(N,k,χ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_11694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace Cason, William Jim, Akash Medlock, Charlie Ross, Erick Vilardi, Trevor Xue, Hui Number Theory 11F25, 11F72, 11F11 For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $χ$ be a Dirichlet character modulo $N$ with $χ(-1)=(-1)^k$. Then let $T_m^{\text{new}}(N,k,χ)$ denote the restriction of the $m$-th Hecke operator to the space $S_k^{\text{new}}(Γ_0(N), χ)$. We demonstrate that for fixed $m$ and trivial character $χ$, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k)$ 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 $S_k^{\text{new}}(Γ_0(N), χ)$ is trivial, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k,χ)$ vanishes for only finitely many triples $(N,k,χ)$. |
| title | Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace |
| topic | Number Theory 11F25, 11F72, 11F11 |
| url | https://arxiv.org/abs/2407.11694 |