Non-repetition of second coefficients of Hecke polynomials

Archer Clayton; Helen Dai; Tianyu Ni; Erick Ross; Hui Xue; Jake Zummo

Non-repetition of second coefficients of Hecke polynomials

Number Theory 2025-06-04 v2

Authors: Archer Clayton , Helen Dai , Tianyu Ni , Erick Ross , Hui Xue , Jake Zummo

Abstract

Let $T_m(N,2k)$ denote the $m$ -th Hecke operator on the space $S_{2k}(\Gamma_0(N))$ of cuspidal modular forms of weight $2k$ and level $N$ . In this paper, we study the non-repetition of the second coefficient of the characteristic polynomial of $T_m(N,2k)$ . We obtain results in the horizontal aspect (where $m$ varies), the vertical aspect (where $k$ varies), and the level aspect (where $N$ varies). Finally, we use these non-repetition results to extend a result of Vilardi and Xue on distinguishing Hecke eigenforms.

Keywords

Hecke operator modular forms characteristic polynomial non-repetition cuspidal forms level aspect

Cite

@article{arxiv.2411.18419,
  title  = {Non-repetition of second coefficients of Hecke polynomials},
  author = {Archer Clayton and Helen Dai and Tianyu Ni and Erick Ross and Hui Xue and Jake Zummo},
  journal= {arXiv preprint arXiv:2411.18419},
  year   = {2025}
}

Comments

minor revisions based on referee feedback

Non-repetition of second coefficients of Hecke polynomials

Abstract

Keywords

Cite

Comments

Related papers