Trace formulas of the Hecke operator on the spaces of newforms

TL;DR

This paper derives explicit trace formulas for Hecke operators acting on spaces of newforms of modular forms, providing detailed decompositions and formulas for various subspaces.

Contribution

It introduces new trace formulas for Hecke operators on spaces of newforms and their subspaces, enhancing understanding of their spectral properties.

Findings

Explicit trace formulas for T(l) on S_k^0(N)

Decomposition of S_k^0(N) into subspaces with individual trace formulas

Enhanced understanding of Hecke operator actions on newform spaces

Abstract

In this paper, for a square-free integer l>1, a even positive integer k and a positive integer N, we give a trace formula of the Hecke operator T(l) on the space S_k^0(N) of all newforms of weight k and level \Gamma_0(N). Moreover, we give a decomposition S_k^0(N)=\oplus_{i|N}S_k^0(N;i) and also give a trace formula of T(l) on S_k^0(N;i).