# Effective joint distribution of eigenvalues of Hecke operators

## TL;DR

This paper extends the effective joint distribution results of eigenvalues of Hecke operators to higher level cusp forms using the Eichler-Selberg trace formula, building on prior work with Kuznetsov trace formula.

## Contribution

It introduces a new method employing the Eichler-Selberg trace formula to analyze eigenvalue distributions for higher level cusp forms, expanding previous results.

## Key findings

- Effective joint distribution of eigenvalues for higher level cusp forms
- Use of Eichler-Selberg trace formula in this context
- Extension of Lau and Wang's results to higher levels

## Abstract

In 1997, Serre proved that the eigenvalues of normalised $p$-th Hecke operator $T^{'}_p$ acting on the space of cusp forms of weight $k$ and level $N$ are equidistributed in $[-2,2]$ with respect to a measure that converge to the Sato-Tate measure, whenever $N+k \to \infty$. In 2009, Murty and Sinha proved the effective version of Serre's theorem. In 2011, using Kuznetsov trace formula, Lau and Wang derived the effective joint distribution of eigenvalues of normalized Hecke operators acting on the space of primitive cusp forms of weight $k$ and level $1$. In this paper, we extend the result of Lau and Wang to space of cusp forms of higher level. Here we use Eichler-Selberg trace formula instead of Kuznetsov trace formula to deduce our result.