# On the trace formula for Hecke operators on congruence subgroups, II

## TL;DR

This paper derives explicit, simple trace formulas for Hecke operators on congruence subgroups, enabling precise calculations of traces and their limits as the level grows, with applications to specific subgroups.

## Contribution

It specializes a general trace formula to $ ext{Gamma}_0(N)$ and $ ext{Gamma}_1(N)$, providing explicit formulas in terms of class numbers that are among the simplest in the literature.

## Key findings

- Explicit trace formulas for $ ext{Gamma}_0(N)$ and $ ext{Gamma}_1(N)$
- Determination of trace forms for $ ext{Gamma}_0(4)$ with Nebentypus
- Calculation of the limit of traces as level $N$ tends to infinity

## Abstract

In a previous paper, we obtained a general trace formula for double coset operators acting on modular forms for congruence subgroups, expressed as a sum over conjugacy classes. Here we specialize it to the congruence subgroups $\Gamma_0(N)$ and $\Gamma_1(N)$, obtaining explicit formulas in terms of class numbers for the trace of a composition of Hecke and Atkin-Lehner operators. The formulas are among the simplest in the literature, and hold without any restriction on the index of the operators. We give two applications of the trace formula for $\Gamma_1(N)$: we determine explicit trace forms for $\Gamma_0(4)$ with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level $N$ tends to infinity.