Twist-minimal trace formula for holomorphic cusp forms

TL;DR

This paper develops an explicit trace formula for twist-minimal holomorphic cusp forms, enabling efficient computation of basis elements for spaces with various levels, characters, and weights, extending previous non-holomorphic and holomorphic results.

Contribution

It provides a new explicit trace formula for twist-minimal holomorphic cusp forms, generalizing prior work and facilitating computational applications.

Findings

Derived an explicit trace formula for Hecke operators on twist-minimal cusp forms.

Showed the formula's efficiency in computing basis elements.

Extended previous non-holomorphic and holomorphic trace formulas.

Abstract

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Str\"ombergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Str\"omberg.