Analytic evaluation of Hecke eigenvalues for classical modular forms

TL;DR

This paper introduces an analytic method for approximating Hecke eigenvalues of classical modular forms, offering high precision, error control, and improved performance over existing exact computation techniques.

Contribution

It presents a novel analytic approach to compute Hecke eigenvalues with arbitrary precision and better efficiency than prior exact methods.

Findings

Method achieves high-precision eigenvalue approximations.

Error control is strictly maintained during computations.

Outperforms current exact computation methods.

Abstract

We propose a method for computing approximations to the Hecke eigenvalues of a classical modular eigenform $f$, based on the analytic evaluation of $f$ at points in the upper half plane. Our approach works with arbitrary precision, allows for a strict control of the error in the approximation, and outperforms current exact computation methods.