Murmurations using Petersson trace formula

TL;DR

This paper demonstrates the murmuration phenomenon in modular forms using a novel approach with the Petersson trace formula, establishing correlations between functional equation signs and Fourier coefficients, and extending the observable range.

Contribution

First to apply a relative trace formula to study murmuration phenomena in modular forms, improving the visibility range and robustness of the results.

Findings

Established murmuration phenomenon in the weight aspect of modular forms.

Extended the range of visibility for murmurations.

Introduced a new approach using the Petersson trace formula.

Abstract

We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations compared to previous results. This is the first approach to the murmuration phenomenon using a relative trace formula, showing its robustness.