Murmurations of modular forms in the weight aspect

Bober, Jonathan; Booker, Andrew R.; Lee, Min; Lowry-Duda, David

doi:10.2140/ant.2026.20.299

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Murmurations of modular forms in the weight aspect

Jonathan Bober, Andrew R. Booker, Min Lee and David Lowry-Duda

Vol. 20 (2026), No. 2, 299–332

DOI: 10.2140/ant.2026.20.299

Abstract

We prove the existence of “murmurations” in the family of holomorphic modular forms of level $1$ and weight $k \to \infty$ , that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.

Keywords

murmurations, weight aspect, modular forms

Mathematical Subject Classification

Primary: 11F30, 11N60

Secondary: 11F72

Milestones

Received: 1 February 2024

Revised: 7 November 2024

Accepted: 11 February 2025

Published: 16 February 2026

Authors

Jonathan Bober
School of Mathematics University of Bristol Bristol BS8 1UG United Kingdom

Andrew R. Booker
School of Mathematics University of Bristol Bristol BS8 1UG United Kingdom

Min Lee
School of Mathematics University of Bristol Bristol BS8 1UG United Kingdom

David Lowry-Duda
ICERM Brown University Providence, RI 02903 United States

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Vol. 20, No. 2, 2026