Murmurations of Modular Forms and $p$-power Coefficients

TL;DR

This paper extends the concept of murmuration patterns in modular forms to include prime-square coefficients, showing that the overall density shape remains consistent.

Contribution

It introduces a new case for murmuration analysis in modular forms by considering prime squares, expanding prior work on prime-indexed coefficients.

Findings

Murmuration density shape remains unchanged with prime-square coefficients.

The extension confirms the robustness of murmuration patterns in modular forms.

Supports broader applicability of murmuration analysis in number theory.

Abstract

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.