Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients

TL;DR

This paper establishes an effective joint Sato-Tate distribution for Fourier coefficients of two non-CM newforms, extending previous results to more general measurable regions and deriving applications to sign changes and distribution properties.

Contribution

It generalizes the joint Sato-Tate distribution to wider regions and develops a framework for studying sign changes and distribution of symmetric power coefficients.

Findings

Proves an unconditional, effective joint Sato-Tate distribution for two newforms.

Extends distribution results to measurable regions with complex boundaries.

Provides bounds for the first sign change of Fourier coefficients.

Abstract

We prove an unconditional, effective joint Sato-Tate distribution for the Fourier coefficients of two twist-inequivalent, non-CM newforms $f$ and $f'$. Our result generalises a result of Thorner, which holds for rectangular regions, by extending it to a wide range of measurable subsets of $[-2,2]^2$. Indeed, our theorem applies to any measurable region whose boundary consists of a finite number of continuous curves of finite length. As a consequence, we develop a unified framework to study various arithmetic properties of Fourier coefficients of symmetric power $L$-functions attached to $f$ and $f'$. In particular, for these coefficients (and their polynomial expressions), we obtain effective distribution results, quantitative statements on simultaneous sign behaviour, and bounds for the first sign change.