The Sato-Tate conjecture for modular forms of weight 3

TL;DR

This paper proves an analogue of the Sato-Tate conjecture for certain modular forms of weight 3, expanding understanding of their distribution of Frobenius eigenvalues in number theory.

Contribution

It establishes a new case of the Sato-Tate conjecture for modular forms of weight 3 with specific automorphic representation properties.

Findings

Proves the Sato-Tate conjecture analogue for weight 3 modular forms.

Identifies conditions involving Steinberg twists at finite places.

Enhances understanding of automorphic representations and eigenvalue distributions.

Abstract

We prove a natural analogue of the Sato-Tate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place.