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arXiv · 2604.17532

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

Arvind Kumar·Moni Kumari·Prabhat Kumar Mishra
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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 any measurable region of $[-2,2]^2$ whose boundary consists of finitely many 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.

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Arvind Kumar, Moni Kumari, Prabhat Kumar Mishra. 2026-04-19. Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients. https://arxiv.org/abs/2604.17532

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