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

2026/04/30 by Arvind Kumar, Moni Kumari, Prabhat Kumar Mishra · 1 citation
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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.

Citations

• Exceptional zeros of Rankin-Selberg L-functions and joint Sato-Tate distributions
• On Effective Sato-Tate Distributions for Surfaces Arising from Products of Elliptic Curves
• An unconditional explicit bound on the error term in the Sato-Tate conjecture
• Symmetric power functoriality for holomorphic modular forms, II
• Symmetric power functoriality for holomorphic modular forms
• Simultaneous non-vanishing and sign changes of Fourier coefficients of\n modular forms
• The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newforms
• On the coefficients of symmetric power L-functions
• Distinguishing Galois representations by their normalized traces
• On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms
• Distinguishing Hecke eigenforms

Cited by

• Determining newforms via arithmetic relations among Fourier coefficients

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