Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients
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| Format: | Preprint |
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2026
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| author | Kumar, Arvind Kumari, Moni Mishra, Prabhat Kumar |
| author_facet | Kumar, Arvind Kumari, Moni Mishra, Prabhat Kumar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17532 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients Kumar, Arvind Kumari, Moni Mishra, Prabhat Kumar Number Theory Primary: 11F11 (Primary) 11F30 (Secondary) 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. |
| title | Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients |
| topic | Number Theory Primary: 11F11 (Primary) 11F30 (Secondary) |
| url | https://arxiv.org/abs/2604.17532 |