Congruence conditions for the mod $λ$ values of the Fourier coefficients of classical eigenforms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915335433617408 |
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| author | Daas, Michael A. |
| author_facet | Daas, Michael A. |
| contents | We classify all instances of the condition $a_{p}(f) \equiv x \bmod λ$ being related to a congruence on the prime $p$, where $a_{p}(f)$ denotes the $p$th Fourier coefficient of a classical normalised cuspidal eigenform $f$ and $λ$ is a prime in the number field generated by the Fourier coefficients of $f$. This classification is done in terms of the (projective) image of the mod $λ$ Galois representation associated with $f$ and extends work by Swinnerton-Dyer. We highlight that for $x = 0$, this condition is more often implied by a congruence on the prime $p$ than the general value of $a_{p}(f) \bmod λ$. Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Congruence conditions for the mod $λ$ values of the Fourier coefficients of classical eigenforms Daas, Michael A. Number Theory 11F30, 11F33, 11F80, 20G40 We classify all instances of the condition $a_{p}(f) \equiv x \bmod λ$ being related to a congruence on the prime $p$, where $a_{p}(f)$ denotes the $p$th Fourier coefficient of a classical normalised cuspidal eigenform $f$ and $λ$ is a prime in the number field generated by the Fourier coefficients of $f$. This classification is done in terms of the (projective) image of the mod $λ$ Galois representation associated with $f$ and extends work by Swinnerton-Dyer. We highlight that for $x = 0$, this condition is more often implied by a congruence on the prime $p$ than the general value of $a_{p}(f) \bmod λ$. Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves. |
| title | Congruence conditions for the mod $λ$ values of the Fourier coefficients of classical eigenforms |
| topic | Number Theory 11F30, 11F33, 11F80, 20G40 |
| url | https://arxiv.org/abs/2506.08865 |