Root number bias for newforms
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866914122213359616 |
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| author | Martin, Kimball |
| author_facet | Martin, Kimball |
| contents | Previously we observed that newforms obey a strict bias towards root number $+1$ in squarefree levels: at least half of the newforms in $S_k(Γ_0(N))$ with root number $+1$ for $N$ squarefree, and it is strictly more than half outside of a few special cases. Subsequently, other authors treated levels which are cubes of squarefree numbers. Here we treat arbitrary levels, and find that if the level is not the square of a squarefree number, this strict bias still holds for any weight. In fact the number of such exceptional levels is finite for fixed weight, and 0 if $k < 12$. We also investigate some variants of this question to better understand the exceptional levels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08121 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Root number bias for newforms Martin, Kimball Number Theory Previously we observed that newforms obey a strict bias towards root number $+1$ in squarefree levels: at least half of the newforms in $S_k(Γ_0(N))$ with root number $+1$ for $N$ squarefree, and it is strictly more than half outside of a few special cases. Subsequently, other authors treated levels which are cubes of squarefree numbers. Here we treat arbitrary levels, and find that if the level is not the square of a squarefree number, this strict bias still holds for any weight. In fact the number of such exceptional levels is finite for fixed weight, and 0 if $k < 12$. We also investigate some variants of this question to better understand the exceptional levels. |
| title | Root number bias for newforms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2207.08121 |