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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.05067 |
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| _version_ | 1866916511036211200 |
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| author | Kumar, Anmol |
| author_facet | Kumar, Anmol |
| contents | K{ö}hler, in [1], presented a weight 1 newform on $Γ_0(576)$ constructed from a linear combination of weight 1 eta quotients and asked, ``What would be a suitable $L$ and representation $ρ$ such that Deligne\text{-}Serre correspondence holds?" In this paper, we find the Galois field extension $L$ and representation $ρ$ such that the Deligne\text{-}Serre correspondence holds for this newform, and also study the splitting of primes in $L$ using the coefficients $a(p)$ of the newform. We also discuss an exotic newform on $Γ_0(1080)$ constructed from a linear combination of weight 1 eta quotients, find the corresponding Galois extension and representation, and study the splitting of primes in this extension. Furthermore, we find all such newforms that can be constructed from a linear combination of weight 1 eta quotients listed in [2] with $q$-expansion of the form $q+\sum_{k=2}^{\infty}a(k)q^k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exotic newforms constructed from a linear combination of eta quotients Kumar, Anmol Number Theory K{ö}hler, in [1], presented a weight 1 newform on $Γ_0(576)$ constructed from a linear combination of weight 1 eta quotients and asked, ``What would be a suitable $L$ and representation $ρ$ such that Deligne\text{-}Serre correspondence holds?" In this paper, we find the Galois field extension $L$ and representation $ρ$ such that the Deligne\text{-}Serre correspondence holds for this newform, and also study the splitting of primes in $L$ using the coefficients $a(p)$ of the newform. We also discuss an exotic newform on $Γ_0(1080)$ constructed from a linear combination of weight 1 eta quotients, find the corresponding Galois extension and representation, and study the splitting of primes in this extension. Furthermore, we find all such newforms that can be constructed from a linear combination of weight 1 eta quotients listed in [2] with $q$-expansion of the form $q+\sum_{k=2}^{\infty}a(k)q^k$. |
| title | Exotic newforms constructed from a linear combination of eta quotients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.05067 |