{
  "id": "2412.05067",
  "version": "v1",
  "published": "2024-12-06T14:24:52.000Z",
  "updated": "2024-12-06T14:24:52.000Z",
  "title": "Exotic newforms constructed from a linear combination of eta quotients",
  "authors": [
    "Anmol Kumar"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "K{\\\"o}hler, in [1], presented a weight 1 newform on $\\Gamma_0(576)$ constructed from a linear combination of weight 1 eta quotients and asked, ``What would be a suitable $L$ and representation $\\rho$ such that Deligne\\text{-}Serre correspondence holds?\" In this paper, we find the Galois field extension $L$ and representation $\\rho$ 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 $\\Gamma_0(1080)$ 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 $q+\\sum_{k=2}^{\\infty}a(k)q^k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-06T14:24:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear combination",
      "eta quotients",
      "exotic newform",
      "correspondence holds",
      "representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}