Newforms with rational coefficients

David P. Roberts

    Research output: Contribution to journalArticle

    Abstract

    We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic twist-classes of these forms with respect to weight k and minimal level N. We conjecture that for each weight k≥ 6 , there are only finitely many classes. In large weights, we make this conjecture effective: in weights 18 ≤ k≤ 24 , all classes have N≤ 30 ; in weights 26 ≤ k≤ 50 , all classes have N∈ { 2 , 6 } ; and in weights k≥ 52 , there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases N= 2 , 3, 4, 6, and 8, where formulas can be kept nearly as simple as those for the classical case N= 1.

    Original languageEnglish (US)
    Pages (from-to)835-862
    Number of pages28
    JournalRamanujan Journal
    Volume46
    Issue number3
    DOIs
    StatePublished - Aug 1 2018

    Keywords

    • Level
    • Maeda conjecture
    • Modular form
    • Newform
    • Weight

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