The Ramanujan Journal

, Volume 46, Issue 3, pp 835–862 | Cite as

Newforms with rational coefficients

  • David P. RobertsEmail author


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 \(k \ge 6\), there are only finitely many classes. In large weights, we make this conjecture effective: in weights \(18 \le k \le 24\), all classes have \(N \le 30\); in weights \(26 \le k \le 50\), all classes have \(N \in \{2,6\}\); and in weights \(k \ge 52\), 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 \(N=1\).


Modular form Newform Weight Level Maeda conjecture 

Mathematics Subject Classification

11F30 (Primary) 11F80 (Secondary) 



The author thanks the conference organizers for the opportunity to speak at Automorphic forms: theory and computation at King’s College London, in September 2016. This paper grew out of the first half of the author’s talk. The list of newforms drawn up here is applied in [19], which is an expanded version of the second half.


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© Springer Science+Business Media New York 2018

Authors and Affiliations

  1. 1.Division of Science and MathematicsUniversity of Minnesota-MorrisMorrisUSA

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