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Newforms with rational coefficients

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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 \(k \ge 6\), 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 \(N=1\).

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Acknowledgements

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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  1. Division of Science and Mathematics, University of Minnesota-Morris, Morris, MN, 56267, USA

    David P. Roberts

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  1. David P. Roberts
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Correspondence to David P. Roberts.

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The author’s research was supported by Grant #209472 from the Simons Foundation and Grant DMS-1601350 from the National Science Foundation and Directorate for Mathematical and Physical Sciences.

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Roberts, D.P. Newforms with rational coefficients. Ramanujan J 46, 835–862 (2018). https://doi.org/10.1007/s11139-017-9914-5

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