Newforms with rational coefficients

TL;DR

This paper investigates classical newforms with rational coefficients and no complex multiplication, proposing conjectures on their finite classification across weights and levels, and providing detailed analysis for certain cases.

Contribution

It introduces conjectures on the finiteness and distribution of rational newforms without CM across weights and levels, with effective bounds in large weights.

Findings

Conjecture of finitely many classes for each weight k ≥ 6.

Effective bounds on levels for large weights (18-24, 26-50, ≥52).

Detailed formulas for newforms at specific levels N=2,3,4,6,8.

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