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Newforms with rational coefficients

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 $N=1$.