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 $k\ge 6$, 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 $N \in {2,6}$,andinweights$k \geq 52$, 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 $N=1$.