  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 $N=1$.
