Higher Level

The results in Section 1.1 strongly suggest that if $f=\sum a_n q^n\in S_2(\Gamma_0(23))$ is a newform, then asymptotically 0% of $a_p$ are in $\mathbb{Q}$ (for $p$ prime). If you actually look at the data, a large proportion of $a_p$ do lie in $\mathbb{Q}$ for relatively small $p$ ; as $p$ gets larger the proportion (appears to) tend to 0 . In contrast for every newform $f\in S_2(\Gamma_0(N))$ of degree $\geq 3$ that I (=Stein) investigated the proportion of $a_p\in \mathbb{Q}$ was very very small.

Problem 1.2.1   Explain the above observations about the proportion of $a_p$ that lie in $\mathbb{Q}$ . Gather further numerical data.