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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.

William Stein 2006-10-20