The distribution of Hecke eigenvalues, part II

Here are some numbers from KB promised in my last post.

“For the first 61595 newforms of squarefree level coprime to 15 here’s
the field extension of Z/3Z generated by the \(a_5\) field extensions:”

I’ve presented the numbers KB send me in various ways. The first column simply counts the field generated by \(a_5\). The second column normalizes by the order of the field. This is a little like counting two representations which differ by an automorphism of the coefficient field as being “the same.” The final two comments are then the proportion of the first two columns overall.

I’m really not quite sure what to make of this data. It does suggest that A is false, which is perhaps not surprising. It’s not terribly overwhelming evidence for B, but then, law of smaller numbers and all.

AV’s suggestion in the comments that the constants \(C_q\) should be independent of \(q\) must refer to the constants in the second last column, I believe. Of course, it might be the case that \(\mathbf{F}_3(a_5)\) is smaller than \(\mathbf{F}_3(a_2,a_5,a_7,a_{11},\ldots)\), so these numbers aren’t exactly the same as the fields generated by the mod-\(p\) reductions of the eigenforms. If you squint, the numbers in this column do look somewhat constant for \(n < 10\) or so. One can even argue that \(n = 1\) might be artificially inflated exactly because the phenomenon of "slipping into a subfield" mentioned above. So I'm giving the points to AV. (Yes, that's right, there were points available and you missed out because you didn’t play the game.)