## 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:”

 $[\mathbf{F}_3(a_5):\mathbf{F}_3]$ Total Number Number of Galois conjugacy classes Density of forms Density of conjugacy classes Totals: 61595 10740 1 1 $1$ 4623 4623 0.07505 0.4304 $2$ 2492 1246 0.04046 0.1160 $3$ 2397 799 0.03892 0.07439 $4$ 2476 619 0.04020 0.05764 $5$ 2600 520 0.04221 0.04842 $6$ 2142 357 0.03478 0.03324 $7$ 2289 327 0.03716 0.03045 $8$ 2008 251 0.03260 0.02337 $9$ 1962 218 0.03185 0.02030 $10$ 1530 153 0.02484 0.01425 $11$ 1837 167 0.02982 0.01555 $12$ 1656 138 0.02689 0.01285 $13$ 1612 124 0.02617 0.01155 $14$ 1638 117 0.02659 0.01089 $15$ 1455 97 0.02362 0.009032 $16$ 1088 68 0.01766 0.006331 $17$ 1292 76 0.02098 0.007076 $18$ 1008 56 0.01636 0.005214 $19$ 1159 61 0.01882 0.005680 $20$ 1120 56 0.01818 0.005214 $21$ 987 47 0.01602 0.004376 $22$ 990 45 0.01607 0.004190 $23$ 966 42 0.01568 0.003911 $24$ 1056 44 0.01714 0.004097 $25$ 1100 44 0.01786 0.004097 $26$ 650 25 0.01055 0.002328 $27$ 783 29 0.01271 0.002700 $28$ 868 31 0.01409 0.002886 $29$ 551 19 0.008946 0.001769 $30$ 420 14 0.006819 0.001304 $31$ 775 25 0.01258 0.002328 $32$ 800 25 0.01299 0.002328 $33$ 759 23 0.01232 0.002142 $34$ 374 11 0.006072 0.001024 $35$ 490 14 0.007955 0.001304 $36$ 576 16 0.009351 0.001490 $37$ 592 16 0.009611 0.001490 $38$ 380 10 0.006169 0.0009311 $39$ 429 11 0.006965 0.001024 $40$ 680 17 0.01104 0.001583 $41$ 492 12 0.007988 0.001117 $42$ 294 7 0.004773 0.0006518 $43$ 258 6 0.004189 0.0005587 $44$ 308 7 0.005000 0.0006518 $45$ 180 4 0.002922 0.0003724 $46$ 322 7 0.005228 0.0006518 $47$ 282 6 0.004578 0.0005587 $48$ 144 3 0.002338 0.0002793 $49$ 147 3 0.002387 0.0002793 $50$ 350 7 0.005682 0.0006518 $51$ 561 11 0.009108 0.001024 $52$ 260 5 0.004221 0.0004655 $53$ 106 2 0.001721 0.0001862 $54$ 378 7 0.006137 0.0006518 $55$ 0 0 0 0 $56$ 112 2 0.001818 0.0001862 $57$ 171 3 0.002776 0.0002793 $58$ 406 7 0.006591 0.0006518 $59$ 236 4 0.003831 0.0003724 $60$ 120 2 0.001948 0.0001862 $61$ 183 3 0.002971 0.0002793 $62$ 62 1 0.001007 0.00009311 $63$ 378 6 0.006137 0.0005587 $64$ 320 5 0.005195 0.0004655 $65$ 130 2 0.002111 0.0001862 $66$ 132 2 0.002143 0.0001862 $67$ 201 3 0.003263 0.0002793 $68$ 68 1 0.001104 0.00009311 $69$ 276 4 0.004481 0.0003724 $70$ 140 2 0.002273 0.0001862 $71$ 284 4 0.004611 0.0003724 $72$ 144 2 0.002338 0.0001862 $73$ 292 4 0.004741 0.0003724 $74$ 74 1 0.001201 0.00009311 $75$ 0 0 0 0 $76$ 152 2 0.002468 0.0001862 $77$ 0 0 0 0 $78$ 78 1 0.001266 0.00009311 $79$ 79 1 0.001283 0.00009311 $80$ 160 2 0.002598 0.0001862 $81$ 81 1 0.001315 0.00009311 $82$ 0 0 0 0 $83$ 83 1 0.001348 0.00009311 $84$ 168 2 0.002727 0.0001862 $85$ 85 1 0.001380 0.00009311 $86$ 0 0 0 0 $87$ 0 0 0 0 $88$ 0 0 0 0 $89$ 89 1 0.001445 0.00009311 $90$ 0 0 0 0 $91$ 0 0 0 0 $92$ 0 0 0 0 $93$ 0 0 0 0 $94$ 0 0 0 0 $95$ 95 1 0.001542 0.00009311 $96$ 0 0 0 0 $97$ 0 0 0 0 $98$ 0 0 0 0 $99$ 0 0 0 0 $100$ 0 0 0 0 $101$ 0 0 0 0 $102$ 0 0 0 0 $103$ 0 0 0 0 $104$ 104 1 0.001688 0.00009311

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.)

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