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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