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Fricke traces of singular moduli

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Numbers
$N$
$m$
$d$ 
$t_m^{(N*)}(d)$
2
1
4:
-26
comment: The full starred stabilizer has order four, and the Hauptmodul value is $-104$.
2
1
7:
-23
2
1
8:
76
comment: The full starred stabilizer has order two, and the Hauptmodul value is $152$.
2
1
12:
-248
2
1
15:
-1
2
1
16:
518
2
1
20:
-1128
2
1
23:
-94
2
1
24:
2200
2
1
28:
-4096
2
1
31:
93
2
1
32:
7180
2
1
36:
-12418
2
1
39:
-236
2
1
40:
20632
2
1
44:
-33512
2
1
47:
235
2
1
48:
53256
2
1
52:
-83048
2
1
55:
-497
2
1
56:
127280
2
1
60:
-192512
2
1
63:
710
2
1
64:
286726
2
1
68:
-422096
2
1
71:
-1163
2
1
72:
614628
2
1
76:
-885480
2
1
79:
1424
2
1
80:
1263640
2
1
84:
-1788112
2
1
87:
-2163
2
1
88:
2508952
2
1
92:
-3493888
2
1
95:
2996
2
1
96:
4831256
2
1
100:
-6635650
2
1
103:
-4165
2
1
104:
9056712
2
1
108:
-12288992
2
1
111:
5333
2
1
112:
16580608
2
1
116:
-22252856
2
1
119:
-7505
2
1
120:
29716784
2
1
124:
-39493632
2
1
127:
10010
2
1
128:
52248588
2
1
132:
-68825296
2
1
135:
-13114
2
1
136:
90284336
2
1
140:
-117966288
2
1
143:
16695
2
1
144:
153553438
2
1
148:
-199148648
2
1
151:
-21950
2
1
152:
257382856
2
1
156:
-331534336
2
1
159:
28399
2
1
160:
425670680
2
1
164:
-544842144
2
1
167:
-36428
2
1
168:
695296304
2
1
172:
-884736744
2
1
175:
45604
2
1
176:
1122660376
2
1
180:
-1420741944
2
1
183:
-58009
2
1
184:
1793290544
2
1
188:
-2257838080
2
1
191:
73570
2
1
192:
2835808264
2
1
196:
-3553321194
2
1
199:
-91980
2
1
200:
4442194452
2
2
4:
518
2
2
7:
-8215
2
2
8:
7180
2
2
12:
52760
2
2
15:
-385025
2
2
16:
287710
2
2
20:
1263640
2
2
23:
-6987870
2
2
24:
4831256
2
2
28:
16572370
2
2
31:
-78987171
2
2
32:
52263100
2
2
36:
153553438
2
2
39:
-663068908
2
2
40:
425670680
2
2
44:
1122593352
2
2
47:
-4515675925
2
2
48:
2835914280
2
2
52:
6896961560
2
2
55:
-26273374705
2
2
56:
16220254256
2
2
60:
37017690110
2
2
63:
-135030414650
2
2
64:
82226889758
2
2
68:
178211459120
2
2
71:
-627291628683
2
2
72:
377674159140
2
2
76:
784072665672
2
2
79:
-2678380575344
2
2
80:
1597179692920
2
2
84:
3196802732080
2
2
87:
-10643523430515
2
2
88:
6294840129560
2
2
92:
12207816859460
2
2
95:
-39748955853900
2
2
96:
23340153962872
2
2
100:
44031505861150
2
2
103:
-140584572563525
2
2
104:
82028223113288
2
2
108:
151013204179040
2
2
111:
-473834685270827
2
2
112:
274917357117440
2
2
116:
495202751076424
2
2
119:
-1529744342441297
2
2
120:
883067941384240
2
2
124:
1559739457388730
2
2
127:
-4750842461214950
2
2
128:
2729960522823740
2
2
132:
4736863567286320
2
2
135:
-14244613986956090
2
2
136:
8151279246143536
2
2
140:
13916137770505360
2
2
143:
-41361681552858825
2
2
144:
23578504275923798
2
2
148:
39660184000219160
2
2
151:
-116618465173714366
2
2
152:
66246265661892680
2
2
156:
109914552223882792
2
2
159:
-320008274044727569
2
2
160:
181195520676041080
2
2
164:
296853791705276512
2
2
167:
-856363618150116940
2
2
168:
483435711381532720
2
2
172:
782759104413854280
2
2
175:
-2238889349968047580
2
2
176:
1260369122297436792
2
2
180:
2018504140031121480
2
2
183:
-5727580536845845145
2
2
184:
3215890893283618864
2
2
188:
5097838271600148950
2
2
191:
-14357748979111633054
2
2
192:
8041801043050209320
2
2
196:
12626092124920484406
2
2
199:
-35312380559541798732
2
2
200:
19733105502833919060
3
1
3:
-7
3
1
8:
-34
3
1
11:
22
3
1
12:
26
3
1
15:
-69
3
1
20:
-116
3
1
23:
115
3
1
24:
174
3
1
27:
-241
3
1
32:
-410
3
1
35:
492
3
1
36:
492
3
1
39:
-705
3
1
44:
-1060
3
1
47:
1272
3
1
48:
1442
3
1
51:
-1770
3
1
56:
-2600
3
1
59:
3074
3
1
60:
3264
3
1
63:
-4119
3
1
68:
-5544
3
1
71:
6755
3
1
72:
7290
3
1
75:
-8689
3
1
80:
-11892
3
1
83:
13890
3
1
84:
14604
3
1
87:
-17460
3
1
92:
-22912
3
1
95:
27007
3
1
96:
28710
3
1
99:
-33534
3
1
104:
-43660
3
1
107:
50690
3
1
108:
52982
3
1
111:
-61917
3
1
116:
-78940
3
1
119:
91339
3
1
120:
96252
3
1
123:
-110634
3
1
128:
-140154
3
1
131:
160302
3
1
132:
167532
3
1
135:
-192444
3
1
140:
-240008
3
1
143:
274569
3
1
144:
287244
3
1
147:
-326641
3
1
152:
-405420
3
1
155:
459288
3
1
156:
478272
3
1
159:
-542964
3
1
164:
-666320
3
1
167:
753477
3
1
168:
785724
3
1
171:
-885480
3
1
176:
-1081140
3
1
179:
1215534
3
1
180:
1262628
3
1
183:
-1420044
3
1
188:
-1720064
3
1
191:
1928982
3
1
192:
2004866
3
1
195:
-2243028
3
1
200:
-2703918
3
2
3:
33
3
2
8:
-410
3
2
11:
-1082
3
2
12:
1428
3
2
15:
3195
3
2
20:
-11892
3
2
23:
-22797
3
2
24:
28710
3
2
27:
53223
3
2
32:
-140222
3
2
35:
-240500
3
2
36:
287244
3
2
39:
477567
3
2
44:
-1081096
3
2
47:
-1718792
3
2
48:
2004918
3
2
51:
3131334
3
2
56:
-6410728
3
2
59:
-9697198
3
2
60:
11110518
3
2
63:
16572393
3
2
68:
-31673832
3
2
71:
-46148893
3
2
72:
52256178
3
2
75:
75375591
3
2
80:
-136648412
3
2
83:
-193512622
3
2
84:
217023372
3
2
87:
304780428
3
2
92:
-530207002
3
2
95:
-733666049
3
2
96:
816741090
3
2
99:
1122661458
3
2
104:
-1887927964
3
2
107:
-2563375022
3
2
108:
2835859596
3
2
111:
3828692835
3
2
116:
-6259322716
3
2
119:
-8363407925
3
2
120:
9204508620
3
2
123:
12239880390
3
2
128:
-19533233454
3
2
131:
-25743489058
3
2
132:
28205242380
3
2
135:
37017686916
3
2
140:
-57855893904
3
2
143:
-75342887159
3
2
144:
82226602980
3
2
147:
106689768423
3
2
152:
-163722607644
3
2
155:
-210975699944
3
2
156:
229458615678
3
2
159:
294726513996
3
2
164:
-445002446288
3
2
167:
-568090666811
3
2
168:
615970205388
3
2
171:
784074436632
3
2
176:
-1166806673212
3
2
179:
-1477092442146
3
2
180:
1597178443428
3
2
183:
2016632959668
3
2
188:
-2962040220176
3
2
191:
-3721378065898
3
2
192:
4013923579206
3
2
195:
5030980222860
3
2
200:
-7302369225270
5
1
4:
-8
5
1
11:
-12
5
1
15:
-19
5
1
16:
-6
5
1
19:
20
5
1
20:
6
5
1
24:
-44
5
1
31:
-39
5
1
35:
-44
5
1
36:
20
5
1
39:
53
5
1
40:
56
5
1
44:
-136
5
1
51:
-56
5
1
55:
-113
5
1
56:
36
5
1
59:
188
5
1
60:
112
5
1
64:
-254
5
1
71:
-173
5
1
75:
-248
5
1
76:
152
5
1
79:
415
5
1
80:
268
5
1
84:
-532
5
1
91:
-248
5
1
95:
-476
5
1
96:
216
5
1
99:
828
5
1
100:
500
5
1
104:
-1000
5
1
111:
-497
5
1
115:
-844
5
1
116:
568
5
1
119:
1474
5
1
120:
1012
5
1
124:
-1824
5
1
131:
-860
5
1
135:
-1501
5
1
136:
872
5
1
139:
2652
5
1
140:
1624
5
1
144:
-3054
5
1
151:
-1408
5
1
155:
-2488
5
1
156:
1600
5
1
159:
4415
5
1
160:
2868
5
1
164:
-5132
5
1
171:
-2300
5
1
175:
-4119
5
1
176:
2440
5
1
179:
7236
5
1
180:
4518
5
1
184:
-8092
5
1
191:
-3742
5
1
195:
-6488
5
1
196:
4212
5
1
199:
11394
5
1
200:
7256
5
2
4:
-6
5
2
11:
-124
5
2
15:
93
5
2
16:
-270
5
2
19:
132
5
2
20:
268
5
2
24:
216
5
2
31:
-1863
5
2
35:
1668
5
2
36:
-3054
5
2
39:
1653
5
2
40:
2868
5
2
44:
2416
5
2
51:
-12792
5
2
55:
10911
5
2
56:
-19648
5
2
59:
9612
5
2
60:
16986
5
2
64:
14070
5
2
71:
-64589
5
2
75:
53256
5
2
76:
-92496
5
2
79:
44607
5
2
80:
76304
5
2
84:
61440
5
2
91:
-260088
5
2
95:
207572
5
2
96:
-359760
5
2
99:
166284
5
2
100:
287250
5
2
104:
226328
5
2
111:
-910545
5
2
115:
712068
5
2
116:
-1221096
5
2
119:
558242
5
2
120:
952536
5
2
124:
736434
5
2
131:
-2853740
5
2
135:
2190147
5
2
136:
-3744336
5
2
139:
1680012
5
2
140:
2868272
5
2
144:
2184498
5
2
151:
-8230656
5
2
155:
6230536
5
2
156:
-10600854
5
2
159:
4714623
5
2
160:
8006004
5
2
164:
6021168
5
2
171:
-22161516
5
2
175:
16572393
5
2
176:
-28129032
5
2
179:
12378516
5
2
180:
20994804
5
2
184:
15620928
5
2
191:
-56424734
5
2
195:
41782536
5
2
196:
-70708230
5
2
199:
30909474
5
2
200:
52255768
6
1
8:
-5
6
1
12:
-7
6
1
15:
-5
6
1
20:
6
6
1
23:
-13
6
1
24:
11
6
1
32:
-29
6
1
36:
-26
6
1
39:
-1
6
1
44:
22
6
1
47:
-8
6
1
48:
33
6
1
56:
-52
6
1
60:
-64
6
1
63:
-23
6
1
68:
76
6
1
71:
-29
6
1
72:
81
6
1
80:
-122
6
1
84:
-122
6
1
87:
12
6
1
92:
128
6
1
95:
-1
6
1
96:
163
6
1
104:
-222
6
1
108:
-241
6
1
111:
-29
6
1
116:
274
6
1
119:
-53
6
1
120:
310
6
1
128:
-381
6
1
132:
-410
6
1
135:
4
6
1
140:
492
6
1
143:
9
6
1
144:
518
6
1
152:
-638
6
1
156:
-704
6
1
159:
-52
6
1
164:
792
6
1
167:
-59
6
1
168:
886
6
1
176:
-1082
6
1
180:
-1134
6
1
183:
52
6
1
188:
1280
6
1
191:
22
6
1
192:
1409
6
1
200:
-1635
6
2
8:
-29
6
2
12:
19
6
2
15:
-133
6
2
20:
-122
6
2
23:
243
6
2
24:
163
6
2
32:
-449
6
2
36:
518
6
2
39:
-1409
6
2
44:
-1038
6
2
47:
2552
6
2
48:
1461
6
2
56:
-2548
6
2
60:
3190
6
2
63:
-8215
6
2
68:
-5620
6
2
71:
13539
6
2
72:
7209
6
2
80:
-12002
6
2
84:
14726
6
2
87:
-34932
6
2
92:
-22810
6
2
95:
54015
6
2
96:
28895
6
2
104:
-43438
6
2
108:
52741
6
2
111:
-123805
6
2
116:
-79214
6
2
119:
182731
6
2
120:
95942
6
2
128:
-140593
6
2
132:
167942
6
2
135:
-384892
6
2
140:
-239516
6
2
143:
549129
6
2
144:
287710
6
2
152:
-404782
6
2
156:
477566
6
2
159:
-1085876
6
2
164:
-667112
6
2
167:
1507013
6
2
168:
784838
6
2
176:
-1082178
6
2
180:
1263762
6
2
183:
-2840140
6
2
188:
-1718800
6
2
191:
3857942
6
2
192:
2006341
6
2
200:
-2702283
7
1
3:
-3
7
1
7:
-5
7
1
12:
-10
7
1
19:
-1
7
1
20:
-22
7
1
24:
-2
7
1
27:
12
7
1
28:
4
7
1
31:
-29
7
1
35:
-17
7
1
40:
-10
7
1
47:
-18
7
1
48:
-54
7
1
52:
34
7
1
55:
33
7
1
56:
22
7
1
59:
-51
7
1
63:
-46
7
1
68:
-36
7
1
75:
19
7
1
76:
-102
7
1
80:
30
7
1
83:
101
7
1
84:
50
7
1
87:
-103
7
1
91:
-73
7
1
96:
-58
7
1
103:
32
7
1
104:
-206
7
1
108:
128
7
1
111:
168
7
1
112:
120
7
1
115:
-154
7
1
119:
-155
7
1
124:
-104
7
1
131:
51
7
1
132:
-308
7
1
136:
204
7
1
139:
349
7
1
140:
170
7
1
143:
-310
7
1
147:
-245
7
1
152:
-102
7
1
159:
91
7
1
160:
-554
7
1
164:
392
7
1
167:
555
7
1
168:
358
7
1
171:
-461
7
1
175:
-385
7
1
180:
-234
7
1
187:
206
7
1
188:
-888
7
1
192:
562
7
1
195:
932
7
1
196:
492
7
1
199:
-727
7
2
3:
-7
7
2
7:
-1
7
2
12:
-60
7
2
19:
-101
7
2
20:
30
7
2
24:
-58
7
2
27:
116
7
2
28:
110
7
2
31:
-133
7
2
35:
187
7
2
40:
-554
7
2
47:
-906
7
2
48:
542
7
2
52:
-274
7
2
55:
1017
7
2
56:
822
7
2
59:
-463
7
2
63:
1150
7
2
68:
-3028
7
2
75:
-4289
7
2
76:
2988
7
2
80:
-1366
7
2
83:
4417
7
2
84:
3790
7
2
87:
-2183
7
2
91:
5227
7
2
96:
-11966
7
2
103:
-16256
7
2
104:
11554
7
2
108:
-4944
7
2
111:
16248
7
2
112:
13464
7
2
115:
-6746
7
2
119:
17525
7
2
124:
-39658
7
2
131:
-52265
7
2
132:
37196
7
2
136:
-15444
7
2
139:
49169
7
2
140:
41068
7
2
143:
-21006
7
2
147:
53263
7
2
152:
-115502
7
2
159:
-148117
7
2
160:
105746
7
2
164:
-43336
7
2
167:
137067
7
2
168:
113270
7
2
171:
-55849
7
2
175:
142723
7
2
180:
-306198
7
2
187:
-385898
7
2
188:
275020
7
2
192:
-111818
7
2
195:
346204
7
2
196:
287244
7
2
199:
-141959
10
1
4:
-1
10
1
15:
-3
10
1
16:
-7
10
1
20:
-4
10
1
24:
0
10
1
31:
-7
10
1
36:
7
10
1
39:
-11
10
1
40:
6
10
1
44:
-12
10
1
55:
-1
10
1
56:
-20
10
1
60:
-16
10
1
64:
1
10
1
71:
-13
10
1
76:
20
10
1
79:
-1
10
1
80:
10
10
1
84:
-12
10
1
95:
-12
10
1
96:
-44
10
1
100:
-25
10
1
104:
12
10
1
111:
-17
10
1
116:
44
10
1
119:
-30
10
1
120:
32
10
1
124:
-32
10
1
135:
3
10
1
136:
-64
10
1
140:
-44
10
1
144:
13
10
1
151:
0
10
1
156:
64
10
1
159:
-1
10
1
160:
50
10
1
164:
-44
10
1
175:
-23
10
1
176:
-124
10
1
180:
-72
10
1
184:
44
10
1
191:
-30
10
1
196:
131
10
1
199:
-30
10
1
200:
76
10
2
4:
-7
10
2
15:
-35
10
2
16:
-15
10
2
20:
10
10
2
24:
-44
10
2
31:
-71
10
2
36:
13
10
2
39:
117
10
2
40:
50
10
2
44:
-148
10
2
55:
-225
10
2
56:
56
10
2
60:
90
10
2
64:
-267
10
2
71:
-333
10
2
76:
172
10
2
79:
831
10
2
80:
270
10
2
84:
-520
10
2
95:
-940
10
2
96:
172
10
2
100:
525
10
2
104:
-1012
10
2
111:
-977
10
2
116:
524
10
2
119:
2978
10
2
120:
980
10
2
124:
-1870
10
2
135:
-3005
10
2
136:
936
10
2
140:
1580
10
2
144:
-3027
10
2
151:
-2816
10
2
156:
1642
10
2
159:
8831
10
2
160:
2930
10
2
164:
-5088
10
2
175:
-8215
10
2
176:
2292
10
2
180:
4590
10
2
184:
-8136
10
2
191:
-7454
10
2
196:
4081
10
2
199:
22818
10
2
200:
7180
13
1
3:
-1
13
1
4:
-2
13
1
12:
-6
13
1
16:
-4
13
1
23:
-11
13
1
27:
-1
13
1
35:
-11
13
1
36:
-4
13
1
39:
-9
13
1
40:
-6
13
1
43:
5
13
1
48:
-2
13
1
51:
5
13
1
52:
2
13
1
55:
-11
13
1
56:
-18
13
1
64:
-20
13
1
68:
-6
13
1
75:
-12
13
1
79:
-6
13
1
87:
-29
13
1
88:
12
13
1
91:
-11
13
1
92:
-24
13
1
95:
-7
13
1
100:
20
13
1
103:
11
13
1
104:
6
13
1
107:
-6
13
1
108:
-30
13
1
116:
-40
13
1
120:
-6
13
1
127:
-24
13
1
131:
-1
13
1
139:
-35
13
1
140:
-2
13
1
143:
-29
13
1
144:
-26
13
1
147:
28
13
1
152:
16
13
1
155:
34
13
1
156:
8
13
1
159:
-30
13
1
160:
-46
13
1
168:
-46
13
1
172:
6
13
1
179:
-41
13
1
183:
5
13
1
191:
-82
13
1
192:
22
13
1
195:
-35
13
1
196:
-32
13
1
199:
27
13
2
3:
-5
13
2
4:
-4
13
2
12:
-4
13
2
16:
-24
13
2
23:
-35
13
2
27:
-29
13
2
35:
9
13
2
36:
-26
13
2
39:
-1
13
2
40:
-46
13
2
43:
1
13
2
48:
10
13
2
51:
41
13
2
52:
32
13
2
55:
-99
13
2
56:
-30
13
2
64:
52
13
2
68:
-134
13
2
75:
-60
13
2
79:
-134
13
2
87:
27
13
2
88:
-64
13
2
91:
97
13
2
92:
-198
13
2
95:
-7
13
2
100:
150
13
2
103:
251
13
2
104:
122
13
2
107:
-310
13
2
108:
-4
13
2
116:
260
13
2
120:
-398
13
2
127:
-224
13
2
131:
-397
13
2
139:
321
13
2
140:
-268
13
2
143:
251
13
2
144:
-570
13
2
147:
84
13
2
152:
452
13
2
155:
706
13
2
156:
414
13
2
159:
-934
13
2
160:
38
13
2
168:
866
13
2
172:
-1100
13
2
179:
-477
13
2
183:
-1059
13
2
191:
726
13
2
192:
-574
13
2
195:
857
13
2
196:
-1338
13
2
199:
251
Definition
The Fricke trace is $t_m^{(N*)}(d)=\sum_{[Q]\in\mathcal Q_{d,N}/\Gamma_N^*}P_{m,N}(j_N^*(\alpha_Q))/|\overline{\Gamma_N^*}_Q|$ [1]. Here $\mathcal Q_{d,N}$ consists of all positive definite integral binary quadratic forms $[a,b,c]$ with $b^2-4ac=-d$ and $N\mid a$, including imprimitive forms.
Parameters
$N$
—   modular level ($N>1$ is square-free and $X_0(N)$ has genus zero)
$m$
—   Faber order ($m>0$)
$d$
—   discriminant magnitude ($d>0$ and $-d$ is a square modulo $4N$)
Formulas
(1)
$P_{1,N}(X)=X$ and $P_{2,N}(X)=X^2-2c_N(1)$, where $c_N(1)$ is the coefficient of $q$ in $j_N^*$. In increasing order of $N=2,3,5,6,7,10,13$, these coefficients are $4372,783,134,79,51,22,12$ [1].
(2)
For $h\pmod{2N}$ satisfying $h^2\equiv-d\pmod{4N}$, let $u_m^{(N,h)}(d)$ be the sum over $\Gamma_0(N)$-classes in $\mathcal Q_{d,N}$ with $b\equiv h\pmod{2N}$, using the reciprocal projective $\Gamma_0(N)$ stabilizer order as weight. Then $u_m^{(N,h)}(d)=2^{\omega(\gcd(N,d))}t_m^{(N*)}(d)$, independently of $h$, where $\omega(k)$ counts the distinct prime divisors of $k$ [1], Lemma 4.2 and its remark.
Comments
(3)
Use $q=e^{2\pi i\tau}$ and $\Gamma_N^*=\langle\Gamma_0(N),W_e:e\mid N\rangle$, where $W_e$ are determinant-one Atkin–Lehner representatives. All Atkin–Lehner involutions are included at composite levels. The projective group is $\overline{\Gamma_N^*}=\Gamma_N^*/\{\pm I\}$, and the denominator is the order of its stabilizer of $Q$, rather than the stabilizer in the full modular group. Put $\alpha_Q=(-b+i\sqrt d)/(2a)$ with the positive square root [1], section 1.
(4)
The normalized Hauptmodul $j_N^*=q^{-1}+O(q)$ has constant coefficient zero. At levels $N=2,3,5,6,7,10,13$ it is the Monster series $T_{2A},T_{3A},T_{5A},T_{6A},T_{7A},T_{10A},T_{13A}$, respectively [1]. The monic Faber polynomial $P_{m,N}(X)$ is determined by $P_{m,N}(j_N^*)=q^{-m}+O(q)$, with no division by $m$. At prime levels these are the Hecke-type Faber polynomials in the same variable $X$ and normalization.
(5)
Each entry includes every level-structured CM class, with its stabilizer weight. There is no genus character, primitive-only restriction, or class-number average. The rectangular CM values give specified individual evaluations; the trace sums the entire orbit. The classical Zagier traces use level one, while the genus-character twisted traces insert a character into a level-one sum.
(6)
The second Faber traces enter the Fourier-coefficient identities of [1], Theorem 1.1. The McKay–Thompson coefficient table holds the relevant coefficients through exponent $30$ for the levels here other than $13$. The Jacobi-form construction [2], Part II, provides exact rational coefficient arithmetic for the traces.
(7)
The convention follows [1], including level-ramified and imprimitive forms. For example, the individual CM values in [3], section 8, give $j_2^*((1+i)/2)=-104$ and $j_2^*(i/\sqrt2)=152$, while the stabilizers in the full starred group give the traces $t_1^{(2*)}(4)=-26$ and $t_1^{(2*)}(8)=76$.
Programs
(P1)
Sage
# Save the attached generate.py beside this worksheet.
# Compute another admissible discriminant using exact Jacobi coefficients.
from generate import trace_coefficients
trace_coefficients(2, 1, bound=204)[204]
References
[1]
Toshiki Matsusaka, The Fourier coefficients of the McKay–Thompson series and the traces of CM values, Research in Number Theory 3 (2017), article 23, sections 1, 3 and 4. (doi)
[2]
Malik Amir, On Traces of Singular Moduli (2019), Part II, Proposition 13 and the Jacobi-basis construction. (arXiv)
[3]
Don Zagier, Traces of Singular Moduli, in Motives, Polylogarithms and Hodge Theory, Part I, International Press (2002), section 8. https://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext.pdf
Links
Similar tables
Hecke-type Faber polynomials —   The prime-level polynomials transform the same zero-constant Hauptmodul before its CM values are summed.
McKay–Thompson values at rectangular CM points —   These are individual principal rectangular evaluations of the prime-level functions used in the orbit sums.
Zagier traces of singular moduli —   These are the level-one analogue, with all imprimitive forms and projective stabilizer weights.
Genus-character twisted traces of singular moduli —   These weight level-one CM classes by a genus character, a different trace construction.
Fourier coefficients of monstrous McKay–Thompson series —   Specified combinations of the second Faber traces give these coefficients by Matsusaka’s identities.
Minimal polynomials of monstrous McKay–Thompson CM values —   For the shared prime-level Fricke series, these are minimal polynomials over $\mathbb Q$ of individual principal rectangular values $T_{pA}(i\sqrt r)$. Their roots alone do not specify the complete level-structured CM orbit and stabilizer weights used in the present trace sums.
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds $m=1,2$ and every admissible $1\leq d\leq200$ at all seven square-free levels $N>1$ with genus-zero $X_0(N)$: $2,3,5,6,7,10,13$. The numbers of discriminants are $75,66,60,49,57,45,53$, respectively, giving $810$ entries. This includes prime and composite levels before increasing discriminant depth, and retains level-ramified cases, nonmaximal orders and all imprimitive contributions. The two Faber orders include the Hauptmodul trace and the order used in Fourier-coefficient identities. The noncanonical discriminant cutoff bounds orbit cost and value growth while supporting those identities for the unstarred Hauptmodul through exponent $50$; the longest value has $21$ characters)
How they were obtained:

Computed by exact rational arithmetic in the complete weight-two, index-$N$ weak Jacobi-form basis, with the singular coefficients fixed by [1], Theorems 3.2 and 3.4. Exact elimination checks full column rank and consistency at every level. Lemma 4.2 then converts the fixed-residue coefficients to the starred traces. No exact value is inferred from rounded CM evaluations. The attached mathematical-method.md explains the basis, singular coefficients and uniqueness calculation.

more

An independent computation enumerates reduced binary quadratic forms, every projective level structure and its stabilizer orbit, then sums eta-quotient CM evaluations using Arb at 640 bits [4]. All 810 sums contain the corresponding exact rational, with finite component radii less than $5.381\times10^{-170}$. All 1392 fixed-residue sums also satisfy (2). Known CM values and deliberately wrong, wide and nonfinite balls test the checking method first. The attachment orbit-evidence.json preserves the congruence-subgroup representatives and their effective weights in the full starred trace.

Exact Euler-product expansions independently verify 424 Fourier-coefficient identities of [1], Theorem 1.1, and agree with all 360 overlapping published coefficients from the coefficient companion. The attached checker repeats the CM and identity comparisons on values read from the database.



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