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Mathieu and umbral moonshine coefficients

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Numbers
$X$
$r$
$s$ 
$C_X(r,s)$
A1^24
1
7:
90
A1^24
1
15:
462
A1^24
1
23:
1540
A1^24
1
31:
4554
A1^24
1
39:
11592
A1^24
1
47:
27830
A1^24
1
55:
61686
A1^24
1
63:
131100
A1^24
1
71:
265650
A1^24
1
79:
521136
A1^24
1
87:
988770
A1^24
1
95:
1830248
A1^24
1
103:
3303630
A1^24
1
111:
5844762
A1^24
1
119:
10139734
A1^24
1
127:
17301060
A1^24
1
135:
29051484
A1^24
1
143:
48106430
A1^24
1
151:
78599556
A1^24
1
159:
126894174
A1^24
1
167:
202537080
A1^24
1
175:
319927608
A1^24
1
183:
500376870
A1^24
1
191:
775492564
A1^24
1
199:
1191453912
A1^24
1
207:
1815754710
A1^24
1
215:
2745870180
A1^24
1
223:
4122417420
A1^24
1
231:
6146311620
A1^24
1
239:
9104078592
A2^12
1
11:
32
A2^12
1
23:
110
A2^12
1
35:
288
A2^12
1
47:
660
A2^12
1
59:
1408
A2^12
1
71:
2794
A2^12
1
83:
5280
A2^12
1
95:
9638
A2^12
1
107:
16960
A2^12
1
119:
29018
A2^12
1
131:
48576
A2^12
1
143:
79530
A2^12
1
155:
127776
A2^12
1
167:
202050
A2^12
1
179:
314688
A2^12
1
191:
483516
A2^12
1
203:
733920
A2^12
1
215:
1101364
A2^12
1
227:
1635680
A2^12
1
239:
2406116
A2^12
1
251:
3507680
A2^12
1
263:
5071000
A2^12
1
275:
7274464
A2^12
1
287:
10359030
A2^12
1
299:
14650176
A2^12
1
311:
20585334
A2^12
1
323:
28747840
A2^12
1
335:
39914402
A2^12
1
347:
55114400
A2^12
1
359:
75704904
A2^12
2
8:
20
A2^12
2
20:
88
A2^12
2
32:
220
A2^12
2
44:
560
A2^12
2
56:
1144
A2^12
2
68:
2400
A2^12
2
80:
4488
A2^12
2
92:
8360
A2^12
2
104:
14696
A2^12
2
116:
25544
A2^12
2
128:
42660
A2^12
2
140:
70576
A2^12
2
152:
113520
A2^12
2
164:
180640
A2^12
2
176:
281808
A2^12
2
188:
435160
A2^12
2
200:
661476
A2^12
2
212:
996600
A2^12
2
224:
1482536
A2^12
2
236:
2187328
A2^12
2
248:
3193960
A2^12
2
260:
4629152
A2^12
2
272:
6650400
A2^12
2
284:
9490536
A2^12
2
296:
13441032
A2^12
2
308:
18920240
A2^12
2
320:
26457464
A2^12
2
332:
36792560
A2^12
2
344:
50865232
A2^12
2
356:
69966336
A3^8
1
15:
14
A3^8
1
31:
42
A3^8
1
47:
86
A3^8
1
63:
188
A3^8
1
79:
336
A3^8
1
95:
616
A3^8
1
111:
1050
A3^8
1
127:
1764
A3^8
1
143:
2814
A3^8
1
159:
4510
A3^8
1
175:
6936
A3^8
1
191:
10612
A3^8
1
207:
15862
A3^8
1
223:
23532
A3^8
1
239:
34272
A3^8
1
255:
49618
A3^8
1
271:
70758
A3^8
1
287:
100310
A3^8
1
303:
140616
A3^8
1
319:
195888
A3^8
1
335:
270296
A3^8
1
351:
371070
A3^8
1
367:
505260
A3^8
1
383:
684518
A3^8
1
399:
921142
A3^8
1
415:
1233708
A3^8
1
431:
1642592
A3^8
1
447:
2177684
A3^8
1
463:
2871918
A3^8
1
479:
3772468
A3^8
2
12:
16
A3^8
2
28:
48
A3^8
2
44:
112
A3^8
2
60:
224
A3^8
2
76:
432
A3^8
2
92:
784
A3^8
2
108:
1344
A3^8
2
124:
2256
A3^8
2
140:
3680
A3^8
2
156:
5824
A3^8
2
172:
9072
A3^8
2
188:
13872
A3^8
2
204:
20832
A3^8
2
220:
30912
A3^8
2
236:
45264
A3^8
2
252:
65456
A3^8
2
268:
93744
A3^8
2
284:
132944
A3^8
2
300:
186800
A3^8
2
316:
260400
A3^8
2
332:
360208
A3^8
2
348:
494624
A3^8
2
364:
674784
A3^8
2
380:
914816
A3^8
2
396:
1232784
A3^8
2
412:
1652208
A3^8
2
428:
2202704
A3^8
2
444:
2921856
A3^8
2
460:
3857760
A3^8
2
476:
5070560
A3^8
3
7:
6
A3^8
3
23:
28
A3^8
3
39:
56
A3^8
3
55:
138
A3^8
3
71:
238
A3^8
3
87:
478
A3^8
3
103:
786
A3^8
3
119:
1386
A3^8
3
135:
2212
A3^8
3
151:
3612
A3^8
3
167:
5544
A3^8
3
183:
8666
A3^8
3
199:
12936
A3^8
3
215:
19420
A3^8
3
231:
28348
A3^8
3
247:
41412
A3^8
3
263:
59178
A3^8
3
279:
84530
A3^8
3
295:
118692
A3^8
3
311:
166320
A3^8
3
327:
230092
A3^8
3
343:
317274
A3^8
3
359:
432964
A3^8
3
375:
588966
A3^8
3
391:
794178
A3^8
3
407:
1067220
A3^8
3
423:
1423884
A3^8
3
439:
1893138
A3^8
3
455:
2501434
A3^8
3
471:
3294256
A4^6
1
19:
8
A4^6
1
39:
18
A4^6
1
59:
40
A4^6
1
79:
70
A4^6
1
99:
120
A4^6
1
119:
208
A4^6
1
139:
328
A4^6
1
159:
510
A4^6
1
179:
792
A4^6
1
199:
1180
A4^6
1
219:
1728
A4^6
1
239:
2518
A4^6
1
259:
3600
A4^6
1
279:
5082
A4^6
1
299:
7120
A4^6
1
319:
9838
A4^6
1
339:
13488
A4^6
1
359:
18380
A4^6
1
379:
24792
A4^6
1
399:
33210
A4^6
1
419:
44248
A4^6
1
439:
58538
A4^6
1
459:
76992
A4^6
1
479:
100772
A4^6
1
499:
131160
A4^6
1
519:
169896
A4^6
1
539:
219128
A4^6
1
559:
281322
A4^6
1
579:
359712
A4^6
1
599:
458220
A4^6
2
16:
10
A4^6
2
36:
30
A4^6
2
56:
52
A4^6
2
76:
108
A4^6
2
96:
180
A4^6
2
116:
312
A4^6
2
136:
488
A4^6
2
156:
792
A4^6
2
176:
1180
A4^6
2
196:
1810
A4^6
2
216:
2640
A4^6
2
236:
3868
A4^6
2
256:
5502
A4^6
2
276:
7848
A4^6
2
296:
10912
A4^6
2
316:
15212
A4^6
2
336:
20808
A4^6
2
356:
28432
A4^6
2
376:
38308
A4^6
2
396:
51540
A4^6
2
416:
68520
A4^6
2
436:
90928
A4^6
2
456:
119544
A4^6
2
476:
156728
A4^6
2
496:
203940
A4^6
2
516:
264672
A4^6
2
536:
341188
A4^6
2
556:
438732
A4^6
2
576:
560958
A4^6
2
596:
715312
A4^6
3
11:
8
A4^6
3
31:
22
A4^6
3
51:
48
A4^6
3
71:
90
A4^6
3
91:
160
A4^6
3
111:
270
A4^6
3
131:
440
A4^6
3
151:
700
A4^6
3
171:
1080
A4^6
3
191:
1620
A4^6
3
211:
2408
A4^6
3
231:
3522
A4^6
3
251:
5048
A4^6
3
271:
7172
A4^6
3
291:
10080
A4^6
3
311:
13998
A4^6
3
331:
19272
A4^6
3
351:
26298
A4^6
3
371:
35600
A4^6
3
391:
47862
A4^6
3
411:
63888
A4^6
3
431:
84722
A4^6
3
451:
111728
A4^6
3
471:
146520
A4^6
3
491:
191080
A4^6
3
511:
248008
A4^6
3
531:
320424
A4^6
3
551:
412088
A4^6
3
571:
527800
A4^6
3
591:
673302
A4^6
4
4:
2
A4^6
4
24:
12
A4^6
4
44:
20
A4^6
4
64:
50
A4^6
4
84:
72
A4^6
4
104:
152
A4^6
4
124:
220
A4^6
4
144:
378
A4^6
4
164:
560
A4^6
4
184:
892
A4^6
4
204:
1272
A4^6
4
224:
1940
A4^6
4
244:
2720
A4^6
4
264:
3960
A4^6
4
284:
5500
A4^6
4
304:
7772
A4^6
4
324:
10590
A4^6
4
344:
14668
A4^6
4
364:
19728
A4^6
4
384:
26772
A4^6
4
404:
35624
A4^6
4
424:
47592
A4^6
4
444:
62568
A4^6
4
464:
82568
A4^6
4
484:
107502
A4^6
4
504:
140172
A4^6
4
524:
180940
A4^6
4
544:
233576
A4^6
4
564:
298968
A4^6
4
584:
382632
A6^4
1
27:
4
A6^4
1
55:
6
A6^4
1
83:
10
A6^4
1
111:
20
A6^4
1
139:
30
A6^4
1
167:
42
A6^4
1
195:
68
A6^4
1
223:
96
A6^4
1
251:
130
A6^4
1
279:
188
A6^4
1
307:
258
A6^4
1
335:
350
A6^4
1
363:
474
A6^4
1
391:
624
A6^4
1
419:
826
A6^4
1
447:
1090
A6^4
1
475:
1410
A6^4
1
503:
1814
A6^4
1
531:
2338
A6^4
1
559:
2982
A6^4
1
587:
3774
A6^4
1
615:
4774
A6^4
1
643:
5994
A6^4
1
671:
7494
A6^4
1
699:
9348
A6^4
1
727:
11586
A6^4
1
755:
14320
A6^4
1
783:
17654
A6^4
1
811:
21654
A6^4
1
839:
26488
A6^4
2
24:
4
A6^4
2
52:
12
A6^4
2
80:
20
A6^4
2
108:
32
A6^4
2
136:
48
A6^4
2
164:
80
A6^4
2
192:
108
A6^4
2
220:
168
A6^4
2
248:
232
A6^4
2
276:
328
A6^4
2
304:
444
A6^4
2
332:
620
A6^4
2
360:
812
A6^4
2
388:
1104
A6^4
2
416:
1444
A6^4
2
444:
1904
A6^4
2
472:
2460
A6^4
2
500:
3208
A6^4
2
528:
4080
A6^4
2
556:
5244
A6^4
2
584:
6632
A6^4
2
612:
8400
A6^4
2
640:
10524
A6^4
2
668:
13224
A6^4
2
696:
16408
A6^4
2
724:
20436
A6^4
2
752:
25216
A6^4
2
780:
31120
A6^4
2
808:
38148
A6^4
2
836:
46784
A6^4
3
19:
6
A6^4
3
47:
12
A6^4
3
75:
22
A6^4
3
103:
36
A6^4
3
131:
58
A6^4
3
159:
90
A6^4
3
187:
132
A6^4
3
215:
190
A6^4
3
243:
274
A6^4
3
271:
384
A6^4
3
299:
528
A6^4
3
327:
722
A6^4
3
355:
972
A6^4
3
383:
1300
A6^4
3
411:
1724
A6^4
3
439:
2256
A6^4
3
467:
2938
A6^4
3
495:
3806
A6^4
3
523:
4890
A6^4
3
551:
6244
A6^4
3
579:
7940
A6^4
3
607:
10038
A6^4
3
635:
12620
A6^4
3
663:
15814
A6^4
3
691:
19722
A6^4
3
719:
24490
A6^4
3
747:
30310
A6^4
3
775:
37362
A6^4
3
803:
45908
A6^4
3
831:
56236
A6^4
4
12:
4
A6^4
4
40:
12
A6^4
4
68:
16
A6^4
4
96:
36
A6^4
4
124:
48
A6^4
4
152:
84
A6^4
4
180:
116
A6^4
4
208:
180
A6^4
4
236:
244
A6^4
4
264:
360
A6^4
4
292:
480
A6^4
4
320:
676
A6^4
4
348:
896
A6^4
4
376:
1224
A6^4
4
404:
1588
A6^4
4
432:
2128
A6^4
4
460:
2736
A6^4
4
488:
3588
A6^4
4
516:
4576
A6^4
4
544:
5904
A6^4
4
572:
7448
A6^4
4
600:
9500
A6^4
4
628:
11892
A6^4
4
656:
14992
A6^4
4
684:
18628
A6^4
4
712:
23256
A6^4
4
740:
28688
A6^4
4
768:
35532
A6^4
4
796:
43560
A6^4
4
824:
53528
A6^4
5
3:
2
A6^4
5
31:
6
A6^4
5
59:
14
A6^4
5
87:
22
A6^4
5
115:
36
A6^4
5
143:
56
A6^4
5
171:
82
A6^4
5
199:
126
A6^4
5
227:
182
A6^4
5
255:
250
A6^4
5
283:
354
A6^4
5
311:
490
A6^4
5
339:
656
A6^4
5
367:
882
A6^4
5
395:
1180
A6^4
5
423:
1550
A6^4
5
451:
2028
A6^4
5
479:
2638
A6^4
5
507:
3394
A6^4
5
535:
4362
A6^4
5
563:
5562
A6^4
5
591:
7032
A6^4
5
619:
8886
A6^4
5
647:
11166
A6^4
5
675:
13940
A6^4
5
703:
17358
A6^4
5
731:
21536
A6^4
5
759:
26594
A6^4
5
787:
32742
A6^4
5
815:
40180
A6^4
6
20:
4
A6^4
6
48:
4
A6^4
6
76:
12
A6^4
6
104:
12
A6^4
6
132:
32
A6^4
6
160:
36
A6^4
6
188:
64
A6^4
6
216:
80
A6^4
6
244:
132
A6^4
6
272:
160
A6^4
6
300:
252
A6^4
6
328:
312
A6^4
6
356:
448
A6^4
6
384:
572
A6^4
6
412:
792
A6^4
6
440:
992
A6^4
6
468:
1348
A6^4
6
496:
1680
A6^4
6
524:
2220
A6^4
6
552:
2776
A6^4
6
580:
3600
A6^4
6
608:
4460
A6^4
6
636:
5712
A6^4
6
664:
7044
A6^4
6
692:
8892
A6^4
6
720:
10932
A6^4
6
748:
13656
A6^4
6
776:
16672
A6^4
6
804:
20672
A6^4
6
832:
25116
A12^2
1
51:
2
A12^2
1
103:
2
A12^2
1
155:
0
A12^2
1
207:
2
A12^2
1
259:
2
A12^2
1
311:
4
A12^2
1
363:
6
A12^2
1
415:
6
A12^2
1
467:
8
A12^2
1
519:
12
A12^2
1
571:
14
A12^2
1
623:
14
A12^2
1
675:
20
A12^2
1
727:
24
A12^2
1
779:
28
A12^2
1
831:
36
A12^2
1
883:
42
A12^2
1
935:
50
A12^2
1
987:
62
A12^2
1
1039:
70
A12^2
1
1091:
84
A12^2
1
1143:
102
A12^2
1
1195:
118
A12^2
1
1247:
136
A12^2
1
1299:
162
A12^2
1
1351:
190
A12^2
1
1403:
216
A12^2
1
1455:
254
A12^2
1
1507:
292
A12^2
1
1559:
336
A12^2
2
48:
0
A12^2
2
100:
2
A12^2
2
152:
4
A12^2
2
204:
4
A12^2
2
256:
6
A12^2
2
308:
8
A12^2
2
360:
8
A12^2
2
412:
12
A12^2
2
464:
16
A12^2
2
516:
20
A12^2
2
568:
24
A12^2
2
620:
32
A12^2
2
672:
36
A12^2
2
724:
48
A12^2
2
776:
56
A12^2
2
828:
68
A12^2
2
880:
80
A12^2
2
932:
100
A12^2
2
984:
112
A12^2
2
1036:
140
A12^2
2
1088:
164
A12^2
2
1140:
192
A12^2
2
1192:
224
A12^2
2
1244:
268
A12^2
2
1296:
306
A12^2
2
1348:
364
A12^2
2
1400:
420
A12^2
2
1452:
488
A12^2
2
1504:
560
A12^2
2
1556:
656
A12^2
3
43:
2
A12^2
3
95:
2
A12^2
3
147:
4
A12^2
3
199:
6
A12^2
3
251:
8
A12^2
3
303:
10
A12^2
3
355:
14
A12^2
3
407:
18
A12^2
3
459:
22
A12^2
3
511:
26
A12^2
3
563:
34
A12^2
3
615:
44
A12^2
3
667:
52
A12^2
3
719:
64
A12^2
3
771:
78
A12^2
3
823:
96
A12^2
3
875:
114
A12^2
3
927:
136
A12^2
3
979:
164
A12^2
3
1031:
194
A12^2
3
1083:
230
A12^2
3
1135:
270
A12^2
3
1187:
318
A12^2
3
1239:
374
A12^2
3
1291:
434
A12^2
3
1343:
506
A12^2
3
1395:
592
A12^2
3
1447:
686
A12^2
3
1499:
792
A12^2
3
1551:
914
A12^2
4
36:
2
A12^2
4
88:
4
A12^2
4
140:
4
A12^2
4
192:
8
A12^2
4
244:
8
A12^2
4
296:
12
A12^2
4
348:
16
A12^2
4
400:
22
A12^2
4
452:
24
A12^2
4
504:
36
A12^2
4
556:
40
A12^2
4
608:
52
A12^2
4
660:
64
A12^2
4
712:
80
A12^2
4
764:
92
A12^2
4
816:
116
A12^2
4
868:
136
A12^2
4
920:
168
A12^2
4
972:
196
A12^2
4
1024:
238
A12^2
4
1076:
272
A12^2
4
1128:
332
A12^2
4
1180:
384
A12^2
4
1232:
456
A12^2
4
1284:
528
A12^2
4
1336:
620
A12^2
4
1388:
712
A12^2
4
1440:
840
A12^2
4
1492:
960
A12^2
4
1544:
1120
A12^2
5
27:
2
A12^2
5
79:
4
A12^2
5
131:
6
A12^2
5
183:
6
A12^2
5
235:
10
A12^2
5
287:
14
A12^2
5
339:
16
A12^2
5
391:
22
A12^2
5
443:
30
A12^2
5
495:
36
A12^2
5
547:
46
A12^2
5
599:
58
A12^2
5
651:
68
A12^2
5
703:
86
A12^2
5
755:
106
A12^2
5
807:
124
A12^2
5
859:
152
A12^2
5
911:
184
A12^2
5
963:
216
A12^2
5
1015:
258
A12^2
5
1067:
308
A12^2
5
1119:
362
A12^2
5
1171:
426
A12^2
5
1223:
502
A12^2
5
1275:
584
A12^2
5
1327:
684
A12^2
5
1379:
798
A12^2
5
1431:
920
A12^2
5
1483:
1070
A12^2
5
1535:
1238
A12^2
6
16:
2
A12^2
6
68:
4
A12^2
6
120:
4
A12^2
6
172:
8
A12^2
6
224:
8
A12^2
6
276:
16
A12^2
6
328:
16
A12^2
6
380:
24
A12^2
6
432:
28
A12^2
6
484:
38
A12^2
6
536:
44
A12^2
6
588:
60
A12^2
6
640:
68
A12^2
6
692:
88
A12^2
6
744:
104
A12^2
6
796:
132
A12^2
6
848:
152
A12^2
6
900:
190
A12^2
6
952:
220
A12^2
6
1004:
268
A12^2
6
1056:
312
A12^2
6
1108:
376
A12^2
6
1160:
432
A12^2
6
1212:
520
A12^2
6
1264:
596
A12^2
6
1316:
708
A12^2
6
1368:
812
A12^2
6
1420:
956
A12^2
6
1472:
1092
A12^2
6
1524:
1280
A12^2
7
3:
2
A12^2
7
55:
2
A12^2
7
107:
4
A12^2
7
159:
8
A12^2
7
211:
10
A12^2
7
263:
12
A12^2
7
315:
16
A12^2
7
367:
22
A12^2
7
419:
26
A12^2
7
471:
34
A12^2
7
523:
44
A12^2
7
575:
54
A12^2
7
627:
68
A12^2
7
679:
82
A12^2
7
731:
102
A12^2
7
783:
124
A12^2
7
835:
148
A12^2
7
887:
176
A12^2
7
939:
214
A12^2
7
991:
256
A12^2
7
1043:
300
A12^2
7
1095:
356
A12^2
7
1147:
420
A12^2
7
1199:
494
A12^2
7
1251:
580
A12^2
7
1303:
674
A12^2
7
1355:
786
A12^2
7
1407:
918
A12^2
7
1459:
1060
A12^2
7
1511:
1226
A12^2
8
40:
4
A12^2
8
92:
4
A12^2
8
144:
6
A12^2
8
196:
6
A12^2
8
248:
12
A12^2
8
300:
12
A12^2
8
352:
20
A12^2
8
404:
24
A12^2
8
456:
32
A12^2
8
508:
36
A12^2
8
560:
52
A12^2
8
612:
56
A12^2
8
664:
76
A12^2
8
716:
88
A12^2
8
768:
112
A12^2
8
820:
128
A12^2
8
872:
164
A12^2
8
924:
184
A12^2
8
976:
232
A12^2
8
1028:
268
A12^2
8
1080:
324
A12^2
8
1132:
372
A12^2
8
1184:
452
A12^2
8
1236:
512
A12^2
8
1288:
616
A12^2
8
1340:
704
A12^2
8
1392:
832
A12^2
8
1444:
950
A12^2
8
1496:
1120
A12^2
8
1548:
1268
A12^2
9
23:
2
A12^2
9
75:
4
A12^2
9
127:
4
A12^2
9
179:
6
A12^2
9
231:
8
A12^2
9
283:
12
A12^2
9
335:
14
A12^2
9
387:
20
A12^2
9
439:
26
A12^2
9
491:
30
A12^2
9
543:
40
A12^2
9
595:
50
A12^2
9
647:
60
A12^2
9
699:
74
A12^2
9
751:
90
A12^2
9
803:
108
A12^2
9
855:
134
A12^2
9
907:
158
A12^2
9
959:
188
A12^2
9
1011:
226
A12^2
9
1063:
266
A12^2
9
1115:
314
A12^2
9
1167:
372
A12^2
9
1219:
436
A12^2
9
1271:
508
A12^2
9
1323:
596
A12^2
9
1375:
692
A12^2
9
1427:
802
A12^2
9
1479:
932
A12^2
9
1531:
1074
A12^2
10
4:
2
A12^2
10
56:
0
A12^2
10
108:
4
A12^2
10
160:
4
A12^2
10
212:
8
A12^2
10
264:
8
A12^2
10
316:
12
A12^2
10
368:
12
A12^2
10
420:
20
A12^2
10
472:
20
A12^2
10
524:
32
A12^2
10
576:
34
A12^2
10
628:
48
A12^2
10
680:
52
A12^2
10
732:
72
A12^2
10
784:
78
A12^2
10
836:
104
A12^2
10
888:
116
A12^2
10
940:
148
A12^2
10
992:
164
A12^2
10
1044:
208
A12^2
10
1096:
232
A12^2
10
1148:
288
A12^2
10
1200:
324
A12^2
10
1252:
396
A12^2
10
1304:
444
A12^2
10
1356:
536
A12^2
10
1408:
604
A12^2
10
1460:
720
A12^2
10
1512:
812
A12^2
11
35:
2
A12^2
11
87:
2
A12^2
11
139:
2
A12^2
11
191:
4
A12^2
11
243:
4
A12^2
11
295:
8
A12^2
11
347:
10
A12^2
11
399:
10
A12^2
11
451:
16
A12^2
11
503:
20
A12^2
11
555:
22
A12^2
11
607:
28
A12^2
11
659:
36
A12^2
11
711:
44
A12^2
11
763:
54
A12^2
11
815:
64
A12^2
11
867:
76
A12^2
11
919:
94
A12^2
11
971:
114
A12^2
11
1023:
130
A12^2
11
1075:
156
A12^2
11
1127:
188
A12^2
11
1179:
216
A12^2
11
1231:
254
A12^2
11
1283:
300
A12^2
11
1335:
346
A12^2
11
1387:
404
A12^2
11
1439:
470
A12^2
11
1491:
542
A12^2
11
1543:
630
A12^2
12
12:
0
A12^2
12
64:
2
A12^2
12
116:
0
A12^2
12
168:
4
A12^2
12
220:
0
A12^2
12
272:
4
A12^2
12
324:
2
A12^2
12
376:
8
A12^2
12
428:
4
A12^2
12
480:
12
A12^2
12
532:
8
A12^2
12
584:
16
A12^2
12
636:
16
A12^2
12
688:
24
A12^2
12
740:
20
A12^2
12
792:
36
A12^2
12
844:
32
A12^2
12
896:
48
A12^2
12
948:
48
A12^2
12
1000:
68
A12^2
12
1052:
68
A12^2
12
1104:
96
A12^2
12
1156:
98
A12^2
12
1208:
128
A12^2
12
1260:
136
A12^2
12
1312:
176
A12^2
12
1364:
184
A12^2
12
1416:
240
A12^2
12
1468:
252
A12^2
12
1520:
312
Definition
The integer $C_X(r,s)$ is the coefficient of $q^{s/(4m_X)}$ in the identity-class component $H^X_{1A,r}$ of the umbral mock modular form, in the module normalization of [3], equation (1.4).
Parameters
$X$
—   Niemeier root system (a nonempty Niemeier root system)
$r$
—   vector component ($r\in I^X$, the independent component set specified in (6))
$s$
—   positive exponent numerator ($s>0$ and $s\equiv-r^2\pmod{4m_X}$)
Formulas
(1)
For the six selected root systems, $H^X_{1A,r}(\tau)=-2\delta_{r,1}q^{-1/(4m_X)}+\sum_{s>0,\ s\equiv-r^2\ (\mathrm{mod}\ 4m_X)}C_X(r,s)q^{s/(4m_X)}$ [1], equation (2.64). The polar coefficient is part of this normalization and is not a positive-exponent entry.
(2)
Write $f_j(\tau,z)=\theta_j(\tau,z)/\theta_j(\tau,0)$ for $j=2,3,4$, and abbreviate $\varphi_m=\varphi^{(m)}_1$. Then $\varphi_2=4(f_2^2+f_3^2+f_4^2)$, $\varphi_3=2(f_2^2f_3^2+f_3^2f_4^2+f_4^2f_2^2)$, $\varphi_4=4f_2^2f_3^2f_4^2$, $\varphi_5=(\varphi_4\varphi_2-\varphi_3^2)/4$, $\varphi_7=\varphi_3\varphi_5-\varphi_4^2$, $\varphi_9=\varphi_3\varphi_7-\varphi_5^2$, and $\varphi_{13}=\varphi_5\varphi_9-2\varphi_7^2$. Set $Z^{(m)}=2\varphi_m$ for $m\in\{2,3,4,5,7,13\}$; then $Z^{(m)}(\tau,0)=24/(m-1)$ [1], equations (2.49)–(2.51) and (2.62). The auxiliary $\varphi_9$ is used only in this construction.
(3)
Put $\mathcal A_m(\tau,z)=\sum_{k\in\mathbb Z}q^{mk^2}y^{2mk}(yq^k+1)/(yq^k-1)$ and $\theta_{m,r}(\tau,z)=\sum_{k\in\mathbb Z}q^{(2mk+r)^2/(4m)}y^{2mk+r}$. For $X=A_{m-1}^{24/(m-1)}$ in the selected six cases, $\mathcal A_1 Z^{(m)}-\dfrac{24}{m-1}\mathcal A_m=\sum_{r=1}^{m-1}H^X_{1A,r}(\theta_{m,-r}-\theta_{m,r})$ [1], equations (2.19)–(2.25) and (2.63). This identity removes the meromorphic polar part before taking the theta coefficients.
(4)
$Z_{K3}$ satisfies $Z_{K3}(\tau,z)=\dfrac{\theta_1(\tau,z)^2}{\eta(\tau)^3}\bigl(24\mu(\tau,z)+H^{A_1^{24}}_{1A,1}(\tau)\bigr)$, where $\mu(\tau,z)=\dfrac{-iy^{1/2}}{\theta_1(\tau,z)}\sum_{j\in\mathbb Z}\dfrac{(-1)^j q^{j(j+1)/2}y^j}{1-yq^j}$ and $\eta(\tau)=q^{1/24}\prod_{n\geq1}(1-q^n)$. The identity is evaluated away from the poles of $\mu$ and extends holomorphically after multiplication [4], equations (1.3)–(1.9).
Comments
(5)
The common Coxeter number of the simple components of $X$ is denoted by $m_X$. The acting class is fixed to the identity $1A$. By [3], Theorem 1.1, $C_X(r,s)$ is the dimension of the graded umbral module component $\check K^X_{r,s/(4m_X)}$. These are full dimensions: the Mathieu component begins $90,462,1540$, rather than $45,231,770$. The negative discriminant $D$ used in [3] corresponds to $s=-D$ here.
(6)
For a root system $X$ with an $A$-type component, $I^X=\{1,\ldots,m_X-1\}$. If $X$ has a $D$-type component and no $A$-type component, $I^X=\{1,3,\ldots,m_X/2\}$. For $X=E_6^4$ and $E_8^3$, the independent sets are $\{1,4,5\}$ and $\{1,7\}$, respectively [3], following equation (1.4). For the six pure $A$ cases in this table, $H^X_{1A,-r}=-H^X_{1A,r}$ and the components at residues $0$ and $m_X$ vanish, so each independent component is listed once.
(7)
The six root systems $A_1^{24},A_2^{12},A_3^8,A_4^6,A_6^4,A_{12}^2$ have Coxeter numbers $2,3,4,5,7,13$ and umbral groups $M_{24}$, $2.M_{12}$, $2.\operatorname{AGL}_3(2)$, $\operatorname{GL}_2(5)/\{\pm I\}$, $\operatorname{SL}_2(3)$ and $C_4$, respectively [2], Table 3. The group $G^X=\operatorname{Aut}(N^X)/W(X)$ is the quotient of the Niemeier lattice automorphism group by its root reflection group. The Niemeier automorphism-order table concerns the full automorphism groups.
(8)
These six cases are precisely the nonzero extremal weight-zero Jacobi cases with $m_X-1$ dividing $12$ [1], Theorem 2.2. The other $17$ Niemeier root systems require additional constructions. Twined traces, with a nonidentity acting class, are not included. The labels such as A1^24 name the root system $A_1^{24}$; the exponent denotes the number of copies.
(9)
In [1], equation (2.18), the antisymmetric theta basis is $\theta_{m,-r}-\theta_{m,r}$. This is the basis used in (3). The later construction in [2], equation (4.31), uses the opposite sign for the meromorphic Jacobi form and the basis $\theta_{m,r}$; it gives the same $H^X_{1A,r}$ and the same coefficients.
(10)
The $A_1^{24}$ member is Mathieu moonshine. Its relation to the K3 elliptic genus is the character decomposition in (4). The Fourier coefficients of the genus and the coefficients of $H^{A_1^{24}}_{1A,1}$ are different quantities related by this decomposition. The first five positive Mathieu coefficients are twice the corresponding $M_{24}$ irreducible character degrees [4], equation (1.12); later graded pieces need not be irreducible.
(11)
In the formulas, $\operatorname{Im}\tau>0$, $y=e^{2\pi iz}$, and the standard Jacobi theta functions use $(\tau,z)$ with $q=e^{2\pi i\tau}$. Thus the theta nome is $q^{1/2}$, not $q$. Fractional powers mean $q^a=e^{2\pi ia\tau}$ and $y^a=e^{2\pi iaz}$. The odd theta convention is $\theta_1(\tau,z)=-i\sum_{j\in\mathbb Z}(-1)^j q^{(j+1/2)^2/2}y^{j+1/2}$ [4]; it agrees with the K3 companion.
Programs
(P1)
Sage
# After downloading generate.py from this table
from generate import UmbralCoefficients, coefficient
generator = UmbralCoefficients()
[(p, generator.value(p, 100)) for p in generator.enumerate()]
# One positive coefficient: X=A1^24, r=1, s=7
coefficient(2, 1, 7)
References
[1]
Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A. Harvey, Umbral Moonshine, Communications in Number Theory and Physics 8 (2014), 101–242. Sections 2.3 and 2.5; Appendix C, Tables 20–47. (arXiv)
[2]
Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A. Harvey, Umbral Moonshine and the Niemeier Lattices, Research in the Mathematical Sciences 1 (2014), article 3. Table 3, Section 4.3 and Appendix A.4. (arXiv)
[3]
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, Research in the Mathematical Sciences 2 (2015), article 26. Equation (1.4) and Theorem 1.1. (arXiv) (doi)
[4]
Tohru Eguchi, Hirosi Ooguri and Yuji Tachikawa, Notes on the K3 Surface and the Mathieu group M24, Experimental Mathematics 20 (2011), 91–96. Equations (1.3)–(1.12). (arXiv) (doi)
Links
Similar tables
K3 elliptic genus coefficients —   The Mathieu member is the mock modular coefficient series in the character decomposition of this Jacobi form.
Degrees of the irreducible characters of the sporadic simple groups —   These give dimensions of irreducible constituents; the present coefficients are dimensions of whole graded module components.
Orders of Niemeier lattice automorphism groups —   The umbral groups are the quotients of these full groups by their root reflection groups.
Fourier coefficients of monstrous McKay–Thompson series —   This is the analogous Monster graded-trace family, with different modular objects and acting groups.
Data properties
Entries are of type: integer
How well the digits are known: exact
Table is complete: no (it holds all $1\leq r<m_X$ and all admissible $0<s\leq120m_X$ for $X=A_1^{24},A_2^{12},A_3^8,A_4^6,A_6^4,A_{12}^2$, namely every positive exponent $s/(4m_X)\leq30$. This is $30$ successive positive grades in each of the $28$ independent components, including structural zeros. The six cases form the complete extremal-Jacobi subfamily with $m_X-1$ dividing $12$; the other $17$ Niemeier cases are deferred. The noncanonical grade bound gives a shared initial reference segment across the six systems within the published coefficient controls, rather than extending only the familiar Mathieu component. All entries are exact and the longest integer has $10$ characters)
How they were obtained:

Computed with exact rational Laurent polynomials and finite formal series using (2) and (3). Rational division is performed in Sage over $\mathbb Q$; every resulting coefficient is checked to be an integer. The generator uses no floating-point approximation or rounding.

more

All $840$ values agree with the identity columns of [1], Appendix C, Tables 20–47, also available in [5]. The source table positions and extraction are attached. An independent scalar implementation of the Eguchi–Ooguri–Tachikawa character relation at $y=-1$, using the stored K3 coefficients, reproduces all $30$ positive Mathieu grades. The six finite-part expansions are reconstructed from their component coefficients at $6174$ Fourier positions through $q^{30}$, including polar and vanishing terms. The $m=2$ Jacobi form agrees with the K3 companion at $525$ Fourier positions, and the first five Mathieu coefficients give twice the relevant $M_{24}$ irreducible degrees. Known-answer controls and deliberate sign, scale and coefficient errors precede the comparisons. The attached checker can repeat these tests on values read from the database.



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