Properties

Label 12.17.d.a
Level 12
Weight 17
Character orbit 12.d
Analytic conductor 19.479
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(19.4789452628\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{106}\cdot 3^{51} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -12 - \beta_{1} ) q^{2} \) \( + ( \beta_{1} - \beta_{3} ) q^{3} \) \( + ( 8542 + 11 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{4} \) \( + ( 22150 + 44 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 97595 - 11 \beta_{1} + 12 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{6} \) \( + ( -186 - 689 \beta_{1} + 171 \beta_{3} - 22 \beta_{4} + \beta_{8} ) q^{7} \) \( + ( 914153 - 8991 \beta_{1} - 10 \beta_{2} - 550 \beta_{3} + 9 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{8} \) \( -14348907 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -12 - \beta_{1} ) q^{2} \) \( + ( \beta_{1} - \beta_{3} ) q^{3} \) \( + ( 8542 + 11 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{4} \) \( + ( 22150 + 44 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 97595 - 11 \beta_{1} + 12 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{6} \) \( + ( -186 - 689 \beta_{1} + 171 \beta_{3} - 22 \beta_{4} + \beta_{8} ) q^{7} \) \( + ( 914153 - 8991 \beta_{1} - 10 \beta_{2} - 550 \beta_{3} + 9 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{8} \) \( -14348907 q^{9} \) \( + ( -3117758 - 19120 \beta_{1} - 63 \beta_{2} + 5114 \beta_{3} + 76 \beta_{4} - 2 \beta_{6} - 4 \beta_{8} + \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{10} \) \( + ( 80143 + 209254 \beta_{1} + \beta_{2} + 4401 \beta_{3} - 91 \beta_{4} + 61 \beta_{5} + 3 \beta_{6} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + \beta_{11} + 4 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} + 3 \beta_{15} ) q^{11} \) \( + ( 26536542 - 107099 \beta_{1} + 102 \beta_{2} - 9355 \beta_{3} - 42 \beta_{4} - 30 \beta_{5} + 18 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{15} ) q^{12} \) \( + ( -57027026 - 1001481 \beta_{1} - 529 \beta_{2} - 6368 \beta_{3} - 4753 \beta_{4} + 167 \beta_{5} - 196 \beta_{6} + 2 \beta_{7} - 7 \beta_{8} + 2 \beta_{9} - 7 \beta_{11} - 17 \beta_{12} + 10 \beta_{13} + 2 \beta_{14} + 8 \beta_{15} ) q^{13} \) \( + ( -50363244 + 20032 \beta_{1} + 783 \beta_{2} + 22337 \beta_{3} - 884 \beta_{4} - 291 \beta_{5} + 160 \beta_{6} + 30 \beta_{7} + 27 \beta_{8} - 2 \beta_{9} - 57 \beta_{10} - 7 \beta_{11} - 16 \beta_{12} + 7 \beta_{13} + 20 \beta_{14} - 15 \beta_{15} ) q^{14} \) \( + ( 417364 + 1132565 \beta_{1} + 66 \beta_{2} - 15003 \beta_{3} + 4204 \beta_{4} + 402 \beta_{5} + 127 \beta_{6} + 30 \beta_{7} + 36 \beta_{8} - 6 \beta_{9} - 24 \beta_{10} + 27 \beta_{11} - 21 \beta_{12} - 12 \beta_{13} - 15 \beta_{15} ) q^{15} \) \( + ( -132044862 - 836669 \beta_{1} - 3684 \beta_{2} + 125090 \beta_{3} - 8504 \beta_{4} - 1454 \beta_{5} - 1036 \beta_{6} - 50 \beta_{7} + 6 \beta_{8} - 9 \beta_{9} + 92 \beta_{10} + 52 \beta_{11} + 7 \beta_{12} + 40 \beta_{13} - 2 \beta_{14} + 64 \beta_{15} ) q^{16} \) \( + ( 761554538 - 9218382 \beta_{1} + 4645 \beta_{2} - 67235 \beta_{3} + 20878 \beta_{4} + 2398 \beta_{5} + 1553 \beta_{6} + 37 \beta_{7} + 13 \beta_{8} + 96 \beta_{9} - 8 \beta_{10} + 3 \beta_{11} - 16 \beta_{12} - 48 \beta_{13} + 31 \beta_{14} - 61 \beta_{15} ) q^{17} \) \( + ( 172186884 + 14348907 \beta_{1} ) q^{18} \) \( + ( 8581911 + 22460278 \beta_{1} + 41 \beta_{2} + 369157 \beta_{3} - 57491 \beta_{4} + 5829 \beta_{5} - 47 \beta_{6} - 271 \beta_{7} - 861 \beta_{8} + 98 \beta_{9} + 92 \beta_{10} - 113 \beta_{11} + 6 \beta_{12} + 174 \beta_{13} - 159 \beta_{14} - 83 \beta_{15} ) q^{19} \) \( + ( 64499524 + 3701318 \beta_{1} + 28656 \beta_{2} + 1193676 \beta_{3} - 18110 \beta_{4} - 5648 \beta_{5} + 6856 \beta_{6} - 320 \beta_{7} - 432 \beta_{8} + 332 \beta_{9} - 96 \beta_{10} - 320 \beta_{11} + 300 \beta_{12} + 248 \beta_{13} - 136 \beta_{14} - 184 \beta_{15} ) q^{20} \) \( + ( 2485890144 - 5337465 \beta_{1} - 7800 \beta_{2} - 34866 \beta_{3} + 12291 \beta_{4} + 987 \beta_{5} - 798 \beta_{6} - 96 \beta_{7} + 135 \beta_{8} + 66 \beta_{9} + 336 \beta_{10} + 459 \beta_{11} + 15 \beta_{12} + 42 \beta_{13} - 324 \beta_{14} + 282 \beta_{15} ) q^{21} \) \( + ( 13542974764 - 2536516 \beta_{1} + 14962 \beta_{2} - 325538 \beta_{3} + 237152 \beta_{4} + 1686 \beta_{5} + 7428 \beta_{6} + 356 \beta_{7} + 2074 \beta_{8} + 532 \beta_{9} + 738 \beta_{10} + 1630 \beta_{11} - 912 \beta_{12} + 130 \beta_{13} - 616 \beta_{14} - 1010 \beta_{15} ) q^{22} \) \( + ( 4500616 + 10367190 \beta_{1} - 3148 \beta_{2} + 1753114 \beta_{3} + 133096 \beta_{4} + 5444 \beta_{5} - 8756 \beta_{6} - 460 \beta_{7} + 3998 \beta_{8} + 440 \beta_{9} - 272 \beta_{10} + 516 \beta_{11} + 128 \beta_{12} + 2168 \beta_{13} + 68 \beta_{14} - 1396 \beta_{15} ) q^{23} \) \( + ( -7005780091 - 25944992 \beta_{1} - 29154 \beta_{2} - 1080300 \beta_{3} - 121873 \beta_{4} + 2847 \beta_{5} - 12685 \beta_{6} - 606 \beta_{7} - 117 \beta_{8} + 861 \beta_{9} + 933 \beta_{10} - 1188 \beta_{11} - 1347 \beta_{12} + 1344 \beta_{13} - 486 \beta_{14} - 264 \beta_{15} ) q^{24} \) \( + ( 12946423803 + 125772574 \beta_{1} + 219608 \beta_{2} + 820682 \beta_{3} - 685514 \beta_{4} - 21354 \beta_{5} + 45834 \beta_{6} + 2070 \beta_{7} - 1952 \beta_{8} + 2540 \beta_{9} + 464 \beta_{10} + 204 \beta_{11} - 1750 \beta_{12} + 908 \beta_{13} - 198 \beta_{14} - 474 \beta_{15} ) q^{25} \) \( + ( 65928698412 + 49265754 \beta_{1} + 54250 \beta_{2} + 8705396 \beta_{3} - 818728 \beta_{4} + 78064 \beta_{5} + 17420 \beta_{6} - 32 \beta_{7} - 11176 \beta_{8} + 1288 \beta_{9} - 3206 \beta_{10} + 112 \beta_{11} - 1970 \beta_{12} + 2284 \beta_{13} - 502 \beta_{14} + 2112 \beta_{15} ) q^{26} \) \( + ( -14348907 \beta_{1} + 14348907 \beta_{3} ) q^{27} \) \( + ( -95354770040 + 66542520 \beta_{1} + 113512 \beta_{2} - 1867244 \beta_{3} - 164568 \beta_{4} + 133672 \beta_{5} + 38736 \beta_{6} + 764 \beta_{7} - 13088 \beta_{8} + 1264 \beta_{9} - 84 \beta_{10} + 5280 \beta_{11} - 968 \beta_{12} - 20 \beta_{13} + 6184 \beta_{14} - 492 \beta_{15} ) q^{28} \) \( + ( 20547209430 + 178354874 \beta_{1} - 483929 \beta_{2} + 1319718 \beta_{3} + 1070942 \beta_{4} - 34002 \beta_{5} - 12666 \beta_{6} + 1346 \beta_{7} + 1084 \beta_{8} + 8524 \beta_{9} + 7760 \beta_{10} + 1488 \beta_{11} - 9030 \beta_{12} - 2180 \beta_{13} + 3182 \beta_{14} - 246 \beta_{15} ) q^{29} \) \( + ( 74517805026 - 11662870 \beta_{1} - 123693 \beta_{2} + 2986693 \beta_{3} + 1375728 \beta_{4} + 82617 \beta_{5} - 9978 \beta_{6} - 1050 \beta_{7} + 31023 \beta_{8} + 1758 \beta_{9} - 1509 \beta_{10} - 2619 \beta_{11} + 2616 \beta_{12} + 1131 \beta_{13} + 5508 \beta_{14} + 813 \beta_{15} ) q^{30} \) \( + ( -133333600 - 386512097 \beta_{1} - 20194 \beta_{2} + 28834949 \beta_{3} - 1963608 \beta_{4} - 122554 \beta_{5} - 75534 \beta_{6} + 3582 \beta_{7} - 64669 \beta_{8} - 1132 \beta_{9} - 25400 \beta_{10} + 12886 \beta_{11} + 1952 \beta_{12} - 2204 \beta_{13} + 3270 \beta_{14} - 10622 \beta_{15} ) q^{31} \) \( + ( -322327084176 + 208882958 \beta_{1} + 260064 \beta_{2} + 39858100 \beta_{3} - 1324396 \beta_{4} + 113032 \beta_{5} + 115724 \beta_{6} + 1652 \beta_{7} + 72960 \beta_{8} - 2270 \beta_{9} - 10876 \beta_{10} + 9400 \beta_{11} - 14142 \beta_{12} + 2944 \beta_{13} - 15900 \beta_{14} + 8816 \beta_{15} ) q^{32} \) \( + ( 42494903820 + 54463092 \beta_{1} + 416175 \beta_{2} + 184257 \beta_{3} - 785472 \beta_{4} + 19920 \beta_{5} + 228957 \beta_{6} + 7269 \beta_{7} - 3897 \beta_{8} + 3660 \beta_{9} - 50376 \beta_{10} - 15147 \beta_{11} + 5466 \beta_{12} + 10236 \beta_{13} - 4617 \beta_{14} - 1137 \beta_{15} ) q^{33} \) \( + ( 591730602324 - 895166918 \beta_{1} + 1195246 \beta_{2} - 46055924 \beta_{3} - 8725272 \beta_{4} - 405344 \beta_{5} + 37220 \beta_{6} - 27072 \beta_{7} - 7736 \beta_{8} - 656 \beta_{9} + 37710 \beta_{10} - 13408 \beta_{11} - 13278 \beta_{12} + 10788 \beta_{13} + 1198 \beta_{14} + 512 \beta_{15} ) q^{34} \) \( + ( 612366581 + 1701723686 \beta_{1} - 528821 \beta_{2} - 67561233 \beta_{3} + 3341159 \beta_{4} + 357087 \beta_{5} - 1172059 \beta_{6} + 7675 \beta_{7} + 126075 \beta_{8} + 18466 \beta_{9} + 41684 \beta_{10} - 23441 \beta_{11} + 12472 \beta_{12} + 15466 \beta_{13} - 8593 \beta_{14} - 13491 \beta_{15} ) q^{35} \) \( + ( -122568363594 - 157837977 \beta_{1} - 28697814 \beta_{3} + 14348907 \beta_{4} ) q^{36} \) \( + ( -509622538674 - 750178063 \beta_{1} + 533123 \beta_{2} - 5089420 \beta_{3} - 12851807 \beta_{4} + 197321 \beta_{5} + 983656 \beta_{6} + 10338 \beta_{7} - 30205 \beta_{8} - 29170 \beta_{9} + 159392 \beta_{10} + 68827 \beta_{11} + 57481 \beta_{12} - 31066 \beta_{13} - 11222 \beta_{14} - 36956 \beta_{15} ) q^{37} \) \( + ( 1457308184804 - 277084076 \beta_{1} - 174934 \beta_{2} - 42281370 \beta_{3} + 25251856 \beta_{4} - 968162 \beta_{5} + 509308 \beta_{6} - 45740 \beta_{7} - 249678 \beta_{8} - 48684 \beta_{9} - 23046 \beta_{10} - 13050 \beta_{11} + 37856 \beta_{12} - 30342 \beta_{13} - 15944 \beta_{14} + 64982 \beta_{15} ) q^{38} \) \( + ( 721270248 + 1857959384 \beta_{1} - 456444 \beta_{2} + 69873658 \beta_{3} + 6187632 \beta_{4} + 437868 \beta_{5} - 944283 \beta_{6} + 7200 \beta_{7} + 103815 \beta_{8} + 15030 \beta_{9} + 83232 \beta_{10} - 49059 \beta_{11} + 7731 \beta_{12} - 6768 \beta_{13} - 14742 \beta_{14} + 18999 \beta_{15} ) q^{39} \) \( + ( -1792053625954 + 111262390 \beta_{1} - 583116 \beta_{2} - 292120548 \beta_{3} - 3163250 \beta_{4} - 2309966 \beta_{5} + 755198 \beta_{6} - 63760 \beta_{7} - 482478 \beta_{8} - 97608 \beta_{9} + 100610 \beta_{10} + 15648 \beta_{11} - 32328 \beta_{12} - 50112 \beta_{13} - 9104 \beta_{14} + 43008 \beta_{15} ) q^{40} \) \( + ( -1596806955078 - 2009141070 \beta_{1} - 5246901 \beta_{2} - 14375897 \beta_{3} + 9943462 \beta_{4} + 784726 \beta_{5} + 2878419 \beta_{6} + 51623 \beta_{7} - 26309 \beta_{8} - 32984 \beta_{9} - 320728 \beta_{10} - 126915 \beta_{11} + 158220 \beta_{12} - 10184 \beta_{13} + 21845 \beta_{14} - 121767 \beta_{15} ) q^{41} \) \( + ( 318054583146 - 2534942766 \beta_{1} - 6774663 \beta_{2} + 35946330 \beta_{3} - 4326708 \beta_{4} - 229200 \beta_{5} - 661266 \beta_{6} - 25248 \beta_{7} + 641052 \beta_{8} - 20568 \beta_{9} + 26889 \beta_{10} - 44496 \beta_{11} - 79521 \beta_{12} + 47694 \beta_{13} - 61479 \beta_{14} + 66624 \beta_{15} ) q^{42} \) \( + ( -728569851 - 1217652134 \beta_{1} - 3111509 \beta_{2} - 737638849 \beta_{3} + 1618791 \beta_{4} - 1496257 \beta_{5} - 7203721 \beta_{6} + 251123 \beta_{7} - 679307 \beta_{8} + 17182 \beta_{9} - 183468 \beta_{10} + 76721 \beta_{11} + 69614 \beta_{12} - 209366 \beta_{13} + 37419 \beta_{14} - 111085 \beta_{15} ) q^{43} \) \( + ( -720097371528 - 13369896772 \beta_{1} + 4584088 \beta_{2} + 151229444 \beta_{3} - 2890152 \beta_{4} - 73176 \beta_{5} - 271720 \beta_{6} - 248580 \beta_{7} + 1737792 \beta_{8} - 150572 \beta_{9} - 61796 \beta_{10} - 247104 \beta_{11} + 45404 \beta_{12} - 23372 \beta_{13} + 157488 \beta_{14} + 287564 \beta_{15} ) q^{44} \) \( + ( -317828290050 - 631351908 \beta_{1} - 14348907 \beta_{2} ) q^{45} \) \( + ( 620225025840 - 384253256 \beta_{1} + 27623262 \beta_{2} - 575567902 \beta_{3} + 6003272 \beta_{4} + 2679034 \beta_{5} + 4321704 \beta_{6} + 22716 \beta_{7} + 734326 \beta_{8} - 406596 \beta_{9} - 57522 \beta_{10} + 380210 \beta_{11} - 77408 \beta_{12} - 310962 \beta_{13} + 180008 \beta_{14} + 389282 \beta_{15} ) q^{46} \) \( + ( -2434317380 - 8254215550 \beta_{1} - 7772840 \beta_{2} + 1663519146 \beta_{3} - 64088412 \beta_{4} - 4521800 \beta_{5} - 17777596 \beta_{6} + 347176 \beta_{7} - 375262 \beta_{8} + 88984 \beta_{9} - 224224 \beta_{10} - 42124 \beta_{11} + 323860 \beta_{12} - 397040 \beta_{13} + 80640 \beta_{14} - 135844 \beta_{15} ) q^{47} \) \( + ( 1830525743454 + 6774791671 \beta_{1} + 16717668 \beta_{2} + 174573530 \beta_{3} - 21611772 \beta_{4} + 2051958 \beta_{5} + 1047936 \beta_{6} - 109362 \beta_{7} - 1592406 \beta_{8} - 102801 \beta_{9} - 67296 \beta_{10} - 178956 \beta_{11} - 4929 \beta_{12} + 64920 \beta_{13} + 16686 \beta_{14} + 327120 \beta_{15} ) q^{48} \) \( + ( -5818041363143 + 32670711486 \beta_{1} - 7811370 \beta_{2} + 229675636 \beta_{3} - 168760114 \beta_{4} - 5665522 \beta_{5} + 16957740 \beta_{6} + 585208 \beta_{7} - 643538 \beta_{8} + 254004 \beta_{9} + 15872 \beta_{10} - 302106 \beta_{11} + 90758 \beta_{12} - 33932 \beta_{13} + 287376 \beta_{14} - 690468 \beta_{15} ) q^{49} \) \( + ( -8355735068332 - 12627618411 \beta_{1} - 22793716 \beta_{2} - 2419763048 \beta_{3} + 100528720 \beta_{4} + 9426144 \beta_{5} - 558168 \beta_{6} - 628800 \beta_{7} - 1341744 \beta_{8} - 87536 \beta_{9} + 76588 \beta_{10} - 268320 \beta_{11} - 769180 \beta_{12} + 231016 \beta_{13} + 73548 \beta_{14} + 294272 \beta_{15} ) q^{50} \) \( + ( -3794461601 - 9149895916 \beta_{1} - 4027263 \beta_{2} - 835716945 \beta_{3} + 151595269 \beta_{4} - 2479107 \beta_{5} - 9298055 \beta_{6} + 358473 \beta_{7} + 926379 \beta_{8} + 100722 \beta_{9} + 4476 \beta_{10} + 188487 \beta_{11} - 11514 \beta_{12} + 302910 \beta_{13} + 78489 \beta_{14} - 389931 \beta_{15} ) q^{51} \) \( + ( 18865898525100 - 68897561594 \beta_{1} - 46247840 \beta_{2} + 1033793204 \beta_{3} + 95221622 \beta_{4} + 7658080 \beta_{5} + 9518288 \beta_{6} - 685056 \beta_{7} - 895200 \beta_{8} - 119816 \beta_{9} + 546240 \beta_{10} + 307328 \beta_{11} - 798024 \beta_{12} + 432432 \beta_{13} - 347984 \beta_{14} + 833360 \beta_{15} ) q^{52} \) \( + ( -5413202644010 + 52591983202 \beta_{1} + 11196651 \beta_{2} + 341781962 \beta_{3} + 9784502 \beta_{4} - 6815674 \beta_{5} + 34123322 \beta_{6} + 622086 \beta_{7} - 235360 \beta_{8} + 707020 \beta_{9} - 180816 \beta_{10} + 880044 \beta_{11} + 854522 \beta_{12} - 111812 \beta_{13} - 726534 \beta_{14} - 448282 \beta_{15} ) q^{53} \) \( + ( -1400381578665 + 157837977 \beta_{1} - 172186884 \beta_{3} - 28697814 \beta_{4} + 14348907 \beta_{6} ) q^{54} \) \( + ( 15446705962 + 37677030848 \beta_{1} - 28005682 \beta_{2} + 3614890938 \beta_{3} + 221427886 \beta_{4} + 5491318 \beta_{5} - 63718750 \beta_{6} + 4270 \beta_{7} + 1338866 \beta_{8} + 1140212 \beta_{9} + 1344648 \beta_{10} - 723866 \beta_{11} + 808960 \beta_{12} + 1790276 \beta_{13} - 382794 \beta_{14} - 1556558 \beta_{15} ) q^{55} \) \( + ( 33218233038292 + 91059921688 \beta_{1} + 107953272 \beta_{2} + 2703404320 \beta_{3} + 147906828 \beta_{4} + 6499484 \beta_{5} + 12163900 \beta_{6} - 1563560 \beta_{7} + 3941420 \beta_{8} - 17956 \beta_{9} + 552100 \beta_{10} + 1416144 \beta_{11} - 955044 \beta_{12} - 58816 \beta_{13} - 553160 \beta_{14} + 377184 \beta_{15} ) q^{56} \) \( + ( 3046056760188 + 7959995082 \beta_{1} + 50317569 \beta_{2} + 47463273 \beta_{3} - 340214922 \beta_{4} + 511686 \beta_{5} + 24704397 \beta_{6} + 901521 \beta_{7} - 1076679 \beta_{8} + 573984 \beta_{9} + 1582488 \beta_{10} + 158679 \beta_{11} - 186480 \beta_{12} - 183024 \beta_{13} + 392931 \beta_{14} - 955593 \beta_{15} ) q^{57} \) \( + ( -11870431464578 - 19027195700 \beta_{1} + 210651223 \beta_{2} - 1416048682 \beta_{3} + 135678100 \beta_{4} - 13978016 \beta_{5} + 13969554 \beta_{6} - 2525504 \beta_{7} + 2323620 \beta_{8} + 744016 \beta_{9} + 1295543 \beta_{10} - 1766048 \beta_{11} - 1424871 \beta_{12} + 2930770 \beta_{13} - 1188009 \beta_{14} + 1033600 \beta_{15} ) q^{58} \) \( + ( 64222672632 + 164719972020 \beta_{1} - 31154120 \beta_{2} + 5394843020 \beta_{3} - 1015059704 \beta_{4} + 34668120 \beta_{5} - 76933204 \beta_{6} + 192232 \beta_{7} - 3785308 \beta_{8} + 1016776 \beta_{9} - 3719392 \beta_{10} + 1275548 \beta_{11} + 930604 \beta_{12} + 2704 \beta_{13} - 670320 \beta_{14} - 3309100 \beta_{15} ) q^{59} \) \( + ( 16748903608948 - 77925121702 \beta_{1} + 165607332 \beta_{2} - 652210458 \beta_{3} + 4628548 \beta_{4} - 12751620 \beta_{5} + 16228708 \beta_{6} + 784410 \beta_{7} + 904032 \beta_{8} + 10302 \beta_{9} + 1377546 \beta_{10} + 1281312 \beta_{11} - 1620150 \beta_{12} + 644142 \beta_{13} - 165240 \beta_{14} - 865326 \beta_{15} ) q^{60} \) \( + ( 29684663078670 - 178093606387 \beta_{1} - 102747241 \beta_{2} - 1216537392 \beta_{3} - 281314179 \beta_{4} + 45879685 \beta_{5} + 65658628 \beta_{6} + 1995574 \beta_{7} - 1027125 \beta_{8} + 2923430 \beta_{9} - 2882240 \beta_{10} + 470059 \beta_{11} - 383619 \beta_{12} + 994398 \beta_{13} - 1315274 \beta_{14} - 256088 \beta_{15} ) q^{61} \) \( + ( -26192388020308 + 2997030504 \beta_{1} - 57909849 \beta_{2} - 3199511991 \beta_{3} - 498568516 \beta_{4} - 37584795 \beta_{5} + 14446520 \beta_{6} - 3210674 \beta_{7} + 6547603 \beta_{8} - 610418 \beta_{9} - 793889 \beta_{10} + 2480225 \beta_{11} + 1713808 \beta_{12} - 1736609 \beta_{13} + 347700 \beta_{14} + 3142745 \beta_{15} ) q^{62} \) \( + ( 2668896702 + 9886396923 \beta_{1} - 2453663097 \beta_{3} + 315675954 \beta_{4} - 14348907 \beta_{8} ) q^{63} \) \( + ( 19252448402152 + 329270449232 \beta_{1} - 379442128 \beta_{2} + 19660952032 \beta_{3} + 368750328 \beta_{4} + 4499192 \beta_{5} + 13587640 \beta_{6} + 1311120 \beta_{7} + 12936792 \beta_{8} - 546952 \beta_{9} - 3498776 \beta_{10} - 3477536 \beta_{11} - 1533064 \beta_{12} + 2058784 \beta_{13} + 2686704 \beta_{14} + 2110624 \beta_{15} ) q^{64} \) \( + ( -79828768933108 - 112149611634 \beta_{1} - 25976303 \beta_{2} - 876842707 \beta_{3} + 1761155594 \beta_{4} + 34652954 \beta_{5} + 1982321 \beta_{6} - 418275 \beta_{7} + 2254817 \beta_{8} + 1542760 \beta_{9} - 723144 \beta_{10} - 2248329 \beta_{11} - 213460 \beta_{12} - 3569768 \beta_{13} + 3787863 \beta_{14} - 3280621 \beta_{15} ) q^{65} \) \( + ( -4074211953156 - 48771112992 \beta_{1} - 366947970 \beta_{2} - 13842812724 \beta_{3} - 24019224 \beta_{4} - 2779968 \beta_{5} - 28875132 \beta_{6} + 829056 \beta_{7} - 14397048 \beta_{8} - 1572000 \beta_{9} + 5429694 \beta_{10} + 2918592 \beta_{11} - 1893342 \beta_{12} + 1288068 \beta_{13} + 1713150 \beta_{14} + 1153920 \beta_{15} ) q^{66} \) \( + ( 56745265472 + 151011587468 \beta_{1} + 62352 \beta_{2} + 3082873852 \beta_{3} + 2743546272 \beta_{4} + 66655440 \beta_{5} + 7362740 \beta_{6} + 8049984 \beta_{7} + 14737100 \beta_{8} - 1690344 \beta_{9} + 2572992 \beta_{10} + 1982580 \beta_{11} - 3181812 \beta_{12} - 3114912 \beta_{13} + 1777608 \beta_{14} + 1398108 \beta_{15} ) q^{67} \) \( + ( -87294657649524 - 557074842074 \beta_{1} + 324311840 \beta_{2} + 42090856340 \beta_{3} - 792303130 \beta_{4} - 21155040 \beta_{5} - 19335696 \beta_{6} - 1606272 \beta_{7} - 35583392 \beta_{8} - 1970776 \beta_{9} - 11895872 \beta_{10} - 1622912 \beta_{11} + 3094504 \beta_{12} - 5061232 \beta_{13} + 981904 \beta_{14} + 1133040 \beta_{15} ) q^{68} \) \( + ( 24096746218680 + 118121066598 \beta_{1} + 35083884 \beta_{2} + 1024144416 \beta_{3} - 3415355346 \beta_{4} - 46038018 \beta_{5} - 17361672 \beta_{6} + 1618212 \beta_{7} - 5738814 \beta_{8} - 2507100 \beta_{9} + 6398400 \beta_{10} + 1049922 \beta_{11} - 2253810 \beta_{12} + 4392276 \beta_{13} - 1808892 \beta_{14} + 2449200 \beta_{15} ) q^{69} \) \( + ( 113390180563832 - 56971151552 \beta_{1} + 370834798 \beta_{2} - 75415725166 \beta_{3} + 1515569912 \beta_{4} + 31001802 \beta_{5} - 33990320 \beta_{6} + 9009500 \beta_{7} - 12340922 \beta_{8} + 1276876 \beta_{9} + 2940222 \beta_{10} - 3021566 \beta_{11} - 6379536 \beta_{12} - 1933730 \beta_{13} + 364520 \beta_{14} + 180754 \beta_{15} ) q^{70} \) \( + ( -193628221776 - 511778840866 \beta_{1} + 14621916 \beta_{2} - 9932592078 \beta_{3} - 5384406976 \beta_{4} - 176850228 \beta_{5} + 32598880 \beta_{6} - 9766356 \beta_{7} + 24495802 \beta_{8} - 1646352 \beta_{9} - 5382576 \beta_{10} - 4887832 \beta_{11} + 10121268 \beta_{12} - 4496088 \beta_{13} + 4188500 \beta_{14} + 8180536 \beta_{15} ) q^{71} \) \( + ( -13117096380771 + 129011022837 \beta_{1} + 143489070 \beta_{2} + 7891898850 \beta_{3} - 129140163 \beta_{4} + 14348907 \beta_{5} - 14348907 \beta_{6} + 14348907 \beta_{8} + 14348907 \beta_{10} ) q^{72} \) \( + ( -22899889947622 + 375444793330 \beta_{1} - 14681974 \beta_{2} + 2872055692 \beta_{3} - 2830516702 \beta_{4} - 126059678 \beta_{5} - 187024236 \beta_{6} - 4004824 \beta_{7} - 2543262 \beta_{8} - 12959764 \beta_{9} - 2750976 \beta_{10} - 4255830 \beta_{11} + 2504298 \beta_{12} + 5750572 \beta_{13} - 1487472 \beta_{14} + 5394852 \beta_{15} ) q^{73} \) \( + ( 54880503134608 + 469897469246 \beta_{1} - 263781788 \beta_{2} - 62038361160 \beta_{3} - 1013784752 \beta_{4} + 179489552 \beta_{5} - 27329448 \beta_{6} + 2079776 \beta_{7} + 61719152 \beta_{8} - 5431304 \beta_{9} - 34908076 \beta_{10} - 3861616 \beta_{11} + 17401572 \beta_{12} - 13777544 \beta_{13} - 1727036 \beta_{14} - 14124608 \beta_{15} ) q^{74} \) \( + ( -201793170018 - 515820395913 \beta_{1} + 67579938 \beta_{2} - 16899839181 \beta_{3} + 5129099898 \beta_{4} - 98914086 \beta_{5} + 164187132 \beta_{6} + 8228970 \beta_{7} + 4842720 \beta_{8} - 2070288 \beta_{9} + 7686648 \beta_{10} + 2817288 \beta_{11} - 10045206 \beta_{12} - 362628 \beta_{13} - 1314306 \beta_{14} + 1323288 \beta_{15} ) q^{75} \) \( + ( -272415773219928 - 1367748407644 \beta_{1} - 1073249528 \beta_{2} + 90840094540 \beta_{3} - 156283064 \beta_{4} - 115778440 \beta_{5} - 147201112 \beta_{6} + 1854964 \beta_{7} + 13819712 \beta_{8} + 2676076 \beta_{9} + 33075988 \beta_{10} + 3142080 \beta_{11} + 9145604 \beta_{12} - 965892 \beta_{13} - 9421712 \beta_{14} - 7307004 \beta_{15} ) q^{76} \) \( + ( -4807742251712 + 773546689664 \beta_{1} - 246854250 \beta_{2} + 4728957122 \beta_{3} + 15876947800 \beta_{4} - 169325320 \beta_{5} - 533031398 \beta_{6} - 28297958 \beta_{7} + 37862838 \beta_{8} - 17502872 \beta_{9} - 17800208 \beta_{10} + 4054482 \beta_{11} + 19177036 \beta_{12} - 15659960 \beta_{13} - 2032130 \beta_{14} + 9922846 \beta_{15} ) q^{77} \) \( + ( 119165995810654 - 53913762766 \beta_{1} + 70684830 \beta_{2} - 66908999382 \beta_{3} + 1676514748 \beta_{4} + 74326986 \beta_{5} + 93796702 \beta_{6} + 11320956 \beta_{7} - 33239754 \beta_{8} + 7064244 \beta_{9} + 8717886 \beta_{10} - 10867230 \beta_{11} - 6770808 \beta_{12} + 3859470 \beta_{13} - 3703320 \beta_{14} - 7997598 \beta_{15} ) q^{78} \) \( + ( -482589175298 - 1191611396589 \beta_{1} + 358846272 \beta_{2} - 92399852041 \beta_{3} + 1412910994 \beta_{4} - 305988512 \beta_{5} + 846852580 \beta_{6} - 28572112 \beta_{7} - 79295207 \beta_{8} - 2351112 \beta_{9} + 31437376 \beta_{10} - 17842140 \beta_{11} - 11235956 \beta_{12} + 8614688 \beta_{13} - 11008296 \beta_{14} + 22544364 \beta_{15} ) q^{79} \) \( + ( -511077625042884 + 1986991034122 \beta_{1} - 583865528 \beta_{2} + 203877233948 \beta_{3} + 675082896 \beta_{4} + 159280348 \beta_{5} - 338289032 \beta_{6} - 6542780 \beta_{7} - 23476556 \beta_{8} + 5171266 \beta_{9} - 23384216 \beta_{10} - 7448872 \beta_{11} + 41478946 \beta_{12} - 13509968 \beta_{13} - 4660380 \beta_{14} - 8177920 \beta_{15} ) q^{80} \) \( + 205891132094649 q^{81} \) \( + ( 149875643363036 + 1471065361330 \beta_{1} - 630086174 \beta_{2} - 205557412012 \beta_{3} - 3184473256 \beta_{4} - 291874400 \beta_{5} - 128326276 \beta_{6} + 16611904 \beta_{7} - 90246920 \beta_{8} - 14872976 \beta_{9} + 14034754 \beta_{10} + 28108960 \beta_{11} + 39321646 \beta_{12} + 381756 \beta_{13} + 14283170 \beta_{14} - 26153216 \beta_{15} ) q^{82} \) \( + ( 311158461255 + 742045877918 \beta_{1} + 576589481 \beta_{2} + 76991085865 \beta_{3} - 13306234643 \beta_{4} + 336841029 \beta_{5} + 1286862695 \beta_{6} - 14736583 \beta_{7} + 70655105 \beta_{8} - 14044522 \beta_{9} - 26304164 \beta_{10} + 12996405 \beta_{11} - 27333976 \beta_{12} - 27108946 \beta_{13} - 13445131 \beta_{14} + 15092639 \beta_{15} ) q^{83} \) \( + ( -31999645808520 - 267842376024 \beta_{1} - 115759248 \beta_{2} + 94292812248 \beta_{3} - 2012608836 \beta_{4} + 163405680 \beta_{5} + 36976776 \beta_{6} + 27718464 \beta_{7} + 9693648 \beta_{8} + 1551468 \beta_{9} - 32508384 \beta_{10} - 17354304 \beta_{11} - 3122868 \beta_{12} + 21276600 \beta_{13} + 8817336 \beta_{14} + 128136 \beta_{15} ) q^{84} \) \( + ( 629060796657404 - 3332719377880 \beta_{1} + 2154602846 \beta_{2} - 22584584864 \beta_{3} - 11330758912 \beta_{4} + 589324128 \beta_{5} - 813844928 \beta_{6} - 4106560 \beta_{7} - 15282656 \beta_{8} - 817600 \beta_{9} + 93066432 \beta_{10} + 2883232 \beta_{11} - 52930720 \beta_{12} + 3242464 \beta_{13} + 21511456 \beta_{14} + 12847648 \beta_{15} ) q^{85} \) \( + ( -56168877924852 - 203155485348 \beta_{1} - 1659532714 \beta_{2} - 431281812070 \beta_{3} - 3673898480 \beta_{4} - 246959582 \beta_{5} - 1042885580 \beta_{6} + 2697644 \beta_{7} + 53529166 \beta_{8} + 52029324 \beta_{9} + 30713542 \beta_{10} - 11762502 \beta_{11} - 5994944 \beta_{12} - 2343546 \beta_{13} - 4936888 \beta_{14} + 2006314 \beta_{15} ) q^{86} \) \( + ( -116802063554 - 273963017161 \beta_{1} + 104266308 \beta_{2} - 23485307643 \beta_{3} + 13506336970 \beta_{4} + 29380812 \beta_{5} + 290781781 \beta_{6} - 9417120 \beta_{7} + 67658256 \beta_{8} - 494250 \beta_{9} + 44918112 \beta_{10} - 15739731 \beta_{11} + 5114787 \beta_{12} + 34195632 \beta_{13} + 12328362 \beta_{14} + 16728231 \beta_{15} ) q^{87} \) \( + ( 42421825864004 + 1004360375760 \beta_{1} + 3455441048 \beta_{2} + 599851482736 \beta_{3} - 10272690132 \beta_{4} - 961140404 \beta_{5} + 416171964 \beta_{6} + 86070216 \beta_{7} - 184472964 \beta_{8} + 19245140 \beta_{9} - 38979548 \beta_{10} + 31192304 \beta_{11} - 38575148 \beta_{12} + 3895936 \beta_{13} + 15842344 \beta_{14} - 72279712 \beta_{15} ) q^{88} \) \( + ( -357828204463518 - 2335167313500 \beta_{1} + 958261114 \beta_{2} - 16468754462 \beta_{3} + 12643136540 \beta_{4} + 412490620 \beta_{5} - 1391252758 \beta_{6} - 33676462 \beta_{7} + 49921074 \beta_{8} + 15138080 \beta_{9} - 97460944 \beta_{10} - 6453810 \beta_{11} - 45698128 \beta_{12} + 34576448 \beta_{13} - 30615850 \beta_{14} + 77420510 \beta_{15} ) q^{89} \) \( + ( 44736419590506 + 274351101840 \beta_{1} + 903981141 \beta_{2} - 73380310398 \beta_{3} - 1090516932 \beta_{4} + 28697814 \beta_{6} + 57395628 \beta_{8} - 14348907 \beta_{10} + 14348907 \beta_{12} + 28697814 \beta_{13} - 14348907 \beta_{14} ) q^{90} \) \( + ( 1890122146285 + 5191858703238 \beta_{1} + 1012023699 \beta_{2} - 134485791997 \beta_{3} + 12107894335 \beta_{4} + 1895763719 \beta_{5} + 2206296883 \beta_{6} + 23739019 \beta_{7} - 18005383 \beta_{8} - 41831034 \beta_{9} - 112949068 \beta_{10} + 95147901 \beta_{11} - 26168998 \beta_{12} + 24281050 \beta_{13} + 54335403 \beta_{14} - 11485833 \beta_{15} ) q^{91} \) \( + ( 482066735591136 - 208721667992 \beta_{1} - 3989893280 \beta_{2} + 1014154037664 \beta_{3} + 8044812640 \beta_{4} + 590961280 \beta_{5} - 616089776 \beta_{6} + 38822848 \beta_{7} + 382833088 \beta_{8} + 52651688 \beta_{9} - 55342816 \beta_{10} + 27593024 \beta_{11} - 21704280 \beta_{12} - 32663536 \beta_{13} + 33367632 \beta_{14} - 84929552 \beta_{15} ) q^{92} \) \( + ( 446035302337032 + 634291577907 \beta_{1} - 1904847726 \beta_{2} + 6198857028 \beta_{3} - 14317477881 \beta_{4} - 292714881 \beta_{5} - 682587408 \beta_{6} + 10979142 \beta_{7} - 25832547 \beta_{8} + 15202050 \beta_{9} - 8105376 \beta_{10} - 37970667 \beta_{11} - 69849105 \beta_{12} + 32118474 \beta_{13} + 19474830 \beta_{14} + 20318820 \beta_{15} ) q^{93} \) \( + ( -593496104068232 - 414248252224 \beta_{1} - 1809719230 \beta_{2} - 1049570226082 \beta_{3} - 18096286552 \beta_{4} - 1661525978 \beta_{5} + 959087840 \beta_{6} + 59749252 \beta_{7} - 87866454 \beta_{8} + 105370724 \beta_{9} - 1254894 \beta_{10} - 20773394 \beta_{11} - 57847040 \beta_{12} - 2707630 \beta_{13} - 25560744 \beta_{14} - 25123394 \beta_{15} ) q^{94} \) \( + ( 149082555776 - 106368332984 \beta_{1} + 436483584 \beta_{2} + 467076744472 \beta_{3} - 38747195136 \beta_{4} - 70507488 \beta_{5} + 968951776 \beta_{6} - 128660160 \beta_{7} - 22282240 \beta_{8} - 2889024 \beta_{9} - 21164736 \beta_{10} - 46003136 \beta_{11} + 73287552 \beta_{12} + 29302176 \beta_{13} + 19591072 \beta_{14} + 51781184 \beta_{15} ) q^{95} \) \( + ( 548608197672680 - 1775589503342 \beta_{1} - 5141769360 \beta_{2} + 313048108812 \beta_{3} + 10572354644 \beta_{4} + 982208256 \beta_{5} + 166125788 \beta_{6} - 13034724 \beta_{7} - 287963928 \beta_{8} + 18664374 \beta_{9} + 96648564 \beta_{10} + 38603304 \beta_{11} - 25292010 \beta_{12} - 19162272 \beta_{13} - 39874356 \beta_{14} - 2600592 \beta_{15} ) q^{96} \) \( + ( -849706987112014 - 5016333907296 \beta_{1} + 632758698 \beta_{2} - 35335740290 \beta_{3} + 7482218312 \beta_{4} + 1128370664 \beta_{5} - 16600602 \beta_{6} - 17167322 \beta_{7} + 29510506 \beta_{8} + 31891416 \beta_{9} + 176235152 \beta_{10} + 116726862 \beta_{11} - 10608748 \beta_{12} - 30433352 \beta_{13} - 36858750 \beta_{14} + 27919266 \beta_{15} ) q^{97} \) \( + ( -2061625812009372 + 5583467916911 \beta_{1} + 4713102968 \beta_{2} - 1279701487344 \beta_{3} + 21324345440 \beta_{4} + 2223492576 \beta_{5} + 434654416 \beta_{6} - 65893440 \beta_{7} - 547037152 \beta_{8} - 34698480 \beta_{9} - 184835944 \beta_{10} + 11416800 \beta_{11} + 111719928 \beta_{12} + 43588752 \beta_{13} + 34266552 \beta_{14} - 105391488 \beta_{15} ) q^{98} \) \( + ( -1149964453701 - 3002566185378 \beta_{1} - 14348907 \beta_{2} - 63149539707 \beta_{3} + 1305750537 \beta_{4} - 875283327 \beta_{5} - 43046721 \beta_{6} - 14348907 \beta_{7} + 43046721 \beta_{8} + 28697814 \beta_{9} + 57395628 \beta_{10} - 14348907 \beta_{11} - 57395628 \beta_{12} + 28697814 \beta_{13} - 71744535 \beta_{14} - 43046721 \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 186q^{2} \) \(\mathstrut +\mathstrut 136588q^{4} \) \(\mathstrut +\mathstrut 354144q^{5} \) \(\mathstrut +\mathstrut 1561518q^{6} \) \(\mathstrut +\mathstrut 14683680q^{8} \) \(\mathstrut -\mathstrut 229582512q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 186q^{2} \) \(\mathstrut +\mathstrut 136588q^{4} \) \(\mathstrut +\mathstrut 354144q^{5} \) \(\mathstrut +\mathstrut 1561518q^{6} \) \(\mathstrut +\mathstrut 14683680q^{8} \) \(\mathstrut -\mathstrut 229582512q^{9} \) \(\mathstrut -\mathstrut 49800172q^{10} \) \(\mathstrut +\mathstrut 425284020q^{12} \) \(\mathstrut -\mathstrut 906419296q^{13} \) \(\mathstrut -\mathstrut 806064072q^{14} \) \(\mathstrut -\mathstrut 2108540816q^{16} \) \(\mathstrut +\mathstrut 12240765600q^{17} \) \(\mathstrut +\mathstrut 2668896702q^{18} \) \(\mathstrut +\mathstrut 1002788712q^{20} \) \(\mathstrut +\mathstrut 39806479296q^{21} \) \(\mathstrut +\mathstrut 216706355928q^{22} \) \(\mathstrut -\mathstrut 111931394832q^{24} \) \(\mathstrut +\mathstrut 206381182512q^{25} \) \(\mathstrut +\mathstrut 1054507182588q^{26} \) \(\mathstrut -\mathstrut 1526063922288q^{28} \) \(\mathstrut +\mathstrut 327679573728q^{29} \) \(\mathstrut +\mathstrut 1192344308100q^{30} \) \(\mathstrut -\mathstrut 5158730488416q^{32} \) \(\mathstrut +\mathstrut 679591529280q^{33} \) \(\mathstrut +\mathstrut 9473293385948q^{34} \) \(\mathstrut -\mathstrut 1959888509316q^{36} \) \(\mathstrut -\mathstrut 8149494749152q^{37} \) \(\mathstrut +\mathstrut 23318999782920q^{38} \) \(\mathstrut -\mathstrut 28671795971776q^{40} \) \(\mathstrut -\mathstrut 25536724613472q^{41} \) \(\mathstrut +\mathstrut 5103781482168q^{42} \) \(\mathstrut -\mathstrut 11442227373552q^{44} \) \(\mathstrut -\mathstrut 5081579320608q^{45} \) \(\mathstrut +\mathstrut 9929654732736q^{46} \) \(\mathstrut +\mathstrut 29246734238832q^{48} \) \(\mathstrut -\mathstrut 93287012964080q^{49} \) \(\mathstrut -\mathstrut 133601044957998q^{50} \) \(\mathstrut +\mathstrut 302261844234872q^{52} \) \(\mathstrut -\mathstrut 86928436629792q^{53} \) \(\mathstrut -\mathstrut 22406076560826q^{54} \) \(\mathstrut +\mathstrut 530930989929024q^{56} \) \(\mathstrut +\mathstrut 48687411524544q^{57} \) \(\mathstrut -\mathstrut 189801665049916q^{58} \) \(\mathstrut +\mathstrut 268455359263896q^{60} \) \(\mathstrut +\mathstrut 476028596468000q^{61} \) \(\mathstrut -\mathstrut 419080420491096q^{62} \) \(\mathstrut +\mathstrut 305944925720704q^{64} \) \(\mathstrut -\mathstrut 1276571708976192q^{65} \) \(\mathstrut -\mathstrut 64815073701480q^{66} \) \(\mathstrut -\mathstrut 1393626893610696q^{68} \) \(\mathstrut +\mathstrut 384812600884992q^{69} \) \(\mathstrut +\mathstrut 1815049064679696q^{70} \) \(\mathstrut -\mathstrut 210694758737760q^{72} \) \(\mathstrut -\mathstrut 368686901901280q^{73} \) \(\mathstrut +\mathstrut 875633111941836q^{74} \) \(\mathstrut -\mathstrut 4351002030861264q^{76} \) \(\mathstrut -\mathstrut 81504772370688q^{77} \) \(\mathstrut +\mathstrut 1907392409236620q^{78} \) \(\mathstrut -\mathstrut 8190389938127328q^{80} \) \(\mathstrut +\mathstrut 3294258113514384q^{81} \) \(\mathstrut +\mathstrut 2390391408733340q^{82} \) \(\mathstrut -\mathstrut 510964995024720q^{84} \) \(\mathstrut +\mathstrut 10085047677398464q^{85} \) \(\mathstrut -\mathstrut 894939801424296q^{86} \) \(\mathstrut +\mathstrut 669091386916800q^{88} \) \(\mathstrut -\mathstrut 5711067674201568q^{89} \) \(\mathstrut +\mathstrut 714578036612004q^{90} \) \(\mathstrut +\mathstrut 7708247605922304q^{92} \) \(\mathstrut +\mathstrut 7132614781502016q^{93} \) \(\mathstrut -\mathstrut 9487273535547696q^{94} \) \(\mathstrut +\mathstrut 8786531038601952q^{96} \) \(\mathstrut -\mathstrut 13564951560560608q^{97} \) \(\mathstrut -\mathstrut 33011660914531290q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(x^{15}\mathstrut +\mathstrut \) \(37115\) \(x^{14}\mathstrut +\mathstrut \) \(433616\) \(x^{13}\mathstrut +\mathstrut \) \(965822723\) \(x^{12}\mathstrut +\mathstrut \) \(11579264195\) \(x^{11}\mathstrut +\mathstrut \) \(12661863519850\) \(x^{10}\mathstrut +\mathstrut \) \(162120109773543\) \(x^{9}\mathstrut +\mathstrut \) \(120758933936870280\) \(x^{8}\mathstrut +\mathstrut \) \(1240343233445130327\) \(x^{7}\mathstrut +\mathstrut \) \(553297868296926315210\) \(x^{6}\mathstrut +\mathstrut \) \(3641854073469392130003\) \(x^{5}\mathstrut +\mathstrut \) \(1781463130185670341595803\) \(x^{4}\mathstrut +\mathstrut \) \(12984494303085096280528944\) \(x^{3}\mathstrut +\mathstrut \) \(125765846444242947670566147\) \(x^{2}\mathstrut -\mathstrut \) \(206760575884676576961270033\) \(x\mathstrut +\mathstrut \) \(435779119237071236287541049\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(17\!\cdots\!75\) \(\nu^{15}\mathstrut +\mathstrut \) \(10\!\cdots\!69\) \(\nu^{14}\mathstrut -\mathstrut \) \(64\!\cdots\!42\) \(\nu^{13}\mathstrut -\mathstrut \) \(41\!\cdots\!46\) \(\nu^{12}\mathstrut -\mathstrut \) \(16\!\cdots\!33\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!57\) \(\nu^{10}\mathstrut -\mathstrut \) \(21\!\cdots\!27\) \(\nu^{9}\mathstrut -\mathstrut \) \(16\!\cdots\!23\) \(\nu^{8}\mathstrut -\mathstrut \) \(20\!\cdots\!77\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!81\) \(\nu^{6}\mathstrut -\mathstrut \) \(94\!\cdots\!83\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!79\) \(\nu^{4}\mathstrut -\mathstrut \) \(30\!\cdots\!26\) \(\nu^{3}\mathstrut -\mathstrut \) \(63\!\cdots\!34\) \(\nu^{2}\mathstrut -\mathstrut \) \(45\!\cdots\!57\) \(\nu\mathstrut +\mathstrut \) \(89\!\cdots\!75\)\()/\)\(36\!\cdots\!40\)
\(\beta_{2}\)\(=\)\((\)\(74\!\cdots\!63\) \(\nu^{15}\mathstrut +\mathstrut \) \(14\!\cdots\!91\) \(\nu^{14}\mathstrut +\mathstrut \) \(27\!\cdots\!14\) \(\nu^{13}\mathstrut +\mathstrut \) \(82\!\cdots\!74\) \(\nu^{12}\mathstrut +\mathstrut \) \(72\!\cdots\!65\) \(\nu^{11}\mathstrut +\mathstrut \) \(21\!\cdots\!97\) \(\nu^{10}\mathstrut +\mathstrut \) \(95\!\cdots\!19\) \(\nu^{9}\mathstrut +\mathstrut \) \(25\!\cdots\!23\) \(\nu^{8}\mathstrut +\mathstrut \) \(90\!\cdots\!21\) \(\nu^{7}\mathstrut +\mathstrut \) \(19\!\cdots\!49\) \(\nu^{6}\mathstrut +\mathstrut \) \(41\!\cdots\!39\) \(\nu^{5}\mathstrut +\mathstrut \) \(44\!\cdots\!55\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!06\) \(\nu^{3}\mathstrut +\mathstrut \) \(70\!\cdots\!18\) \(\nu^{2}\mathstrut +\mathstrut \) \(45\!\cdots\!33\) \(\nu\mathstrut -\mathstrut \) \(33\!\cdots\!79\)\()/\)\(84\!\cdots\!80\)
\(\beta_{3}\)\(=\)\((\)\(58\!\cdots\!65\) \(\nu^{15}\mathstrut +\mathstrut \) \(75\!\cdots\!49\) \(\nu^{14}\mathstrut +\mathstrut \) \(21\!\cdots\!98\) \(\nu^{13}\mathstrut +\mathstrut \) \(30\!\cdots\!14\) \(\nu^{12}\mathstrut +\mathstrut \) \(56\!\cdots\!67\) \(\nu^{11}\mathstrut +\mathstrut \) \(80\!\cdots\!03\) \(\nu^{10}\mathstrut +\mathstrut \) \(74\!\cdots\!53\) \(\nu^{9}\mathstrut +\mathstrut \) \(11\!\cdots\!37\) \(\nu^{8}\mathstrut +\mathstrut \) \(70\!\cdots\!43\) \(\nu^{7}\mathstrut +\mathstrut \) \(88\!\cdots\!59\) \(\nu^{6}\mathstrut +\mathstrut \) \(32\!\cdots\!37\) \(\nu^{5}\mathstrut +\mathstrut \) \(28\!\cdots\!81\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!74\) \(\nu^{3}\mathstrut +\mathstrut \) \(99\!\cdots\!26\) \(\nu^{2}\mathstrut +\mathstrut \) \(74\!\cdots\!63\) \(\nu\mathstrut -\mathstrut \) \(35\!\cdots\!65\)\()/\)\(36\!\cdots\!40\)
\(\beta_{4}\)\(=\)\((\)\(25\!\cdots\!13\) \(\nu^{15}\mathstrut -\mathstrut \) \(77\!\cdots\!27\) \(\nu^{14}\mathstrut +\mathstrut \) \(95\!\cdots\!58\) \(\nu^{13}\mathstrut +\mathstrut \) \(11\!\cdots\!22\) \(\nu^{12}\mathstrut +\mathstrut \) \(24\!\cdots\!27\) \(\nu^{11}\mathstrut +\mathstrut \) \(31\!\cdots\!03\) \(\nu^{10}\mathstrut +\mathstrut \) \(32\!\cdots\!89\) \(\nu^{9}\mathstrut +\mathstrut \) \(44\!\cdots\!33\) \(\nu^{8}\mathstrut +\mathstrut \) \(31\!\cdots\!27\) \(\nu^{7}\mathstrut +\mathstrut \) \(34\!\cdots\!47\) \(\nu^{6}\mathstrut +\mathstrut \) \(14\!\cdots\!33\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!61\) \(\nu^{4}\mathstrut +\mathstrut \) \(46\!\cdots\!78\) \(\nu^{3}\mathstrut +\mathstrut \) \(36\!\cdots\!46\) \(\nu^{2}\mathstrut +\mathstrut \) \(40\!\cdots\!27\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!37\)\()/\)\(36\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(51\!\cdots\!03\) \(\nu^{15}\mathstrut -\mathstrut \) \(42\!\cdots\!65\) \(\nu^{14}\mathstrut +\mathstrut \) \(19\!\cdots\!22\) \(\nu^{13}\mathstrut +\mathstrut \) \(85\!\cdots\!94\) \(\nu^{12}\mathstrut +\mathstrut \) \(49\!\cdots\!33\) \(\nu^{11}\mathstrut +\mathstrut \) \(23\!\cdots\!17\) \(\nu^{10}\mathstrut +\mathstrut \) \(64\!\cdots\!63\) \(\nu^{9}\mathstrut +\mathstrut \) \(36\!\cdots\!19\) \(\nu^{8}\mathstrut +\mathstrut \) \(61\!\cdots\!41\) \(\nu^{7}\mathstrut +\mathstrut \) \(19\!\cdots\!69\) \(\nu^{6}\mathstrut +\mathstrut \) \(27\!\cdots\!19\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!41\) \(\nu^{4}\mathstrut +\mathstrut \) \(89\!\cdots\!50\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!34\) \(\nu^{2}\mathstrut +\mathstrut \) \(77\!\cdots\!81\) \(\nu\mathstrut -\mathstrut \) \(33\!\cdots\!87\)\()/\)\(36\!\cdots\!40\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(20\!\cdots\!53\) \(\nu^{15}\mathstrut +\mathstrut \) \(25\!\cdots\!64\) \(\nu^{14}\mathstrut -\mathstrut \) \(76\!\cdots\!76\) \(\nu^{13}\mathstrut -\mathstrut \) \(87\!\cdots\!14\) \(\nu^{12}\mathstrut -\mathstrut \) \(19\!\cdots\!09\) \(\nu^{11}\mathstrut -\mathstrut \) \(23\!\cdots\!86\) \(\nu^{10}\mathstrut -\mathstrut \) \(26\!\cdots\!33\) \(\nu^{9}\mathstrut -\mathstrut \) \(32\!\cdots\!72\) \(\nu^{8}\mathstrut -\mathstrut \) \(24\!\cdots\!01\) \(\nu^{7}\mathstrut -\mathstrut \) \(24\!\cdots\!30\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!05\) \(\nu^{5}\mathstrut -\mathstrut \) \(72\!\cdots\!84\) \(\nu^{4}\mathstrut -\mathstrut \) \(36\!\cdots\!12\) \(\nu^{3}\mathstrut -\mathstrut \) \(25\!\cdots\!70\) \(\nu^{2}\mathstrut -\mathstrut \) \(31\!\cdots\!57\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!14\)\()/\)\(60\!\cdots\!40\)
\(\beta_{7}\)\(=\)\((\)\(33\!\cdots\!39\) \(\nu^{15}\mathstrut -\mathstrut \) \(29\!\cdots\!87\) \(\nu^{14}\mathstrut +\mathstrut \) \(12\!\cdots\!62\) \(\nu^{13}\mathstrut -\mathstrut \) \(91\!\cdots\!58\) \(\nu^{12}\mathstrut +\mathstrut \) \(31\!\cdots\!73\) \(\nu^{11}\mathstrut -\mathstrut \) \(23\!\cdots\!85\) \(\nu^{10}\mathstrut +\mathstrut \) \(40\!\cdots\!59\) \(\nu^{9}\mathstrut -\mathstrut \) \(30\!\cdots\!39\) \(\nu^{8}\mathstrut +\mathstrut \) \(36\!\cdots\!81\) \(\nu^{7}\mathstrut -\mathstrut \) \(29\!\cdots\!65\) \(\nu^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!71\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!95\) \(\nu^{4}\mathstrut +\mathstrut \) \(52\!\cdots\!82\) \(\nu^{3}\mathstrut -\mathstrut \) \(45\!\cdots\!98\) \(\nu^{2}\mathstrut -\mathstrut \) \(18\!\cdots\!31\) \(\nu\mathstrut -\mathstrut \) \(16\!\cdots\!21\)\()/\)\(84\!\cdots\!80\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(21\!\cdots\!81\) \(\nu^{15}\mathstrut +\mathstrut \) \(87\!\cdots\!37\) \(\nu^{14}\mathstrut -\mathstrut \) \(80\!\cdots\!70\) \(\nu^{13}\mathstrut -\mathstrut \) \(69\!\cdots\!46\) \(\nu^{12}\mathstrut -\mathstrut \) \(20\!\cdots\!35\) \(\nu^{11}\mathstrut -\mathstrut \) \(18\!\cdots\!45\) \(\nu^{10}\mathstrut -\mathstrut \) \(27\!\cdots\!57\) \(\nu^{9}\mathstrut -\mathstrut \) \(26\!\cdots\!47\) \(\nu^{8}\mathstrut -\mathstrut \) \(26\!\cdots\!83\) \(\nu^{7}\mathstrut -\mathstrut \) \(19\!\cdots\!01\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!97\) \(\nu^{5}\mathstrut -\mathstrut \) \(43\!\cdots\!35\) \(\nu^{4}\mathstrut -\mathstrut \) \(39\!\cdots\!38\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!50\) \(\nu^{2}\mathstrut -\mathstrut \) \(40\!\cdots\!43\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!79\)\()/\)\(28\!\cdots\!60\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(23\!\cdots\!93\) \(\nu^{15}\mathstrut -\mathstrut \) \(21\!\cdots\!35\) \(\nu^{14}\mathstrut -\mathstrut \) \(88\!\cdots\!90\) \(\nu^{13}\mathstrut -\mathstrut \) \(19\!\cdots\!98\) \(\nu^{12}\mathstrut -\mathstrut \) \(23\!\cdots\!67\) \(\nu^{11}\mathstrut -\mathstrut \) \(51\!\cdots\!53\) \(\nu^{10}\mathstrut -\mathstrut \) \(30\!\cdots\!49\) \(\nu^{9}\mathstrut -\mathstrut \) \(69\!\cdots\!27\) \(\nu^{8}\mathstrut -\mathstrut \) \(29\!\cdots\!63\) \(\nu^{7}\mathstrut -\mathstrut \) \(59\!\cdots\!81\) \(\nu^{6}\mathstrut -\mathstrut \) \(13\!\cdots\!57\) \(\nu^{5}\mathstrut -\mathstrut \) \(22\!\cdots\!07\) \(\nu^{4}\mathstrut -\mathstrut \) \(42\!\cdots\!86\) \(\nu^{3}\mathstrut -\mathstrut \) \(75\!\cdots\!66\) \(\nu^{2}\mathstrut -\mathstrut \) \(49\!\cdots\!07\) \(\nu\mathstrut -\mathstrut \) \(22\!\cdots\!69\)\()/\)\(84\!\cdots\!80\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(37\!\cdots\!31\) \(\nu^{15}\mathstrut +\mathstrut \) \(87\!\cdots\!87\) \(\nu^{14}\mathstrut -\mathstrut \) \(13\!\cdots\!58\) \(\nu^{13}\mathstrut -\mathstrut \) \(14\!\cdots\!30\) \(\nu^{12}\mathstrut -\mathstrut \) \(36\!\cdots\!33\) \(\nu^{11}\mathstrut -\mathstrut \) \(38\!\cdots\!75\) \(\nu^{10}\mathstrut -\mathstrut \) \(47\!\cdots\!19\) \(\nu^{9}\mathstrut -\mathstrut \) \(54\!\cdots\!69\) \(\nu^{8}\mathstrut -\mathstrut \) \(44\!\cdots\!73\) \(\nu^{7}\mathstrut -\mathstrut \) \(40\!\cdots\!87\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!59\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!53\) \(\nu^{4}\mathstrut -\mathstrut \) \(65\!\cdots\!26\) \(\nu^{3}\mathstrut -\mathstrut \) \(39\!\cdots\!42\) \(\nu^{2}\mathstrut -\mathstrut \) \(20\!\cdots\!53\) \(\nu\mathstrut +\mathstrut \) \(17\!\cdots\!77\)\()/\)\(84\!\cdots\!80\)
\(\beta_{11}\)\(=\)\((\)\(57\!\cdots\!03\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!27\) \(\nu^{14}\mathstrut +\mathstrut \) \(21\!\cdots\!62\) \(\nu^{13}\mathstrut +\mathstrut \) \(32\!\cdots\!42\) \(\nu^{12}\mathstrut +\mathstrut \) \(55\!\cdots\!97\) \(\nu^{11}\mathstrut +\mathstrut \) \(86\!\cdots\!61\) \(\nu^{10}\mathstrut +\mathstrut \) \(73\!\cdots\!15\) \(\nu^{9}\mathstrut +\mathstrut \) \(11\!\cdots\!55\) \(\nu^{8}\mathstrut +\mathstrut \) \(69\!\cdots\!29\) \(\nu^{7}\mathstrut +\mathstrut \) \(95\!\cdots\!85\) \(\nu^{6}\mathstrut +\mathstrut \) \(31\!\cdots\!07\) \(\nu^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!87\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!10\) \(\nu^{3}\mathstrut +\mathstrut \) \(10\!\cdots\!26\) \(\nu^{2}\mathstrut +\mathstrut \) \(48\!\cdots\!69\) \(\nu\mathstrut -\mathstrut \) \(70\!\cdots\!11\)\()/\)\(84\!\cdots\!80\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(68\!\cdots\!65\) \(\nu^{15}\mathstrut -\mathstrut \) \(21\!\cdots\!39\) \(\nu^{14}\mathstrut -\mathstrut \) \(25\!\cdots\!90\) \(\nu^{13}\mathstrut -\mathstrut \) \(40\!\cdots\!90\) \(\nu^{12}\mathstrut -\mathstrut \) \(66\!\cdots\!07\) \(\nu^{11}\mathstrut -\mathstrut \) \(10\!\cdots\!97\) \(\nu^{10}\mathstrut -\mathstrut \) \(87\!\cdots\!53\) \(\nu^{9}\mathstrut -\mathstrut \) \(14\!\cdots\!07\) \(\nu^{8}\mathstrut -\mathstrut \) \(83\!\cdots\!91\) \(\nu^{7}\mathstrut -\mathstrut \) \(11\!\cdots\!21\) \(\nu^{6}\mathstrut -\mathstrut \) \(38\!\cdots\!29\) \(\nu^{5}\mathstrut -\mathstrut \) \(40\!\cdots\!03\) \(\nu^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!70\) \(\nu^{3}\mathstrut -\mathstrut \) \(13\!\cdots\!42\) \(\nu^{2}\mathstrut -\mathstrut \) \(11\!\cdots\!03\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!29\)\()/\)\(84\!\cdots\!80\)
\(\beta_{13}\)\(=\)\((\)\(71\!\cdots\!47\) \(\nu^{15}\mathstrut -\mathstrut \) \(92\!\cdots\!57\) \(\nu^{14}\mathstrut +\mathstrut \) \(26\!\cdots\!14\) \(\nu^{13}\mathstrut +\mathstrut \) \(30\!\cdots\!34\) \(\nu^{12}\mathstrut +\mathstrut \) \(69\!\cdots\!29\) \(\nu^{11}\mathstrut +\mathstrut \) \(81\!\cdots\!29\) \(\nu^{10}\mathstrut +\mathstrut \) \(90\!\cdots\!15\) \(\nu^{9}\mathstrut +\mathstrut \) \(11\!\cdots\!51\) \(\nu^{8}\mathstrut +\mathstrut \) \(86\!\cdots\!25\) \(\nu^{7}\mathstrut +\mathstrut \) \(86\!\cdots\!37\) \(\nu^{6}\mathstrut +\mathstrut \) \(39\!\cdots\!15\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!03\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!62\) \(\nu^{3}\mathstrut +\mathstrut \) \(90\!\cdots\!54\) \(\nu^{2}\mathstrut +\mathstrut \) \(53\!\cdots\!97\) \(\nu\mathstrut -\mathstrut \) \(15\!\cdots\!43\)\()/\)\(84\!\cdots\!80\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(50\!\cdots\!37\) \(\nu^{15}\mathstrut +\mathstrut \) \(17\!\cdots\!63\) \(\nu^{14}\mathstrut -\mathstrut \) \(18\!\cdots\!82\) \(\nu^{13}\mathstrut -\mathstrut \) \(17\!\cdots\!30\) \(\nu^{12}\mathstrut -\mathstrut \) \(48\!\cdots\!71\) \(\nu^{11}\mathstrut -\mathstrut \) \(45\!\cdots\!11\) \(\nu^{10}\mathstrut -\mathstrut \) \(63\!\cdots\!17\) \(\nu^{9}\mathstrut -\mathstrut \) \(65\!\cdots\!97\) \(\nu^{8}\mathstrut -\mathstrut \) \(60\!\cdots\!07\) \(\nu^{7}\mathstrut -\mathstrut \) \(47\!\cdots\!31\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!73\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!05\) \(\nu^{4}\mathstrut -\mathstrut \) \(89\!\cdots\!70\) \(\nu^{3}\mathstrut -\mathstrut \) \(42\!\cdots\!66\) \(\nu^{2}\mathstrut -\mathstrut \) \(59\!\cdots\!87\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!29\)\()/\)\(28\!\cdots\!60\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(52\!\cdots\!93\) \(\nu^{15}\mathstrut -\mathstrut \) \(12\!\cdots\!21\) \(\nu^{14}\mathstrut -\mathstrut \) \(19\!\cdots\!14\) \(\nu^{13}\mathstrut -\mathstrut \) \(29\!\cdots\!82\) \(\nu^{12}\mathstrut -\mathstrut \) \(50\!\cdots\!11\) \(\nu^{11}\mathstrut -\mathstrut \) \(77\!\cdots\!95\) \(\nu^{10}\mathstrut -\mathstrut \) \(66\!\cdots\!69\) \(\nu^{9}\mathstrut -\mathstrut \) \(10\!\cdots\!29\) \(\nu^{8}\mathstrut -\mathstrut \) \(63\!\cdots\!51\) \(\nu^{7}\mathstrut -\mathstrut \) \(86\!\cdots\!35\) \(\nu^{6}\mathstrut -\mathstrut \) \(29\!\cdots\!25\) \(\nu^{5}\mathstrut -\mathstrut \) \(28\!\cdots\!49\) \(\nu^{4}\mathstrut -\mathstrut \) \(93\!\cdots\!90\) \(\nu^{3}\mathstrut -\mathstrut \) \(99\!\cdots\!10\) \(\nu^{2}\mathstrut -\mathstrut \) \(88\!\cdots\!79\) \(\nu\mathstrut +\mathstrut \) \(52\!\cdots\!85\)\()/\)\(28\!\cdots\!60\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(8\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(5\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(271\) \(\beta_{6}\mathstrut +\mathstrut \) \(542\) \(\beta_{5}\mathstrut -\mathstrut \) \(1795\) \(\beta_{4}\mathstrut +\mathstrut \) \(14256\) \(\beta_{3}\mathstrut -\mathstrut \) \(179\) \(\beta_{2}\mathstrut +\mathstrut \) \(2063242\) \(\beta_{1}\mathstrut +\mathstrut \) \(1339491\)\()/8957952\)
\(\nu^{2}\)\(=\)\((\)\(1092\) \(\beta_{15}\mathstrut +\mathstrut \) \(108\) \(\beta_{14}\mathstrut -\mathstrut \) \(795\) \(\beta_{13}\mathstrut -\mathstrut \) \(489\) \(\beta_{12}\mathstrut -\mathstrut \) \(396\) \(\beta_{11}\mathstrut +\mathstrut \) \(1290\) \(\beta_{10}\mathstrut -\mathstrut \) \(348\) \(\beta_{9}\mathstrut -\mathstrut \) \(342\) \(\beta_{8}\mathstrut +\mathstrut \) \(705\) \(\beta_{7}\mathstrut -\mathstrut \) \(6631\) \(\beta_{6}\mathstrut +\mathstrut \) \(3633\) \(\beta_{5}\mathstrut -\mathstrut \) \(296344\) \(\beta_{4}\mathstrut +\mathstrut \) \(19099204\) \(\beta_{3}\mathstrut +\mathstrut \) \(5043\) \(\beta_{2}\mathstrut -\mathstrut \) \(7629132\) \(\beta_{1}\mathstrut -\mathstrut \) \(41554325728\)\()/8957952\)
\(\nu^{3}\)\(=\)\((\)\(124658\) \(\beta_{15}\mathstrut -\mathstrut \) \(20682\) \(\beta_{14}\mathstrut +\mathstrut \) \(13465\) \(\beta_{13}\mathstrut -\mathstrut \) \(44291\) \(\beta_{12}\mathstrut +\mathstrut \) \(34362\) \(\beta_{11}\mathstrut +\mathstrut \) \(82130\) \(\beta_{10}\mathstrut -\mathstrut \) \(87190\) \(\beta_{9}\mathstrut +\mathstrut \) \(83928\) \(\beta_{8}\mathstrut -\mathstrut \) \(111433\) \(\beta_{7}\mathstrut -\mathstrut \) \(3957769\) \(\beta_{6}\mathstrut -\mathstrut \) \(185909\) \(\beta_{5}\mathstrut +\mathstrut \) \(12518446\) \(\beta_{4}\mathstrut -\mathstrut \) \(8265390\) \(\beta_{3}\mathstrut +\mathstrut \) \(4667693\) \(\beta_{2}\mathstrut -\mathstrut \) \(1177809094\) \(\beta_{1}\mathstrut -\mathstrut \) \(395773131918\)\()/4478976\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(8165460\) \(\beta_{15}\mathstrut -\mathstrut \) \(1121580\) \(\beta_{14}\mathstrut +\mathstrut \) \(35184171\) \(\beta_{13}\mathstrut -\mathstrut \) \(6733311\) \(\beta_{12}\mathstrut +\mathstrut \) \(679500\) \(\beta_{11}\mathstrut +\mathstrut \) \(12581142\) \(\beta_{10}\mathstrut +\mathstrut \) \(233988\) \(\beta_{9}\mathstrut -\mathstrut \) \(34959834\) \(\beta_{8}\mathstrut -\mathstrut \) \(13552545\) \(\beta_{7}\mathstrut -\mathstrut \) \(9837017\) \(\beta_{6}\mathstrut -\mathstrut \) \(153915585\) \(\beta_{5}\mathstrut -\mathstrut \) \(5408591576\) \(\beta_{4}\mathstrut -\mathstrut \) \(289928027044\) \(\beta_{3}\mathstrut +\mathstrut \) \(574005597\) \(\beta_{2}\mathstrut -\mathstrut \) \(582221957820\) \(\beta_{1}\mathstrut -\mathstrut \) \(621854512408688\)\()/8957952\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(2335437428\) \(\beta_{15}\mathstrut -\mathstrut \) \(206698932\) \(\beta_{14}\mathstrut +\mathstrut \) \(2491387439\) \(\beta_{13}\mathstrut +\mathstrut \) \(147278681\) \(\beta_{12}\mathstrut -\mathstrut \) \(2292424812\) \(\beta_{11}\mathstrut +\mathstrut \) \(1148024734\) \(\beta_{10}\mathstrut +\mathstrut \) \(2595718000\) \(\beta_{9}\mathstrut +\mathstrut \) \(9407849694\) \(\beta_{8}\mathstrut +\mathstrut \) \(742461379\) \(\beta_{7}\mathstrut +\mathstrut \) \(65023840411\) \(\beta_{6}\mathstrut -\mathstrut \) \(133659472408\) \(\beta_{5}\mathstrut +\mathstrut \) \(450277796795\) \(\beta_{4}\mathstrut -\mathstrut \) \(9377545318764\) \(\beta_{3}\mathstrut -\mathstrut \) \(101658614879\) \(\beta_{2}\mathstrut -\mathstrut \) \(443999756167502\) \(\beta_{1}\mathstrut +\mathstrut \) \(13618442476512357\)\()/8957952\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(4189257874\) \(\beta_{15}\mathstrut -\mathstrut \) \(865853118\) \(\beta_{14}\mathstrut -\mathstrut \) \(7721017532\) \(\beta_{13}\mathstrut +\mathstrut \) \(6873420994\) \(\beta_{12}\mathstrut +\mathstrut \) \(2342930958\) \(\beta_{11}\mathstrut -\mathstrut \) \(13774728856\) \(\beta_{10}\mathstrut +\mathstrut \) \(7606383554\) \(\beta_{9}\mathstrut +\mathstrut \) \(10835950638\) \(\beta_{8}\mathstrut -\mathstrut \) \(1411257082\) \(\beta_{7}\mathstrut +\mathstrut \) \(137930572574\) \(\beta_{6}\mathstrut -\mathstrut \) \(29467613291\) \(\beta_{5}\mathstrut +\mathstrut \) \(6656247520195\) \(\beta_{4}\mathstrut +\mathstrut \) \(2179041656646\) \(\beta_{3}\mathstrut -\mathstrut \) \(749294270302\) \(\beta_{2}\mathstrut +\mathstrut \) \(351586970742896\) \(\beta_{1}\mathstrut +\mathstrut \) \(584087058928371021\)\()/248832\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(32865728557664\) \(\beta_{15}\mathstrut +\mathstrut \) \(29503330762536\) \(\beta_{14}\mathstrut -\mathstrut \) \(70554016885345\) \(\beta_{13}\mathstrut +\mathstrut \) \(15349984212821\) \(\beta_{12}\mathstrut +\mathstrut \) \(9213638755608\) \(\beta_{11}\mathstrut -\mathstrut \) \(130976654123330\) \(\beta_{10}\mathstrut +\mathstrut \) \(19976945851300\) \(\beta_{9}\mathstrut -\mathstrut \) \(243928504251390\) \(\beta_{8}\mathstrut +\mathstrut \) \(66942815890879\) \(\beta_{7}\mathstrut +\mathstrut \) \(993223916965639\) \(\beta_{6}\mathstrut +\mathstrut \) \(2351596614232058\) \(\beta_{5}\mathstrut -\mathstrut \) \(11853672894820447\) \(\beta_{4}\mathstrut +\mathstrut \) \(241261168128622536\) \(\beta_{3}\mathstrut -\mathstrut \) \(2183374954646459\) \(\beta_{2}\mathstrut +\mathstrut \) \(9573159950213437762\) \(\beta_{1}\mathstrut +\mathstrut \) \(384838554024321803007\)\()/8957952\)
\(\nu^{8}\)\(=\)\((\)\(4784492040317508\) \(\beta_{15}\mathstrut +\mathstrut \) \(1152394470522972\) \(\beta_{14}\mathstrut -\mathstrut \) \(10291654449546411\) \(\beta_{13}\mathstrut -\mathstrut \) \(1966823336567361\) \(\beta_{12}\mathstrut +\mathstrut \) \(152170375179780\) \(\beta_{11}\mathstrut +\mathstrut \) \(654796718884074\) \(\beta_{10}\mathstrut -\mathstrut \) \(5527286915520804\) \(\beta_{9}\mathstrut +\mathstrut \) \(7856167103474058\) \(\beta_{8}\mathstrut +\mathstrut \) \(7337677563393969\) \(\beta_{7}\mathstrut -\mathstrut \) \(128679413494767127\) \(\beta_{6}\mathstrut +\mathstrut \) \(127210832384297721\) \(\beta_{5}\mathstrut -\mathstrut \) \(3751287088062868192\) \(\beta_{4}\mathstrut +\mathstrut \) \(89542554746507337076\) \(\beta_{3}\mathstrut +\mathstrut \) \(269747676829892595\) \(\beta_{2}\mathstrut +\mathstrut \) \(200106991673642479884\) \(\beta_{1}\mathstrut -\mathstrut \) \(191049513829218139370968\)\()/8957952\)
\(\nu^{9}\)\(=\)\((\)\(695615614994775950\) \(\beta_{15}\mathstrut -\mathstrut \) \(205585065299594790\) \(\beta_{14}\mathstrut +\mathstrut \) \(77935513206454345\) \(\beta_{13}\mathstrut -\mathstrut \) \(62366584416789431\) \(\beta_{12}\mathstrut +\mathstrut \) \(436565090174697270\) \(\beta_{11}\mathstrut +\mathstrut \) \(1134790264250979314\) \(\beta_{10}\mathstrut -\mathstrut \) \(777828857113194082\) \(\beta_{9}\mathstrut +\mathstrut \) \(295014779122349604\) \(\beta_{8}\mathstrut -\mathstrut \) \(645179670957151957\) \(\beta_{7}\mathstrut -\mathstrut \) \(22031714893622151781\) \(\beta_{6}\mathstrut +\mathstrut \) \(2101191527678285461\) \(\beta_{5}\mathstrut -\mathstrut \) \(46660378113671680580\) \(\beta_{4}\mathstrut -\mathstrut \) \(164317101776818592514\) \(\beta_{3}\mathstrut +\mathstrut \) \(45008374456955343497\) \(\beta_{2}\mathstrut -\mathstrut \) \(23858038203139002136294\) \(\beta_{1}\mathstrut -\mathstrut \) \(9580383150647210841747324\)\()/4478976\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(19217573460243097356\) \(\beta_{15}\mathstrut -\mathstrut \) \(17555106221673780132\) \(\beta_{14}\mathstrut +\mathstrut \) \(324143549271734702187\) \(\beta_{13}\mathstrut -\mathstrut \) \(38741167347703680687\) \(\beta_{12}\mathstrut -\mathstrut \) \(44622683973745023996\) \(\beta_{11}\mathstrut +\mathstrut \) \(223707159027584544534\) \(\beta_{10}\mathstrut -\mathstrut \) \(37712500770066911868\) \(\beta_{9}\mathstrut -\mathstrut \) \(258491485084843162098\) \(\beta_{8}\mathstrut -\mathstrut \) \(159338360449793484777\) \(\beta_{7}\mathstrut +\mathstrut \) \(187397194561875279775\) \(\beta_{6}\mathstrut -\mathstrut \) \(2301262593054891180765\) \(\beta_{5}\mathstrut -\mathstrut \) \(8504605312607473734236\) \(\beta_{4}\mathstrut -\mathstrut \) \(1758774684577257157005676\) \(\beta_{3}\mathstrut +\mathstrut \) \(7195964869404611065941\) \(\beta_{2}\mathstrut -\mathstrut \) \(12290923762596683018023956\) \(\beta_{1}\mathstrut -\mathstrut \) \(3640379245542341938639778540\)\()/8957952\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(15576515273664450622292\) \(\beta_{15}\mathstrut -\mathstrut \) \(5377855822106290867620\) \(\beta_{14}\mathstrut +\mathstrut \) \(32352321941016976310063\) \(\beta_{13}\mathstrut -\mathstrut \) \(5185690695358309825615\) \(\beta_{12}\mathstrut -\mathstrut \) \(22347226932273103332348\) \(\beta_{11}\mathstrut +\mathstrut \) \(12809267642911215167134\) \(\beta_{10}\mathstrut +\mathstrut \) \(24142107176888590377640\) \(\beta_{9}\mathstrut +\mathstrut \) \(81346045080436280645694\) \(\beta_{8}\mathstrut -\mathstrut \) \(5646605940852255619805\) \(\beta_{7}\mathstrut +\mathstrut \) \(603570308522819000699995\) \(\beta_{6}\mathstrut -\mathstrut \) \(932561050848412051873384\) \(\beta_{5}\mathstrut +\mathstrut \) \(7468262765824324699724891\) \(\beta_{4}\mathstrut -\mathstrut \) \(123384118075703281853948508\) \(\beta_{3}\mathstrut -\mathstrut \) \(886364567767522482634127\) \(\beta_{2}\mathstrut -\mathstrut \) \(2616246416762175683981848262\) \(\beta_{1}\mathstrut +\mathstrut \) \(227616776968306834051021076037\)\()/8957952\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(23380931138524075517986\) \(\beta_{15}\mathstrut -\mathstrut \) \(1163626843312066557726\) \(\beta_{14}\mathstrut -\mathstrut \) \(27203949059629651115228\) \(\beta_{13}\mathstrut +\mathstrut \) \(15543021686041973524282\) \(\beta_{12}\mathstrut +\mathstrut \) \(5257587205551905933166\) \(\beta_{11}\mathstrut -\mathstrut \) \(63128889878117759915224\) \(\beta_{10}\mathstrut +\mathstrut \) \(50934149899047341681114\) \(\beta_{9}\mathstrut +\mathstrut \) \(33207214920449841133662\) \(\beta_{8}\mathstrut +\mathstrut \) \(8725398682803331996022\) \(\beta_{7}\mathstrut +\mathstrut \) \(889863192717531088507118\) \(\beta_{6}\mathstrut -\mathstrut \) \(237086517000790508192519\) \(\beta_{5}\mathstrut +\mathstrut \) \(24440561299769993697643663\) \(\beta_{4}\mathstrut +\mathstrut \) \(13894076528807219311055430\) \(\beta_{3}\mathstrut -\mathstrut \) \(3851138959413000777407614\) \(\beta_{2}\mathstrut +\mathstrut \) \(2123946414009329716559447936\) \(\beta_{1}\mathstrut +\mathstrut \) \(1977181720395231213704552465121\)\()/124416\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(209158324550564376294919928\) \(\beta_{15}\mathstrut +\mathstrut \) \(270902570007301714804642704\) \(\beta_{14}\mathstrut -\mathstrut \) \(790708741991629907095281409\) \(\beta_{13}\mathstrut +\mathstrut \) \(151017983497977589914810917\) \(\beta_{12}\mathstrut +\mathstrut \) \(112803399851790372364107504\) \(\beta_{11}\mathstrut -\mathstrut \) \(1281499313383376418598109954\) \(\beta_{10}\mathstrut +\mathstrut \) \(170225727554344660442611204\) \(\beta_{9}\mathstrut -\mathstrut \) \(1716602632899521581072636086\) \(\beta_{8}\mathstrut +\mathstrut \) \(636647559392169381751509367\) \(\beta_{7}\mathstrut +\mathstrut \) \(4296951890495206610245700095\) \(\beta_{6}\mathstrut +\mathstrut \) \(16008547913499403311867503126\) \(\beta_{5}\mathstrut -\mathstrut \) \(84362003148849945685199795131\) \(\beta_{4}\mathstrut +\mathstrut \) \(2959374516673854343551438011808\) \(\beta_{3}\mathstrut -\mathstrut \) \(21912354638306140245825209795\) \(\beta_{2}\mathstrut +\mathstrut \) \(75141917911109879124482619192634\) \(\beta_{1}\mathstrut +\mathstrut \) \(5328498935572703820390304631525019\)\()/8957952\)
\(\nu^{14}\)\(=\)\((\)\(39099854420784832667266324740\) \(\beta_{15}\mathstrut +\mathstrut \) \(12655054690553983152817319436\) \(\beta_{14}\mathstrut -\mathstrut \) \(102902939757098112608604931323\) \(\beta_{13}\mathstrut -\mathstrut \) \(3072940060710679722497349849\) \(\beta_{12}\mathstrut +\mathstrut \) \(18061255589944322885401820628\) \(\beta_{11}\mathstrut -\mathstrut \) \(18894456232399774492999490742\) \(\beta_{10}\mathstrut -\mathstrut \) \(60439382996269945550039533932\) \(\beta_{9}\mathstrut +\mathstrut \) \(17810381705957773177281755562\) \(\beta_{8}\mathstrut +\mathstrut \) \(57019293512197034044547344161\) \(\beta_{7}\mathstrut -\mathstrut \) \(1475621709640624326105824550343\) \(\beta_{6}\mathstrut +\mathstrut \) \(1602663117135235777276867764897\) \(\beta_{5}\mathstrut -\mathstrut \) \(35964744051791627793212014864072\) \(\beta_{4}\mathstrut +\mathstrut \) \(677356701304683860265748569959140\) \(\beta_{3}\mathstrut +\mathstrut \) \(2624742917712809934120572047923\) \(\beta_{2}\mathstrut +\mathstrut \) \(3211777401089462157041293014772260\) \(\beta_{1}\mathstrut -\mathstrut \) \(1418599400812729725342948943034991664\)\()/8957952\)
\(\nu^{15}\)\(=\)\((\)\(52\!\cdots\!22\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\!\cdots\!02\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\!\cdots\!17\) \(\beta_{13}\mathstrut -\mathstrut \) \(39\!\cdots\!87\) \(\beta_{12}\mathstrut +\mathstrut \) \(33\!\cdots\!02\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\!\cdots\!22\) \(\beta_{10}\mathstrut -\mathstrut \) \(70\!\cdots\!66\) \(\beta_{9}\mathstrut +\mathstrut \) \(98\!\cdots\!16\) \(\beta_{8}\mathstrut -\mathstrut \) \(46\!\cdots\!49\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\!\cdots\!53\) \(\beta_{6}\mathstrut +\mathstrut \) \(30\!\cdots\!51\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\!\cdots\!38\) \(\beta_{4}\mathstrut -\mathstrut \) \(18\!\cdots\!90\) \(\beta_{3}\mathstrut +\mathstrut \) \(42\!\cdots\!01\) \(\beta_{2}\mathstrut -\mathstrut \) \(26\!\cdots\!42\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\!\cdots\!78\)\()/4478976\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.857908 1.48594i
0.857908 + 1.48594i
36.4487 + 63.1310i
36.4487 63.1310i
−54.1541 + 93.7976i
−54.1541 93.7976i
72.8587 + 126.195i
72.8587 126.195i
−66.3279 + 114.883i
−66.3279 114.883i
54.8669 + 95.0323i
54.8669 95.0323i
−39.4461 + 68.3227i
−39.4461 68.3227i
−4.60413 + 7.97459i
−4.60413 7.97459i
−254.964 23.0089i 3788.00i 64477.2 + 11732.9i −363465. 87157.5 965802.i 2.07768e6i −1.61694e7 4.47501e6i −1.43489e7 9.26704e7 + 8.36292e6i
7.2 −254.964 + 23.0089i 3788.00i 64477.2 11732.9i −363465. 87157.5 + 965802.i 2.07768e6i −1.61694e7 + 4.47501e6i −1.43489e7 9.26704e7 8.36292e6i
7.3 −235.541 100.281i 3788.00i 45423.4 + 47240.7i 699204. −379865. + 892229.i 4.77594e6i −5.96172e6 1.56822e7i −1.43489e7 −1.64691e8 7.01170e7i
7.4 −235.541 + 100.281i 3788.00i 45423.4 47240.7i 699204. −379865. 892229.i 4.77594e6i −5.96172e6 + 1.56822e7i −1.43489e7 −1.64691e8 + 7.01170e7i
7.5 −141.143 213.576i 3788.00i −25693.4 + 60289.4i 110106. 809025. 534648.i 7.35151e6i 1.65028e7 3.02193e6i −1.43489e7 −1.55406e7 2.35159e7i
7.6 −141.143 + 213.576i 3788.00i −25693.4 60289.4i 110106. 809025. + 534648.i 7.35151e6i 1.65028e7 + 3.02193e6i −1.43489e7 −1.55406e7 + 2.35159e7i
7.7 −119.478 226.409i 3788.00i −36986.2 + 54101.7i −481576. −857637. + 452581.i 515341.i 1.66681e7 + 1.91007e6i −1.43489e7 5.75375e7 + 1.09033e8i
7.8 −119.478 + 226.409i 3788.00i −36986.2 54101.7i −481576. −857637. 452581.i 515341.i 1.66681e7 1.91007e6i −1.43489e7 5.75375e7 1.09033e8i
7.9 11.3734 255.747i 3788.00i −65277.3 5817.42i 86591.3 968769. + 43082.3i 1.07917e7i −2.23021e6 + 1.66283e7i −1.43489e7 984836. 2.21455e7i
7.10 11.3734 + 255.747i 3788.00i −65277.3 + 5817.42i 86591.3 968769. 43082.3i 1.07917e7i −2.23021e6 1.66283e7i −1.43489e7 984836. + 2.21455e7i
7.11 196.501 164.084i 3788.00i 11689.0 64485.1i −45026.4 −621548. 744344.i 3.24149e6i −8.28406e6 1.45894e7i −1.43489e7 −8.84773e6 + 7.38811e6i
7.12 196.501 + 164.084i 3788.00i 11689.0 + 64485.1i −45026.4 −621548. + 744344.i 3.24149e6i −8.28406e6 + 1.45894e7i −1.43489e7 −8.84773e6 7.38811e6i
7.13 197.709 162.626i 3788.00i 12641.5 64305.2i 551815. 616027. + 748920.i 6.23463e6i −7.95836e6 1.47695e7i −1.43489e7 1.09099e8 8.97395e7i
7.14 197.709 + 162.626i 3788.00i 12641.5 + 64305.2i 551815. 616027. 748920.i 6.23463e6i −7.95836e6 + 1.47695e7i −1.43489e7 1.09099e8 + 8.97395e7i
7.15 252.543 41.9299i 3788.00i 62019.8 21178.2i −380576. 158830. + 956631.i 8.07663e6i 1.47746e7 7.94889e6i −1.43489e7 −9.61118e7 + 1.59575e7i
7.16 252.543 + 41.9299i 3788.00i 62019.8 + 21178.2i −380576. 158830. 956631.i 8.07663e6i 1.47746e7 + 7.94889e6i −1.43489e7 −9.61118e7 1.59575e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{17}^{\mathrm{new}}(12, [\chi])\).