Properties

Label 8.15.d.b
Level 8
Weight 15
Character orbit 8.d
Analytic conductor 9.946
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 15 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.94631745215\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{66}\cdot 3^{6}\cdot 5^{2}\cdot 7 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 18 + \beta_{1} ) q^{2} \) \( + ( -252 + 3 \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -2573 + 21 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( -13 + 75 \beta_{1} - \beta_{3} + \beta_{6} ) q^{5} \) \( + ( 43283 - 306 \beta_{1} + 3 \beta_{2} + 40 \beta_{3} - \beta_{7} ) q^{6} \) \( + ( 173 - 990 \beta_{1} - 7 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{7} \) \( + ( -90994 - 2526 \beta_{1} + 21 \beta_{2} + 124 \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{8} \) \( + ( 1097776 + 6150 \beta_{1} - 49 \beta_{2} - 452 \beta_{3} + 19 \beta_{4} + 4 \beta_{6} - 6 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 18 + \beta_{1} ) q^{2} \) \( + ( -252 + 3 \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -2573 + 21 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( -13 + 75 \beta_{1} - \beta_{3} + \beta_{6} ) q^{5} \) \( + ( 43283 - 306 \beta_{1} + 3 \beta_{2} + 40 \beta_{3} - \beta_{7} ) q^{6} \) \( + ( 173 - 990 \beta_{1} - 7 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{7} \) \( + ( -90994 - 2526 \beta_{1} + 21 \beta_{2} + 124 \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{8} \) \( + ( 1097776 + 6150 \beta_{1} - 49 \beta_{2} - 452 \beta_{3} + 19 \beta_{4} + 4 \beta_{6} - 6 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{9} \) \( + ( -1211100 + 1346 \beta_{1} + 57 \beta_{2} + 539 \beta_{3} - 18 \beta_{4} + 4 \beta_{5} + 13 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 4 \beta_{10} + 4 \beta_{11} ) q^{10} \) \( + ( -2358815 + 18535 \beta_{1} - 236 \beta_{2} - 2216 \beta_{3} + 63 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} - 6 \beta_{10} + 4 \beta_{11} ) q^{11} \) \( + ( 2901070 + 40911 \beta_{1} - 277 \beta_{2} - 4534 \beta_{3} + 120 \beta_{4} + 24 \beta_{5} - 21 \beta_{6} - 62 \beta_{7} + 9 \beta_{8} + 4 \beta_{9} - \beta_{10} - 9 \beta_{11} ) q^{12} \) \( + ( -2353 + 14435 \beta_{1} - 254 \beta_{2} - 85 \beta_{3} - 22 \beta_{4} + 8 \beta_{5} + 45 \beta_{6} + 92 \beta_{7} + 26 \beta_{8} + 4 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} ) q^{13} \) \( + ( 15966992 - 17562 \beta_{1} - 704 \beta_{2} - 518 \beta_{3} + 216 \beta_{4} - 72 \beta_{5} - 372 \beta_{6} + 24 \beta_{7} + 10 \beta_{8} - 30 \beta_{9} - 10 \beta_{10} - 10 \beta_{11} ) q^{14} \) \( + ( 38309 - 217962 \beta_{1} - 1391 \beta_{2} + 2102 \beta_{3} + 478 \beta_{4} + 27 \beta_{5} + 55 \beta_{6} - 584 \beta_{7} + 52 \beta_{8} + 72 \beta_{9} - 12 \beta_{10} + 4 \beta_{11} ) q^{15} \) \( + ( -15488484 - 75122 \beta_{1} - 3416 \beta_{2} - 34924 \beta_{3} + 982 \beta_{4} - 176 \beta_{5} + 1308 \beta_{6} - 316 \beta_{7} - 32 \beta_{8} + 46 \beta_{9} - 50 \beta_{10} - 2 \beta_{11} ) q^{16} \) \( + ( 22562377 - 26638 \beta_{1} + 10237 \beta_{2} + 68780 \beta_{3} - 511 \beta_{4} - 4 \beta_{6} - 530 \beta_{7} - 29 \beta_{8} - 147 \beta_{9} - 29 \beta_{10} + 90 \beta_{11} ) q^{17} \) \( + ( 118259182 + 983439 \beta_{1} + 5380 \beta_{2} + 49624 \beta_{3} - 1608 \beta_{4} + 288 \beta_{5} + 4884 \beta_{6} + 544 \beta_{7} + 60 \beta_{8} - 136 \beta_{9} + 28 \beta_{10} - 132 \beta_{11} ) q^{18} \) \( + ( -206988079 + 761311 \beta_{1} + 9076 \beta_{2} - 127408 \beta_{3} + 2735 \beta_{4} - 350 \beta_{6} + 764 \beta_{7} + 186 \beta_{8} + 284 \beta_{9} + 186 \beta_{10} - 60 \beta_{11} ) q^{19} \) \( + ( -139735288 - 1074258 \beta_{1} + 4270 \beta_{2} + 69916 \beta_{3} - 2600 \beta_{4} + 144 \beta_{5} - 10062 \beta_{6} - 164 \beta_{7} - 346 \beta_{8} + 304 \beta_{9} - 206 \beta_{10} + 66 \beta_{11} ) q^{20} \) \( + ( -67038 + 597270 \beta_{1} - 59810 \beta_{2} + 8778 \beta_{3} - 1674 \beta_{4} - 264 \beta_{5} - 1314 \beta_{6} + 4068 \beta_{7} - 186 \beta_{8} - 580 \beta_{9} + 358 \beta_{10} + 222 \beta_{11} ) q^{21} \) \( + ( 253938811 - 2703594 \beta_{1} + 32787 \beta_{2} - 54600 \beta_{3} - 10800 \beta_{4} + 448 \beta_{5} - 18312 \beta_{6} + 3479 \beta_{7} - 472 \beta_{8} - 112 \beta_{9} - 152 \beta_{10} - 88 \beta_{11} ) q^{22} \) \( + ( 835315 - 4425070 \beta_{1} - 129369 \beta_{2} + 66626 \beta_{3} + 7418 \beta_{4} - 315 \beta_{5} - 2895 \beta_{6} - 2216 \beta_{7} - 476 \beta_{8} - 344 \beta_{9} + 740 \beta_{10} + 436 \beta_{11} ) q^{23} \) \( + ( 631029332 + 2199768 \beta_{1} + 40050 \beta_{2} - 727856 \beta_{3} - 19322 \beta_{4} + 2016 \beta_{5} + 31714 \beta_{6} + 2104 \beta_{7} - 518 \beta_{8} + 286 \beta_{9} + 68 \beta_{10} - 1052 \beta_{11} ) q^{24} \) \( + ( -1321429857 - 621148 \beta_{1} + 220986 \beta_{2} + 32976 \beta_{3} - 1926 \beta_{4} - 3416 \beta_{6} + 1276 \beta_{7} + 1910 \beta_{8} + 1274 \beta_{9} + 1910 \beta_{10} + 116 \beta_{11} ) q^{25} \) \( + ( -230504196 + 285470 \beta_{1} - 8657 \beta_{2} + 1336973 \beta_{3} + 3970 \beta_{4} - 4644 \beta_{5} + 28315 \beta_{6} + 2702 \beta_{7} - 2782 \beta_{8} + 265 \beta_{9} - 1548 \beta_{10} - 212 \beta_{11} ) q^{26} \) \( + ( -1382061735 + 8178354 \beta_{1} + 451332 \beta_{2} + 1526493 \beta_{3} + 9819 \beta_{4} - 5814 \beta_{6} - 6228 \beta_{7} + 450 \beta_{8} - 1332 \beta_{9} + 450 \beta_{10} + 3348 \beta_{11} ) q^{27} \) \( + ( 2109814608 + 13955388 \beta_{1} - 3492 \beta_{2} + 2726008 \beta_{3} - 33904 \beta_{4} - 5600 \beta_{5} - 12508 \beta_{6} + 10744 \beta_{7} + 76 \beta_{8} - 2624 \beta_{9} - 316 \beta_{10} - 4124 \beta_{11} ) q^{28} \) \( + ( 670169 - 1732159 \beta_{1} - 569520 \beta_{2} + 142461 \beta_{3} + 12048 \beta_{4} + 4032 \beta_{5} - 829 \beta_{6} - 49312 \beta_{7} + 4496 \beta_{8} + 5024 \beta_{9} + 1168 \beta_{10} + 2000 \beta_{11} ) q^{29} \) \( + ( 3514082512 - 4052610 \beta_{1} - 181088 \beta_{2} - 7983806 \beta_{3} + 102520 \beta_{4} + 3864 \beta_{5} + 49180 \beta_{6} - 26568 \beta_{7} - 2126 \beta_{8} + 1866 \beta_{9} - 3570 \beta_{10} - 4850 \beta_{11} ) q^{30} \) \( + ( 9195088 - 51332708 \beta_{1} - 752240 \beta_{2} + 591780 \beta_{3} + 82868 \beta_{4} + 1980 \beta_{5} - 4256 \beta_{6} + 46536 \beta_{7} + 9996 \beta_{8} - 968 \beta_{9} + 844 \beta_{10} + 3132 \beta_{11} ) q^{31} \) \( + ( 3191341352 - 19662116 \beta_{1} + 63784 \beta_{2} - 8703256 \beta_{3} - 149804 \beta_{4} - 4320 \beta_{5} - 115504 \beta_{6} + 26984 \beta_{7} - 3848 \beta_{8} - 7068 \beta_{9} - 9300 \beta_{10} + 2188 \beta_{11} ) q^{32} \) \( + ( -11352032091 - 18822738 \beta_{1} + 107187 \beta_{2} - 5634692 \beta_{3} - 3945 \beta_{4} - 6060 \beta_{6} - 750 \beta_{7} - 6339 \beta_{8} - 3357 \beta_{9} - 6339 \beta_{10} + 7878 \beta_{11} ) q^{33} \) \( + ( 31679260 + 23454990 \beta_{1} + 3084 \beta_{2} + 12799624 \beta_{3} + 235560 \beta_{4} + 24416 \beta_{5} - 225348 \beta_{6} - 42784 \beta_{7} + 10804 \beta_{8} + 11112 \beta_{9} - 2604 \beta_{10} - 8204 \beta_{11} ) q^{34} \) \( + ( 12467185102 + 168678088 \beta_{1} + 49688 \beta_{2} - 867814 \beta_{3} + 427674 \beta_{4} + 26956 \beta_{6} - 55384 \beta_{7} + 8636 \beta_{8} - 9528 \beta_{9} + 8636 \beta_{10} - 13032 \beta_{11} ) q^{35} \) \( + ( -12560745519 + 134955951 \beta_{1} + 536243 \beta_{2} + 22221264 \beta_{3} - 509824 \beta_{4} + 38592 \beta_{5} + 474200 \beta_{6} - 22640 \beta_{7} - 9016 \beta_{8} + 480 \beta_{9} - 13512 \beta_{10} + 15608 \beta_{11} ) q^{36} \) \( + ( -49208399 + 287780221 \beta_{1} + 160206 \beta_{2} - 2410027 \beta_{3} - 475866 \beta_{4} - 37576 \beta_{5} - 35821 \beta_{6} + 61700 \beta_{7} - 5002 \beta_{8} - 6756 \beta_{9} - 106 \beta_{10} - 1330 \beta_{11} ) q^{37} \) \( + ( 8594201523 - 221570362 \beta_{1} + 812491 \beta_{2} - 16050952 \beta_{3} + 458704 \beta_{4} - 46656 \beta_{5} + 494456 \beta_{6} + 92335 \beta_{7} + 11048 \beta_{8} + 9616 \beta_{9} + 5224 \beta_{10} + 15528 \beta_{11} ) q^{38} \) \( + ( 80800967 - 480729402 \beta_{1} + 1982891 \beta_{2} + 3492542 \beta_{3} + 820006 \beta_{4} - 6243 \beta_{5} + 138685 \beta_{6} - 150464 \beta_{7} - 30368 \beta_{8} + 3008 \beta_{9} - 1184 \beta_{10} - 8480 \beta_{11} ) q^{39} \) \( + ( -5582007488 - 137088392 \beta_{1} - 697368 \beta_{2} - 44295024 \beta_{3} - 695792 \beta_{4} - 41280 \beta_{5} - 679560 \beta_{6} - 182096 \beta_{7} + 21608 \beta_{8} + 26896 \beta_{9} + 45160 \beta_{10} - 20824 \beta_{11} ) q^{40} \) \( + ( 22165180052 - 783829652 \beta_{1} - 993810 \beta_{2} + 19491824 \beta_{3} - 2082002 \beta_{4} + 40440 \beta_{6} - 10636 \beta_{7} + 25378 \beta_{8} + 10030 \beta_{9} + 25378 \beta_{10} - 37924 \beta_{11} ) q^{41} \) \( + ( -9165496560 + 12268152 \beta_{1} + 444964 \beta_{2} + 64860732 \beta_{3} + 1478648 \beta_{4} - 29232 \beta_{5} - 502444 \beta_{6} + 210760 \beta_{7} - 45304 \beta_{8} - 81908 \beta_{9} + 29792 \beta_{10} + 38528 \beta_{11} ) q^{42} \) \( + ( 2609846406 + 499504659 \beta_{1} - 5583352 \beta_{2} + 5957215 \beta_{3} + 1142902 \beta_{4} - 126188 \beta_{6} + 478296 \beta_{7} - 62204 \beta_{8} + 88472 \beta_{9} - 62204 \beta_{10} + 49960 \beta_{11} ) q^{43} \) \( + ( -48844392578 + 315063447 \beta_{1} - 3165613 \beta_{2} + 3811770 \beta_{3} - 1622984 \beta_{4} - 83880 \beta_{5} - 468429 \beta_{6} + 44146 \beta_{7} + 55457 \beta_{8} + 53220 \beta_{9} + 97959 \beta_{10} - 31777 \beta_{11} ) q^{44} \) \( + ( -438559465 + 2525131443 \beta_{1} + 12409854 \beta_{2} - 23770669 \beta_{3} - 4419434 \beta_{4} + 236664 \beta_{5} + 339925 \beta_{6} + 711844 \beta_{7} - 65690 \beta_{8} - 67844 \beta_{9} - 26746 \beta_{10} - 36482 \beta_{11} ) q^{45} \) \( + ( 72072130096 - 78602558 \beta_{1} - 3214688 \beta_{2} - 13980994 \beta_{3} + 4144136 \beta_{4} + 139496 \beta_{5} - 1173596 \beta_{6} - 6840 \beta_{7} - 36338 \beta_{8} - 144714 \beta_{9} + 59570 \beta_{10} + 22450 \beta_{11} ) q^{46} \) \( + ( 431624378 - 2578353256 \beta_{1} + 13828434 \beta_{2} + 18751680 \beta_{3} + 4108512 \beta_{4} + 162 \beta_{5} - 1242514 \beta_{6} - 319720 \beta_{7} - 61756 \beta_{8} + 30440 \beta_{9} - 48764 \beta_{10} - 52012 \beta_{11} ) q^{47} \) \( + ( -199138291384 + 866060916 \beta_{1} - 623520 \beta_{2} - 46110792 \beta_{3} - 5359564 \beta_{4} + 263136 \beta_{5} + 2473304 \beta_{6} + 483864 \beta_{7} - 10768 \beta_{8} + 23684 \beta_{9} - 47852 \beta_{10} + 166964 \beta_{11} ) q^{48} \) \( + ( -53461085111 - 2653894768 \beta_{1} - 15251992 \beta_{2} - 59160416 \beta_{3} - 6109880 \beta_{4} + 212576 \beta_{6} - 32144 \beta_{7} - 133224 \beta_{8} - 74648 \beta_{9} - 133224 \beta_{10} - 2352 \beta_{11} ) q^{49} \) \( + ( -32172385966 - 1309118399 \beta_{1} - 2518952 \beta_{2} + 16145808 \beta_{3} + 7850192 \beta_{4} - 187200 \beta_{5} + 3600952 \beta_{6} - 247232 \beta_{7} + 266920 \beta_{8} + 224592 \beta_{9} + 31720 \beta_{10} + 55848 \beta_{11} ) q^{50} \) \( + ( 395634753739 + 4134178746 \beta_{1} - 17587348 \beta_{2} - 11990605 \beta_{3} + 10370113 \beta_{4} + 761086 \beta_{6} - 940284 \beta_{7} + 76678 \beta_{8} - 196732 \beta_{9} + 76678 \beta_{10} - 320516 \beta_{11} ) q^{51} \) \( + ( 66523706808 - 314233822 \beta_{1} + 3040802 \beta_{2} - 59534204 \beta_{3} - 8943256 \beta_{4} - 159504 \beta_{5} - 5000322 \beta_{6} - 1313724 \beta_{7} + 8554 \beta_{8} - 142256 \beta_{9} - 130050 \beta_{10} + 168046 \beta_{11} ) q^{52} \) \( + ( -846524487 + 4969339645 \beta_{1} - 194478 \beta_{2} - 40888003 \beta_{3} - 7475206 \beta_{4} - 1049400 \beta_{5} - 844069 \beta_{6} - 1898052 \beta_{7} + 108714 \beta_{8} + 233892 \beta_{9} - 78966 \beta_{10} - 32046 \beta_{11} ) q^{53} \) \( + ( 109120591590 - 1524822732 \beta_{1} + 10824462 \beta_{2} + 237806688 \beta_{3} + 13205520 \beta_{4} + 163008 \beta_{5} - 4778280 \beta_{6} - 1049970 \beta_{7} + 433224 \beta_{8} + 530640 \beta_{9} - 45432 \beta_{10} - 37944 \beta_{11} ) q^{54} \) \( + ( 1228364425 - 7201587094 \beta_{1} - 4935067 \beta_{2} + 57611474 \beta_{3} + 12683690 \beta_{4} + 77475 \beta_{5} + 7935987 \beta_{6} + 2390176 \beta_{7} + 331888 \beta_{8} - 57504 \beta_{9} - 62096 \beta_{10} + 36400 \beta_{11} ) q^{55} \) \( + ( -87702377472 + 2252076208 \beta_{1} + 7981520 \beta_{2} + 372761504 \beta_{3} - 15094816 \beta_{4} - 379008 \beta_{5} + 4596592 \beta_{6} - 2197024 \beta_{7} + 124368 \beta_{8} - 295712 \beta_{9} + 43664 \beta_{10} - 226032 \beta_{11} ) q^{56} \) \( + ( -622460306515 - 5632776546 \beta_{1} + 2761099 \beta_{2} - 231980996 \beta_{3} - 12248977 \beta_{4} - 328012 \beta_{6} + 639906 \beta_{7} - 30811 \beta_{8} + 144571 \beta_{9} - 30811 \beta_{10} + 107126 \beta_{11} ) q^{57} \) \( + ( 32133167276 - 42579738 \beta_{1} - 1102085 \beta_{2} - 610561855 \beta_{3} + 22759210 \beta_{4} + 637260 \beta_{5} + 2845111 \beta_{6} - 2078810 \beta_{7} - 524582 \beta_{8} - 262739 \beta_{9} - 170956 \beta_{10} - 316724 \beta_{11} ) q^{58} \) \( + ( 100881984372 + 7301851723 \beta_{1} + 66789344 \beta_{2} + 421471401 \beta_{3} + 15021488 \beta_{4} - 1139168 \beta_{6} - 514816 \beta_{7} - 105280 \beta_{8} - 181344 \beta_{9} - 105280 \beta_{10} + 712896 \beta_{11} ) q^{59} \) \( + ( 361621739920 + 3143081196 \beta_{1} + 4841100 \beta_{2} - 821383208 \beta_{3} - 28431152 \beta_{4} + 957600 \beta_{5} + 4844788 \beta_{6} + 7040856 \beta_{7} - 231620 \beta_{8} - 160064 \beta_{9} - 174316 \beta_{10} - 153932 \beta_{11} ) q^{60} \) \( + ( -2370361189 + 14099521711 \beta_{1} - 51898318 \beta_{2} - 105531865 \beta_{3} - 23877030 \beta_{4} + 3276616 \beta_{5} + 2254193 \beta_{6} - 3385860 \beta_{7} + 215562 \beta_{8} + 218980 \beta_{9} + 270058 \beta_{10} + 256434 \beta_{11} ) q^{61} \) \( + ( 830320482496 - 895364648 \beta_{1} - 51510624 \beta_{2} + 751957960 \beta_{3} + 21131808 \beta_{4} - 1800864 \beta_{5} + 7570512 \beta_{6} + 1805216 \beta_{7} - 1543704 \beta_{8} - 611352 \beta_{9} - 383912 \beta_{10} + 43352 \beta_{11} ) q^{62} \) \( + ( 3191281067 - 18299072742 \beta_{1} - 104931969 \beta_{2} + 186720986 \beta_{3} + 26530930 \beta_{4} - 260955 \beta_{5} - 31222919 \beta_{6} - 164312 \beta_{7} - 383972 \beta_{8} - 418856 \beta_{9} + 609116 \beta_{10} + 360844 \beta_{11} ) q^{63} \) \( + ( -1401489382480 + 4853133416 \beta_{1} - 26000496 \beta_{2} + 227324400 \beta_{3} - 26900840 \beta_{4} - 1173824 \beta_{5} - 17356288 \beta_{6} + 10375280 \beta_{7} - 357904 \beta_{8} + 149368 \beta_{9} + 40968 \beta_{10} - 735928 \beta_{11} ) q^{64} \) \( + ( -670521434310 - 11669103940 \beta_{1} + 151056438 \beta_{2} + 94447184 \beta_{3} - 29544554 \beta_{4} - 1907880 \beta_{6} + 45604 \beta_{7} + 1450202 \beta_{8} + 736502 \beta_{9} + 1450202 \beta_{10} - 139412 \beta_{11} ) q^{65} \) \( + ( -503836692208 - 11006384616 \beta_{1} - 7607052 \beta_{2} - 154234760 \beta_{3} - 242472 \beta_{4} + 189600 \beta_{5} - 25044540 \beta_{6} + 6742688 \beta_{7} - 240564 \beta_{8} + 364440 \beta_{9} - 197460 \beta_{10} + 42636 \beta_{11} ) q^{66} \) \( + ( -777603295255 + 10361266959 \beta_{1} + 100543092 \beta_{2} - 78152472 \beta_{3} + 25852791 \beta_{4} - 2805102 \beta_{6} + 3255900 \beta_{7} + 594186 \beta_{8} + 1111068 \beta_{9} + 594186 \beta_{10} + 549924 \beta_{11} ) q^{67} \) \( + ( 2692046159622 - 3056926462 \beta_{1} + 37312122 \beta_{2} - 551762320 \beta_{3} - 19997312 \beta_{4} - 188352 \beta_{5} + 30563464 \beta_{6} - 15920208 \beta_{7} - 200488 \beta_{8} + 960416 \beta_{9} + 721704 \beta_{10} - 1547160 \beta_{11} ) q^{68} \) \( + ( -1036699458 + 6641789226 \beta_{1} - 169859726 \beta_{2} - 14818794 \beta_{3} - 11535462 \beta_{4} - 6789816 \beta_{5} - 9213054 \beta_{6} + 15981948 \beta_{7} - 661686 \beta_{8} - 1993756 \beta_{9} + 882730 \beta_{10} + 496626 \beta_{11} ) q^{69} \) \( + ( 2931829640800 + 9347165328 \beta_{1} + 189147856 \beta_{2} + 644270816 \beta_{3} + 21984 \beta_{4} + 2522752 \beta_{5} + 35213136 \beta_{6} + 3409984 \beta_{7} + 924272 \beta_{8} - 583584 \beta_{9} + 183792 \beta_{10} - 1119376 \beta_{11} ) q^{70} \) \( + ( 780640753 - 4342517770 \beta_{1} - 82587987 \beta_{2} + 22871846 \beta_{3} + 19945038 \beta_{4} + 65511 \beta_{5} + 84033243 \beta_{6} - 14977432 \beta_{7} - 277636 \beta_{8} + 959384 \beta_{9} + 392380 \beta_{10} + 224876 \beta_{11} ) q^{71} \) \( + ( -3639461796070 - 8691759786 \beta_{1} + 98508679 \beta_{2} + 810784148 \beta_{3} - 13923553 \beta_{4} + 4412160 \beta_{5} - 25528597 \beta_{6} - 24933696 \beta_{7} - 1840981 \beta_{8} + 1683787 \beta_{9} - 2033248 \beta_{10} + 2146976 \beta_{11} ) q^{72} \) \( + ( 301192597711 + 6490218070 \beta_{1} + 17635991 \beta_{2} + 1338199132 \beta_{3} + 7145659 \beta_{4} + 1611492 \beta_{6} - 8201302 \beta_{7} - 1929031 \beta_{8} - 3014841 \beta_{9} - 1929031 \beta_{10} + 1666190 \beta_{11} ) q^{73} \) \( + ( -4625578877884 + 5141587714 \beta_{1} + 247784737 \beta_{2} + 790828003 \beta_{3} - 11236898 \beta_{4} - 3093660 \beta_{5} - 16485899 \beta_{6} + 3502866 \beta_{7} + 710846 \beta_{8} - 965465 \beta_{9} - 70324 \beta_{10} + 104084 \beta_{11} ) q^{74} \) \( + ( 755363595274 + 2025028131 \beta_{1} - 243818632 \beta_{2} - 2015566837 \beta_{3} + 19853602 \beta_{4} + 6587452 \beta_{6} - 2500152 \beta_{7} + 1393900 \beta_{8} + 71912 \beta_{9} + 1393900 \beta_{10} - 4026632 \beta_{11} ) q^{75} \) \( + ( 2793559697454 + 5684269103 \beta_{1} - 185125045 \beta_{2} + 104509642 \beta_{3} + 67291384 \beta_{4} - 5243368 \beta_{5} - 15823573 \beta_{6} + 21395522 \beta_{7} - 580919 \beta_{8} - 291964 \beta_{9} - 3064897 \beta_{10} + 3090871 \beta_{11} ) q^{76} \) \( + ( 3044142218 - 18764987410 \beta_{1} + 254072070 \beta_{2} + 94527186 \beta_{3} + 29828926 \beta_{4} + 6976152 \beta_{5} + 12901430 \beta_{6} + 188436 \beta_{7} + 1661646 \beta_{8} + 1283340 \beta_{9} - 1490322 \beta_{10} - 702330 \beta_{11} ) q^{77} \) \( + ( 7710079290160 - 8613029262 \beta_{1} - 383268160 \beta_{2} - 2300994386 \beta_{3} - 56345208 \beta_{4} + 6250536 \beta_{5} - 22802652 \beta_{6} - 5460152 \beta_{7} + 4272606 \beta_{8} + 1434982 \beta_{9} + 707618 \beta_{10} + 386082 \beta_{11} ) q^{78} \) \( + ( -6536547894 + 37185681716 \beta_{1} + 384615074 \beta_{2} - 320917148 \beta_{3} - 85089900 \beta_{4} + 1771806 \beta_{5} - 192825842 \beta_{6} + 3429344 \beta_{7} + 4442192 \beta_{8} + 2757664 \beta_{9} - 3125424 \beta_{10} - 1233520 \beta_{11} ) q^{79} \) \( + ( -7238184729408 + 2483508736 \beta_{1} - 218823200 \beta_{2} - 4653615360 \beta_{3} + 140535584 \beta_{4} - 1598976 \beta_{5} + 64149728 \beta_{6} + 29439872 \beta_{7} + 5208416 \beta_{8} - 1721696 \beta_{9} + 3909696 \beta_{10} - 2938304 \beta_{11} ) q^{80} \) \( + ( 4697897380950 + 38001837294 \beta_{1} - 535668141 \beta_{2} - 3212228580 \beta_{3} + 117069015 \beta_{4} + 6973716 \beta_{6} + 3866706 \beta_{7} - 2484771 \beta_{8} - 275709 \beta_{9} - 2484771 \beta_{10} - 2106618 \beta_{11} ) q^{81} \) \( + ( -12191249187244 + 36712483386 \beta_{1} - 1026552824 \beta_{2} + 999353008 \beta_{3} - 17615760 \beta_{4} + 99904 \beta_{5} + 109878696 \beta_{6} - 34355520 \beta_{7} + 617720 \beta_{8} - 2272784 \beta_{9} + 839864 \beta_{10} - 543880 \beta_{11} ) q^{82} \) \( + ( -1546473851124 - 98608005165 \beta_{1} - 196030720 \beta_{2} + 2962688969 \beta_{3} - 265191528 \beta_{4} - 3574448 \beta_{6} + 4334880 \beta_{7} - 8201360 \beta_{8} - 3016960 \beta_{9} - 8201360 \beta_{10} + 7396384 \beta_{11} ) q^{83} \) \( + ( 15583324317344 - 26762159976 \beta_{1} + 190129560 \beta_{2} + 9075710000 \beta_{3} + 169554912 \beta_{4} + 3806016 \beta_{5} - 119374872 \beta_{6} - 33870544 \beta_{7} + 4345080 \beta_{8} - 1310016 \beta_{9} + 4793064 \beta_{10} + 1028136 \beta_{11} ) q^{84} \) \( + ( 19945894360 - 119715859580 \beta_{1} + 811475450 \beta_{2} + 805226776 \beta_{3} + 205869634 \beta_{4} + 6949096 \beta_{5} + 19130976 \beta_{6} - 60662036 \beta_{7} + 581554 \beta_{8} + 7337972 \beta_{9} - 4177902 \beta_{10} - 2988038 \beta_{11} ) q^{85} \) \( + ( 8018067429067 - 6802192210 \beta_{1} + 712026571 \beta_{2} - 7703003000 \beta_{3} - 149451744 \beta_{4} - 20575872 \beta_{5} - 179556432 \beta_{6} - 9181385 \beta_{7} - 11456880 \beta_{8} - 2085472 \beta_{9} - 638192 \beta_{10} + 8613264 \beta_{11} ) q^{86} \) \( + ( -37184150881 + 214787970690 \beta_{1} + 719580323 \beta_{2} - 2090712286 \beta_{3} - 319715558 \beta_{4} - 4095423 \beta_{5} + 400947637 \beta_{6} + 59019112 \beta_{7} - 9718532 \beta_{8} - 8050024 \beta_{9} - 215492 \beta_{10} - 2591252 \beta_{11} ) q^{87} \) \( + ( -13902749976332 - 34106645416 \beta_{1} + 96961346 \beta_{2} - 1918203440 \beta_{3} + 290697782 \beta_{4} - 10205984 \beta_{5} + 149691570 \beta_{6} - 23407240 \beta_{7} + 4182282 \beta_{8} - 6331506 \beta_{9} + 4313764 \beta_{10} + 1410372 \beta_{11} ) q^{88} \) \( + ( 4547550220327 + 123903487270 \beta_{1} - 565843169 \beta_{2} + 5733672028 \beta_{3} + 257565699 \beta_{4} - 2382460 \beta_{6} + 46435418 \beta_{7} + 7787889 \beta_{8} + 15502799 \beta_{9} + 7787889 \beta_{10} - 10454114 \beta_{11} ) q^{89} \) \( + ( -40672501912244 + 45088734822 \beta_{1} + 2094182211 \beta_{2} + 8680062889 \beta_{3} - 458544550 \beta_{4} + 11233836 \beta_{5} + 77286527 \beta_{6} + 22000502 \beta_{7} + 6144986 \beta_{8} + 14794997 \beta_{9} + 4919044 \beta_{10} + 7188956 \beta_{11} ) q^{90} \) \( + ( 3083918817538 - 151877180936 \beta_{1} - 328732248 \beta_{2} + 168537174 \beta_{3} - 374053290 \beta_{4} + 19625684 \beta_{6} - 27757288 \beta_{7} + 3417700 \beta_{8} - 5230472 \beta_{9} + 3417700 \beta_{10} - 8906456 \beta_{11} ) q^{91} \) \( + ( 34428952912496 + 34710692372 \beta_{1} + 309421172 \beta_{2} + 8739962408 \beta_{3} + 246860592 \beta_{4} + 25455456 \beta_{5} - 29255668 \beta_{6} + 62062248 \beta_{7} + 3225540 \beta_{8} - 2181824 \beta_{9} + 11198956 \beta_{10} - 4548020 \beta_{11} ) q^{92} \) \( + ( 42223899136 - 247560276768 \beta_{1} - 54044272 \beta_{2} + 2082016768 \beta_{3} + 404406224 \beta_{4} - 41737920 \beta_{5} - 64582720 \beta_{6} + 46967264 \beta_{7} - 18034864 \beta_{8} - 12281568 \beta_{9} + 4712016 \beta_{10} - 974704 \beta_{11} ) q^{93} \) \( + ( 41339984334176 - 46095681612 \beta_{1} - 2133474464 \beta_{2} - 7087366420 \beta_{3} - 388077680 \beta_{4} + 2890512 \beta_{5} - 20559416 \beta_{6} - 19780976 \beta_{7} + 18696556 \beta_{8} + 14964124 \beta_{9} + 5802836 \beta_{10} - 7585196 \beta_{11} ) q^{94} \) \( + ( -36961310447 + 218615840090 \beta_{1} - 326578995 \beta_{2} - 1484562238 \beta_{3} - 455267366 \beta_{4} - 2439909 \beta_{5} - 663615285 \beta_{6} + 35874144 \beta_{7} - 12671856 \beta_{8} - 10014048 \beta_{9} + 5601168 \beta_{10} + 1032912 \beta_{11} ) q^{95} \) \( + ( -51317811651536 - 144941810616 \beta_{1} + 430346224 \beta_{2} + 2327261232 \beta_{3} + 257127448 \beta_{4} + 7298496 \beta_{5} - 185461088 \beta_{6} + 32856624 \beta_{7} - 19785392 \beta_{8} + 8921592 \beta_{9} - 11249112 \beta_{10} + 19647592 \beta_{11} ) q^{96} \) \( + ( 6120367928201 + 155186006514 \beta_{1} + 383454333 \beta_{2} - 13555559220 \beta_{3} + 487781793 \beta_{4} - 7585476 \beta_{6} - 24229266 \beta_{7} - 14561373 \beta_{8} - 13338003 \beta_{9} - 14561373 \beta_{10} + 17742426 \beta_{11} ) q^{97} \) \( + ( -43637038566030 - 4393221455 \beta_{1} - 3090471328 \beta_{2} - 2946797504 \beta_{3} - 536380096 \beta_{4} + 10144512 \beta_{5} - 257759264 \beta_{6} + 41167616 \beta_{7} - 18931808 \beta_{8} - 15112384 \beta_{9} - 2167648 \beta_{10} - 2757216 \beta_{11} ) q^{98} \) \( + ( -18616970820572 - 138297353271 \beta_{1} + 366430784 \beta_{2} - 11904639797 \beta_{3} - 260075384 \beta_{4} - 14655632 \beta_{6} + 51487968 \beta_{7} + 18792208 \beta_{8} + 22268096 \beta_{9} + 18792208 \beta_{10} - 13202336 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 218q^{2} \) \(\mathstrut -\mathstrut 3024q^{3} \) \(\mathstrut -\mathstrut 30828q^{4} \) \(\mathstrut +\mathstrut 518556q^{6} \) \(\mathstrut -\mathstrut 1097608q^{8} \) \(\mathstrut +\mathstrut 13188036q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 218q^{2} \) \(\mathstrut -\mathstrut 3024q^{3} \) \(\mathstrut -\mathstrut 30828q^{4} \) \(\mathstrut +\mathstrut 518556q^{6} \) \(\mathstrut -\mathstrut 1097608q^{8} \) \(\mathstrut +\mathstrut 13188036q^{9} \) \(\mathstrut -\mathstrut 14533440q^{10} \) \(\mathstrut -\mathstrut 28256720q^{11} \) \(\mathstrut +\mathstrut 34920024q^{12} \) \(\mathstrut +\mathstrut 191568384q^{14} \) \(\mathstrut -\mathstrut 185822448q^{16} \) \(\mathstrut +\mathstrut 270339544q^{17} \) \(\mathstrut +\mathstrut 1420811358q^{18} \) \(\mathstrut -\mathstrut 2481505872q^{19} \) \(\mathstrut -\mathstrut 1679371200q^{20} \) \(\mathstrut +\mathstrut 3042383484q^{22} \) \(\mathstrut +\mathstrut 7581335184q^{24} \) \(\mathstrut -\mathstrut 15857276820q^{25} \) \(\mathstrut -\mathstrut 2773507776q^{26} \) \(\mathstrut -\mathstrut 16574868000q^{27} \) \(\mathstrut +\mathstrut 25329333120q^{28} \) \(\mathstrut +\mathstrut 42207767040q^{30} \) \(\mathstrut +\mathstrut 38309251808q^{32} \) \(\mathstrut -\mathstrut 136227597840q^{33} \) \(\mathstrut +\mathstrut 350437044q^{34} \) \(\mathstrut +\mathstrut 149949623040q^{35} \) \(\mathstrut -\mathstrut 150590403492q^{36} \) \(\mathstrut +\mathstrut 102789916636q^{38} \) \(\mathstrut -\mathstrut 66999085440q^{40} \) \(\mathstrut +\mathstrut 264287409880q^{41} \) \(\mathstrut -\mathstrut 110343609600q^{42} \) \(\mathstrut +\mathstrut 32253127344q^{43} \) \(\mathstrut -\mathstrut 585547356392q^{44} \) \(\mathstrut +\mathstrut 864780977664q^{46} \) \(\mathstrut -\mathstrut 2387663418144q^{48} \) \(\mathstrut -\mathstrut 646589230644q^{49} \) \(\mathstrut -\mathstrut 388785556630q^{50} \) \(\mathstrut +\mathstrut 4755867895776q^{51} \) \(\mathstrut +\mathstrut 798005307840q^{52} \) \(\mathstrut +\mathstrut 1305053764344q^{54} \) \(\mathstrut -\mathstrut 1050155264256q^{56} \) \(\mathstrut -\mathstrut 7479401742480q^{57} \) \(\mathstrut +\mathstrut 389204742720q^{58} \) \(\mathstrut +\mathstrut 1223083947184q^{59} \) \(\mathstrut +\mathstrut 4350689397120q^{60} \) \(\mathstrut +\mathstrut 9957296947200q^{62} \) \(\mathstrut -\mathstrut 16809671099328q^{64} \) \(\mathstrut -\mathstrut 8069319822720q^{65} \) \(\mathstrut -\mathstrut 6067132925784q^{66} \) \(\mathstrut -\mathstrut 9309378171216q^{67} \) \(\mathstrut +\mathstrut 32301846360616q^{68} \) \(\mathstrut +\mathstrut 35197935521280q^{70} \) \(\mathstrut -\mathstrut 43695386222808q^{72} \) \(\mathstrut +\mathstrut 3619334364696q^{73} \) \(\mathstrut -\mathstrut 55499920147776q^{74} \) \(\mathstrut +\mathstrut 9079078926000q^{75} \) \(\mathstrut +\mathstrut 33532610502360q^{76} \) \(\mathstrut +\mathstrut 92515055193600q^{78} \) \(\mathstrut -\mathstrut 86826189154560q^{80} \) \(\mathstrut +\mathstrut 56467107312444q^{81} \) \(\mathstrut -\mathstrut 146233962574956q^{82} \) \(\mathstrut -\mathstrut 18774355695824q^{83} \) \(\mathstrut +\mathstrut 186893160787200q^{84} \) \(\mathstrut +\mathstrut 96253393476220q^{86} \) \(\mathstrut -\mathstrut 166888683024624q^{88} \) \(\mathstrut +\mathstrut 54781416936088q^{89} \) \(\mathstrut -\mathstrut 488020221650880q^{90} \) \(\mathstrut +\mathstrut 36699395136768q^{91} \) \(\mathstrut +\mathstrut 413167093560960q^{92} \) \(\mathstrut +\mathstrut 496016398930944q^{94} \) \(\mathstrut -\mathstrut 616114307580864q^{96} \) \(\mathstrut +\mathstrut 73839238696536q^{97} \) \(\mathstrut -\mathstrut 523654870565638q^{98} \) \(\mathstrut -\mathstrut 223606851712368q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(x^{11}\mathstrut +\mathstrut \) \(4349\) \(x^{10}\mathstrut -\mathstrut \) \(33891\) \(x^{9}\mathstrut +\mathstrut \) \(12151288\) \(x^{8}\mathstrut -\mathstrut \) \(474141530\) \(x^{7}\mathstrut +\mathstrut \) \(82897017850\) \(x^{6}\mathstrut -\mathstrut \) \(1813403462870\) \(x^{5}\mathstrut +\mathstrut \) \(371618750972305\) \(x^{4}\mathstrut -\mathstrut \) \(24344904673267125\) \(x^{3}\mathstrut +\mathstrut \) \(1896685485375873305\) \(x^{2}\mathstrut -\mathstrut \) \(84437535339775221175\) \(x\mathstrut +\mathstrut \) \(3788457560343891547750\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} + 30 \nu + 2897 \)
\(\beta_{3}\)\(=\)\((\)\(636923\) \(\nu^{11}\mathstrut -\mathstrut \) \(78759886\) \(\nu^{10}\mathstrut -\mathstrut \) \(245335731\) \(\nu^{9}\mathstrut -\mathstrut \) \(64007142854\) \(\nu^{8}\mathstrut +\mathstrut \) \(12787711625758\) \(\nu^{7}\mathstrut -\mathstrut \) \(822864962230860\) \(\nu^{6}\mathstrut +\mathstrut \) \(38168394038046922\) \(\nu^{5}\mathstrut -\mathstrut \) \(4507406695463155436\) \(\nu^{4}\mathstrut +\mathstrut \) \(196713423019633802871\) \(\nu^{3}\mathstrut -\mathstrut \) \(27580525290189265452454\) \(\nu^{2}\mathstrut +\mathstrut \) \(2265819948914359980944729\) \(\nu\mathstrut -\mathstrut \) \(82011248452843967275766670\)\()/\)\(22\!\cdots\!56\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(5206759\) \(\nu^{11}\mathstrut +\mathstrut \) \(414020870\) \(\nu^{10}\mathstrut -\mathstrut \) \(7749618561\) \(\nu^{9}\mathstrut +\mathstrut \) \(469354871518\) \(\nu^{8}\mathstrut -\mathstrut \) \(89854084665494\) \(\nu^{7}\mathstrut +\mathstrut \) \(5306347000024956\) \(\nu^{6}\mathstrut -\mathstrut \) \(369199877153340914\) \(\nu^{5}\mathstrut +\mathstrut \) \(27809217572005527196\) \(\nu^{4}\mathstrut -\mathstrut \) \(1782483399516517486419\) \(\nu^{3}\mathstrut +\mathstrut \) \(194321775926330540726846\) \(\nu^{2}\mathstrut -\mathstrut \) \(15253324623605251498253245\) \(\nu\mathstrut +\mathstrut \) \(598672526304648296090644422\)\()/\)\(22\!\cdots\!56\)
\(\beta_{5}\)\(=\)\((\)\(360140747\) \(\nu^{11}\mathstrut +\mathstrut \) \(10850839058\) \(\nu^{10}\mathstrut +\mathstrut \) \(582416384253\) \(\nu^{9}\mathstrut -\mathstrut \) \(21486353930342\) \(\nu^{8}\mathstrut +\mathstrut \) \(485958641556286\) \(\nu^{7}\mathstrut -\mathstrut \) \(205804244689519500\) \(\nu^{6}\mathstrut +\mathstrut \) \(8061288714189591850\) \(\nu^{5}\mathstrut -\mathstrut \) \(605231613961170249260\) \(\nu^{4}\mathstrut +\mathstrut \) \(52649340440165565281415\) \(\nu^{3}\mathstrut -\mathstrut \) \(6332850867802448927992390\) \(\nu^{2}\mathstrut +\mathstrut \) \(197050364851263859276882505\) \(\nu\mathstrut -\mathstrut \) \(4061769132841273325626680750\)\()/\)\(11\!\cdots\!80\)
\(\beta_{6}\)\(=\)\((\)\(376164913\) \(\nu^{11}\mathstrut +\mathstrut \) \(12640903702\) \(\nu^{10}\mathstrut +\mathstrut \) \(313658894247\) \(\nu^{9}\mathstrut -\mathstrut \) \(19280426207698\) \(\nu^{8}\mathstrut +\mathstrut \) \(2938344023187674\) \(\nu^{7}\mathstrut +\mathstrut \) \(32957617914724380\) \(\nu^{6}\mathstrut +\mathstrut \) \(15325664195736620510\) \(\nu^{5}\mathstrut +\mathstrut \) \(73464750376148841020\) \(\nu^{4}\mathstrut +\mathstrut \) \(63586983269347749419685\) \(\nu^{3}\mathstrut -\mathstrut \) \(5426328849705924542157170\) \(\nu^{2}\mathstrut +\mathstrut \) \(194271218906035271071941835\) \(\nu\mathstrut +\mathstrut \) \(8236891518202354624091557590\)\()/\)\(11\!\cdots\!80\)
\(\beta_{7}\)\(=\)\((\)\(21282279\) \(\nu^{11}\mathstrut +\mathstrut \) \(537238778\) \(\nu^{10}\mathstrut +\mathstrut \) \(9930623105\) \(\nu^{9}\mathstrut -\mathstrut \) \(1438088847582\) \(\nu^{8}\mathstrut +\mathstrut \) \(165384536100502\) \(\nu^{7}\mathstrut +\mathstrut \) \(1394777169372292\) \(\nu^{6}\mathstrut +\mathstrut \) \(943065164025030642\) \(\nu^{5}\mathstrut -\mathstrut \) \(2400591736048773788\) \(\nu^{4}\mathstrut +\mathstrut \) \(3559635806048480056915\) \(\nu^{3}\mathstrut -\mathstrut \) \(340290779426096136970302\) \(\nu^{2}\mathstrut +\mathstrut \) \(13252906367497316123141309\) \(\nu\mathstrut +\mathstrut \) \(393790269268466722674975290\)\()/\)\(28\!\cdots\!32\)
\(\beta_{8}\)\(=\)\((\)\(820762567\) \(\nu^{11}\mathstrut +\mathstrut \) \(105005693498\) \(\nu^{10}\mathstrut +\mathstrut \) \(1459283923873\) \(\nu^{9}\mathstrut +\mathstrut \) \(30272617601378\) \(\nu^{8}\mathstrut +\mathstrut \) \(4229904890704726\) \(\nu^{7}\mathstrut +\mathstrut \) \(879525037032530180\) \(\nu^{6}\mathstrut +\mathstrut \) \(30354398449499581490\) \(\nu^{5}\mathstrut +\mathstrut \) \(3628448353700553379300\) \(\nu^{4}\mathstrut +\mathstrut \) \(94665584366638009672115\) \(\nu^{3}\mathstrut +\mathstrut \) \(3405367897627263492075010\) \(\nu^{2}\mathstrut -\mathstrut \) \(952180020063505682487309475\) \(\nu\mathstrut +\mathstrut \) \(95367616757518272988621522810\)\()/\)\(56\!\cdots\!40\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(622670923\) \(\nu^{11}\mathstrut -\mathstrut \) \(41902712082\) \(\nu^{10}\mathstrut -\mathstrut \) \(799090716797\) \(\nu^{9}\mathstrut +\mathstrut \) \(1364985661798\) \(\nu^{8}\mathstrut -\mathstrut \) \(5168586570637374\) \(\nu^{7}\mathstrut -\mathstrut \) \(300559052599069300\) \(\nu^{6}\mathstrut -\mathstrut \) \(26309215888906247210\) \(\nu^{5}\mathstrut -\mathstrut \) \(1029657760752504080340\) \(\nu^{4}\mathstrut -\mathstrut \) \(77637113904662411551495\) \(\nu^{3}\mathstrut +\mathstrut \) \(5033247080617616288224070\) \(\nu^{2}\mathstrut +\mathstrut \) \(37237238715775551067335735\) \(\nu\mathstrut -\mathstrut \) \(34683844151182914221888535890\)\()/\)\(28\!\cdots\!20\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(3005874881\) \(\nu^{11}\mathstrut +\mathstrut \) \(197184062346\) \(\nu^{10}\mathstrut -\mathstrut \) \(915675936919\) \(\nu^{9}\mathstrut +\mathstrut \) \(191680338864626\) \(\nu^{8}\mathstrut -\mathstrut \) \(46509445082619258\) \(\nu^{7}\mathstrut +\mathstrut \) \(2551035903996273700\) \(\nu^{6}\mathstrut -\mathstrut \) \(177877629101997158590\) \(\nu^{5}\mathstrut +\mathstrut \) \(11818343045624571355140\) \(\nu^{4}\mathstrut -\mathstrut \) \(855272312491856255357045\) \(\nu^{3}\mathstrut +\mathstrut \) \(96151805278962788007336850\) \(\nu^{2}\mathstrut -\mathstrut \) \(7466917062940221526804590075\) \(\nu\mathstrut +\mathstrut \) \(233222657372625932600091601610\)\()/\)\(11\!\cdots\!80\)
\(\beta_{11}\)\(=\)\((\)\(94370893\) \(\nu^{11}\mathstrut -\mathstrut \) \(10467446018\) \(\nu^{10}\mathstrut -\mathstrut \) \(118995218133\) \(\nu^{9}\mathstrut -\mathstrut \) \(16384430595658\) \(\nu^{8}\mathstrut +\mathstrut \) \(1514521522443314\) \(\nu^{7}\mathstrut -\mathstrut \) \(120332685879686100\) \(\nu^{6}\mathstrut +\mathstrut \) \(5303221063690968230\) \(\nu^{5}\mathstrut -\mathstrut \) \(596104088455717667380\) \(\nu^{4}\mathstrut +\mathstrut \) \(26131313172188141719665\) \(\nu^{3}\mathstrut -\mathstrut \) \(3984869906043932426877290\) \(\nu^{2}\mathstrut +\mathstrut \) \(298231825939385452674963775\) \(\nu\mathstrut -\mathstrut \) \(10392967654102722820898782530\)\()/\)\(35\!\cdots\!40\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(15\) \(\beta_{1}\mathstrut -\mathstrut \) \(2897\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(124\) \(\beta_{3}\mathstrut -\mathstrut \) \(33\) \(\beta_{2}\mathstrut -\mathstrut \) \(2688\) \(\beta_{1}\mathstrut +\mathstrut \) \(59612\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(25\) \(\beta_{10}\mathstrut -\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(52\) \(\beta_{8}\mathstrut -\mathstrut \) \(158\) \(\beta_{7}\mathstrut +\mathstrut \) \(618\) \(\beta_{6}\mathstrut -\mathstrut \) \(88\) \(\beta_{5}\mathstrut +\mathstrut \) \(599\) \(\beta_{4}\mathstrut -\mathstrut \) \(21926\) \(\beta_{3}\mathstrut -\mathstrut \) \(1492\) \(\beta_{2}\mathstrut +\mathstrut \) \(62123\) \(\beta_{1}\mathstrut -\mathstrut \) \(7126878\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(148\) \(\beta_{11}\mathstrut -\mathstrut \) \(300\) \(\beta_{10}\mathstrut -\mathstrut \) \(498\) \(\beta_{9}\mathstrut +\mathstrut \) \(142\) \(\beta_{8}\mathstrut +\mathstrut \) \(3464\) \(\beta_{7}\mathstrut -\mathstrut \) \(14374\) \(\beta_{6}\mathstrut +\mathstrut \) \(720\) \(\beta_{5}\mathstrut -\mathstrut \) \(15494\) \(\beta_{4}\mathstrut -\mathstrut \) \(322396\) \(\beta_{3}\mathstrut +\mathstrut \) \(23809\) \(\beta_{2}\mathstrut -\mathstrut \) \(1361576\) \(\beta_{1}\mathstrut +\mathstrut \) \(278006249\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(61372\) \(\beta_{11}\mathstrut +\mathstrut \) \(50148\) \(\beta_{10}\mathstrut +\mathstrut \) \(63727\) \(\beta_{9}\mathstrut -\mathstrut \) \(13405\) \(\beta_{8}\mathstrut +\mathstrut \) \(370328\) \(\beta_{7}\mathstrut +\mathstrut \) \(84899\) \(\beta_{6}\mathstrut -\mathstrut \) \(97664\) \(\beta_{5}\mathstrut -\mathstrut \) \(349973\) \(\beta_{4}\mathstrut +\mathstrut \) \(61442628\) \(\beta_{3}\mathstrut -\mathstrut \) \(3147858\) \(\beta_{2}\mathstrut +\mathstrut \) \(432994481\) \(\beta_{1}\mathstrut -\mathstrut \) \(113439771901\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(8045658\) \(\beta_{11}\mathstrut +\mathstrut \) \(2413130\) \(\beta_{10}\mathstrut -\mathstrut \) \(8887157\) \(\beta_{9}\mathstrut +\mathstrut \) \(1964433\) \(\beta_{8}\mathstrut -\mathstrut \) \(80674228\) \(\beta_{7}\mathstrut +\mathstrut \) \(99876373\) \(\beta_{6}\mathstrut -\mathstrut \) \(12830544\) \(\beta_{5}\mathstrut -\mathstrut \) \(88704881\) \(\beta_{4}\mathstrut -\mathstrut \) \(2821150776\) \(\beta_{3}\mathstrut +\mathstrut \) \(421886243\) \(\beta_{2}\mathstrut -\mathstrut \) \(121052558538\) \(\beta_{1}\mathstrut -\mathstrut \) \(344011645228\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(326554601\) \(\beta_{11}\mathstrut -\mathstrut \) \(409692737\) \(\beta_{10}\mathstrut +\mathstrut \) \(146557429\) \(\beta_{9}\mathstrut +\mathstrut \) \(388951174\) \(\beta_{8}\mathstrut +\mathstrut \) \(2602917362\) \(\beta_{7}\mathstrut -\mathstrut \) \(8717235932\) \(\beta_{6}\mathstrut +\mathstrut \) \(1164605096\) \(\beta_{5}\mathstrut +\mathstrut \) \(24912996257\) \(\beta_{4}\mathstrut +\mathstrut \) \(242936048722\) \(\beta_{3}\mathstrut -\mathstrut \) \(66968939622\) \(\beta_{2}\mathstrut +\mathstrut \) \(1361878039507\) \(\beta_{1}\mathstrut -\mathstrut \) \(362165003016430\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(9853827781\) \(\beta_{11}\mathstrut +\mathstrut \) \(6711836611\) \(\beta_{10}\mathstrut -\mathstrut \) \(7664756327\) \(\beta_{9}\mathstrut -\mathstrut \) \(32926070282\) \(\beta_{8}\mathstrut -\mathstrut \) \(42772439830\) \(\beta_{7}\mathstrut +\mathstrut \) \(299064219756\) \(\beta_{6}\mathstrut -\mathstrut \) \(27529757496\) \(\beta_{5}\mathstrut -\mathstrut \) \(659888956667\) \(\beta_{4}\mathstrut +\mathstrut \) \(483638245178\) \(\beta_{3}\mathstrut +\mathstrut \) \(1115988848286\) \(\beta_{2}\mathstrut -\mathstrut \) \(157847828452211\) \(\beta_{1}\mathstrut +\mathstrut \) \(53555545470848654\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(1038882617494\) \(\beta_{11}\mathstrut +\mathstrut \) \(171139828006\) \(\beta_{10}\mathstrut -\mathstrut \) \(245567624140\) \(\beta_{9}\mathstrut +\mathstrut \) \(2070000797374\) \(\beta_{8}\mathstrut -\mathstrut \) \(1972482864684\) \(\beta_{7}\mathstrut -\mathstrut \) \(14384845713126\) \(\beta_{6}\mathstrut +\mathstrut \) \(2597775388112\) \(\beta_{5}\mathstrut -\mathstrut \) \(18137652440268\) \(\beta_{4}\mathstrut -\mathstrut \) \(298302590947300\) \(\beta_{3}\mathstrut -\mathstrut \) \(112088970903349\) \(\beta_{2}\mathstrut +\mathstrut \) \(30956442780772239\) \(\beta_{1}\mathstrut -\mathstrut \) \(2417704660209493613\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(146144280031564\) \(\beta_{11}\mathstrut -\mathstrut \) \(17558620143468\) \(\beta_{10}\mathstrut +\mathstrut \) \(9708632887341\) \(\beta_{9}\mathstrut -\mathstrut \) \(259299801373831\) \(\beta_{8}\mathstrut +\mathstrut \) \(368930225175448\) \(\beta_{7}\mathstrut +\mathstrut \) \(1521068892873409\) \(\beta_{6}\mathstrut -\mathstrut \) \(51805429333920\) \(\beta_{5}\mathstrut +\mathstrut \) \(7265799510970905\) \(\beta_{4}\mathstrut +\mathstrut \) \(78825491288537076\) \(\beta_{3}\mathstrut +\mathstrut \) \(30336348517891583\) \(\beta_{2}\mathstrut -\mathstrut \) \(2712246345548709828\) \(\beta_{1}\mathstrut -\mathstrut \) \(235565938081267641604\)\()/8\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−64.5010 31.8690i
−64.5010 + 31.8690i
−37.2836 57.4111i
−37.2836 + 57.4111i
4.85446 62.4824i
4.85446 + 62.4824i
5.50725 62.3341i
5.50725 + 62.3341i
41.6607 39.1087i
41.6607 + 39.1087i
50.2622 24.1659i
50.2622 + 24.1659i
−111.002 63.7381i −2044.69 8258.92 + 14150.1i 54892.6i 226964. + 130324.i 432198.i −14856.1 2.09710e6i −602230. 3.49875e6 6.09319e6i
3.2 −111.002 + 63.7381i −2044.69 8258.92 14150.1i 54892.6i 226964. 130324.i 432198.i −14856.1 + 2.09710e6i −602230. 3.49875e6 + 6.09319e6i
3.3 −56.5672 114.822i 1525.47 −9984.31 + 12990.3i 71626.9i −86291.8 175158.i 647418.i 2.05637e6 + 411594.i −2.45590e6 −8.22436e6 + 4.05173e6i
3.4 −56.5672 + 114.822i 1525.47 −9984.31 12990.3i 71626.9i −86291.8 + 175158.i 647418.i 2.05637e6 411594.i −2.45590e6 −8.22436e6 4.05173e6i
3.5 27.7089 124.965i 563.873 −14848.4 6925.28i 134561.i 15624.3 70464.4i 255103.i −1.27685e6 + 1.66364e6i −4.46502e6 1.68154e7 + 3.72854e6i
3.6 27.7089 + 124.965i 563.873 −14848.4 + 6925.28i 134561.i 15624.3 + 70464.4i 255103.i −1.27685e6 1.66364e6i −4.46502e6 1.68154e7 3.72854e6i
3.7 29.0145 124.668i −3879.84 −14700.3 7234.37i 109885.i −112571. + 483692.i 642807.i −1.32842e6 + 1.62276e6i 1.02702e7 −1.36992e7 3.18827e6i
3.8 29.0145 + 124.668i −3879.84 −14700.3 + 7234.37i 109885.i −112571. 483692.i 642807.i −1.32842e6 1.62276e6i 1.02702e7 −1.36992e7 + 3.18827e6i
3.9 101.321 78.2175i 3476.15 4148.05 15850.2i 78301.9i 352208. 271895.i 1.22276e6i −819477. 1.93042e6i 7.30063e6 −6.12457e6 7.93365e6i
3.10 101.321 + 78.2175i 3476.15 4148.05 + 15850.2i 78301.9i 352208. + 271895.i 1.22276e6i −819477. + 1.93042e6i 7.30063e6 −6.12457e6 + 7.93365e6i
3.11 118.524 48.3317i −1152.97 11712.1 11457.0i 9668.61i −136655. + 55725.2i 1.34658e6i 834433. 1.92400e6i −3.45362e6 467301. + 1.14597e6i
3.12 118.524 + 48.3317i −1152.97 11712.1 + 11457.0i 9668.61i −136655. 55725.2i 1.34658e6i 834433. + 1.92400e6i −3.45362e6 467301. 1.14597e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{6} \) \(\mathstrut +\mathstrut 1512 T_{3}^{5} \) \(\mathstrut -\mathstrut 16502844 T_{3}^{4} \) \(\mathstrut -\mathstrut 18520805952 T_{3}^{3} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!44\)\( T_{3}^{2} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!84\)\( T_{3} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\( \) acting on \(S_{15}^{\mathrm{new}}(8, [\chi])\).