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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.94631745215\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} + 4349 x^{10} - 33891 x^{9} + 12151288 x^{8} - 474141530 x^{7} + 82897017850 x^{6} - 1813403462870 x^{5} + 371618750972305 x^{4} - 24344904673267125 x^{3} + 1896685485375873305 x^{2} - 84437535339775221175 x + 3788457560343891547750\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{66}\cdot 3^{6}\cdot 5^{2}\cdot 7 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 218q^{2} - 3024q^{3} - 30828q^{4} + 518556q^{6} - 1097608q^{8} + 13188036q^{9} + O(q^{10}) \) \( 12q + 218q^{2} - 3024q^{3} - 30828q^{4} + 518556q^{6} - 1097608q^{8} + 13188036q^{9} - 14533440q^{10} - 28256720q^{11} + 34920024q^{12} + 191568384q^{14} - 185822448q^{16} + 270339544q^{17} + 1420811358q^{18} - 2481505872q^{19} - 1679371200q^{20} + 3042383484q^{22} + 7581335184q^{24} - 15857276820q^{25} - 2773507776q^{26} - 16574868000q^{27} + 25329333120q^{28} + 42207767040q^{30} + 38309251808q^{32} - 136227597840q^{33} + 350437044q^{34} + 149949623040q^{35} - 150590403492q^{36} + 102789916636q^{38} - 66999085440q^{40} + 264287409880q^{41} - 110343609600q^{42} + 32253127344q^{43} - 585547356392q^{44} + 864780977664q^{46} - 2387663418144q^{48} - 646589230644q^{49} - 388785556630q^{50} + 4755867895776q^{51} + 798005307840q^{52} + 1305053764344q^{54} - 1050155264256q^{56} - 7479401742480q^{57} + 389204742720q^{58} + 1223083947184q^{59} + 4350689397120q^{60} + 9957296947200q^{62} - 16809671099328q^{64} - 8069319822720q^{65} - 6067132925784q^{66} - 9309378171216q^{67} + 32301846360616q^{68} + 35197935521280q^{70} - 43695386222808q^{72} + 3619334364696q^{73} - 55499920147776q^{74} + 9079078926000q^{75} + 33532610502360q^{76} + 92515055193600q^{78} - 86826189154560q^{80} + 56467107312444q^{81} - 146233962574956q^{82} - 18774355695824q^{83} + 186893160787200q^{84} + 96253393476220q^{86} - 166888683024624q^{88} + 54781416936088q^{89} - 488020221650880q^{90} + 36699395136768q^{91} + 413167093560960q^{92} + 496016398930944q^{94} - 616114307580864q^{96} + 73839238696536q^{97} - 523654870565638q^{98} - 223606851712368q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.15.d.b 12
3.b odd 2 1 72.15.b.b 12
4.b odd 2 1 32.15.d.b 12
8.b even 2 1 32.15.d.b 12
8.d odd 2 1 inner 8.15.d.b 12
24.f even 2 1 72.15.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.15.d.b 12 1.a even 1 1 trivial
8.15.d.b 12 8.d odd 2 1 inner
32.15.d.b 12 4.b odd 2 1
32.15.d.b 12 8.b even 2 1
72.15.b.b 12 3.b odd 2 1
72.15.b.b 12 24.f even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 218 T + 39176 T^{2} - 4721088 T^{3} + 545864704 T^{4} - 60071772160 T^{5} + 9530421280768 T^{6} - 984215915069440 T^{7} + 146529440732545024 T^{8} - 20763564607014961152 T^{9} + \)\(28\!\cdots\!36\)\( T^{10} - \)\(25\!\cdots\!32\)\( T^{11} + \)\(19\!\cdots\!16\)\( T^{12} \)
$3$ \( ( 1 + 1512 T + 12194970 T^{2} + 17638439688 T^{3} + 75281481066015 T^{4} + 110296673788274640 T^{5} + \)\(35\!\cdots\!48\)\( T^{6} + \)\(52\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!15\)\( T^{8} + \)\(19\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!70\)\( T^{10} + \)\(37\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!81\)\( T^{12} )^{2} \)
$5$ \( 1 - 28692455340 T^{2} + \)\(45\!\cdots\!50\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{6} + \)\(46\!\cdots\!75\)\( T^{8} - \)\(35\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(64\!\cdots\!75\)\( T^{16} - \)\(26\!\cdots\!00\)\( T^{18} + \)\(88\!\cdots\!50\)\( T^{20} - \)\(20\!\cdots\!00\)\( T^{22} + \)\(26\!\cdots\!25\)\( T^{24} \)
$7$ \( 1 - 3746043821772 T^{2} + \)\(72\!\cdots\!26\)\( T^{4} - \)\(97\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(89\!\cdots\!88\)\( T^{10} + \)\(66\!\cdots\!96\)\( T^{12} - \)\(41\!\cdots\!88\)\( T^{14} + \)\(21\!\cdots\!35\)\( T^{16} - \)\(94\!\cdots\!80\)\( T^{18} + \)\(32\!\cdots\!26\)\( T^{20} - \)\(77\!\cdots\!72\)\( T^{22} + \)\(94\!\cdots\!01\)\( T^{24} \)
$11$ \( ( 1 + 14128360 T + 1563532555046330 T^{2} + \)\(15\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(77\!\cdots\!40\)\( T^{5} + \)\(46\!\cdots\!76\)\( T^{6} + \)\(29\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} + \)\(85\!\cdots\!76\)\( T^{9} + \)\(32\!\cdots\!30\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} + \)\(29\!\cdots\!41\)\( T^{12} )^{2} \)
$13$ \( 1 - 25049990458333932 T^{2} + \)\(31\!\cdots\!66\)\( T^{4} - \)\(25\!\cdots\!60\)\( T^{6} + \)\(15\!\cdots\!75\)\( T^{8} - \)\(77\!\cdots\!88\)\( T^{10} + \)\(32\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!48\)\( T^{14} + \)\(37\!\cdots\!75\)\( T^{16} - \)\(95\!\cdots\!60\)\( T^{18} + \)\(18\!\cdots\!46\)\( T^{20} - \)\(22\!\cdots\!32\)\( T^{22} + \)\(13\!\cdots\!21\)\( T^{24} \)
$17$ \( ( 1 - 135169772 T + 447311642391670850 T^{2} - \)\(69\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} - \)\(29\!\cdots\!40\)\( T^{7} + \)\(36\!\cdots\!75\)\( T^{8} - \)\(33\!\cdots\!72\)\( T^{9} + \)\(35\!\cdots\!50\)\( T^{10} - \)\(18\!\cdots\!28\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \)
$19$ \( ( 1 + 1240752936 T + 3803834107333602138 T^{2} + \)\(36\!\cdots\!32\)\( T^{3} + \)\(64\!\cdots\!95\)\( T^{4} + \)\(49\!\cdots\!52\)\( T^{5} + \)\(64\!\cdots\!32\)\( T^{6} + \)\(39\!\cdots\!92\)\( T^{7} + \)\(41\!\cdots\!95\)\( T^{8} + \)\(18\!\cdots\!52\)\( T^{9} + \)\(15\!\cdots\!78\)\( T^{10} + \)\(40\!\cdots\!36\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \)
$23$ \( 1 - 64183863256723383372 T^{2} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(50\!\cdots\!80\)\( T^{6} + \)\(93\!\cdots\!95\)\( T^{8} - \)\(14\!\cdots\!88\)\( T^{10} + \)\(17\!\cdots\!56\)\( T^{12} - \)\(19\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!95\)\( T^{16} - \)\(12\!\cdots\!80\)\( T^{18} + \)\(69\!\cdots\!86\)\( T^{20} - \)\(28\!\cdots\!72\)\( T^{22} + \)\(58\!\cdots\!81\)\( T^{24} \)
$29$ \( 1 - \)\(12\!\cdots\!72\)\( T^{2} + \)\(75\!\cdots\!46\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(29\!\cdots\!12\)\( T^{10} + \)\(82\!\cdots\!44\)\( T^{12} - \)\(26\!\cdots\!32\)\( T^{14} + \)\(82\!\cdots\!15\)\( T^{16} - \)\(22\!\cdots\!40\)\( T^{18} + \)\(46\!\cdots\!86\)\( T^{20} - \)\(65\!\cdots\!72\)\( T^{22} + \)\(48\!\cdots\!61\)\( T^{24} \)
$31$ \( 1 - \)\(39\!\cdots\!32\)\( T^{2} + \)\(75\!\cdots\!06\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(10\!\cdots\!92\)\( T^{10} + \)\(83\!\cdots\!64\)\( T^{12} - \)\(59\!\cdots\!72\)\( T^{14} + \)\(37\!\cdots\!95\)\( T^{16} - \)\(19\!\cdots\!20\)\( T^{18} + \)\(81\!\cdots\!66\)\( T^{20} - \)\(24\!\cdots\!32\)\( T^{22} + \)\(35\!\cdots\!41\)\( T^{24} \)
$37$ \( 1 - \)\(86\!\cdots\!12\)\( T^{2} + \)\(35\!\cdots\!66\)\( T^{4} - \)\(93\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!35\)\( T^{8} - \)\(23\!\cdots\!68\)\( T^{10} + \)\(24\!\cdots\!16\)\( T^{12} - \)\(19\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!35\)\( T^{16} - \)\(50\!\cdots\!80\)\( T^{18} + \)\(15\!\cdots\!46\)\( T^{20} - \)\(30\!\cdots\!12\)\( T^{22} + \)\(28\!\cdots\!21\)\( T^{24} \)
$41$ \( ( 1 - 132143704940 T + \)\(12\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!44\)\( T^{3} + \)\(77\!\cdots\!35\)\( T^{4} - \)\(82\!\cdots\!20\)\( T^{5} + \)\(33\!\cdots\!96\)\( T^{6} - \)\(31\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!35\)\( T^{8} - \)\(84\!\cdots\!64\)\( T^{9} + \)\(26\!\cdots\!50\)\( T^{10} - \)\(10\!\cdots\!40\)\( T^{11} + \)\(29\!\cdots\!61\)\( T^{12} )^{2} \)
$43$ \( ( 1 - 16126563672 T + \)\(11\!\cdots\!66\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!19\)\( T^{4} + \)\(87\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!28\)\( T^{6} + \)\(64\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!19\)\( T^{8} + \)\(43\!\cdots\!92\)\( T^{9} + \)\(34\!\cdots\!66\)\( T^{10} - \)\(35\!\cdots\!28\)\( T^{11} + \)\(16\!\cdots\!01\)\( T^{12} )^{2} \)
$47$ \( 1 - \)\(12\!\cdots\!72\)\( T^{2} + \)\(79\!\cdots\!46\)\( T^{4} - \)\(36\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!95\)\( T^{8} - \)\(46\!\cdots\!28\)\( T^{10} + \)\(12\!\cdots\!96\)\( T^{12} - \)\(30\!\cdots\!08\)\( T^{14} + \)\(62\!\cdots\!95\)\( T^{16} - \)\(10\!\cdots\!20\)\( T^{18} + \)\(14\!\cdots\!86\)\( T^{20} - \)\(15\!\cdots\!72\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \)
$53$ \( 1 - \)\(46\!\cdots\!32\)\( T^{2} + \)\(72\!\cdots\!66\)\( T^{4} + \)\(59\!\cdots\!40\)\( T^{6} - \)\(10\!\cdots\!45\)\( T^{8} - \)\(23\!\cdots\!08\)\( T^{10} + \)\(25\!\cdots\!76\)\( T^{12} - \)\(45\!\cdots\!88\)\( T^{14} - \)\(39\!\cdots\!45\)\( T^{16} + \)\(41\!\cdots\!40\)\( T^{18} + \)\(95\!\cdots\!06\)\( T^{20} - \)\(11\!\cdots\!32\)\( T^{22} + \)\(47\!\cdots\!61\)\( T^{24} \)
$59$ \( ( 1 - 611541973592 T + \)\(18\!\cdots\!46\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!75\)\( T^{4} - \)\(21\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!36\)\( T^{6} - \)\(13\!\cdots\!88\)\( T^{7} + \)\(69\!\cdots\!75\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(26\!\cdots\!86\)\( T^{10} - \)\(55\!\cdots\!92\)\( T^{11} + \)\(56\!\cdots\!61\)\( T^{12} )^{2} \)
$61$ \( 1 - \)\(30\!\cdots\!72\)\( T^{2} + \)\(77\!\cdots\!46\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(20\!\cdots\!35\)\( T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(26\!\cdots\!44\)\( T^{12} - \)\(23\!\cdots\!52\)\( T^{14} + \)\(19\!\cdots\!35\)\( T^{16} - \)\(12\!\cdots\!60\)\( T^{18} + \)\(70\!\cdots\!66\)\( T^{20} - \)\(26\!\cdots\!72\)\( T^{22} + \)\(86\!\cdots\!81\)\( T^{24} \)
$67$ \( ( 1 + 4654689085608 T + \)\(18\!\cdots\!90\)\( T^{2} + \)\(53\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!95\)\( T^{4} + \)\(26\!\cdots\!60\)\( T^{5} + \)\(62\!\cdots\!08\)\( T^{6} + \)\(97\!\cdots\!40\)\( T^{7} + \)\(18\!\cdots\!95\)\( T^{8} + \)\(26\!\cdots\!68\)\( T^{9} + \)\(33\!\cdots\!90\)\( T^{10} + \)\(31\!\cdots\!92\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \)
$71$ \( 1 - \)\(59\!\cdots\!72\)\( T^{2} + \)\(18\!\cdots\!66\)\( T^{4} - \)\(36\!\cdots\!80\)\( T^{6} + \)\(55\!\cdots\!55\)\( T^{8} - \)\(64\!\cdots\!32\)\( T^{10} + \)\(60\!\cdots\!44\)\( T^{12} - \)\(44\!\cdots\!52\)\( T^{14} + \)\(25\!\cdots\!55\)\( T^{16} - \)\(11\!\cdots\!80\)\( T^{18} + \)\(39\!\cdots\!06\)\( T^{20} - \)\(88\!\cdots\!72\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \)
$73$ \( ( 1 - 1809667182348 T + \)\(55\!\cdots\!70\)\( T^{2} - \)\(98\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!35\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!48\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!35\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!70\)\( T^{10} - \)\(49\!\cdots\!52\)\( T^{11} + \)\(33\!\cdots\!41\)\( T^{12} )^{2} \)
$79$ \( 1 - \)\(13\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!06\)\( T^{4} - \)\(99\!\cdots\!20\)\( T^{6} + \)\(57\!\cdots\!95\)\( T^{8} - \)\(27\!\cdots\!72\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(37\!\cdots\!92\)\( T^{14} + \)\(10\!\cdots\!95\)\( T^{16} - \)\(24\!\cdots\!20\)\( T^{18} + \)\(47\!\cdots\!46\)\( T^{20} - \)\(64\!\cdots\!72\)\( T^{22} + \)\(63\!\cdots\!61\)\( T^{24} \)
$83$ \( ( 1 + 9387177847912 T + \)\(17\!\cdots\!90\)\( T^{2} + \)\(60\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!35\)\( T^{4} + \)\(36\!\cdots\!60\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} + \)\(26\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!35\)\( T^{8} + \)\(24\!\cdots\!12\)\( T^{9} + \)\(51\!\cdots\!90\)\( T^{10} + \)\(20\!\cdots\!88\)\( T^{11} + \)\(15\!\cdots\!21\)\( T^{12} )^{2} \)
$89$ \( ( 1 - 27390708468044 T + \)\(56\!\cdots\!78\)\( T^{2} + \)\(18\!\cdots\!92\)\( T^{3} + \)\(92\!\cdots\!95\)\( T^{4} + \)\(50\!\cdots\!32\)\( T^{5} + \)\(96\!\cdots\!92\)\( T^{6} + \)\(98\!\cdots\!12\)\( T^{7} + \)\(35\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!32\)\( T^{9} + \)\(82\!\cdots\!58\)\( T^{10} - \)\(78\!\cdots\!44\)\( T^{11} + \)\(56\!\cdots\!41\)\( T^{12} )^{2} \)
$97$ \( ( 1 - 36919619348268 T + \)\(26\!\cdots\!46\)\( T^{2} - \)\(35\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!59\)\( T^{4} + \)\(51\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!48\)\( T^{6} + \)\(33\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!99\)\( T^{8} - \)\(99\!\cdots\!52\)\( T^{9} + \)\(48\!\cdots\!66\)\( T^{10} - \)\(43\!\cdots\!32\)\( T^{11} + \)\(77\!\cdots\!81\)\( T^{12} )^{2} \)
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