Properties

Label 3.39.b.a
Level 3
Weight 39
Character orbit 3.b
Analytic conductor 27.439
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 39 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(27.4390407101\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{75}\cdot 3^{91}\cdot 5^{7} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -9561867 + 97 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -141606210704 - \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -1705115 \beta_{1} - 340 \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -40300933723344 - 5737864 \beta_{1} + 17 \beta_{2} + 707 \beta_{3} + 6 \beta_{4} - \beta_{7} ) q^{6} \) \( + ( 675656019711554 + 486451 \beta_{1} + 2066454 \beta_{2} + 346 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{7} \) \( + ( -106891803083 \beta_{1} - 45304936 \beta_{2} - 714 \beta_{3} + 1334 \beta_{4} - 9 \beta_{5} - 20 \beta_{7} + \beta_{8} ) q^{8} \) \( + ( -35359929288459495 - 284217479375 \beta_{1} - 1262720 \beta_{2} + 387217 \beta_{3} + 10480 \beta_{4} - 49 \beta_{5} - 10 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(-9561867 + 97 \beta_{1} + \beta_{2}) q^{3}\) \(+(-141606210704 - \beta_{2} + \beta_{3}) q^{4}\) \(+(-1705115 \beta_{1} - 340 \beta_{2} + \beta_{4}) q^{5}\) \(+(-40300933723344 - 5737864 \beta_{1} + 17 \beta_{2} + 707 \beta_{3} + 6 \beta_{4} - \beta_{7}) q^{6}\) \(+(675656019711554 + 486451 \beta_{1} + 2066454 \beta_{2} + 346 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7}) q^{7}\) \(+(-106891803083 \beta_{1} - 45304936 \beta_{2} - 714 \beta_{3} + 1334 \beta_{4} - 9 \beta_{5} - 20 \beta_{7} + \beta_{8}) q^{8}\) \(+(-35359929288459495 - 284217479375 \beta_{1} - 1262720 \beta_{2} + 387217 \beta_{3} + 10480 \beta_{4} - 49 \beta_{5} - 10 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + \beta_{9}) q^{9}\) \(+(710119790424444960 - 455618446 \beta_{1} - 1927629089 \beta_{2} - 8176408 \beta_{3} - 3179 \beta_{4} + 462 \beta_{5} + 194 \beta_{6} + 2527 \beta_{7} - \beta_{8} - 8 \beta_{9} + \beta_{11}) q^{10}\) \(+(27390754663596 \beta_{1} + 11652995750 \beta_{2} + 185972 \beta_{3} + 76601 \beta_{4} + 2341 \beta_{5} + 8 \beta_{6} + 4995 \beta_{7} - 293 \beta_{8} + 30 \beta_{9} + \beta_{10} + 3 \beta_{11}) q^{11}\) \(+(-238547044532722128 - 183484830441206 \beta_{1} - 136564546404 \beta_{2} - 150439761 \beta_{3} + 3388702 \beta_{4} - 11135 \beta_{5} - 14442 \beta_{6} + 3832 \beta_{7} + 1722 \beta_{8} - 72 \beta_{9} + 15 \beta_{10} + 9 \beta_{11}) q^{12}\) \(+(89091564444452625146 + 55263541281 \beta_{1} + 235496387523 \beta_{2} - 697666076 \beta_{3} - 134562 \beta_{4} - 27096 \beta_{5} + 84600 \beta_{6} + 668568 \beta_{7} + 329 \beta_{8} + 472 \beta_{9} + 135 \beta_{10} - 59 \beta_{11}) q^{13}\) \(+(600532489034223 \beta_{1} - 249327308600 \beta_{2} - 3440546 \beta_{3} - 20034253 \beta_{4} + 47672 \beta_{5} - 4076 \beta_{6} - 5541125 \beta_{7} - 1266 \beta_{8} - 1920 \beta_{9} + 827 \beta_{10} - 192 \beta_{11}) q^{14}\) \(+(\)\(52\!\cdots\!40\)\( + 6245149471880296 \beta_{1} - 186133268171 \beta_{2} - 17104287632 \beta_{3} + 347123922 \beta_{4} + 365178 \beta_{5} - 947259 \beta_{6} + 1557463 \beta_{7} - 54909 \beta_{8} + 2538 \beta_{9} + 3825 \beta_{10} - 621 \beta_{11}) q^{15}\) \(+(\)\(55\!\cdots\!88\)\( - 2494188410618 \beta_{1} - 10479902478018 \beta_{2} - 117396179496 \beta_{3} - 55711960 \beta_{4} + 787524 \beta_{5} + 479380 \beta_{6} + 94176272 \beta_{7} + 25078 \beta_{8} - 12784 \beta_{9} + 13338 \beta_{10} + 1598 \beta_{11}) q^{16}\) \(+(-56025599754603189 \beta_{1} - 42748812501027 \beta_{2} - 672767318 \beta_{3} - 2889669704 \beta_{4} - 5179330 \beta_{5} - 114200 \beta_{6} - 218023564 \beta_{7} + 608119 \beta_{8} + 57750 \beta_{9} + 34325 \beta_{10} + 5775 \beta_{11}) q^{17}\) \(+(\)\(11\!\cdots\!60\)\( - 128353304280714909 \beta_{1} - 35888093280609 \beta_{2} - 866153505228 \beta_{3} + 28821558693 \beta_{4} + 1244670 \beta_{5} + 34011114 \beta_{6} + 11978979 \beta_{7} + 802851 \beta_{8} - 58344 \beta_{9} + 52380 \beta_{10} + 20493 \beta_{11}) q^{18}\) \(+(-\)\(38\!\cdots\!30\)\( - 28220154698801 \beta_{1} - 122468212881275 \beta_{2} + 2568729226482 \beta_{3} - 4231165 \beta_{4} - 4345587 \beta_{5} - 77021570 \beta_{6} - 155809681 \beta_{7} - 24564 \beta_{8} + 206112 \beta_{9} - 25164 \beta_{10} - 25764 \beta_{11}) q^{19}\) \(+(2197892044416010190 \beta_{1} + 1259633825834640 \beta_{2} + 18372761060 \beta_{3} - 251726070196 \beta_{4} + 187040770 \beta_{5} + 1488800 \beta_{6} + 3200020480 \beta_{7} - 16660930 \beta_{8} - 1075200 \beta_{9} - 479720 \beta_{10} - 107520 \beta_{11}) q^{20}\) \(+(\)\(27\!\cdots\!46\)\( - 747998542258694032 \beta_{1} + 717046895828107 \beta_{2} - 15312896577090 \beta_{3} + 248149340371 \beta_{4} - 22506098 \beta_{5} - 344144652 \beta_{6} - 974793740 \beta_{7} + 1763409 \beta_{8} + 981378 \beta_{9} - 1739181 \beta_{10} - 429399 \beta_{11}) q^{21}\) \(+(-\)\(11\!\cdots\!80\)\( + 440436341457781 \beta_{1} + 1763408432235986 \beta_{2} + 107928692108020 \beta_{3} + 13560597245 \beta_{4} + 664168632 \beta_{5} + 1377885028 \beta_{6} - 25842855229 \beta_{7} - 7865138 \beta_{8} - 2101984 \beta_{9} - 3801195 \beta_{10} + 262748 \beta_{11}) q^{22}\) \(+(-26787086683905899794 \beta_{1} - 87257900318298 \beta_{2} - 828702692 \beta_{3} + 665830890518 \beta_{4} - 442538926 \beta_{5} + 21788800 \beta_{6} + 25425965682 \beta_{7} + 234600272 \beta_{8} + 13668000 \beta_{9} - 4080400 \beta_{10} + 1366800 \beta_{11}) q^{23}\) \(+(\)\(65\!\cdots\!32\)\( + 33791945754583867135 \beta_{1} - 638290082267978 \beta_{2} - 492514336252346 \beta_{3} - 3233275975530 \beta_{4} + 10096449639 \beta_{5} - 460338156 \beta_{6} + 27638759212 \beta_{7} - 252024597 \beta_{8} - 12820464 \beta_{9} + 5030298 \beta_{10} + 6392574 \beta_{11}) q^{24}\) \(+(-\)\(10\!\cdots\!35\)\( - 28443364351272709 \beta_{1} - 121377002786677411 \beta_{2} + 528314122722748 \beta_{3} - 243688213946 \beta_{4} + 17122514628 \beta_{5} - 9680661364 \beta_{6} + 273595097968 \beta_{7} + 75496811 \beta_{8} + 12266248 \beta_{9} + 36981765 \beta_{10} - 1533281 \beta_{11}) q^{25}\) \(+(\)\(25\!\cdots\!82\)\( \beta_{1} + 288297764264519200 \beta_{2} + 4788185111448 \beta_{3} + 22771577555084 \beta_{4} + 71216921424 \beta_{5} - 425984304 \beta_{6} - 525459889140 \beta_{7} - 2140307832 \beta_{8} - 122008320 \beta_{9} + 94295244 \beta_{10} - 12200832 \beta_{11}) q^{26}\) \(+(\)\(29\!\cdots\!17\)\( - \)\(24\!\cdots\!23\)\( \beta_{1} - 26344517184065691 \beta_{2} - 2010091010267556 \beta_{3} - 23150651113323 \beta_{4} + 70887934017 \beta_{5} + 41182835742 \beta_{6} - 132762058947 \beta_{7} + 4244825997 \beta_{8} + 134463330 \beta_{9} + 140225175 \beta_{10} - 71623035 \beta_{11}) q^{27}\) \(+(-\)\(64\!\cdots\!64\)\( - 336556120571587042 \beta_{1} - 1430394628658130668 \beta_{2} + 456785202946266 \beta_{3} - 1646137011896 \beta_{4} + 300597846612 \beta_{5} - 2484509212 \beta_{6} + 548972805328 \beta_{7} + 71382094 \beta_{8} - 6194608 \beta_{9} + 36078210 \beta_{10} + 774326 \beta_{11}) q^{28}\) \(+(-\)\(10\!\cdots\!27\)\( \beta_{1} + 2905747846092856550 \beta_{2} + 45665656678372 \beta_{3} - 65627849239509 \beta_{4} + 543397999436 \beta_{5} + 1900316944 \beta_{6} + 3087523112040 \beta_{7} + 12749585702 \beta_{8} + 731245020 \beta_{9} - 401954734 \beta_{10} + 73124502 \beta_{11}) q^{29}\) \(+(-\)\(26\!\cdots\!20\)\( + \)\(46\!\cdots\!87\)\( \beta_{1} + 558586540417140898 \beta_{2} + 18860294575218816 \beta_{3} + 174017843828123 \beta_{4} + 1902563908256 \beta_{5} - 402613028148 \beta_{6} + 3716965156241 \beta_{7} - 38850812238 \beta_{8} - 1151856864 \beta_{9} - 1323199005 \beta_{10} + 623197548 \beta_{11}) q^{30}\) \(+(\)\(52\!\cdots\!82\)\( - 2524872460226435945 \beta_{1} - 10707541626430795140 \beta_{2} - 19945971849264902 \beta_{3} - 4627687572952 \beta_{4} + 2812946199462 \beta_{5} + 561725838487 \beta_{6} - 12540081785041 \beta_{7} - 4504315596 \beta_{8} - 636667680 \beta_{9} - 2212366068 \beta_{10} + 79583460 \beta_{11}) q^{31}\) \(+(\)\(43\!\cdots\!28\)\( \beta_{1} + 27716359898457440000 \beta_{2} + 434473199020480 \beta_{3} - 1089116262972352 \beta_{4} + 5367848715488 \beta_{5} + 7055526400 \beta_{6} + 17841577607936 \beta_{7} - 33094777568 \beta_{8} - 2145792000 \beta_{9} - 1978460800 \beta_{10} - 214579200 \beta_{11}) q^{32}\) \(+(-\)\(16\!\cdots\!20\)\( - \)\(59\!\cdots\!04\)\( \beta_{1} + 82469908697798411 \beta_{2} + 136345866128782373 \beta_{3} + 873830119658700 \beta_{4} + 5493557745051 \beta_{5} + 1653083816922 \beta_{6} - 6985489227154 \beta_{7} + 225514914471 \beta_{8} + 8100337257 \beta_{9} + 2053426635 \beta_{10} - 4273162479 \beta_{11}) q^{33}\) \(+(\)\(23\!\cdots\!92\)\( - 17656281543064870852 \beta_{1} - 74786314788783110904 \beta_{2} - 230637631317283728 \beta_{3} - 111435459222836 \beta_{4} + 12931834300032 \beta_{5} - 4289186198704 \beta_{6} + 85171093040500 \beta_{7} + 21481371224 \beta_{8} + 6709291264 \beta_{9} + 10321354908 \beta_{10} - 838661408 \beta_{11}) q^{34}\) \(+(-\)\(26\!\cdots\!35\)\( \beta_{1} + 87410197297432307715 \beta_{2} + 1434849438702790 \beta_{3} + 4537769335210254 \beta_{4} + 20797400700980 \beta_{5} - 96758372600 \beta_{6} - 125784393587630 \beta_{7} - 224004072645 \beta_{8} - 9185974050 \beta_{9} + 23270995745 \beta_{10} - 918597405 \beta_{11}) q^{35}\) \(+(\)\(43\!\cdots\!12\)\( + \)\(24\!\cdots\!78\)\( \beta_{1} - 7436031932689446869 \beta_{2} - 434823822556177595 \beta_{3} - 7610473251121052 \beta_{4} + 34929411081014 \beta_{5} + 744774337688 \beta_{6} + 97193913857120 \beta_{7} - 843299988922 \beta_{8} - 46428255392 \beta_{9} + 27246173796 \beta_{10} + 23077495188 \beta_{11}) q^{36}\) \(+(-\)\(87\!\cdots\!46\)\( - 15158815315448853689 \beta_{1} - 64984693657887025427 \beta_{2} + 578717592374995740 \beta_{3} - 125984631923926 \beta_{4} + 16718934734928 \beta_{5} + 11330062646320 \beta_{6} + 121238547255800 \beta_{7} + 26724583887 \beta_{8} - 36050751384 \beta_{9} + 15615463905 \beta_{10} + 4506343923 \beta_{11}) q^{37}\) \(+(-\)\(10\!\cdots\!63\)\( \beta_{1} - \)\(18\!\cdots\!76\)\( \beta_{2} - 2876924533857614 \beta_{3} + 18660478413653325 \beta_{4} - 42141675757408 \beta_{5} + 267392848300 \beta_{6} + 272195666340597 \beta_{7} + 3225129786170 \beta_{8} + 166179552000 \beta_{9} - 50230256875 \beta_{10} + 16617955200 \beta_{11}) q^{38}\) \(+(\)\(31\!\cdots\!14\)\( + \)\(95\!\cdots\!05\)\( \beta_{1} + 62623026919039607065 \beta_{2} - 1830692252264024610 \beta_{3} - 11694291806969726 \beta_{4} - 115962178410020 \beta_{5} - 40185901161870 \beta_{6} - 323912344487636 \beta_{7} + 1819867478475 \beta_{8} + 211321005066 \beta_{9} - 135706363671 \beta_{10} - 96033816261 \beta_{11}) q^{39}\) \(+(-\)\(72\!\cdots\!20\)\( + \)\(26\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!48\)\( \beta_{2} + 6756095367820761616 \beta_{3} + 2125702709474928 \beta_{4} - 192549350989224 \beta_{5} + 40368305011512 \beta_{6} - 2158813038055584 \beta_{7} - 553444275228 \beta_{8} + 75592984416 \beta_{9} - 281446699140 \beta_{10} - 9449123052 \beta_{11}) q^{40}\) \(+(-\)\(19\!\cdots\!46\)\( \beta_{1} - \)\(79\!\cdots\!00\)\( \beta_{2} - 13613493799806032 \beta_{3} - 141011002948988106 \beta_{4} - 198567824538496 \beta_{5} + 493056035392 \beta_{6} + 1322572308335280 \beta_{7} - 19780150505992 \beta_{8} - 1187163904080 \beta_{9} - 241980399256 \beta_{10} - 118716390408 \beta_{11}) q^{41}\) \(+(\)\(31\!\cdots\!40\)\( + \)\(64\!\cdots\!16\)\( \beta_{1} - \)\(19\!\cdots\!43\)\( \beta_{2} - 2148665495400002360 \beta_{3} + 164745771961641903 \beta_{4} - 583820068344654 \beta_{5} + 153050465846286 \beta_{6} - 167607787451075 \beta_{7} - 1341244240263 \beta_{8} - 711458034552 \beta_{9} - 136829229660 \beta_{10} + 286711315119 \beta_{11}) q^{42}\) \(+(\)\(83\!\cdots\!66\)\( + \)\(86\!\cdots\!87\)\( \beta_{1} + \)\(36\!\cdots\!53\)\( \beta_{2} - 12495565869512929502 \beta_{3} + 1908490221361135 \beta_{4} - 1069547529267111 \beta_{5} - 461895699745198 \beta_{6} + 3867020126885095 \beta_{7} + 1419058886188 \beta_{8} + 463218481184 \beta_{9} + 680578288020 \beta_{10} - 57902310148 \beta_{11}) q^{43}\) \(+(-\)\(29\!\cdots\!46\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2} - 173910237892624236 \beta_{3} + 106283723373376732 \beta_{4} - 2110216876523238 \beta_{5} - 2724094726368 \beta_{6} - 9619173592042240 \beta_{7} + 72536523618854 \beta_{8} + 5445352857600 \beta_{9} + 1225558967352 \beta_{10} + 544535285760 \beta_{11}) q^{44}\) \(+(-\)\(51\!\cdots\!60\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} - \)\(48\!\cdots\!86\)\( \beta_{2} + 30033770122900592898 \beta_{3} + 19034514037355343 \beta_{4} - 1775596873836642 \beta_{5} + 196046033543076 \beta_{6} - 2306025181110372 \beta_{7} + 4271281132386 \beta_{8} + 1352999538618 \beta_{9} + 2787878238930 \beta_{10} - 457147899306 \beta_{11}) q^{45}\) \(+(\)\(11\!\cdots\!72\)\( + \)\(18\!\cdots\!14\)\( \beta_{1} + \)\(80\!\cdots\!40\)\( \beta_{2} - 81376745937518586760 \beta_{3} + 2709638422055318 \beta_{4} - 1139454577885440 \beta_{5} + 1764374780357032 \beta_{6} + 8706136014327818 \beta_{7} + 2324383068460 \beta_{8} - 5310291906688 \beta_{9} + 1494084778398 \beta_{10} + 663786488336 \beta_{11}) q^{46}\) \(+(-\)\(64\!\cdots\!34\)\( \beta_{1} + \)\(42\!\cdots\!54\)\( \beta_{2} + 74758531848771276 \beta_{3} + 892410525149630932 \beta_{4} + 1157216128763904 \beta_{5} - 12886450772400 \beta_{6} - 10537138412171604 \beta_{7} - 147126022209522 \beta_{8} - 16050909388500 \beta_{9} + 1616521754250 \beta_{10} - 1605090938850 \beta_{11}) q^{47}\) \(+(-\)\(14\!\cdots\!72\)\( + \)\(13\!\cdots\!86\)\( \beta_{1} + \)\(15\!\cdots\!30\)\( \beta_{2} + 99531776001858523128 \beta_{3} - 2124875254989240312 \beta_{4} + 2718801568311348 \beta_{5} - 3975252245642844 \beta_{6} + 4746264541199184 \beta_{7} - 98840760709554 \beta_{8} + 1903238532432 \beta_{9} - 6794078328990 \beta_{10} - 718191011754 \beta_{11}) q^{48}\) \(+(-\)\(61\!\cdots\!21\)\( - \)\(61\!\cdots\!23\)\( \beta_{1} - \)\(26\!\cdots\!53\)\( \beta_{2} + 90093475731030478084 \beta_{3} - 466009434656150 \beta_{4} + 5213840999004204 \beta_{5} - 3068805972181180 \beta_{6} - 48270638648648528 \beta_{7} - 11170513450187 \beta_{8} + 26595095593976 \beta_{9} - 7247450199717 \beta_{10} - 3324386949247 \beta_{11}) q^{49}\) \(+(-\)\(23\!\cdots\!75\)\( \beta_{1} + \)\(76\!\cdots\!00\)\( \beta_{2} + 1200328930483382600 \beta_{3} - 588114373631702140 \beta_{4} + 12882556994765200 \beta_{5} + 102440129942000 \beta_{6} + 165992297466879300 \beta_{7} + 39215756774200 \beta_{8} + 19380317088000 \beta_{9} - 23672000776700 \beta_{10} + 1938031708800 \beta_{11}) q^{50}\) \(+(\)\(59\!\cdots\!56\)\( + \)\(24\!\cdots\!49\)\( \beta_{1} - \)\(68\!\cdots\!29\)\( \beta_{2} - \)\(34\!\cdots\!18\)\( \beta_{3} + 1831987056703869444 \beta_{4} + 19426258943982210 \beta_{5} + 16464632256433476 \beta_{6} + 8923669398503160 \beta_{7} + 709210238748345 \beta_{8} - 27412496825022 \beta_{9} - 5889017078613 \beta_{10} + 7519233795201 \beta_{11}) q^{51}\) \(+(-\)\(82\!\cdots\!36\)\( - \)\(39\!\cdots\!16\)\( \beta_{1} - \)\(16\!\cdots\!62\)\( \beta_{2} + \)\(71\!\cdots\!82\)\( \beta_{3} - 124629025234012000 \beta_{4} + 39054288517870608 \beta_{5} + 1511031414630736 \beta_{6} - 82570578907201984 \beta_{7} - 51104182856616 \beta_{8} - 74856417098688 \beta_{9} - 20873565359640 \beta_{10} + 9357052137336 \beta_{11}) q^{52}\) \(+(-\)\(46\!\cdots\!59\)\( \beta_{1} + \)\(16\!\cdots\!68\)\( \beta_{2} + 2703413397641146592 \beta_{3} - 4160417948812300011 \beta_{4} + 40129043841417088 \beta_{5} - 187182042428800 \beta_{6} - 296904500480807328 \beta_{7} + 567735159735856 \beta_{8} + 78520460268000 \beta_{9} + 54647556634000 \beta_{10} + 7852046026800 \beta_{11}) q^{53}\) \(+(\)\(10\!\cdots\!76\)\( + \)\(77\!\cdots\!43\)\( \beta_{1} + \)\(96\!\cdots\!83\)\( \beta_{2} - \)\(91\!\cdots\!81\)\( \beta_{3} + 14575059318925273077 \beta_{4} + 65817985452264792 \beta_{5} - 30830997264616692 \beta_{6} + 82871160255572976 \beta_{7} - 2677065627425526 \beta_{8} + 114930327265632 \beta_{9} + 24773713192215 \beta_{10} - 24969189777228 \beta_{11}) q^{54}\) \(+(-\)\(12\!\cdots\!20\)\( - \)\(72\!\cdots\!68\)\( \beta_{1} - \)\(30\!\cdots\!22\)\( \beta_{2} - \)\(76\!\cdots\!04\)\( \beta_{3} - 876370688186817442 \beta_{4} + 39596480344456206 \beta_{5} - 12107524707375578 \beta_{6} + 1215585412129465736 \beta_{7} + 354782478934372 \beta_{8} + 51785712569696 \beta_{9} + 174154632431580 \beta_{10} - 6473214071212 \beta_{11}) q^{55}\) \(+(-\)\(14\!\cdots\!26\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} + 1944033416166409452 \beta_{3} - 22940044212684888724 \beta_{4} + 26810074507111086 \beta_{5} - 218881997004288 \beta_{6} - 79406889571609000 \beta_{7} - 832578525812158 \beta_{8} - 523816538664960 \beta_{9} + 2338845384576 \beta_{10} - 52381653866496 \beta_{11}) q^{56}\) \(+(-\)\(16\!\cdots\!26\)\( - \)\(69\!\cdots\!37\)\( \beta_{1} - \)\(36\!\cdots\!66\)\( \beta_{2} + \)\(13\!\cdots\!47\)\( \beta_{3} - 10720770471791443806 \beta_{4} - 69339373608465879 \beta_{5} + 10139080081292514 \beta_{6} - 15753481488856734 \beta_{7} + 4634021768157720 \beta_{8} - 241699131335889 \beta_{9} + 200407318391430 \beta_{10} + 37636363499058 \beta_{11}) q^{57}\) \(+(\)\(45\!\cdots\!60\)\( + \)\(30\!\cdots\!90\)\( \beta_{1} + \)\(12\!\cdots\!29\)\( \beta_{2} + 63172056881629030744 \beta_{3} + 2535677486567438607 \beta_{4} - 179607368891188422 \beta_{5} + 114498399724472214 \beta_{6} - 2802201655865445459 \beta_{7} - 623442040703403 \beta_{8} + 565252625651496 \beta_{9} - 347049309454920 \beta_{10} - 70656578206437 \beta_{11}) q^{58}\) \(+(\)\(19\!\cdots\!41\)\( \beta_{1} - \)\(14\!\cdots\!75\)\( \beta_{2} - 22031066367683383086 \beta_{3} + \)\(13\!\cdots\!37\)\( \beta_{4} - 285930444497395293 \beta_{5} + 499626168471648 \beta_{6} - 90695937773248245 \beta_{7} - 2235889322236476 \beta_{8} + 1405973646824040 \beta_{9} + 15690822564492 \beta_{10} + 140597364682404 \beta_{11}) q^{59}\) \(+(-\)\(17\!\cdots\!40\)\( - \)\(54\!\cdots\!66\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3} - 91227594899698143732 \beta_{4} - 539875008744244878 \beta_{5} + 23113649893586184 \beta_{6} - 787016368293871328 \beta_{7} + 6445790383381794 \beta_{8} - 74964336984288 \beta_{9} - 1048730428146420 \beta_{10} + 37600148077596 \beta_{11}) q^{60}\) \(+(\)\(16\!\cdots\!02\)\( + \)\(35\!\cdots\!79\)\( \beta_{1} + \)\(15\!\cdots\!01\)\( \beta_{2} - \)\(13\!\cdots\!36\)\( \beta_{3} + 1019935112357887458 \beta_{4} - 506645254346652000 \beta_{5} - 357521963541645408 \beta_{6} + 1279136699199772728 \beta_{7} - 166140380919285 \beta_{8} - 2822005804299768 \beta_{9} + 93305172309093 \beta_{10} + 352750725537471 \beta_{11}) q^{61}\) \(+(\)\(99\!\cdots\!47\)\( \beta_{1} - \)\(34\!\cdots\!00\)\( \beta_{2} - 57556895200063011530 \beta_{3} - \)\(15\!\cdots\!13\)\( \beta_{4} - 877452289986409048 \beta_{5} + 5120793845827300 \beta_{6} + 7678687812883043719 \beta_{7} + 3538052233391478 \beta_{8} - 1117447527504000 \beta_{9} - 1391943214207225 \beta_{10} - 111744752750400 \beta_{11}) q^{62}\) \(+(-\)\(57\!\cdots\!74\)\( - \)\(94\!\cdots\!03\)\( \beta_{1} + \)\(30\!\cdots\!22\)\( \beta_{2} - \)\(49\!\cdots\!86\)\( \beta_{3} - \)\(10\!\cdots\!58\)\( \beta_{4} - 482285613665874152 \beta_{5} + 222334394104982407 \beta_{6} - 325467892732554839 \beta_{7} - 58921230724949744 \beta_{8} + 2439961354397624 \beta_{9} + 1622694819546720 \beta_{10} - 342088777592928 \beta_{11}) q^{63}\) \(+(-\)\(27\!\cdots\!12\)\( + \)\(14\!\cdots\!12\)\( \beta_{1} + \)\(60\!\cdots\!56\)\( \beta_{2} + \)\(19\!\cdots\!84\)\( \beta_{3} + 9158277165005273344 \beta_{4} - 1088968962053109120 \beta_{5} + 219979573345533056 \beta_{6} - 7150080416835155456 \beta_{7} - 503371017401920 \beta_{8} + 5866869003536896 \beta_{9} - 618364821422016 \beta_{10} - 733358625442112 \beta_{11}) q^{64}\) \(+(\)\(24\!\cdots\!60\)\( \beta_{1} - \)\(54\!\cdots\!90\)\( \beta_{2} - 81756758019520175140 \beta_{3} + \)\(14\!\cdots\!46\)\( \beta_{4} - 526948722811241180 \beta_{5} - 21498728881056400 \beta_{6} - 32418437617294808920 \beta_{7} + 41150227727358570 \beta_{8} - 5846058930752700 \beta_{9} + 4790076327188830 \beta_{10} - 584605893075270 \beta_{11}) q^{65}\) \(+(\)\(24\!\cdots\!60\)\( - \)\(49\!\cdots\!06\)\( \beta_{1} - \)\(99\!\cdots\!59\)\( \beta_{2} - \)\(48\!\cdots\!60\)\( \beta_{3} + \)\(13\!\cdots\!35\)\( \beta_{4} - 197782199295584534 \beta_{5} - 755691724975692066 \beta_{6} + 244831326568005277 \beta_{7} + 134824150269941817 \beta_{8} - 8430656186013048 \beta_{9} - 208343883664056 \beta_{10} + 780155291329695 \beta_{11}) q^{66}\) \(+(-\)\(10\!\cdots\!46\)\( - \)\(22\!\cdots\!49\)\( \beta_{1} - \)\(94\!\cdots\!53\)\( \beta_{2} + \)\(18\!\cdots\!34\)\( \beta_{3} - 31705948987609575179 \beta_{4} + 1899292757953481391 \beta_{5} + 1582737385835595704 \beta_{6} + 45465232985726136631 \beta_{7} + 13578292804390184 \beta_{8} + 1124999330149312 \beta_{9} + 6718833944060760 \beta_{10} - 140624916268664 \beta_{11}) q^{67}\) \(+(\)\(63\!\cdots\!28\)\( \beta_{1} + \)\(30\!\cdots\!96\)\( \beta_{2} + \)\(46\!\cdots\!04\)\( \beta_{3} - \)\(16\!\cdots\!84\)\( \beta_{4} + 5452009054847675544 \beta_{5} + 27060809732707200 \beta_{6} + 39689938566465141760 \beta_{7} - 206882408252316056 \beta_{8} + 24619512766464000 \beta_{9} - 4303251156530400 \beta_{10} + 2461951276646400 \beta_{11}) q^{68}\) \(+(\)\(11\!\cdots\!36\)\( - \)\(25\!\cdots\!86\)\( \beta_{1} + \)\(76\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!98\)\( \beta_{3} - \)\(14\!\cdots\!16\)\( \beta_{4} + 6005268658627722366 \beta_{5} - 974937900775454532 \beta_{6} + 7985070294640904332 \beta_{7} - 24325846511114268 \beta_{8} + 10838488398257250 \beta_{9} + 3098690042367456 \beta_{10} - 441952937029152 \beta_{11}) q^{69}\) \(+(\)\(10\!\cdots\!40\)\( - \)\(12\!\cdots\!94\)\( \beta_{1} - \)\(54\!\cdots\!76\)\( \beta_{2} + \)\(93\!\cdots\!68\)\( \beta_{3} - 15763879093163102286 \beta_{4} + 11235689431928230248 \beta_{5} - 4346784729603047424 \beta_{6} - 71677185794394518562 \beta_{7} - 34953216382218624 \beta_{8} - 43958596612580832 \beta_{9} - 14729195902823010 \beta_{10} + 5494824576572604 \beta_{11}) q^{70}\) \(+(-\)\(39\!\cdots\!08\)\( \beta_{1} + \)\(36\!\cdots\!00\)\( \beta_{2} + \)\(60\!\cdots\!84\)\( \beta_{3} + \)\(45\!\cdots\!42\)\( \beta_{4} + 7509892586223450802 \beta_{5} + 2656369230596816 \beta_{6} + 23988027142116280190 \beta_{7} + 364915602799650654 \beta_{8} - 36945261899698740 \beta_{9} - 4358618497619078 \beta_{10} - 3694526189969874 \beta_{11}) q^{71}\) \(+(-\)\(70\!\cdots\!20\)\( + \)\(11\!\cdots\!01\)\( \beta_{1} + \)\(61\!\cdots\!24\)\( \beta_{2} + \)\(34\!\cdots\!90\)\( \beta_{3} - \)\(15\!\cdots\!58\)\( \beta_{4} + 15179344269259437255 \beta_{5} + 9188354804625235272 \beta_{6} + 3041023411307784108 \beta_{7} - 566271965983289739 \beta_{8} + 23367021289685664 \beta_{9} - 21592375516945980 \beta_{10} - 2082999760641108 \beta_{11}) q^{72}\) \(+(\)\(75\!\cdots\!06\)\( - \)\(76\!\cdots\!24\)\( \beta_{1} - \)\(32\!\cdots\!20\)\( \beta_{2} - \)\(36\!\cdots\!64\)\( \beta_{3} + 20939277311717259504 \beta_{4} + 2681964765545325768 \beta_{5} + 2411080195838277912 \beta_{6} - 52613010242253875184 \beta_{7} + 10087878439308060 \beta_{8} + 123372592306757280 \beta_{9} - 2666847799518300 \beta_{10} - 15421574038344660 \beta_{11}) q^{73}\) \(+(-\)\(22\!\cdots\!62\)\( \beta_{1} + \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(31\!\cdots\!76\)\( \beta_{3} - \)\(31\!\cdots\!32\)\( \beta_{4} + 3556377166064866928 \beta_{5} + 12887592775144624 \beta_{6} + 31750645977040560660 \beta_{7} + 294310271633419256 \beta_{8} - 25040345045295360 \beta_{9} - 5725932698315692 \beta_{10} - 2504034504529536 \beta_{11}) q^{74}\) \(+(-\)\(16\!\cdots\!55\)\( + \)\(22\!\cdots\!18\)\( \beta_{1} - \)\(11\!\cdots\!78\)\( \beta_{2} + \)\(16\!\cdots\!54\)\( \beta_{3} - \)\(75\!\cdots\!18\)\( \beta_{4} - 8426695866525378056 \beta_{5} - 16039576997140990722 \beta_{6} + 2504164414031285164 \beta_{7} + 1160806382715533853 \beta_{8} - 140263792625727846 \beta_{9} + 18295864382834295 \beta_{10} + 5786446917852837 \beta_{11}) q^{75}\) \(+(\)\(31\!\cdots\!04\)\( + \)\(66\!\cdots\!06\)\( \beta_{1} + \)\(29\!\cdots\!48\)\( \beta_{2} - \)\(13\!\cdots\!82\)\( \beta_{3} - \)\(11\!\cdots\!72\)\( \beta_{4} - 10405544156186550516 \beta_{5} + 2408785413924912892 \beta_{6} + \)\(28\!\cdots\!60\)\( \beta_{7} + 69588135609007938 \beta_{8} - 91669491523779792 \beta_{9} + 40523411024740206 \beta_{10} + 11458686440472474 \beta_{11}) q^{76}\) \(+(-\)\(38\!\cdots\!50\)\( \beta_{1} - \)\(74\!\cdots\!72\)\( \beta_{2} - \)\(11\!\cdots\!68\)\( \beta_{3} + \)\(62\!\cdots\!74\)\( \beta_{4} - 11658887362955365872 \beta_{5} - 73073343527140800 \beta_{6} - \)\(19\!\cdots\!28\)\( \beta_{7} - 2961643936073331304 \beta_{8} + 217040902725798000 \beta_{9} + 39972426154365000 \beta_{10} + 21704090272579800 \beta_{11}) q^{77}\) \(+(-\)\(39\!\cdots\!40\)\( + \)\(75\!\cdots\!36\)\( \beta_{1} - \)\(25\!\cdots\!66\)\( \beta_{2} + \)\(35\!\cdots\!94\)\( \beta_{3} + \)\(36\!\cdots\!64\)\( \beta_{4} - 61078282624573004424 \beta_{5} - 5063965185948868296 \beta_{6} - 15072815295054677342 \beta_{7} - 176734958034565836 \beta_{8} + 260958836585360160 \beta_{9} + 86749142556798000 \beta_{10} - 4786657700354820 \beta_{11}) q^{78}\) \(+(\)\(28\!\cdots\!10\)\( + \)\(74\!\cdots\!35\)\( \beta_{1} + \)\(31\!\cdots\!84\)\( \beta_{2} + \)\(93\!\cdots\!10\)\( \beta_{3} + \)\(29\!\cdots\!04\)\( \beta_{4} - 61660497701867049834 \beta_{5} + 17157072733501385751 \beta_{6} + 3066987996005151327 \beta_{7} - 57627220182269428 \beta_{8} - 392544096295021280 \beta_{9} - 4279604072695884 \beta_{10} + 49068012036877660 \beta_{11}) q^{79}\) \(+(-\)\(17\!\cdots\!40\)\( \beta_{1} - \)\(76\!\cdots\!40\)\( \beta_{2} - \)\(12\!\cdots\!40\)\( \beta_{3} - \)\(35\!\cdots\!64\)\( \beta_{4} - \)\(14\!\cdots\!80\)\( \beta_{5} - 435801327012275200 \beta_{6} - \)\(74\!\cdots\!20\)\( \beta_{7} + 6205492116181312320 \beta_{8} - 364254272767795200 \beta_{9} + 72524904476289280 \beta_{10} - 36425427276779520 \beta_{11}) q^{80}\) \(+(-\)\(32\!\cdots\!99\)\( + \)\(12\!\cdots\!91\)\( \beta_{1} + \)\(25\!\cdots\!83\)\( \beta_{2} - \)\(27\!\cdots\!26\)\( \beta_{3} - \)\(22\!\cdots\!98\)\( \beta_{4} - \)\(11\!\cdots\!70\)\( \beta_{5} + 42657455380906010304 \beta_{6} - \)\(21\!\cdots\!44\)\( \beta_{7} - 1657859039803007955 \beta_{8} + 68071671254003526 \beta_{9} - 172956392440634241 \beta_{10} - 3368231980945875 \beta_{11}) q^{81}\) \(+(\)\(80\!\cdots\!80\)\( + \)\(17\!\cdots\!24\)\( \beta_{1} + \)\(75\!\cdots\!54\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} + \)\(10\!\cdots\!10\)\( \beta_{4} - \)\(17\!\cdots\!52\)\( \beta_{5} - 48402833926886053428 \beta_{6} - \)\(51\!\cdots\!26\)\( \beta_{7} + 164338500670149978 \beta_{8} + 1337727884598538704 \beta_{9} - 1438742452333680 \beta_{10} - 167215985574817338 \beta_{11}) q^{82}\) \(+(-\)\(87\!\cdots\!94\)\( \beta_{1} - \)\(19\!\cdots\!96\)\( \beta_{2} - \)\(31\!\cdots\!24\)\( \beta_{3} + \)\(30\!\cdots\!67\)\( \beta_{4} - 84072274568348558981 \beta_{5} + 1661438543014077800 \beta_{6} + \)\(25\!\cdots\!01\)\( \beta_{7} - 2000413324208028737 \beta_{8} - 3333721986152250 \beta_{9} - 415693007952134675 \beta_{10} - 333372198615225 \beta_{11}) q^{83}\) \(+(-\)\(19\!\cdots\!96\)\( + \)\(62\!\cdots\!14\)\( \beta_{1} - \)\(96\!\cdots\!54\)\( \beta_{2} - \)\(41\!\cdots\!94\)\( \beta_{3} - \)\(19\!\cdots\!48\)\( \beta_{4} + 62957472971940386214 \beta_{5} - 4347123589805681376 \beta_{6} - 14660291841428469504 \beta_{7} - 1760789278045235142 \beta_{8} - 1481256978939562752 \beta_{9} + 2541238745994216 \beta_{10} + 3681080010103392 \beta_{11}) q^{84}\) \(+(\)\(13\!\cdots\!80\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} - \)\(50\!\cdots\!72\)\( \beta_{2} + \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(94\!\cdots\!08\)\( \beta_{4} + \)\(12\!\cdots\!96\)\( \beta_{5} - 86342973182505462848 \beta_{6} - \)\(28\!\cdots\!44\)\( \beta_{7} - 1183333381558445468 \beta_{8} - 1287478595333896864 \beta_{9} - 511199278570854180 \beta_{10} + 160934824416737108 \beta_{11}) q^{85}\) \(+(\)\(38\!\cdots\!77\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2} + \)\(30\!\cdots\!86\)\( \beta_{3} + \)\(55\!\cdots\!53\)\( \beta_{4} + \)\(41\!\cdots\!48\)\( \beta_{5} - 1023584600263944724 \beta_{6} - \)\(14\!\cdots\!75\)\( \beta_{7} - 18690744099459295494 \beta_{8} + 922302013645989120 \beta_{9} + 348126351430585093 \beta_{10} + 92230201364598912 \beta_{11}) q^{86}\) \(+(-\)\(38\!\cdots\!60\)\( - \)\(41\!\cdots\!06\)\( \beta_{1} + \)\(96\!\cdots\!61\)\( \beta_{2} - \)\(24\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!74\)\( \beta_{4} + \)\(16\!\cdots\!34\)\( \beta_{5} - 39893969886115808529 \beta_{6} + \)\(62\!\cdots\!57\)\( \beta_{7} + 7751791536844796811 \beta_{8} + 3163606543386373410 \beta_{9} - 221083588311755175 \beta_{10} + 8302337076094155 \beta_{11}) q^{87}\) \(+(\)\(92\!\cdots\!60\)\( - \)\(78\!\cdots\!60\)\( \beta_{1} - \)\(33\!\cdots\!36\)\( \beta_{2} - \)\(30\!\cdots\!96\)\( \beta_{3} - \)\(73\!\cdots\!88\)\( \beta_{4} + \)\(77\!\cdots\!48\)\( \beta_{5} + \)\(45\!\cdots\!24\)\( \beta_{6} + \)\(80\!\cdots\!56\)\( \beta_{7} + 1817439218666837652 \beta_{8} - 2263203642083909664 \beta_{9} + 1050169836963663180 \beta_{10} + 282900455260488708 \beta_{11}) q^{88}\) \(+(\)\(40\!\cdots\!29\)\( \beta_{1} + \)\(10\!\cdots\!25\)\( \beta_{2} + \)\(16\!\cdots\!66\)\( \beta_{3} + \)\(81\!\cdots\!58\)\( \beta_{4} + \)\(25\!\cdots\!58\)\( \beta_{5} - 1538371133881750488 \beta_{6} - \)\(21\!\cdots\!80\)\( \beta_{7} + 45751752574214014831 \beta_{8} - 1107769609956002490 \beta_{9} + 273815822474837373 \beta_{10} - 110776960995600249 \beta_{11}) q^{89}\) \(+(-\)\(13\!\cdots\!60\)\( - \)\(12\!\cdots\!54\)\( \beta_{1} - \)\(23\!\cdots\!21\)\( \beta_{2} + \)\(26\!\cdots\!68\)\( \beta_{3} + \)\(19\!\cdots\!29\)\( \beta_{4} + \)\(94\!\cdots\!98\)\( \beta_{5} - \)\(35\!\cdots\!74\)\( \beta_{6} + \)\(72\!\cdots\!43\)\( \beta_{7} + 16568600347234737531 \beta_{8} - 684637618646821032 \beta_{9} + 1552204917009964080 \beta_{10} + 47889780034672629 \beta_{11}) q^{90}\) \(+(\)\(10\!\cdots\!52\)\( - \)\(46\!\cdots\!18\)\( \beta_{1} - \)\(19\!\cdots\!76\)\( \beta_{2} + \)\(69\!\cdots\!88\)\( \beta_{3} - \)\(19\!\cdots\!24\)\( \beta_{4} + \)\(28\!\cdots\!88\)\( \beta_{5} - \)\(30\!\cdots\!86\)\( \beta_{6} + \)\(34\!\cdots\!50\)\( \beta_{7} + 1589586621437926916 \beta_{8} + 7978462216265620576 \beta_{9} + 296139422202362172 \beta_{10} - 997307777033202572 \beta_{11}) q^{91}\) \(+(\)\(23\!\cdots\!88\)\( \beta_{1} + \)\(70\!\cdots\!52\)\( \beta_{2} + \)\(11\!\cdots\!68\)\( \beta_{3} - \)\(27\!\cdots\!76\)\( \beta_{4} + \)\(15\!\cdots\!40\)\( \beta_{5} - 3321015155834075200 \beta_{6} - \)\(32\!\cdots\!16\)\( \beta_{7} - 42727588872565877804 \beta_{8} - 108498958356480000 \beta_{9} + 819403893122870800 \beta_{10} - 10849895835648000 \beta_{11}) q^{92}\) \(+(-\)\(14\!\cdots\!94\)\( - \)\(10\!\cdots\!78\)\( \beta_{1} + \)\(54\!\cdots\!29\)\( \beta_{2} - \)\(18\!\cdots\!36\)\( \beta_{3} + \)\(28\!\cdots\!33\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(99\!\cdots\!48\)\( \beta_{6} - \)\(39\!\cdots\!24\)\( \beta_{7} - 86359228748735269353 \beta_{8} - 10801252449283229856 \beta_{9} - 883297079755976055 \beta_{10} - 245879019004889493 \beta_{11}) q^{93}\) \(+(\)\(26\!\cdots\!72\)\( - \)\(12\!\cdots\!44\)\( \beta_{1} - \)\(54\!\cdots\!92\)\( \beta_{2} - \)\(33\!\cdots\!16\)\( \beta_{3} - \)\(14\!\cdots\!36\)\( \beta_{4} + \)\(87\!\cdots\!48\)\( \beta_{5} - \)\(10\!\cdots\!24\)\( \beta_{6} - \)\(71\!\cdots\!12\)\( \beta_{7} - 4017477871813732624 \beta_{8} - 6420246176507959232 \beta_{9} - 1607473549875118860 \beta_{10} + 802530772063494904 \beta_{11}) q^{94}\) \(+(\)\(23\!\cdots\!30\)\( \beta_{1} - \)\(39\!\cdots\!70\)\( \beta_{2} - \)\(69\!\cdots\!40\)\( \beta_{3} - \)\(82\!\cdots\!02\)\( \beta_{4} - \)\(11\!\cdots\!30\)\( \beta_{5} + 9280077863292564800 \beta_{6} + \)\(14\!\cdots\!30\)\( \beta_{7} - 28614154357394751080 \beta_{8} - 1346001090646621200 \beta_{9} - 2454619574887803320 \beta_{10} - 134600109064662120 \beta_{11}) q^{95}\) \(+(-\)\(37\!\cdots\!52\)\( - \)\(28\!\cdots\!20\)\( \beta_{1} + \)\(60\!\cdots\!24\)\( \beta_{2} + \)\(45\!\cdots\!36\)\( \beta_{3} - \)\(33\!\cdots\!40\)\( \beta_{4} - \)\(24\!\cdots\!88\)\( \beta_{5} + 39749769779997057408 \beta_{6} + \)\(33\!\cdots\!00\)\( \beta_{7} + 81708084566305350624 \beta_{8} + 22961930417607292416 \beta_{9} - 4299813119615350464 \beta_{10} + 254912493521003328 \beta_{11}) q^{96}\) \(+(\)\(20\!\cdots\!94\)\( + \)\(47\!\cdots\!17\)\( \beta_{1} + \)\(20\!\cdots\!55\)\( \beta_{2} + \)\(14\!\cdots\!36\)\( \beta_{3} + \)\(26\!\cdots\!02\)\( \beta_{4} - \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(12\!\cdots\!64\)\( \beta_{6} - \)\(14\!\cdots\!20\)\( \beta_{7} - 4640269810231590983 \beta_{8} - 7943436718327595944 \beta_{9} - 1823670110220320745 \beta_{10} + 992929589790949493 \beta_{11}) q^{97}\) \(+(-\)\(27\!\cdots\!49\)\( \beta_{1} - \)\(22\!\cdots\!56\)\( \beta_{2} - \)\(34\!\cdots\!64\)\( \beta_{3} + \)\(17\!\cdots\!32\)\( \beta_{4} - \)\(48\!\cdots\!56\)\( \beta_{5} + 21774413176593232400 \beta_{6} + \)\(23\!\cdots\!56\)\( \beta_{7} + \)\(20\!\cdots\!08\)\( \beta_{8} + 8283881081096544000 \beta_{9} - 4615215186038653700 \beta_{10} + 828388108109654400 \beta_{11}) q^{98}\) \(+(-\)\(28\!\cdots\!20\)\( - \)\(50\!\cdots\!29\)\( \beta_{1} - \)\(16\!\cdots\!27\)\( \beta_{2} - \)\(21\!\cdots\!38\)\( \beta_{3} + \)\(21\!\cdots\!11\)\( \beta_{4} - \)\(83\!\cdots\!51\)\( \beta_{5} - \)\(28\!\cdots\!86\)\( \beta_{6} - \)\(18\!\cdots\!09\)\( \beta_{7} + \)\(15\!\cdots\!98\)\( \beta_{8} - 6535349611762124004 \beta_{9} + 4049332131129180498 \beta_{10} + 539071471116633654 \beta_{11}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 114742404q^{3} \) \(\mathstrut -\mathstrut 1699274528448q^{4} \) \(\mathstrut -\mathstrut 483611204680128q^{6} \) \(\mathstrut +\mathstrut 8107872236538648q^{7} \) \(\mathstrut -\mathstrut 424319151461513940q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 114742404q^{3} \) \(\mathstrut -\mathstrut 1699274528448q^{4} \) \(\mathstrut -\mathstrut 483611204680128q^{6} \) \(\mathstrut +\mathstrut 8107872236538648q^{7} \) \(\mathstrut -\mathstrut 424319151461513940q^{9} \) \(\mathstrut +\mathstrut 8521437485093339520q^{10} \) \(\mathstrut -\mathstrut 2862564534392665536q^{12} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!52\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!80\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!56\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!20\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!60\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!52\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!84\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!04\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!68\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!40\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!84\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!40\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!04\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!44\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!52\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!68\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!40\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!80\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!92\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!20\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!64\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!64\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!52\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!72\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!32\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!12\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!12\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!80\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!24\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!88\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!44\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!20\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!52\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!32\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!40\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!72\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!60\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!48\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!80\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!88\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!60\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!52\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!60\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!24\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!28\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!64\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!24\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!28\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!40\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut +\mathstrut \) \(17353504902\) \(x^{10}\mathstrut +\mathstrut \) \(111006258614054318328\) \(x^{8}\mathstrut +\mathstrut \) \(323765701965839203118204176384\) \(x^{6}\mathstrut +\mathstrut \) \(420150309279704216298413492838082805760\) \(x^{4}\mathstrut +\mathstrut \) \(190068212511425710374530430459662273636990976000\) \(x^{2}\mathstrut +\mathstrut \) \(27342285412416035125187079526375866471795145886924800000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(64\!\cdots\!25\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!92\) \(\nu^{10}\mathstrut -\mathstrut \) \(89\!\cdots\!50\) \(\nu^{9}\mathstrut +\mathstrut \) \(30\!\cdots\!44\) \(\nu^{8}\mathstrut -\mathstrut \) \(37\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!56\) \(\nu^{6}\mathstrut -\mathstrut \) \(44\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(37\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(32\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(74\!\cdots\!00\)\()/\)\(21\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(64\!\cdots\!25\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!92\) \(\nu^{10}\mathstrut -\mathstrut \) \(89\!\cdots\!50\) \(\nu^{9}\mathstrut +\mathstrut \) \(30\!\cdots\!44\) \(\nu^{8}\mathstrut -\mathstrut \) \(37\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!56\) \(\nu^{6}\mathstrut -\mathstrut \) \(44\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(37\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(63\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(96\!\cdots\!00\)\()/\)\(21\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(12\!\cdots\!73\) \(\nu^{11}\mathstrut +\mathstrut \) \(66\!\cdots\!28\) \(\nu^{10}\mathstrut -\mathstrut \) \(19\!\cdots\!66\) \(\nu^{9}\mathstrut +\mathstrut \) \(10\!\cdots\!96\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!24\) \(\nu^{7}\mathstrut +\mathstrut \) \(55\!\cdots\!04\) \(\nu^{6}\mathstrut -\mathstrut \) \(24\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(12\!\cdots\!92\) \(\nu^{4}\mathstrut -\mathstrut \) \(20\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(46\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(25\!\cdots\!00\)\()/\)\(21\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(54\!\cdots\!91\) \(\nu^{11}\mathstrut -\mathstrut \) \(93\!\cdots\!48\) \(\nu^{10}\mathstrut -\mathstrut \) \(32\!\cdots\!62\) \(\nu^{9}\mathstrut -\mathstrut \) \(14\!\cdots\!36\) \(\nu^{8}\mathstrut +\mathstrut \) \(30\!\cdots\!12\) \(\nu^{7}\mathstrut -\mathstrut \) \(78\!\cdots\!64\) \(\nu^{6}\mathstrut +\mathstrut \) \(26\!\cdots\!76\) \(\nu^{5}\mathstrut -\mathstrut \) \(17\!\cdots\!72\) \(\nu^{4}\mathstrut +\mathstrut \) \(53\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(15\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(19\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(36\!\cdots\!00\)\()/\)\(21\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(38\!\cdots\!31\) \(\nu^{11}\mathstrut +\mathstrut \) \(25\!\cdots\!96\) \(\nu^{10}\mathstrut -\mathstrut \) \(64\!\cdots\!42\) \(\nu^{9}\mathstrut +\mathstrut \) \(29\!\cdots\!52\) \(\nu^{8}\mathstrut -\mathstrut \) \(40\!\cdots\!08\) \(\nu^{7}\mathstrut +\mathstrut \) \(70\!\cdots\!68\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!84\) \(\nu^{5}\mathstrut -\mathstrut \) \(16\!\cdots\!76\) \(\nu^{4}\mathstrut -\mathstrut \) \(13\!\cdots\!60\) \(\nu^{3}\mathstrut -\mathstrut \) \(59\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(38\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(18\!\cdots\!00\)\()/\)\(53\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(19\!\cdots\!49\) \(\nu^{11}\mathstrut -\mathstrut \) \(17\!\cdots\!32\) \(\nu^{10}\mathstrut -\mathstrut \) \(30\!\cdots\!18\) \(\nu^{9}\mathstrut -\mathstrut \) \(25\!\cdots\!24\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!32\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!76\) \(\nu^{6}\mathstrut -\mathstrut \) \(36\!\cdots\!36\) \(\nu^{5}\mathstrut -\mathstrut \) \(21\!\cdots\!48\) \(\nu^{4}\mathstrut -\mathstrut \) \(32\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!20\) \(\nu^{2}\mathstrut -\mathstrut \) \(73\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(86\!\cdots\!00\)\()/\)\(13\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(75\!\cdots\!71\) \(\nu^{11}\mathstrut -\mathstrut \) \(14\!\cdots\!92\) \(\nu^{10}\mathstrut +\mathstrut \) \(12\!\cdots\!22\) \(\nu^{9}\mathstrut -\mathstrut \) \(19\!\cdots\!44\) \(\nu^{8}\mathstrut +\mathstrut \) \(76\!\cdots\!28\) \(\nu^{7}\mathstrut -\mathstrut \) \(81\!\cdots\!56\) \(\nu^{6}\mathstrut +\mathstrut \) \(20\!\cdots\!44\) \(\nu^{5}\mathstrut -\mathstrut \) \(81\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(23\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(85\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(83\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(45\!\cdots\!00\)\()/\)\(21\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(25\!\cdots\!11\) \(\nu^{11}\mathstrut -\mathstrut \) \(10\!\cdots\!92\) \(\nu^{10}\mathstrut +\mathstrut \) \(47\!\cdots\!82\) \(\nu^{9}\mathstrut -\mathstrut \) \(15\!\cdots\!44\) \(\nu^{8}\mathstrut +\mathstrut \) \(31\!\cdots\!88\) \(\nu^{7}\mathstrut -\mathstrut \) \(77\!\cdots\!56\) \(\nu^{6}\mathstrut +\mathstrut \) \(93\!\cdots\!84\) \(\nu^{5}\mathstrut -\mathstrut \) \(16\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(11\!\cdots\!60\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(29\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(24\!\cdots\!00\)\()/\)\(10\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(10\!\cdots\!23\) \(\nu^{11}\mathstrut -\mathstrut \) \(31\!\cdots\!28\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!86\) \(\nu^{9}\mathstrut -\mathstrut \) \(41\!\cdots\!56\) \(\nu^{8}\mathstrut +\mathstrut \) \(84\!\cdots\!64\) \(\nu^{7}\mathstrut -\mathstrut \) \(15\!\cdots\!84\) \(\nu^{6}\mathstrut +\mathstrut \) \(19\!\cdots\!72\) \(\nu^{5}\mathstrut -\mathstrut \) \(90\!\cdots\!52\) \(\nu^{4}\mathstrut +\mathstrut \) \(16\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(33\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(38\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(14\!\cdots\!00\)\()/\)\(10\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(24\!\cdots\!09\) \(\nu^{11}\mathstrut +\mathstrut \) \(10\!\cdots\!92\) \(\nu^{10}\mathstrut +\mathstrut \) \(43\!\cdots\!78\) \(\nu^{9}\mathstrut +\mathstrut \) \(15\!\cdots\!04\) \(\nu^{8}\mathstrut +\mathstrut \) \(28\!\cdots\!32\) \(\nu^{7}\mathstrut +\mathstrut \) \(77\!\cdots\!36\) \(\nu^{6}\mathstrut +\mathstrut \) \(81\!\cdots\!16\) \(\nu^{5}\mathstrut +\mathstrut \) \(16\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(96\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(24\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(24\!\cdots\!00\)\()/\)\(10\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(416484117648\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(1334\) \(\beta_{4}\mathstrut -\mathstrut \) \(714\) \(\beta_{3}\mathstrut -\mathstrut \) \(45304936\) \(\beta_{2}\mathstrut -\mathstrut \) \(656647616971\) \(\beta_{1}\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(799\) \(\beta_{11}\mathstrut +\mathstrut \) \(6669\) \(\beta_{10}\mathstrut -\mathstrut \) \(6392\) \(\beta_{9}\mathstrut +\mathstrut \) \(12539\) \(\beta_{8}\mathstrut +\mathstrut \) \(47088136\) \(\beta_{7}\mathstrut +\mathstrut \) \(239690\) \(\beta_{6}\mathstrut +\mathstrut \) \(393762\) \(\beta_{5}\mathstrut -\mathstrut \) \(27855980\) \(\beta_{4}\mathstrut -\mathstrut \) \(471014950164\) \(\beta_{3}\mathstrut -\mathstrut \) \(4827634378593\) \(\beta_{2}\mathstrut -\mathstrut \) \(1247094205309\) \(\beta_{1}\mathstrut +\mathstrut \) \(136739430988927591014144\)\()/10368\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(6705600\) \(\beta_{11}\mathstrut -\mathstrut \) \(61826900\) \(\beta_{10}\mathstrut -\mathstrut \) \(67056000\) \(\beta_{9}\mathstrut -\mathstrut \) \(35393950167\) \(\beta_{8}\mathstrut +\mathstrut \) \(1244744067608\) \(\beta_{7}\mathstrut +\mathstrut \) \(220485200\) \(\beta_{6}\mathstrut +\mathstrut \) \(476982917671\) \(\beta_{5}\mathstrut -\mathstrut \) \(79870774200798\) \(\beta_{4}\mathstrut +\mathstrut \) \(38110140664142\) \(\beta_{3}\mathstrut +\mathstrut \) \(2422801994565779448\) \(\beta_{2}\mathstrut +\mathstrut \) \(15615161793150985633113\) \(\beta_{1}\)\()/7776\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(15258505739191\) \(\beta_{11}\mathstrut -\mathstrut \) \(98697773101653\) \(\beta_{10}\mathstrut +\mathstrut \) \(122068045913528\) \(\beta_{9}\mathstrut -\mathstrut \) \(182137040464115\) \(\beta_{8}\mathstrut -\mathstrut \) \(711380016003017288\) \(\beta_{7}\mathstrut -\mathstrut \) \(2285808759419482\) \(\beta_{6}\mathstrut -\mathstrut \) \(11309029718718450\) \(\beta_{5}\mathstrut +\mathstrut \) \(446501103894502732\) \(\beta_{4}\mathstrut +\mathstrut \) \(4483183904191038657684\) \(\beta_{3}\mathstrut +\mathstrut \) \(102758841196879549578601\) \(\beta_{2}\mathstrut +\mathstrut \) \(25240662805316933219141\) \(\beta_{1}\mathstrut -\mathstrut \) \(1083871560824682019823285236088064\)\()/15552\)
\(\nu^{7}\)\(=\)\((\)\(99182555428440000\) \(\beta_{11}\mathstrut +\mathstrut \) \(1295065687396294900\) \(\beta_{10}\mathstrut +\mathstrut \) \(991825554284400000\) \(\beta_{9}\mathstrut +\mathstrut \) \(370338136631142908741\) \(\beta_{8}\mathstrut -\mathstrut \) \(17757837781737448706576\) \(\beta_{7}\mathstrut -\mathstrut \) \(4783532527871419600\) \(\beta_{6}\mathstrut -\mathstrut \) \(5678078049686331088325\) \(\beta_{5}\mathstrut +\mathstrut \) \(1096167248320355444491314\) \(\beta_{4}\mathstrut -\mathstrut \) \(457513123140565387678042\) \(\beta_{3}\mathstrut -\mathstrut \) \(29154280914062865880015909288\) \(\beta_{2}\mathstrut -\mathstrut \) \(135012583980859288782195363234163\) \(\beta_{1}\)\()/11664\)
\(\nu^{8}\)\(=\)\((\)\(69148968752239466894431\) \(\beta_{11}\mathstrut +\mathstrut \) \(374894423822411950483149\) \(\beta_{10}\mathstrut -\mathstrut \) \(553191750017915735155448\) \(\beta_{9}\mathstrut +\mathstrut \) \(680639878892584434071867\) \(\beta_{8}\mathstrut +\mathstrut \) \(2749462141706947022269585160\) \(\beta_{7}\mathstrut +\mathstrut \) \(4157713660772753478694858\) \(\beta_{6}\mathstrut +\mathstrut \) \(60501768950149151114291106\) \(\beta_{5}\mathstrut -\mathstrut \) \(1806837164947397906187781228\) \(\beta_{4}\mathstrut -\mathstrut \) \(14407333215307329975466032759348\) \(\beta_{3}\mathstrut -\mathstrut \) \(486246632118159252660015069138433\) \(\beta_{2}\mathstrut -\mathstrut \) \(117834985733986370272472760793661\) \(\beta_{1}\mathstrut +\mathstrut \) \(3123763265555227088202800914376907762102528\)\()/7776\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(371589767032584553967870400\) \(\beta_{11}\mathstrut -\mathstrut \) \(6164654658599298190175218900\) \(\beta_{10}\mathstrut -\mathstrut \) \(3715897670325845539678704000\) \(\beta_{9}\mathstrut -\mathstrut \) \(1249291362867536527678100670425\) \(\beta_{8}\mathstrut +\mathstrut \) \(72196003546279773341550204010688\) \(\beta_{7}\mathstrut +\mathstrut \) \(23172259566266854544829394000\) \(\beta_{6}\mathstrut +\mathstrut \) \(20151151573164475635090447385913\) \(\beta_{5}\mathstrut -\mathstrut \) \(4339552662811867207569098126290890\) \(\beta_{4}\mathstrut +\mathstrut \) \(1635499767933901056655909541062114\) \(\beta_{3}\mathstrut +\mathstrut \) \(104428226042747080567456360838090341576\) \(\beta_{2}\mathstrut +\mathstrut \) \(411383115161410030859426714103269736329071\) \(\beta_{1}\)\()/5832\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(273041063278624275941834669396875\) \(\beta_{11}\mathstrut -\mathstrut \) \(1313014395647663854412599843497489\) \(\beta_{10}\mathstrut +\mathstrut \) \(2184328506228994207534677355175000\) \(\beta_{9}\mathstrut -\mathstrut \) \(2352987728016703432883365017598103\) \(\beta_{8}\mathstrut -\mathstrut \) \(9760741401779474047109913658691068328\) \(\beta_{7}\mathstrut -\mathstrut \) \(1478140447780421818629323710322194\) \(\beta_{6}\mathstrut -\mathstrut \) \(260251364933100926765875323286341834\) \(\beta_{5}\mathstrut +\mathstrut \) \(6634906906959240143382121353263887324\) \(\beta_{4}\mathstrut +\mathstrut \) \(46796177626378833306322987770502871970756\) \(\beta_{3}\mathstrut +\mathstrut \) \(1965079398013998004116379965297211633849813\) \(\beta_{2}\mathstrut +\mathstrut \) \(473519845695187263487624596803892524782145\) \(\beta_{1}\mathstrut -\mathstrut \) \(9518016447981516609234775917271798602015576485350656\)\()/3888\)
\(\nu^{11}\)\(=\)\((\)\(1279958726284170746771894545002820800\) \(\beta_{11}\mathstrut +\mathstrut \) \(25055191514492707769075876664609561700\) \(\beta_{10}\mathstrut +\mathstrut \) \(12799587262841707467718945450028208000\) \(\beta_{9}\mathstrut +\mathstrut \) \(4164213936055426159129739292336300830741\) \(\beta_{8}\mathstrut -\mathstrut \) \(270597421775670663354467628769238163610208\) \(\beta_{7}\mathstrut -\mathstrut \) \(95100931152834148089215928478426963600\) \(\beta_{6}\mathstrut -\mathstrut \) \(68674282138302235967361310669630354408757\) \(\beta_{5}\mathstrut +\mathstrut \) \(15979945773202597880720600545664681609947074\) \(\beta_{4}\mathstrut -\mathstrut \) \(5605180436972689518626551019810478093365338\) \(\beta_{3}\mathstrut -\mathstrut \) \(358445096981947526973357701700936725761055459752\) \(\beta_{2}\mathstrut -\mathstrut \) \(1296852824569760987584387021355100020786032868160307\) \(\beta_{1}\)\()/2916\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
81459.4i
65373.8i
54008.5i
53338.0i
18842.7i
18089.7i
18089.7i
18842.7i
53338.0i
54008.5i
65373.8i
81459.4i
977513.i −4.43499e8 1.07432e9i −6.80653e11 1.94075e13i −1.05016e15 + 4.33526e14i −1.85443e15 3.96650e17i −9.57469e17 + 9.52919e17i 1.89711e19
2.2 784486.i 1.15997e9 + 7.30136e7i −3.40540e11 3.07374e13i 5.72781e13 9.09976e14i 1.07708e16 5.15107e16i 1.34019e18 + 1.69386e17i −2.41131e19
2.3 648102.i −9.89523e8 + 6.09669e8i −1.45158e11 1.81665e13i 3.95128e14 + 6.41312e14i −1.77541e16 8.40715e16i 6.07458e17 1.20656e18i −1.17737e19
2.4 640056.i 2.88152e8 + 1.12598e9i −1.34793e11 3.15820e13i 7.20687e14 1.84433e14i 9.50531e15 8.96619e16i −1.18479e18 + 6.48904e17i 2.02142e19
2.5 226112.i −8.54601e8 7.87723e8i 2.23751e11 6.89065e12i −1.78114e14 + 1.93236e14i 1.19822e16 1.12746e17i 1.09835e17 + 1.34638e18i −1.55806e18
2.6 217076.i 7.82134e8 8.59720e8i 2.27756e11 1.16100e13i −1.86625e14 1.69783e14i −8.59590e15 1.09110e17i −1.27385e17 1.34483e18i 2.52026e18
2.7 217076.i 7.82134e8 + 8.59720e8i 2.27756e11 1.16100e13i −1.86625e14 + 1.69783e14i −8.59590e15 1.09110e17i −1.27385e17 + 1.34483e18i 2.52026e18
2.8 226112.i −8.54601e8 + 7.87723e8i 2.23751e11 6.89065e12i −1.78114e14 1.93236e14i 1.19822e16 1.12746e17i 1.09835e17 1.34638e18i −1.55806e18
2.9 640056.i 2.88152e8 1.12598e9i −1.34793e11 3.15820e13i 7.20687e14 + 1.84433e14i 9.50531e15 8.96619e16i −1.18479e18 6.48904e17i 2.02142e19
2.10 648102.i −9.89523e8 6.09669e8i −1.45158e11 1.81665e13i 3.95128e14 6.41312e14i −1.77541e16 8.40715e16i 6.07458e17 + 1.20656e18i −1.17737e19
2.11 784486.i 1.15997e9 7.30136e7i −3.40540e11 3.07374e13i 5.72781e13 + 9.09976e14i 1.07708e16 5.15107e16i 1.34019e18 1.69386e17i −2.41131e19
2.12 977513.i −4.43499e8 + 1.07432e9i −6.80653e11 1.94075e13i −1.05016e15 4.33526e14i −1.85443e15 3.96650e17i −9.57469e17 9.52919e17i 1.89711e19
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{39}^{\mathrm{new}}(3, [\chi])\).