Properties

Label 6.31.b.a
Level $6$
Weight $31$
Character orbit 6.b
Analytic conductor $34.209$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.2085571416\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 8552062877595970390 x^{8} + 25392807796994293834069533242559594025 x^{6} + 31363214047445929775687291161211122208644643150843750000 x^{4} + 14997479733093492468171154918719535444710088825890735887189931640625000000 x^{2} + 1779211874045772876005913493072356377360261727229346754892124420427761840820312500000000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{104}\cdot 3^{53}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1413976 - 27 \beta_{1} + \beta_{2} ) q^{3} -536870912 q^{4} + ( -116 - 206950 \beta_{1} + 579 \beta_{2} + \beta_{3} ) q^{5} + ( -14578876416 - 1413976 \beta_{1} - \beta_{3} - \beta_{4} ) q^{6} + ( -145933478929 + 14603 \beta_{1} + 94058 \beta_{2} - 13 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{9} ) q^{7} + 536870912 \beta_{1} q^{8} + ( 4171311735384 + 815871281 \beta_{1} + 1460005 \beta_{2} - 2537 \beta_{3} - 17 \beta_{4} - 54 \beta_{5} - 25 \beta_{6} + 13 \beta_{7} - 4 \beta_{8} - 14 \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} +(1413976 - 27 \beta_{1} + \beta_{2}) q^{3} -536870912 q^{4} +(-116 - 206950 \beta_{1} + 579 \beta_{2} + \beta_{3}) q^{5} +(-14578876416 - 1413976 \beta_{1} - \beta_{3} - \beta_{4}) q^{6} +(-145933478929 + 14603 \beta_{1} + 94058 \beta_{2} - 13 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{9}) q^{7} +536870912 \beta_{1} q^{8} +(4171311735384 + 815871281 \beta_{1} + 1460005 \beta_{2} - 2537 \beta_{3} - 17 \beta_{4} - 54 \beta_{5} - 25 \beta_{6} + 13 \beta_{7} - 4 \beta_{8} - 14 \beta_{9}) q^{9} +(-111153818093696 + 396140 \beta_{1} + 2550730 \beta_{2} - 556 \beta_{3} - 586 \beta_{4} + 134 \beta_{5} - 132 \beta_{6} - 18 \beta_{7} + 28 \beta_{8} - 26 \beta_{9}) q^{10} +(1813832 - 32777796167 \beta_{1} - 9164972 \beta_{2} + 99869 \beta_{3} + 3156 \beta_{4} - 4 \beta_{5} + 123 \beta_{6} + 546 \beta_{7} + 123 \beta_{8} - 182 \beta_{9}) q^{11} +(-759122584666112 + 14495514624 \beta_{1} - 536870912 \beta_{2}) q^{12} +(1124185766448477 + 18939265 \beta_{1} + 122016603 \beta_{2} - 3926 \beta_{3} + 26562 \beta_{4} - 806 \beta_{5} - 8946 \beta_{6} - 13314 \beta_{7} + 1201 \beta_{8} - 10953 \beta_{9}) q^{13} +(248443392 + 145740663552 \beta_{1} - 1243452826 \beta_{2} + 1111302 \beta_{3} - 96748 \beta_{4} + 3402 \beta_{5} - 7516 \beta_{6} - 6238 \beta_{7} + 21892 \beta_{8} + 11882 \beta_{9}) q^{14} +(-120837075235712946 + 674180355111 \beta_{1} + 888029244 \beta_{2} + 2197128 \beta_{3} - 176484 \beta_{4} + 22134 \beta_{5} - 50010 \beta_{6} - 74283 \beta_{7} + 98889 \beta_{8} - 16227 \beta_{9}) q^{15} +288230376151711744 q^{16} +(9969856515 - 4100799237375 \beta_{1} - 49807962611 \beta_{2} - 37971390 \beta_{3} + 2660964 \beta_{4} + 74930 \beta_{5} + 418632 \beta_{6} + 374100 \beta_{7} - 328125 \beta_{8} - 373619 \beta_{9}) q^{17} +(437896148929571712 - 4174110144265 \beta_{1} - 18090595958 \beta_{2} + 48583744 \beta_{3} - 1973534 \beta_{4} + 190662 \beta_{5} + 655868 \beta_{6} + 1072942 \beta_{7} + 710300 \beta_{8} + 972838 \beta_{9}) q^{18} +(-40643627591296415 + 103633909337 \beta_{1} + 667533182990 \beta_{2} - 93132086 \beta_{3} - 11984290 \beta_{4} + 2023805 \beta_{5} + 1443331 \beta_{6} + 3051211 \beta_{7} + 2388370 \beta_{8} + 1078766 \beta_{9}) q^{19} +(62277025792 + 111105435238400 \beta_{1} - 310848258048 \beta_{2} - 536870912 \beta_{3}) q^{20} +(18934137396867615947 + 475259115007365 \beta_{1} - 245189054068 \beta_{2} - 820697967 \beta_{3} - 160968 \beta_{4} - 5496582 \beta_{5} + 12942852 \beta_{6} + 945564 \beta_{7} + 5418597 \beta_{8} - 1045413 \beta_{9}) q^{21} +(-17596692484439432704 + 307915516092 \beta_{1} + 1983442647138 \beta_{2} - 257785084 \beta_{3} + 12194974 \beta_{4} + 20225070 \beta_{5} - 9529076 \beta_{6} - 9064970 \beta_{7} + 12503212 \beta_{8} - 1807218 \beta_{9}) q^{22} +(1588030997134 - 2777677638943234 \beta_{1} - 7945465845340 \beta_{2} + 5088674212 \beta_{3} - 163712004 \beta_{4} + 27486394 \beta_{5} + 23115126 \beta_{6} + 8475114 \beta_{7} + 50407428 \beta_{8} + 6272396 \beta_{9}) q^{23} +(7826974677393211392 + 759122584666112 \beta_{1} + 536870912 \beta_{3} + 536870912 \beta_{4}) q^{24} +(-53875068221422464894 - 513868349365 \beta_{1} - 3311227476011 \beta_{2} + 250813842 \beta_{3} - 555494414 \beta_{4} + 31388282 \beta_{5} - 166701418 \beta_{6} - 73614982 \beta_{7} - 16667037 \beta_{8} - 118646099 \beta_{9}) q^{25} +(-2323609619456 - 1122382825345714 \beta_{1} + 11633861320184 \beta_{2} - 15791671688 \beta_{3} - 29772400 \beta_{4} - 98498616 \beta_{5} - 212295472 \beta_{6} - 50911576 \beta_{7} + 63907408 \beta_{8} + 109038152 \beta_{9}) q^{26} +(-50968622641552390836 + 34714167201340356 \beta_{1} + 7061387912529 \beta_{2} + 15969108201 \beta_{3} + 553218324 \beta_{4} - 605531628 \beta_{5} + 495982263 \beta_{6} - 42942684 \beta_{7} - 450218823 \beta_{8} - 125276988 \beta_{9}) q^{27} +(78347439923945013248 - 7839925927936 \beta_{1} - 50497004240896 \beta_{2} + 6979321856 \beta_{3} + 1073741824 \beta_{4} + 536870912 \beta_{5} + 536870912 \beta_{9}) q^{28} +(-47627459977886 - 191231032228432672 \beta_{1} + 238111943904411 \beta_{2} + 18397639063 \beta_{3} - 5266968168 \beta_{4} + 179352588 \beta_{5} + 662043552 \beta_{6} - 1245444672 \beta_{7} - 1604034834 \beta_{8} - 340211238 \beta_{9}) q^{29} +(\)\(36\!\cdots\!44\)\( + 120820553663624592 \beta_{1} - 106320984965202 \beta_{2} - 58172700096 \beta_{3} - 791154570 \beta_{4} + 1408194114 \beta_{5} - 881956524 \beta_{6} - 1128365382 \beta_{7} + 468544692 \beta_{8} + 2394695394 \beta_{9}) q^{30} +(-\)\(38\!\cdots\!05\)\( - 74632595035525 \beta_{1} - 480687855056002 \beta_{2} + 73973467869 \beta_{3} + 28382328702 \beta_{4} - 538139341 \beta_{5} + 1692579314 \beta_{6} - 2248368454 \beta_{7} - 935986504 \beta_{8} + 2090426477 \beta_{9}) q^{31} -288230376151711744 \beta_{1} q^{32} +(\)\(13\!\cdots\!78\)\( + 705465125175794454 \beta_{1} + 40907798694666 \beta_{2} + 131392973643 \beta_{3} - 35218283787 \beta_{4} + 6838106700 \beta_{5} - 5652102183 \beta_{6} - 11545377441 \beta_{7} + 9123052929 \beta_{8} - 3979367055 \beta_{9}) q^{33} +(-\)\(21\!\cdots\!72\)\( + 213677278561808 \beta_{1} + 1376448386194936 \beta_{2} - 157783133456 \beta_{3} + 57243690504 \beta_{4} - 10748312632 \beta_{5} + 1120057552 \beta_{6} - 11162895704 \beta_{7} + 3620178512 \beta_{8} - 13248433592 \beta_{9}) q^{34} +(821654382726049 - 1619544881421622696 \beta_{1} - 4108201376353396 \beta_{2} - 457437026675 \beta_{3} - 295668086670 \beta_{4} + 17581255741 \beta_{5} + 3598943652 \beta_{6} - 25038812349 \beta_{7} + 33293087997 \beta_{8} + 18244318898 \beta_{9}) q^{35} +(-\)\(22\!\cdots\!08\)\( - 438017558705078272 \beta_{1} - 783834215874560 \beta_{2} + 1362041503744 \beta_{3} + 9126805504 \beta_{4} + 28991029248 \beta_{5} + 13421772800 \beta_{6} - 6979321856 \beta_{7} + 2147483648 \beta_{8} + 7516192768 \beta_{9}) q^{36} +(\)\(69\!\cdots\!87\)\( + 335524336281799 \beta_{1} + 2163057988535829 \beta_{2} + 46949169806 \beta_{3} + 925941428678 \beta_{4} + 88556849710 \beta_{5} + 52985025338 \beta_{6} - 70372968526 \beta_{7} + 57339113895 \beta_{8} + 84202761153 \beta_{9}) q^{37} +(1255137807592448 + 39669079342854148 \beta_{1} - 6274117584853974 \beta_{2} - 2427399622006 \beta_{3} - 647532741172 \beta_{4} + 47765166758 \beta_{5} + 45652588220 \beta_{6} - 65474852722 \beta_{7} + 20487476956 \beta_{8} + 13436580486 \beta_{9}) q^{38} +(\)\(26\!\cdots\!79\)\( - 4440277106772111852 \beta_{1} + 635231022296690 \beta_{2} - 16244783935965 \beta_{3} + 134667643422 \beta_{4} + 68031682851 \beta_{5} + 71013526428 \beta_{6} - 175840383777 \beta_{7} + 43400811525 \beta_{8} + 83549552868 \beta_{9}) q^{39} +(\)\(59\!\cdots\!52\)\( - 212676043079680 \beta_{1} - 1369412741365760 \beta_{2} + 298500227072 \beta_{3} + 314606354432 \beta_{4} - 71940702208 \beta_{5} + 70866960384 \beta_{6} + 9663676416 \beta_{7} - 15032385536 \beta_{8} + 13958643712 \beta_{9}) q^{40} +(-5795533374643216 + 3410110952402987200 \beta_{1} + 28965541879263742 \beta_{2} + 8233373916986 \beta_{3} - 2991700415112 \beta_{4} + 97491041948 \beta_{5} + 17805432072 \beta_{6} - 418753935492 \beta_{7} - 7018180560 \beta_{8} + 131310107620 \beta_{9}) q^{41} +(\)\(25\!\cdots\!36\)\( - 18934615195536106022 \beta_{1} - 3128648787515106 \beta_{2} + 36464634355228 \beta_{3} + 138193147522 \beta_{4} - 26125445166 \beta_{5} + 43846151412 \beta_{6} - 276741355254 \beta_{7} + 50699026644 \beta_{8} + 86194181106 \beta_{9}) q^{42} +(-\)\(10\!\cdots\!67\)\( - 12443182337273303 \beta_{1} - 80140948581602370 \beta_{2} + 12934847710182 \beta_{3} + 6182621569006 \beta_{4} + 506127451069 \beta_{5} - 401352952457 \beta_{6} - 1155940173857 \beta_{7} + 55871941466 \beta_{8} + 48902557146 \beta_{9}) q^{43} +(-973793640054784 + 17597445321527394304 \beta_{1} + 4920406876094464 \beta_{2} - 53616761110528 \beta_{3} - 1694364598272 \beta_{4} + 2147483648 \beta_{5} - 66035122176 \beta_{6} - 293131517952 \beta_{7} - 66035122176 \beta_{8} + 97710505984 \beta_{9}) q^{44} +(\)\(21\!\cdots\!16\)\( - 79401516410969097486 \beta_{1} - 143206077924152751 \beta_{2} + 17865473674353 \beta_{3} + 1214603082978 \beta_{4} + 35803907748 \beta_{5} + 371264550618 \beta_{6} - 1606810593630 \beta_{7} + 336068709582 \beta_{8} + 594016991826 \beta_{9}) q^{45} +(-\)\(14\!\cdots\!52\)\( - 11790201225354168 \beta_{1} - 75930320277707172 \beta_{2} + 13025229980728 \beta_{3} + 8094698496932 \beta_{4} + 1443889120644 \beta_{5} - 681695208280 \beta_{6} - 1532962481548 \beta_{7} + 375975328616 \beta_{8} + 386218583748 \beta_{9}) q^{46} +(9241935873049394 - 15047284174436787824 \beta_{1} - 46385350810094896 \beta_{2} + 151335094329722 \beta_{3} - 18450494626380 \beta_{4} + 1284294478354 \beta_{5} + 1100328037248 \beta_{6} - 1871130501906 \beta_{7} + 727724352738 \beta_{8} + 499508939132 \beta_{9}) q^{47} +(\)\(40\!\cdots\!44\)\( - 7782220156096217088 \beta_{1} + 288230376151711744 \beta_{2}) q^{48} +(-\)\(14\!\cdots\!14\)\( + 89020173385308181 \beta_{1} + 573409548693352635 \beta_{2} - 76776254375938 \beta_{3} - 2605934325250 \beta_{4} - 3612285569930 \beta_{5} + 4104865248218 \beta_{6} + 3202636940678 \beta_{7} + 848480669085 \beta_{8} - 355900990797 \beta_{9}) q^{49} +(61857455898616832 + 53827015946280193743 \beta_{1} - 308837243478504088 \beta_{2} - 442629546461464 \beta_{3} + 5434661757360 \beta_{4} - 639049381928 \beta_{5} - 1361606019216 \beta_{6} + 829031772792 \beta_{7} + 1194671319024 \beta_{8} + 575748521816 \beta_{9}) q^{50} +(\)\(10\!\cdots\!81\)\( + \)\(51\!\cdots\!70\)\( \beta_{1} - 33068076120758052 \beta_{2} - 885501030584925 \beta_{3} - 7096924217190 \beta_{4} + 6241081103997 \beta_{5} + 530564519382 \beta_{6} + 6761123065635 \beta_{7} - 191070890637 \beta_{8} - 1509859299438 \beta_{9}) q^{51} +(-\)\(60\!\cdots\!24\)\( - 10167940473159680 \beta_{1} - 65507164931751936 \beta_{2} + 2107755200512 \beta_{3} - 14260365164544 \beta_{4} + 432717955072 \beta_{5} + 4802847178752 \beta_{6} + 7147899322368 \beta_{7} - 644781965312 \beta_{8} + 5880347099136 \beta_{9}) q^{52} +(62565447684218764 - \)\(73\!\cdots\!54\)\( \beta_{1} - 313586407707206217 \beta_{2} + 889862430363997 \beta_{3} + 100337692696944 \beta_{4} - 3497519302536 \beta_{5} - 1295468154480 \beta_{6} + 14113824343224 \beta_{7} + 928799650848 \beta_{8} - 3963185512632 \beta_{9}) q^{53} +(\)\(18\!\cdots\!88\)\( + 51018301479551774628 \beta_{1} + 317263222838610870 \beta_{2} + 1912698682954539 \beta_{3} - 15028454737335 \beta_{4} - 5255502582150 \beta_{5} - 6343489800060 \beta_{6} + 1555305216018 \beta_{7} - 112838907804 \beta_{8} - 7394424867174 \beta_{9}) q^{54} +(-\)\(90\!\cdots\!84\)\( + 53086593955095560 \beta_{1} + 341660870716765780 \beta_{2} - 97883608895334 \beta_{3} - 148180136254544 \beta_{4} - 5904514785124 \beta_{5} - 16015558966678 \beta_{6} + 4768652538578 \beta_{7} - 4559186926248 \beta_{8} - 17360886825554 \beta_{9}) q^{55} +(-133382030443413504 - 78243922956647399424 \beta_{1} + 667573652723597312 \beta_{2} - 596625718247424 \beta_{3} + 51941186994176 \beta_{4} - 1826434842624 \beta_{5} + 4035121774592 \beta_{6} + 3349000749056 \beta_{7} - 11753178005504 \beta_{8} - 6379100176384 \beta_{9}) q^{56} +(\)\(13\!\cdots\!57\)\( + \)\(23\!\cdots\!33\)\( \beta_{1} - 814133342602854745 \beta_{2} + 109651301861967 \beta_{3} - 37700081571291 \beta_{4} - 34383700670586 \beta_{5} - 28788850530987 \beta_{6} + 25925244404835 \beta_{7} - 3193744979370 \beta_{8} - 2050749635832 \beta_{9}) q^{57} +(-\)\(10\!\cdots\!36\)\( - 433539401051455996 \beta_{1} - 2793134882650338434 \beta_{2} + 291041943016892 \beta_{3} - 233379669111550 \beta_{4} - 45144061596046 \beta_{5} - 29522797354188 \beta_{6} - 1189603157718 \beta_{7} - 27620406727916 \beta_{8} - 47046452222318 \beta_{9}) q^{58} +(-1398404075442690783 + \)\(20\!\cdots\!91\)\( \beta_{1} + 6993258139729400224 \beta_{2} - 256004900422908 \beta_{3} + 744965308192782 \beta_{4} - 51983987856031 \beta_{5} - 39088572862299 \beta_{6} + 70810943916009 \beta_{7} - 43802353459098 \beta_{8} - 25174908170936 \beta_{9}) q^{59} +(\)\(64\!\cdots\!52\)\( - \)\(36\!\cdots\!32\)\( \beta_{1} - 476757070108950528 \beta_{2} - 1179574113140736 \beta_{3} + 94749126033408 \beta_{4} - 11883100766208 \beta_{5} + 26848914309120 \beta_{6} + 39880381956096 \beta_{7} - 53090627616768 \beta_{8} + 8711804289024 \beta_{9}) q^{60} +(-\)\(29\!\cdots\!33\)\( - 311259844778676829 \beta_{1} - 2006493393430610567 \beta_{2} - 147084298225674 \beta_{3} - 1041199201610258 \beta_{4} + 63807880869494 \beta_{5} + 3077168346482 \beta_{6} + 176397790766714 \beta_{7} - 22214696796189 \beta_{8} + 89099746012165 \beta_{9}) q^{61} +(-3053254235796763136 + \)\(38\!\cdots\!48\)\( \beta_{1} + 15258934596664579294 \beta_{2} + 7911166850770942 \beta_{3} + 435660871359140 \beta_{4} - 35982884088686 \beta_{5} - 17161110040972 \beta_{6} + 24102117367754 \beta_{7} - 67549815253292 \beta_{8} - 24830274193358 \beta_{9}) q^{62} +(\)\(50\!\cdots\!41\)\( - \)\(11\!\cdots\!07\)\( \beta_{1} + 14762779333851523514 \beta_{2} - 785620568722291 \beta_{3} + 490456930488338 \beta_{4} - 89789535738279 \beta_{5} + 132107532558232 \beta_{6} + 149192369416664 \beta_{7} - 33938074026080 \beta_{8} - 111400689333079 \beta_{9}) q^{63} -\)\(15\!\cdots\!28\)\( q^{64} +(13354452661695858810 + \)\(23\!\cdots\!30\)\( \beta_{1} - 66757016889698787440 \beta_{2} - 17570059508093430 \beta_{3} - 1743326657894400 \beta_{4} + 188482359257120 \beta_{5} + 148339227953640 \beta_{6} - 62723768694180 \beta_{7} + 281406691555290 \beta_{8} + 65263744098610 \beta_{9}) q^{65} +(\)\(37\!\cdots\!96\)\( - \)\(13\!\cdots\!04\)\( \beta_{1} - 20535341298405334602 \beta_{2} - 17140802744089152 \beta_{3} + 57440704468830 \beta_{4} + 39568990891002 \beta_{5} - 166511595467388 \beta_{6} - 199135040299374 \beta_{7} - 3022360708380 \beta_{8} + 158728701893274 \beta_{9}) q^{66} +(-\)\(10\!\cdots\!11\)\( + 14543689981228657761 \beta_{1} + 93686833240629129162 \beta_{2} - 11112397247386608 \beta_{3} + 3227391734731038 \beta_{4} + 3419434927137 \beta_{5} + 77501764899369 \beta_{6} - 501255281416455 \beta_{7} + 415373229667002 \beta_{8} - 334452029840496 \beta_{9}) q^{67} +(-5352525959717191680 + \)\(22\!\cdots\!00\)\( \beta_{1} + 26740446311829471232 \beta_{2} + 20385734779207680 \beta_{3} - 1428594169479168 \beta_{4} - 40227737436160 \beta_{5} - 224751343632384 \beta_{6} - 200843408179200 \beta_{7} + 176160768000000 \beta_{8} + 200585173270528 \beta_{9}) q^{68} +(\)\(15\!\cdots\!86\)\( - \)\(32\!\cdots\!74\)\( \beta_{1} - 13295067913203199362 \beta_{2} + 52840890487683858 \beta_{3} - 2912998595327082 \beta_{4} + 781240917829488 \beta_{5} + 33003719369862 \beta_{6} - 570080705562282 \beta_{7} + 782313883716144 \beta_{8} + 170392959609408 \beta_{9}) q^{69} +(-\)\(86\!\cdots\!48\)\( - 25281848595562893480 \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2} + 22944679668074536 \beta_{3} + 3956195028642092 \beta_{4} + 223546578281676 \beta_{5} - 55409539157896 \beta_{6} - 450321117852004 \beta_{7} - 372360410609992 \beta_{8} + 540497449733772 \beta_{9}) q^{70} +(24108226846335115296 - \)\(42\!\cdots\!54\)\( \beta_{1} - \)\(12\!\cdots\!12\)\( \beta_{2} - 45538499125383666 \beta_{3} - 6109596683593968 \beta_{4} + 600677797512380 \beta_{5} + 694400504445366 \beta_{6} - 499974685050360 \beta_{7} + 243453443364870 \beta_{8} + 16342541323288 \beta_{9}) q^{71} +(-\)\(23\!\cdots\!44\)\( + \)\(22\!\cdots\!80\)\( \beta_{1} + 9712314750594973696 \beta_{2} - 26083198949654528 \beta_{3} + 1059532998443008 \beta_{4} - 102360881823744 \beta_{5} - 352116451311616 \beta_{6} - 576031350063104 \beta_{7} - 381339408793600 \beta_{8} - 522288424288256 \beta_{9}) q^{72} +(\)\(12\!\cdots\!38\)\( + 3865988600325308716 \beta_{1} + 24924105384704769452 \beta_{2} + 720273786549552 \beta_{3} + 11231451441136832 \beta_{4} - 71784721288160 \beta_{5} + 1649816859946672 \beta_{6} - 82331276338136 \beta_{7} + 375008035140252 \beta_{8} + 1203024103518260 \beta_{9}) q^{73} +(-98521753339280180224 - \)\(69\!\cdots\!42\)\( \beta_{1} + \)\(49\!\cdots\!52\)\( \beta_{2} + 56773671997232456 \beta_{3} - 2407117647460624 \beta_{4} - 510715789173128 \beta_{5} - 194715967457872 \beta_{6} - 1488330062645608 \beta_{7} - 2337761213207504 \beta_{8} - 218238394368008 \beta_{9}) q^{74} +(-\)\(75\!\cdots\!83\)\( + \)\(11\!\cdots\!59\)\( \beta_{1} - 40810253690300792255 \beta_{2} - 119249188450346295 \beta_{3} - 181745182089054 \beta_{4} + 1751278891440573 \beta_{5} + 538663459370904 \beta_{6} - 1581835028297211 \beta_{7} + 949934367777195 \beta_{8} + 2508093275854644 \beta_{9}) q^{75} +(\)\(21\!\cdots\!80\)\( - 55638031419880505344 \beta_{1} - \)\(35\!\cdots\!80\)\( \beta_{2} + 49999907947282432 \beta_{3} + 6434016701972480 \beta_{4} - 1086522036060160 \beta_{5} - 774882430287872 \beta_{6} - 1638106432274432 \beta_{7} - 1282246380093440 \beta_{8} - 579158086254592 \beta_{9}) q^{76} +(68861697664488095760 - \)\(49\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!66\)\( \beta_{2} - 169544879898238590 \beta_{3} - 20230172966755536 \beta_{4} + 926884463135960 \beta_{5} - 475286267019408 \beta_{6} - 1733057436835320 \beta_{7} + 2796151056602760 \beta_{8} + 1668164920152496 \beta_{9}) q^{77} +(-\)\(23\!\cdots\!68\)\( - \)\(26\!\cdots\!24\)\( \beta_{1} + 14674227168717498456 \beta_{2} + 200393259709380478 \beta_{3} - 1489464989387402 \beta_{4} - 3935039817941400 \beta_{5} + 580771496146704 \beta_{6} - 1220902118431032 \beta_{7} - 4015805808371568 \beta_{8} + 39344868144360 \beta_{9}) q^{78} +(\)\(68\!\cdots\!91\)\( + \)\(14\!\cdots\!43\)\( \beta_{1} + \)\(90\!\cdots\!34\)\( \beta_{2} - 122220691882509587 \beta_{3} - 6091769840592514 \beta_{4} + 6742534705716707 \beta_{5} - 2695794702622094 \beta_{6} - 1235910978114038 \beta_{7} + 4678826590234768 \beta_{8} - 632086587140155 \beta_{9}) q^{79} +(-33434723633598562304 - \)\(59\!\cdots\!00\)\( \beta_{1} + \)\(16\!\cdots\!76\)\( \beta_{2} + 288230376151711744 \beta_{3}) q^{80} +(-\)\(11\!\cdots\!62\)\( - \)\(26\!\cdots\!69\)\( \beta_{1} + 7007902123004882001 \beta_{2} - 55567272409777476 \beta_{3} + 38981592931485600 \beta_{4} - 2625988548764718 \beta_{5} + 4631913378680220 \beta_{6} + 7599672180082980 \beta_{7} + 2997851638421889 \beta_{8} - 6826064198221917 \beta_{9}) q^{81} +(\)\(18\!\cdots\!44\)\( - \)\(25\!\cdots\!84\)\( \beta_{1} - \)\(16\!\cdots\!88\)\( \beta_{2} + 202157244940304312 \beta_{3} - 30836269252816412 \beta_{4} - 3649719365042748 \beta_{5} - 5009605581593816 \beta_{6} + 423243183712180 \beta_{7} - 7246555417205528 \beta_{8} - 1412769529431036 \beta_{9}) q^{82} +(\)\(43\!\cdots\!78\)\( + \)\(94\!\cdots\!39\)\( \beta_{1} - \)\(21\!\cdots\!36\)\( \beta_{2} + 207685384888400609 \beta_{3} + 87388800602202216 \beta_{4} - 178053067470902 \beta_{5} + 1696811059174149 \beta_{6} + 16391371507670808 \beta_{7} + 7492762814367291 \beta_{8} - 3531806584159222 \beta_{9}) q^{83} +(-\)\(10\!\cdots\!64\)\( - \)\(25\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2} + 440608866019835904 \beta_{3} + 86419036962816 \beta_{4} + 2950954991222784 \beta_{5} - 6948640757121024 \beta_{6} - 507645807034368 \beta_{7} - 2909087113150464 \beta_{8} + 561251830726656 \beta_{9}) q^{84} +(\)\(40\!\cdots\!44\)\( + \)\(60\!\cdots\!40\)\( \beta_{1} + \)\(39\!\cdots\!64\)\( \beta_{2} - 574481543599359480 \beta_{3} - 157400702118140056 \beta_{4} - 7821789060069560 \beta_{5} + 5837072021955928 \beta_{6} + 24292501361378872 \beta_{7} + 6284559311020164 \beta_{8} - 8269276349133796 \beta_{9}) q^{85} +(-\)\(64\!\cdots\!68\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(32\!\cdots\!90\)\( \beta_{2} - 896528257829139990 \beta_{3} + 83590392330035468 \beta_{4} - 11983032793026490 \beta_{5} - 11999253514892548 \beta_{6} + 452219213524910 \beta_{7} - 15228947737734884 \beta_{8} - 1227304478789082 \beta_{9}) q^{86} +(-\)\(51\!\cdots\!96\)\( - \)\(13\!\cdots\!23\)\( \beta_{1} + \)\(26\!\cdots\!32\)\( \beta_{2} - 2251505507947993506 \beta_{3} - 172053322668824184 \beta_{4} + 125568667722252 \beta_{5} - 2774111513768466 \beta_{6} + 28327671261741549 \beta_{7} + 14969176277475489 \beta_{8} + 2951009310230523 \beta_{9}) q^{87} +(\)\(94\!\cdots\!48\)\( - \)\(16\!\cdots\!04\)\( \beta_{1} - \)\(10\!\cdots\!56\)\( \beta_{2} + 138397313147076608 \beta_{3} - 6547126813196288 \beta_{4} - 10858251776163840 \beta_{5} + 5115883722637312 \beta_{6} + 4866718711152640 \beta_{7} - 6712610829369344 \beta_{8} + 970242775842816 \beta_{9}) q^{88} +(\)\(20\!\cdots\!79\)\( + \)\(34\!\cdots\!49\)\( \beta_{1} - \)\(10\!\cdots\!29\)\( \beta_{2} + 1482440293470815028 \beta_{3} + 143686974757674972 \beta_{4} + 3962265414322766 \beta_{5} + 21044110832112720 \beta_{6} + 21313124563314576 \beta_{7} - 14093558602640109 \beta_{8} - 18816931332689135 \beta_{9}) q^{89} +(-\)\(42\!\cdots\!64\)\( - \)\(21\!\cdots\!88\)\( \beta_{1} + \)\(50\!\cdots\!66\)\( \beta_{2} + 2586347795456328012 \beta_{3} + 128210900797164246 \beta_{4} - 9784670231507202 \beta_{5} - 14921872925156052 \beta_{6} - 3105016355399034 \beta_{7} - 25045073044724148 \beta_{8} + 5588503311138270 \beta_{9}) q^{90} +(\)\(12\!\cdots\!54\)\( - \)\(33\!\cdots\!90\)\( \beta_{1} - \)\(21\!\cdots\!24\)\( \beta_{2} + 252377503507464738 \beta_{3} - 71068626171384836 \beta_{4} - 9292011663752402 \beta_{5} + 20268945189540628 \beta_{6} + 31266145631064652 \beta_{7} - 12247900123394068 \beta_{8} + 23224833649182294 \beta_{9}) q^{91} +(-\)\(85\!\cdots\!08\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(42\!\cdots\!80\)\( \beta_{2} - 2731961165067321344 \beta_{3} + 87892212892827648 \beta_{4} - 14756645414371328 \beta_{5} - 12409838776614912 \beta_{6} - 4550042182483968 \beta_{7} - 27062281841934336 \beta_{8} - 3367466960945152 \beta_{9}) q^{92} +(-\)\(98\!\cdots\!33\)\( - \)\(57\!\cdots\!03\)\( \beta_{1} - \)\(11\!\cdots\!02\)\( \beta_{2} + 1607270211246767625 \beta_{3} + 49617832396304142 \beta_{4} - 589828668194754 \beta_{5} - 4655989804215582 \beta_{6} - 7428009343403142 \beta_{7} - 22874702213541825 \beta_{8} - 44240478020696007 \beta_{9}) q^{93} +(-\)\(80\!\cdots\!64\)\( - \)\(14\!\cdots\!40\)\( \beta_{1} - \)\(96\!\cdots\!16\)\( \beta_{2} + 1299123165129790512 \beta_{3} + 46833534835089256 \beta_{4} + 2918413061212712 \beta_{5} - 47867637202588528 \beta_{6} - 45276569002842232 \beta_{7} - 29567093055443952 \beta_{8} - 15382131085931864 \beta_{9}) q^{94} +(\)\(18\!\cdots\!90\)\( + \)\(49\!\cdots\!62\)\( \beta_{1} - \)\(94\!\cdots\!80\)\( \beta_{2} + 1878122443521962052 \beta_{3} - 186905210979637260 \beta_{4} - 9734904516828802 \beta_{5} - 55145410045241094 \beta_{6} - 14887612412588322 \beta_{7} + 67744306496135316 \beta_{8} + 45925776317988244 \beta_{9}) q^{95} +(-\)\(42\!\cdots\!04\)\( - \)\(40\!\cdots\!44\)\( \beta_{1} - 288230376151711744 \beta_{3} - 288230376151711744 \beta_{4}) q^{96} +(-\)\(13\!\cdots\!27\)\( + \)\(15\!\cdots\!53\)\( \beta_{1} + \)\(98\!\cdots\!75\)\( \beta_{2} - 1119263679374742962 \beta_{3} + 436873386563874750 \beta_{4} + 9218537428130518 \beta_{5} - 39831486013496406 \beta_{6} - 113846917615068642 \beta_{7} + 49835854751695969 \beta_{8} - 80448803337061857 \beta_{9}) q^{97} +(\)\(31\!\cdots\!24\)\( + \)\(14\!\cdots\!93\)\( \beta_{1} - \)\(15\!\cdots\!12\)\( \beta_{2} + 12646117932342633752 \beta_{3} - 623481576888162736 \beta_{4} + 44606997465283624 \beta_{5} + 26186418728186000 \beta_{6} - 51348266494967416 \beta_{7} + 58886467632586768 \beta_{8} + 28016105133122728 \beta_{9}) q^{98} +(\)\(27\!\cdots\!71\)\( - \)\(95\!\cdots\!65\)\( \beta_{1} - \)\(11\!\cdots\!44\)\( \beta_{2} - 4447445657312028876 \beta_{3} + 789399013324323666 \beta_{4} + 75496307196271887 \beta_{5} + 32990461148520915 \beta_{6} - 144471760047810267 \beta_{7} + 79105649247233160 \beta_{8} + 128363008057356054 \beta_{9}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 14139762q^{3} - 5368709120q^{4} - 145788764160q^{6} - 1459334601196q^{7} + 41713120268730q^{9} + O(q^{10}) \) \( 10q + 14139762q^{3} - 5368709120q^{4} - 145788764160q^{6} - 1459334601196q^{7} + 41713120268730q^{9} - 1111538175836160q^{10} - 7591226920402944q^{12} + 11241857908456580q^{13} - 1208370750576105600q^{15} + 2882303761517117440q^{16} + 4378961453216956416q^{18} - 406434941012438620q^{19} + \)\(18\!\cdots\!20\)\(q^{21} - \)\(17\!\cdots\!56\)\(q^{22} + 78269746773932113920q^{24} - \)\(53\!\cdots\!30\)\(q^{25} - \)\(50\!\cdots\!26\)\(q^{27} + \)\(78\!\cdots\!52\)\(q^{28} + \)\(36\!\cdots\!60\)\(q^{30} - \)\(38\!\cdots\!20\)\(q^{31} + \)\(13\!\cdots\!52\)\(q^{33} - \)\(21\!\cdots\!40\)\(q^{34} - \)\(22\!\cdots\!60\)\(q^{36} + \)\(69\!\cdots\!16\)\(q^{37} + \)\(26\!\cdots\!00\)\(q^{39} + \)\(59\!\cdots\!20\)\(q^{40} + \)\(25\!\cdots\!56\)\(q^{42} - \)\(10\!\cdots\!72\)\(q^{43} + \)\(21\!\cdots\!20\)\(q^{45} - \)\(14\!\cdots\!60\)\(q^{46} + \)\(40\!\cdots\!28\)\(q^{48} - \)\(14\!\cdots\!50\)\(q^{49} + \)\(10\!\cdots\!80\)\(q^{51} - \)\(60\!\cdots\!60\)\(q^{52} + \)\(18\!\cdots\!00\)\(q^{54} - \)\(90\!\cdots\!40\)\(q^{55} + \)\(13\!\cdots\!76\)\(q^{57} - \)\(10\!\cdots\!00\)\(q^{58} + \)\(64\!\cdots\!00\)\(q^{60} - \)\(29\!\cdots\!60\)\(q^{61} + \)\(50\!\cdots\!92\)\(q^{63} - \)\(15\!\cdots\!80\)\(q^{64} + \)\(37\!\cdots\!20\)\(q^{66} - \)\(10\!\cdots\!44\)\(q^{67} + \)\(15\!\cdots\!20\)\(q^{69} - \)\(86\!\cdots\!40\)\(q^{70} - \)\(23\!\cdots\!92\)\(q^{72} + \)\(12\!\cdots\!00\)\(q^{73} - \)\(75\!\cdots\!30\)\(q^{75} + \)\(21\!\cdots\!40\)\(q^{76} - \)\(23\!\cdots\!80\)\(q^{78} + \)\(68\!\cdots\!20\)\(q^{79} - \)\(11\!\cdots\!30\)\(q^{81} + \)\(18\!\cdots\!96\)\(q^{82} - \)\(10\!\cdots\!40\)\(q^{84} + \)\(40\!\cdots\!00\)\(q^{85} - \)\(51\!\cdots\!80\)\(q^{87} + \)\(94\!\cdots\!72\)\(q^{88} - \)\(42\!\cdots\!00\)\(q^{90} + \)\(12\!\cdots\!00\)\(q^{91} - \)\(98\!\cdots\!84\)\(q^{93} - \)\(80\!\cdots\!00\)\(q^{94} - \)\(42\!\cdots\!40\)\(q^{96} - \)\(13\!\cdots\!12\)\(q^{97} + \)\(27\!\cdots\!20\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 8552062877595970390 x^{8} + 25392807796994293834069533242559594025 x^{6} + 31363214047445929775687291161211122208644643150843750000 x^{4} + 14997479733093492468171154918719535444710088825890735887189931640625000000 x^{2} + 1779211874045772876005913493072356377360261727229346754892124420427761840820312500000000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(8886143899548276569983795825304840554511243036416 \nu^{9} + 63807351588639611333582266654816168054304075373539137304167027722240 \nu^{7} + 148836379851207852013787940603322383684295073638684072752964348314423945346759251014400 \nu^{5} + 152449523961055697469437065813550492446923143489602050575194018698388352592964598468060078541411000000000 \nu^{3} + 84823538798775959845876906902799440481768886279043132737113864174535443730952486382285314385527450229736188125000000000000 \nu\)\()/ \)\(11\!\cdots\!75\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(15\!\cdots\!32\)\( \nu^{9} - \)\(42\!\cdots\!00\)\( \nu^{8} + \)\(11\!\cdots\!80\)\( \nu^{7} - \)\(31\!\cdots\!00\)\( \nu^{6} + \)\(28\!\cdots\!00\)\( \nu^{5} - \)\(77\!\cdots\!00\)\( \nu^{4} + \)\(24\!\cdots\!00\)\( \nu^{3} - \)\(82\!\cdots\!00\)\( \nu^{2} + \)\(52\!\cdots\!00\)\( \nu - \)\(25\!\cdots\!25\)\(\)\()/ \)\(95\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(24\!\cdots\!76\)\( \nu^{9} + \)\(82\!\cdots\!00\)\( \nu^{8} - \)\(19\!\cdots\!40\)\( \nu^{7} + \)\(59\!\cdots\!00\)\( \nu^{6} - \)\(46\!\cdots\!00\)\( \nu^{5} + \)\(15\!\cdots\!00\)\( \nu^{4} - \)\(37\!\cdots\!00\)\( \nu^{3} + \)\(15\!\cdots\!00\)\( \nu^{2} + \)\(25\!\cdots\!00\)\( \nu + \)\(48\!\cdots\!25\)\(\)\()/ \)\(31\!\cdots\!25\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(72\!\cdots\!52\)\( \nu^{9} - \)\(29\!\cdots\!00\)\( \nu^{8} - \)\(75\!\cdots\!80\)\( \nu^{7} - \)\(22\!\cdots\!00\)\( \nu^{6} - \)\(25\!\cdots\!00\)\( \nu^{5} - \)\(54\!\cdots\!00\)\( \nu^{4} - \)\(33\!\cdots\!00\)\( \nu^{3} - \)\(43\!\cdots\!00\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu - \)\(79\!\cdots\!75\)\(\)\()/ \)\(95\!\cdots\!75\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(12\!\cdots\!88\)\( \nu^{9} + \)\(81\!\cdots\!00\)\( \nu^{8} - \)\(91\!\cdots\!20\)\( \nu^{7} + \)\(39\!\cdots\!00\)\( \nu^{6} - \)\(18\!\cdots\!00\)\( \nu^{5} + \)\(14\!\cdots\!00\)\( \nu^{4} - \)\(81\!\cdots\!00\)\( \nu^{3} - \)\(74\!\cdots\!00\)\( \nu^{2} + \)\(31\!\cdots\!00\)\( \nu - \)\(19\!\cdots\!75\)\(\)\()/ \)\(95\!\cdots\!75\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(55\!\cdots\!08\)\( \nu^{9} - \)\(41\!\cdots\!00\)\( \nu^{8} - \)\(36\!\cdots\!20\)\( \nu^{7} - \)\(28\!\cdots\!00\)\( \nu^{6} - \)\(63\!\cdots\!00\)\( \nu^{5} - \)\(56\!\cdots\!00\)\( \nu^{4} - \)\(37\!\cdots\!00\)\( \nu^{3} - \)\(41\!\cdots\!00\)\( \nu^{2} + \)\(28\!\cdots\!00\)\( \nu - \)\(14\!\cdots\!50\)\(\)\()/ \)\(35\!\cdots\!25\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(15\!\cdots\!24\)\( \nu^{9} + \)\(26\!\cdots\!00\)\( \nu^{8} - \)\(13\!\cdots\!60\)\( \nu^{7} + \)\(19\!\cdots\!00\)\( \nu^{6} - \)\(37\!\cdots\!00\)\( \nu^{5} + \)\(43\!\cdots\!00\)\( \nu^{4} - \)\(39\!\cdots\!00\)\( \nu^{3} + \)\(30\!\cdots\!00\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu + \)\(37\!\cdots\!50\)\(\)\()/ \)\(95\!\cdots\!75\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(23\!\cdots\!12\)\( \nu^{9} - \)\(25\!\cdots\!00\)\( \nu^{8} - \)\(18\!\cdots\!80\)\( \nu^{7} - \)\(17\!\cdots\!00\)\( \nu^{6} - \)\(47\!\cdots\!00\)\( \nu^{5} - \)\(39\!\cdots\!00\)\( \nu^{4} - \)\(48\!\cdots\!00\)\( \nu^{3} - \)\(30\!\cdots\!00\)\( \nu^{2} - \)\(14\!\cdots\!00\)\( \nu - \)\(57\!\cdots\!25\)\(\)\()/ \)\(95\!\cdots\!75\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(98\!\cdots\!12\)\( \nu^{9} + \)\(66\!\cdots\!00\)\( \nu^{8} + \)\(74\!\cdots\!80\)\( \nu^{7} + \)\(45\!\cdots\!00\)\( \nu^{6} + \)\(17\!\cdots\!00\)\( \nu^{5} + \)\(93\!\cdots\!00\)\( \nu^{4} + \)\(13\!\cdots\!00\)\( \nu^{3} + \)\(58\!\cdots\!00\)\( \nu^{2} + \)\(13\!\cdots\!00\)\( \nu + \)\(40\!\cdots\!25\)\(\)\()/ \)\(31\!\cdots\!25\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 579 \beta_{2} - 206950 \beta_{1} - 116\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(-118646099 \beta_{9} - 16667037 \beta_{8} - 73614982 \beta_{7} - 166701418 \beta_{6} + 31388282 \beta_{5} - 555494414 \beta_{4} + 250813842 \beta_{3} - 3311227476011 \beta_{2} - 513868349365 \beta_{1} - 985197642836900980519\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(175124989972378365 \beta_{9} + 387273479206681735 \beta_{8} - 107344220600437120 \beta_{7} - 30757270110016240 \beta_{6} + 121484816183925830 \beta_{5} - 1950317354108562100 \beta_{4} - 232519287329037460830 \beta_{3} - 181237123868939074826175 \beta_{2} + 168883559432653341441379145 \beta_{1} + 36293419750272005549775\)\()/2304\)
\(\nu^{4}\)\(=\)\((\)\(899951940253520573850114135 \beta_{9} - 108897863205191808401248595 \beta_{8} + 350188405875126500303958680 \beta_{7} + 1182732628092652283795541695 \beta_{6} - 391678551044323518346676155 \beta_{5} + 5262248179844417064550225985 \beta_{4} + 7457142907341981548957522445 \beta_{3} - 42580370684447536668493739188260 \beta_{2} - 6612052396974632671707045760525 \beta_{1} + 5149938563786703036722880699890605393935\)\()/1152\)
\(\nu^{5}\)\(=\)\((\)\(-4463671232822438983179327111969736900 \beta_{9} - 8998291981872476401022995630910053475 \beta_{8} + 2520958454933614320895716317448516575 \beta_{7} + 1871763261661226227619269387550640650 \beta_{6} - 2127101932425319384973169175158404175 \beta_{5} + 42560354500519703008486742965666207250 \beta_{4} + 2734688349979604500432032498774967548575 \beta_{3} + 2516994382029043050463639847489411810184350 \beta_{2} - 2588642627442519316096055272499368857613021200 \beta_{1} - 503934775354570664177775522239558718877525\)\()/9216\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(54\!\cdots\!25\)\( \beta_{9} + \)\(12\!\cdots\!75\)\( \beta_{8} - \)\(11\!\cdots\!00\)\( \beta_{7} - \)\(63\!\cdots\!00\)\( \beta_{6} + \)\(21\!\cdots\!50\)\( \beta_{5} - \)\(35\!\cdots\!50\)\( \beta_{4} - \)\(75\!\cdots\!00\)\( \beta_{3} + \)\(48\!\cdots\!25\)\( \beta_{2} + \)\(74\!\cdots\!75\)\( \beta_{1} - \)\(25\!\cdots\!75\)\(\)\()/18432\)
\(\nu^{7}\)\(=\)\((\)\(\)\(16\!\cdots\!75\)\( \beta_{9} + \)\(32\!\cdots\!50\)\( \beta_{8} - \)\(10\!\cdots\!25\)\( \beta_{7} - \)\(75\!\cdots\!50\)\( \beta_{6} + \)\(74\!\cdots\!75\)\( \beta_{5} - \)\(16\!\cdots\!50\)\( \beta_{4} - \)\(70\!\cdots\!75\)\( \beta_{3} - \)\(72\!\cdots\!75\)\( \beta_{2} + \)\(72\!\cdots\!75\)\( \beta_{1} + \)\(14\!\cdots\!00\)\(\)\()/73728\)
\(\nu^{8}\)\(=\)\((\)\(\)\(20\!\cdots\!25\)\( \beta_{9} - \)\(55\!\cdots\!00\)\( \beta_{8} + \)\(19\!\cdots\!75\)\( \beta_{7} + \)\(21\!\cdots\!50\)\( \beta_{6} - \)\(62\!\cdots\!25\)\( \beta_{5} + \)\(13\!\cdots\!75\)\( \beta_{4} + \)\(34\!\cdots\!50\)\( \beta_{3} - \)\(22\!\cdots\!25\)\( \beta_{2} - \)\(35\!\cdots\!00\)\( \beta_{1} + \)\(82\!\cdots\!50\)\(\)\()/18432\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(29\!\cdots\!75\)\( \beta_{9} - \)\(54\!\cdots\!25\)\( \beta_{8} + \)\(20\!\cdots\!00\)\( \beta_{7} + \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(13\!\cdots\!50\)\( \beta_{5} + \)\(28\!\cdots\!00\)\( \beta_{4} + \)\(98\!\cdots\!50\)\( \beta_{3} + \)\(10\!\cdots\!25\)\( \beta_{2} - \)\(10\!\cdots\!75\)\( \beta_{1} - \)\(21\!\cdots\!25\)\(\)\()/3072\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
4.15807e8i
1.91213e9i
8.86301e8i
1.19483e9i
1.58422e9i
4.15807e8i
1.91213e9i
8.86301e8i
1.19483e9i
1.58422e9i
23170.5i −1.23817e7 + 7.25146e6i −5.36871e8 9.97937e9i 1.68020e11 + 2.86891e11i 1.72818e12 1.24396e13i 1.00724e14 1.79571e14i 2.31227e14
5.2 23170.5i −8.56717e6 1.15106e7i −5.36871e8 4.58912e10i −2.66707e11 + 1.98505e11i −5.02434e12 1.24396e13i −5.90983e13 + 1.97227e14i −1.06332e15
5.3 23170.5i 6.21475e6 1.29332e7i −5.36871e8 2.12712e10i −2.99669e11 1.43999e11i −1.34382e12 1.24396e13i −1.28645e14 1.60753e14i 4.92865e14
5.4 23170.5i 7.57629e6 + 1.21857e7i −5.36871e8 2.86759e10i 2.82348e11 1.75546e11i −3.86992e12 1.24396e13i −9.10907e13 + 1.84645e14i 6.64434e14
5.5 23170.5i 1.42278e7 + 1.86071e6i −5.36871e8 3.80214e10i 4.31134e10 3.29664e11i 7.78024e12 1.24396e13i 1.98967e14 + 5.29473e13i −8.80973e14
5.6 23170.5i −1.23817e7 7.25146e6i −5.36871e8 9.97937e9i 1.68020e11 2.86891e11i 1.72818e12 1.24396e13i 1.00724e14 + 1.79571e14i 2.31227e14
5.7 23170.5i −8.56717e6 + 1.15106e7i −5.36871e8 4.58912e10i −2.66707e11 1.98505e11i −5.02434e12 1.24396e13i −5.90983e13 1.97227e14i −1.06332e15
5.8 23170.5i 6.21475e6 + 1.29332e7i −5.36871e8 2.12712e10i −2.99669e11 + 1.43999e11i −1.34382e12 1.24396e13i −1.28645e14 + 1.60753e14i 4.92865e14
5.9 23170.5i 7.57629e6 1.21857e7i −5.36871e8 2.86759e10i 2.82348e11 + 1.75546e11i −3.86992e12 1.24396e13i −9.10907e13 1.84645e14i 6.64434e14
5.10 23170.5i 1.42278e7 1.86071e6i −5.36871e8 3.80214e10i 4.31134e10 + 3.29664e11i 7.78024e12 1.24396e13i 1.98967e14 5.29473e13i −8.80973e14
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.31.b.a 10
3.b odd 2 1 inner 6.31.b.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.31.b.a 10 1.a even 1 1 trivial
6.31.b.a 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{31}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 536870912 + T^{2} )^{5} \)
$3$ \( \)\(36\!\cdots\!49\)\( - \)\(25\!\cdots\!62\)\( T + \)\(69\!\cdots\!93\)\( T^{2} - \)\(26\!\cdots\!16\)\( T^{3} + \)\(52\!\cdots\!50\)\( T^{4} - \)\(10\!\cdots\!76\)\( T^{5} + \)\(25\!\cdots\!50\)\( T^{6} - 6364997729561534616 T^{7} + 79109874573957 T^{8} - 14139762 T^{9} + T^{10} \)
$5$ \( \)\(11\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{4} + \)\(84\!\cdots\!00\)\( T^{6} + \)\(49\!\cdots\!40\)\( T^{8} + T^{10} \)
$7$ \( ( \)\(35\!\cdots\!68\)\( + \)\(17\!\cdots\!76\)\( T - \)\(13\!\cdots\!64\)\( T^{2} - \)\(52\!\cdots\!08\)\( T^{3} + 729667300598 T^{4} + T^{5} )^{2} \)
$11$ \( \)\(27\!\cdots\!32\)\( + \)\(31\!\cdots\!80\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{6} + \)\(65\!\cdots\!60\)\( T^{8} + T^{10} \)
$13$ \( ( -\)\(12\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T + \)\(44\!\cdots\!00\)\( T^{2} - \)\(70\!\cdots\!00\)\( T^{3} - 5620928954228290 T^{4} + T^{5} )^{2} \)
$17$ \( \)\(29\!\cdots\!68\)\( + \)\(61\!\cdots\!80\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{4} + \)\(90\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!40\)\( T^{8} + T^{10} \)
$19$ \( ( -\)\(83\!\cdots\!68\)\( + \)\(59\!\cdots\!80\)\( T - \)\(40\!\cdots\!20\)\( T^{2} - \)\(57\!\cdots\!60\)\( T^{3} + 203217470506219310 T^{4} + T^{5} )^{2} \)
$23$ \( \)\(16\!\cdots\!68\)\( + \)\(48\!\cdots\!80\)\( T^{2} + \)\(87\!\cdots\!20\)\( T^{4} + \)\(36\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{8} + T^{10} \)
$29$ \( \)\(22\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(50\!\cdots\!00\)\( T^{4} + \)\(72\!\cdots\!00\)\( T^{6} + \)\(45\!\cdots\!00\)\( T^{8} + T^{10} \)
$31$ \( ( -\)\(20\!\cdots\!68\)\( - \)\(32\!\cdots\!20\)\( T - \)\(96\!\cdots\!20\)\( T^{2} - \)\(78\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} + T^{5} )^{2} \)
$37$ \( ( \)\(39\!\cdots\!32\)\( + \)\(57\!\cdots\!56\)\( T + \)\(10\!\cdots\!04\)\( T^{2} - \)\(51\!\cdots\!28\)\( T^{3} - \)\(34\!\cdots\!58\)\( T^{4} + T^{5} )^{2} \)
$41$ \( \)\(96\!\cdots\!32\)\( + \)\(15\!\cdots\!80\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{4} + \)\(46\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{8} + T^{10} \)
$43$ \( ( \)\(71\!\cdots\!76\)\( + \)\(61\!\cdots\!76\)\( T - \)\(11\!\cdots\!52\)\( T^{2} - \)\(23\!\cdots\!92\)\( T^{3} + \)\(52\!\cdots\!86\)\( T^{4} + T^{5} )^{2} \)
$47$ \( \)\(42\!\cdots\!00\)\( + \)\(37\!\cdots\!00\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{4} + \)\(83\!\cdots\!00\)\( T^{6} + \)\(53\!\cdots\!00\)\( T^{8} + T^{10} \)
$53$ \( \)\(29\!\cdots\!68\)\( + \)\(68\!\cdots\!80\)\( T^{2} + \)\(18\!\cdots\!20\)\( T^{4} + \)\(99\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!40\)\( T^{8} + T^{10} \)
$59$ \( \)\(64\!\cdots\!32\)\( + \)\(71\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!60\)\( T^{8} + T^{10} \)
$61$ \( ( -\)\(90\!\cdots\!24\)\( - \)\(67\!\cdots\!20\)\( T - \)\(14\!\cdots\!40\)\( T^{2} - \)\(38\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!30\)\( T^{4} + T^{5} )^{2} \)
$67$ \( ( \)\(30\!\cdots\!32\)\( + \)\(36\!\cdots\!96\)\( T - \)\(80\!\cdots\!76\)\( T^{2} - \)\(12\!\cdots\!68\)\( T^{3} + \)\(52\!\cdots\!22\)\( T^{4} + T^{5} )^{2} \)
$71$ \( \)\(37\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{8} + T^{10} \)
$73$ \( ( -\)\(22\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T + \)\(24\!\cdots\!00\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} - \)\(60\!\cdots\!50\)\( T^{4} + T^{5} )^{2} \)
$79$ \( ( -\)\(82\!\cdots\!32\)\( + \)\(40\!\cdots\!80\)\( T + \)\(43\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} - \)\(34\!\cdots\!10\)\( T^{4} + T^{5} )^{2} \)
$83$ \( \)\(66\!\cdots\!32\)\( + \)\(12\!\cdots\!80\)\( T^{2} + \)\(30\!\cdots\!80\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!60\)\( T^{8} + T^{10} \)
$89$ \( \)\(24\!\cdots\!32\)\( + \)\(12\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{4} + \)\(80\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!60\)\( T^{8} + T^{10} \)
$97$ \( ( \)\(82\!\cdots\!76\)\( + \)\(35\!\cdots\!36\)\( T - \)\(26\!\cdots\!92\)\( T^{2} - \)\(42\!\cdots\!72\)\( T^{3} + \)\(69\!\cdots\!06\)\( T^{4} + T^{5} )^{2} \)
show more
show less