Properties

Label 3.21.b.a
Level 3
Weight 21
Character orbit 3.b
Analytic conductor 7.605
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(7.60541295308\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{14}\cdot 3^{19}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -687 + 11 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -354200 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 1285 \beta_{1} + 18 \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -16066872 - 8111 \beta_{1} + 30 \beta_{2} + 47 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{6} \) \( + ( 93267986 - 718 \beta_{1} + 1371 \beta_{2} - 254 \beta_{3} - 9 \beta_{5} ) q^{7} \) \( + ( -220818 \beta_{1} - 13734 \beta_{2} + 108 \beta_{3} - 34 \beta_{4} - 54 \beta_{5} ) q^{8} \) \( + ( -181247247 - 1138179 \beta_{1} + 4626 \beta_{2} + 840 \beta_{3} + 117 \beta_{4} - 204 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -687 + 11 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -354200 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 1285 \beta_{1} + 18 \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -16066872 - 8111 \beta_{1} + 30 \beta_{2} + 47 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{6} \) \( + ( 93267986 - 718 \beta_{1} + 1371 \beta_{2} - 254 \beta_{3} - 9 \beta_{5} ) q^{7} \) \( + ( -220818 \beta_{1} - 13734 \beta_{2} + 108 \beta_{3} - 34 \beta_{4} - 54 \beta_{5} ) q^{8} \) \( + ( -181247247 - 1138179 \beta_{1} + 4626 \beta_{2} + 840 \beta_{3} + 117 \beta_{4} - 204 \beta_{5} ) q^{9} \) \( + ( -1813384080 - 67814 \beta_{1} + 153330 \beta_{2} + 5606 \beta_{3} - 576 \beta_{5} ) q^{10} \) \( + ( 8548244 \beta_{1} - 233757 \beta_{2} + 1998 \beta_{3} + 500 \beta_{4} - 999 \beta_{5} ) q^{11} \) \( + ( 10659647688 - 46894375 \beta_{1} - 138551 \beta_{2} - 26055 \beta_{3} - 2106 \beta_{4} - 702 \beta_{5} ) q^{12} \) \( + ( 2974646546 + 353528 \beta_{1} - 794364 \beta_{2} - 23048 \beta_{3} + 3060 \beta_{5} ) q^{13} \) \( + ( 310115088 \beta_{1} + 2856474 \beta_{2} - 24084 \beta_{3} - 3874 \beta_{4} + 12042 \beta_{5} ) q^{14} \) \( + ( -81294104160 - 364324048 \beta_{1} + 1693545 \beta_{2} + 309202 \beta_{3} + 22980 \beta_{4} + 27343 \beta_{5} ) q^{15} \) \( + ( -52912140608 + 3461064 \beta_{1} - 7777368 \beta_{2} - 226248 \beta_{3} + 29952 \beta_{5} ) q^{16} \) \( + ( 898272248 \beta_{1} + 1561212 \beta_{2} - 11016 \beta_{3} + 12376 \beta_{4} + 5508 \beta_{5} ) q^{17} \) \( + ( 1594025885520 - 952151013 \beta_{1} - 11573334 \beta_{2} - 2183670 \beta_{3} - 167076 \beta_{4} - 102348 \beta_{5} ) q^{18} \) \( + ( -2493003365854 - 26579442 \beta_{1} + 60264333 \beta_{2} + 2404830 \beta_{3} - 223839 \beta_{5} ) q^{19} \) \( + ( -6595720620 \beta_{1} - 98564004 \beta_{2} + 819720 \beta_{3} + 57332 \beta_{4} - 409860 \beta_{5} ) q^{20} \) \( + ( 4329833314674 + 10265308291 \beta_{1} + 47733458 \beta_{2} + 8924040 \beta_{3} + 828711 \beta_{4} - 255204 \beta_{5} ) q^{21} \) \( + ( -11841881003280 - 4316278 \beta_{1} + 4580130 \beta_{2} - 6072458 \beta_{3} - 96192 \beta_{5} ) q^{22} \) \( + ( -18110483664 \beta_{1} + 265230810 \beta_{2} - 2315196 \beta_{3} - 892528 \beta_{4} + 1157598 \beta_{5} ) q^{23} \) \( + ( 49008117633984 + 26369192414 \beta_{1} - 114758022 \beta_{2} - 16013036 \beta_{3} - 2669082 \beta_{4} + 1629730 \beta_{5} ) q^{24} \) \( + ( -66200795836175 + 475755560 \beta_{1} - 1063907700 \beta_{2} - 24689240 \beta_{3} + 4176540 \beta_{5} ) q^{25} \) \( + ( 29376177018 \beta_{1} + 531408312 \beta_{2} - 3627504 \beta_{3} + 5037032 \beta_{4} + 1813752 \beta_{5} ) q^{26} \) \( + ( 179504265417633 - 58252064193 \beta_{1} + 128602242 \beta_{2} - 8540694 \beta_{3} + 4123548 \beta_{4} + 3379995 \beta_{5} ) q^{27} \) \( + ( -339047481545584 - 1257243738 \beta_{1} + 2943680526 \beta_{2} + 229318746 \beta_{3} - 9517824 \beta_{5} ) q^{28} \) \( + ( 195662895343 \beta_{1} - 1845354258 \beta_{2} + 12939696 \beta_{3} - 15176733 \beta_{4} - 6469848 \beta_{5} ) q^{29} \) \( + ( 510177353050800 - 368654258250 \beta_{1} - 13075434 \beta_{2} + 1524150 \beta_{3} + 6996132 \beta_{4} - 31680180 \beta_{5} ) q^{30} \) \( + ( -363817527580558 + 276519002 \beta_{1} - 1212642105 \beta_{2} - 752072918 \beta_{3} - 4403277 \beta_{5} ) q^{31} \) \( + ( -25438468592 \beta_{1} - 9190683216 \beta_{2} + 77669280 \beta_{3} + 13674128 \beta_{4} - 38834640 \beta_{5} ) q^{32} \) \( + ( 635199537002400 - 14193956465 \beta_{1} + 124467258 \beta_{2} + 578430656 \beta_{3} - 56981433 \beta_{4} + 49837640 \beta_{5} ) q^{33} \) \( + ( -1261062107441856 - 1928372424 \beta_{1} + 5072925528 \beta_{2} + 1044272328 \beta_{3} - 8186112 \beta_{5} ) q^{34} \) \( + ( 195593445220 \beta_{1} + 43560294210 \beta_{2} - 345950460 \beta_{3} + 84850740 \beta_{4} + 172975230 \beta_{5} ) q^{35} \) \( + ( 1151685279339624 + 2428307547387 \beta_{1} - 70149609 \beta_{2} - 1927947111 \beta_{3} + 133478748 \beta_{4} + 7479156 \beta_{5} ) q^{36} \) \( + ( -322390243694734 + 26283007176 \beta_{1} - 56837616804 \beta_{2} + 1041495240 \beta_{3} + 253004652 \beta_{5} ) q^{37} \) \( + ( -5082179732428 \beta_{1} - 48745462122 \beta_{2} + 342989748 \beta_{3} - 392900430 \beta_{4} - 171494874 \beta_{5} ) q^{38} \) \( + ( -2635600340141646 + 1595682891106 \beta_{1} - 5137339726 \beta_{2} - 183548160 \beta_{3} - 51348492 \beta_{4} + 163573776 \beta_{5} ) q^{39} \) \( + ( 7413736827922560 - 52942967632 \beta_{1} + 110255942640 \beta_{2} - 7354500272 \beta_{3} - 558309888 \beta_{5} ) q^{40} \) \( + ( 4202403991286 \beta_{1} + 9305272332 \beta_{2} + 14457312 \beta_{3} + 614546430 \beta_{4} - 7228656 \beta_{5} ) q^{41} \) \( + ( -14425432616614320 - 3941973579808 \beta_{1} + 14457596406 \beta_{2} + 12981589714 \beta_{3} - 494834496 \beta_{4} - 952264832 \beta_{5} ) q^{42} \) \( + ( 9699164583697346 - 7946929378 \beta_{1} + 27437114181 \beta_{2} + 12411334126 \beta_{3} + 41337081 \beta_{5} ) q^{43} \) \( + ( 2512216708404 \beta_{1} - 168467251716 \beta_{2} + 1475012808 \beta_{3} + 597044692 \beta_{4} - 737506404 \beta_{5} ) q^{44} \) \( + ( -13642370507736000 - 21350363214015 \beta_{1} + 18381429522 \beta_{2} - 24385334400 \beta_{3} + 1198723509 \beta_{4} + 806147280 \beta_{5} ) q^{45} \) \( + ( 25235231869819296 + 34040195644 \beta_{1} - 76730740788 \beta_{2} - 2521765948 \beta_{3} + 291837312 \beta_{5} ) q^{46} \) \( + ( 36886807350312 \beta_{1} + 332805906108 \beta_{2} - 3389688648 \beta_{3} - 4391181368 \beta_{4} + 1694844324 \beta_{5} ) q^{47} \) \( + ( -25748280180005952 + 14744499295352 \beta_{1} - 132454073864 \beta_{2} - 1779709320 \beta_{3} - 501244848 \beta_{4} + 1601554032 \beta_{5} ) q^{48} \) \( + ( 31360859714575587 + 236324521496 \beta_{1} - 538178610732 \beta_{2} - 24303199592 \beta_{3} + 1963160388 \beta_{5} ) q^{49} \) \( + ( -36360366057175 \beta_{1} + 632350307880 \beta_{2} - 4220110800 \beta_{3} + 6644824760 \beta_{4} + 2110055400 \beta_{5} ) q^{50} \) \( + ( -19609625996007552 - 12945410027364 \beta_{1} + 52720478064 \beta_{2} + 45009896592 \beta_{3} - 2115953892 \beta_{4} - 786963048 \beta_{5} ) q^{51} \) \( + ( -38424586299492784 - 61861270386 \beta_{1} + 182023052886 \beta_{2} + 57441018738 \beta_{3} - 40928256 \beta_{5} ) q^{52} \) \( + ( -35485218666663 \beta_{1} - 1466165547846 \beta_{2} + 12571022016 \beta_{3} + 3400757061 \beta_{4} - 6285511008 \beta_{5} ) q^{53} \) \( + ( 81633393050858712 + 186436885042719 \beta_{1} + 792481606740 \beta_{2} - 11269412631 \beta_{3} + 2345398821 \beta_{4} - 2166346377 \beta_{5} ) q^{54} \) \( + ( -65534768078356800 - 1200654524120 \beta_{1} + 2553213043500 \beta_{2} - 101242684120 \beta_{3} - 12054603780 \beta_{5} ) q^{55} \) \( + ( -240249819500884 \beta_{1} - 822556142172 \beta_{2} + 3053150712 \beta_{3} - 25088796148 \beta_{4} - 1526575356 \beta_{5} ) q^{56} \) \( + ( 201845332072347906 - 172663737629999 \beta_{1} - 1744588232206 \beta_{2} - 5303578680 \beta_{3} + 2242204497 \beta_{4} - 12470067612 \beta_{5} ) q^{57} \) \( + ( -273304519758028080 + 1056869249246 \beta_{1} - 2301998962650 \beta_{2} + 21403725346 \beta_{3} + 9984009024 \beta_{5} ) q^{58} \) \( + ( 311917339598656 \beta_{1} - 519339050181 \beta_{2} + 8014643982 \beta_{3} + 25246677424 \beta_{4} - 4007321991 \beta_{5} ) q^{59} \) \( + ( 431888047042135680 + 128428280005364 \beta_{1} - 496632563940 \beta_{2} - 274293669896 \beta_{3} + 2964347940 \beta_{4} + 26318404876 \beta_{5} ) q^{60} \) \( + ( -499851590896924078 - 15016706392 \beta_{1} + 447038332140 \beta_{2} + 514036556968 \beta_{3} + 4620554172 \beta_{5} ) q^{61} \) \( + ( 316836791569512 \beta_{1} + 10019780151426 \beta_{2} - 79585856100 \beta_{3} + 19449924182 \beta_{4} + 39792928050 \beta_{5} ) q^{62} \) \( + ( 74091239349532146 - 703936268820294 \beta_{1} + 7925609885913 \beta_{2} + 505664756574 \beta_{3} - 27296148672 \beta_{4} + 5371891737 \beta_{5} ) q^{63} \) \( + ( -13928259160065536 + 4202520429504 \beta_{1} - 9911700857664 \beta_{2} - 855935067072 \beta_{3} + 30986901504 \beta_{5} ) q^{64} \) \( + ( 649897311137170 \beta_{1} - 4100007414420 \beta_{2} + 27939230640 \beta_{3} - 39188382870 \beta_{4} - 13969615320 \beta_{5} ) q^{65} \) \( + ( 20122551186208560 + 112864823586630 \beta_{1} - 13052639979738 \beta_{2} + 182865169350 \beta_{3} + 15928592004 \beta_{4} - 370510740 \beta_{5} ) q^{66} \) \( + ( 748025994981699746 - 3362690393530 \beta_{1} + 6555496959921 \beta_{2} - 1022574156266 \beta_{3} - 40604301387 \beta_{5} ) q^{67} \) \( + ( -1289067489555472 \beta_{1} - 13279799131056 \beta_{2} + 104275531872 \beta_{3} - 33906778256 \beta_{4} - 52137765936 \beta_{5} ) q^{68} \) \( + ( -585218737364908608 + 193823393342930 \beta_{1} - 908642779356 \beta_{2} - 1133816212880 \beta_{3} + 82239694914 \beta_{4} - 58509401288 \beta_{5} ) q^{69} \) \( + ( -302094807097456800 - 12333080609500 \beta_{1} + 29857823303700 \beta_{2} + 3467871406300 \beta_{3} - 82085270400 \beta_{5} ) q^{70} \) \( + ( -882000267190312 \beta_{1} - 5902832524554 \beta_{2} + 50107575996 \beta_{3} + 10290997720 \beta_{4} - 25053787998 \beta_{5} ) q^{71} \) \( + ( -1735755420935214720 + 1881711434202750 \beta_{1} + 42081277234986 \beta_{2} + 684599865948 \beta_{3} - 114496557858 \beta_{4} - 83786782518 \beta_{5} ) q^{72} \) \( + ( 2623067634043766306 + 6446424530016 \beta_{1} - 13864797508464 \beta_{2} + 349476812640 \beta_{3} + 62925012432 \beta_{5} ) q^{73} \) \( + ( -836676810612022 \beta_{1} + 3461585027976 \beta_{2} + 18363755376 \beta_{3} + 316265628120 \beta_{4} - 9181877688 \beta_{5} ) q^{74} \) \( + ( -3478511674742973375 + 1114014104819375 \beta_{1} - 75813067724015 \beta_{2} - 426876696000 \beta_{3} - 84339255780 \beta_{4} + 218507425200 \beta_{5} ) q^{75} \) \( + ( 4545880037232095888 + 9787776066022 \beta_{1} - 27229613158578 \beta_{2} - 7139026361830 \beta_{3} + 24525460224 \beta_{5} ) q^{76} \) \( + ( -755899956663118 \beta_{1} + 65905123962996 \beta_{2} - 603416808576 \beta_{3} - 411667682166 \beta_{4} + 301708404288 \beta_{5} ) q^{77} \) \( + ( -2235129655009497840 - 2442260817730726 \beta_{1} + 10959544428756 \beta_{2} + 2539048146046 \beta_{3} + 128294498514 \beta_{4} + 34808491846 \beta_{5} ) q^{78} \) \( + ( 2946656791850633906 + 38430845316698 \beta_{1} - 82816142246457 \beta_{2} + 1884677901610 \beta_{3} + 373291881651 \beta_{5} ) q^{79} \) \( + ( 6259924658643040 \beta_{1} - 41548546038240 \beta_{2} + 272943967680 \beta_{3} - 465880775840 \beta_{4} - 136471983840 \beta_{5} ) q^{80} \) \( + ( -2168604175871501919 + 1072177911866106 \beta_{1} + 183648226029912 \beta_{2} - 2088383867928 \beta_{3} + 326344759650 \beta_{4} + 175433302764 \beta_{5} ) q^{81} \) \( + ( -5900558852150615520 - 44816243058452 \beta_{1} + 103773325945020 \beta_{2} + 6736432151828 \beta_{3} - 352590841728 \beta_{5} ) q^{82} \) \( + ( 4668747826236388 \beta_{1} + 33211441735497 \beta_{2} - 79368666102 \beta_{3} + 1309341600228 \beta_{4} + 39684333051 \beta_{5} ) q^{83} \) \( + ( 10066266436498046544 - 15368771551913654 \beta_{1} - 279823154352982 \beta_{2} - 3536147883510 \beta_{3} - 172254950244 \beta_{4} - 619436306124 \beta_{5} ) q^{84} \) \( + ( -3762762471951886080 - 48973564771104 \beta_{1} + 111800034625680 \beta_{2} + 5375504469216 \beta_{3} - 403685743536 \beta_{5} ) q^{85} \) \( + ( -1587907084721532 \beta_{1} - 167554655732826 \beta_{2} + 1325046691476 \beta_{3} - 364526817694 \beta_{4} - 662523345738 \beta_{5} ) q^{86} \) \( + ( 2976378599982382560 + 5287145706136744 \beta_{1} - 25461418244469 \beta_{2} + 5758529362166 \beta_{3} - 1284099036636 \beta_{4} - 344718593299 \beta_{5} ) q^{87} \) \( + ( -15833368184788241280 - 19077383174864 \beta_{1} + 30857130726000 \beta_{2} - 13664019471664 \beta_{3} - 303161135616 \beta_{5} ) q^{88} \) \( + ( -12091673108792186 \beta_{1} - 104697531834984 \beta_{2} + 740780534808 \beta_{3} - 816260936434 \beta_{4} - 370390267404 \beta_{5} ) q^{89} \) \( + ( 29927054217485216880 + 8080144204314114 \beta_{1} + 505524939555690 \beta_{2} - 9819007738506 \beta_{3} + 1166594144040 \beta_{4} + 515269395576 \beta_{5} ) q^{90} \) \( + ( -1076835080860952252 + 60121969780916 \beta_{1} - 144656713876722 \beta_{2} - 15793313340332 \beta_{3} + 410450522598 \beta_{5} ) q^{91} \) \( + ( 9059164515774776 \beta_{1} + 333240827730408 \beta_{2} - 2808573163344 \beta_{3} - 444489534216 \beta_{4} + 1404286581672 \beta_{5} ) q^{92} \) \( + ( -4139713001957082414 + 27512760480564247 \beta_{1} - 521480446971886 \beta_{2} + 20723511544200 \beta_{3} + 1727601736419 \beta_{4} + 315858141900 \beta_{5} ) q^{93} \) \( + ( -51959051808989178816 + 191639811024344 \beta_{1} - 383179231947912 \beta_{2} + 46382507613736 \beta_{3} + 2203910357760 \beta_{5} ) q^{94} \) \( + ( -54878154341958160 \beta_{1} + 192686831529174 \beta_{2} - 1454483588580 \beta_{3} + 887059750928 \beta_{4} + 727241794290 \beta_{5} ) q^{95} \) \( + ( 30789673356293558784 + 4389252480044688 \beta_{1} - 15430804560720 \beta_{2} + 4238128429152 \beta_{3} - 1296492406704 \beta_{4} + 2130130383216 \beta_{5} ) q^{96} \) \( + ( 37206722707591399106 - 101589021149896 \beta_{1} + 187419319206276 \beta_{2} - 44084040669512 \beta_{3} - 1348824646476 \beta_{5} ) q^{97} \) \( + ( 57008469002794603 \beta_{1} + 471629339321112 \beta_{2} - 3355041220272 \beta_{3} + 3555101725448 \beta_{4} + 1677520610136 \beta_{5} ) q^{98} \) \( + ( -29688338594808838080 + 59566120341984 \beta_{1} + 669836159109171 \beta_{2} - 39285079363602 \beta_{3} - 2633173561224 \beta_{4} - 1784433170439 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 4122q^{3} \) \(\mathstrut -\mathstrut 2125200q^{4} \) \(\mathstrut -\mathstrut 96401232q^{6} \) \(\mathstrut +\mathstrut 559607916q^{7} \) \(\mathstrut -\mathstrut 1087483482q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 4122q^{3} \) \(\mathstrut -\mathstrut 2125200q^{4} \) \(\mathstrut -\mathstrut 96401232q^{6} \) \(\mathstrut +\mathstrut 559607916q^{7} \) \(\mathstrut -\mathstrut 1087483482q^{9} \) \(\mathstrut -\mathstrut 10880304480q^{10} \) \(\mathstrut +\mathstrut 63957886128q^{12} \) \(\mathstrut +\mathstrut 17847879276q^{13} \) \(\mathstrut -\mathstrut 487764624960q^{15} \) \(\mathstrut -\mathstrut 317472843648q^{16} \) \(\mathstrut +\mathstrut 9564155313120q^{18} \) \(\mathstrut -\mathstrut 14958020195124q^{19} \) \(\mathstrut +\mathstrut 25978999888044q^{21} \) \(\mathstrut -\mathstrut 71051286019680q^{22} \) \(\mathstrut +\mathstrut 294048705803904q^{24} \) \(\mathstrut -\mathstrut 397204775017050q^{25} \) \(\mathstrut +\mathstrut 1077025592505798q^{27} \) \(\mathstrut -\mathstrut 2034284889273504q^{28} \) \(\mathstrut +\mathstrut 3061064118304800q^{30} \) \(\mathstrut -\mathstrut 2182905165483348q^{31} \) \(\mathstrut +\mathstrut 3811197222014400q^{33} \) \(\mathstrut -\mathstrut 7566372644651136q^{34} \) \(\mathstrut +\mathstrut 6910111676037744q^{36} \) \(\mathstrut -\mathstrut 1934341462168404q^{37} \) \(\mathstrut -\mathstrut 15813602040849876q^{39} \) \(\mathstrut +\mathstrut 44482420967535360q^{40} \) \(\mathstrut -\mathstrut 86552595699685920q^{42} \) \(\mathstrut +\mathstrut 58194987502184076q^{43} \) \(\mathstrut -\mathstrut 81854223046416000q^{45} \) \(\mathstrut +\mathstrut 151411391218915776q^{46} \) \(\mathstrut -\mathstrut 154489681080035712q^{48} \) \(\mathstrut +\mathstrut 188165158287453522q^{49} \) \(\mathstrut -\mathstrut 117657755976045312q^{51} \) \(\mathstrut -\mathstrut 230547517796956704q^{52} \) \(\mathstrut +\mathstrut 489800358305152272q^{54} \) \(\mathstrut -\mathstrut 393208608470140800q^{55} \) \(\mathstrut +\mathstrut 1211071992434087436q^{57} \) \(\mathstrut -\mathstrut 1639827118548168480q^{58} \) \(\mathstrut +\mathstrut 2591328282252814080q^{60} \) \(\mathstrut -\mathstrut 2999109545381544468q^{61} \) \(\mathstrut +\mathstrut 444547436097192876q^{63} \) \(\mathstrut -\mathstrut 83569554960393216q^{64} \) \(\mathstrut +\mathstrut 120735307117251360q^{66} \) \(\mathstrut +\mathstrut 4488155969890198476q^{67} \) \(\mathstrut -\mathstrut 3511312424189451648q^{69} \) \(\mathstrut -\mathstrut 1812568842584740800q^{70} \) \(\mathstrut -\mathstrut 10414532525611288320q^{72} \) \(\mathstrut +\mathstrut 15738405804262597836q^{73} \) \(\mathstrut -\mathstrut 20871070048457840250q^{75} \) \(\mathstrut +\mathstrut 27275280223392575328q^{76} \) \(\mathstrut -\mathstrut 13410777930056987040q^{78} \) \(\mathstrut +\mathstrut 17679940751103803436q^{79} \) \(\mathstrut -\mathstrut 13011625055229011514q^{81} \) \(\mathstrut -\mathstrut 35403353112903693120q^{82} \) \(\mathstrut +\mathstrut 60397598618988279264q^{84} \) \(\mathstrut -\mathstrut 22576574831711316480q^{85} \) \(\mathstrut +\mathstrut 17858271599894295360q^{87} \) \(\mathstrut -\mathstrut 95000209108729447680q^{88} \) \(\mathstrut +\mathstrut 179562325304911301280q^{90} \) \(\mathstrut -\mathstrut 6461010485165713512q^{91} \) \(\mathstrut -\mathstrut 24838278011742494484q^{93} \) \(\mathstrut -\mathstrut 311754310853935072896q^{94} \) \(\mathstrut +\mathstrut 184738040137761352704q^{96} \) \(\mathstrut +\mathstrut 223240336245548394636q^{97} \) \(\mathstrut -\mathstrut 178130031568853028480q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(116898\) \(x^{4}\mathstrut +\mathstrut \) \(3059043456\) \(x^{2}\mathstrut +\mathstrut \) \(18153107947520\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -17 \nu^{5} - 5728 \nu^{4} - 2170562 \nu^{3} - 526987456 \nu^{2} - 55742610560 \nu - 6124748591104 \)\()/64336896\)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{5} + 5728 \nu^{4} + 2170562 \nu^{3} + 1299030208 \nu^{2} + 55871284352 \nu + 36208166465536 \)\()/21445632\)
\(\beta_{4}\)\(=\)\((\)\( 373 \nu^{5} + 2864 \nu^{4} + 36271114 \nu^{3} + 263493728 \nu^{2} + 493897505920 \nu + 3062374295552 \)\()/1787136\)
\(\beta_{5}\)\(=\)\((\)\( -1343 \nu^{5} + 475424 \nu^{4} - 171474398 \nu^{3} + 45284044352 \nu^{2} - 4417563003776 \nu + 568520968810496 \)\()/21445632\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1402776\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(54\) \(\beta_{3}\mathstrut -\mathstrut \) \(6867\) \(\beta_{2}\mathstrut -\mathstrut \) \(1158985\) \(\beta_{1}\)\()/108\)
\(\nu^{4}\)\(=\)\((\)\(416\) \(\beta_{5}\mathstrut -\mathstrut \) \(46833\) \(\beta_{3}\mathstrut -\mathstrut \) \(239091\) \(\beta_{2}\mathstrut +\mathstrut \) \(91761\) \(\beta_{1}\mathstrut +\mathstrut \) \(45282332952\)\()/18\)
\(\nu^{5}\)\(=\)\((\)\(1303179\) \(\beta_{5}\mathstrut +\mathstrut \) \(1085281\) \(\beta_{4}\mathstrut -\mathstrut \) \(2606358\) \(\beta_{3}\mathstrut +\mathstrut \) \(336207555\) \(\beta_{2}\mathstrut +\mathstrut \) \(44432621465\) \(\beta_{1}\)\()/54\)

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
287.188i
161.041i
92.1239i
92.1239i
161.041i
287.188i
1723.13i −25942.2 53045.1i −1.92059e6 9.20415e6i −9.14036e7 + 4.47017e7i 4.08861e8 1.50260e9i −2.14079e9 + 2.75221e9i −1.58599e10
2.2 966.247i 56662.8 + 16616.6i 114943. 1.69009e7i 1.60557e7 5.47503e7i 2.16509e8 1.12425e9i 2.93456e9 + 1.88308e9i 1.63305e10
2.3 552.743i −32781.6 + 49113.6i 743051. 1.06933e7i 2.71472e7 + 1.81198e7i −3.45566e8 9.90310e8i −1.33751e9 3.22005e9i −5.91067e9
2.4 552.743i −32781.6 49113.6i 743051. 1.06933e7i 2.71472e7 1.81198e7i −3.45566e8 9.90310e8i −1.33751e9 + 3.22005e9i −5.91067e9
2.5 966.247i 56662.8 16616.6i 114943. 1.69009e7i 1.60557e7 + 5.47503e7i 2.16509e8 1.12425e9i 2.93456e9 1.88308e9i 1.63305e10
2.6 1723.13i −25942.2 + 53045.1i −1.92059e6 9.20415e6i −9.14036e7 4.47017e7i 4.08861e8 1.50260e9i −2.14079e9 2.75221e9i −1.58599e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
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 \(S_{21}^{\mathrm{new}}(3, [\chi])\).