Properties

Label 3.34.a.b
Level 3
Weight 34
Character orbit 3.a
Self dual Yes
Analytic conductor 20.695
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(20.6948486643\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6}\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 45540 - \beta_{1} ) q^{2} \) \( -43046721 q^{3} \) \( + ( 4863185200 - 67826 \beta_{1} - 10 \beta_{2} ) q^{4} \) \( + ( -86829345378 + 1305823 \beta_{1} + 297 \beta_{2} ) q^{5} \) \( + ( -1960347674340 + 43046721 \beta_{1} ) q^{6} \) \( + ( 3586899630944 + 291628981 \beta_{1} + 147875 \beta_{2} ) q^{7} \) \( + ( 602117349936192 - 2933686424 \beta_{1} - 1366200 \beta_{2} ) q^{8} \) \( + 1853020188851841 q^{9} \) \(+O(q^{10})\) \( q\) \(+(45540 - \beta_{1}) q^{2}\) \(-43046721 q^{3}\) \(+(4863185200 - 67826 \beta_{1} - 10 \beta_{2}) q^{4}\) \(+(-86829345378 + 1305823 \beta_{1} + 297 \beta_{2}) q^{5}\) \(+(-1960347674340 + 43046721 \beta_{1}) q^{6}\) \(+(3586899630944 + 291628981 \beta_{1} + 147875 \beta_{2}) q^{7}\) \(+(602117349936192 - 2933686424 \beta_{1} - 1366200 \beta_{2}) q^{8}\) \(+1853020188851841 q^{9}\) \(+(-18814238065025352 + 268852224482 \beta_{1} + 33490048 \beta_{2}) q^{10}\) \(+(-115160789733595452 + 1374009141218 \beta_{1} - 117158130 \beta_{2}) q^{11}\) \(+(-209344176475729200 + 2919686898546 \beta_{1} + 430467210 \beta_{2}) q^{12}\) \(+(243349234327264574 + 9029058791434 \beta_{1} - 3407034970 \beta_{2}) q^{13}\) \(+(-3155549579462270592 + 79051193603872 \beta_{1} + 13089202560 \beta_{2}) q^{14}\) \(+(3737718605099405538 - 56211398356383 \beta_{1} - 12784876137 \beta_{2}) q^{15}\) \(+(19032618795066604288 - 788314597484064 \beta_{1} - 37423881120 \beta_{2}) q^{16}\) \(+(-80567381117531622078 - 401349169596570 \beta_{1} + 85047521130 \beta_{2}) q^{17}\) \(+(84386539400312839140 - 1853020188851841 \beta_{1}) q^{18}\) \(+(-\)\(10\!\cdots\!44\)\( - 852472207041702 \beta_{1} - 394062747690 \beta_{2}) q^{19}\) \(+(-\)\(31\!\cdots\!92\)\( + 30832520123076772 \beta_{1} + 2441226033108 \beta_{2}) q^{20}\) \(+(-\)\(15\!\cdots\!24\)\( - 12553671380621301 \beta_{1} - 6365533867875 \beta_{2}) q^{21}\) \(+(-\)\(20\!\cdots\!96\)\( + 85458811157363260 \beta_{1} + 5680315016960 \beta_{2}) q^{22}\) \(+(-\)\(20\!\cdots\!08\)\( - 138440615323674422 \beta_{1} - 12222579683610 \beta_{2}) q^{23}\) \(+(-\)\(25\!\cdots\!32\)\( + 126285580995415704 \beta_{1} + 58810430230200 \beta_{2}) q^{24}\) \(+(-\)\(37\!\cdots\!73\)\( - 692128799755206332 \beta_{1} - 41247911010148 \beta_{2}) q^{25}\) \(+(-\)\(91\!\cdots\!08\)\( - 1796364138790299710 \beta_{1} - 144092975811840 \beta_{2}) q^{26}\) \(-\)\(79\!\cdots\!61\)\( q^{27}\) \(+(-\)\(10\!\cdots\!32\)\( + 9151665061279739712 \beta_{1} + 420733959159360 \beta_{2}) q^{28}\) \(+(-\)\(41\!\cdots\!98\)\( + 7356185636766476021 \beta_{1} + 650382979885155 \beta_{2}) q^{29}\) \(+(\)\(80\!\cdots\!92\)\( - 11573206697506023522 \beta_{1} - 1441636752532608 \beta_{2}) q^{30}\) \(+(\)\(37\!\cdots\!84\)\( + 5996767654496208601 \beta_{1} - 1942248070367905 \beta_{2}) q^{31}\) \(+(\)\(46\!\cdots\!04\)\( - 30669876755876060544 \beta_{1} + 1277884186980480 \beta_{2}) q^{32}\) \(+(\)\(49\!\cdots\!92\)\( - 59146588153460846178 \beta_{1} + 5043273334991730 \beta_{2}) q^{33}\) \(+(\)\(89\!\cdots\!80\)\( + \)\(11\!\cdots\!98\)\( \beta_{1} + 1837267472651520 \beta_{2}) q^{34}\) \(+(\)\(29\!\cdots\!60\)\( - \)\(20\!\cdots\!10\)\( \beta_{1} + 526068048546810 \beta_{2}) q^{35}\) \(+(\)\(90\!\cdots\!00\)\( - \)\(12\!\cdots\!66\)\( \beta_{1} - 18530201888518410 \beta_{2}) q^{36}\) \(+(-\)\(15\!\cdots\!06\)\( + \)\(99\!\cdots\!56\)\( \beta_{1} - 13657279933217280 \beta_{2}) q^{37}\) \(+(\)\(51\!\cdots\!44\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} - 35633874735002880 \beta_{2}) q^{38}\) \(+(-\)\(10\!\cdots\!54\)\( - \)\(38\!\cdots\!14\)\( \beta_{1} + 146661683790833370 \beta_{2}) q^{39}\) \(+(-\)\(33\!\cdots\!44\)\( + \)\(28\!\cdots\!04\)\( \beta_{1} + 188589583149459056 \beta_{2}) q^{40}\) \(+(-\)\(17\!\cdots\!06\)\( - \)\(44\!\cdots\!94\)\( \beta_{1} - 276313934012595570 \beta_{2}) q^{41}\) \(+(\)\(13\!\cdots\!32\)\( - \)\(34\!\cdots\!12\)\( \beta_{1} - 563447250712805760 \beta_{2}) q^{42}\) \(+(-\)\(80\!\cdots\!92\)\( - \)\(76\!\cdots\!34\)\( \beta_{1} - 374805537341295310 \beta_{2}) q^{43}\) \(+(-\)\(93\!\cdots\!56\)\( + \)\(13\!\cdots\!80\)\( \beta_{1} + 2251740376471411800 \beta_{2}) q^{44}\) \(+(-\)\(16\!\cdots\!98\)\( + \)\(24\!\cdots\!43\)\( \beta_{1} + 550346996088996777 \beta_{2}) q^{45}\) \(+(\)\(63\!\cdots\!84\)\( + \)\(11\!\cdots\!36\)\( \beta_{1} - 2225246299991010560 \beta_{2}) q^{46}\) \(+(\)\(64\!\cdots\!76\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} - 3162089172287251710 \beta_{2}) q^{47}\) \(+(-\)\(81\!\cdots\!48\)\( + \)\(33\!\cdots\!44\)\( \beta_{1} + 1610975369309807520 \beta_{2}) q^{48}\) \(+(\)\(60\!\cdots\!41\)\( - \)\(66\!\cdots\!28\)\( \beta_{1} + 9463742798274712640 \beta_{2}) q^{49}\) \(+(\)\(61\!\cdots\!68\)\( + \)\(11\!\cdots\!37\)\( \beta_{1} - 9758896787584184832 \beta_{2}) q^{50}\) \(+(\)\(34\!\cdots\!38\)\( + \)\(17\!\cdots\!70\)\( \beta_{1} - 3661016913824714730 \beta_{2}) q^{51}\) \(+(\)\(14\!\cdots\!12\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + 1389833979053964180 \beta_{2}) q^{52}\) \(+(\)\(26\!\cdots\!22\)\( - \)\(43\!\cdots\!87\)\( \beta_{1} + 25951231157220031815 \beta_{2}) q^{53}\) \(+(-\)\(36\!\cdots\!40\)\( + \)\(79\!\cdots\!61\)\( \beta_{1}) q^{54}\) \(+(\)\(10\!\cdots\!52\)\( - \)\(31\!\cdots\!32\)\( \beta_{1} - 65820023746468742648 \beta_{2}) q^{55}\) \(+(-\)\(12\!\cdots\!00\)\( + \)\(81\!\cdots\!20\)\( \beta_{1} + 8025228747367453440 \beta_{2}) q^{56}\) \(+(\)\(43\!\cdots\!24\)\( + \)\(36\!\cdots\!42\)\( \beta_{1} + 16963109156304824490 \beta_{2}) q^{57}\) \(+(-\)\(10\!\cdots\!92\)\( + \)\(91\!\cdots\!94\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2}) q^{58}\) \(+(-\)\(82\!\cdots\!56\)\( - \)\(75\!\cdots\!28\)\( \beta_{1} - 20648918645650648440 \beta_{2}) q^{59}\) \(+(\)\(13\!\cdots\!32\)\( - \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2}) q^{60}\) \(+(\)\(12\!\cdots\!06\)\( + \)\(11\!\cdots\!92\)\( \beta_{1} - 75857882953725781660 \beta_{2}) q^{61}\) \(+(\)\(10\!\cdots\!08\)\( - \)\(45\!\cdots\!88\)\( \beta_{1} - 73647337207927570560 \beta_{2}) q^{62}\) \(+(\)\(66\!\cdots\!04\)\( + \)\(54\!\cdots\!21\)\( \beta_{1} + \)\(27\!\cdots\!75\)\( \beta_{2}) q^{63}\) \(+(\)\(39\!\cdots\!72\)\( + \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!20\)\( \beta_{2}) q^{64}\) \(+(-\)\(47\!\cdots\!04\)\( + \)\(19\!\cdots\!14\)\( \beta_{1} - \)\(22\!\cdots\!54\)\( \beta_{2}) q^{65}\) \(+(\)\(89\!\cdots\!16\)\( - \)\(36\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!60\)\( \beta_{2}) q^{66}\) \(+(\)\(33\!\cdots\!32\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(28\!\cdots\!00\)\( \beta_{2}) q^{67}\) \(+(-\)\(58\!\cdots\!00\)\( + \)\(60\!\cdots\!48\)\( \beta_{1} + \)\(54\!\cdots\!00\)\( \beta_{2}) q^{68}\) \(+(\)\(89\!\cdots\!68\)\( + \)\(59\!\cdots\!62\)\( \beta_{1} + \)\(52\!\cdots\!10\)\( \beta_{2}) q^{69}\) \(+(\)\(37\!\cdots\!40\)\( - \)\(34\!\cdots\!40\)\( \beta_{1} - \)\(20\!\cdots\!60\)\( \beta_{2}) q^{70}\) \(+(-\)\(43\!\cdots\!28\)\( + \)\(17\!\cdots\!30\)\( \beta_{1} - 32207871855713784210 \beta_{2}) q^{71}\) \(+(\)\(11\!\cdots\!72\)\( - \)\(54\!\cdots\!84\)\( \beta_{1} - \)\(25\!\cdots\!00\)\( \beta_{2}) q^{72}\) \(+(-\)\(70\!\cdots\!98\)\( - \)\(29\!\cdots\!92\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2}) q^{73}\) \(+(-\)\(12\!\cdots\!52\)\( + \)\(31\!\cdots\!82\)\( \beta_{1} + \)\(90\!\cdots\!40\)\( \beta_{2}) q^{74}\) \(+(\)\(16\!\cdots\!33\)\( + \)\(29\!\cdots\!72\)\( \beta_{1} + \)\(17\!\cdots\!08\)\( \beta_{2}) q^{75}\) \(+(\)\(24\!\cdots\!64\)\( - \)\(18\!\cdots\!28\)\( \beta_{1} - \)\(27\!\cdots\!20\)\( \beta_{2}) q^{76}\) \(+(-\)\(60\!\cdots\!92\)\( - \)\(85\!\cdots\!48\)\( \beta_{1} - \)\(34\!\cdots\!80\)\( \beta_{2}) q^{77}\) \(+(\)\(39\!\cdots\!68\)\( + \)\(77\!\cdots\!10\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2}) q^{78}\) \(+(\)\(74\!\cdots\!00\)\( + \)\(62\!\cdots\!05\)\( \beta_{1} + \)\(23\!\cdots\!35\)\( \beta_{2}) q^{79}\) \(+(-\)\(19\!\cdots\!32\)\( + \)\(22\!\cdots\!12\)\( \beta_{1} + \)\(20\!\cdots\!68\)\( \beta_{2}) q^{80}\) \(+\)\(34\!\cdots\!81\)\( q^{81}\) \(+(\)\(42\!\cdots\!68\)\( - \)\(71\!\cdots\!38\)\( \beta_{1} - \)\(63\!\cdots\!20\)\( \beta_{2}) q^{82}\) \(+(-\)\(12\!\cdots\!20\)\( - \)\(24\!\cdots\!58\)\( \beta_{1} + \)\(74\!\cdots\!50\)\( \beta_{2}) q^{83}\) \(+(\)\(46\!\cdots\!72\)\( - \)\(39\!\cdots\!52\)\( \beta_{1} - \)\(18\!\cdots\!60\)\( \beta_{2}) q^{84}\) \(+(\)\(15\!\cdots\!24\)\( - \)\(10\!\cdots\!34\)\( \beta_{1} - \)\(12\!\cdots\!26\)\( \beta_{2}) q^{85}\) \(+(\)\(83\!\cdots\!28\)\( - \)\(28\!\cdots\!12\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2}) q^{86}\) \(+(\)\(18\!\cdots\!58\)\( - \)\(31\!\cdots\!41\)\( \beta_{1} - \)\(27\!\cdots\!55\)\( \beta_{2}) q^{87}\) \(+(-\)\(21\!\cdots\!68\)\( + \)\(16\!\cdots\!16\)\( \beta_{1} + \)\(24\!\cdots\!80\)\( \beta_{2}) q^{88}\) \(+(-\)\(10\!\cdots\!74\)\( + \)\(87\!\cdots\!48\)\( \beta_{1} - \)\(18\!\cdots\!20\)\( \beta_{2}) q^{89}\) \(+(-\)\(34\!\cdots\!32\)\( + \)\(49\!\cdots\!62\)\( \beta_{1} + \)\(62\!\cdots\!68\)\( \beta_{2}) q^{90}\) \(+(-\)\(26\!\cdots\!36\)\( + \)\(59\!\cdots\!66\)\( \beta_{1} - \)\(26\!\cdots\!90\)\( \beta_{2}) q^{91}\) \(+(\)\(77\!\cdots\!64\)\( - \)\(33\!\cdots\!44\)\( \beta_{1} + \)\(65\!\cdots\!40\)\( \beta_{2}) q^{92}\) \(+(-\)\(15\!\cdots\!64\)\( - \)\(25\!\cdots\!21\)\( \beta_{1} + \)\(83\!\cdots\!05\)\( \beta_{2}) q^{93}\) \(+(\)\(44\!\cdots\!16\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2}) q^{94}\) \(+(-\)\(72\!\cdots\!72\)\( + \)\(45\!\cdots\!52\)\( \beta_{1} - \)\(26\!\cdots\!72\)\( \beta_{2}) q^{95}\) \(+(-\)\(20\!\cdots\!84\)\( + \)\(13\!\cdots\!24\)\( \beta_{1} - \)\(55\!\cdots\!80\)\( \beta_{2}) q^{96}\) \(+(-\)\(29\!\cdots\!98\)\( - \)\(32\!\cdots\!20\)\( \beta_{1} + \)\(75\!\cdots\!80\)\( \beta_{2}) q^{97}\) \(+(\)\(10\!\cdots\!96\)\( - \)\(26\!\cdots\!29\)\( \beta_{1} - \)\(14\!\cdots\!20\)\( \beta_{2}) q^{98}\) \(+(-\)\(21\!\cdots\!32\)\( + \)\(25\!\cdots\!38\)\( \beta_{1} - \)\(21\!\cdots\!30\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 136620q^{2} \) \(\mathstrut -\mathstrut 129140163q^{3} \) \(\mathstrut +\mathstrut 14589555600q^{4} \) \(\mathstrut -\mathstrut 260488036134q^{5} \) \(\mathstrut -\mathstrut 5881043023020q^{6} \) \(\mathstrut +\mathstrut 10760698892832q^{7} \) \(\mathstrut +\mathstrut 1806352049808576q^{8} \) \(\mathstrut +\mathstrut 5559060566555523q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 136620q^{2} \) \(\mathstrut -\mathstrut 129140163q^{3} \) \(\mathstrut +\mathstrut 14589555600q^{4} \) \(\mathstrut -\mathstrut 260488036134q^{5} \) \(\mathstrut -\mathstrut 5881043023020q^{6} \) \(\mathstrut +\mathstrut 10760698892832q^{7} \) \(\mathstrut +\mathstrut 1806352049808576q^{8} \) \(\mathstrut +\mathstrut 5559060566555523q^{9} \) \(\mathstrut -\mathstrut 56442714195076056q^{10} \) \(\mathstrut -\mathstrut 345482369200786356q^{11} \) \(\mathstrut -\mathstrut 628032529427187600q^{12} \) \(\mathstrut +\mathstrut 730047702981793722q^{13} \) \(\mathstrut -\mathstrut 9466648738386811776q^{14} \) \(\mathstrut +\mathstrut 11213155815298216614q^{15} \) \(\mathstrut +\mathstrut 57097856385199812864q^{16} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!34\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!20\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!32\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!76\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!88\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!96\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!19\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!24\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!83\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!96\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!94\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!76\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!52\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!12\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!76\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!18\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!32\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!62\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!32\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!18\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!96\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!76\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!68\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!94\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!52\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!28\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!44\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!23\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!04\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!14\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!36\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!66\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!56\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!72\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!76\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!68\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!96\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!18\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!24\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!12\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!16\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!12\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!48\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!04\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!84\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!16\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!94\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!56\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!99\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!92\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!76\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!04\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!96\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!43\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!04\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!60\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!16\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!72\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!84\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!74\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!04\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!22\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!96\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!08\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!92\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!92\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!48\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!16\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!52\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!94\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!88\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!96\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(5357605\) \(x\mathstrut +\mathstrut \) \(842871622\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4 \nu^{2} + 1788 \nu + 14286352 \)\()/105\)
\(\beta_{2}\)\(=\)\((\)\( 14876 \nu^{2} + 41250588 \nu - 53146909808 \)\()/105\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(3719\) \(\beta_{1}\mathstrut +\mathstrut \) \(152064\)\()/456192\)
\(\nu^{2}\)\(=\)\((\)\(149\) \(\beta_{2}\mathstrut -\mathstrut \) \(3437549\) \(\beta_{1}\mathstrut +\mathstrut \) \(543132615168\)\()/152064\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
158.055
2232.09
−2389.14
−92260.3 −4.30467e7 −7.79766e7 −3.77234e10 3.97150e12 −2.13696e13 7.99704e14 1.85302e15 3.48037e15
1.2 61269.1 −4.30467e7 −4.83603e9 2.12383e11 −2.63744e12 1.58203e14 −8.22597e14 1.85302e15 1.30125e16
1.3 167611. −4.30467e7 1.95036e10 −4.35148e11 −7.21511e12 −1.26073e14 1.82925e15 1.85302e15 −7.29356e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut 136620 T_{2}^{2} \) \(\mathstrut -\mathstrut 10847167488 T_{2} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!08\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(3))\).