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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 5357605 x + 842871622\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6}\cdot 11 \)
Twist minimal: yes
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.

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

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

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

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:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.34.a.b 3
3.b odd 2 1 9.34.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.34.a.b 3 1.a even 1 1 trivial
9.34.a.c 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 136620 T_{2}^{2} - 10847167488 T_{2} + \)947456640811008

'>\(94\!\cdots\!08\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 136620 T + 14922636288 T^{2} - 1399657087107072 T^{3} + \)\(12\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + \)\(63\!\cdots\!88\)\( T^{6} \)
$3$ \( ( 1 + 43046721 T )^{3} \)
$5$ \( 1 + 260488036134 T + \)\(26\!\cdots\!75\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!75\)\( T^{4} + \)\(35\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 10760698892832 T + \)\(25\!\cdots\!61\)\( T^{2} - \)\(59\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!27\)\( T^{4} - \)\(64\!\cdots\!68\)\( T^{5} + \)\(46\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + 345482369200786356 T + \)\(65\!\cdots\!53\)\( T^{2} + \)\(91\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!43\)\( T^{4} + \)\(18\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} \)
$13$ \( 1 - 730047702981793722 T + \)\(58\!\cdots\!79\)\( T^{2} + \)\(55\!\cdots\!04\)\( T^{3} + \)\(33\!\cdots\!87\)\( T^{4} - \)\(24\!\cdots\!98\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \)
$17$ \( 1 + \)\(24\!\cdots\!34\)\( T + \)\(13\!\cdots\!63\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(52\!\cdots\!31\)\( T^{4} + \)\(39\!\cdots\!46\)\( T^{5} + \)\(65\!\cdots\!53\)\( T^{6} \)
$19$ \( 1 + \)\(30\!\cdots\!32\)\( T + \)\(46\!\cdots\!13\)\( T^{2} + \)\(94\!\cdots\!56\)\( T^{3} + \)\(73\!\cdots\!67\)\( T^{4} + \)\(75\!\cdots\!92\)\( T^{5} + \)\(39\!\cdots\!79\)\( T^{6} \)
$23$ \( 1 + \)\(62\!\cdots\!24\)\( T + \)\(34\!\cdots\!69\)\( T^{2} + \)\(10\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!27\)\( T^{4} + \)\(46\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!87\)\( T^{6} \)
$29$ \( 1 + \)\(12\!\cdots\!94\)\( T + \)\(46\!\cdots\!51\)\( T^{2} + \)\(46\!\cdots\!32\)\( T^{3} + \)\(85\!\cdots\!39\)\( T^{4} + \)\(41\!\cdots\!74\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \)
$31$ \( 1 - \)\(11\!\cdots\!52\)\( T + \)\(86\!\cdots\!73\)\( T^{2} - \)\(40\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!43\)\( T^{4} - \)\(29\!\cdots\!12\)\( T^{5} + \)\(44\!\cdots\!71\)\( T^{6} \)
$37$ \( 1 + \)\(47\!\cdots\!18\)\( T + \)\(52\!\cdots\!51\)\( T^{2} - \)\(22\!\cdots\!08\)\( T^{3} + \)\(29\!\cdots\!47\)\( T^{4} + \)\(15\!\cdots\!62\)\( T^{5} + \)\(17\!\cdots\!73\)\( T^{6} \)
$41$ \( 1 + \)\(51\!\cdots\!18\)\( T + \)\(18\!\cdots\!83\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!43\)\( T^{4} + \)\(14\!\cdots\!38\)\( T^{5} + \)\(46\!\cdots\!61\)\( T^{6} \)
$43$ \( 1 + \)\(24\!\cdots\!76\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} + \)\(15\!\cdots\!24\)\( T^{5} + \)\(51\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(19\!\cdots\!28\)\( T + \)\(37\!\cdots\!17\)\( T^{2} - \)\(43\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!59\)\( T^{4} - \)\(43\!\cdots\!12\)\( T^{5} + \)\(34\!\cdots\!83\)\( T^{6} \)
$53$ \( 1 - \)\(80\!\cdots\!66\)\( T + \)\(17\!\cdots\!19\)\( T^{2} - \)\(16\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!87\)\( T^{4} - \)\(50\!\cdots\!14\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \)
$59$ \( 1 + \)\(24\!\cdots\!68\)\( T + \)\(92\!\cdots\!73\)\( T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!67\)\( T^{4} + \)\(18\!\cdots\!88\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} \)
$61$ \( 1 - \)\(37\!\cdots\!18\)\( T + \)\(22\!\cdots\!99\)\( T^{2} - \)\(75\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!19\)\( T^{4} - \)\(25\!\cdots\!98\)\( T^{5} + \)\(55\!\cdots\!41\)\( T^{6} \)
$67$ \( 1 - \)\(10\!\cdots\!96\)\( T + \)\(24\!\cdots\!33\)\( T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(45\!\cdots\!71\)\( T^{4} - \)\(33\!\cdots\!24\)\( T^{5} + \)\(60\!\cdots\!03\)\( T^{6} \)
$71$ \( 1 + \)\(13\!\cdots\!84\)\( T + \)\(94\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} + \)\(20\!\cdots\!64\)\( T^{5} + \)\(18\!\cdots\!31\)\( T^{6} \)
$73$ \( 1 + \)\(21\!\cdots\!94\)\( T + \)\(76\!\cdots\!79\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(23\!\cdots\!07\)\( T^{4} + \)\(20\!\cdots\!66\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$79$ \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(70\!\cdots\!17\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!63\)\( T^{4} - \)\(38\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!19\)\( T^{6} \)
$83$ \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(97\!\cdots\!57\)\( T^{2} - \)\(13\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!91\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(97\!\cdots\!47\)\( T^{6} \)
$89$ \( 1 + \)\(30\!\cdots\!22\)\( T + \)\(93\!\cdots\!23\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(20\!\cdots\!87\)\( T^{4} + \)\(13\!\cdots\!42\)\( T^{5} + \)\(97\!\cdots\!09\)\( T^{6} \)
$97$ \( 1 + \)\(87\!\cdots\!94\)\( T + \)\(67\!\cdots\!43\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(24\!\cdots\!11\)\( T^{4} + \)\(11\!\cdots\!26\)\( T^{5} + \)\(49\!\cdots\!33\)\( T^{6} \)
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