Properties

Label 3.36.a.b
Level $3$
Weight $36$
Character orbit 3.a
Self dual yes
Analytic conductor $23.279$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.2785391901\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 1847580440 x + 20051963761200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{5}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q + ( -29110 + \beta_{1} ) q^{2} -129140163 q^{3} + ( 10829584300 - 155888 \beta_{1} + 23 \beta_{2} ) q^{4} + ( 922892078470 - 7872460 \beta_{1} - 1434 \beta_{2} ) q^{5} + ( 3759270144930 - 129140163 \beta_{1} ) q^{6} + ( 162745949512688 - 1530829676 \beta_{1} + 705590 \beta_{2} ) q^{7} + ( -6227715559376984 + 8864941856 \beta_{1} - 2008590 \beta_{2} ) q^{8} + 16677181699666569 q^{9} +O(q^{10})\) \( q +(-29110 + \beta_{1}) q^{2} -129140163 q^{3} +(10829584300 - 155888 \beta_{1} + 23 \beta_{2}) q^{4} +(922892078470 - 7872460 \beta_{1} - 1434 \beta_{2}) q^{5} +(3759270144930 - 129140163 \beta_{1}) q^{6} +(162745949512688 - 1530829676 \beta_{1} + 705590 \beta_{2}) q^{7} +(-6227715559376984 + 8864941856 \beta_{1} - 2008590 \beta_{2}) q^{8} +16677181699666569 q^{9} +(-375926542309622340 + 1133373620870 \beta_{1} - 279378752 \beta_{2}) q^{10} +(1142118725041408268 - 172276958744 \beta_{1} + 2103404556 \beta_{2}) q^{11} +(-1398534281724240900 + 20131401729744 \beta_{1} - 2970223749 \beta_{2}) q^{12} +(16695906143752846502 + 104639888932984 \beta_{1} - 12217424188 \beta_{2}) q^{13} +(-72626787653481350048 + 744341535666416 \beta_{1} + 13164756672 \beta_{2}) q^{14} +(-\)\(11\!\cdots\!10\)\( + 1016650767610980 \beta_{1} + 185186993742 \beta_{2}) q^{15} +(\)\(20\!\cdots\!48\)\( - 3098470453435968 \beta_{1} - 724085232996 \beta_{2}) q^{16} +(\)\(91\!\cdots\!14\)\( - 7154824633530840 \beta_{1} + 1416590862828 \beta_{2}) q^{17} +(-\)\(48\!\cdots\!90\)\( + 16677181699666569 \beta_{1}) q^{18} +(\)\(95\!\cdots\!64\)\( - 59286509654693496 \beta_{1} - 15574423845828 \beta_{2}) q^{19} +(\)\(29\!\cdots\!20\)\( - 402556500667987360 \beta_{1} + 56185809620106 \beta_{2}) q^{20} +(-\)\(21\!\cdots\!44\)\( + 197691593883877188 \beta_{1} - 91120007611170 \beta_{2}) q^{21} +(-\)\(40\!\cdots\!32\)\( + 2319179345885041420 \beta_{1} + 140242839499136 \beta_{2}) q^{22} +(\)\(60\!\cdots\!36\)\( + 786887228305973288 \beta_{1} - 307740421145364 \beta_{2}) q^{23} +(\)\(80\!\cdots\!92\)\( - 1144820036269362528 \beta_{1} + 259389640000170 \beta_{2}) q^{24} +(\)\(28\!\cdots\!75\)\( - 3388961875629683600 \beta_{1} - 145625899280440 \beta_{2}) q^{25} +(\)\(41\!\cdots\!72\)\( - 3280112791214552410 \beta_{1} + 1569115277977728 \beta_{2}) q^{26} -\)\(21\!\cdots\!47\)\( q^{27} +(\)\(29\!\cdots\!64\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} - 6221483086830576 \beta_{2}) q^{28} +(\)\(26\!\cdots\!38\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} - 14297266947051054 \beta_{2}) q^{29} +(\)\(48\!\cdots\!20\)\( - \)\(14\!\cdots\!10\)\( \beta_{1} + 36079017572016576 \beta_{2}) q^{30} +(-\)\(53\!\cdots\!16\)\( + \)\(76\!\cdots\!92\)\( \beta_{1} + 58006914115659806 \beta_{2}) q^{31} +(\)\(70\!\cdots\!68\)\( - \)\(10\!\cdots\!76\)\( \beta_{1} - 51892028944185912 \beta_{2}) q^{32} +(-\)\(14\!\cdots\!84\)\( + 22247874533344435272 \beta_{1} - 271634007216782628 \beta_{2}) q^{33} +(-\)\(34\!\cdots\!40\)\( + \)\(25\!\cdots\!94\)\( \beta_{1} - 67442330197447296 \beta_{2}) q^{34} +(-\)\(38\!\cdots\!00\)\( - \)\(80\!\cdots\!00\)\( \beta_{1} + 922574061615363300 \beta_{2}) q^{35} +(\)\(18\!\cdots\!00\)\( - \)\(25\!\cdots\!72\)\( \beta_{1} + 383575179092331087 \beta_{2}) q^{36} +(-\)\(17\!\cdots\!62\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} - 1231959827831347488 \beta_{2}) q^{37} +(-\)\(29\!\cdots\!88\)\( + \)\(85\!\cdots\!92\)\( \beta_{1} - 2431341072080226432 \beta_{2}) q^{38} +(-\)\(21\!\cdots\!26\)\( - \)\(13\!\cdots\!92\)\( \beta_{1} + 1577760151078462644 \beta_{2}) q^{39} +(-\)\(57\!\cdots\!20\)\( + \)\(72\!\cdots\!60\)\( \beta_{1} + 4192568044869874604 \beta_{2}) q^{40} +(-\)\(67\!\cdots\!86\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} - 632694542246452476 \beta_{2}) q^{41} +(\)\(93\!\cdots\!24\)\( - \)\(96\!\cdots\!08\)\( \beta_{1} - 1700098822477417536 \beta_{2}) q^{42} +(\)\(15\!\cdots\!24\)\( + \)\(15\!\cdots\!56\)\( \beta_{1} + 4060988346763694516 \beta_{2}) q^{43} +(\)\(64\!\cdots\!96\)\( - \)\(25\!\cdots\!80\)\( \beta_{1} - 9316536680481486060 \beta_{2}) q^{44} +(\)\(15\!\cdots\!30\)\( - \)\(13\!\cdots\!40\)\( \beta_{1} - 23915078557321859946 \beta_{2}) q^{45} +(\)\(17\!\cdots\!64\)\( + \)\(33\!\cdots\!92\)\( \beta_{1} - 2999661541846479488 \beta_{2}) q^{46} +(\)\(97\!\cdots\!72\)\( - \)\(69\!\cdots\!12\)\( \beta_{1} + 87782805605485124484 \beta_{2}) q^{47} +(-\)\(26\!\cdots\!24\)\( + \)\(40\!\cdots\!84\)\( \beta_{1} + 93508485014996418348 \beta_{2}) q^{48} +(\)\(27\!\cdots\!69\)\( - \)\(75\!\cdots\!44\)\( \beta_{1} - \)\(20\!\cdots\!52\)\( \beta_{2}) q^{49} +(-\)\(23\!\cdots\!50\)\( + \)\(32\!\cdots\!75\)\( \beta_{1} - 87929943542351128320 \beta_{2}) q^{50} +(-\)\(11\!\cdots\!82\)\( + \)\(92\!\cdots\!20\)\( \beta_{1} - \)\(18\!\cdots\!64\)\( \beta_{2}) q^{51} +(-\)\(84\!\cdots\!16\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} + \)\(45\!\cdots\!78\)\( \beta_{2}) q^{52} +(-\)\(36\!\cdots\!94\)\( - \)\(33\!\cdots\!12\)\( \beta_{1} + \)\(24\!\cdots\!26\)\( \beta_{2}) q^{53} +(\)\(62\!\cdots\!70\)\( - \)\(21\!\cdots\!47\)\( \beta_{1}) q^{54} +(-\)\(20\!\cdots\!80\)\( - \)\(25\!\cdots\!60\)\( \beta_{1} + \)\(39\!\cdots\!56\)\( \beta_{2}) q^{55} +(-\)\(31\!\cdots\!00\)\( + \)\(14\!\cdots\!20\)\( \beta_{1} - \)\(33\!\cdots\!28\)\( \beta_{2}) q^{56} +(-\)\(12\!\cdots\!32\)\( + \)\(76\!\cdots\!48\)\( \beta_{1} + \)\(20\!\cdots\!64\)\( \beta_{2}) q^{57} +(\)\(37\!\cdots\!24\)\( + \)\(54\!\cdots\!14\)\( \beta_{1} + \)\(13\!\cdots\!72\)\( \beta_{2}) q^{58} +(\)\(37\!\cdots\!16\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(55\!\cdots\!52\)\( \beta_{2}) q^{59} +(-\)\(38\!\cdots\!60\)\( + \)\(51\!\cdots\!80\)\( \beta_{1} - \)\(72\!\cdots\!78\)\( \beta_{2}) q^{60} +(\)\(42\!\cdots\!06\)\( - \)\(33\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!08\)\( \beta_{2}) q^{61} +(\)\(33\!\cdots\!56\)\( - \)\(65\!\cdots\!72\)\( \beta_{1} + \)\(21\!\cdots\!64\)\( \beta_{2}) q^{62} +(\)\(27\!\cdots\!72\)\( - \)\(25\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!10\)\( \beta_{2}) q^{63} +(-\)\(13\!\cdots\!92\)\( + \)\(16\!\cdots\!80\)\( \beta_{1} + \)\(18\!\cdots\!84\)\( \beta_{2}) q^{64} +(-\)\(25\!\cdots\!20\)\( + \)\(56\!\cdots\!60\)\( \beta_{1} - \)\(69\!\cdots\!56\)\( \beta_{2}) q^{65} +(\)\(52\!\cdots\!16\)\( - \)\(29\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!68\)\( \beta_{2}) q^{66} +(\)\(58\!\cdots\!04\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} + \)\(67\!\cdots\!00\)\( \beta_{2}) q^{67} +(\)\(93\!\cdots\!00\)\( - \)\(46\!\cdots\!72\)\( \beta_{1} + \)\(63\!\cdots\!90\)\( \beta_{2}) q^{68} +(-\)\(77\!\cdots\!68\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} + \)\(39\!\cdots\!32\)\( \beta_{2}) q^{69} +(-\)\(34\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2}) q^{70} +(-\)\(91\!\cdots\!08\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!32\)\( \beta_{2}) q^{71} +(-\)\(10\!\cdots\!96\)\( + \)\(14\!\cdots\!64\)\( \beta_{1} - \)\(33\!\cdots\!10\)\( \beta_{2}) q^{72} +(-\)\(19\!\cdots\!74\)\( + \)\(39\!\cdots\!08\)\( \beta_{1} - \)\(18\!\cdots\!80\)\( \beta_{2}) q^{73} +(\)\(44\!\cdots\!72\)\( - \)\(23\!\cdots\!34\)\( \beta_{1} - \)\(86\!\cdots\!12\)\( \beta_{2}) q^{74} +(-\)\(37\!\cdots\!25\)\( + \)\(43\!\cdots\!00\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2}) q^{75} +(\)\(13\!\cdots\!04\)\( - \)\(32\!\cdots\!76\)\( \beta_{1} + \)\(56\!\cdots\!64\)\( \beta_{2}) q^{76} +(\)\(17\!\cdots\!36\)\( - \)\(17\!\cdots\!52\)\( \beta_{1} + \)\(12\!\cdots\!72\)\( \beta_{2}) q^{77} +(-\)\(53\!\cdots\!36\)\( + \)\(42\!\cdots\!30\)\( \beta_{1} - \)\(20\!\cdots\!64\)\( \beta_{2}) q^{78} +(-\)\(88\!\cdots\!00\)\( + \)\(96\!\cdots\!20\)\( \beta_{1} - \)\(38\!\cdots\!78\)\( \beta_{2}) q^{79} +(\)\(23\!\cdots\!80\)\( + \)\(11\!\cdots\!60\)\( \beta_{1} + \)\(23\!\cdots\!04\)\( \beta_{2}) q^{80} +\)\(27\!\cdots\!61\)\( q^{81} +(\)\(30\!\cdots\!96\)\( - \)\(73\!\cdots\!02\)\( \beta_{1} + \)\(13\!\cdots\!28\)\( \beta_{2}) q^{82} +(\)\(21\!\cdots\!60\)\( + \)\(25\!\cdots\!32\)\( \beta_{1} - \)\(15\!\cdots\!60\)\( \beta_{2}) q^{83} +(-\)\(38\!\cdots\!32\)\( + \)\(13\!\cdots\!44\)\( \beta_{1} + \)\(80\!\cdots\!88\)\( \beta_{2}) q^{84} +(\)\(11\!\cdots\!80\)\( - \)\(24\!\cdots\!40\)\( \beta_{1} + \)\(23\!\cdots\!64\)\( \beta_{2}) q^{85} +(\)\(62\!\cdots\!08\)\( - \)\(13\!\cdots\!24\)\( \beta_{1} + \)\(37\!\cdots\!16\)\( \beta_{2}) q^{86} +(-\)\(33\!\cdots\!94\)\( - \)\(13\!\cdots\!24\)\( \beta_{1} + \)\(18\!\cdots\!02\)\( \beta_{2}) q^{87} +(-\)\(11\!\cdots\!24\)\( + \)\(11\!\cdots\!76\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2}) q^{88} +(\)\(38\!\cdots\!74\)\( - \)\(57\!\cdots\!36\)\( \beta_{1} - \)\(75\!\cdots\!84\)\( \beta_{2}) q^{89} +(-\)\(62\!\cdots\!60\)\( + \)\(18\!\cdots\!30\)\( \beta_{1} - \)\(46\!\cdots\!88\)\( \beta_{2}) q^{90} +(-\)\(13\!\cdots\!16\)\( + \)\(37\!\cdots\!12\)\( \beta_{1} + \)\(19\!\cdots\!68\)\( \beta_{2}) q^{91} +(-\)\(64\!\cdots\!52\)\( - \)\(53\!\cdots\!56\)\( \beta_{1} + \)\(18\!\cdots\!64\)\( \beta_{2}) q^{92} +(\)\(68\!\cdots\!08\)\( - \)\(98\!\cdots\!96\)\( \beta_{1} - \)\(74\!\cdots\!78\)\( \beta_{2}) q^{93} +(-\)\(33\!\cdots\!76\)\( + \)\(23\!\cdots\!88\)\( \beta_{1} - \)\(10\!\cdots\!04\)\( \beta_{2}) q^{94} +(\)\(53\!\cdots\!20\)\( - \)\(97\!\cdots\!60\)\( \beta_{1} - \)\(93\!\cdots\!04\)\( \beta_{2}) q^{95} +(-\)\(91\!\cdots\!84\)\( + \)\(13\!\cdots\!88\)\( \beta_{1} + \)\(67\!\cdots\!56\)\( \beta_{2}) q^{96} +(\)\(70\!\cdots\!14\)\( + \)\(44\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!92\)\( \beta_{2}) q^{97} +(-\)\(41\!\cdots\!62\)\( + \)\(26\!\cdots\!61\)\( \beta_{1} - \)\(31\!\cdots\!28\)\( \beta_{2}) q^{98} +(\)\(19\!\cdots\!92\)\( - \)\(28\!\cdots\!36\)\( \beta_{1} + \)\(35\!\cdots\!64\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 87330q^{2} - 387420489q^{3} + 32488752900q^{4} + 2768676235410q^{5} + 11277810434790q^{6} + 488237848538064q^{7} - 18683146678130952q^{8} + 50031545098999707q^{9} + O(q^{10}) \) \( 3q - 87330q^{2} - 387420489q^{3} + 32488752900q^{4} + 2768676235410q^{5} + 11277810434790q^{6} + 488237848538064q^{7} - 18683146678130952q^{8} + 50031545098999707q^{9} - 1127779626928867020q^{10} + 3426356175124224804q^{11} - 4195602845172722700q^{12} + 50087718431258539506q^{13} - \)\(21\!\cdots\!44\)\(q^{14} - \)\(35\!\cdots\!30\)\(q^{15} + \)\(60\!\cdots\!44\)\(q^{16} + \)\(27\!\cdots\!42\)\(q^{17} - \)\(14\!\cdots\!70\)\(q^{18} + \)\(28\!\cdots\!92\)\(q^{19} + \)\(88\!\cdots\!60\)\(q^{20} - \)\(63\!\cdots\!32\)\(q^{21} - \)\(12\!\cdots\!96\)\(q^{22} + \)\(18\!\cdots\!08\)\(q^{23} + \)\(24\!\cdots\!76\)\(q^{24} + \)\(85\!\cdots\!25\)\(q^{25} + \)\(12\!\cdots\!16\)\(q^{26} - \)\(64\!\cdots\!41\)\(q^{27} + \)\(88\!\cdots\!92\)\(q^{28} + \)\(78\!\cdots\!14\)\(q^{29} + \)\(14\!\cdots\!60\)\(q^{30} - \)\(15\!\cdots\!48\)\(q^{31} + \)\(21\!\cdots\!04\)\(q^{32} - \)\(44\!\cdots\!52\)\(q^{33} - \)\(10\!\cdots\!20\)\(q^{34} - \)\(11\!\cdots\!00\)\(q^{35} + \)\(54\!\cdots\!00\)\(q^{36} - \)\(51\!\cdots\!86\)\(q^{37} - \)\(87\!\cdots\!64\)\(q^{38} - \)\(64\!\cdots\!78\)\(q^{39} - \)\(17\!\cdots\!60\)\(q^{40} - \)\(20\!\cdots\!58\)\(q^{41} + \)\(28\!\cdots\!72\)\(q^{42} + \)\(46\!\cdots\!72\)\(q^{43} + \)\(19\!\cdots\!88\)\(q^{44} + \)\(46\!\cdots\!90\)\(q^{45} + \)\(52\!\cdots\!92\)\(q^{46} + \)\(29\!\cdots\!16\)\(q^{47} - \)\(78\!\cdots\!72\)\(q^{48} + \)\(83\!\cdots\!07\)\(q^{49} - \)\(70\!\cdots\!50\)\(q^{50} - \)\(35\!\cdots\!46\)\(q^{51} - \)\(25\!\cdots\!48\)\(q^{52} - \)\(10\!\cdots\!82\)\(q^{53} + \)\(18\!\cdots\!10\)\(q^{54} - \)\(62\!\cdots\!40\)\(q^{55} - \)\(93\!\cdots\!00\)\(q^{56} - \)\(37\!\cdots\!96\)\(q^{57} + \)\(11\!\cdots\!72\)\(q^{58} + \)\(11\!\cdots\!48\)\(q^{59} - \)\(11\!\cdots\!80\)\(q^{60} + \)\(12\!\cdots\!18\)\(q^{61} + \)\(10\!\cdots\!68\)\(q^{62} + \)\(81\!\cdots\!16\)\(q^{63} - \)\(41\!\cdots\!76\)\(q^{64} - \)\(77\!\cdots\!60\)\(q^{65} + \)\(15\!\cdots\!48\)\(q^{66} + \)\(17\!\cdots\!12\)\(q^{67} + \)\(28\!\cdots\!00\)\(q^{68} - \)\(23\!\cdots\!04\)\(q^{69} - \)\(10\!\cdots\!00\)\(q^{70} - \)\(27\!\cdots\!24\)\(q^{71} - \)\(31\!\cdots\!88\)\(q^{72} - \)\(58\!\cdots\!22\)\(q^{73} + \)\(13\!\cdots\!16\)\(q^{74} - \)\(11\!\cdots\!75\)\(q^{75} + \)\(40\!\cdots\!12\)\(q^{76} + \)\(53\!\cdots\!08\)\(q^{77} - \)\(16\!\cdots\!08\)\(q^{78} - \)\(26\!\cdots\!00\)\(q^{79} + \)\(71\!\cdots\!40\)\(q^{80} + \)\(83\!\cdots\!83\)\(q^{81} + \)\(91\!\cdots\!88\)\(q^{82} + \)\(65\!\cdots\!80\)\(q^{83} - \)\(11\!\cdots\!96\)\(q^{84} + \)\(35\!\cdots\!40\)\(q^{85} + \)\(18\!\cdots\!24\)\(q^{86} - \)\(10\!\cdots\!82\)\(q^{87} - \)\(34\!\cdots\!72\)\(q^{88} + \)\(11\!\cdots\!22\)\(q^{89} - \)\(18\!\cdots\!80\)\(q^{90} - \)\(40\!\cdots\!48\)\(q^{91} - \)\(19\!\cdots\!56\)\(q^{92} + \)\(20\!\cdots\!24\)\(q^{93} - \)\(10\!\cdots\!28\)\(q^{94} + \)\(15\!\cdots\!60\)\(q^{95} - \)\(27\!\cdots\!52\)\(q^{96} + \)\(21\!\cdots\!42\)\(q^{97} - \)\(12\!\cdots\!86\)\(q^{98} + \)\(57\!\cdots\!76\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 1847580440 x + 20051963761200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 2 \)
\(\beta_{2}\)\(=\)\((\)\( 36 \nu^{2} + 585984 \nu - 44342125900 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(23 \beta_{2} - 97664 \beta_{1} + 44341930572\)\()/36\)

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
−47629.1
11725.6
35904.5
−314887. −1.29140e8 6.47940e10 2.58564e12 4.06645e13 8.89060e14 −9.58334e15 1.66772e16 −8.14184e17
1.2 41241.5 −1.29140e8 −3.26589e10 2.39670e12 −5.32594e12 −9.42639e14 −2.76395e15 1.66772e16 9.88435e16
1.3 186315. −1.29140e8 3.53648e8 −2.21366e12 −2.40608e13 5.41817e14 −6.33585e15 1.66772e16 −4.12439e17
\(n\): 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 3.36.a.b 3
3.b odd 2 1 9.36.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.36.a.b 3 1.a even 1 1 trivial
9.36.a.c 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 87330 T_{2}^{2} - 63970719552 T_{2} + \)2419568332406784

'>\(24\!\cdots\!84\)\( \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 87330 T + 39108495552 T^{2} + 8420840235761664 T^{3} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} \)
$3$ \( ( 1 + 129140163 T )^{3} \)
$5$ \( 1 - 2768676235410 T + \)\(38\!\cdots\!75\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(24\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 488237848538064 T + \)\(26\!\cdots\!09\)\( T^{2} + \)\(84\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!87\)\( T^{4} - \)\(70\!\cdots\!36\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 - 3426356175124224804 T + \)\(53\!\cdots\!33\)\( T^{2} - \)\(70\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!83\)\( T^{4} - \)\(27\!\cdots\!04\)\( T^{5} + \)\(22\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 - 50087718431258539506 T + \)\(27\!\cdots\!11\)\( T^{2} - \)\(77\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!27\)\( T^{4} - \)\(47\!\cdots\!94\)\( T^{5} + \)\(92\!\cdots\!93\)\( T^{6} \)
$17$ \( 1 - \)\(27\!\cdots\!42\)\( T + \)\(30\!\cdots\!67\)\( T^{2} - \)\(59\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!31\)\( T^{4} - \)\(36\!\cdots\!58\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} \)
$19$ \( 1 - \)\(28\!\cdots\!92\)\( T + \)\(13\!\cdots\!93\)\( T^{2} - \)\(22\!\cdots\!96\)\( T^{3} + \)\(78\!\cdots\!07\)\( T^{4} - \)\(93\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 - \)\(18\!\cdots\!08\)\( T + \)\(22\!\cdots\!61\)\( T^{2} - \)\(17\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} - \)\(37\!\cdots\!92\)\( T^{5} + \)\(95\!\cdots\!43\)\( T^{6} \)
$29$ \( 1 - \)\(78\!\cdots\!14\)\( T + \)\(56\!\cdots\!11\)\( T^{2} - \)\(22\!\cdots\!52\)\( T^{3} + \)\(85\!\cdots\!39\)\( T^{4} - \)\(18\!\cdots\!14\)\( T^{5} + \)\(35\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + \)\(15\!\cdots\!48\)\( T + \)\(33\!\cdots\!53\)\( T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(52\!\cdots\!03\)\( T^{4} + \)\(39\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + \)\(51\!\cdots\!86\)\( T + \)\(29\!\cdots\!19\)\( T^{2} + \)\(78\!\cdots\!96\)\( T^{3} + \)\(22\!\cdots\!67\)\( T^{4} + \)\(30\!\cdots\!14\)\( T^{5} + \)\(45\!\cdots\!57\)\( T^{6} \)
$41$ \( 1 + \)\(20\!\cdots\!58\)\( T + \)\(97\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(27\!\cdots\!83\)\( T^{4} + \)\(15\!\cdots\!58\)\( T^{5} + \)\(21\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 - \)\(46\!\cdots\!72\)\( T + \)\(36\!\cdots\!57\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!99\)\( T^{4} - \)\(10\!\cdots\!28\)\( T^{5} + \)\(32\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(29\!\cdots\!16\)\( T + \)\(84\!\cdots\!53\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!79\)\( T^{4} - \)\(32\!\cdots\!84\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \)
$53$ \( 1 + \)\(10\!\cdots\!82\)\( T + \)\(58\!\cdots\!31\)\( T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!67\)\( T^{4} + \)\(54\!\cdots\!18\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} \)
$59$ \( 1 - \)\(11\!\cdots\!48\)\( T + \)\(26\!\cdots\!33\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(25\!\cdots\!67\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(86\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 - \)\(12\!\cdots\!18\)\( T + \)\(44\!\cdots\!99\)\( T^{2} - \)\(42\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!18\)\( T^{5} + \)\(28\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 - \)\(17\!\cdots\!12\)\( T + \)\(26\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!11\)\( T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$71$ \( 1 + \)\(27\!\cdots\!24\)\( T + \)\(62\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!95\)\( T^{4} + \)\(10\!\cdots\!24\)\( T^{5} + \)\(24\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 + \)\(58\!\cdots\!22\)\( T + \)\(54\!\cdots\!71\)\( T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(89\!\cdots\!47\)\( T^{4} + \)\(15\!\cdots\!78\)\( T^{5} + \)\(44\!\cdots\!93\)\( T^{6} \)
$79$ \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(99\!\cdots\!97\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!03\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 - \)\(65\!\cdots\!80\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!11\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(31\!\cdots\!43\)\( T^{6} \)
$89$ \( 1 - \)\(11\!\cdots\!22\)\( T + \)\(24\!\cdots\!03\)\( T^{2} - \)\(97\!\cdots\!16\)\( T^{3} + \)\(40\!\cdots\!47\)\( T^{4} - \)\(32\!\cdots\!22\)\( T^{5} + \)\(48\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - \)\(21\!\cdots\!42\)\( T + \)\(24\!\cdots\!67\)\( T^{2} - \)\(17\!\cdots\!56\)\( T^{3} + \)\(83\!\cdots\!31\)\( T^{4} - \)\(25\!\cdots\!58\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} \)
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