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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{5}\cdot 5 \)
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.

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

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

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

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:

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 \(T_{2}^{3} \) \(\mathstrut +\mathstrut 87330 T_{2}^{2} \) \(\mathstrut -\mathstrut 63970719552 T_{2} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!84\)\( \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\).