Properties

Label 2.66.a.b
Level 2
Weight 66
Character orbit 2.a
Self dual Yes
Analytic conductor 53.514
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 66 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(53.5144712945\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11\cdot 13 \)
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.

\(f(q)\) \(=\) \( q\) \(+4294967296 q^{2}\) \(+(994852866070404 - \beta_{1}) q^{3}\) \(+18446744073709551616 q^{4}\) \(+(\)\(13\!\cdots\!50\)\( - 6038434 \beta_{1} - 13 \beta_{2}) q^{5}\) \(+(\)\(42\!\cdots\!84\)\( - 4294967296 \beta_{1}) q^{6}\) \(+(\)\(14\!\cdots\!88\)\( - 603582908362 \beta_{1} + 653660 \beta_{2}) q^{7}\) \(+\)\(79\!\cdots\!36\)\( q^{8}\) \(+(\)\(28\!\cdots\!73\)\( - 3475318834685628 \beta_{1} + 1845485370 \beta_{2}) q^{9}\) \(+O(q^{10})\) \( q\) \(+4294967296 q^{2}\) \(+(994852866070404 - \beta_{1}) q^{3}\) \(+18446744073709551616 q^{4}\) \(+(\)\(13\!\cdots\!50\)\( - 6038434 \beta_{1} - 13 \beta_{2}) q^{5}\) \(+(\)\(42\!\cdots\!84\)\( - 4294967296 \beta_{1}) q^{6}\) \(+(\)\(14\!\cdots\!88\)\( - 603582908362 \beta_{1} + 653660 \beta_{2}) q^{7}\) \(+\)\(79\!\cdots\!36\)\( q^{8}\) \(+(\)\(28\!\cdots\!73\)\( - 3475318834685628 \beta_{1} + 1845485370 \beta_{2}) q^{9}\) \(+(\)\(56\!\cdots\!00\)\( - 25934876549054464 \beta_{1} - 55834574848 \beta_{2}) q^{10}\) \(+(-\)\(65\!\cdots\!28\)\( - 73952434140039763 \beta_{1} - 1635827959720 \beta_{2}) q^{11}\) \(+(\)\(18\!\cdots\!64\)\( - 18446744073709551616 \beta_{1}) q^{12}\) \(+(\)\(19\!\cdots\!14\)\( + \)\(23\!\cdots\!58\)\( \beta_{1} - 347185116853685 \beta_{2}) q^{13}\) \(+(\)\(61\!\cdots\!48\)\( - \)\(25\!\cdots\!52\)\( \beta_{1} + 2807448322703360 \beta_{2}) q^{14}\) \(+(\)\(74\!\cdots\!00\)\( + \)\(10\!\cdots\!34\)\( \beta_{1} + 17523724678877988 \beta_{2}) q^{15}\) \(+\)\(34\!\cdots\!56\)\( q^{16}\) \(+(-\)\(15\!\cdots\!82\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} - 2224277365667729270 \beta_{2}) q^{17}\) \(+(\)\(12\!\cdots\!08\)\( - \)\(14\!\cdots\!88\)\( \beta_{1} + 7926299309396459520 \beta_{2}) q^{18}\) \(+(-\)\(25\!\cdots\!60\)\( - \)\(51\!\cdots\!93\)\( \beta_{1} + 37348845711370692520 \beta_{2}) q^{19}\) \(+(\)\(24\!\cdots\!00\)\( - \)\(11\!\cdots\!44\)\( \beta_{1} - \)\(23\!\cdots\!08\)\( \beta_{2}) q^{20}\) \(+(\)\(87\!\cdots\!52\)\( - \)\(42\!\cdots\!76\)\( \beta_{1} + \)\(79\!\cdots\!80\)\( \beta_{2}) q^{21}\) \(+(-\)\(28\!\cdots\!88\)\( - \)\(31\!\cdots\!48\)\( \beta_{1} - \)\(70\!\cdots\!20\)\( \beta_{2}) q^{22}\) \(+(-\)\(82\!\cdots\!56\)\( + \)\(67\!\cdots\!74\)\( \beta_{1} + \)\(30\!\cdots\!60\)\( \beta_{2}) q^{23}\) \(+(\)\(78\!\cdots\!44\)\( - \)\(79\!\cdots\!36\)\( \beta_{1}) q^{24}\) \(+(\)\(64\!\cdots\!75\)\( + \)\(54\!\cdots\!00\)\( \beta_{1} - \)\(87\!\cdots\!00\)\( \beta_{2}) q^{25}\) \(+(\)\(85\!\cdots\!44\)\( + \)\(10\!\cdots\!68\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{26}\) \(+(\)\(34\!\cdots\!20\)\( - \)\(50\!\cdots\!02\)\( \beta_{1} + \)\(55\!\cdots\!40\)\( \beta_{2}) q^{27}\) \(+(\)\(26\!\cdots\!08\)\( - \)\(11\!\cdots\!92\)\( \beta_{1} + \)\(12\!\cdots\!60\)\( \beta_{2}) q^{28}\) \(+(\)\(17\!\cdots\!70\)\( - \)\(60\!\cdots\!86\)\( \beta_{1} - \)\(84\!\cdots\!05\)\( \beta_{2}) q^{29}\) \(+(\)\(32\!\cdots\!00\)\( + \)\(47\!\cdots\!64\)\( \beta_{1} + \)\(75\!\cdots\!48\)\( \beta_{2}) q^{30}\) \(+(\)\(20\!\cdots\!92\)\( + \)\(62\!\cdots\!04\)\( \beta_{1} + \)\(15\!\cdots\!80\)\( \beta_{2}) q^{31}\) \(+\)\(14\!\cdots\!76\)\( q^{32}\) \(+(\)\(83\!\cdots\!88\)\( + \)\(33\!\cdots\!16\)\( \beta_{1} + \)\(93\!\cdots\!30\)\( \beta_{2}) q^{33}\) \(+(-\)\(65\!\cdots\!72\)\( - \)\(72\!\cdots\!68\)\( \beta_{1} - \)\(95\!\cdots\!20\)\( \beta_{2}) q^{34}\) \(+(-\)\(71\!\cdots\!00\)\( - \)\(21\!\cdots\!52\)\( \beta_{1} + \)\(12\!\cdots\!36\)\( \beta_{2}) q^{35}\) \(+(\)\(52\!\cdots\!68\)\( - \)\(64\!\cdots\!48\)\( \beta_{1} + \)\(34\!\cdots\!20\)\( \beta_{2}) q^{36}\) \(+(-\)\(54\!\cdots\!02\)\( + \)\(23\!\cdots\!02\)\( \beta_{1} - \)\(99\!\cdots\!45\)\( \beta_{2}) q^{37}\) \(+(-\)\(10\!\cdots\!60\)\( - \)\(21\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2}) q^{38}\) \(+(-\)\(26\!\cdots\!44\)\( + \)\(11\!\cdots\!78\)\( \beta_{1} - \)\(26\!\cdots\!00\)\( \beta_{2}) q^{39}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(47\!\cdots\!24\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2}) q^{40}\) \(+(\)\(12\!\cdots\!62\)\( + \)\(14\!\cdots\!16\)\( \beta_{1} + \)\(40\!\cdots\!60\)\( \beta_{2}) q^{41}\) \(+(\)\(37\!\cdots\!92\)\( - \)\(18\!\cdots\!96\)\( \beta_{1} + \)\(34\!\cdots\!80\)\( \beta_{2}) q^{42}\) \(+(\)\(66\!\cdots\!64\)\( + \)\(12\!\cdots\!97\)\( \beta_{1} - \)\(18\!\cdots\!40\)\( \beta_{2}) q^{43}\) \(+(-\)\(12\!\cdots\!48\)\( - \)\(13\!\cdots\!08\)\( \beta_{1} - \)\(30\!\cdots\!20\)\( \beta_{2}) q^{44}\) \(+(-\)\(72\!\cdots\!50\)\( - \)\(21\!\cdots\!22\)\( \beta_{1} + \)\(10\!\cdots\!71\)\( \beta_{2}) q^{45}\) \(+(-\)\(35\!\cdots\!76\)\( + \)\(28\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!60\)\( \beta_{2}) q^{46}\) \(+(-\)\(51\!\cdots\!52\)\( + \)\(13\!\cdots\!08\)\( \beta_{1} - \)\(48\!\cdots\!80\)\( \beta_{2}) q^{47}\) \(+(\)\(33\!\cdots\!24\)\( - \)\(34\!\cdots\!56\)\( \beta_{1}) q^{48}\) \(+(\)\(38\!\cdots\!37\)\( - \)\(32\!\cdots\!92\)\( \beta_{1} + \)\(47\!\cdots\!40\)\( \beta_{2}) q^{49}\) \(+(\)\(27\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(37\!\cdots\!00\)\( \beta_{2}) q^{50}\) \(+(\)\(19\!\cdots\!72\)\( + \)\(19\!\cdots\!90\)\( \beta_{1} + \)\(42\!\cdots\!80\)\( \beta_{2}) q^{51}\) \(+(\)\(36\!\cdots\!24\)\( + \)\(43\!\cdots\!28\)\( \beta_{1} - \)\(64\!\cdots\!60\)\( \beta_{2}) q^{52}\) \(+(\)\(12\!\cdots\!94\)\( + \)\(60\!\cdots\!18\)\( \beta_{1} - \)\(92\!\cdots\!45\)\( \beta_{2}) q^{53}\) \(+(\)\(14\!\cdots\!20\)\( - \)\(21\!\cdots\!92\)\( \beta_{1} + \)\(23\!\cdots\!40\)\( \beta_{2}) q^{54}\) \(+(\)\(29\!\cdots\!00\)\( + \)\(58\!\cdots\!62\)\( \beta_{1} - \)\(28\!\cdots\!16\)\( \beta_{2}) q^{55}\) \(+(\)\(11\!\cdots\!68\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} + \)\(51\!\cdots\!60\)\( \beta_{2}) q^{56}\) \(+(\)\(59\!\cdots\!60\)\( - \)\(17\!\cdots\!72\)\( \beta_{1} + \)\(76\!\cdots\!90\)\( \beta_{2}) q^{57}\) \(+(\)\(76\!\cdots\!20\)\( - \)\(26\!\cdots\!56\)\( \beta_{1} - \)\(36\!\cdots\!80\)\( \beta_{2}) q^{58}\) \(+(\)\(33\!\cdots\!40\)\( + \)\(56\!\cdots\!73\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2}) q^{59}\) \(+(\)\(13\!\cdots\!00\)\( + \)\(20\!\cdots\!44\)\( \beta_{1} + \)\(32\!\cdots\!08\)\( \beta_{2}) q^{60}\) \(+(\)\(10\!\cdots\!82\)\( + \)\(21\!\cdots\!38\)\( \beta_{1} - \)\(33\!\cdots\!65\)\( \beta_{2}) q^{61}\) \(+(\)\(86\!\cdots\!32\)\( + \)\(26\!\cdots\!84\)\( \beta_{1} + \)\(67\!\cdots\!80\)\( \beta_{2}) q^{62}\) \(+(\)\(46\!\cdots\!24\)\( - \)\(14\!\cdots\!10\)\( \beta_{1} + \)\(80\!\cdots\!60\)\( \beta_{2}) q^{63}\) \(+\)\(62\!\cdots\!96\)\( q^{64}\) \(+(\)\(45\!\cdots\!00\)\( + \)\(67\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2}) q^{65}\) \(+(\)\(35\!\cdots\!48\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(40\!\cdots\!80\)\( \beta_{2}) q^{66}\) \(+(\)\(57\!\cdots\!88\)\( - \)\(95\!\cdots\!61\)\( \beta_{1} + \)\(75\!\cdots\!20\)\( \beta_{2}) q^{67}\) \(+(-\)\(27\!\cdots\!12\)\( - \)\(31\!\cdots\!28\)\( \beta_{1} - \)\(41\!\cdots\!20\)\( \beta_{2}) q^{68}\) \(+(-\)\(89\!\cdots\!24\)\( + \)\(18\!\cdots\!32\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2}) q^{69}\) \(+(-\)\(30\!\cdots\!00\)\( - \)\(90\!\cdots\!92\)\( \beta_{1} + \)\(54\!\cdots\!56\)\( \beta_{2}) q^{70}\) \(+(-\)\(68\!\cdots\!68\)\( - \)\(10\!\cdots\!82\)\( \beta_{1} + \)\(86\!\cdots\!80\)\( \beta_{2}) q^{71}\) \(+(\)\(22\!\cdots\!28\)\( - \)\(27\!\cdots\!08\)\( \beta_{1} + \)\(14\!\cdots\!20\)\( \beta_{2}) q^{72}\) \(+(-\)\(49\!\cdots\!66\)\( - \)\(64\!\cdots\!72\)\( \beta_{1} + \)\(17\!\cdots\!30\)\( \beta_{2}) q^{73}\) \(+(-\)\(23\!\cdots\!92\)\( + \)\(99\!\cdots\!92\)\( \beta_{1} - \)\(42\!\cdots\!20\)\( \beta_{2}) q^{74}\) \(+(-\)\(65\!\cdots\!00\)\( + \)\(14\!\cdots\!25\)\( \beta_{1} - \)\(96\!\cdots\!00\)\( \beta_{2}) q^{75}\) \(+(-\)\(47\!\cdots\!60\)\( - \)\(94\!\cdots\!88\)\( \beta_{1} + \)\(68\!\cdots\!20\)\( \beta_{2}) q^{76}\) \(+(-\)\(14\!\cdots\!64\)\( + \)\(31\!\cdots\!72\)\( \beta_{1} + \)\(13\!\cdots\!80\)\( \beta_{2}) q^{77}\) \(+(-\)\(11\!\cdots\!24\)\( + \)\(48\!\cdots\!88\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2}) q^{78}\) \(+(\)\(20\!\cdots\!00\)\( - \)\(70\!\cdots\!28\)\( \beta_{1} + \)\(47\!\cdots\!40\)\( \beta_{2}) q^{79}\) \(+(\)\(44\!\cdots\!00\)\( - \)\(20\!\cdots\!04\)\( \beta_{1} - \)\(44\!\cdots\!28\)\( \beta_{2}) q^{80}\) \(+(\)\(66\!\cdots\!41\)\( - \)\(22\!\cdots\!64\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2}) q^{81}\) \(+(\)\(52\!\cdots\!52\)\( + \)\(62\!\cdots\!36\)\( \beta_{1} + \)\(17\!\cdots\!60\)\( \beta_{2}) q^{82}\) \(+(\)\(68\!\cdots\!24\)\( + \)\(70\!\cdots\!15\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2}) q^{83}\) \(+(\)\(16\!\cdots\!32\)\( - \)\(79\!\cdots\!16\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2}) q^{84}\) \(+(\)\(52\!\cdots\!00\)\( + \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(27\!\cdots\!86\)\( \beta_{2}) q^{85}\) \(+(\)\(28\!\cdots\!44\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} - \)\(81\!\cdots\!40\)\( \beta_{2}) q^{86}\) \(+(\)\(91\!\cdots\!80\)\( - \)\(15\!\cdots\!34\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2}) q^{87}\) \(+(-\)\(51\!\cdots\!08\)\( - \)\(58\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2}) q^{88}\) \(+(-\)\(37\!\cdots\!10\)\( - \)\(56\!\cdots\!40\)\( \beta_{1} - \)\(39\!\cdots\!50\)\( \beta_{2}) q^{89}\) \(+(-\)\(31\!\cdots\!00\)\( - \)\(94\!\cdots\!12\)\( \beta_{1} + \)\(45\!\cdots\!16\)\( \beta_{2}) q^{90}\) \(+(-\)\(45\!\cdots\!68\)\( + \)\(81\!\cdots\!56\)\( \beta_{1} + \)\(38\!\cdots\!40\)\( \beta_{2}) q^{91}\) \(+(-\)\(15\!\cdots\!96\)\( + \)\(12\!\cdots\!84\)\( \beta_{1} + \)\(55\!\cdots\!60\)\( \beta_{2}) q^{92}\) \(+(-\)\(55\!\cdots\!32\)\( - \)\(79\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!60\)\( \beta_{2}) q^{93}\) \(+(-\)\(22\!\cdots\!92\)\( + \)\(56\!\cdots\!68\)\( \beta_{1} - \)\(20\!\cdots\!80\)\( \beta_{2}) q^{94}\) \(+(-\)\(29\!\cdots\!00\)\( - \)\(26\!\cdots\!50\)\( \beta_{1} + \)\(26\!\cdots\!00\)\( \beta_{2}) q^{95}\) \(+(\)\(14\!\cdots\!04\)\( - \)\(14\!\cdots\!76\)\( \beta_{1}) q^{96}\) \(+(-\)\(24\!\cdots\!82\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!30\)\( \beta_{2}) q^{97}\) \(+(\)\(16\!\cdots\!52\)\( - \)\(13\!\cdots\!32\)\( \beta_{1} + \)\(20\!\cdots\!40\)\( \beta_{2}) q^{98}\) \(+(-\)\(38\!\cdots\!44\)\( + \)\(61\!\cdots\!05\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 12884901888q^{2} \) \(\mathstrut +\mathstrut 2984558598211212q^{3} \) \(\mathstrut +\mathstrut 55340232221128654848q^{4} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!52\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!64\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!08\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!19\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 12884901888q^{2} \) \(\mathstrut +\mathstrut 2984558598211212q^{3} \) \(\mathstrut +\mathstrut 55340232221128654848q^{4} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!52\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!64\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!08\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!19\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!84\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!92\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!42\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!44\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!68\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!46\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!24\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!80\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!56\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!64\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!68\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!32\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!32\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!24\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!10\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!76\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!28\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!64\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!16\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!04\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!06\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!80\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!32\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!86\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!76\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!92\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!44\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!56\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!11\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!16\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!82\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!04\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!60\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!46\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!96\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!72\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!88\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!64\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!36\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!72\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!04\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!84\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!98\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!76\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!80\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!92\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!72\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!23\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!56\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!72\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!96\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!32\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!24\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!30\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!04\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!88\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!96\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!76\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!12\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!46\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!56\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!32\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(4862367805520722608042\) \(x\mathstrut +\mathstrut \) \(130125819203569060903952569933488\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 8640 \nu^{2} + 341742997270080 \nu - 28007238559913276554748160 \)\()/\)\(439529881\)
\(\beta_{2}\)\(=\)\((\)\( 253376640 \nu^{2} + 14192563459783927680 \nu - 821340278009406951018491727360 \)\()/62789983\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(205282\) \(\beta_{1}\mathstrut +\mathstrut \) \(22140518400\)\()/\)\(66421555200\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(5650512521\) \(\beta_{2}\mathstrut +\mathstrut \) \(1642657807845362\) \(\beta_{1}\mathstrut +\mathstrut \) \(30758669675914051567045988352000\)\()/\)\(9488793600\)

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
4.20131e10
−8.04922e10
3.84791e10
4.29497e9 −2.64736e15 1.84467e19 −6.66845e22 −1.13703e25 1.54537e27 7.92282e28 −3.29254e30 −2.86408e32
1.2 4.29497e9 −5.99699e13 1.84467e19 6.16250e22 −2.57569e23 −2.55894e27 7.92282e28 −1.02975e31 2.64677e32
1.3 4.29497e9 5.69189e15 1.84467e19 8.97760e21 2.44465e25 5.30637e27 7.92282e28 2.20965e31 3.85585e31
\(n\): 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
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{3} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!12\)\( T_{3}^{2} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!52\)\( T_{3} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!64\)\( \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\).