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

Related objects

Downloads

Learn more about

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: \( 2^{22}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11\cdot 13 \)
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.

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.66.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.66.a.b 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4294967296 T )^{3} \)
$3$ \( 1 - 2984558598211212 T + \)\(15\!\cdots\!77\)\( T^{2} - \)\(62\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} - \)\(31\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \)
$5$ \( 1 - \)\(39\!\cdots\!50\)\( T + \)\(39\!\cdots\!75\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(19\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - \)\(42\!\cdots\!64\)\( T + \)\(16\!\cdots\!53\)\( T^{2} - \)\(52\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!71\)\( T^{4} - \)\(31\!\cdots\!36\)\( T^{5} + \)\(62\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + \)\(19\!\cdots\!84\)\( T + \)\(91\!\cdots\!05\)\( T^{2} - \)\(97\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!55\)\( T^{4} + \)\(47\!\cdots\!84\)\( T^{5} + \)\(11\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 - \)\(59\!\cdots\!42\)\( T + \)\(42\!\cdots\!67\)\( T^{2} - \)\(35\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(38\!\cdots\!58\)\( T^{5} + \)\(16\!\cdots\!57\)\( T^{6} \)
$17$ \( 1 + \)\(45\!\cdots\!46\)\( T + \)\(13\!\cdots\!43\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!51\)\( T^{4} + \)\(41\!\cdots\!54\)\( T^{5} + \)\(86\!\cdots\!93\)\( T^{6} \)
$19$ \( 1 + \)\(76\!\cdots\!80\)\( T + \)\(31\!\cdots\!97\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!03\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{5} + \)\(22\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 + \)\(24\!\cdots\!68\)\( T + \)\(16\!\cdots\!37\)\( T^{2} + \)\(33\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!91\)\( T^{4} + \)\(26\!\cdots\!32\)\( T^{5} + \)\(34\!\cdots\!07\)\( T^{6} \)
$29$ \( 1 - \)\(53\!\cdots\!10\)\( T + \)\(22\!\cdots\!47\)\( T^{2} - \)\(59\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!03\)\( T^{4} - \)\(68\!\cdots\!10\)\( T^{5} + \)\(14\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 - \)\(60\!\cdots\!76\)\( T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(98\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!95\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + \)\(16\!\cdots\!06\)\( T + \)\(22\!\cdots\!83\)\( T^{2} + \)\(27\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(11\!\cdots\!94\)\( T^{5} + \)\(62\!\cdots\!93\)\( T^{6} \)
$41$ \( 1 - \)\(36\!\cdots\!86\)\( T + \)\(20\!\cdots\!35\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!35\)\( T^{4} - \)\(16\!\cdots\!86\)\( T^{5} + \)\(31\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 - \)\(19\!\cdots\!92\)\( T + \)\(47\!\cdots\!17\)\( T^{2} - \)\(57\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!31\)\( T^{4} - \)\(44\!\cdots\!08\)\( T^{5} + \)\(33\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + \)\(15\!\cdots\!56\)\( T + \)\(10\!\cdots\!33\)\( T^{2} + \)\(81\!\cdots\!92\)\( T^{3} + \)\(49\!\cdots\!31\)\( T^{4} + \)\(36\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!43\)\( T^{6} \)
$53$ \( 1 - \)\(38\!\cdots\!82\)\( T + \)\(83\!\cdots\!87\)\( T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!91\)\( T^{4} - \)\(55\!\cdots\!18\)\( T^{5} + \)\(17\!\cdots\!57\)\( T^{6} \)
$59$ \( 1 - \)\(99\!\cdots\!20\)\( T + \)\(64\!\cdots\!97\)\( T^{2} - \)\(27\!\cdots\!60\)\( T^{3} + \)\(82\!\cdots\!03\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 - \)\(30\!\cdots\!46\)\( T + \)\(56\!\cdots\!75\)\( T^{2} - \)\(67\!\cdots\!60\)\( T^{3} + \)\(62\!\cdots\!75\)\( T^{4} - \)\(38\!\cdots\!46\)\( T^{5} + \)\(13\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 - \)\(17\!\cdots\!64\)\( T + \)\(28\!\cdots\!53\)\( T^{2} + \)\(14\!\cdots\!32\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(41\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$71$ \( 1 + \)\(20\!\cdots\!04\)\( T + \)\(75\!\cdots\!25\)\( T^{2} + \)\(89\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!75\)\( T^{4} + \)\(95\!\cdots\!04\)\( T^{5} + \)\(98\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 + \)\(14\!\cdots\!98\)\( T + \)\(31\!\cdots\!47\)\( T^{2} + \)\(25\!\cdots\!24\)\( T^{3} + \)\(41\!\cdots\!71\)\( T^{4} + \)\(25\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!57\)\( T^{6} \)
$79$ \( 1 - \)\(61\!\cdots\!00\)\( T + \)\(52\!\cdots\!97\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(30\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 - \)\(20\!\cdots\!72\)\( T + \)\(84\!\cdots\!57\)\( T^{2} - \)\(58\!\cdots\!16\)\( T^{3} + \)\(46\!\cdots\!51\)\( T^{4} - \)\(62\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!07\)\( T^{6} \)
$89$ \( 1 + \)\(11\!\cdots\!30\)\( T + \)\(68\!\cdots\!47\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!03\)\( T^{4} + \)\(29\!\cdots\!30\)\( T^{5} + \)\(13\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 + \)\(74\!\cdots\!46\)\( T + \)\(59\!\cdots\!43\)\( T^{2} + \)\(21\!\cdots\!12\)\( T^{3} + \)\(81\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!54\)\( T^{5} + \)\(26\!\cdots\!93\)\( T^{6} \)
show more
show less