Properties

Label 2.88.a.a
Level 2
Weight 88
Character orbit 2.a
Self dual Yes
Analytic conductor 95.867
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 88 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(95.8667262922\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{9}\cdot 5^{3}\cdot 7\cdot 11\cdot 29 \)
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.

\(f(q)\) \(=\) \( q +8796093022208 q^{2} +(-\)\(10\!\cdots\!72\)\( - \beta_{1}) q^{3} +\)\(77\!\cdots\!64\)\( q^{4} +(\)\(37\!\cdots\!50\)\( - 1603680785 \beta_{1} + 2245 \beta_{2}) q^{5} +(-\)\(94\!\cdots\!76\)\( - 8796093022208 \beta_{1}) q^{6} +(-\)\(96\!\cdots\!96\)\( + 3965744603380730 \beta_{1} - 1882438036 \beta_{2}) q^{7} +\)\(68\!\cdots\!12\)\( q^{8} +(\)\(19\!\cdots\!97\)\( + \)\(37\!\cdots\!82\)\( \beta_{1} + 333196522499934 \beta_{2}) q^{9} +O(q^{10})\) \( q +8796093022208 q^{2} +(-\)\(10\!\cdots\!72\)\( - \beta_{1}) q^{3} +\)\(77\!\cdots\!64\)\( q^{4} +(\)\(37\!\cdots\!50\)\( - 1603680785 \beta_{1} + 2245 \beta_{2}) q^{5} +(-\)\(94\!\cdots\!76\)\( - 8796093022208 \beta_{1}) q^{6} +(-\)\(96\!\cdots\!96\)\( + 3965744603380730 \beta_{1} - 1882438036 \beta_{2}) q^{7} +\)\(68\!\cdots\!12\)\( q^{8} +(\)\(19\!\cdots\!97\)\( + \)\(37\!\cdots\!82\)\( \beta_{1} + 333196522499934 \beta_{2}) q^{9} +(\)\(32\!\cdots\!00\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} + 19747228834856960 \beta_{2}) q^{10} +(-\)\(64\!\cdots\!48\)\( - \)\(78\!\cdots\!51\)\( \beta_{1} - 4274259712418070056 \beta_{2}) q^{11} +(-\)\(83\!\cdots\!08\)\( - \)\(77\!\cdots\!64\)\( \beta_{1}) q^{12} +(-\)\(30\!\cdots\!02\)\( - \)\(21\!\cdots\!09\)\( \beta_{1} + \)\(21\!\cdots\!09\)\( \beta_{2}) q^{13} +(-\)\(84\!\cdots\!68\)\( + \)\(34\!\cdots\!40\)\( \beta_{1} - \)\(16\!\cdots\!88\)\( \beta_{2}) q^{14} +(\)\(49\!\cdots\!00\)\( - \)\(88\!\cdots\!30\)\( \beta_{1} + \)\(65\!\cdots\!60\)\( \beta_{2}) q^{15} +\)\(59\!\cdots\!96\)\( q^{16} +(-\)\(69\!\cdots\!26\)\( + \)\(59\!\cdots\!62\)\( \beta_{1} - \)\(20\!\cdots\!58\)\( \beta_{2}) q^{17} +(\)\(17\!\cdots\!76\)\( + \)\(32\!\cdots\!56\)\( \beta_{1} + \)\(29\!\cdots\!72\)\( \beta_{2}) q^{18} +(-\)\(30\!\cdots\!00\)\( + \)\(11\!\cdots\!27\)\( \beta_{1} - \)\(69\!\cdots\!96\)\( \beta_{2}) q^{19} +(\)\(28\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(17\!\cdots\!80\)\( \beta_{2}) q^{20} +(-\)\(12\!\cdots\!88\)\( + \)\(69\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2}) q^{21} +(-\)\(56\!\cdots\!84\)\( - \)\(68\!\cdots\!08\)\( \beta_{1} - \)\(37\!\cdots\!48\)\( \beta_{2}) q^{22} +(\)\(62\!\cdots\!28\)\( + \)\(25\!\cdots\!66\)\( \beta_{1} + \)\(17\!\cdots\!56\)\( \beta_{2}) q^{23} +(-\)\(73\!\cdots\!64\)\( - \)\(68\!\cdots\!12\)\( \beta_{1}) q^{24} +(-\)\(33\!\cdots\!25\)\( - \)\(48\!\cdots\!00\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2}) q^{25} +(-\)\(26\!\cdots\!16\)\( - \)\(18\!\cdots\!72\)\( \beta_{1} + \)\(19\!\cdots\!72\)\( \beta_{2}) q^{26} +(-\)\(91\!\cdots\!20\)\( + \)\(63\!\cdots\!14\)\( \beta_{1} - \)\(10\!\cdots\!44\)\( \beta_{2}) q^{27} +(-\)\(74\!\cdots\!44\)\( + \)\(30\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2}) q^{28} +(\)\(11\!\cdots\!10\)\( + \)\(39\!\cdots\!11\)\( \beta_{1} - \)\(11\!\cdots\!71\)\( \beta_{2}) q^{29} +(\)\(43\!\cdots\!00\)\( - \)\(78\!\cdots\!40\)\( \beta_{1} + \)\(57\!\cdots\!80\)\( \beta_{2}) q^{30} +(\)\(27\!\cdots\!72\)\( + \)\(44\!\cdots\!20\)\( \beta_{1} + \)\(22\!\cdots\!24\)\( \beta_{2}) q^{31} +\)\(52\!\cdots\!68\)\( q^{32} +(\)\(32\!\cdots\!56\)\( + \)\(26\!\cdots\!62\)\( \beta_{1} + \)\(34\!\cdots\!38\)\( \beta_{2}) q^{33} +(-\)\(61\!\cdots\!08\)\( + \)\(52\!\cdots\!96\)\( \beta_{1} - \)\(18\!\cdots\!64\)\( \beta_{2}) q^{34} +(-\)\(42\!\cdots\!00\)\( + \)\(78\!\cdots\!60\)\( \beta_{1} - \)\(51\!\cdots\!20\)\( \beta_{2}) q^{35} +(\)\(15\!\cdots\!08\)\( + \)\(29\!\cdots\!48\)\( \beta_{1} + \)\(25\!\cdots\!76\)\( \beta_{2}) q^{36} +(\)\(77\!\cdots\!54\)\( + \)\(21\!\cdots\!27\)\( \beta_{1} + \)\(14\!\cdots\!77\)\( \beta_{2}) q^{37} +(-\)\(26\!\cdots\!00\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} - \)\(61\!\cdots\!68\)\( \beta_{2}) q^{38} +(\)\(10\!\cdots\!44\)\( - \)\(55\!\cdots\!14\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2}) q^{39} +(\)\(25\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2}) q^{40} +(-\)\(39\!\cdots\!78\)\( - \)\(29\!\cdots\!96\)\( \beta_{1} - \)\(39\!\cdots\!52\)\( \beta_{2}) q^{41} +(-\)\(10\!\cdots\!04\)\( + \)\(61\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!68\)\( \beta_{2}) q^{42} +(-\)\(32\!\cdots\!32\)\( - \)\(58\!\cdots\!71\)\( \beta_{1} + \)\(16\!\cdots\!76\)\( \beta_{2}) q^{43} +(-\)\(49\!\cdots\!72\)\( - \)\(60\!\cdots\!64\)\( \beta_{1} - \)\(33\!\cdots\!84\)\( \beta_{2}) q^{44} +(\)\(12\!\cdots\!50\)\( - \)\(99\!\cdots\!45\)\( \beta_{1} - \)\(39\!\cdots\!35\)\( \beta_{2}) q^{45} +(\)\(54\!\cdots\!24\)\( + \)\(22\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!48\)\( \beta_{2}) q^{46} +(-\)\(16\!\cdots\!36\)\( + \)\(97\!\cdots\!88\)\( \beta_{1} - \)\(96\!\cdots\!32\)\( \beta_{2}) q^{47} +(-\)\(64\!\cdots\!12\)\( - \)\(59\!\cdots\!96\)\( \beta_{1}) q^{48} +(-\)\(25\!\cdots\!27\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} + \)\(97\!\cdots\!28\)\( \beta_{2}) q^{49} +(-\)\(29\!\cdots\!00\)\( - \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2}) q^{50} +(-\)\(19\!\cdots\!28\)\( - \)\(37\!\cdots\!22\)\( \beta_{1} - \)\(21\!\cdots\!36\)\( \beta_{2}) q^{51} +(-\)\(23\!\cdots\!28\)\( - \)\(16\!\cdots\!76\)\( \beta_{1} + \)\(16\!\cdots\!76\)\( \beta_{2}) q^{52} +(-\)\(22\!\cdots\!02\)\( - \)\(17\!\cdots\!21\)\( \beta_{1} + \)\(76\!\cdots\!13\)\( \beta_{2}) q^{53} +(-\)\(80\!\cdots\!60\)\( + \)\(55\!\cdots\!12\)\( \beta_{1} - \)\(94\!\cdots\!52\)\( \beta_{2}) q^{54} +(-\)\(38\!\cdots\!00\)\( + \)\(50\!\cdots\!30\)\( \beta_{1} + \)\(60\!\cdots\!40\)\( \beta_{2}) q^{55} +(-\)\(65\!\cdots\!52\)\( + \)\(26\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!32\)\( \beta_{2}) q^{56} +(-\)\(34\!\cdots\!00\)\( + \)\(29\!\cdots\!94\)\( \beta_{1} - \)\(74\!\cdots\!54\)\( \beta_{2}) q^{57} +(\)\(99\!\cdots\!80\)\( + \)\(34\!\cdots\!88\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2}) q^{58} +(\)\(48\!\cdots\!20\)\( - \)\(16\!\cdots\!03\)\( \beta_{1} + \)\(88\!\cdots\!28\)\( \beta_{2}) q^{59} +(\)\(38\!\cdots\!00\)\( - \)\(68\!\cdots\!20\)\( \beta_{1} + \)\(50\!\cdots\!40\)\( \beta_{2}) q^{60} +(\)\(18\!\cdots\!42\)\( - \)\(43\!\cdots\!25\)\( \beta_{1} - \)\(42\!\cdots\!27\)\( \beta_{2}) q^{61} +(\)\(24\!\cdots\!76\)\( + \)\(38\!\cdots\!60\)\( \beta_{1} + \)\(19\!\cdots\!92\)\( \beta_{2}) q^{62} +(\)\(21\!\cdots\!88\)\( + \)\(33\!\cdots\!42\)\( \beta_{1} + \)\(30\!\cdots\!16\)\( \beta_{2}) q^{63} +\)\(46\!\cdots\!44\)\( q^{64} +(\)\(20\!\cdots\!00\)\( - \)\(21\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!40\)\( \beta_{2}) q^{65} +(\)\(28\!\cdots\!48\)\( + \)\(23\!\cdots\!96\)\( \beta_{1} + \)\(30\!\cdots\!04\)\( \beta_{2}) q^{66} +(\)\(33\!\cdots\!64\)\( - \)\(10\!\cdots\!33\)\( \beta_{1} + \)\(18\!\cdots\!68\)\( \beta_{2}) q^{67} +(-\)\(54\!\cdots\!64\)\( + \)\(46\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!12\)\( \beta_{2}) q^{68} +(-\)\(85\!\cdots\!16\)\( - \)\(14\!\cdots\!92\)\( \beta_{1} - \)\(76\!\cdots\!48\)\( \beta_{2}) q^{69} +(-\)\(37\!\cdots\!00\)\( + \)\(69\!\cdots\!80\)\( \beta_{1} - \)\(45\!\cdots\!60\)\( \beta_{2}) q^{70} +(-\)\(20\!\cdots\!88\)\( - \)\(56\!\cdots\!78\)\( \beta_{1} + \)\(44\!\cdots\!84\)\( \beta_{2}) q^{71} +(\)\(13\!\cdots\!64\)\( + \)\(25\!\cdots\!84\)\( \beta_{1} + \)\(22\!\cdots\!08\)\( \beta_{2}) q^{72} +(-\)\(19\!\cdots\!02\)\( - \)\(10\!\cdots\!86\)\( \beta_{1} - \)\(18\!\cdots\!22\)\( \beta_{2}) q^{73} +(\)\(68\!\cdots\!32\)\( + \)\(19\!\cdots\!16\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2}) q^{74} +(\)\(19\!\cdots\!00\)\( + \)\(39\!\cdots\!25\)\( \beta_{1} + \)\(16\!\cdots\!00\)\( \beta_{2}) q^{75} +(-\)\(23\!\cdots\!00\)\( + \)\(88\!\cdots\!28\)\( \beta_{1} - \)\(54\!\cdots\!44\)\( \beta_{2}) q^{76} +(\)\(29\!\cdots\!08\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(33\!\cdots\!12\)\( \beta_{2}) q^{77} +(\)\(90\!\cdots\!52\)\( - \)\(48\!\cdots\!12\)\( \beta_{1} + \)\(16\!\cdots\!00\)\( \beta_{2}) q^{78} +(\)\(87\!\cdots\!40\)\( - \)\(57\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!28\)\( \beta_{2}) q^{79} +(\)\(22\!\cdots\!00\)\( - \)\(96\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!20\)\( \beta_{2}) q^{80} +(-\)\(17\!\cdots\!99\)\( - \)\(13\!\cdots\!58\)\( \beta_{1} - \)\(13\!\cdots\!38\)\( \beta_{2}) q^{81} +(-\)\(34\!\cdots\!24\)\( - \)\(26\!\cdots\!68\)\( \beta_{1} - \)\(34\!\cdots\!16\)\( \beta_{2}) q^{82} +(-\)\(28\!\cdots\!92\)\( + \)\(50\!\cdots\!91\)\( \beta_{1} + \)\(41\!\cdots\!08\)\( \beta_{2}) q^{83} +(-\)\(93\!\cdots\!32\)\( + \)\(53\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!44\)\( \beta_{2}) q^{84} +(-\)\(53\!\cdots\!00\)\( + \)\(98\!\cdots\!10\)\( \beta_{1} - \)\(62\!\cdots\!70\)\( \beta_{2}) q^{85} +(-\)\(28\!\cdots\!56\)\( - \)\(51\!\cdots\!68\)\( \beta_{1} + \)\(14\!\cdots\!08\)\( \beta_{2}) q^{86} +(-\)\(13\!\cdots\!20\)\( - \)\(68\!\cdots\!06\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2}) q^{87} +(-\)\(43\!\cdots\!76\)\( - \)\(53\!\cdots\!12\)\( \beta_{1} - \)\(29\!\cdots\!72\)\( \beta_{2}) q^{88} +(\)\(18\!\cdots\!10\)\( - \)\(75\!\cdots\!30\)\( \beta_{1} + \)\(65\!\cdots\!50\)\( \beta_{2}) q^{89} +(\)\(10\!\cdots\!00\)\( - \)\(87\!\cdots\!60\)\( \beta_{1} - \)\(34\!\cdots\!80\)\( \beta_{2}) q^{90} +(-\)\(16\!\cdots\!08\)\( + \)\(34\!\cdots\!52\)\( \beta_{1} - \)\(15\!\cdots\!28\)\( \beta_{2}) q^{91} +(\)\(48\!\cdots\!92\)\( + \)\(19\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!84\)\( \beta_{2}) q^{92} +(-\)\(17\!\cdots\!84\)\( - \)\(49\!\cdots\!88\)\( \beta_{1} - \)\(13\!\cdots\!96\)\( \beta_{2}) q^{93} +(-\)\(14\!\cdots\!88\)\( + \)\(86\!\cdots\!04\)\( \beta_{1} - \)\(85\!\cdots\!56\)\( \beta_{2}) q^{94} +(-\)\(72\!\cdots\!00\)\( + \)\(95\!\cdots\!50\)\( \beta_{1} + \)\(49\!\cdots\!00\)\( \beta_{2}) q^{95} +(-\)\(56\!\cdots\!96\)\( - \)\(52\!\cdots\!68\)\( \beta_{1}) q^{96} +(-\)\(29\!\cdots\!46\)\( - \)\(34\!\cdots\!54\)\( \beta_{1} - \)\(99\!\cdots\!82\)\( \beta_{2}) q^{97} +(-\)\(22\!\cdots\!16\)\( - \)\(10\!\cdots\!84\)\( \beta_{1} + \)\(85\!\cdots\!24\)\( \beta_{2}) q^{98} +(-\)\(70\!\cdots\!56\)\( - \)\(79\!\cdots\!31\)\( \beta_{1} + \)\(50\!\cdots\!72\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 26388279066624q^{2} - \)\(32\!\cdots\!16\)\(q^{3} + \)\(23\!\cdots\!92\)\(q^{4} + \)\(11\!\cdots\!50\)\(q^{5} - \)\(28\!\cdots\!28\)\(q^{6} - \)\(28\!\cdots\!88\)\(q^{7} + \)\(20\!\cdots\!36\)\(q^{8} + \)\(58\!\cdots\!91\)\(q^{9} + O(q^{10}) \) \( 3q + 26388279066624q^{2} - \)\(32\!\cdots\!16\)\(q^{3} + \)\(23\!\cdots\!92\)\(q^{4} + \)\(11\!\cdots\!50\)\(q^{5} - \)\(28\!\cdots\!28\)\(q^{6} - \)\(28\!\cdots\!88\)\(q^{7} + \)\(20\!\cdots\!36\)\(q^{8} + \)\(58\!\cdots\!91\)\(q^{9} + \)\(98\!\cdots\!00\)\(q^{10} - \)\(19\!\cdots\!44\)\(q^{11} - \)\(24\!\cdots\!24\)\(q^{12} - \)\(90\!\cdots\!06\)\(q^{13} - \)\(25\!\cdots\!04\)\(q^{14} + \)\(14\!\cdots\!00\)\(q^{15} + \)\(17\!\cdots\!88\)\(q^{16} - \)\(20\!\cdots\!78\)\(q^{17} + \)\(51\!\cdots\!28\)\(q^{18} - \)\(91\!\cdots\!00\)\(q^{19} + \)\(86\!\cdots\!00\)\(q^{20} - \)\(36\!\cdots\!64\)\(q^{21} - \)\(16\!\cdots\!52\)\(q^{22} + \)\(18\!\cdots\!84\)\(q^{23} - \)\(21\!\cdots\!92\)\(q^{24} - \)\(10\!\cdots\!75\)\(q^{25} - \)\(79\!\cdots\!48\)\(q^{26} - \)\(27\!\cdots\!60\)\(q^{27} - \)\(22\!\cdots\!32\)\(q^{28} + \)\(34\!\cdots\!30\)\(q^{29} + \)\(12\!\cdots\!00\)\(q^{30} + \)\(83\!\cdots\!16\)\(q^{31} + \)\(15\!\cdots\!04\)\(q^{32} + \)\(98\!\cdots\!68\)\(q^{33} - \)\(18\!\cdots\!24\)\(q^{34} - \)\(12\!\cdots\!00\)\(q^{35} + \)\(45\!\cdots\!24\)\(q^{36} + \)\(23\!\cdots\!62\)\(q^{37} - \)\(80\!\cdots\!00\)\(q^{38} + \)\(30\!\cdots\!32\)\(q^{39} + \)\(76\!\cdots\!00\)\(q^{40} - \)\(11\!\cdots\!34\)\(q^{41} - \)\(31\!\cdots\!12\)\(q^{42} - \)\(98\!\cdots\!96\)\(q^{43} - \)\(14\!\cdots\!16\)\(q^{44} + \)\(36\!\cdots\!50\)\(q^{45} + \)\(16\!\cdots\!72\)\(q^{46} - \)\(50\!\cdots\!08\)\(q^{47} - \)\(19\!\cdots\!36\)\(q^{48} - \)\(77\!\cdots\!81\)\(q^{49} - \)\(88\!\cdots\!00\)\(q^{50} - \)\(57\!\cdots\!84\)\(q^{51} - \)\(69\!\cdots\!84\)\(q^{52} - \)\(66\!\cdots\!06\)\(q^{53} - \)\(24\!\cdots\!80\)\(q^{54} - \)\(11\!\cdots\!00\)\(q^{55} - \)\(19\!\cdots\!56\)\(q^{56} - \)\(10\!\cdots\!00\)\(q^{57} + \)\(29\!\cdots\!40\)\(q^{58} + \)\(14\!\cdots\!60\)\(q^{59} + \)\(11\!\cdots\!00\)\(q^{60} + \)\(54\!\cdots\!26\)\(q^{61} + \)\(73\!\cdots\!28\)\(q^{62} + \)\(63\!\cdots\!64\)\(q^{63} + \)\(13\!\cdots\!32\)\(q^{64} + \)\(60\!\cdots\!00\)\(q^{65} + \)\(86\!\cdots\!44\)\(q^{66} + \)\(99\!\cdots\!92\)\(q^{67} - \)\(16\!\cdots\!92\)\(q^{68} - \)\(25\!\cdots\!48\)\(q^{69} - \)\(11\!\cdots\!00\)\(q^{70} - \)\(62\!\cdots\!64\)\(q^{71} + \)\(40\!\cdots\!92\)\(q^{72} - \)\(57\!\cdots\!06\)\(q^{73} + \)\(20\!\cdots\!96\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{75} - \)\(70\!\cdots\!00\)\(q^{76} + \)\(88\!\cdots\!24\)\(q^{77} + \)\(27\!\cdots\!56\)\(q^{78} + \)\(26\!\cdots\!20\)\(q^{79} + \)\(66\!\cdots\!00\)\(q^{80} - \)\(52\!\cdots\!97\)\(q^{81} - \)\(10\!\cdots\!72\)\(q^{82} - \)\(84\!\cdots\!76\)\(q^{83} - \)\(28\!\cdots\!96\)\(q^{84} - \)\(16\!\cdots\!00\)\(q^{85} - \)\(86\!\cdots\!68\)\(q^{86} - \)\(39\!\cdots\!60\)\(q^{87} - \)\(13\!\cdots\!28\)\(q^{88} + \)\(55\!\cdots\!30\)\(q^{89} + \)\(31\!\cdots\!00\)\(q^{90} - \)\(50\!\cdots\!24\)\(q^{91} + \)\(14\!\cdots\!76\)\(q^{92} - \)\(52\!\cdots\!52\)\(q^{93} - \)\(44\!\cdots\!64\)\(q^{94} - \)\(21\!\cdots\!00\)\(q^{95} - \)\(16\!\cdots\!88\)\(q^{96} - \)\(87\!\cdots\!38\)\(q^{97} - \)\(67\!\cdots\!48\)\(q^{98} - \)\(21\!\cdots\!68\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 11473904362186221800312196301729 x - 156905659743614387346100645850205598702591560\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 77760 \nu^{2} + 116872286813505034560 \nu - 594807202135733738128184256281631360 \)\()/ 926145375996031 \)
\(\beta_{2}\)\(=\)\((\)\(-68041581120 \nu^{2} + 172190444978619227343516480 \nu + 520468396281877114114836497326305166504320\)\()/ 926145375996031 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{2} + 2625061 \beta_{1}\)\()/ 889026969600 \)
\(\nu^{2}\)\(=\)\((\)\(-22211633904531 \beta_{2} + 32724876273075782123 \beta_{1} + 33499541624284524131379337313781478195200\)\()/ 4379443200 \)

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
3.39413e15
−3.38046e15
−1.36752e13
8.79609e12 −8.60735e20 7.73713e25 −5.73526e28 −7.57111e33 1.36940e36 6.80565e38 4.17607e41 −5.04479e41
1.2 8.79609e12 1.94339e18 7.73713e25 −1.48628e30 1.70942e31 3.06439e35 6.80565e38 −3.23254e41 −1.30735e43
1.3 8.79609e12 5.36528e20 7.73713e25 2.66105e30 4.71935e33 −4.57229e36 6.80565e38 −3.53951e40 2.34068e43
\(n\): 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 \( T_{3}^{3} + \)\(32\!\cdots\!16\)\( T_{3}^{2} - \)\(46\!\cdots\!48\)\( T_{3} + \)\(89\!\cdots\!48\)\( \) acting on \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\).