Properties

Label 9.90.a.b
Level 9
Weight 90
Character orbit 9.a
Self dual yes
Analytic conductor 451.462
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 90 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(451.461862736\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{43}\cdot 5^{9}\cdot 7^{5}\cdot 11^{2}\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(4486761478773 + \beta_{1}) q^{2} +(\)\(32\!\cdots\!44\)\( + 4221830337502 \beta_{1} + \beta_{2}) q^{4} +(-\)\(14\!\cdots\!83\)\( - 45551948860804901 \beta_{1} - 2871 \beta_{2} - 469 \beta_{3} - \beta_{4}) q^{5} +(\)\(55\!\cdots\!89\)\( + \)\(48\!\cdots\!89\)\( \beta_{1} + 2995501728 \beta_{2} + 23830291530 \beta_{3} + 86387 \beta_{4} + 2 \beta_{5} + \beta_{6}) q^{7} +(\)\(25\!\cdots\!64\)\( + \)\(38\!\cdots\!64\)\( \beta_{1} + 4899941400496 \beta_{2} + 25754724451968 \beta_{3} + 164226144 \beta_{4} - 5056 \beta_{5} + 672 \beta_{6}) q^{8} +O(q^{10})\) \( q +(4486761478773 + \beta_{1}) q^{2} +(\)\(32\!\cdots\!44\)\( + 4221830337502 \beta_{1} + \beta_{2}) q^{4} +(-\)\(14\!\cdots\!83\)\( - 45551948860804901 \beta_{1} - 2871 \beta_{2} - 469 \beta_{3} - \beta_{4}) q^{5} +(\)\(55\!\cdots\!89\)\( + \)\(48\!\cdots\!89\)\( \beta_{1} + 2995501728 \beta_{2} + 23830291530 \beta_{3} + 86387 \beta_{4} + 2 \beta_{5} + \beta_{6}) q^{7} +(\)\(25\!\cdots\!64\)\( + \)\(38\!\cdots\!64\)\( \beta_{1} + 4899941400496 \beta_{2} + 25754724451968 \beta_{3} + 164226144 \beta_{4} - 5056 \beta_{5} + 672 \beta_{6}) q^{8} +(-\)\(48\!\cdots\!14\)\( - \)\(34\!\cdots\!58\)\( \beta_{1} - 78659792691981368 \beta_{2} - 164940315218663252 \beta_{3} - 15974505275908 \beta_{4} + 14329100 \beta_{5} - 5062400 \beta_{6}) q^{10} +(\)\(47\!\cdots\!79\)\( - \)\(11\!\cdots\!40\)\( \beta_{1} + 2499715728858212480 \beta_{2} - 22518890942083928359 \beta_{3} + 1431847043601986 \beta_{4} + 11828400364 \beta_{5} - 196384650 \beta_{6}) q^{11} +(-\)\(14\!\cdots\!93\)\( + \)\(31\!\cdots\!85\)\( \beta_{1} - \)\(37\!\cdots\!41\)\( \beta_{2} + \)\(23\!\cdots\!21\)\( \beta_{3} + 2257472477606418721 \beta_{4} - 12427137972464 \beta_{5} + 80246616968 \beta_{6}) q^{13} +(\)\(47\!\cdots\!80\)\( + \)\(79\!\cdots\!24\)\( \beta_{1} + \)\(15\!\cdots\!32\)\( \beta_{2} - \)\(48\!\cdots\!30\)\( \beta_{3} + 26133122027180691950 \beta_{4} - 59105083748810 \beta_{5} + 228630604800 \beta_{6}) q^{14} +(\)\(16\!\cdots\!48\)\( + \)\(37\!\cdots\!84\)\( \beta_{1} + \)\(55\!\cdots\!32\)\( \beta_{2} + \)\(75\!\cdots\!48\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4} - 140431807767999488 \beta_{5} + 1148090844454400 \beta_{6}) q^{16} +(-\)\(11\!\cdots\!00\)\( + \)\(53\!\cdots\!10\)\( \beta_{1} - \)\(20\!\cdots\!22\)\( \beta_{2} - \)\(89\!\cdots\!22\)\( \beta_{3} + \)\(17\!\cdots\!62\)\( \beta_{4} - 2556139588806721848 \beta_{5} + 26652581864159076 \beta_{6}) q^{17} +(\)\(80\!\cdots\!11\)\( - \)\(74\!\cdots\!92\)\( \beta_{1} - \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(13\!\cdots\!73\)\( \beta_{3} + \)\(29\!\cdots\!58\)\( \beta_{4} + \)\(24\!\cdots\!92\)\( \beta_{5} - 7001867487502589450 \beta_{6}) q^{19} +(-\)\(24\!\cdots\!16\)\( - \)\(76\!\cdots\!52\)\( \beta_{1} - \)\(77\!\cdots\!42\)\( \beta_{2} - \)\(51\!\cdots\!88\)\( \beta_{3} + \)\(23\!\cdots\!48\)\( \beta_{4} + \)\(94\!\cdots\!00\)\( \beta_{5} - 87621594645561984000 \beta_{6}) q^{20} +(-\)\(81\!\cdots\!32\)\( + \)\(59\!\cdots\!90\)\( \beta_{1} - \)\(46\!\cdots\!54\)\( \beta_{2} - \)\(11\!\cdots\!41\)\( \beta_{3} + \)\(16\!\cdots\!99\)\( \beta_{4} + \)\(16\!\cdots\!59\)\( \beta_{5} + \)\(90\!\cdots\!92\)\( \beta_{6}) q^{22} +(\)\(16\!\cdots\!29\)\( - \)\(95\!\cdots\!51\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!38\)\( \beta_{3} + \)\(13\!\cdots\!95\)\( \beta_{4} + \)\(15\!\cdots\!70\)\( \beta_{5} + \)\(75\!\cdots\!85\)\( \beta_{6}) q^{23} +(-\)\(30\!\cdots\!25\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} - \)\(95\!\cdots\!00\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!00\)\( \beta_{4} - \)\(36\!\cdots\!00\)\( \beta_{5} - \)\(36\!\cdots\!00\)\( \beta_{6}) q^{25} +(-\)\(36\!\cdots\!42\)\( - \)\(39\!\cdots\!86\)\( \beta_{1} + \)\(37\!\cdots\!72\)\( \beta_{2} + \)\(61\!\cdots\!76\)\( \beta_{3} + \)\(31\!\cdots\!56\)\( \beta_{4} + \)\(97\!\cdots\!24\)\( \beta_{5} - \)\(20\!\cdots\!00\)\( \beta_{6}) q^{26} +(\)\(60\!\cdots\!08\)\( + \)\(11\!\cdots\!16\)\( \beta_{1} + \)\(84\!\cdots\!76\)\( \beta_{2} + \)\(31\!\cdots\!16\)\( \beta_{3} + \)\(59\!\cdots\!64\)\( \beta_{4} - \)\(86\!\cdots\!36\)\( \beta_{5} + \)\(46\!\cdots\!32\)\( \beta_{6}) q^{28} +(\)\(20\!\cdots\!57\)\( - \)\(57\!\cdots\!77\)\( \beta_{1} + \)\(12\!\cdots\!09\)\( \beta_{2} - \)\(12\!\cdots\!01\)\( \beta_{3} + \)\(48\!\cdots\!19\)\( \beta_{4} - \)\(32\!\cdots\!24\)\( \beta_{5} + \)\(16\!\cdots\!00\)\( \beta_{6}) q^{29} +(\)\(97\!\cdots\!84\)\( - \)\(26\!\cdots\!60\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2} - \)\(26\!\cdots\!28\)\( \beta_{3} + \)\(43\!\cdots\!52\)\( \beta_{4} - \)\(67\!\cdots\!32\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{31} +(\)\(26\!\cdots\!80\)\( + \)\(32\!\cdots\!52\)\( \beta_{1} + \)\(74\!\cdots\!08\)\( \beta_{2} + \)\(20\!\cdots\!32\)\( \beta_{3} + \)\(38\!\cdots\!32\)\( \beta_{4} - \)\(49\!\cdots\!28\)\( \beta_{5} - \)\(85\!\cdots\!64\)\( \beta_{6}) q^{32} +(\)\(44\!\cdots\!46\)\( - \)\(15\!\cdots\!86\)\( \beta_{1} + \)\(69\!\cdots\!32\)\( \beta_{2} + \)\(15\!\cdots\!28\)\( \beta_{3} + \)\(23\!\cdots\!88\)\( \beta_{4} + \)\(54\!\cdots\!12\)\( \beta_{5} - \)\(17\!\cdots\!00\)\( \beta_{6}) q^{34} +(-\)\(47\!\cdots\!36\)\( - \)\(72\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!32\)\( \beta_{2} - \)\(15\!\cdots\!48\)\( \beta_{3} + \)\(17\!\cdots\!08\)\( \beta_{4} + \)\(83\!\cdots\!00\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6}) q^{35} +(-\)\(77\!\cdots\!01\)\( + \)\(14\!\cdots\!33\)\( \beta_{1} - \)\(33\!\cdots\!57\)\( \beta_{2} - \)\(33\!\cdots\!67\)\( \beta_{3} + \)\(11\!\cdots\!77\)\( \beta_{4} - \)\(10\!\cdots\!48\)\( \beta_{5} + \)\(29\!\cdots\!76\)\( \beta_{6}) q^{37} +(-\)\(64\!\cdots\!52\)\( - \)\(10\!\cdots\!58\)\( \beta_{1} - \)\(95\!\cdots\!18\)\( \beta_{2} - \)\(10\!\cdots\!21\)\( \beta_{3} + \)\(40\!\cdots\!63\)\( \beta_{4} - \)\(80\!\cdots\!57\)\( \beta_{5} + \)\(36\!\cdots\!84\)\( \beta_{6}) q^{38} +(-\)\(51\!\cdots\!60\)\( - \)\(56\!\cdots\!20\)\( \beta_{1} - \)\(13\!\cdots\!20\)\( \beta_{2} - \)\(20\!\cdots\!80\)\( \beta_{3} + \)\(98\!\cdots\!80\)\( \beta_{4} + \)\(58\!\cdots\!00\)\( \beta_{5} - \)\(90\!\cdots\!00\)\( \beta_{6}) q^{40} +(\)\(93\!\cdots\!46\)\( + \)\(93\!\cdots\!40\)\( \beta_{1} + \)\(58\!\cdots\!00\)\( \beta_{2} + \)\(10\!\cdots\!28\)\( \beta_{3} + \)\(28\!\cdots\!08\)\( \beta_{4} - \)\(84\!\cdots\!48\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6}) q^{41} +(\)\(45\!\cdots\!71\)\( + \)\(35\!\cdots\!74\)\( \beta_{1} - \)\(19\!\cdots\!60\)\( \beta_{2} + \)\(11\!\cdots\!79\)\( \beta_{3} - \)\(80\!\cdots\!00\)\( \beta_{4} + \)\(29\!\cdots\!80\)\( \beta_{5} - \)\(40\!\cdots\!60\)\( \beta_{6}) q^{43} +(\)\(22\!\cdots\!04\)\( - \)\(35\!\cdots\!44\)\( \beta_{1} + \)\(64\!\cdots\!88\)\( \beta_{2} - \)\(24\!\cdots\!88\)\( \beta_{3} - \)\(58\!\cdots\!08\)\( \beta_{4} + \)\(39\!\cdots\!28\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6}) q^{44} +(-\)\(80\!\cdots\!36\)\( - \)\(97\!\cdots\!80\)\( \beta_{1} + \)\(41\!\cdots\!80\)\( \beta_{2} - \)\(20\!\cdots\!98\)\( \beta_{3} + \)\(34\!\cdots\!22\)\( \beta_{4} + \)\(30\!\cdots\!18\)\( \beta_{5} - \)\(55\!\cdots\!00\)\( \beta_{6}) q^{46} +(\)\(84\!\cdots\!18\)\( + \)\(31\!\cdots\!30\)\( \beta_{1} - \)\(53\!\cdots\!20\)\( \beta_{2} - \)\(13\!\cdots\!52\)\( \beta_{3} - \)\(28\!\cdots\!50\)\( \beta_{4} + \)\(43\!\cdots\!60\)\( \beta_{5} - \)\(36\!\cdots\!70\)\( \beta_{6}) q^{47} +(-\)\(44\!\cdots\!15\)\( + \)\(48\!\cdots\!60\)\( \beta_{1} + \)\(96\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4} - \)\(41\!\cdots\!48\)\( \beta_{5} + \)\(54\!\cdots\!00\)\( \beta_{6}) q^{49} +(\)\(15\!\cdots\!75\)\( - \)\(92\!\cdots\!25\)\( \beta_{1} - \)\(42\!\cdots\!00\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!00\)\( \beta_{4} - \)\(99\!\cdots\!00\)\( \beta_{5} - \)\(50\!\cdots\!00\)\( \beta_{6}) q^{50} +(-\)\(27\!\cdots\!48\)\( + \)\(22\!\cdots\!48\)\( \beta_{1} - \)\(32\!\cdots\!30\)\( \beta_{2} + \)\(69\!\cdots\!68\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(26\!\cdots\!60\)\( \beta_{5} + \)\(35\!\cdots\!20\)\( \beta_{6}) q^{52} +(\)\(25\!\cdots\!13\)\( + \)\(37\!\cdots\!79\)\( \beta_{1} - \)\(37\!\cdots\!19\)\( \beta_{2} - \)\(43\!\cdots\!33\)\( \beta_{3} - \)\(15\!\cdots\!41\)\( \beta_{4} - \)\(24\!\cdots\!16\)\( \beta_{5} - \)\(23\!\cdots\!08\)\( \beta_{6}) q^{53} +(-\)\(17\!\cdots\!39\)\( - \)\(45\!\cdots\!83\)\( \beta_{1} + \)\(21\!\cdots\!32\)\( \beta_{2} - \)\(48\!\cdots\!02\)\( \beta_{3} - \)\(50\!\cdots\!33\)\( \beta_{4} + \)\(41\!\cdots\!50\)\( \beta_{5} + \)\(71\!\cdots\!25\)\( \beta_{6}) q^{55} +(\)\(83\!\cdots\!28\)\( + \)\(72\!\cdots\!32\)\( \beta_{1} + \)\(93\!\cdots\!96\)\( \beta_{2} + \)\(13\!\cdots\!16\)\( \beta_{3} + \)\(68\!\cdots\!56\)\( \beta_{4} - \)\(19\!\cdots\!96\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6}) q^{56} +(-\)\(44\!\cdots\!98\)\( + \)\(10\!\cdots\!14\)\( \beta_{1} + \)\(98\!\cdots\!48\)\( \beta_{2} + \)\(51\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!12\)\( \beta_{4} + \)\(27\!\cdots\!92\)\( \beta_{5} + \)\(90\!\cdots\!96\)\( \beta_{6}) q^{58} +(\)\(12\!\cdots\!81\)\( + \)\(40\!\cdots\!26\)\( \beta_{1} - \)\(18\!\cdots\!52\)\( \beta_{2} - \)\(46\!\cdots\!95\)\( \beta_{3} + \)\(32\!\cdots\!20\)\( \beta_{4} - \)\(36\!\cdots\!00\)\( \beta_{5} + \)\(11\!\cdots\!00\)\( \beta_{6}) q^{59} +(\)\(12\!\cdots\!67\)\( - \)\(46\!\cdots\!35\)\( \beta_{1} - \)\(80\!\cdots\!85\)\( \beta_{2} + \)\(17\!\cdots\!65\)\( \beta_{3} + \)\(88\!\cdots\!45\)\( \beta_{4} - \)\(60\!\cdots\!80\)\( \beta_{5} + \)\(78\!\cdots\!00\)\( \beta_{6}) q^{61} +(-\)\(24\!\cdots\!56\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(38\!\cdots\!88\)\( \beta_{2} + \)\(93\!\cdots\!48\)\( \beta_{3} + \)\(50\!\cdots\!48\)\( \beta_{4} + \)\(40\!\cdots\!08\)\( \beta_{5} + \)\(13\!\cdots\!04\)\( \beta_{6}) q^{62} +(\)\(20\!\cdots\!52\)\( + \)\(57\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} + \)\(43\!\cdots\!36\)\( \beta_{3} - \)\(40\!\cdots\!04\)\( \beta_{4} - \)\(10\!\cdots\!76\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6}) q^{64} +(-\)\(18\!\cdots\!28\)\( - \)\(22\!\cdots\!16\)\( \beta_{1} + \)\(50\!\cdots\!64\)\( \beta_{2} - \)\(25\!\cdots\!04\)\( \beta_{3} - \)\(22\!\cdots\!16\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6}) q^{65} +(\)\(83\!\cdots\!15\)\( + \)\(74\!\cdots\!60\)\( \beta_{1} + \)\(53\!\cdots\!52\)\( \beta_{2} + \)\(96\!\cdots\!97\)\( \beta_{3} - \)\(42\!\cdots\!82\)\( \beta_{4} - \)\(45\!\cdots\!52\)\( \beta_{5} + \)\(13\!\cdots\!74\)\( \beta_{6}) q^{67} +(-\)\(48\!\cdots\!08\)\( + \)\(62\!\cdots\!08\)\( \beta_{1} - \)\(13\!\cdots\!86\)\( \beta_{2} + \)\(40\!\cdots\!92\)\( \beta_{3} - \)\(47\!\cdots\!04\)\( \beta_{4} + \)\(38\!\cdots\!96\)\( \beta_{5} + \)\(49\!\cdots\!48\)\( \beta_{6}) q^{68} +(-\)\(69\!\cdots\!88\)\( - \)\(16\!\cdots\!36\)\( \beta_{1} - \)\(85\!\cdots\!56\)\( \beta_{2} - \)\(12\!\cdots\!84\)\( \beta_{3} + \)\(23\!\cdots\!64\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!00\)\( \beta_{6}) q^{70} +(\)\(78\!\cdots\!43\)\( - \)\(11\!\cdots\!45\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(23\!\cdots\!70\)\( \beta_{3} + \)\(10\!\cdots\!65\)\( \beta_{4} + \)\(34\!\cdots\!90\)\( \beta_{5} + \)\(11\!\cdots\!75\)\( \beta_{6}) q^{71} +(-\)\(28\!\cdots\!32\)\( + \)\(93\!\cdots\!22\)\( \beta_{1} + \)\(27\!\cdots\!78\)\( \beta_{2} + \)\(30\!\cdots\!10\)\( \beta_{3} + \)\(19\!\cdots\!02\)\( \beta_{4} + \)\(24\!\cdots\!72\)\( \beta_{5} - \)\(62\!\cdots\!64\)\( \beta_{6}) q^{73} +(\)\(13\!\cdots\!54\)\( - \)\(34\!\cdots\!30\)\( \beta_{1} + \)\(14\!\cdots\!20\)\( \beta_{2} + \)\(70\!\cdots\!28\)\( \beta_{3} + \)\(13\!\cdots\!28\)\( \beta_{4} + \)\(20\!\cdots\!92\)\( \beta_{5} - \)\(90\!\cdots\!00\)\( \beta_{6}) q^{74} +(-\)\(89\!\cdots\!56\)\( - \)\(86\!\cdots\!04\)\( \beta_{1} + \)\(17\!\cdots\!88\)\( \beta_{2} - \)\(43\!\cdots\!92\)\( \beta_{3} - \)\(14\!\cdots\!72\)\( \beta_{4} - \)\(60\!\cdots\!48\)\( \beta_{5} - \)\(25\!\cdots\!00\)\( \beta_{6}) q^{76} +(-\)\(19\!\cdots\!12\)\( + \)\(81\!\cdots\!00\)\( \beta_{1} - \)\(42\!\cdots\!52\)\( \beta_{2} - \)\(28\!\cdots\!64\)\( \beta_{3} - \)\(38\!\cdots\!08\)\( \beta_{4} + \)\(14\!\cdots\!32\)\( \beta_{5} - \)\(66\!\cdots\!84\)\( \beta_{6}) q^{77} +(\)\(38\!\cdots\!50\)\( + \)\(33\!\cdots\!42\)\( \beta_{1} - \)\(22\!\cdots\!64\)\( \beta_{2} + \)\(34\!\cdots\!28\)\( \beta_{3} - \)\(76\!\cdots\!02\)\( \beta_{4} - \)\(16\!\cdots\!68\)\( \beta_{5} + \)\(59\!\cdots\!50\)\( \beta_{6}) q^{79} +(-\)\(39\!\cdots\!88\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} - \)\(30\!\cdots\!56\)\( \beta_{2} - \)\(69\!\cdots\!84\)\( \beta_{3} - \)\(48\!\cdots\!36\)\( \beta_{4} + \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(55\!\cdots\!00\)\( \beta_{6}) q^{80} +(\)\(90\!\cdots\!26\)\( + \)\(15\!\cdots\!94\)\( \beta_{1} + \)\(14\!\cdots\!08\)\( \beta_{2} + \)\(22\!\cdots\!32\)\( \beta_{3} + \)\(50\!\cdots\!12\)\( \beta_{4} - \)\(75\!\cdots\!88\)\( \beta_{5} + \)\(11\!\cdots\!56\)\( \beta_{6}) q^{82} +(\)\(50\!\cdots\!39\)\( + \)\(27\!\cdots\!50\)\( \beta_{1} + \)\(98\!\cdots\!00\)\( \beta_{2} + \)\(33\!\cdots\!99\)\( \beta_{3} + \)\(40\!\cdots\!00\)\( \beta_{4} + \)\(92\!\cdots\!00\)\( \beta_{5} - \)\(17\!\cdots\!00\)\( \beta_{6}) q^{83} +(-\)\(20\!\cdots\!14\)\( - \)\(36\!\cdots\!58\)\( \beta_{1} + \)\(23\!\cdots\!82\)\( \beta_{2} - \)\(62\!\cdots\!02\)\( \beta_{3} + \)\(32\!\cdots\!42\)\( \beta_{4} - \)\(36\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!00\)\( \beta_{6}) q^{85} +(\)\(35\!\cdots\!64\)\( - \)\(84\!\cdots\!18\)\( \beta_{1} - \)\(29\!\cdots\!14\)\( \beta_{2} - \)\(31\!\cdots\!31\)\( \beta_{3} - \)\(21\!\cdots\!71\)\( \beta_{4} + \)\(82\!\cdots\!61\)\( \beta_{5} - \)\(86\!\cdots\!00\)\( \beta_{6}) q^{86} +(-\)\(26\!\cdots\!48\)\( + \)\(25\!\cdots\!52\)\( \beta_{1} - \)\(34\!\cdots\!32\)\( \beta_{2} + \)\(21\!\cdots\!44\)\( \beta_{3} - \)\(23\!\cdots\!28\)\( \beta_{4} + \)\(26\!\cdots\!12\)\( \beta_{5} - \)\(19\!\cdots\!44\)\( \beta_{6}) q^{88} +(\)\(23\!\cdots\!76\)\( - \)\(69\!\cdots\!26\)\( \beta_{1} + \)\(10\!\cdots\!62\)\( \beta_{2} + \)\(10\!\cdots\!02\)\( \beta_{3} - \)\(89\!\cdots\!58\)\( \beta_{4} - \)\(93\!\cdots\!92\)\( \beta_{5} - \)\(55\!\cdots\!00\)\( \beta_{6}) q^{89} +(-\)\(43\!\cdots\!56\)\( - \)\(35\!\cdots\!72\)\( \beta_{1} + \)\(55\!\cdots\!84\)\( \beta_{2} - \)\(72\!\cdots\!16\)\( \beta_{3} - \)\(53\!\cdots\!56\)\( \beta_{4} - \)\(67\!\cdots\!04\)\( \beta_{5} - \)\(19\!\cdots\!00\)\( \beta_{6}) q^{91} +(-\)\(10\!\cdots\!24\)\( + \)\(22\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!08\)\( \beta_{2} - \)\(26\!\cdots\!16\)\( \beta_{3} - \)\(51\!\cdots\!12\)\( \beta_{4} + \)\(21\!\cdots\!88\)\( \beta_{5} - \)\(23\!\cdots\!56\)\( \beta_{6}) q^{92} +(\)\(32\!\cdots\!20\)\( + \)\(43\!\cdots\!68\)\( \beta_{1} - \)\(13\!\cdots\!76\)\( \beta_{2} - \)\(64\!\cdots\!72\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4} + \)\(94\!\cdots\!32\)\( \beta_{5} - \)\(27\!\cdots\!00\)\( \beta_{6}) q^{94} +(-\)\(32\!\cdots\!55\)\( - \)\(66\!\cdots\!35\)\( \beta_{1} + \)\(47\!\cdots\!40\)\( \beta_{2} + \)\(20\!\cdots\!10\)\( \beta_{3} - \)\(43\!\cdots\!85\)\( \beta_{4} + \)\(18\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!25\)\( \beta_{6}) q^{95} +(\)\(10\!\cdots\!48\)\( + \)\(43\!\cdots\!38\)\( \beta_{1} + \)\(15\!\cdots\!94\)\( \beta_{2} + \)\(22\!\cdots\!78\)\( \beta_{3} - \)\(15\!\cdots\!34\)\( \beta_{4} + \)\(71\!\cdots\!16\)\( \beta_{5} + \)\(38\!\cdots\!08\)\( \beta_{6}) q^{97} +(\)\(25\!\cdots\!53\)\( + \)\(25\!\cdots\!53\)\( \beta_{1} + \)\(18\!\cdots\!68\)\( \beta_{2} + \)\(45\!\cdots\!12\)\( \beta_{3} + \)\(44\!\cdots\!52\)\( \beta_{4} - \)\(15\!\cdots\!48\)\( \beta_{5} + \)\(63\!\cdots\!76\)\( \beta_{6}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 31407330351408q^{2} + \)\(22\!\cdots\!04\)\(q^{4} - \)\(10\!\cdots\!50\)\(q^{5} + \)\(38\!\cdots\!92\)\(q^{7} + \)\(17\!\cdots\!20\)\(q^{8} + O(q^{10}) \) \( 7q + 31407330351408q^{2} + \)\(22\!\cdots\!04\)\(q^{4} - \)\(10\!\cdots\!50\)\(q^{5} + \)\(38\!\cdots\!92\)\(q^{7} + \)\(17\!\cdots\!20\)\(q^{8} - \)\(33\!\cdots\!00\)\(q^{10} + \)\(33\!\cdots\!56\)\(q^{11} - \)\(10\!\cdots\!34\)\(q^{13} + \)\(33\!\cdots\!32\)\(q^{14} + \)\(11\!\cdots\!32\)\(q^{16} - \)\(83\!\cdots\!42\)\(q^{17} + \)\(56\!\cdots\!80\)\(q^{19} - \)\(17\!\cdots\!00\)\(q^{20} - \)\(57\!\cdots\!36\)\(q^{22} + \)\(11\!\cdots\!04\)\(q^{23} - \)\(21\!\cdots\!75\)\(q^{25} - \)\(25\!\cdots\!24\)\(q^{26} + \)\(42\!\cdots\!84\)\(q^{28} + \)\(14\!\cdots\!30\)\(q^{29} + \)\(68\!\cdots\!04\)\(q^{31} + \)\(18\!\cdots\!48\)\(q^{32} + \)\(31\!\cdots\!28\)\(q^{34} - \)\(33\!\cdots\!00\)\(q^{35} - \)\(54\!\cdots\!58\)\(q^{37} - \)\(45\!\cdots\!20\)\(q^{38} - \)\(36\!\cdots\!00\)\(q^{40} + \)\(65\!\cdots\!66\)\(q^{41} + \)\(32\!\cdots\!56\)\(q^{43} + \)\(15\!\cdots\!32\)\(q^{44} - \)\(56\!\cdots\!76\)\(q^{46} + \)\(58\!\cdots\!08\)\(q^{47} - \)\(30\!\cdots\!01\)\(q^{49} + \)\(11\!\cdots\!00\)\(q^{50} - \)\(19\!\cdots\!68\)\(q^{52} + \)\(18\!\cdots\!14\)\(q^{53} - \)\(12\!\cdots\!00\)\(q^{55} + \)\(58\!\cdots\!20\)\(q^{56} - \)\(30\!\cdots\!20\)\(q^{58} + \)\(90\!\cdots\!60\)\(q^{59} + \)\(87\!\cdots\!94\)\(q^{61} - \)\(16\!\cdots\!24\)\(q^{62} + \)\(14\!\cdots\!44\)\(q^{64} - \)\(13\!\cdots\!00\)\(q^{65} + \)\(58\!\cdots\!92\)\(q^{67} - \)\(33\!\cdots\!84\)\(q^{68} - \)\(48\!\cdots\!00\)\(q^{70} + \)\(54\!\cdots\!76\)\(q^{71} - \)\(19\!\cdots\!54\)\(q^{73} + \)\(96\!\cdots\!52\)\(q^{74} - \)\(62\!\cdots\!40\)\(q^{76} - \)\(13\!\cdots\!64\)\(q^{77} + \)\(26\!\cdots\!20\)\(q^{79} - \)\(27\!\cdots\!00\)\(q^{80} + \)\(63\!\cdots\!04\)\(q^{82} + \)\(35\!\cdots\!24\)\(q^{83} - \)\(14\!\cdots\!00\)\(q^{85} + \)\(24\!\cdots\!76\)\(q^{86} - \)\(18\!\cdots\!40\)\(q^{88} + \)\(16\!\cdots\!90\)\(q^{89} - \)\(30\!\cdots\!36\)\(q^{91} - \)\(72\!\cdots\!92\)\(q^{92} + \)\(22\!\cdots\!08\)\(q^{94} - \)\(22\!\cdots\!00\)\(q^{95} + \)\(71\!\cdots\!42\)\(q^{97} + \)\(17\!\cdots\!56\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 1400531600527934473811256 x^{5} + 92429106535860966322690362643440028 x^{4} + 486502004825754823566786579226467181483733375376 x^{3} - 41390338158988484679355574715314473323669246141474080139600 x^{2} - 47785461930919140795588898989186212855196409324706742802409577734342400 x + 5612439960923763868733925256800794059272997589318959539312206365735127554315560000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 21 \)
\(\beta_{2}\)\(=\)\( 2304 \nu^{2} + 228081245760096 \nu - 921949945033343757008509810 \)
\(\beta_{3}\)\(=\)\((\)\(\)\(33\!\cdots\!39\)\( \nu^{6} - \)\(10\!\cdots\!47\)\( \nu^{5} - \)\(38\!\cdots\!42\)\( \nu^{4} + \)\(15\!\cdots\!00\)\( \nu^{3} + \)\(58\!\cdots\!04\)\( \nu^{2} - \)\(31\!\cdots\!76\)\( \nu + \)\(29\!\cdots\!16\)\(\)\()/ \)\(35\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(25\!\cdots\!71\)\( \nu^{6} - \)\(39\!\cdots\!89\)\( \nu^{5} + \)\(33\!\cdots\!98\)\( \nu^{4} + \)\(29\!\cdots\!28\)\( \nu^{3} - \)\(98\!\cdots\!20\)\( \nu^{2} - \)\(73\!\cdots\!20\)\( \nu + \)\(60\!\cdots\!60\)\(\)\()/ \)\(17\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(12\!\cdots\!91\)\( \nu^{6} - \)\(86\!\cdots\!91\)\( \nu^{5} - \)\(14\!\cdots\!38\)\( \nu^{4} + \)\(11\!\cdots\!72\)\( \nu^{3} + \)\(19\!\cdots\!80\)\( \nu^{2} - \)\(20\!\cdots\!00\)\( \nu + \)\(19\!\cdots\!00\)\(\)\()/ \)\(44\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(19\!\cdots\!91\)\( \nu^{6} - \)\(45\!\cdots\!29\)\( \nu^{5} + \)\(21\!\cdots\!98\)\( \nu^{4} + \)\(52\!\cdots\!28\)\( \nu^{3} - \)\(23\!\cdots\!80\)\( \nu^{2} - \)\(20\!\cdots\!60\)\( \nu - \)\(23\!\cdots\!40\)\(\)\()/ \)\(17\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 21\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 4751692620002 \beta_{1} + 921949945033243971463489768\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(42 \beta_{6} - 316 \beta_{5} + 10264134 \beta_{4} + 1609670278248 \beta_{3} - 535021439735 \beta_{2} + 101634125630290025052700568 \beta_{1} - 273801421862883403640682105610377156722\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(-42626265665746 \beta_{6} - 194107592272340 \beta_{5} + 67269459070670055586 \beta_{4} + 1127231570943432842363384 \beta_{3} + 9562404746094213156233329 \beta_{2} - 67905865187793886023239782848417368372 \beta_{1} + 5856348533646391890503911099481507908146528772438186\)\()/20736\)
\(\nu^{5}\)\(=\)\((\)\(430862942480855839403083782 \beta_{6} - 3738450509163176341926096356 \beta_{5} + 90584446641651031217225464502794 \beta_{4} + 17769475571802567536738602663767049816 \beta_{3} - 8435220614394625548590449868037095299 \beta_{2} + 795084583061473588894881299210202882450053746332044 \beta_{1} - 3912863042324753767205138395238430512483846107084730728679888270\)\()/62208\)
\(\nu^{6}\)\(=\)\((\)\(-190042400085323939661321936001188823878 \beta_{6} - 516160410681425951411831404459542062748 \beta_{5} + 218266581913983184206602167004836612775461686 \beta_{4} + 3365001098846429926936923159719053200768215536808 \beta_{3} + 27984158795443254288573758815933859061483398832387 \beta_{2} - 271977902369978894298261576379767963730152106352080077811700428 \beta_{1} + 15271420576046376546288652184838209908197252348968461206679826811043493877710\)\()/62208\)

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
−1.01484e12
−4.94369e11
−4.68207e11
1.22894e11
3.74539e11
5.71170e11
9.08809e11
−4.42254e13 0 1.33691e27 −3.67006e30 0 −4.35255e37 −3.17513e40 0 1.62310e44
1.2 −1.92430e13 0 −2.48679e26 3.02831e30 0 1.70526e37 1.66961e40 0 −5.82737e43
1.3 −1.79872e13 0 −2.95431e26 4.75717e30 0 3.90338e37 1.64475e40 0 −8.55680e43
1.4 1.03857e13 0 −5.11108e26 −2.06848e31 0 −1.95248e37 −1.17366e40 0 −2.14826e44
1.5 2.24646e13 0 −1.14310e26 1.81181e31 0 −2.76134e37 −1.64729e40 0 4.07017e44
1.6 3.19029e13 0 3.98828e26 −1.00222e30 0 1.60489e37 −7.02319e39 0 −3.19739e43
1.7 4.81096e13 0 1.69556e27 −1.07790e31 0 5.70296e37 5.17945e40 0 −5.18572e44
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 9.90.a.b 7
3.b odd 2 1 1.90.a.a 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.90.a.a 7 3.b odd 2 1
9.90.a.b 7 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - \)\(31\!\cdots\!08\)\( T_{2}^{6} - \)\(28\!\cdots\!12\)\( T_{2}^{5} + \)\(79\!\cdots\!96\)\( T_{2}^{4} + \)\(17\!\cdots\!48\)\( T_{2}^{3} - \)\(41\!\cdots\!84\)\( T_{2}^{2} - \)\(34\!\cdots\!64\)\( T_{2} + \)\(54\!\cdots\!12\)\( \) acting on \(S_{90}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 31407330351408 T + \)\(15\!\cdots\!72\)\( T^{2} - \)\(37\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!52\)\( T^{4} - \)\(24\!\cdots\!56\)\( T^{5} + \)\(48\!\cdots\!64\)\( T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!68\)\( T^{8} - \)\(95\!\cdots\!64\)\( T^{9} + \)\(26\!\cdots\!56\)\( T^{10} - \)\(54\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!04\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(34\!\cdots\!08\)\( T^{14} \)
$3$ 1
$5$ \( 1 + \)\(10\!\cdots\!50\)\( T + \)\(72\!\cdots\!75\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!25\)\( T^{4} + \)\(20\!\cdots\!50\)\( T^{5} + \)\(50\!\cdots\!75\)\( T^{6} + \)\(38\!\cdots\!00\)\( T^{7} + \)\(81\!\cdots\!75\)\( T^{8} + \)\(52\!\cdots\!50\)\( T^{9} + \)\(10\!\cdots\!25\)\( T^{10} + \)\(46\!\cdots\!00\)\( T^{11} + \)\(79\!\cdots\!75\)\( T^{12} + \)\(18\!\cdots\!50\)\( T^{13} + \)\(28\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 - \)\(38\!\cdots\!92\)\( T + \)\(80\!\cdots\!57\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!97\)\( T^{4} - \)\(93\!\cdots\!24\)\( T^{5} + \)\(76\!\cdots\!29\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!76\)\( T^{9} + \)\(13\!\cdots\!71\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} + \)\(93\!\cdots\!99\)\( T^{12} - \)\(73\!\cdots\!08\)\( T^{13} + \)\(31\!\cdots\!43\)\( T^{14} \)
$11$ \( 1 - \)\(33\!\cdots\!56\)\( T + \)\(18\!\cdots\!81\)\( T^{2} - \)\(51\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!81\)\( T^{4} - \)\(43\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!13\)\( T^{6} - \)\(25\!\cdots\!88\)\( T^{7} + \)\(62\!\cdots\!83\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{9} + \)\(21\!\cdots\!51\)\( T^{10} - \)\(27\!\cdots\!56\)\( T^{11} + \)\(47\!\cdots\!31\)\( T^{12} - \)\(42\!\cdots\!96\)\( T^{13} + \)\(61\!\cdots\!31\)\( T^{14} \)
$13$ \( 1 + \)\(10\!\cdots\!34\)\( T + \)\(79\!\cdots\!63\)\( T^{2} + \)\(36\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!57\)\( T^{4} + \)\(24\!\cdots\!58\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!83\)\( T^{8} + \)\(47\!\cdots\!82\)\( T^{9} + \)\(34\!\cdots\!69\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(40\!\cdots\!59\)\( T^{12} + \)\(72\!\cdots\!26\)\( T^{13} + \)\(96\!\cdots\!97\)\( T^{14} \)
$17$ \( 1 + \)\(83\!\cdots\!42\)\( T + \)\(16\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!37\)\( T^{4} + \)\(83\!\cdots\!34\)\( T^{5} + \)\(58\!\cdots\!59\)\( T^{6} + \)\(34\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!23\)\( T^{8} + \)\(87\!\cdots\!06\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(57\!\cdots\!39\)\( T^{12} + \)\(95\!\cdots\!18\)\( T^{13} + \)\(37\!\cdots\!13\)\( T^{14} \)
$19$ \( 1 - \)\(56\!\cdots\!80\)\( T + \)\(27\!\cdots\!53\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!61\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!65\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!35\)\( T^{8} - \)\(86\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!79\)\( T^{10} - \)\(27\!\cdots\!20\)\( T^{11} + \)\(30\!\cdots\!47\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{13} + \)\(46\!\cdots\!59\)\( T^{14} \)
$23$ \( 1 - \)\(11\!\cdots\!04\)\( T + \)\(63\!\cdots\!73\)\( T^{2} - \)\(40\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!37\)\( T^{4} - \)\(50\!\cdots\!28\)\( T^{5} + \)\(41\!\cdots\!21\)\( T^{6} - \)\(38\!\cdots\!20\)\( T^{7} + \)\(65\!\cdots\!23\)\( T^{8} - \)\(12\!\cdots\!32\)\( T^{9} + \)\(75\!\cdots\!39\)\( T^{10} - \)\(23\!\cdots\!40\)\( T^{11} + \)\(58\!\cdots\!39\)\( T^{12} - \)\(16\!\cdots\!36\)\( T^{13} + \)\(22\!\cdots\!67\)\( T^{14} \)
$29$ \( 1 - \)\(14\!\cdots\!30\)\( T + \)\(42\!\cdots\!83\)\( T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + \)\(88\!\cdots\!81\)\( T^{4} - \)\(54\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!15\)\( T^{6} - \)\(72\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(11\!\cdots\!50\)\( T^{9} + \)\(25\!\cdots\!29\)\( T^{10} - \)\(16\!\cdots\!20\)\( T^{11} + \)\(24\!\cdots\!67\)\( T^{12} - \)\(11\!\cdots\!30\)\( T^{13} + \)\(11\!\cdots\!89\)\( T^{14} \)
$31$ \( 1 - \)\(68\!\cdots\!04\)\( T + \)\(21\!\cdots\!61\)\( T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!41\)\( T^{4} - \)\(29\!\cdots\!52\)\( T^{5} + \)\(13\!\cdots\!33\)\( T^{6} - \)\(21\!\cdots\!72\)\( T^{7} + \)\(75\!\cdots\!43\)\( T^{8} - \)\(86\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!51\)\( T^{10} - \)\(19\!\cdots\!04\)\( T^{11} + \)\(96\!\cdots\!11\)\( T^{12} - \)\(16\!\cdots\!84\)\( T^{13} + \)\(13\!\cdots\!91\)\( T^{14} \)
$37$ \( 1 + \)\(54\!\cdots\!58\)\( T + \)\(17\!\cdots\!67\)\( T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!17\)\( T^{4} + \)\(15\!\cdots\!46\)\( T^{5} + \)\(78\!\cdots\!19\)\( T^{6} + \)\(75\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!63\)\( T^{8} + \)\(21\!\cdots\!34\)\( T^{9} + \)\(73\!\cdots\!61\)\( T^{10} + \)\(29\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!19\)\( T^{12} + \)\(14\!\cdots\!62\)\( T^{13} + \)\(97\!\cdots\!53\)\( T^{14} \)
$41$ \( 1 - \)\(65\!\cdots\!66\)\( T + \)\(14\!\cdots\!51\)\( T^{2} - \)\(11\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!21\)\( T^{4} - \)\(83\!\cdots\!98\)\( T^{5} + \)\(64\!\cdots\!43\)\( T^{6} - \)\(35\!\cdots\!48\)\( T^{7} + \)\(22\!\cdots\!23\)\( T^{8} - \)\(98\!\cdots\!58\)\( T^{9} + \)\(49\!\cdots\!01\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(72\!\cdots\!51\)\( T^{12} - \)\(11\!\cdots\!26\)\( T^{13} + \)\(58\!\cdots\!21\)\( T^{14} \)
$43$ \( 1 - \)\(32\!\cdots\!56\)\( T + \)\(10\!\cdots\!93\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!97\)\( T^{4} - \)\(92\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!03\)\( T^{8} - \)\(53\!\cdots\!68\)\( T^{9} + \)\(68\!\cdots\!79\)\( T^{10} - \)\(79\!\cdots\!00\)\( T^{11} + \)\(79\!\cdots\!99\)\( T^{12} - \)\(60\!\cdots\!44\)\( T^{13} + \)\(44\!\cdots\!07\)\( T^{14} \)
$47$ \( 1 - \)\(58\!\cdots\!08\)\( T + \)\(50\!\cdots\!37\)\( T^{2} - \)\(21\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} - \)\(33\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!49\)\( T^{6} - \)\(28\!\cdots\!80\)\( T^{7} + \)\(76\!\cdots\!83\)\( T^{8} - \)\(14\!\cdots\!04\)\( T^{9} + \)\(29\!\cdots\!91\)\( T^{10} - \)\(39\!\cdots\!60\)\( T^{11} + \)\(61\!\cdots\!59\)\( T^{12} - \)\(46\!\cdots\!52\)\( T^{13} + \)\(52\!\cdots\!23\)\( T^{14} \)
$53$ \( 1 - \)\(18\!\cdots\!14\)\( T + \)\(25\!\cdots\!03\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!77\)\( T^{4} - \)\(17\!\cdots\!38\)\( T^{5} + \)\(11\!\cdots\!71\)\( T^{6} - \)\(65\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!43\)\( T^{8} - \)\(14\!\cdots\!82\)\( T^{9} + \)\(56\!\cdots\!49\)\( T^{10} - \)\(18\!\cdots\!80\)\( T^{11} + \)\(50\!\cdots\!79\)\( T^{12} - \)\(10\!\cdots\!66\)\( T^{13} + \)\(16\!\cdots\!77\)\( T^{14} \)
$59$ \( 1 - \)\(90\!\cdots\!60\)\( T + \)\(21\!\cdots\!73\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} - \)\(99\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!65\)\( T^{6} - \)\(47\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!35\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!79\)\( T^{10} - \)\(36\!\cdots\!40\)\( T^{11} + \)\(22\!\cdots\!27\)\( T^{12} - \)\(39\!\cdots\!60\)\( T^{13} + \)\(17\!\cdots\!79\)\( T^{14} \)
$61$ \( 1 - \)\(87\!\cdots\!94\)\( T + \)\(29\!\cdots\!31\)\( T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(44\!\cdots\!81\)\( T^{4} - \)\(42\!\cdots\!02\)\( T^{5} + \)\(45\!\cdots\!63\)\( T^{6} - \)\(43\!\cdots\!12\)\( T^{7} + \)\(35\!\cdots\!83\)\( T^{8} - \)\(26\!\cdots\!62\)\( T^{9} + \)\(21\!\cdots\!01\)\( T^{10} - \)\(93\!\cdots\!44\)\( T^{11} + \)\(88\!\cdots\!31\)\( T^{12} - \)\(20\!\cdots\!54\)\( T^{13} + \)\(18\!\cdots\!81\)\( T^{14} \)
$67$ \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(34\!\cdots\!77\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!37\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{5} + \)\(24\!\cdots\!09\)\( T^{6} - \)\(45\!\cdots\!80\)\( T^{7} + \)\(80\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(14\!\cdots\!51\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!39\)\( T^{12} - \)\(77\!\cdots\!68\)\( T^{13} + \)\(44\!\cdots\!63\)\( T^{14} \)
$71$ \( 1 - \)\(54\!\cdots\!76\)\( T + \)\(38\!\cdots\!21\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(62\!\cdots\!61\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{5} + \)\(57\!\cdots\!73\)\( T^{6} - \)\(13\!\cdots\!08\)\( T^{7} + \)\(33\!\cdots\!63\)\( T^{8} - \)\(61\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!51\)\( T^{10} - \)\(16\!\cdots\!76\)\( T^{11} + \)\(24\!\cdots\!71\)\( T^{12} - \)\(20\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!11\)\( T^{14} \)
$73$ \( 1 + \)\(19\!\cdots\!54\)\( T + \)\(53\!\cdots\!23\)\( T^{2} + \)\(79\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!37\)\( T^{4} + \)\(13\!\cdots\!78\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(94\!\cdots\!23\)\( T^{8} + \)\(62\!\cdots\!82\)\( T^{9} + \)\(37\!\cdots\!89\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(81\!\cdots\!39\)\( T^{12} + \)\(20\!\cdots\!86\)\( T^{13} + \)\(70\!\cdots\!17\)\( T^{14} \)
$79$ \( 1 - \)\(26\!\cdots\!20\)\( T + \)\(19\!\cdots\!33\)\( T^{2} - \)\(22\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!65\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(93\!\cdots\!00\)\( T^{9} + \)\(64\!\cdots\!79\)\( T^{10} - \)\(79\!\cdots\!80\)\( T^{11} + \)\(54\!\cdots\!67\)\( T^{12} - \)\(57\!\cdots\!20\)\( T^{13} + \)\(16\!\cdots\!39\)\( T^{14} \)
$83$ \( 1 - \)\(35\!\cdots\!24\)\( T + \)\(34\!\cdots\!33\)\( T^{2} - \)\(80\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!17\)\( T^{4} - \)\(78\!\cdots\!48\)\( T^{5} + \)\(37\!\cdots\!21\)\( T^{6} - \)\(51\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!63\)\( T^{8} - \)\(30\!\cdots\!32\)\( T^{9} + \)\(11\!\cdots\!59\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(33\!\cdots\!19\)\( T^{12} - \)\(21\!\cdots\!96\)\( T^{13} + \)\(38\!\cdots\!87\)\( T^{14} \)
$89$ \( 1 - \)\(16\!\cdots\!90\)\( T + \)\(25\!\cdots\!63\)\( T^{2} - \)\(22\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!01\)\( T^{4} - \)\(13\!\cdots\!50\)\( T^{5} + \)\(88\!\cdots\!15\)\( T^{6} - \)\(47\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!35\)\( T^{8} - \)\(12\!\cdots\!50\)\( T^{9} + \)\(62\!\cdots\!29\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} + \)\(75\!\cdots\!87\)\( T^{12} - \)\(15\!\cdots\!90\)\( T^{13} + \)\(29\!\cdots\!69\)\( T^{14} \)
$97$ \( 1 - \)\(71\!\cdots\!42\)\( T + \)\(48\!\cdots\!87\)\( T^{2} - \)\(20\!\cdots\!40\)\( T^{3} + \)\(79\!\cdots\!57\)\( T^{4} - \)\(24\!\cdots\!14\)\( T^{5} + \)\(72\!\cdots\!99\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(47\!\cdots\!83\)\( T^{8} - \)\(11\!\cdots\!46\)\( T^{9} + \)\(23\!\cdots\!41\)\( T^{10} - \)\(40\!\cdots\!40\)\( T^{11} + \)\(62\!\cdots\!59\)\( T^{12} - \)\(61\!\cdots\!98\)\( T^{13} + \)\(57\!\cdots\!73\)\( T^{14} \)
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