Properties

Label 3.72.a.a
Level 3
Weight 72
Character orbit 3.a
Self dual yes
Analytic conductor 95.774
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(95.7738481683\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-5010255538 - \beta_{1}) q^{2} +50031545098999707 q^{3} +(\)\(38\!\cdots\!24\)\( + 14833524748 \beta_{1} + \beta_{2}) q^{4} +(\)\(28\!\cdots\!47\)\( + 22205806960703 \beta_{1} - 1156 \beta_{2} - \beta_{3}) q^{5} +(-\)\(25\!\cdots\!66\)\( - 50031545098999707 \beta_{1}) q^{6} +(-\)\(25\!\cdots\!77\)\( - 12693346119954501029 \beta_{1} - 105331541 \beta_{2} + 63091 \beta_{3} - 3983 \beta_{4}) q^{7} +(-\)\(30\!\cdots\!56\)\( - 72150789618238048416 \beta_{1} - 41145611768 \beta_{2} + 1969408 \beta_{3} - 991104 \beta_{4}) q^{8} +\)\(25\!\cdots\!49\)\( q^{9} +O(q^{10})\) \( q +(-5010255538 - \beta_{1}) q^{2} +50031545098999707 q^{3} +(\)\(38\!\cdots\!24\)\( + 14833524748 \beta_{1} + \beta_{2}) q^{4} +(\)\(28\!\cdots\!47\)\( + 22205806960703 \beta_{1} - 1156 \beta_{2} - \beta_{3}) q^{5} +(-\)\(25\!\cdots\!66\)\( - 50031545098999707 \beta_{1}) q^{6} +(-\)\(25\!\cdots\!77\)\( - 12693346119954501029 \beta_{1} - 105331541 \beta_{2} + 63091 \beta_{3} - 3983 \beta_{4}) q^{7} +(-\)\(30\!\cdots\!56\)\( - 72150789618238048416 \beta_{1} - 41145611768 \beta_{2} + 1969408 \beta_{3} - 991104 \beta_{4}) q^{8} +\)\(25\!\cdots\!49\)\( q^{9} +(-\)\(61\!\cdots\!84\)\( + \)\(17\!\cdots\!34\)\( \beta_{1} - 7653174060568 \beta_{2} + 13194541072 \beta_{3} + 3346421800 \beta_{4}) q^{10} +(\)\(27\!\cdots\!26\)\( - \)\(72\!\cdots\!66\)\( \beta_{1} + 941677448344864 \beta_{2} - 653169989582 \beta_{3} - 113546592984 \beta_{4}) q^{11} +(\)\(19\!\cdots\!68\)\( + \)\(74\!\cdots\!36\)\( \beta_{1} + 50031545098999707 \beta_{2}) q^{12} +(-\)\(74\!\cdots\!36\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} + 131383178781509317 \beta_{2} - 59886625166368 \beta_{3} + 72687561798059 \beta_{4}) q^{13} +(\)\(35\!\cdots\!92\)\( + \)\(57\!\cdots\!64\)\( \beta_{1} + 19345982584020187480 \beta_{2} - 350761150967696 \beta_{3} - 160175633400552 \beta_{4}) q^{14} +(\)\(14\!\cdots\!29\)\( + \)\(11\!\cdots\!21\)\( \beta_{1} - 57836466134443661292 \beta_{2} - 50031545098999707 \beta_{3}) q^{15} +(-\)\(55\!\cdots\!84\)\( + \)\(74\!\cdots\!48\)\( \beta_{1} - \)\(45\!\cdots\!44\)\( \beta_{2} + 95719960882767872 \beta_{3} + 5160789222769664 \beta_{4}) q^{16} +(-\)\(55\!\cdots\!08\)\( + \)\(45\!\cdots\!10\)\( \beta_{1} + \)\(11\!\cdots\!82\)\( \beta_{2} + 2400820664738926274 \beta_{3} - 21924450278509662 \beta_{4}) q^{17} +(-\)\(12\!\cdots\!62\)\( - \)\(25\!\cdots\!49\)\( \beta_{1}) q^{18} +(\)\(58\!\cdots\!94\)\( + \)\(56\!\cdots\!86\)\( \beta_{1} + \)\(58\!\cdots\!78\)\( \beta_{2} + \)\(24\!\cdots\!26\)\( \beta_{3} - 7744471202131557338 \beta_{4}) q^{19} +(-\)\(49\!\cdots\!08\)\( + \)\(72\!\cdots\!08\)\( \beta_{1} - \)\(97\!\cdots\!66\)\( \beta_{2} + \)\(14\!\cdots\!64\)\( \beta_{3} + 84179304783469209600 \beta_{4}) q^{20} +(-\)\(12\!\cdots\!39\)\( - \)\(63\!\cdots\!03\)\( \beta_{1} - \)\(52\!\cdots\!87\)\( \beta_{2} + \)\(31\!\cdots\!37\)\( \beta_{3} - \)\(19\!\cdots\!81\)\( \beta_{4}) q^{21} +(\)\(19\!\cdots\!84\)\( - \)\(13\!\cdots\!80\)\( \beta_{1} + \)\(11\!\cdots\!24\)\( \beta_{2} + \)\(35\!\cdots\!48\)\( \beta_{3} - \)\(30\!\cdots\!24\)\( \beta_{4}) q^{22} +(-\)\(84\!\cdots\!74\)\( - \)\(14\!\cdots\!98\)\( \beta_{1} - \)\(35\!\cdots\!10\)\( \beta_{2} + \)\(12\!\cdots\!82\)\( \beta_{3} + \)\(29\!\cdots\!34\)\( \beta_{4}) q^{23} +(-\)\(15\!\cdots\!92\)\( - \)\(36\!\cdots\!12\)\( \beta_{1} - \)\(20\!\cdots\!76\)\( \beta_{2} + \)\(98\!\cdots\!56\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4}) q^{24} +(\)\(39\!\cdots\!35\)\( + \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(13\!\cdots\!30\)\( \beta_{2} - \)\(36\!\cdots\!80\)\( \beta_{3} - \)\(48\!\cdots\!50\)\( \beta_{4}) q^{25} +(-\)\(32\!\cdots\!08\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(70\!\cdots\!64\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4}) q^{26} +\)\(12\!\cdots\!43\)\( q^{27} +(-\)\(11\!\cdots\!32\)\( - \)\(48\!\cdots\!12\)\( \beta_{1} - \)\(72\!\cdots\!32\)\( \beta_{2} - \)\(71\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!40\)\( \beta_{4}) q^{28} +(-\)\(73\!\cdots\!19\)\( + \)\(30\!\cdots\!77\)\( \beta_{1} - \)\(15\!\cdots\!94\)\( \beta_{2} + \)\(11\!\cdots\!41\)\( \beta_{3} - \)\(87\!\cdots\!58\)\( \beta_{4}) q^{29} +(-\)\(30\!\cdots\!88\)\( + \)\(85\!\cdots\!38\)\( \beta_{1} - \)\(38\!\cdots\!76\)\( \beta_{2} + \)\(66\!\cdots\!04\)\( \beta_{3} + \)\(16\!\cdots\!00\)\( \beta_{4}) q^{30} +(\)\(14\!\cdots\!63\)\( + \)\(70\!\cdots\!03\)\( \beta_{1} + \)\(34\!\cdots\!27\)\( \beta_{2} - \)\(23\!\cdots\!25\)\( \beta_{3} - \)\(63\!\cdots\!75\)\( \beta_{4}) q^{31} +(-\)\(12\!\cdots\!68\)\( + \)\(86\!\cdots\!16\)\( \beta_{1} + \)\(32\!\cdots\!48\)\( \beta_{2} - \)\(81\!\cdots\!92\)\( \beta_{3} + \)\(27\!\cdots\!96\)\( \beta_{4}) q^{32} +(\)\(13\!\cdots\!82\)\( - \)\(36\!\cdots\!62\)\( \beta_{1} + \)\(47\!\cdots\!48\)\( \beta_{2} - \)\(32\!\cdots\!74\)\( \beta_{3} - \)\(56\!\cdots\!88\)\( \beta_{4}) q^{33} +(-\)\(12\!\cdots\!64\)\( - \)\(67\!\cdots\!94\)\( \beta_{1} - \)\(43\!\cdots\!16\)\( \beta_{2} - \)\(30\!\cdots\!96\)\( \beta_{3} - \)\(71\!\cdots\!52\)\( \beta_{4}) q^{34} +(-\)\(26\!\cdots\!70\)\( + \)\(26\!\cdots\!70\)\( \beta_{1} + \)\(70\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!10\)\( \beta_{3} + \)\(33\!\cdots\!00\)\( \beta_{4}) q^{35} +(\)\(95\!\cdots\!76\)\( + \)\(37\!\cdots\!52\)\( \beta_{1} + \)\(25\!\cdots\!49\)\( \beta_{2}) q^{36} +(-\)\(32\!\cdots\!14\)\( + \)\(12\!\cdots\!66\)\( \beta_{1} - \)\(18\!\cdots\!61\)\( \beta_{2} + \)\(39\!\cdots\!10\)\( \beta_{3} + \)\(14\!\cdots\!45\)\( \beta_{4}) q^{37} +(-\)\(18\!\cdots\!32\)\( - \)\(17\!\cdots\!52\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} - \)\(10\!\cdots\!24\)\( \beta_{3} - \)\(13\!\cdots\!88\)\( \beta_{4}) q^{38} +(-\)\(37\!\cdots\!52\)\( + \)\(67\!\cdots\!52\)\( \beta_{1} + \)\(65\!\cdots\!19\)\( \beta_{2} - \)\(29\!\cdots\!76\)\( \beta_{3} + \)\(36\!\cdots\!13\)\( \beta_{4}) q^{39} +(\)\(15\!\cdots\!08\)\( + \)\(27\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!84\)\( \beta_{2} - \)\(72\!\cdots\!64\)\( \beta_{3} - \)\(74\!\cdots\!00\)\( \beta_{4}) q^{40} +(\)\(45\!\cdots\!88\)\( + \)\(14\!\cdots\!98\)\( \beta_{1} + \)\(42\!\cdots\!78\)\( \beta_{2} + \)\(18\!\cdots\!30\)\( \beta_{3} + \)\(15\!\cdots\!10\)\( \beta_{4}) q^{41} +(\)\(17\!\cdots\!44\)\( + \)\(28\!\cdots\!48\)\( \beta_{1} + \)\(96\!\cdots\!60\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(80\!\cdots\!64\)\( \beta_{4}) q^{42} +(\)\(13\!\cdots\!34\)\( + \)\(41\!\cdots\!14\)\( \beta_{1} - \)\(12\!\cdots\!42\)\( \beta_{2} + \)\(61\!\cdots\!74\)\( \beta_{3} - \)\(27\!\cdots\!62\)\( \beta_{4}) q^{43} +(\)\(20\!\cdots\!84\)\( - \)\(23\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(26\!\cdots\!12\)\( \beta_{4}) q^{44} +(\)\(71\!\cdots\!03\)\( + \)\(55\!\cdots\!47\)\( \beta_{1} - \)\(28\!\cdots\!44\)\( \beta_{2} - \)\(25\!\cdots\!49\)\( \beta_{3}) q^{45} +(\)\(43\!\cdots\!36\)\( + \)\(16\!\cdots\!28\)\( \beta_{1} + \)\(43\!\cdots\!60\)\( \beta_{2} - \)\(86\!\cdots\!16\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4}) q^{46} +(-\)\(93\!\cdots\!58\)\( + \)\(11\!\cdots\!82\)\( \beta_{1} - \)\(83\!\cdots\!14\)\( \beta_{2} - \)\(15\!\cdots\!58\)\( \beta_{3} - \)\(20\!\cdots\!46\)\( \beta_{4}) q^{47} +(-\)\(27\!\cdots\!88\)\( + \)\(37\!\cdots\!36\)\( \beta_{1} - \)\(22\!\cdots\!08\)\( \beta_{2} + \)\(47\!\cdots\!04\)\( \beta_{3} + \)\(25\!\cdots\!48\)\( \beta_{4}) q^{48} +(-\)\(23\!\cdots\!23\)\( + \)\(12\!\cdots\!44\)\( \beta_{1} + \)\(30\!\cdots\!18\)\( \beta_{2} + \)\(61\!\cdots\!88\)\( \beta_{3} - \)\(99\!\cdots\!94\)\( \beta_{4}) q^{49} +(-\)\(13\!\cdots\!70\)\( + \)\(20\!\cdots\!45\)\( \beta_{1} + \)\(60\!\cdots\!60\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3} + \)\(58\!\cdots\!00\)\( \beta_{4}) q^{50} +(-\)\(27\!\cdots\!56\)\( + \)\(22\!\cdots\!70\)\( \beta_{1} + \)\(59\!\cdots\!74\)\( \beta_{2} + \)\(12\!\cdots\!18\)\( \beta_{3} - \)\(10\!\cdots\!34\)\( \beta_{4}) q^{51} +(\)\(93\!\cdots\!28\)\( + \)\(13\!\cdots\!20\)\( \beta_{1} - \)\(70\!\cdots\!66\)\( \beta_{2} - \)\(26\!\cdots\!48\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4}) q^{52} +(\)\(33\!\cdots\!57\)\( + \)\(36\!\cdots\!37\)\( \beta_{1} - \)\(32\!\cdots\!18\)\( \beta_{2} - \)\(87\!\cdots\!15\)\( \beta_{3} - \)\(34\!\cdots\!30\)\( \beta_{4}) q^{53} +(-\)\(62\!\cdots\!34\)\( - \)\(12\!\cdots\!43\)\( \beta_{1}) q^{54} +(\)\(13\!\cdots\!92\)\( + \)\(75\!\cdots\!08\)\( \beta_{1} - \)\(21\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4}) q^{55} +(\)\(52\!\cdots\!04\)\( + \)\(16\!\cdots\!60\)\( \beta_{1} + \)\(30\!\cdots\!92\)\( \beta_{2} + \)\(34\!\cdots\!12\)\( \beta_{3} + \)\(81\!\cdots\!44\)\( \beta_{4}) q^{56} +(\)\(29\!\cdots\!58\)\( + \)\(28\!\cdots\!02\)\( \beta_{1} + \)\(29\!\cdots\!46\)\( \beta_{2} + \)\(12\!\cdots\!82\)\( \beta_{3} - \)\(38\!\cdots\!66\)\( \beta_{4}) q^{57} +(-\)\(78\!\cdots\!08\)\( + \)\(74\!\cdots\!26\)\( \beta_{1} - \)\(20\!\cdots\!16\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4}) q^{58} +(-\)\(78\!\cdots\!00\)\( + \)\(34\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!68\)\( \beta_{2} + \)\(22\!\cdots\!96\)\( \beta_{3} + \)\(98\!\cdots\!52\)\( \beta_{4}) q^{59} +(-\)\(24\!\cdots\!56\)\( + \)\(36\!\cdots\!56\)\( \beta_{1} - \)\(49\!\cdots\!62\)\( \beta_{2} + \)\(72\!\cdots\!48\)\( \beta_{3} + \)\(42\!\cdots\!00\)\( \beta_{4}) q^{60} +(\)\(14\!\cdots\!78\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + \)\(78\!\cdots\!87\)\( \beta_{2} - \)\(41\!\cdots\!38\)\( \beta_{3} + \)\(17\!\cdots\!69\)\( \beta_{4}) q^{61} +(-\)\(19\!\cdots\!76\)\( - \)\(86\!\cdots\!88\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(23\!\cdots\!16\)\( \beta_{3} - \)\(48\!\cdots\!08\)\( \beta_{4}) q^{62} +(-\)\(63\!\cdots\!73\)\( - \)\(31\!\cdots\!21\)\( \beta_{1} - \)\(26\!\cdots\!09\)\( \beta_{2} + \)\(15\!\cdots\!59\)\( \beta_{3} - \)\(99\!\cdots\!67\)\( \beta_{4}) q^{63} +(-\)\(40\!\cdots\!24\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} - \)\(27\!\cdots\!36\)\( \beta_{2} - \)\(61\!\cdots\!56\)\( \beta_{3} + \)\(60\!\cdots\!28\)\( \beta_{4}) q^{64} +(\)\(26\!\cdots\!58\)\( - \)\(36\!\cdots\!58\)\( \beta_{1} + \)\(69\!\cdots\!66\)\( \beta_{2} + \)\(27\!\cdots\!86\)\( \beta_{3} + \)\(15\!\cdots\!50\)\( \beta_{4}) q^{65} +(\)\(98\!\cdots\!88\)\( - \)\(67\!\cdots\!60\)\( \beta_{1} + \)\(59\!\cdots\!68\)\( \beta_{2} + \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(15\!\cdots\!68\)\( \beta_{4}) q^{66} +(\)\(13\!\cdots\!72\)\( - \)\(52\!\cdots\!40\)\( \beta_{1} - \)\(69\!\cdots\!24\)\( \beta_{2} - \)\(76\!\cdots\!16\)\( \beta_{3} - \)\(58\!\cdots\!92\)\( \beta_{4}) q^{67} +(\)\(24\!\cdots\!12\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} + \)\(20\!\cdots\!42\)\( \beta_{2} - \)\(45\!\cdots\!72\)\( \beta_{3} + \)\(32\!\cdots\!36\)\( \beta_{4}) q^{68} +(-\)\(42\!\cdots\!18\)\( - \)\(71\!\cdots\!86\)\( \beta_{1} - \)\(17\!\cdots\!70\)\( \beta_{2} + \)\(60\!\cdots\!74\)\( \beta_{3} + \)\(14\!\cdots\!38\)\( \beta_{4}) q^{69} +(-\)\(58\!\cdots\!60\)\( + \)\(10\!\cdots\!60\)\( \beta_{1} - \)\(67\!\cdots\!20\)\( \beta_{2} - \)\(73\!\cdots\!20\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4}) q^{70} +(\)\(36\!\cdots\!50\)\( + \)\(31\!\cdots\!50\)\( \beta_{1} + \)\(10\!\cdots\!50\)\( \beta_{2} + \)\(69\!\cdots\!54\)\( \beta_{3} - \)\(34\!\cdots\!02\)\( \beta_{4}) q^{71} +(-\)\(76\!\cdots\!44\)\( - \)\(18\!\cdots\!84\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(49\!\cdots\!92\)\( \beta_{3} - \)\(24\!\cdots\!96\)\( \beta_{4}) q^{72} +(-\)\(62\!\cdots\!38\)\( + \)\(60\!\cdots\!92\)\( \beta_{1} - \)\(20\!\cdots\!30\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(65\!\cdots\!30\)\( \beta_{4}) q^{73} +(-\)\(18\!\cdots\!52\)\( + \)\(65\!\cdots\!34\)\( \beta_{1} + \)\(30\!\cdots\!64\)\( \beta_{2} - \)\(12\!\cdots\!68\)\( \beta_{3} + \)\(13\!\cdots\!84\)\( \beta_{4}) q^{74} +(\)\(19\!\cdots\!45\)\( + \)\(20\!\cdots\!80\)\( \beta_{1} - \)\(66\!\cdots\!10\)\( \beta_{2} - \)\(18\!\cdots\!60\)\( \beta_{3} - \)\(24\!\cdots\!50\)\( \beta_{4}) q^{75} +(\)\(34\!\cdots\!00\)\( + \)\(44\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!24\)\( \beta_{2} - \)\(29\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!32\)\( \beta_{4}) q^{76} +(\)\(20\!\cdots\!88\)\( - \)\(28\!\cdots\!68\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} + \)\(64\!\cdots\!68\)\( \beta_{3} - \)\(74\!\cdots\!84\)\( \beta_{4}) q^{77} +(-\)\(16\!\cdots\!56\)\( + \)\(18\!\cdots\!10\)\( \beta_{1} - \)\(35\!\cdots\!48\)\( \beta_{2} - \)\(70\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{78} +(-\)\(54\!\cdots\!21\)\( - \)\(14\!\cdots\!65\)\( \beta_{1} - \)\(37\!\cdots\!33\)\( \beta_{2} + \)\(35\!\cdots\!87\)\( \beta_{3} - \)\(15\!\cdots\!31\)\( \beta_{4}) q^{79} +(\)\(34\!\cdots\!08\)\( - \)\(19\!\cdots\!08\)\( \beta_{1} + \)\(38\!\cdots\!16\)\( \beta_{2} - \)\(72\!\cdots\!64\)\( \beta_{3} - \)\(27\!\cdots\!00\)\( \beta_{4}) q^{80} +\)\(62\!\cdots\!01\)\( q^{81} +(-\)\(42\!\cdots\!60\)\( - \)\(14\!\cdots\!38\)\( \beta_{1} - \)\(36\!\cdots\!28\)\( \beta_{2} - \)\(26\!\cdots\!16\)\( \beta_{3} + \)\(29\!\cdots\!08\)\( \beta_{4}) q^{82} +(-\)\(23\!\cdots\!46\)\( - \)\(15\!\cdots\!02\)\( \beta_{1} + \)\(60\!\cdots\!40\)\( \beta_{2} - \)\(27\!\cdots\!50\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(57\!\cdots\!24\)\( - \)\(24\!\cdots\!84\)\( \beta_{1} - \)\(36\!\cdots\!24\)\( \beta_{2} - \)\(35\!\cdots\!40\)\( \beta_{3} - \)\(70\!\cdots\!80\)\( \beta_{4}) q^{84} +(-\)\(70\!\cdots\!62\)\( - \)\(95\!\cdots\!38\)\( \beta_{1} + \)\(37\!\cdots\!76\)\( \beta_{2} + \)\(37\!\cdots\!46\)\( \beta_{3} - \)\(80\!\cdots\!00\)\( \beta_{4}) q^{85} +(-\)\(11\!\cdots\!96\)\( + \)\(70\!\cdots\!64\)\( \beta_{1} + \)\(22\!\cdots\!60\)\( \beta_{2} - \)\(62\!\cdots\!88\)\( \beta_{3} - \)\(95\!\cdots\!56\)\( \beta_{4}) q^{86} +(-\)\(36\!\cdots\!33\)\( + \)\(15\!\cdots\!39\)\( \beta_{1} - \)\(79\!\cdots\!58\)\( \beta_{2} + \)\(56\!\cdots\!87\)\( \beta_{3} - \)\(43\!\cdots\!06\)\( \beta_{4}) q^{87} +(\)\(17\!\cdots\!12\)\( + \)\(58\!\cdots\!64\)\( \beta_{1} + \)\(37\!\cdots\!32\)\( \beta_{2} - \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(35\!\cdots\!44\)\( \beta_{4}) q^{88} +(\)\(18\!\cdots\!50\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(15\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!04\)\( \beta_{3} + \)\(30\!\cdots\!48\)\( \beta_{4}) q^{89} +(-\)\(15\!\cdots\!16\)\( + \)\(42\!\cdots\!66\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(33\!\cdots\!28\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4}) q^{90} +(-\)\(16\!\cdots\!34\)\( + \)\(15\!\cdots\!78\)\( \beta_{1} - \)\(97\!\cdots\!78\)\( \beta_{2} - \)\(15\!\cdots\!86\)\( \beta_{3} - \)\(36\!\cdots\!82\)\( \beta_{4}) q^{91} +(-\)\(27\!\cdots\!60\)\( - \)\(34\!\cdots\!24\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(35\!\cdots\!88\)\( \beta_{3} - \)\(22\!\cdots\!56\)\( \beta_{4}) q^{92} +(\)\(74\!\cdots\!41\)\( + \)\(35\!\cdots\!21\)\( \beta_{1} + \)\(17\!\cdots\!89\)\( \beta_{2} - \)\(11\!\cdots\!75\)\( \beta_{3} - \)\(31\!\cdots\!25\)\( \beta_{4}) q^{93} +(-\)\(26\!\cdots\!16\)\( + \)\(24\!\cdots\!72\)\( \beta_{1} + \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(29\!\cdots\!72\)\( \beta_{3} + \)\(20\!\cdots\!64\)\( \beta_{4}) q^{94} +(-\)\(12\!\cdots\!88\)\( + \)\(27\!\cdots\!88\)\( \beta_{1} + \)\(16\!\cdots\!24\)\( \beta_{2} + \)\(53\!\cdots\!04\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4}) q^{95} +(-\)\(64\!\cdots\!76\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2} - \)\(40\!\cdots\!44\)\( \beta_{3} + \)\(13\!\cdots\!72\)\( \beta_{4}) q^{96} +(-\)\(29\!\cdots\!94\)\( - \)\(63\!\cdots\!80\)\( \beta_{1} + \)\(78\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3} - \)\(18\!\cdots\!68\)\( \beta_{4}) q^{97} +(-\)\(32\!\cdots\!30\)\( - \)\(48\!\cdots\!81\)\( \beta_{1} - \)\(20\!\cdots\!96\)\( \beta_{2} + \)\(61\!\cdots\!20\)\( \beta_{3} - \)\(31\!\cdots\!60\)\( \beta_{4}) q^{98} +(\)\(67\!\cdots\!74\)\( - \)\(18\!\cdots\!34\)\( \beta_{1} + \)\(23\!\cdots\!36\)\( \beta_{2} - \)\(16\!\cdots\!18\)\( \beta_{3} - \)\(28\!\cdots\!16\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 25051277688q^{2} + 250157725494998535q^{3} + \)\(19\!\cdots\!24\)\(q^{4} + \)\(14\!\cdots\!30\)\(q^{5} - \)\(12\!\cdots\!16\)\(q^{6} - \)\(12\!\cdots\!52\)\(q^{7} - \)\(15\!\cdots\!48\)\(q^{8} + \)\(12\!\cdots\!45\)\(q^{9} + O(q^{10}) \) \( 5q - 25051277688q^{2} + 250157725494998535q^{3} + \)\(19\!\cdots\!24\)\(q^{4} + \)\(14\!\cdots\!30\)\(q^{5} - \)\(12\!\cdots\!16\)\(q^{6} - \)\(12\!\cdots\!52\)\(q^{7} - \)\(15\!\cdots\!48\)\(q^{8} + \)\(12\!\cdots\!45\)\(q^{9} - \)\(30\!\cdots\!60\)\(q^{10} + \)\(13\!\cdots\!12\)\(q^{11} + \)\(95\!\cdots\!68\)\(q^{12} - \)\(37\!\cdots\!02\)\(q^{13} + \)\(17\!\cdots\!32\)\(q^{14} + \)\(71\!\cdots\!10\)\(q^{15} - \)\(27\!\cdots\!16\)\(q^{16} - \)\(27\!\cdots\!10\)\(q^{17} - \)\(62\!\cdots\!12\)\(q^{18} + \)\(29\!\cdots\!48\)\(q^{19} - \)\(24\!\cdots\!20\)\(q^{20} - \)\(63\!\cdots\!64\)\(q^{21} + \)\(98\!\cdots\!80\)\(q^{22} - \)\(42\!\cdots\!24\)\(q^{23} - \)\(76\!\cdots\!36\)\(q^{24} + \)\(19\!\cdots\!75\)\(q^{25} - \)\(16\!\cdots\!00\)\(q^{26} + \)\(62\!\cdots\!15\)\(q^{27} - \)\(57\!\cdots\!36\)\(q^{28} - \)\(36\!\cdots\!74\)\(q^{29} - \)\(15\!\cdots\!20\)\(q^{30} + \)\(74\!\cdots\!84\)\(q^{31} - \)\(63\!\cdots\!72\)\(q^{32} + \)\(67\!\cdots\!84\)\(q^{33} - \)\(62\!\cdots\!32\)\(q^{34} - \)\(13\!\cdots\!00\)\(q^{35} + \)\(47\!\cdots\!76\)\(q^{36} - \)\(16\!\cdots\!02\)\(q^{37} - \)\(91\!\cdots\!56\)\(q^{38} - \)\(18\!\cdots\!14\)\(q^{39} + \)\(75\!\cdots\!20\)\(q^{40} + \)\(22\!\cdots\!94\)\(q^{41} + \)\(89\!\cdots\!24\)\(q^{42} + \)\(67\!\cdots\!92\)\(q^{43} + \)\(10\!\cdots\!60\)\(q^{44} + \)\(35\!\cdots\!70\)\(q^{45} + \)\(21\!\cdots\!24\)\(q^{46} - \)\(46\!\cdots\!04\)\(q^{47} - \)\(13\!\cdots\!12\)\(q^{48} - \)\(11\!\cdots\!03\)\(q^{49} - \)\(66\!\cdots\!00\)\(q^{50} - \)\(13\!\cdots\!70\)\(q^{51} + \)\(46\!\cdots\!00\)\(q^{52} + \)\(16\!\cdots\!86\)\(q^{53} - \)\(31\!\cdots\!84\)\(q^{54} + \)\(69\!\cdots\!80\)\(q^{55} + \)\(26\!\cdots\!00\)\(q^{56} + \)\(14\!\cdots\!36\)\(q^{57} - \)\(39\!\cdots\!92\)\(q^{58} - \)\(39\!\cdots\!08\)\(q^{59} - \)\(12\!\cdots\!40\)\(q^{60} + \)\(72\!\cdots\!78\)\(q^{61} - \)\(99\!\cdots\!04\)\(q^{62} - \)\(31\!\cdots\!48\)\(q^{63} - \)\(20\!\cdots\!80\)\(q^{64} + \)\(13\!\cdots\!20\)\(q^{65} + \)\(49\!\cdots\!60\)\(q^{66} + \)\(66\!\cdots\!40\)\(q^{67} + \)\(12\!\cdots\!36\)\(q^{68} - \)\(21\!\cdots\!68\)\(q^{69} - \)\(29\!\cdots\!00\)\(q^{70} + \)\(18\!\cdots\!00\)\(q^{71} - \)\(38\!\cdots\!52\)\(q^{72} - \)\(31\!\cdots\!74\)\(q^{73} - \)\(90\!\cdots\!28\)\(q^{74} + \)\(98\!\cdots\!25\)\(q^{75} + \)\(17\!\cdots\!88\)\(q^{76} + \)\(10\!\cdots\!76\)\(q^{77} - \)\(81\!\cdots\!00\)\(q^{78} - \)\(27\!\cdots\!00\)\(q^{79} + \)\(17\!\cdots\!20\)\(q^{80} + \)\(31\!\cdots\!05\)\(q^{81} - \)\(21\!\cdots\!24\)\(q^{82} - \)\(11\!\cdots\!76\)\(q^{83} - \)\(28\!\cdots\!52\)\(q^{84} - \)\(35\!\cdots\!80\)\(q^{85} - \)\(59\!\cdots\!08\)\(q^{86} - \)\(18\!\cdots\!18\)\(q^{87} + \)\(88\!\cdots\!32\)\(q^{88} + \)\(92\!\cdots\!98\)\(q^{89} - \)\(77\!\cdots\!40\)\(q^{90} - \)\(80\!\cdots\!76\)\(q^{91} - \)\(13\!\cdots\!52\)\(q^{92} + \)\(37\!\cdots\!88\)\(q^{93} - \)\(13\!\cdots\!24\)\(q^{94} - \)\(61\!\cdots\!20\)\(q^{95} - \)\(32\!\cdots\!04\)\(q^{96} - \)\(14\!\cdots\!10\)\(q^{97} - \)\(16\!\cdots\!88\)\(q^{98} + \)\(33\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 11794880527205841026 x^{3} - 1576914464895109132006308600 x^{2} + 21982147283297468753929167380086357881 x - 10487125185557323126488817758602332290924256254\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 10 \)
\(\beta_{2}\)\(=\)\( 576 \nu^{2} - 115512328608 \nu - 2717540473422020841408 \)
\(\beta_{3}\)\(=\)\((\)\(627183 \nu^{4} + 387649018384857 \nu^{3} - 6901727976782003164396941 \nu^{2} - 5609906443063199591123859798720621 \nu + 8323431865157550665308972831447685029379122\)\()/ 200332900414259200 \)
\(\beta_{4}\)\(=\)\((\)\(89019 \nu^{4} + 254610810147501 \nu^{3} - 1196771911116752539811313 \nu^{2} - 2362874168895880655055146438671953 \nu + 2017174951889050070721069247792617717110746\)\()/ 14309492886732800 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 10\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 4813013692 \beta_{1} + 2717540473470150978328\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(15486 \beta_{4} - 30772 \beta_{3} + 408044456 \beta_{2} + 72607281339847992463 \beta_{1} + 204368117736682088695710785870\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(-229717936820656 \beta_{4} + 2112326721223712 \beta_{3} + 92986064678441917467 \beta_{2} + 1331661414327886771669041456900 \beta_{1} + 197313225711958020400085616512940566052648\)\()/5184\)

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
3.23970e9
9.01834e8
6.47992e8
−1.95252e9
−2.83700e9
−8.27630e10 5.00315e16 4.48853e21 −1.31164e24 −4.14076e27 −1.62984e30 −1.76066e32 2.50316e33 1.08555e35
1.2 −2.66543e10 5.00315e16 −1.65073e21 1.17208e25 −1.33355e27 −5.64464e29 1.06935e32 2.50316e33 −3.12409e35
1.3 −2.05621e10 5.00315e16 −1.93838e21 −6.43665e24 −1.02875e27 4.31786e29 8.84080e31 2.50316e33 1.32351e35
1.4 4.18503e10 5.00315e16 −6.09737e20 3.60424e24 2.09383e27 −2.88017e29 −1.24334e32 2.50316e33 1.50838e35
1.5 6.30778e10 5.00315e16 1.61762e21 −6.15414e24 3.15588e27 7.89316e29 −4.69021e31 2.50316e33 −3.88189e35
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 3.72.a.a 5
3.b odd 2 1 9.72.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.72.a.a 5 1.a even 1 1 trivial
9.72.a.a 5 3.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 25051277688 T_{2}^{4} - \)\(65\!\cdots\!60\)\( T_{2}^{3} - \)\(79\!\cdots\!72\)\( T_{2}^{2} + \)\(70\!\cdots\!68\)\( T_{2} + \)\(11\!\cdots\!68\)\( \) acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 25051277688 T + \)\(52\!\cdots\!80\)\( T^{2} + \)\(15\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!68\)\( T^{4} + \)\(58\!\cdots\!68\)\( T^{5} + \)\(38\!\cdots\!64\)\( T^{6} + \)\(87\!\cdots\!96\)\( T^{7} + \)\(69\!\cdots\!60\)\( T^{8} + \)\(77\!\cdots\!08\)\( T^{9} + \)\(73\!\cdots\!68\)\( T^{10} \)
$3$ \( ( 1 - 50031545098999707 T )^{5} \)
$5$ \( 1 - \)\(14\!\cdots\!30\)\( T + \)\(97\!\cdots\!25\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!50\)\( T^{6} - \)\(82\!\cdots\!00\)\( T^{7} + \)\(73\!\cdots\!25\)\( T^{8} - \)\(45\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 + \)\(12\!\cdots\!52\)\( T + \)\(38\!\cdots\!11\)\( T^{2} + \)\(42\!\cdots\!64\)\( T^{3} + \)\(68\!\cdots\!74\)\( T^{4} + \)\(61\!\cdots\!08\)\( T^{5} + \)\(68\!\cdots\!82\)\( T^{6} + \)\(43\!\cdots\!36\)\( T^{7} + \)\(39\!\cdots\!77\)\( T^{8} + \)\(12\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!43\)\( T^{10} \)
$11$ \( 1 - \)\(13\!\cdots\!12\)\( T + \)\(22\!\cdots\!39\)\( T^{2} - \)\(74\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!58\)\( T^{4} - \)\(77\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!38\)\( T^{6} - \)\(56\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!09\)\( T^{8} - \)\(77\!\cdots\!92\)\( T^{9} + \)\(49\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 + \)\(37\!\cdots\!02\)\( T + \)\(15\!\cdots\!09\)\( T^{2} + \)\(27\!\cdots\!96\)\( T^{3} + \)\(68\!\cdots\!10\)\( T^{4} - \)\(43\!\cdots\!68\)\( T^{5} + \)\(84\!\cdots\!70\)\( T^{6} + \)\(41\!\cdots\!24\)\( T^{7} + \)\(28\!\cdots\!77\)\( T^{8} + \)\(85\!\cdots\!22\)\( T^{9} + \)\(28\!\cdots\!57\)\( T^{10} \)
$17$ \( 1 + \)\(27\!\cdots\!10\)\( T + \)\(95\!\cdots\!45\)\( T^{2} - \)\(75\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!64\)\( T^{5} + \)\(92\!\cdots\!30\)\( T^{6} - \)\(40\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!65\)\( T^{8} + \)\(77\!\cdots\!10\)\( T^{9} + \)\(64\!\cdots\!93\)\( T^{10} \)
$19$ \( 1 - \)\(29\!\cdots\!48\)\( T + \)\(23\!\cdots\!23\)\( T^{2} - \)\(50\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!66\)\( T^{4} - \)\(43\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!54\)\( T^{6} - \)\(19\!\cdots\!04\)\( T^{7} + \)\(55\!\cdots\!57\)\( T^{8} - \)\(43\!\cdots\!08\)\( T^{9} + \)\(90\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + \)\(42\!\cdots\!24\)\( T + \)\(18\!\cdots\!39\)\( T^{2} + \)\(47\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!14\)\( T^{4} + \)\(31\!\cdots\!56\)\( T^{5} + \)\(73\!\cdots\!78\)\( T^{6} + \)\(11\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!37\)\( T^{8} + \)\(22\!\cdots\!84\)\( T^{9} + \)\(25\!\cdots\!07\)\( T^{10} \)
$29$ \( 1 + \)\(36\!\cdots\!74\)\( T + \)\(21\!\cdots\!37\)\( T^{2} + \)\(39\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!14\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!06\)\( T^{6} + \)\(17\!\cdots\!36\)\( T^{7} + \)\(67\!\cdots\!93\)\( T^{8} + \)\(77\!\cdots\!94\)\( T^{9} + \)\(14\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 - \)\(74\!\cdots\!84\)\( T + \)\(17\!\cdots\!19\)\( T^{2} - \)\(17\!\cdots\!16\)\( T^{3} + \)\(20\!\cdots\!62\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(16\!\cdots\!22\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} + \)\(81\!\cdots\!29\)\( T^{8} - \)\(26\!\cdots\!64\)\( T^{9} + \)\(27\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + \)\(16\!\cdots\!02\)\( T + \)\(15\!\cdots\!01\)\( T^{2} + \)\(99\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!94\)\( T^{4} + \)\(25\!\cdots\!88\)\( T^{5} + \)\(11\!\cdots\!22\)\( T^{6} + \)\(48\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!97\)\( T^{8} + \)\(37\!\cdots\!22\)\( T^{9} + \)\(51\!\cdots\!93\)\( T^{10} \)
$41$ \( 1 - \)\(22\!\cdots\!94\)\( T + \)\(12\!\cdots\!09\)\( T^{2} - \)\(26\!\cdots\!56\)\( T^{3} + \)\(68\!\cdots\!02\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{5} + \)\(22\!\cdots\!82\)\( T^{6} - \)\(27\!\cdots\!36\)\( T^{7} + \)\(40\!\cdots\!89\)\( T^{8} - \)\(24\!\cdots\!34\)\( T^{9} + \)\(34\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(67\!\cdots\!92\)\( T + \)\(41\!\cdots\!27\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!38\)\( T^{4} - \)\(34\!\cdots\!16\)\( T^{5} + \)\(69\!\cdots\!66\)\( T^{6} - \)\(22\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!61\)\( T^{8} - \)\(54\!\cdots\!92\)\( T^{9} + \)\(76\!\cdots\!07\)\( T^{10} \)
$47$ \( 1 + \)\(46\!\cdots\!04\)\( T + \)\(20\!\cdots\!23\)\( T^{2} + \)\(56\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!78\)\( T^{4} + \)\(39\!\cdots\!72\)\( T^{5} + \)\(93\!\cdots\!34\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(29\!\cdots\!21\)\( T^{8} + \)\(35\!\cdots\!24\)\( T^{9} + \)\(39\!\cdots\!43\)\( T^{10} \)
$53$ \( 1 - \)\(16\!\cdots\!86\)\( T + \)\(31\!\cdots\!49\)\( T^{2} - \)\(18\!\cdots\!88\)\( T^{3} + \)\(62\!\cdots\!14\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!58\)\( T^{6} - \)\(12\!\cdots\!92\)\( T^{7} + \)\(59\!\cdots\!77\)\( T^{8} - \)\(83\!\cdots\!66\)\( T^{9} + \)\(13\!\cdots\!57\)\( T^{10} \)
$59$ \( 1 + \)\(39\!\cdots\!08\)\( T + \)\(23\!\cdots\!43\)\( T^{2} + \)\(69\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!66\)\( T^{4} + \)\(52\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!94\)\( T^{6} + \)\(20\!\cdots\!84\)\( T^{7} + \)\(37\!\cdots\!97\)\( T^{8} + \)\(32\!\cdots\!88\)\( T^{9} + \)\(44\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 - \)\(72\!\cdots\!78\)\( T + \)\(94\!\cdots\!77\)\( T^{2} + \)\(37\!\cdots\!52\)\( T^{3} + \)\(58\!\cdots\!82\)\( T^{4} + \)\(36\!\cdots\!32\)\( T^{5} + \)\(33\!\cdots\!02\)\( T^{6} + \)\(12\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!37\)\( T^{8} - \)\(78\!\cdots\!98\)\( T^{9} + \)\(61\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 - \)\(66\!\cdots\!40\)\( T + \)\(12\!\cdots\!95\)\( T^{2} - \)\(91\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!10\)\( T^{4} - \)\(57\!\cdots\!64\)\( T^{5} + \)\(41\!\cdots\!30\)\( T^{6} - \)\(18\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!65\)\( T^{8} - \)\(26\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!43\)\( T^{10} \)
$71$ \( 1 - \)\(18\!\cdots\!00\)\( T + \)\(11\!\cdots\!15\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!50\)\( T^{4} - \)\(74\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!50\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!65\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 + \)\(31\!\cdots\!74\)\( T + \)\(95\!\cdots\!29\)\( T^{2} + \)\(15\!\cdots\!92\)\( T^{3} + \)\(27\!\cdots\!54\)\( T^{4} + \)\(32\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!58\)\( T^{6} + \)\(59\!\cdots\!68\)\( T^{7} + \)\(73\!\cdots\!57\)\( T^{8} + \)\(47\!\cdots\!34\)\( T^{9} + \)\(30\!\cdots\!57\)\( T^{10} \)
$79$ \( 1 + \)\(27\!\cdots\!00\)\( T + \)\(14\!\cdots\!95\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(77\!\cdots\!10\)\( T^{4} + \)\(47\!\cdots\!00\)\( T^{5} + \)\(41\!\cdots\!90\)\( T^{6} + \)\(51\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!05\)\( T^{8} + \)\(23\!\cdots\!00\)\( T^{9} + \)\(45\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 + \)\(11\!\cdots\!76\)\( T + \)\(34\!\cdots\!75\)\( T^{2} + \)\(56\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!78\)\( T^{4} + \)\(10\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!26\)\( T^{6} + \)\(18\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!25\)\( T^{8} + \)\(12\!\cdots\!96\)\( T^{9} + \)\(18\!\cdots\!07\)\( T^{10} \)
$89$ \( 1 - \)\(92\!\cdots\!98\)\( T + \)\(69\!\cdots\!53\)\( T^{2} - \)\(74\!\cdots\!64\)\( T^{3} + \)\(30\!\cdots\!86\)\( T^{4} - \)\(23\!\cdots\!76\)\( T^{5} + \)\(76\!\cdots\!54\)\( T^{6} - \)\(48\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!57\)\( T^{8} - \)\(39\!\cdots\!18\)\( T^{9} + \)\(10\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + \)\(14\!\cdots\!10\)\( T + \)\(10\!\cdots\!45\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!10\)\( T^{4} + \)\(44\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!30\)\( T^{6} + \)\(61\!\cdots\!40\)\( T^{7} + \)\(16\!\cdots\!65\)\( T^{8} + \)\(26\!\cdots\!10\)\( T^{9} + \)\(20\!\cdots\!93\)\( T^{10} \)
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