Properties

Label 3.70.a.a
Level 3
Weight 70
Character orbit 3.a
Self dual yes
Analytic conductor 90.454
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 70 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(90.4544859877\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 1895395655112138158 x^{4} - 215343254036998938223429254 x^{3} + 753811615313247966666178630793780821 x^{2} + 48156657138617542519489631631312055499331993 x - 14832095927450491582029231424734280560496715550059928\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{51}\cdot 3^{33}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 17\cdot 23^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-144893898 + \beta_{1}) q^{2} -16677181699666569 q^{3} +(\)\(22\!\cdots\!68\)\( + 5845361818 \beta_{1} + \beta_{2}) q^{4} +(\)\(89\!\cdots\!70\)\( - 6200297883215 \beta_{1} - 7 \beta_{2} - \beta_{3}) q^{5} +(\)\(24\!\cdots\!62\)\( - 16677181699666569 \beta_{1}) q^{6} +(-\)\(27\!\cdots\!36\)\( + 1433095886119221334 \beta_{1} + 42214965 \beta_{2} - 29780 \beta_{3} - 6 \beta_{4} + \beta_{5}) q^{7} +(\)\(48\!\cdots\!16\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} + 14298139570 \beta_{2} + 2447652 \beta_{3} - 130 \beta_{4} - 422 \beta_{5}) q^{8} +\)\(27\!\cdots\!61\)\( q^{9} +O(q^{10})\) \( q +(-144893898 + \beta_{1}) q^{2} -16677181699666569 q^{3} +(\)\(22\!\cdots\!68\)\( + 5845361818 \beta_{1} + \beta_{2}) q^{4} +(\)\(89\!\cdots\!70\)\( - 6200297883215 \beta_{1} - 7 \beta_{2} - \beta_{3}) q^{5} +(\)\(24\!\cdots\!62\)\( - 16677181699666569 \beta_{1}) q^{6} +(-\)\(27\!\cdots\!36\)\( + 1433095886119221334 \beta_{1} + 42214965 \beta_{2} - 29780 \beta_{3} - 6 \beta_{4} + \beta_{5}) q^{7} +(\)\(48\!\cdots\!16\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} + 14298139570 \beta_{2} + 2447652 \beta_{3} - 130 \beta_{4} - 422 \beta_{5}) q^{8} +\)\(27\!\cdots\!61\)\( q^{9} +(-\)\(50\!\cdots\!40\)\( + \)\(48\!\cdots\!66\)\( \beta_{1} - 16131015268001 \beta_{2} + 2317118146 \beta_{3} - 430521 \beta_{4} - 279803 \beta_{5}) q^{10} +(-\)\(32\!\cdots\!96\)\( - \)\(11\!\cdots\!74\)\( \beta_{1} + 362326162064878 \beta_{2} - 184110955006 \beta_{3} + 30619360 \beta_{4} + 29235296 \beta_{5}) q^{11} +(-\)\(38\!\cdots\!92\)\( - \)\(97\!\cdots\!42\)\( \beta_{1} - 16677181699666569 \beta_{2}) q^{12} +(-\)\(78\!\cdots\!42\)\( - \)\(33\!\cdots\!77\)\( \beta_{1} - 263119949343678772 \beta_{2} + 41676591338315 \beta_{3} + 1667157702 \beta_{4} + 2939229503 \beta_{5}) q^{13} +(\)\(11\!\cdots\!52\)\( + \)\(50\!\cdots\!12\)\( \beta_{1} + 1449256065877874749 \beta_{2} - 460927730575354 \beta_{3} + 178861238773 \beta_{4} - 9605478289 \beta_{5}) q^{14} +(-\)\(14\!\cdots\!30\)\( + \)\(10\!\cdots\!35\)\( \beta_{1} + 116740271897665983 \beta_{2} + 16677181699666569 \beta_{3}) q^{15} +(\)\(62\!\cdots\!80\)\( + \)\(10\!\cdots\!80\)\( \beta_{1} + 36802971003154347592 \beta_{2} - 17572749416984176 \beta_{3} + 656456568 \beta_{4} - 13291167282712 \beta_{5}) q^{16} +(-\)\(84\!\cdots\!02\)\( - \)\(97\!\cdots\!56\)\( \beta_{1} - \)\(26\!\cdots\!22\)\( \beta_{2} + 536135679791576828 \beta_{3} - 41796688444476 \beta_{4} + 116930963053338 \beta_{5}) q^{17} +(-\)\(40\!\cdots\!78\)\( + \)\(27\!\cdots\!61\)\( \beta_{1}) q^{18} +(-\)\(10\!\cdots\!48\)\( + \)\(42\!\cdots\!84\)\( \beta_{1} - \)\(40\!\cdots\!06\)\( \beta_{2} + 79485256719651864340 \beta_{3} - 1785277348280772 \beta_{4} - 2147665016282138 \beta_{5}) q^{19} +(-\)\(12\!\cdots\!60\)\( - \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(12\!\cdots\!66\)\( \beta_{2} + \)\(47\!\cdots\!84\)\( \beta_{3} + 14497354998977856 \beta_{4} + 2071155722294208 \beta_{5}) q^{20} +(\)\(45\!\cdots\!84\)\( - \)\(23\!\cdots\!46\)\( \beta_{1} - \)\(70\!\cdots\!85\)\( \beta_{2} + \)\(49\!\cdots\!20\)\( \beta_{3} + 100063090197999414 \beta_{4} - 16677181699666569 \beta_{5}) q^{21} +(-\)\(95\!\cdots\!44\)\( - \)\(18\!\cdots\!20\)\( \beta_{1} - \)\(28\!\cdots\!14\)\( \beta_{2} + \)\(71\!\cdots\!84\)\( \beta_{3} - 864370701359578254 \beta_{4} + 353325411726666230 \beta_{5}) q^{22} +(-\)\(85\!\cdots\!48\)\( + \)\(17\!\cdots\!96\)\( \beta_{1} - \)\(11\!\cdots\!26\)\( \beta_{2} - \)\(44\!\cdots\!88\)\( \beta_{3} + 865228787664610724 \beta_{4} - 1616075419168768646 \beta_{5}) q^{23} +(-\)\(80\!\cdots\!04\)\( - \)\(40\!\cdots\!08\)\( \beta_{1} - \)\(23\!\cdots\!30\)\( \beta_{2} - \)\(40\!\cdots\!88\)\( \beta_{3} + 2168033620956653970 \beta_{4} + 7037770677259292118 \beta_{5}) q^{24} +(\)\(22\!\cdots\!75\)\( - \)\(57\!\cdots\!90\)\( \beta_{1} - \)\(26\!\cdots\!40\)\( \beta_{2} - \)\(50\!\cdots\!30\)\( \beta_{3} + 7422034455464443140 \beta_{4} - 49714601211021570230 \beta_{5}) q^{25} +(-\)\(27\!\cdots\!48\)\( - \)\(17\!\cdots\!18\)\( \beta_{1} - \)\(68\!\cdots\!86\)\( \beta_{2} - \)\(32\!\cdots\!56\)\( \beta_{3} + 22548569954006357514 \beta_{4} + \)\(17\!\cdots\!82\)\( \beta_{5}) q^{26} -\)\(46\!\cdots\!09\)\( q^{27} +(\)\(20\!\cdots\!48\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} - \)\(21\!\cdots\!04\)\( \beta_{2} + \)\(42\!\cdots\!92\)\( \beta_{3} - \)\(25\!\cdots\!48\)\( \beta_{4} - \)\(14\!\cdots\!24\)\( \beta_{5}) q^{28} +(\)\(13\!\cdots\!54\)\( + \)\(22\!\cdots\!89\)\( \beta_{1} + \)\(21\!\cdots\!27\)\( \beta_{2} + \)\(32\!\cdots\!75\)\( \beta_{3} + \)\(78\!\cdots\!08\)\( \beta_{4} + \)\(16\!\cdots\!22\)\( \beta_{5}) q^{29} +(\)\(84\!\cdots\!60\)\( - \)\(80\!\cdots\!54\)\( \beta_{1} + \)\(26\!\cdots\!69\)\( \beta_{2} - \)\(38\!\cdots\!74\)\( \beta_{3} + \)\(71\!\cdots\!49\)\( \beta_{4} + \)\(46\!\cdots\!07\)\( \beta_{5}) q^{30} +(\)\(11\!\cdots\!16\)\( - \)\(11\!\cdots\!70\)\( \beta_{1} + \)\(12\!\cdots\!33\)\( \beta_{2} - \)\(91\!\cdots\!52\)\( \beta_{3} - \)\(24\!\cdots\!62\)\( \beta_{4} + \)\(68\!\cdots\!69\)\( \beta_{5}) q^{31} +(\)\(61\!\cdots\!64\)\( + \)\(64\!\cdots\!16\)\( \beta_{1} + \)\(98\!\cdots\!88\)\( \beta_{2} - \)\(23\!\cdots\!20\)\( \beta_{3} - \)\(39\!\cdots\!68\)\( \beta_{4} - \)\(20\!\cdots\!92\)\( \beta_{5}) q^{32} +(\)\(53\!\cdots\!24\)\( + \)\(19\!\cdots\!06\)\( \beta_{1} - \)\(60\!\cdots\!82\)\( \beta_{2} + \)\(30\!\cdots\!14\)\( \beta_{3} - \)\(51\!\cdots\!40\)\( \beta_{4} - \)\(48\!\cdots\!24\)\( \beta_{5}) q^{33} +(\)\(42\!\cdots\!96\)\( - \)\(99\!\cdots\!18\)\( \beta_{1} - \)\(57\!\cdots\!62\)\( \beta_{2} + \)\(19\!\cdots\!08\)\( \beta_{3} + \)\(24\!\cdots\!30\)\( \beta_{4} + \)\(24\!\cdots\!62\)\( \beta_{5}) q^{34} +(\)\(46\!\cdots\!60\)\( + \)\(11\!\cdots\!34\)\( \beta_{1} - \)\(21\!\cdots\!46\)\( \beta_{2} + \)\(87\!\cdots\!18\)\( \beta_{3} - \)\(13\!\cdots\!44\)\( \beta_{4} - \)\(40\!\cdots\!92\)\( \beta_{5}) q^{35} +(\)\(63\!\cdots\!48\)\( + \)\(16\!\cdots\!98\)\( \beta_{1} + \)\(27\!\cdots\!61\)\( \beta_{2}) q^{36} +(-\)\(12\!\cdots\!66\)\( - \)\(13\!\cdots\!45\)\( \beta_{1} - \)\(89\!\cdots\!98\)\( \beta_{2} - \)\(30\!\cdots\!53\)\( \beta_{3} - \)\(34\!\cdots\!18\)\( \beta_{4} + \)\(69\!\cdots\!41\)\( \beta_{5}) q^{37} +(\)\(35\!\cdots\!72\)\( - \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(21\!\cdots\!42\)\( \beta_{2} - \)\(65\!\cdots\!20\)\( \beta_{3} + \)\(79\!\cdots\!06\)\( \beta_{4} - \)\(11\!\cdots\!26\)\( \beta_{5}) q^{38} +(\)\(13\!\cdots\!98\)\( + \)\(55\!\cdots\!13\)\( \beta_{1} + \)\(43\!\cdots\!68\)\( \beta_{2} - \)\(69\!\cdots\!35\)\( \beta_{3} - \)\(27\!\cdots\!38\)\( \beta_{4} - \)\(49\!\cdots\!07\)\( \beta_{5}) q^{39} +(-\)\(54\!\cdots\!00\)\( - \)\(30\!\cdots\!16\)\( \beta_{1} + \)\(26\!\cdots\!00\)\( \beta_{2} - \)\(29\!\cdots\!84\)\( \beta_{3} + \)\(66\!\cdots\!76\)\( \beta_{4} + \)\(28\!\cdots\!68\)\( \beta_{5}) q^{40} +(-\)\(42\!\cdots\!54\)\( + \)\(68\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!82\)\( \beta_{2} + \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(71\!\cdots\!16\)\( \beta_{4} - \)\(70\!\cdots\!42\)\( \beta_{5}) q^{41} +(-\)\(19\!\cdots\!88\)\( - \)\(83\!\cdots\!28\)\( \beta_{1} - \)\(24\!\cdots\!81\)\( \beta_{2} + \)\(76\!\cdots\!26\)\( \beta_{3} - \)\(29\!\cdots\!37\)\( \beta_{4} + \)\(16\!\cdots\!41\)\( \beta_{5}) q^{42} +(-\)\(11\!\cdots\!40\)\( - \)\(47\!\cdots\!36\)\( \beta_{1} - \)\(68\!\cdots\!26\)\( \beta_{2} - \)\(18\!\cdots\!04\)\( \beta_{3} - \)\(20\!\cdots\!20\)\( \beta_{4} + \)\(81\!\cdots\!74\)\( \beta_{5}) q^{43} +(\)\(39\!\cdots\!64\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} - \)\(69\!\cdots\!80\)\( \beta_{2} - \)\(54\!\cdots\!04\)\( \beta_{3} + \)\(18\!\cdots\!88\)\( \beta_{4} + \)\(32\!\cdots\!96\)\( \beta_{5}) q^{44} +(\)\(24\!\cdots\!70\)\( - \)\(17\!\cdots\!15\)\( \beta_{1} - \)\(19\!\cdots\!27\)\( \beta_{2} - \)\(27\!\cdots\!61\)\( \beta_{3}) q^{45} +(\)\(13\!\cdots\!16\)\( - \)\(50\!\cdots\!88\)\( \beta_{1} + \)\(19\!\cdots\!26\)\( \beta_{2} + \)\(56\!\cdots\!68\)\( \beta_{3} - \)\(57\!\cdots\!74\)\( \beta_{4} - \)\(37\!\cdots\!34\)\( \beta_{5}) q^{46} +(\)\(12\!\cdots\!20\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} + \)\(28\!\cdots\!54\)\( \beta_{2} + \)\(52\!\cdots\!24\)\( \beta_{3} + \)\(14\!\cdots\!72\)\( \beta_{4} + \)\(59\!\cdots\!74\)\( \beta_{5}) q^{47} +(-\)\(10\!\cdots\!20\)\( - \)\(18\!\cdots\!20\)\( \beta_{1} - \)\(61\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!92\)\( \beta_{4} + \)\(22\!\cdots\!28\)\( \beta_{5}) q^{48} +(\)\(65\!\cdots\!65\)\( - \)\(73\!\cdots\!82\)\( \beta_{1} + \)\(58\!\cdots\!56\)\( \beta_{2} + \)\(81\!\cdots\!58\)\( \beta_{3} + \)\(29\!\cdots\!52\)\( \beta_{4} - \)\(21\!\cdots\!90\)\( \beta_{5}) q^{49} +(-\)\(47\!\cdots\!50\)\( - \)\(13\!\cdots\!45\)\( \beta_{1} - \)\(70\!\cdots\!20\)\( \beta_{2} - \)\(82\!\cdots\!40\)\( \beta_{3} - \)\(49\!\cdots\!80\)\( \beta_{4} + \)\(61\!\cdots\!60\)\( \beta_{5}) q^{50} +(\)\(14\!\cdots\!38\)\( + \)\(16\!\cdots\!64\)\( \beta_{1} + \)\(43\!\cdots\!18\)\( \beta_{2} - \)\(89\!\cdots\!32\)\( \beta_{3} + \)\(69\!\cdots\!44\)\( \beta_{4} - \)\(19\!\cdots\!22\)\( \beta_{5}) q^{51} +(-\)\(14\!\cdots\!12\)\( - \)\(57\!\cdots\!20\)\( \beta_{1} - \)\(18\!\cdots\!82\)\( \beta_{2} - \)\(24\!\cdots\!08\)\( \beta_{3} + \)\(44\!\cdots\!80\)\( \beta_{4} + \)\(44\!\cdots\!28\)\( \beta_{5}) q^{52} +(-\)\(74\!\cdots\!38\)\( - \)\(56\!\cdots\!63\)\( \beta_{1} + \)\(12\!\cdots\!99\)\( \beta_{2} - \)\(68\!\cdots\!89\)\( \beta_{3} - \)\(62\!\cdots\!84\)\( \beta_{4} - \)\(90\!\cdots\!42\)\( \beta_{5}) q^{53} +(\)\(67\!\cdots\!82\)\( - \)\(46\!\cdots\!09\)\( \beta_{1}) q^{54} +(\)\(37\!\cdots\!40\)\( - \)\(16\!\cdots\!12\)\( \beta_{1} + \)\(43\!\cdots\!76\)\( \beta_{2} + \)\(86\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!52\)\( \beta_{4} - \)\(10\!\cdots\!64\)\( \beta_{5}) q^{55} +(\)\(27\!\cdots\!60\)\( + \)\(11\!\cdots\!72\)\( \beta_{1} + \)\(15\!\cdots\!36\)\( \beta_{2} - \)\(29\!\cdots\!68\)\( \beta_{3} - \)\(14\!\cdots\!24\)\( \beta_{4} - \)\(11\!\cdots\!48\)\( \beta_{5}) q^{56} +(\)\(18\!\cdots\!12\)\( - \)\(71\!\cdots\!96\)\( \beta_{1} + \)\(68\!\cdots\!14\)\( \beta_{2} - \)\(13\!\cdots\!60\)\( \beta_{3} + \)\(29\!\cdots\!68\)\( \beta_{4} + \)\(35\!\cdots\!22\)\( \beta_{5}) q^{57} +(\)\(18\!\cdots\!04\)\( + \)\(26\!\cdots\!82\)\( \beta_{1} + \)\(28\!\cdots\!73\)\( \beta_{2} + \)\(15\!\cdots\!86\)\( \beta_{3} - \)\(24\!\cdots\!59\)\( \beta_{4} - \)\(50\!\cdots\!17\)\( \beta_{5}) q^{58} +(\)\(16\!\cdots\!88\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} - \)\(17\!\cdots\!96\)\( \beta_{2} - \)\(45\!\cdots\!12\)\( \beta_{3} + \)\(34\!\cdots\!96\)\( \beta_{4} + \)\(96\!\cdots\!76\)\( \beta_{5}) q^{59} +(\)\(20\!\cdots\!40\)\( + \)\(17\!\cdots\!44\)\( \beta_{1} - \)\(21\!\cdots\!54\)\( \beta_{2} - \)\(78\!\cdots\!96\)\( \beta_{3} - \)\(24\!\cdots\!64\)\( \beta_{4} - \)\(34\!\cdots\!52\)\( \beta_{5}) q^{60} +(-\)\(98\!\cdots\!30\)\( + \)\(60\!\cdots\!11\)\( \beta_{1} + \)\(21\!\cdots\!26\)\( \beta_{2} + \)\(27\!\cdots\!75\)\( \beta_{3} + \)\(56\!\cdots\!54\)\( \beta_{4} - \)\(59\!\cdots\!39\)\( \beta_{5}) q^{61} +(-\)\(97\!\cdots\!52\)\( + \)\(17\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!31\)\( \beta_{2} + \)\(27\!\cdots\!14\)\( \beta_{3} + \)\(28\!\cdots\!93\)\( \beta_{4} - \)\(66\!\cdots\!77\)\( \beta_{5}) q^{62} +(-\)\(76\!\cdots\!96\)\( + \)\(39\!\cdots\!74\)\( \beta_{1} + \)\(11\!\cdots\!65\)\( \beta_{2} - \)\(82\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!66\)\( \beta_{4} + \)\(27\!\cdots\!61\)\( \beta_{5}) q^{63} +(-\)\(32\!\cdots\!28\)\( + \)\(53\!\cdots\!36\)\( \beta_{1} + \)\(59\!\cdots\!88\)\( \beta_{2} + \)\(33\!\cdots\!76\)\( \beta_{3} - \)\(11\!\cdots\!32\)\( \beta_{4} + \)\(26\!\cdots\!36\)\( \beta_{5}) q^{64} +(-\)\(61\!\cdots\!20\)\( + \)\(15\!\cdots\!36\)\( \beta_{1} + \)\(14\!\cdots\!02\)\( \beta_{2} - \)\(11\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!42\)\( \beta_{5}) q^{65} +(\)\(15\!\cdots\!36\)\( + \)\(30\!\cdots\!80\)\( \beta_{1} + \)\(48\!\cdots\!66\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} + \)\(14\!\cdots\!26\)\( \beta_{4} - \)\(58\!\cdots\!70\)\( \beta_{5}) q^{66} +(\)\(21\!\cdots\!08\)\( - \)\(22\!\cdots\!84\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(13\!\cdots\!64\)\( \beta_{3} + \)\(23\!\cdots\!40\)\( \beta_{4} - \)\(90\!\cdots\!04\)\( \beta_{5}) q^{67} +(-\)\(31\!\cdots\!36\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} - \)\(28\!\cdots\!50\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3} - \)\(26\!\cdots\!56\)\( \beta_{4} + \)\(21\!\cdots\!20\)\( \beta_{5}) q^{68} +(\)\(14\!\cdots\!12\)\( - \)\(28\!\cdots\!24\)\( \beta_{1} + \)\(19\!\cdots\!94\)\( \beta_{2} + \)\(74\!\cdots\!72\)\( \beta_{3} - \)\(14\!\cdots\!56\)\( \beta_{4} + \)\(26\!\cdots\!74\)\( \beta_{5}) q^{69} +(\)\(94\!\cdots\!80\)\( - \)\(69\!\cdots\!68\)\( \beta_{1} + \)\(25\!\cdots\!22\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} + \)\(75\!\cdots\!38\)\( \beta_{4} + \)\(38\!\cdots\!34\)\( \beta_{5}) q^{70} +(-\)\(13\!\cdots\!28\)\( - \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(56\!\cdots\!94\)\( \beta_{2} + \)\(25\!\cdots\!76\)\( \beta_{3} - \)\(65\!\cdots\!56\)\( \beta_{4} - \)\(25\!\cdots\!30\)\( \beta_{5}) q^{71} +(\)\(13\!\cdots\!76\)\( + \)\(67\!\cdots\!52\)\( \beta_{1} + \)\(39\!\cdots\!70\)\( \beta_{2} + \)\(68\!\cdots\!72\)\( \beta_{3} - \)\(36\!\cdots\!30\)\( \beta_{4} - \)\(11\!\cdots\!42\)\( \beta_{5}) q^{72} +(\)\(39\!\cdots\!02\)\( - \)\(50\!\cdots\!82\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} - \)\(69\!\cdots\!66\)\( \beta_{3} - \)\(24\!\cdots\!96\)\( \beta_{4} - \)\(28\!\cdots\!98\)\( \beta_{5}) q^{73} +(-\)\(11\!\cdots\!08\)\( - \)\(71\!\cdots\!30\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2} - \)\(50\!\cdots\!24\)\( \beta_{3} + \)\(11\!\cdots\!52\)\( \beta_{4} + \)\(40\!\cdots\!92\)\( \beta_{5}) q^{74} +(-\)\(37\!\cdots\!75\)\( + \)\(96\!\cdots\!10\)\( \beta_{1} + \)\(44\!\cdots\!60\)\( \beta_{2} + \)\(84\!\cdots\!70\)\( \beta_{3} - \)\(12\!\cdots\!60\)\( \beta_{4} + \)\(82\!\cdots\!70\)\( \beta_{5}) q^{75} +(-\)\(19\!\cdots\!92\)\( + \)\(11\!\cdots\!92\)\( \beta_{1} - \)\(15\!\cdots\!28\)\( \beta_{2} - \)\(30\!\cdots\!12\)\( \beta_{3} - \)\(20\!\cdots\!44\)\( \beta_{4} + \)\(16\!\cdots\!16\)\( \beta_{5}) q^{76} +(-\)\(61\!\cdots\!92\)\( + \)\(30\!\cdots\!68\)\( \beta_{1} - \)\(21\!\cdots\!68\)\( \beta_{2} + \)\(83\!\cdots\!08\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} - \)\(23\!\cdots\!08\)\( \beta_{5}) q^{77} +(\)\(45\!\cdots\!12\)\( + \)\(29\!\cdots\!42\)\( \beta_{1} + \)\(11\!\cdots\!34\)\( \beta_{2} + \)\(54\!\cdots\!64\)\( \beta_{3} - \)\(37\!\cdots\!66\)\( \beta_{4} - \)\(29\!\cdots\!58\)\( \beta_{5}) q^{78} +(\)\(25\!\cdots\!40\)\( + \)\(39\!\cdots\!86\)\( \beta_{1} + \)\(40\!\cdots\!89\)\( \beta_{2} + \)\(51\!\cdots\!60\)\( \beta_{3} + \)\(12\!\cdots\!86\)\( \beta_{4} + \)\(63\!\cdots\!89\)\( \beta_{5}) q^{79} +(-\)\(17\!\cdots\!80\)\( + \)\(19\!\cdots\!76\)\( \beta_{1} - \)\(24\!\cdots\!92\)\( \beta_{2} - \)\(25\!\cdots\!72\)\( \beta_{3} - \)\(67\!\cdots\!76\)\( \beta_{4} + \)\(30\!\cdots\!32\)\( \beta_{5}) q^{80} +\)\(77\!\cdots\!21\)\( q^{81} +(\)\(62\!\cdots\!08\)\( + \)\(77\!\cdots\!50\)\( \beta_{1} + \)\(68\!\cdots\!78\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(22\!\cdots\!54\)\( \beta_{4} - \)\(16\!\cdots\!86\)\( \beta_{5}) q^{82} +(-\)\(36\!\cdots\!84\)\( + \)\(11\!\cdots\!46\)\( \beta_{1} + \)\(10\!\cdots\!90\)\( \beta_{2} - \)\(13\!\cdots\!98\)\( \beta_{3} - \)\(75\!\cdots\!48\)\( \beta_{4} + \)\(65\!\cdots\!16\)\( \beta_{5}) q^{83} +(-\)\(33\!\cdots\!12\)\( - \)\(19\!\cdots\!60\)\( \beta_{1} + \)\(36\!\cdots\!76\)\( \beta_{2} - \)\(71\!\cdots\!48\)\( \beta_{3} + \)\(42\!\cdots\!12\)\( \beta_{4} + \)\(24\!\cdots\!56\)\( \beta_{5}) q^{84} +(-\)\(10\!\cdots\!40\)\( + \)\(18\!\cdots\!70\)\( \beta_{1} + \)\(82\!\cdots\!94\)\( \beta_{2} + \)\(30\!\cdots\!02\)\( \beta_{3} + \)\(49\!\cdots\!60\)\( \beta_{4} - \)\(46\!\cdots\!20\)\( \beta_{5}) q^{85} +(-\)\(38\!\cdots\!76\)\( - \)\(18\!\cdots\!00\)\( \beta_{1} - \)\(57\!\cdots\!02\)\( \beta_{2} - \)\(30\!\cdots\!40\)\( \beta_{3} + \)\(73\!\cdots\!34\)\( \beta_{4} + \)\(38\!\cdots\!46\)\( \beta_{5}) q^{86} +(-\)\(21\!\cdots\!26\)\( - \)\(37\!\cdots\!41\)\( \beta_{1} - \)\(36\!\cdots\!63\)\( \beta_{2} - \)\(54\!\cdots\!75\)\( \beta_{3} - \)\(13\!\cdots\!52\)\( \beta_{4} - \)\(26\!\cdots\!18\)\( \beta_{5}) q^{87} +(-\)\(10\!\cdots\!64\)\( - \)\(36\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!52\)\( \beta_{2} - \)\(33\!\cdots\!24\)\( \beta_{3} + \)\(90\!\cdots\!28\)\( \beta_{4} + \)\(13\!\cdots\!76\)\( \beta_{5}) q^{88} +(-\)\(11\!\cdots\!38\)\( - \)\(10\!\cdots\!28\)\( \beta_{1} + \)\(27\!\cdots\!12\)\( \beta_{2} - \)\(17\!\cdots\!76\)\( \beta_{3} - \)\(67\!\cdots\!44\)\( \beta_{4} - \)\(21\!\cdots\!20\)\( \beta_{5}) q^{89} +(-\)\(14\!\cdots\!40\)\( + \)\(13\!\cdots\!26\)\( \beta_{1} - \)\(44\!\cdots\!61\)\( \beta_{2} + \)\(64\!\cdots\!06\)\( \beta_{3} - \)\(11\!\cdots\!81\)\( \beta_{4} - \)\(77\!\cdots\!83\)\( \beta_{5}) q^{90} +(-\)\(13\!\cdots\!24\)\( - \)\(15\!\cdots\!60\)\( \beta_{1} + \)\(48\!\cdots\!74\)\( \beta_{2} - \)\(11\!\cdots\!68\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4} + \)\(55\!\cdots\!98\)\( \beta_{5}) q^{91} +(\)\(72\!\cdots\!84\)\( + \)\(14\!\cdots\!52\)\( \beta_{1} + \)\(44\!\cdots\!24\)\( \beta_{2} + \)\(21\!\cdots\!48\)\( \beta_{3} + \)\(78\!\cdots\!56\)\( \beta_{4} - \)\(58\!\cdots\!44\)\( \beta_{5}) q^{92} +(-\)\(19\!\cdots\!04\)\( + \)\(19\!\cdots\!30\)\( \beta_{1} - \)\(20\!\cdots\!77\)\( \beta_{2} + \)\(15\!\cdots\!88\)\( \beta_{3} + \)\(41\!\cdots\!78\)\( \beta_{4} - \)\(11\!\cdots\!61\)\( \beta_{5}) q^{93} +(-\)\(93\!\cdots\!92\)\( + \)\(22\!\cdots\!20\)\( \beta_{1} - \)\(88\!\cdots\!98\)\( \beta_{2} + \)\(80\!\cdots\!44\)\( \beta_{3} - \)\(56\!\cdots\!50\)\( \beta_{4} - \)\(94\!\cdots\!94\)\( \beta_{5}) q^{94} +(-\)\(15\!\cdots\!00\)\( + \)\(17\!\cdots\!68\)\( \beta_{1} + \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(15\!\cdots\!88\)\( \beta_{3} - \)\(22\!\cdots\!48\)\( \beta_{4} + \)\(43\!\cdots\!36\)\( \beta_{5}) q^{95} +(-\)\(10\!\cdots\!16\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!72\)\( \beta_{2} + \)\(38\!\cdots\!80\)\( \beta_{3} + \)\(66\!\cdots\!92\)\( \beta_{4} + \)\(33\!\cdots\!48\)\( \beta_{5}) q^{96} +(-\)\(19\!\cdots\!22\)\( + \)\(45\!\cdots\!20\)\( \beta_{1} + \)\(25\!\cdots\!08\)\( \beta_{2} - \)\(17\!\cdots\!88\)\( \beta_{3} + \)\(23\!\cdots\!36\)\( \beta_{4} - \)\(90\!\cdots\!88\)\( \beta_{5}) q^{97} +(-\)\(60\!\cdots\!98\)\( + \)\(54\!\cdots\!65\)\( \beta_{1} - \)\(51\!\cdots\!72\)\( \beta_{2} + \)\(14\!\cdots\!36\)\( \beta_{3} - \)\(83\!\cdots\!24\)\( \beta_{4} - \)\(41\!\cdots\!52\)\( \beta_{5}) q^{98} +(-\)\(89\!\cdots\!56\)\( - \)\(32\!\cdots\!14\)\( \beta_{1} + \)\(10\!\cdots\!58\)\( \beta_{2} - \)\(51\!\cdots\!66\)\( \beta_{3} + \)\(85\!\cdots\!60\)\( \beta_{4} + \)\(81\!\cdots\!56\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 869363388q^{2} - 100063090197999414q^{3} + \)\(13\!\cdots\!08\)\(q^{4} + \)\(53\!\cdots\!20\)\(q^{5} + \)\(14\!\cdots\!72\)\(q^{6} - \)\(16\!\cdots\!16\)\(q^{7} + \)\(29\!\cdots\!96\)\(q^{8} + \)\(16\!\cdots\!66\)\(q^{9} + O(q^{10}) \) \( 6q - 869363388q^{2} - 100063090197999414q^{3} + \)\(13\!\cdots\!08\)\(q^{4} + \)\(53\!\cdots\!20\)\(q^{5} + \)\(14\!\cdots\!72\)\(q^{6} - \)\(16\!\cdots\!16\)\(q^{7} + \)\(29\!\cdots\!96\)\(q^{8} + \)\(16\!\cdots\!66\)\(q^{9} - \)\(30\!\cdots\!40\)\(q^{10} - \)\(19\!\cdots\!76\)\(q^{11} - \)\(22\!\cdots\!52\)\(q^{12} - \)\(47\!\cdots\!52\)\(q^{13} + \)\(70\!\cdots\!12\)\(q^{14} - \)\(89\!\cdots\!80\)\(q^{15} + \)\(37\!\cdots\!80\)\(q^{16} - \)\(50\!\cdots\!12\)\(q^{17} - \)\(24\!\cdots\!68\)\(q^{18} - \)\(65\!\cdots\!88\)\(q^{19} - \)\(74\!\cdots\!60\)\(q^{20} + \)\(27\!\cdots\!04\)\(q^{21} - \)\(57\!\cdots\!64\)\(q^{22} - \)\(51\!\cdots\!88\)\(q^{23} - \)\(48\!\cdots\!24\)\(q^{24} + \)\(13\!\cdots\!50\)\(q^{25} - \)\(16\!\cdots\!88\)\(q^{26} - \)\(27\!\cdots\!54\)\(q^{27} + \)\(12\!\cdots\!88\)\(q^{28} + \)\(78\!\cdots\!24\)\(q^{29} + \)\(50\!\cdots\!60\)\(q^{30} + \)\(69\!\cdots\!96\)\(q^{31} + \)\(36\!\cdots\!84\)\(q^{32} + \)\(32\!\cdots\!44\)\(q^{33} + \)\(25\!\cdots\!76\)\(q^{34} + \)\(27\!\cdots\!60\)\(q^{35} + \)\(38\!\cdots\!88\)\(q^{36} - \)\(75\!\cdots\!96\)\(q^{37} + \)\(21\!\cdots\!32\)\(q^{38} + \)\(78\!\cdots\!88\)\(q^{39} - \)\(32\!\cdots\!00\)\(q^{40} - \)\(25\!\cdots\!24\)\(q^{41} - \)\(11\!\cdots\!28\)\(q^{42} - \)\(67\!\cdots\!40\)\(q^{43} + \)\(23\!\cdots\!84\)\(q^{44} + \)\(14\!\cdots\!20\)\(q^{45} + \)\(83\!\cdots\!96\)\(q^{46} + \)\(75\!\cdots\!20\)\(q^{47} - \)\(62\!\cdots\!20\)\(q^{48} + \)\(39\!\cdots\!90\)\(q^{49} - \)\(28\!\cdots\!00\)\(q^{50} + \)\(84\!\cdots\!28\)\(q^{51} - \)\(84\!\cdots\!72\)\(q^{52} - \)\(44\!\cdots\!28\)\(q^{53} + \)\(40\!\cdots\!92\)\(q^{54} + \)\(22\!\cdots\!40\)\(q^{55} + \)\(16\!\cdots\!60\)\(q^{56} + \)\(10\!\cdots\!72\)\(q^{57} + \)\(10\!\cdots\!24\)\(q^{58} + \)\(97\!\cdots\!28\)\(q^{59} + \)\(12\!\cdots\!40\)\(q^{60} - \)\(59\!\cdots\!80\)\(q^{61} - \)\(58\!\cdots\!12\)\(q^{62} - \)\(46\!\cdots\!76\)\(q^{63} - \)\(19\!\cdots\!68\)\(q^{64} - \)\(36\!\cdots\!20\)\(q^{65} + \)\(95\!\cdots\!16\)\(q^{66} + \)\(12\!\cdots\!48\)\(q^{67} - \)\(18\!\cdots\!16\)\(q^{68} + \)\(85\!\cdots\!72\)\(q^{69} + \)\(56\!\cdots\!80\)\(q^{70} - \)\(80\!\cdots\!68\)\(q^{71} + \)\(80\!\cdots\!56\)\(q^{72} + \)\(23\!\cdots\!12\)\(q^{73} - \)\(66\!\cdots\!48\)\(q^{74} - \)\(22\!\cdots\!50\)\(q^{75} - \)\(11\!\cdots\!52\)\(q^{76} - \)\(37\!\cdots\!52\)\(q^{77} + \)\(27\!\cdots\!72\)\(q^{78} + \)\(15\!\cdots\!40\)\(q^{79} - \)\(10\!\cdots\!80\)\(q^{80} + \)\(46\!\cdots\!26\)\(q^{81} + \)\(37\!\cdots\!48\)\(q^{82} - \)\(21\!\cdots\!04\)\(q^{83} - \)\(20\!\cdots\!72\)\(q^{84} - \)\(64\!\cdots\!40\)\(q^{85} - \)\(23\!\cdots\!56\)\(q^{86} - \)\(13\!\cdots\!56\)\(q^{87} - \)\(65\!\cdots\!84\)\(q^{88} - \)\(66\!\cdots\!28\)\(q^{89} - \)\(84\!\cdots\!40\)\(q^{90} - \)\(82\!\cdots\!44\)\(q^{91} + \)\(43\!\cdots\!04\)\(q^{92} - \)\(11\!\cdots\!24\)\(q^{93} - \)\(56\!\cdots\!52\)\(q^{94} - \)\(91\!\cdots\!00\)\(q^{95} - \)\(61\!\cdots\!96\)\(q^{96} - \)\(11\!\cdots\!32\)\(q^{97} - \)\(36\!\cdots\!88\)\(q^{98} - \)\(53\!\cdots\!36\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 1895395655112138158 x^{4} - 215343254036998938223429254 x^{3} + 753811615313247966666178630793780821 x^{2} + 48156657138617542519489631631312055499331993 x - 14832095927450491582029231424734280560496715550059928\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 36 \nu - 18 \)
\(\beta_{2}\)\(=\)\( 1296 \nu^{2} - 220865387400 \nu - 818810922898010992500 \)
\(\beta_{3}\)\(=\)\((\)\(-389268019957851 \nu^{5} - 80126357303727372397716 \nu^{4} + 801988336401927982640150134591854 \nu^{3} + 159180356525896081601063625054976551676980 \nu^{2} - 329678419411991881658681318111579245357798234553451 \nu - 14454413561189401943886929764293204214804481730085257624312\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-692185838694481107 \nu^{5} + 506263692069652280892691788 \nu^{4} + 848058434330038878745560438241698078 \nu^{3} - 389491516796098133350741618241410403873696940 \nu^{2} - 143852259454971814355253236818319948195991585083275107 \nu + 10589775620890392018138484592716369758008143208475605107590216\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(514221373952130699 \nu^{5} - 2492196868021078827518159916 \nu^{4} - 1017960810863448347338202755563687246 \nu^{3} + 3413430388883414619244182413269827612297495180 \nu^{2} + 1005996293224171396273071047208892195342930774089958299 \nu - 469268048502937353876188106064749300263805878254747782182379912\)\()/ \)\(20\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 18\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 6135149650 \beta_{1} + 818810923008443686200\)\()/1296\)
\(\nu^{3}\)\(=\)\((\)\(-211 \beta_{5} - 65 \beta_{4} + 1223826 \beta_{3} + 7366410659 \beta_{2} + 712788457850080193444 \beta_{1} + 2511763781411240379122017348368\)\()/23328\)
\(\nu^{4}\)\(=\)\((\)\(-1691968526615 \beta_{5} - 9336047469 \beta_{4} - 2019268735480406 \beta_{3} + 227012902661985202193 \beta_{2} + 2769268032750145899398571173568 \beta_{1} + 145909743782486113781881645195879342995312\)\()/209952\)
\(\nu^{5}\)\(=\)\((\)\(-16038616768987910502877 \beta_{5} - 5414952942107689259087 \beta_{4} + 54080426588258298274282894 \beta_{3} + 702481695151289010691805352957 \beta_{2} + 36512353669759345040248671595201976075688 \beta_{1} + 283438364583106460934744938168007771946992518182152\)\()/944784\)

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
−9.72827e8
−8.84844e8
−1.78886e8
1.14909e8
6.69381e8
1.25227e9
−3.51667e10 −1.66772e16 6.46400e20 9.85410e23 5.86481e26 −2.48063e29 −1.97299e30 2.78128e32 −3.46536e34
1.2 −3.19993e10 −1.66772e16 4.33658e20 −1.19891e23 5.33658e26 2.02540e29 5.01228e30 2.78128e32 3.83642e33
1.3 −6.58478e9 −1.66772e16 −5.46936e20 −2.05454e24 1.09816e26 −2.33794e29 7.48843e30 2.78128e32 1.35287e34
1.4 3.99182e9 −1.66772e16 −5.74361e20 2.45458e24 −6.65724e25 3.62318e28 −4.64911e30 2.78128e32 9.79827e33
1.5 2.39528e10 −1.66772e16 −1.65573e19 −4.62338e23 −3.99466e26 1.66345e28 −1.45359e31 2.78128e32 −1.10743e34
1.6 4.49367e10 −1.66772e16 1.42901e21 −2.66474e23 −7.49418e26 6.09654e28 3.76892e31 2.78128e32 −1.19745e34
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 3.70.a.a 6
3.b odd 2 1 9.70.a.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.70.a.a 6 1.a even 1 1 trivial
9.70.a.d 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 869363388 T_{2}^{5} - \)\(24\!\cdots\!68\)\( T_{2}^{4} - \)\(11\!\cdots\!36\)\( T_{2}^{3} + \)\(12\!\cdots\!96\)\( T_{2}^{2} + \)\(32\!\cdots\!92\)\( T_{2} - \)\(31\!\cdots\!48\)\( \) acting on \(S_{70}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 869363388 T + \)\(10\!\cdots\!04\)\( T^{2} - \)\(89\!\cdots\!56\)\( T^{3} + \)\(68\!\cdots\!92\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{5} + \)\(43\!\cdots\!64\)\( T^{6} - \)\(82\!\cdots\!08\)\( T^{7} + \)\(24\!\cdots\!48\)\( T^{8} - \)\(18\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(62\!\cdots\!16\)\( T^{11} + \)\(42\!\cdots\!84\)\( T^{12} \)
$3$ \( ( 1 + 16677181699666569 T )^{6} \)
$5$ \( 1 - \)\(53\!\cdots\!20\)\( T + \)\(45\!\cdots\!50\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!75\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(37\!\cdots\!50\)\( T^{10} - \)\(74\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 + \)\(16\!\cdots\!16\)\( T + \)\(55\!\cdots\!54\)\( T^{2} + \)\(10\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} + \)\(27\!\cdots\!88\)\( T^{5} + \)\(50\!\cdots\!48\)\( T^{6} + \)\(55\!\cdots\!16\)\( T^{7} + \)\(81\!\cdots\!39\)\( T^{8} + \)\(92\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!54\)\( T^{10} + \)\(59\!\cdots\!12\)\( T^{11} + \)\(74\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(19\!\cdots\!76\)\( T + \)\(23\!\cdots\!74\)\( T^{2} + \)\(99\!\cdots\!92\)\( T^{3} + \)\(86\!\cdots\!43\)\( T^{4} - \)\(66\!\cdots\!20\)\( T^{5} - \)\(38\!\cdots\!92\)\( T^{6} - \)\(47\!\cdots\!20\)\( T^{7} + \)\(44\!\cdots\!83\)\( T^{8} + \)\(36\!\cdots\!32\)\( T^{9} + \)\(63\!\cdots\!14\)\( T^{10} + \)\(36\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$13$ \( 1 + \)\(47\!\cdots\!52\)\( T + \)\(28\!\cdots\!10\)\( T^{2} + \)\(35\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!55\)\( T^{4} + \)\(65\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!16\)\( T^{6} + \)\(47\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!52\)\( T^{9} + \)\(80\!\cdots\!10\)\( T^{10} + \)\(96\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!89\)\( T^{12} \)
$17$ \( 1 + \)\(50\!\cdots\!12\)\( T + \)\(25\!\cdots\!42\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!55\)\( T^{4} + \)\(52\!\cdots\!32\)\( T^{5} + \)\(94\!\cdots\!24\)\( T^{6} + \)\(42\!\cdots\!04\)\( T^{7} + \)\(84\!\cdots\!95\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!84\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} \)
$19$ \( 1 + \)\(65\!\cdots\!88\)\( T + \)\(54\!\cdots\!62\)\( T^{2} + \)\(98\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} - \)\(33\!\cdots\!28\)\( T^{5} + \)\(20\!\cdots\!48\)\( T^{6} - \)\(57\!\cdots\!12\)\( T^{7} + \)\(36\!\cdots\!83\)\( T^{8} + \)\(49\!\cdots\!64\)\( T^{9} + \)\(46\!\cdots\!22\)\( T^{10} + \)\(97\!\cdots\!12\)\( T^{11} + \)\(25\!\cdots\!21\)\( T^{12} \)
$23$ \( 1 + \)\(51\!\cdots\!88\)\( T + \)\(30\!\cdots\!06\)\( T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + \)\(46\!\cdots\!71\)\( T^{4} + \)\(15\!\cdots\!64\)\( T^{5} + \)\(49\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!32\)\( T^{7} + \)\(38\!\cdots\!99\)\( T^{8} + \)\(81\!\cdots\!52\)\( T^{9} + \)\(20\!\cdots\!66\)\( T^{10} + \)\(32\!\cdots\!84\)\( T^{11} + \)\(56\!\cdots\!09\)\( T^{12} \)
$29$ \( 1 - \)\(78\!\cdots\!24\)\( T + \)\(55\!\cdots\!06\)\( T^{2} - \)\(25\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(30\!\cdots\!56\)\( T^{7} + \)\(72\!\cdots\!27\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(23\!\cdots\!26\)\( T^{10} - \)\(26\!\cdots\!76\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \)
$31$ \( 1 - \)\(69\!\cdots\!96\)\( T + \)\(59\!\cdots\!98\)\( T^{2} - \)\(25\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} - \)\(39\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!32\)\( T^{6} - \)\(31\!\cdots\!24\)\( T^{7} + \)\(80\!\cdots\!83\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(24\!\cdots\!38\)\( T^{10} - \)\(22\!\cdots\!96\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \)
$37$ \( 1 + \)\(75\!\cdots\!96\)\( T + \)\(54\!\cdots\!74\)\( T^{2} + \)\(33\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!91\)\( T^{4} + \)\(88\!\cdots\!08\)\( T^{5} + \)\(32\!\cdots\!88\)\( T^{6} + \)\(14\!\cdots\!16\)\( T^{7} + \)\(41\!\cdots\!39\)\( T^{8} + \)\(13\!\cdots\!04\)\( T^{9} + \)\(36\!\cdots\!34\)\( T^{10} + \)\(81\!\cdots\!72\)\( T^{11} + \)\(17\!\cdots\!89\)\( T^{12} \)
$41$ \( 1 + \)\(25\!\cdots\!24\)\( T + \)\(72\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!03\)\( T^{4} + \)\(58\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} + \)\(12\!\cdots\!72\)\( T^{9} + \)\(97\!\cdots\!58\)\( T^{10} + \)\(66\!\cdots\!24\)\( T^{11} + \)\(49\!\cdots\!61\)\( T^{12} \)
$43$ \( 1 + \)\(67\!\cdots\!40\)\( T + \)\(42\!\cdots\!10\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!47\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + \)\(44\!\cdots\!80\)\( T^{6} + \)\(91\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!03\)\( T^{8} + \)\(23\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(75\!\cdots\!20\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(76\!\cdots\!40\)\( T^{3} + \)\(67\!\cdots\!67\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} - \)\(78\!\cdots\!20\)\( T^{7} + \)\(37\!\cdots\!63\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{9} + \)\(38\!\cdots\!30\)\( T^{10} - \)\(56\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!69\)\( T^{12} \)
$53$ \( 1 + \)\(44\!\cdots\!28\)\( T + \)\(48\!\cdots\!26\)\( T^{2} + \)\(15\!\cdots\!96\)\( T^{3} + \)\(97\!\cdots\!31\)\( T^{4} + \)\(24\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!52\)\( T^{7} + \)\(86\!\cdots\!59\)\( T^{8} + \)\(12\!\cdots\!52\)\( T^{9} + \)\(38\!\cdots\!46\)\( T^{10} + \)\(33\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!69\)\( T^{12} \)
$59$ \( 1 - \)\(97\!\cdots\!28\)\( T + \)\(23\!\cdots\!02\)\( T^{2} - \)\(57\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!63\)\( T^{4} - \)\(50\!\cdots\!32\)\( T^{5} + \)\(86\!\cdots\!68\)\( T^{6} - \)\(78\!\cdots\!48\)\( T^{7} + \)\(61\!\cdots\!23\)\( T^{8} - \)\(21\!\cdots\!04\)\( T^{9} + \)\(13\!\cdots\!82\)\( T^{10} - \)\(85\!\cdots\!72\)\( T^{11} + \)\(13\!\cdots\!61\)\( T^{12} \)
$61$ \( 1 + \)\(59\!\cdots\!80\)\( T + \)\(49\!\cdots\!34\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!95\)\( T^{4} + \)\(61\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!60\)\( T^{6} + \)\(94\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!95\)\( T^{8} + \)\(96\!\cdots\!60\)\( T^{9} + \)\(27\!\cdots\!74\)\( T^{10} + \)\(51\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$67$ \( 1 - \)\(12\!\cdots\!48\)\( T + \)\(50\!\cdots\!42\)\( T^{2} - \)\(50\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(91\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!04\)\( T^{6} - \)\(90\!\cdots\!96\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(50\!\cdots\!60\)\( T^{9} + \)\(50\!\cdots\!02\)\( T^{10} - \)\(12\!\cdots\!36\)\( T^{11} + \)\(98\!\cdots\!29\)\( T^{12} \)
$71$ \( 1 + \)\(80\!\cdots\!68\)\( T + \)\(22\!\cdots\!46\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!08\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} + \)\(74\!\cdots\!48\)\( T^{7} + \)\(66\!\cdots\!95\)\( T^{8} + \)\(24\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} + \)\(38\!\cdots\!68\)\( T^{11} + \)\(26\!\cdots\!81\)\( T^{12} \)
$73$ \( 1 - \)\(23\!\cdots\!12\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(63\!\cdots\!91\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!12\)\( T^{6} - \)\(37\!\cdots\!08\)\( T^{7} + \)\(87\!\cdots\!79\)\( T^{8} - \)\(84\!\cdots\!88\)\( T^{9} + \)\(20\!\cdots\!06\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(26\!\cdots\!09\)\( T^{12} \)
$79$ \( 1 - \)\(15\!\cdots\!40\)\( T + \)\(15\!\cdots\!14\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!15\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(60\!\cdots\!15\)\( T^{8} + \)\(61\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!94\)\( T^{10} - \)\(72\!\cdots\!60\)\( T^{11} + \)\(41\!\cdots\!81\)\( T^{12} \)
$83$ \( 1 + \)\(21\!\cdots\!04\)\( T + \)\(15\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{3} + \)\(99\!\cdots\!27\)\( T^{4} + \)\(13\!\cdots\!88\)\( T^{5} + \)\(34\!\cdots\!76\)\( T^{6} + \)\(34\!\cdots\!64\)\( T^{7} + \)\(67\!\cdots\!43\)\( T^{8} + \)\(47\!\cdots\!76\)\( T^{9} + \)\(71\!\cdots\!06\)\( T^{10} + \)\(26\!\cdots\!72\)\( T^{11} + \)\(31\!\cdots\!29\)\( T^{12} \)
$89$ \( 1 + \)\(66\!\cdots\!28\)\( T + \)\(74\!\cdots\!42\)\( T^{2} + \)\(50\!\cdots\!56\)\( T^{3} + \)\(46\!\cdots\!43\)\( T^{4} + \)\(25\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} + \)\(81\!\cdots\!48\)\( T^{7} + \)\(47\!\cdots\!83\)\( T^{8} + \)\(16\!\cdots\!24\)\( T^{9} + \)\(80\!\cdots\!62\)\( T^{10} + \)\(23\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$97$ \( 1 + \)\(11\!\cdots\!32\)\( T + \)\(92\!\cdots\!62\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!32\)\( T^{5} + \)\(43\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{7} + \)\(40\!\cdots\!95\)\( T^{8} + \)\(99\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!02\)\( T^{10} + \)\(32\!\cdots\!24\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \)
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