Properties

Label 3.66.a.b
Level 3
Weight 66
Character orbit 3.a
Self dual yes
Analytic conductor 80.272
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(80.2717069417\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{43}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11\cdot 13 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(1035163827 + \beta_{1}) q^{2} +1853020188851841 q^{3} +(15223484176714109398 + 2766449869 \beta_{1} + \beta_{2}) q^{4} +(\)\(58\!\cdots\!50\)\( + 4058291886090 \beta_{1} - 115 \beta_{2} + \beta_{3}) q^{5} +(\)\(19\!\cdots\!07\)\( + 1853020188851841 \beta_{1}) q^{6} +(\)\(21\!\cdots\!84\)\( - 79972783738454934 \beta_{1} + 20360674 \beta_{2} - 13013 \beta_{3} + 2 \beta_{4} - 11 \beta_{5}) q^{7} +(\)\(11\!\cdots\!36\)\( + 21078861912981178230 \beta_{1} + 5208391705 \beta_{2} + 1535423 \beta_{3} + 957 \beta_{4} + 902 \beta_{5}) q^{8} +\)\(34\!\cdots\!81\)\( q^{9} +O(q^{10})\) \( q +(1035163827 + \beta_{1}) q^{2} +1853020188851841 q^{3} +(15223484176714109398 + 2766449869 \beta_{1} + \beta_{2}) q^{4} +(\)\(58\!\cdots\!50\)\( + 4058291886090 \beta_{1} - 115 \beta_{2} + \beta_{3}) q^{5} +(\)\(19\!\cdots\!07\)\( + 1853020188851841 \beta_{1}) q^{6} +(\)\(21\!\cdots\!84\)\( - 79972783738454934 \beta_{1} + 20360674 \beta_{2} - 13013 \beta_{3} + 2 \beta_{4} - 11 \beta_{5}) q^{7} +(\)\(11\!\cdots\!36\)\( + 21078861912981178230 \beta_{1} + 5208391705 \beta_{2} + 1535423 \beta_{3} + 957 \beta_{4} + 902 \beta_{5}) q^{8} +\)\(34\!\cdots\!81\)\( q^{9} +(\)\(21\!\cdots\!50\)\( + \)\(85\!\cdots\!74\)\( \beta_{1} + 5991131641124 \beta_{2} + 914975956 \beta_{3} - 2231044 \beta_{4} + 538648 \beta_{5}) q^{10} +(-\)\(38\!\cdots\!96\)\( + \)\(36\!\cdots\!40\)\( \beta_{1} + 27075971338498 \beta_{2} + 12902997746 \beta_{3} + 83434704 \beta_{4} + 3069064 \beta_{5}) q^{11} +(\)\(28\!\cdots\!18\)\( + \)\(51\!\cdots\!29\)\( \beta_{1} + 1853020188851841 \beta_{2}) q^{12} +(\)\(16\!\cdots\!58\)\( - \)\(44\!\cdots\!48\)\( \beta_{1} + 4434013985876935 \beta_{2} - 2482793608804 \beta_{3} - 8774881934 \beta_{4} - 5647118323 \beta_{5}) q^{13} +(-\)\(38\!\cdots\!68\)\( + \)\(85\!\cdots\!04\)\( \beta_{1} - 242504447544261548 \beta_{2} - 97016561333692 \beta_{3} + 62640614220 \beta_{4} - 29679396296 \beta_{5}) q^{14} +(\)\(10\!\cdots\!50\)\( + \)\(75\!\cdots\!90\)\( \beta_{1} - 213097321717961715 \beta_{2} + 1853020188851841 \beta_{3}) q^{15} +(\)\(63\!\cdots\!60\)\( + \)\(25\!\cdots\!92\)\( \beta_{1} + 21369434725528966530 \beta_{2} + 9632904364827470 \beta_{3} - 3565193201942 \beta_{4} + 8242560826412 \beta_{5}) q^{16} +(-\)\(22\!\cdots\!22\)\( - \)\(59\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} + 150414742067361454 \beta_{3} + 29629607763420 \beta_{4} - 40885437728058 \beta_{5}) q^{17} +(\)\(35\!\cdots\!87\)\( + \)\(34\!\cdots\!81\)\( \beta_{1}) q^{18} +(\)\(93\!\cdots\!32\)\( - \)\(17\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!12\)\( \beta_{2} + 5776526584106469454 \beta_{3} - 1710096887588356 \beta_{4} - 778455099992042 \beta_{5}) q^{19} +(\)\(43\!\cdots\!00\)\( + \)\(30\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!26\)\( \beta_{2} - 3233815714682863936 \beta_{3} + 10675902499527744 \beta_{4} + 9259352738477952 \beta_{5}) q^{20} +(\)\(40\!\cdots\!44\)\( - \)\(14\!\cdots\!94\)\( \beta_{1} + \)\(37\!\cdots\!34\)\( \beta_{2} - 24113351717529006933 \beta_{3} + 3706040377703682 \beta_{4} - 20383222077370251 \beta_{5}) q^{21} +(\)\(18\!\cdots\!36\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + \)\(96\!\cdots\!12\)\( \beta_{2} - \)\(61\!\cdots\!76\)\( \beta_{3} - 306812746499247368 \beta_{4} - 8360672652871120 \beta_{5}) q^{22} +(\)\(79\!\cdots\!32\)\( - \)\(29\!\cdots\!40\)\( \beta_{1} + \)\(26\!\cdots\!28\)\( \beta_{2} - \)\(47\!\cdots\!50\)\( \beta_{3} + 1084590165223127604 \beta_{4} - 283796188057375774 \beta_{5}) q^{23} +(\)\(22\!\cdots\!76\)\( + \)\(39\!\cdots\!30\)\( \beta_{1} + \)\(96\!\cdots\!05\)\( \beta_{2} + \)\(28\!\cdots\!43\)\( \beta_{3} + 1773340320731211837 \beta_{4} + 1671424210344360582 \beta_{5}) q^{24} +(\)\(98\!\cdots\!75\)\( + \)\(81\!\cdots\!00\)\( \beta_{1} - \)\(33\!\cdots\!50\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3} - 21856701789537085300 \beta_{4} + 424227633178820350 \beta_{5}) q^{25} +(-\)\(21\!\cdots\!38\)\( + \)\(25\!\cdots\!94\)\( \beta_{1} - \)\(18\!\cdots\!32\)\( \beta_{2} + \)\(27\!\cdots\!68\)\( \beta_{3} + 52622463852183003336 \beta_{4} - 11883871272460988592 \beta_{5}) q^{26} +\)\(63\!\cdots\!21\)\( q^{27} +(\)\(31\!\cdots\!88\)\( - \)\(86\!\cdots\!00\)\( \beta_{1} - \)\(89\!\cdots\!32\)\( \beta_{2} - \)\(78\!\cdots\!76\)\( \beta_{3} - \)\(26\!\cdots\!12\)\( \beta_{4} - 17846740719351029376 \beta_{5}) q^{28} +(\)\(30\!\cdots\!34\)\( - \)\(20\!\cdots\!50\)\( \beta_{1} - \)\(41\!\cdots\!75\)\( \beta_{2} - \)\(17\!\cdots\!01\)\( \beta_{3} + \)\(10\!\cdots\!84\)\( \beta_{4} + \)\(30\!\cdots\!98\)\( \beta_{5}) q^{29} +(\)\(39\!\cdots\!50\)\( + \)\(15\!\cdots\!34\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(16\!\cdots\!96\)\( \beta_{3} - \)\(41\!\cdots\!04\)\( \beta_{4} + \)\(99\!\cdots\!68\)\( \beta_{5}) q^{30} +(\)\(43\!\cdots\!16\)\( + \)\(10\!\cdots\!54\)\( \beta_{1} - \)\(17\!\cdots\!14\)\( \beta_{2} + \)\(24\!\cdots\!11\)\( \beta_{3} + \)\(12\!\cdots\!22\)\( \beta_{4} - \)\(10\!\cdots\!59\)\( \beta_{5}) q^{31} +(\)\(90\!\cdots\!24\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} + \)\(89\!\cdots\!56\)\( \beta_{3} - \)\(39\!\cdots\!64\)\( \beta_{4} + \)\(23\!\cdots\!12\)\( \beta_{5}) q^{32} +(-\)\(71\!\cdots\!36\)\( + \)\(67\!\cdots\!40\)\( \beta_{1} + \)\(50\!\cdots\!18\)\( \beta_{2} + \)\(23\!\cdots\!86\)\( \beta_{3} + \)\(15\!\cdots\!64\)\( \beta_{4} + \)\(56\!\cdots\!24\)\( \beta_{5}) q^{33} +(-\)\(32\!\cdots\!14\)\( - \)\(76\!\cdots\!18\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} - \)\(62\!\cdots\!08\)\( \beta_{3} - \)\(45\!\cdots\!92\)\( \beta_{4} - \)\(16\!\cdots\!92\)\( \beta_{5}) q^{34} +(-\)\(56\!\cdots\!00\)\( - \)\(91\!\cdots\!44\)\( \beta_{1} - \)\(14\!\cdots\!74\)\( \beta_{2} - \)\(31\!\cdots\!10\)\( \beta_{3} + \)\(59\!\cdots\!04\)\( \beta_{4} + \)\(44\!\cdots\!32\)\( \beta_{5}) q^{35} +(\)\(52\!\cdots\!38\)\( + \)\(94\!\cdots\!89\)\( \beta_{1} + \)\(34\!\cdots\!81\)\( \beta_{2}) q^{36} +(-\)\(32\!\cdots\!46\)\( + \)\(37\!\cdots\!16\)\( \beta_{1} - \)\(71\!\cdots\!61\)\( \beta_{2} + \)\(28\!\cdots\!94\)\( \beta_{3} + \)\(17\!\cdots\!58\)\( \beta_{4} - \)\(27\!\cdots\!21\)\( \beta_{5}) q^{37} +(-\)\(78\!\cdots\!48\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2} + \)\(21\!\cdots\!48\)\( \beta_{3} + \)\(51\!\cdots\!68\)\( \beta_{4} + \)\(63\!\cdots\!76\)\( \beta_{5}) q^{38} +(\)\(29\!\cdots\!78\)\( - \)\(83\!\cdots\!68\)\( \beta_{1} + \)\(82\!\cdots\!35\)\( \beta_{2} - \)\(46\!\cdots\!64\)\( \beta_{3} - \)\(16\!\cdots\!94\)\( \beta_{4} - \)\(10\!\cdots\!43\)\( \beta_{5}) q^{39} +(\)\(81\!\cdots\!00\)\( + \)\(16\!\cdots\!56\)\( \beta_{1} + \)\(36\!\cdots\!66\)\( \beta_{2} - \)\(34\!\cdots\!18\)\( \beta_{3} + \)\(57\!\cdots\!34\)\( \beta_{4} + \)\(28\!\cdots\!72\)\( \beta_{5}) q^{40} +(\)\(74\!\cdots\!86\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!98\)\( \beta_{3} - \)\(83\!\cdots\!16\)\( \beta_{4} - \)\(28\!\cdots\!78\)\( \beta_{5}) q^{41} +(-\)\(71\!\cdots\!88\)\( + \)\(15\!\cdots\!64\)\( \beta_{1} - \)\(44\!\cdots\!68\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4} - \)\(54\!\cdots\!36\)\( \beta_{5}) q^{42} +(\)\(25\!\cdots\!40\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(13\!\cdots\!66\)\( \beta_{3} - \)\(26\!\cdots\!64\)\( \beta_{4} - \)\(58\!\cdots\!54\)\( \beta_{5}) q^{43} +(\)\(97\!\cdots\!04\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} - \)\(54\!\cdots\!92\)\( \beta_{2} + \)\(28\!\cdots\!88\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4} + \)\(51\!\cdots\!44\)\( \beta_{5}) q^{44} +(\)\(20\!\cdots\!50\)\( + \)\(13\!\cdots\!90\)\( \beta_{1} - \)\(39\!\cdots\!15\)\( \beta_{2} + \)\(34\!\cdots\!81\)\( \beta_{3}) q^{45} +(-\)\(14\!\cdots\!24\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(33\!\cdots\!32\)\( \beta_{2} - \)\(80\!\cdots\!20\)\( \beta_{3} + \)\(28\!\cdots\!36\)\( \beta_{4} + \)\(44\!\cdots\!04\)\( \beta_{5}) q^{46} +(-\)\(39\!\cdots\!80\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(20\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!18\)\( \beta_{3} + \)\(16\!\cdots\!40\)\( \beta_{4} - \)\(92\!\cdots\!94\)\( \beta_{5}) q^{47} +(\)\(11\!\cdots\!60\)\( + \)\(46\!\cdots\!72\)\( \beta_{1} + \)\(39\!\cdots\!30\)\( \beta_{2} + \)\(17\!\cdots\!70\)\( \beta_{3} - \)\(66\!\cdots\!22\)\( \beta_{4} + \)\(15\!\cdots\!92\)\( \beta_{5}) q^{48} +(-\)\(22\!\cdots\!55\)\( - \)\(58\!\cdots\!08\)\( \beta_{1} - \)\(26\!\cdots\!02\)\( \beta_{2} - \)\(92\!\cdots\!52\)\( \beta_{3} - \)\(64\!\cdots\!56\)\( \beta_{4} - \)\(33\!\cdots\!10\)\( \beta_{5}) q^{49} +(\)\(14\!\cdots\!25\)\( + \)\(87\!\cdots\!75\)\( \beta_{1} - \)\(56\!\cdots\!00\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4} + \)\(27\!\cdots\!00\)\( \beta_{5}) q^{50} +(-\)\(41\!\cdots\!02\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} - \)\(21\!\cdots\!68\)\( \beta_{2} + \)\(27\!\cdots\!14\)\( \beta_{3} + \)\(54\!\cdots\!20\)\( \beta_{4} - \)\(75\!\cdots\!78\)\( \beta_{5}) q^{51} +(\)\(46\!\cdots\!68\)\( - \)\(71\!\cdots\!54\)\( \beta_{1} - \)\(23\!\cdots\!22\)\( \beta_{2} - \)\(71\!\cdots\!92\)\( \beta_{3} - \)\(77\!\cdots\!28\)\( \beta_{4} - \)\(16\!\cdots\!88\)\( \beta_{5}) q^{52} +(\)\(16\!\cdots\!22\)\( - \)\(96\!\cdots\!02\)\( \beta_{1} - \)\(95\!\cdots\!35\)\( \beta_{2} + \)\(45\!\cdots\!07\)\( \beta_{3} - \)\(25\!\cdots\!56\)\( \beta_{4} + \)\(55\!\cdots\!02\)\( \beta_{5}) q^{53} +(\)\(65\!\cdots\!67\)\( + \)\(63\!\cdots\!21\)\( \beta_{1}) q^{54} +(\)\(10\!\cdots\!00\)\( - \)\(32\!\cdots\!48\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(43\!\cdots\!84\)\( \beta_{3} + \)\(15\!\cdots\!08\)\( \beta_{4} + \)\(58\!\cdots\!64\)\( \beta_{5}) q^{55} +(-\)\(26\!\cdots\!40\)\( - \)\(48\!\cdots\!16\)\( \beta_{1} - \)\(53\!\cdots\!28\)\( \beta_{2} + \)\(34\!\cdots\!76\)\( \beta_{3} - \)\(51\!\cdots\!20\)\( \beta_{4} - \)\(11\!\cdots\!72\)\( \beta_{5}) q^{56} +(\)\(17\!\cdots\!12\)\( - \)\(31\!\cdots\!28\)\( \beta_{1} + \)\(88\!\cdots\!92\)\( \beta_{2} + \)\(10\!\cdots\!14\)\( \beta_{3} - \)\(31\!\cdots\!96\)\( \beta_{4} - \)\(14\!\cdots\!22\)\( \beta_{5}) q^{57} +(-\)\(10\!\cdots\!26\)\( - \)\(16\!\cdots\!78\)\( \beta_{1} - \)\(23\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!68\)\( \beta_{3} - \)\(47\!\cdots\!96\)\( \beta_{4} - \)\(43\!\cdots\!28\)\( \beta_{5}) q^{58} +(-\)\(11\!\cdots\!92\)\( - \)\(30\!\cdots\!56\)\( \beta_{1} - \)\(47\!\cdots\!00\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!76\)\( \beta_{4} + \)\(54\!\cdots\!44\)\( \beta_{5}) q^{59} +(\)\(81\!\cdots\!00\)\( + \)\(56\!\cdots\!66\)\( \beta_{1} + \)\(49\!\cdots\!66\)\( \beta_{2} - \)\(59\!\cdots\!76\)\( \beta_{3} + \)\(19\!\cdots\!04\)\( \beta_{4} + \)\(17\!\cdots\!32\)\( \beta_{5}) q^{60} +(\)\(67\!\cdots\!50\)\( + \)\(36\!\cdots\!44\)\( \beta_{1} - \)\(36\!\cdots\!45\)\( \beta_{2} - \)\(11\!\cdots\!98\)\( \beta_{3} - \)\(39\!\cdots\!78\)\( \beta_{4} + \)\(23\!\cdots\!59\)\( \beta_{5}) q^{61} +(\)\(58\!\cdots\!48\)\( - \)\(45\!\cdots\!16\)\( \beta_{1} - \)\(43\!\cdots\!88\)\( \beta_{2} - \)\(22\!\cdots\!40\)\( \beta_{3} - \)\(47\!\cdots\!32\)\( \beta_{4} - \)\(58\!\cdots\!48\)\( \beta_{5}) q^{62} +(\)\(75\!\cdots\!04\)\( - \)\(27\!\cdots\!54\)\( \beta_{1} + \)\(69\!\cdots\!94\)\( \beta_{2} - \)\(44\!\cdots\!53\)\( \beta_{3} + \)\(68\!\cdots\!62\)\( \beta_{4} - \)\(37\!\cdots\!91\)\( \beta_{5}) q^{63} +(\)\(42\!\cdots\!32\)\( + \)\(11\!\cdots\!32\)\( \beta_{1} + \)\(13\!\cdots\!96\)\( \beta_{2} + \)\(67\!\cdots\!88\)\( \beta_{3} + \)\(28\!\cdots\!00\)\( \beta_{4} + \)\(92\!\cdots\!84\)\( \beta_{5}) q^{64} +(-\)\(16\!\cdots\!00\)\( + \)\(34\!\cdots\!24\)\( \beta_{1} - \)\(18\!\cdots\!56\)\( \beta_{2} + \)\(48\!\cdots\!42\)\( \beta_{3} - \)\(76\!\cdots\!04\)\( \beta_{4} + \)\(14\!\cdots\!18\)\( \beta_{5}) q^{65} +(\)\(33\!\cdots\!76\)\( + \)\(23\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!92\)\( \beta_{2} - \)\(11\!\cdots\!16\)\( \beta_{3} - \)\(56\!\cdots\!88\)\( \beta_{4} - \)\(15\!\cdots\!20\)\( \beta_{5}) q^{66} +(\)\(24\!\cdots\!68\)\( + \)\(13\!\cdots\!52\)\( \beta_{1} + \)\(62\!\cdots\!20\)\( \beta_{2} + \)\(22\!\cdots\!96\)\( \beta_{3} - \)\(21\!\cdots\!76\)\( \beta_{4} - \)\(20\!\cdots\!16\)\( \beta_{5}) q^{67} +(-\)\(34\!\cdots\!36\)\( - \)\(66\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!06\)\( \beta_{2} - \)\(56\!\cdots\!04\)\( \beta_{3} + \)\(11\!\cdots\!88\)\( \beta_{4} - \)\(19\!\cdots\!00\)\( \beta_{5}) q^{68} +(\)\(14\!\cdots\!12\)\( - \)\(54\!\cdots\!40\)\( \beta_{1} + \)\(49\!\cdots\!48\)\( \beta_{2} - \)\(88\!\cdots\!50\)\( \beta_{3} + \)\(20\!\cdots\!64\)\( \beta_{4} - \)\(52\!\cdots\!34\)\( \beta_{5}) q^{69} +(-\)\(62\!\cdots\!00\)\( - \)\(10\!\cdots\!32\)\( \beta_{1} + \)\(67\!\cdots\!28\)\( \beta_{2} - \)\(35\!\cdots\!20\)\( \beta_{3} - \)\(37\!\cdots\!88\)\( \beta_{4} - \)\(28\!\cdots\!04\)\( \beta_{5}) q^{70} +(\)\(76\!\cdots\!52\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(21\!\cdots\!86\)\( \beta_{3} - \)\(37\!\cdots\!52\)\( \beta_{4} + \)\(38\!\cdots\!70\)\( \beta_{5}) q^{71} +(\)\(40\!\cdots\!16\)\( + \)\(72\!\cdots\!30\)\( \beta_{1} + \)\(17\!\cdots\!05\)\( \beta_{2} + \)\(52\!\cdots\!63\)\( \beta_{3} + \)\(32\!\cdots\!17\)\( \beta_{4} + \)\(30\!\cdots\!62\)\( \beta_{5}) q^{72} +(\)\(88\!\cdots\!42\)\( - \)\(39\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!26\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3} + \)\(14\!\cdots\!56\)\( \beta_{4} - \)\(66\!\cdots\!02\)\( \beta_{5}) q^{73} +(\)\(15\!\cdots\!02\)\( - \)\(53\!\cdots\!94\)\( \beta_{1} - \)\(27\!\cdots\!92\)\( \beta_{2} - \)\(35\!\cdots\!60\)\( \beta_{3} - \)\(17\!\cdots\!88\)\( \beta_{4} - \)\(15\!\cdots\!52\)\( \beta_{5}) q^{74} +(\)\(18\!\cdots\!75\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(62\!\cdots\!50\)\( \beta_{2} + \)\(45\!\cdots\!00\)\( \beta_{3} - \)\(40\!\cdots\!00\)\( \beta_{4} + \)\(78\!\cdots\!50\)\( \beta_{5}) q^{75} +(\)\(82\!\cdots\!88\)\( - \)\(31\!\cdots\!16\)\( \beta_{1} + \)\(18\!\cdots\!28\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(99\!\cdots\!24\)\( \beta_{4} + \)\(50\!\cdots\!04\)\( \beta_{5}) q^{76} +(-\)\(30\!\cdots\!72\)\( + \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(33\!\cdots\!88\)\( \beta_{2} + \)\(81\!\cdots\!52\)\( \beta_{3} + \)\(72\!\cdots\!08\)\( \beta_{4} - \)\(10\!\cdots\!12\)\( \beta_{5}) q^{77} +(-\)\(39\!\cdots\!58\)\( + \)\(46\!\cdots\!54\)\( \beta_{1} - \)\(34\!\cdots\!12\)\( \beta_{2} + \)\(51\!\cdots\!88\)\( \beta_{3} + \)\(97\!\cdots\!76\)\( \beta_{4} - \)\(22\!\cdots\!72\)\( \beta_{5}) q^{78} +(\)\(91\!\cdots\!40\)\( + \)\(26\!\cdots\!98\)\( \beta_{1} - \)\(54\!\cdots\!90\)\( \beta_{2} + \)\(15\!\cdots\!23\)\( \beta_{3} - \)\(11\!\cdots\!22\)\( \beta_{4} - \)\(50\!\cdots\!19\)\( \beta_{5}) q^{79} +(\)\(77\!\cdots\!00\)\( + \)\(13\!\cdots\!04\)\( \beta_{1} + \)\(23\!\cdots\!64\)\( \beta_{2} + \)\(44\!\cdots\!84\)\( \beta_{3} - \)\(30\!\cdots\!04\)\( \beta_{4} + \)\(47\!\cdots\!68\)\( \beta_{5}) q^{80} +\)\(11\!\cdots\!61\)\( q^{81} +(\)\(84\!\cdots\!38\)\( + \)\(57\!\cdots\!26\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2} + \)\(56\!\cdots\!28\)\( \beta_{3} + \)\(65\!\cdots\!08\)\( \beta_{4} - \)\(27\!\cdots\!84\)\( \beta_{5}) q^{82} +(\)\(20\!\cdots\!76\)\( - \)\(90\!\cdots\!20\)\( \beta_{1} + \)\(17\!\cdots\!82\)\( \beta_{2} - \)\(14\!\cdots\!06\)\( \beta_{3} + \)\(46\!\cdots\!48\)\( \beta_{4} + \)\(66\!\cdots\!44\)\( \beta_{5}) q^{83} +(\)\(58\!\cdots\!08\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3} - \)\(48\!\cdots\!92\)\( \beta_{4} - \)\(33\!\cdots\!16\)\( \beta_{5}) q^{84} +(\)\(30\!\cdots\!00\)\( - \)\(70\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!90\)\( \beta_{2} + \)\(16\!\cdots\!98\)\( \beta_{3} - \)\(19\!\cdots\!80\)\( \beta_{4} - \)\(85\!\cdots\!40\)\( \beta_{5}) q^{85} +(\)\(63\!\cdots\!44\)\( - \)\(18\!\cdots\!60\)\( \beta_{1} + \)\(50\!\cdots\!32\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} - \)\(27\!\cdots\!68\)\( \beta_{4} - \)\(16\!\cdots\!36\)\( \beta_{5}) q^{86} +(\)\(57\!\cdots\!94\)\( - \)\(38\!\cdots\!50\)\( \beta_{1} - \)\(76\!\cdots\!75\)\( \beta_{2} - \)\(31\!\cdots\!41\)\( \beta_{3} + \)\(19\!\cdots\!44\)\( \beta_{4} + \)\(57\!\cdots\!18\)\( \beta_{5}) q^{87} +(\)\(16\!\cdots\!16\)\( + \)\(10\!\cdots\!36\)\( \beta_{1} + \)\(23\!\cdots\!36\)\( \beta_{2} + \)\(27\!\cdots\!48\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} + \)\(32\!\cdots\!64\)\( \beta_{5}) q^{88} +(\)\(10\!\cdots\!02\)\( - \)\(23\!\cdots\!32\)\( \beta_{1} - \)\(70\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!76\)\( \beta_{3} + \)\(44\!\cdots\!12\)\( \beta_{4} - \)\(43\!\cdots\!20\)\( \beta_{5}) q^{89} +(\)\(73\!\cdots\!50\)\( + \)\(29\!\cdots\!94\)\( \beta_{1} + \)\(20\!\cdots\!44\)\( \beta_{2} + \)\(31\!\cdots\!36\)\( \beta_{3} - \)\(76\!\cdots\!64\)\( \beta_{4} + \)\(18\!\cdots\!88\)\( \beta_{5}) q^{90} +(\)\(26\!\cdots\!16\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!84\)\( \beta_{2} - \)\(56\!\cdots\!02\)\( \beta_{3} - \)\(68\!\cdots\!28\)\( \beta_{4} + \)\(16\!\cdots\!22\)\( \beta_{5}) q^{91} +(\)\(49\!\cdots\!84\)\( + \)\(27\!\cdots\!76\)\( \beta_{1} + \)\(57\!\cdots\!40\)\( \beta_{2} + \)\(15\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!64\)\( \beta_{4} + \)\(97\!\cdots\!24\)\( \beta_{5}) q^{92} +(\)\(81\!\cdots\!56\)\( + \)\(19\!\cdots\!14\)\( \beta_{1} - \)\(32\!\cdots\!74\)\( \beta_{2} + \)\(45\!\cdots\!51\)\( \beta_{3} + \)\(23\!\cdots\!02\)\( \beta_{4} - \)\(18\!\cdots\!19\)\( \beta_{5}) q^{93} +(\)\(93\!\cdots\!88\)\( + \)\(69\!\cdots\!08\)\( \beta_{1} + \)\(50\!\cdots\!72\)\( \beta_{2} - \)\(81\!\cdots\!24\)\( \beta_{3} + \)\(66\!\cdots\!44\)\( \beta_{4} - \)\(25\!\cdots\!96\)\( \beta_{5}) q^{94} +(\)\(12\!\cdots\!00\)\( + \)\(18\!\cdots\!32\)\( \beta_{1} - \)\(49\!\cdots\!48\)\( \beta_{2} + \)\(18\!\cdots\!84\)\( \beta_{3} + \)\(52\!\cdots\!48\)\( \beta_{4} + \)\(39\!\cdots\!84\)\( \beta_{5}) q^{95} +(\)\(16\!\cdots\!84\)\( + \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(47\!\cdots\!40\)\( \beta_{2} + \)\(16\!\cdots\!96\)\( \beta_{3} - \)\(73\!\cdots\!24\)\( \beta_{4} + \)\(43\!\cdots\!92\)\( \beta_{5}) q^{96} +(\)\(21\!\cdots\!38\)\( - \)\(68\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(86\!\cdots\!04\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4} - \)\(24\!\cdots\!32\)\( \beta_{5}) q^{97} +(-\)\(53\!\cdots\!33\)\( - \)\(33\!\cdots\!35\)\( \beta_{1} - \)\(42\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!48\)\( \beta_{3} + \)\(20\!\cdots\!04\)\( \beta_{4} - \)\(90\!\cdots\!48\)\( \beta_{5}) q^{98} +(-\)\(13\!\cdots\!76\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(92\!\cdots\!38\)\( \beta_{2} + \)\(44\!\cdots\!26\)\( \beta_{3} + \)\(28\!\cdots\!24\)\( \beta_{4} + \)\(10\!\cdots\!84\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6210982962q^{2} + 11118121133111046q^{3} + 91340905060284656388q^{4} + \)\(35\!\cdots\!00\)\(q^{5} + \)\(11\!\cdots\!42\)\(q^{6} + \)\(13\!\cdots\!04\)\(q^{7} + \)\(71\!\cdots\!16\)\(q^{8} + \)\(20\!\cdots\!86\)\(q^{9} + O(q^{10}) \) \( 6q + 6210982962q^{2} + 11118121133111046q^{3} + 91340905060284656388q^{4} + \)\(35\!\cdots\!00\)\(q^{5} + \)\(11\!\cdots\!42\)\(q^{6} + \)\(13\!\cdots\!04\)\(q^{7} + \)\(71\!\cdots\!16\)\(q^{8} + \)\(20\!\cdots\!86\)\(q^{9} + \)\(12\!\cdots\!00\)\(q^{10} - \)\(23\!\cdots\!76\)\(q^{11} + \)\(16\!\cdots\!08\)\(q^{12} + \)\(96\!\cdots\!48\)\(q^{13} - \)\(23\!\cdots\!08\)\(q^{14} + \)\(65\!\cdots\!00\)\(q^{15} + \)\(38\!\cdots\!60\)\(q^{16} - \)\(13\!\cdots\!32\)\(q^{17} + \)\(21\!\cdots\!22\)\(q^{18} + \)\(56\!\cdots\!92\)\(q^{19} + \)\(26\!\cdots\!00\)\(q^{20} + \)\(24\!\cdots\!64\)\(q^{21} + \)\(10\!\cdots\!16\)\(q^{22} + \)\(47\!\cdots\!92\)\(q^{23} + \)\(13\!\cdots\!56\)\(q^{24} + \)\(59\!\cdots\!50\)\(q^{25} - \)\(12\!\cdots\!28\)\(q^{26} + \)\(38\!\cdots\!26\)\(q^{27} + \)\(18\!\cdots\!28\)\(q^{28} + \)\(18\!\cdots\!04\)\(q^{29} + \)\(23\!\cdots\!00\)\(q^{30} + \)\(26\!\cdots\!96\)\(q^{31} + \)\(54\!\cdots\!44\)\(q^{32} - \)\(43\!\cdots\!16\)\(q^{33} - \)\(19\!\cdots\!84\)\(q^{34} - \)\(33\!\cdots\!00\)\(q^{35} + \)\(31\!\cdots\!28\)\(q^{36} - \)\(19\!\cdots\!76\)\(q^{37} - \)\(46\!\cdots\!88\)\(q^{38} + \)\(17\!\cdots\!68\)\(q^{39} + \)\(49\!\cdots\!00\)\(q^{40} + \)\(44\!\cdots\!16\)\(q^{41} - \)\(42\!\cdots\!28\)\(q^{42} + \)\(15\!\cdots\!40\)\(q^{43} + \)\(58\!\cdots\!24\)\(q^{44} + \)\(12\!\cdots\!00\)\(q^{45} - \)\(84\!\cdots\!44\)\(q^{46} - \)\(23\!\cdots\!80\)\(q^{47} + \)\(70\!\cdots\!60\)\(q^{48} - \)\(13\!\cdots\!30\)\(q^{49} + \)\(86\!\cdots\!50\)\(q^{50} - \)\(24\!\cdots\!12\)\(q^{51} + \)\(28\!\cdots\!08\)\(q^{52} + \)\(96\!\cdots\!32\)\(q^{53} + \)\(39\!\cdots\!02\)\(q^{54} + \)\(61\!\cdots\!00\)\(q^{55} - \)\(15\!\cdots\!40\)\(q^{56} + \)\(10\!\cdots\!72\)\(q^{57} - \)\(62\!\cdots\!56\)\(q^{58} - \)\(71\!\cdots\!52\)\(q^{59} + \)\(48\!\cdots\!00\)\(q^{60} + \)\(40\!\cdots\!00\)\(q^{61} + \)\(34\!\cdots\!88\)\(q^{62} + \)\(45\!\cdots\!24\)\(q^{63} + \)\(25\!\cdots\!92\)\(q^{64} - \)\(97\!\cdots\!00\)\(q^{65} + \)\(20\!\cdots\!56\)\(q^{66} + \)\(14\!\cdots\!08\)\(q^{67} - \)\(20\!\cdots\!16\)\(q^{68} + \)\(88\!\cdots\!72\)\(q^{69} - \)\(37\!\cdots\!00\)\(q^{70} + \)\(45\!\cdots\!12\)\(q^{71} + \)\(24\!\cdots\!96\)\(q^{72} + \)\(52\!\cdots\!52\)\(q^{73} + \)\(93\!\cdots\!12\)\(q^{74} + \)\(10\!\cdots\!50\)\(q^{75} + \)\(49\!\cdots\!28\)\(q^{76} - \)\(18\!\cdots\!32\)\(q^{77} - \)\(23\!\cdots\!48\)\(q^{78} + \)\(54\!\cdots\!40\)\(q^{79} + \)\(46\!\cdots\!00\)\(q^{80} + \)\(70\!\cdots\!66\)\(q^{81} + \)\(50\!\cdots\!28\)\(q^{82} + \)\(12\!\cdots\!56\)\(q^{83} + \)\(35\!\cdots\!48\)\(q^{84} + \)\(18\!\cdots\!00\)\(q^{85} + \)\(38\!\cdots\!64\)\(q^{86} + \)\(34\!\cdots\!64\)\(q^{87} + \)\(98\!\cdots\!96\)\(q^{88} + \)\(64\!\cdots\!12\)\(q^{89} + \)\(43\!\cdots\!00\)\(q^{90} + \)\(15\!\cdots\!96\)\(q^{91} + \)\(29\!\cdots\!04\)\(q^{92} + \)\(48\!\cdots\!36\)\(q^{93} + \)\(56\!\cdots\!28\)\(q^{94} + \)\(72\!\cdots\!00\)\(q^{95} + \)\(10\!\cdots\!04\)\(q^{96} + \)\(12\!\cdots\!28\)\(q^{97} - \)\(31\!\cdots\!98\)\(q^{98} - \)\(79\!\cdots\!56\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 4253784014616993888 x^{4} - 329017052684987797861154944 x^{3} + 3760278921468113125234877358788007936 x^{2} + 22551593102849743113985540842573617614848 x - 272928737776547549611547193931962354302098378900635648\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 3 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} - 4176733326 \nu - 51045408173315560047 \)
\(\beta_{3}\)\(=\)\((\)\(34220984678163 \nu^{5} - 204624719522368468497 \nu^{4} - 129792467407002101837559992998872 \nu^{3} - 15397563421483621154453628474844242720960 \nu^{2} + 83525140826605298574632007994422423386411159674368 \nu + 3994341035062746726204001760515400120170047068531324092416\)\()/ \)\(32\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-85579356314943207 \nu^{5} - 235898923333219397441125587 \nu^{4} + 521491127431162087307044631956063608 \nu^{3} + 737257168921916481180634357769803270549855680 \nu^{2} - 584969923936038046306057858489375490015648207868667392 \nu - 199882640008057964412149983364237685683878312304851746019344384\)\()/ \)\(22\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-435832873032153699 \nu^{5} + 347491719717916643716757121 \nu^{4} + 2105163063868379798861919121276379736 \nu^{3} - 1082415425851397114846133417392816600222610240 \nu^{2} - 2323062035005819942853152802088639418336341889754955264 \nu + 471969830014766643367604179830241315488762618506341209216909312\)\()/ \)\(30\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 696122221 \beta_{1} + 51045408175403926710\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(902 \beta_{5} + 957 \beta_{4} + 1535423 \beta_{3} + 2102900233 \beta_{2} + 89489344159479276454 \beta_{1} + 35533843068204702904910754636\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(2253844874710 \beta_{5} - 3763900160107 \beta_{4} + 1637623494138631 \beta_{3} + 58456564662972687201 \beta_{2} + 88833175400517927855447807830 \beta_{1} + 2284010050042572295739582589377650637868\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(30830171529496779873862 \beta_{5} + 32599645253477211559773 \beta_{4} + 70664756610820794943662495 \beta_{3} + 97128090264824385005614260105 \beta_{2} + 2282417534443336051534027761857044439110 \beta_{1} + 2267262852465788443179627291856979571575121285132\)\()/1944\)

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
−1.64208e9
−1.16886e9
−2.78112e8
2.86568e8
9.77853e8
1.82463e9
−8.81729e9 1.85302e15 4.08511e19 −7.15176e22 −1.63386e25 3.06094e27 −3.48954e28 3.43368e30 6.30591e32
1.2 −5.97799e9 1.85302e15 −1.15708e18 3.17694e22 −1.10773e25 6.82141e26 2.27466e29 3.43368e30 −1.89917e32
1.3 −6.33506e8 1.85302e15 −3.64922e19 −5.06462e22 −1.17390e24 −3.41514e27 4.64902e28 3.43368e30 3.20846e31
1.4 2.75457e9 1.85302e15 −2.93058e19 9.23075e22 5.10428e24 −1.32074e27 −1.82351e29 3.43368e30 2.54268e32
1.5 6.90228e9 1.85302e15 1.07480e19 −3.03033e22 1.27901e25 3.59710e27 −1.80464e29 3.43368e30 −2.09162e32
1.6 1.19829e10 1.85302e15 1.06697e20 6.35463e22 2.22046e25 −1.28649e27 8.36449e29 3.43368e30 7.61471e32
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 3.66.a.b 6
3.b odd 2 1 9.66.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.66.a.b 6 1.a even 1 1 trivial
9.66.a.c 6 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 \( T_{2}^{6} - 6210982962 T_{2}^{5} - \)\(13\!\cdots\!68\)\( T_{2}^{4} + \)\(54\!\cdots\!64\)\( T_{2}^{3} + \)\(41\!\cdots\!96\)\( T_{2}^{2} - \)\(96\!\cdots\!08\)\( T_{2} - \)\(76\!\cdots\!48\)\( \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6210982962 T + 84298166589242128224 T^{2} - \)\(60\!\cdots\!56\)\( T^{3} + \)\(43\!\cdots\!52\)\( T^{4} - \)\(34\!\cdots\!44\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - \)\(12\!\cdots\!08\)\( T^{7} + \)\(58\!\cdots\!48\)\( T^{8} - \)\(30\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!24\)\( T^{10} - \)\(42\!\cdots\!84\)\( T^{11} + \)\(25\!\cdots\!24\)\( T^{12} \)
$3$ \( ( 1 - 1853020188851841 T )^{6} \)
$5$ \( 1 - \)\(35\!\cdots\!00\)\( T + \)\(57\!\cdots\!50\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!75\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(80\!\cdots\!00\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!75\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!50\)\( T^{10} - \)\(51\!\cdots\!00\)\( T^{11} + \)\(39\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(13\!\cdots\!04\)\( T + \)\(33\!\cdots\!94\)\( T^{2} - \)\(42\!\cdots\!52\)\( T^{3} + \)\(55\!\cdots\!11\)\( T^{4} - \)\(58\!\cdots\!12\)\( T^{5} + \)\(58\!\cdots\!28\)\( T^{6} - \)\(50\!\cdots\!84\)\( T^{7} + \)\(40\!\cdots\!39\)\( T^{8} - \)\(26\!\cdots\!36\)\( T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(59\!\cdots\!28\)\( T^{11} + \)\(38\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(23\!\cdots\!76\)\( T + \)\(39\!\cdots\!94\)\( T^{2} + \)\(32\!\cdots\!32\)\( T^{3} + \)\(15\!\cdots\!83\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(37\!\cdots\!83\)\( T^{8} + \)\(37\!\cdots\!32\)\( T^{9} + \)\(23\!\cdots\!94\)\( T^{10} + \)\(65\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} \)
$13$ \( 1 - \)\(96\!\cdots\!48\)\( T + \)\(90\!\cdots\!70\)\( T^{2} - \)\(89\!\cdots\!24\)\( T^{3} + \)\(45\!\cdots\!75\)\( T^{4} - \)\(39\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} - \)\(10\!\cdots\!04\)\( T^{7} + \)\(29\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!68\)\( T^{9} + \)\(38\!\cdots\!70\)\( T^{10} - \)\(10\!\cdots\!64\)\( T^{11} + \)\(27\!\cdots\!49\)\( T^{12} \)
$17$ \( 1 + \)\(13\!\cdots\!32\)\( T + \)\(10\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!55\)\( T^{4} + \)\(17\!\cdots\!32\)\( T^{5} + \)\(21\!\cdots\!64\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(14\!\cdots\!95\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(85\!\cdots\!02\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{11} + \)\(75\!\cdots\!49\)\( T^{12} \)
$19$ \( 1 - \)\(56\!\cdots\!92\)\( T + \)\(29\!\cdots\!22\)\( T^{2} - \)\(45\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!43\)\( T^{4} + \)\(28\!\cdots\!72\)\( T^{5} + \)\(96\!\cdots\!68\)\( T^{6} + \)\(37\!\cdots\!28\)\( T^{7} + \)\(22\!\cdots\!43\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} + \)\(89\!\cdots\!22\)\( T^{10} - \)\(22\!\cdots\!08\)\( T^{11} + \)\(51\!\cdots\!01\)\( T^{12} \)
$23$ \( 1 - \)\(47\!\cdots\!92\)\( T + \)\(10\!\cdots\!86\)\( T^{2} + \)\(21\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!91\)\( T^{4} + \)\(19\!\cdots\!64\)\( T^{5} + \)\(21\!\cdots\!12\)\( T^{6} + \)\(64\!\cdots\!52\)\( T^{7} + \)\(56\!\cdots\!59\)\( T^{8} + \)\(74\!\cdots\!92\)\( T^{9} + \)\(11\!\cdots\!86\)\( T^{10} - \)\(17\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
$29$ \( 1 - \)\(18\!\cdots\!04\)\( T + \)\(45\!\cdots\!26\)\( T^{2} - \)\(71\!\cdots\!72\)\( T^{3} + \)\(97\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!92\)\( T^{6} - \)\(14\!\cdots\!36\)\( T^{7} + \)\(12\!\cdots\!27\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} + \)\(76\!\cdots\!26\)\( T^{10} - \)\(35\!\cdots\!96\)\( T^{11} + \)\(21\!\cdots\!01\)\( T^{12} \)
$31$ \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(33\!\cdots\!78\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(57\!\cdots\!03\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(61\!\cdots\!52\)\( T^{6} - \)\(14\!\cdots\!64\)\( T^{7} + \)\(43\!\cdots\!03\)\( T^{8} - \)\(65\!\cdots\!48\)\( T^{9} + \)\(19\!\cdots\!78\)\( T^{10} - \)\(12\!\cdots\!96\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \)
$37$ \( 1 + \)\(19\!\cdots\!76\)\( T + \)\(53\!\cdots\!74\)\( T^{2} + \)\(68\!\cdots\!08\)\( T^{3} + \)\(11\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} + \)\(90\!\cdots\!36\)\( T^{7} + \)\(81\!\cdots\!59\)\( T^{8} + \)\(43\!\cdots\!44\)\( T^{9} + \)\(29\!\cdots\!74\)\( T^{10} + \)\(91\!\cdots\!32\)\( T^{11} + \)\(39\!\cdots\!49\)\( T^{12} \)
$41$ \( 1 - \)\(44\!\cdots\!16\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + \)\(57\!\cdots\!83\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{5} + \)\(49\!\cdots\!12\)\( T^{6} - \)\(93\!\cdots\!44\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(37\!\cdots\!08\)\( T^{9} + \)\(80\!\cdots\!78\)\( T^{10} - \)\(63\!\cdots\!16\)\( T^{11} + \)\(96\!\cdots\!01\)\( T^{12} \)
$43$ \( 1 - \)\(15\!\cdots\!40\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(62\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{5} + \)\(30\!\cdots\!60\)\( T^{6} - \)\(22\!\cdots\!60\)\( T^{7} + \)\(36\!\cdots\!03\)\( T^{8} - \)\(21\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!70\)\( T^{10} - \)\(11\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + \)\(23\!\cdots\!80\)\( T + \)\(11\!\cdots\!30\)\( T^{2} + \)\(88\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!47\)\( T^{4} - \)\(75\!\cdots\!60\)\( T^{5} - \)\(80\!\cdots\!60\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(53\!\cdots\!03\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(64\!\cdots\!30\)\( T^{10} + \)\(64\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!49\)\( T^{12} \)
$53$ \( 1 - \)\(96\!\cdots\!32\)\( T + \)\(51\!\cdots\!26\)\( T^{2} - \)\(33\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} - \)\(62\!\cdots\!36\)\( T^{5} + \)\(18\!\cdots\!92\)\( T^{6} - \)\(74\!\cdots\!48\)\( T^{7} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(57\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!26\)\( T^{10} - \)\(23\!\cdots\!76\)\( T^{11} + \)\(29\!\cdots\!49\)\( T^{12} \)
$59$ \( 1 + \)\(71\!\cdots\!52\)\( T + \)\(69\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!24\)\( T^{3} + \)\(21\!\cdots\!63\)\( T^{4} + \)\(86\!\cdots\!68\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!32\)\( T^{7} + \)\(34\!\cdots\!63\)\( T^{8} + \)\(76\!\cdots\!76\)\( T^{9} + \)\(18\!\cdots\!22\)\( T^{10} + \)\(24\!\cdots\!48\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \)
$61$ \( 1 - \)\(40\!\cdots\!00\)\( T + \)\(59\!\cdots\!14\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(37\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{9} + \)\(91\!\cdots\!14\)\( T^{10} - \)\(69\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} \)
$67$ \( 1 - \)\(14\!\cdots\!08\)\( T + \)\(22\!\cdots\!02\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!55\)\( T^{4} - \)\(24\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!84\)\( T^{6} - \)\(12\!\cdots\!76\)\( T^{7} + \)\(57\!\cdots\!95\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{9} + \)\(13\!\cdots\!02\)\( T^{10} - \)\(43\!\cdots\!56\)\( T^{11} + \)\(14\!\cdots\!49\)\( T^{12} \)
$71$ \( 1 - \)\(45\!\cdots\!12\)\( T + \)\(14\!\cdots\!66\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(75\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(24\!\cdots\!95\)\( T^{8} - \)\(30\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!66\)\( T^{10} - \)\(20\!\cdots\!12\)\( T^{11} + \)\(97\!\cdots\!01\)\( T^{12} \)
$73$ \( 1 - \)\(52\!\cdots\!52\)\( T + \)\(75\!\cdots\!26\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + \)\(24\!\cdots\!31\)\( T^{4} - \)\(79\!\cdots\!96\)\( T^{5} + \)\(41\!\cdots\!32\)\( T^{6} - \)\(10\!\cdots\!28\)\( T^{7} + \)\(41\!\cdots\!19\)\( T^{8} - \)\(70\!\cdots\!88\)\( T^{9} + \)\(21\!\cdots\!26\)\( T^{10} - \)\(20\!\cdots\!36\)\( T^{11} + \)\(49\!\cdots\!49\)\( T^{12} \)
$79$ \( 1 - \)\(54\!\cdots\!40\)\( T + \)\(10\!\cdots\!94\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!15\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(26\!\cdots\!94\)\( T^{10} - \)\(29\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} \)
$83$ \( 1 - \)\(12\!\cdots\!56\)\( T + \)\(88\!\cdots\!26\)\( T^{2} - \)\(43\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!67\)\( T^{4} - \)\(50\!\cdots\!92\)\( T^{5} + \)\(13\!\cdots\!36\)\( T^{6} - \)\(27\!\cdots\!56\)\( T^{7} + \)\(49\!\cdots\!83\)\( T^{8} - \)\(71\!\cdots\!04\)\( T^{9} + \)\(80\!\cdots\!26\)\( T^{10} - \)\(63\!\cdots\!08\)\( T^{11} + \)\(27\!\cdots\!49\)\( T^{12} \)
$89$ \( 1 - \)\(64\!\cdots\!12\)\( T + \)\(35\!\cdots\!62\)\( T^{2} - \)\(12\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!63\)\( T^{4} - \)\(96\!\cdots\!48\)\( T^{5} + \)\(24\!\cdots\!48\)\( T^{6} - \)\(49\!\cdots\!52\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(16\!\cdots\!36\)\( T^{9} + \)\(24\!\cdots\!62\)\( T^{10} - \)\(22\!\cdots\!88\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} \)
$97$ \( 1 - \)\(12\!\cdots\!28\)\( T + \)\(12\!\cdots\!02\)\( T^{2} - \)\(86\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!55\)\( T^{4} - \)\(23\!\cdots\!68\)\( T^{5} + \)\(95\!\cdots\!44\)\( T^{6} - \)\(32\!\cdots\!76\)\( T^{7} + \)\(95\!\cdots\!95\)\( T^{8} - \)\(22\!\cdots\!60\)\( T^{9} + \)\(46\!\cdots\!02\)\( T^{10} - \)\(64\!\cdots\!96\)\( T^{11} + \)\(69\!\cdots\!49\)\( T^{12} \)
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