Properties

Label 3.68.a.b
Level 3
Weight 68
Character orbit 3.a
Self dual yes
Analytic conductor 85.287
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 68 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(85.2871055790\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{46}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11^{2}\cdot 13\cdot 17 \)
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.

\(f(q)\) \(=\) \( q +(2289225861 - \beta_{1}) q^{2} -5559060566555523 q^{3} +(76856221419709869142 - 871210566 \beta_{1} + \beta_{2}) q^{4} +(-\)\(30\!\cdots\!50\)\( - 5867760487466 \beta_{1} + 352 \beta_{2} - \beta_{3}) q^{5} +(-\)\(12\!\cdots\!03\)\( + 5559060566555523 \beta_{1}) q^{6} +(-\)\(46\!\cdots\!68\)\( + 291321334454968162 \beta_{1} - 80666795 \beta_{2} + 1598 \beta_{3} - \beta_{4} + 2 \beta_{5}) q^{7} +(\)\(29\!\cdots\!32\)\( - 43246795112688713255 \beta_{1} - 2546139725 \beta_{2} - 2166905 \beta_{3} - 110 \beta_{4} - 141 \beta_{5}) q^{8} +\)\(30\!\cdots\!29\)\( q^{9} +O(q^{10})\) \( q +(2289225861 - \beta_{1}) q^{2} -5559060566555523 q^{3} +(76856221419709869142 - 871210566 \beta_{1} + \beta_{2}) q^{4} +(-\)\(30\!\cdots\!50\)\( - 5867760487466 \beta_{1} + 352 \beta_{2} - \beta_{3}) q^{5} +(-\)\(12\!\cdots\!03\)\( + 5559060566555523 \beta_{1}) q^{6} +(-\)\(46\!\cdots\!68\)\( + 291321334454968162 \beta_{1} - 80666795 \beta_{2} + 1598 \beta_{3} - \beta_{4} + 2 \beta_{5}) q^{7} +(\)\(29\!\cdots\!32\)\( - 43246795112688713255 \beta_{1} - 2546139725 \beta_{2} - 2166905 \beta_{3} - 110 \beta_{4} - 141 \beta_{5}) q^{8} +\)\(30\!\cdots\!29\)\( q^{9} +(\)\(12\!\cdots\!50\)\( - \)\(21\!\cdots\!08\)\( \beta_{1} + 11275681472306 \beta_{2} - 11521777498 \beta_{3} - 286060 \beta_{4} + 16190 \beta_{5}) q^{10} +(\)\(20\!\cdots\!04\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} - 238526413697568 \beta_{2} + 101100377470 \beta_{3} - 489760 \beta_{4} - 2842176 \beta_{5}) q^{11} +(-\)\(42\!\cdots\!66\)\( + \)\(48\!\cdots\!18\)\( \beta_{1} - 5559060566555523 \beta_{2}) q^{12} +(-\)\(17\!\cdots\!94\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} - 5436148551484991 \beta_{2} - 18442904453723 \beta_{3} - 2441657309 \beta_{4} + 400101562 \beta_{5}) q^{13} +(-\)\(74\!\cdots\!92\)\( + \)\(13\!\cdots\!18\)\( \beta_{1} - 458625319886245086 \beta_{2} - 211620741738506 \beta_{3} + 11474357492 \beta_{4} + 28966144974 \beta_{5}) q^{14} +(\)\(16\!\cdots\!50\)\( + \)\(32\!\cdots\!18\)\( \beta_{1} - 1956789319427544096 \beta_{2} + 5559060566555523 \beta_{3}) q^{15} +(-\)\(17\!\cdots\!80\)\( + \)\(45\!\cdots\!78\)\( \beta_{1} + 42872511004161547578 \beta_{2} - 49623051005294894 \beta_{3} + 245849360668 \beta_{4} - 859755870086 \beta_{5}) q^{16} +(-\)\(37\!\cdots\!66\)\( - \)\(56\!\cdots\!24\)\( \beta_{1} + \)\(42\!\cdots\!86\)\( \beta_{2} - 484211423013981496 \beta_{3} - 2330416206678 \beta_{4} - 9069535208916 \beta_{5}) q^{17} +(\)\(70\!\cdots\!69\)\( - \)\(30\!\cdots\!29\)\( \beta_{1}) q^{18} +(-\)\(11\!\cdots\!92\)\( - \)\(21\!\cdots\!72\)\( \beta_{1} - \)\(17\!\cdots\!06\)\( \beta_{2} - 18473170732721507368 \beta_{3} + 311331020568626 \beta_{4} + 651351290112284 \beta_{5}) q^{19} +(\)\(76\!\cdots\!00\)\( - \)\(16\!\cdots\!16\)\( \beta_{1} + \)\(12\!\cdots\!62\)\( \beta_{2} - 96920911026180581696 \beta_{3} - 3060130678465920 \beta_{4} - 863367828793920 \beta_{5}) q^{20} +(\)\(26\!\cdots\!64\)\( - \)\(16\!\cdots\!26\)\( \beta_{1} + \)\(44\!\cdots\!85\)\( \beta_{2} - 8883378785355725754 \beta_{3} + 5559060566555523 \beta_{4} - 11118121133111046 \beta_{5}) q^{21} +(\)\(23\!\cdots\!28\)\( + \)\(28\!\cdots\!36\)\( \beta_{1} + \)\(28\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3} + 55067333963906728 \beta_{4} + 624153984764092 \beta_{5}) q^{22} +(\)\(20\!\cdots\!44\)\( + \)\(55\!\cdots\!76\)\( \beta_{1} + \)\(21\!\cdots\!82\)\( \beta_{2} + \)\(48\!\cdots\!96\)\( \beta_{3} - 301038046592161562 \beta_{4} + 120696402704387124 \beta_{5}) q^{23} +(-\)\(16\!\cdots\!36\)\( + \)\(24\!\cdots\!65\)\( \beta_{1} + \)\(14\!\cdots\!75\)\( \beta_{2} + \)\(12\!\cdots\!15\)\( \beta_{3} + 611496662321107530 \beta_{4} + 783827539884328743 \beta_{5}) q^{24} +(\)\(20\!\cdots\!75\)\( - \)\(34\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2} + \)\(12\!\cdots\!50\)\( \beta_{3} - 1651190398107421750 \beta_{4} - 4875320240304740500 \beta_{5}) q^{25} +(\)\(24\!\cdots\!82\)\( + \)\(96\!\cdots\!82\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!48\)\( \beta_{3} + 4639782283551053496 \beta_{4} + 4108393197068209236 \beta_{5}) q^{26} -\)\(17\!\cdots\!67\)\( q^{27} +(-\)\(24\!\cdots\!24\)\( + \)\(65\!\cdots\!96\)\( \beta_{1} - \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3} + 83109627162696689280 \beta_{4} + 56114506768901699520 \beta_{5}) q^{28} +(-\)\(31\!\cdots\!94\)\( - \)\(12\!\cdots\!30\)\( \beta_{1} - \)\(21\!\cdots\!98\)\( \beta_{2} - \)\(64\!\cdots\!17\)\( \beta_{3} - \)\(75\!\cdots\!46\)\( \beta_{4} - 3006089876479886988 \beta_{5}) q^{29} +(-\)\(67\!\cdots\!50\)\( + \)\(12\!\cdots\!84\)\( \beta_{1} - \)\(62\!\cdots\!38\)\( \beta_{2} + \)\(64\!\cdots\!54\)\( \beta_{3} + \)\(15\!\cdots\!80\)\( \beta_{4} - 90001190572533917370 \beta_{5}) q^{30} +(-\)\(37\!\cdots\!84\)\( - \)\(13\!\cdots\!18\)\( \beta_{1} + \)\(29\!\cdots\!97\)\( \beta_{2} + \)\(76\!\cdots\!74\)\( \beta_{3} + \)\(73\!\cdots\!27\)\( \beta_{4} - \)\(36\!\cdots\!18\)\( \beta_{5}) q^{31} +(-\)\(10\!\cdots\!08\)\( + \)\(25\!\cdots\!12\)\( \beta_{1} + \)\(38\!\cdots\!76\)\( \beta_{2} - \)\(19\!\cdots\!32\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!88\)\( \beta_{5}) q^{32} +(-\)\(11\!\cdots\!92\)\( + \)\(59\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!64\)\( \beta_{2} - \)\(56\!\cdots\!10\)\( \beta_{3} + \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(15\!\cdots\!48\)\( \beta_{5}) q^{33} +(\)\(11\!\cdots\!34\)\( - \)\(44\!\cdots\!86\)\( \beta_{1} + \)\(78\!\cdots\!88\)\( \beta_{2} - \)\(65\!\cdots\!76\)\( \beta_{3} - \)\(15\!\cdots\!88\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5}) q^{34} +(-\)\(10\!\cdots\!00\)\( + \)\(23\!\cdots\!80\)\( \beta_{1} - \)\(75\!\cdots\!40\)\( \beta_{2} + \)\(18\!\cdots\!90\)\( \beta_{3} + \)\(47\!\cdots\!60\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5}) q^{35} +(\)\(23\!\cdots\!18\)\( - \)\(26\!\cdots\!14\)\( \beta_{1} + \)\(30\!\cdots\!29\)\( \beta_{2}) q^{36} +(\)\(45\!\cdots\!42\)\( - \)\(71\!\cdots\!48\)\( \beta_{1} - \)\(17\!\cdots\!81\)\( \beta_{2} + \)\(41\!\cdots\!29\)\( \beta_{3} - \)\(50\!\cdots\!03\)\( \beta_{4} - \)\(24\!\cdots\!18\)\( \beta_{5}) q^{37} +(\)\(20\!\cdots\!84\)\( + \)\(11\!\cdots\!36\)\( \beta_{1} - \)\(32\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(61\!\cdots\!80\)\( \beta_{4} + \)\(74\!\cdots\!28\)\( \beta_{5}) q^{38} +(\)\(95\!\cdots\!62\)\( + \)\(63\!\cdots\!60\)\( \beta_{1} + \)\(30\!\cdots\!93\)\( \beta_{2} + \)\(10\!\cdots\!29\)\( \beta_{3} + \)\(13\!\cdots\!07\)\( \beta_{4} - \)\(22\!\cdots\!26\)\( \beta_{5}) q^{39} +(\)\(19\!\cdots\!00\)\( - \)\(19\!\cdots\!54\)\( \beta_{1} + \)\(21\!\cdots\!18\)\( \beta_{2} - \)\(61\!\cdots\!54\)\( \beta_{3} + \)\(15\!\cdots\!40\)\( \beta_{4} - \)\(23\!\cdots\!10\)\( \beta_{5}) q^{40} +(\)\(44\!\cdots\!66\)\( - \)\(44\!\cdots\!56\)\( \beta_{1} + \)\(66\!\cdots\!38\)\( \beta_{2} - \)\(64\!\cdots\!64\)\( \beta_{3} - \)\(46\!\cdots\!82\)\( \beta_{4} + \)\(55\!\cdots\!28\)\( \beta_{5}) q^{41} +(\)\(41\!\cdots\!16\)\( - \)\(75\!\cdots\!14\)\( \beta_{1} + \)\(25\!\cdots\!78\)\( \beta_{2} + \)\(11\!\cdots\!38\)\( \beta_{3} - \)\(63\!\cdots\!16\)\( \beta_{4} - \)\(16\!\cdots\!02\)\( \beta_{5}) q^{42} +(-\)\(18\!\cdots\!00\)\( - \)\(85\!\cdots\!64\)\( \beta_{1} + \)\(70\!\cdots\!82\)\( \beta_{2} + \)\(84\!\cdots\!64\)\( \beta_{3} + \)\(19\!\cdots\!82\)\( \beta_{4} + \)\(34\!\cdots\!64\)\( \beta_{5}) q^{43} +(-\)\(58\!\cdots\!84\)\( - \)\(44\!\cdots\!72\)\( \beta_{1} - \)\(42\!\cdots\!20\)\( \beta_{2} + \)\(13\!\cdots\!20\)\( \beta_{3} - \)\(57\!\cdots\!80\)\( \beta_{4} - \)\(39\!\cdots\!48\)\( \beta_{5}) q^{44} +(-\)\(92\!\cdots\!50\)\( - \)\(18\!\cdots\!14\)\( \beta_{1} + \)\(10\!\cdots\!08\)\( \beta_{2} - \)\(30\!\cdots\!29\)\( \beta_{3}) q^{45} +(-\)\(75\!\cdots\!04\)\( - \)\(46\!\cdots\!56\)\( \beta_{1} - \)\(63\!\cdots\!56\)\( \beta_{2} - \)\(99\!\cdots\!72\)\( \beta_{3} - \)\(19\!\cdots\!16\)\( \beta_{4} - \)\(24\!\cdots\!04\)\( \beta_{5}) q^{46} +(\)\(96\!\cdots\!60\)\( - \)\(80\!\cdots\!40\)\( \beta_{1} - \)\(40\!\cdots\!22\)\( \beta_{2} - \)\(11\!\cdots\!28\)\( \beta_{3} + \)\(86\!\cdots\!46\)\( \beta_{4} + \)\(11\!\cdots\!12\)\( \beta_{5}) q^{47} +(\)\(99\!\cdots\!40\)\( - \)\(25\!\cdots\!94\)\( \beta_{1} - \)\(23\!\cdots\!94\)\( \beta_{2} + \)\(27\!\cdots\!62\)\( \beta_{3} - \)\(13\!\cdots\!64\)\( \beta_{4} + \)\(47\!\cdots\!78\)\( \beta_{5}) q^{48} +(\)\(16\!\cdots\!45\)\( - \)\(18\!\cdots\!32\)\( \beta_{1} + \)\(23\!\cdots\!66\)\( \beta_{2} - \)\(90\!\cdots\!46\)\( \beta_{3} - \)\(10\!\cdots\!78\)\( \beta_{4} + \)\(23\!\cdots\!76\)\( \beta_{5}) q^{49} +(\)\(79\!\cdots\!75\)\( - \)\(29\!\cdots\!75\)\( \beta_{1} + \)\(53\!\cdots\!00\)\( \beta_{2} + \)\(73\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} - \)\(71\!\cdots\!00\)\( \beta_{5}) q^{50} +(\)\(20\!\cdots\!18\)\( + \)\(31\!\cdots\!52\)\( \beta_{1} - \)\(23\!\cdots\!78\)\( \beta_{2} + \)\(26\!\cdots\!08\)\( \beta_{3} + \)\(12\!\cdots\!94\)\( \beta_{4} + \)\(50\!\cdots\!68\)\( \beta_{5}) q^{51} +(-\)\(12\!\cdots\!96\)\( - \)\(16\!\cdots\!44\)\( \beta_{1} - \)\(23\!\cdots\!34\)\( \beta_{2} - \)\(96\!\cdots\!88\)\( \beta_{3} - \)\(57\!\cdots\!44\)\( \beta_{4} - \)\(13\!\cdots\!28\)\( \beta_{5}) q^{52} +(-\)\(22\!\cdots\!06\)\( - \)\(53\!\cdots\!30\)\( \beta_{1} - \)\(21\!\cdots\!58\)\( \beta_{2} + \)\(25\!\cdots\!67\)\( \beta_{3} - \)\(59\!\cdots\!54\)\( \beta_{4} + \)\(31\!\cdots\!16\)\( \beta_{5}) q^{53} +(-\)\(39\!\cdots\!87\)\( + \)\(17\!\cdots\!67\)\( \beta_{1}) q^{54} +(-\)\(85\!\cdots\!00\)\( + \)\(53\!\cdots\!44\)\( \beta_{1} + \)\(22\!\cdots\!12\)\( \beta_{2} - \)\(35\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4} + \)\(41\!\cdots\!40\)\( \beta_{5}) q^{55} +(-\)\(88\!\cdots\!40\)\( + \)\(24\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!16\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(31\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5}) q^{56} +(\)\(62\!\cdots\!16\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(95\!\cdots\!38\)\( \beta_{2} + \)\(10\!\cdots\!64\)\( \beta_{3} - \)\(17\!\cdots\!98\)\( \beta_{4} - \)\(36\!\cdots\!32\)\( \beta_{5}) q^{57} +(-\)\(45\!\cdots\!02\)\( + \)\(56\!\cdots\!52\)\( \beta_{1} + \)\(52\!\cdots\!54\)\( \beta_{2} - \)\(27\!\cdots\!06\)\( \beta_{3} + \)\(35\!\cdots\!52\)\( \beta_{4} + \)\(29\!\cdots\!22\)\( \beta_{5}) q^{58} +(\)\(72\!\cdots\!52\)\( + \)\(64\!\cdots\!44\)\( \beta_{1} + \)\(23\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!12\)\( \beta_{3} - \)\(10\!\cdots\!36\)\( \beta_{4} + \)\(15\!\cdots\!92\)\( \beta_{5}) q^{59} +(-\)\(42\!\cdots\!00\)\( + \)\(91\!\cdots\!68\)\( \beta_{1} - \)\(71\!\cdots\!26\)\( \beta_{2} + \)\(53\!\cdots\!08\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4} + \)\(47\!\cdots\!60\)\( \beta_{5}) q^{60} +(-\)\(50\!\cdots\!90\)\( - \)\(93\!\cdots\!96\)\( \beta_{1} - \)\(20\!\cdots\!21\)\( \beta_{2} - \)\(14\!\cdots\!75\)\( \beta_{3} - \)\(72\!\cdots\!15\)\( \beta_{4} - \)\(45\!\cdots\!34\)\( \beta_{5}) q^{61} +(\)\(21\!\cdots\!64\)\( + \)\(35\!\cdots\!10\)\( \beta_{1} + \)\(21\!\cdots\!18\)\( \beta_{2} + \)\(11\!\cdots\!66\)\( \beta_{3} + \)\(15\!\cdots\!48\)\( \beta_{4} - \)\(42\!\cdots\!54\)\( \beta_{5}) q^{62} +(-\)\(14\!\cdots\!72\)\( + \)\(90\!\cdots\!98\)\( \beta_{1} - \)\(24\!\cdots\!55\)\( \beta_{2} + \)\(49\!\cdots\!42\)\( \beta_{3} - \)\(30\!\cdots\!29\)\( \beta_{4} + \)\(61\!\cdots\!58\)\( \beta_{5}) q^{63} +(-\)\(55\!\cdots\!52\)\( - \)\(70\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(36\!\cdots\!20\)\( \beta_{3} - \)\(13\!\cdots\!80\)\( \beta_{4} + \)\(17\!\cdots\!92\)\( \beta_{5}) q^{64} +(\)\(15\!\cdots\!00\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} + \)\(29\!\cdots\!38\)\( \beta_{2} + \)\(58\!\cdots\!76\)\( \beta_{3} + \)\(41\!\cdots\!10\)\( \beta_{4} - \)\(11\!\cdots\!40\)\( \beta_{5}) q^{65} +(-\)\(13\!\cdots\!44\)\( - \)\(15\!\cdots\!28\)\( \beta_{1} - \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(85\!\cdots\!88\)\( \beta_{3} - \)\(30\!\cdots\!44\)\( \beta_{4} - \)\(34\!\cdots\!16\)\( \beta_{5}) q^{66} +(-\)\(60\!\cdots\!16\)\( - \)\(75\!\cdots\!80\)\( \beta_{1} + \)\(34\!\cdots\!80\)\( \beta_{2} - \)\(89\!\cdots\!96\)\( \beta_{3} + \)\(21\!\cdots\!52\)\( \beta_{4} - \)\(20\!\cdots\!56\)\( \beta_{5}) q^{67} +(\)\(10\!\cdots\!48\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} + \)\(13\!\cdots\!70\)\( \beta_{2} - \)\(62\!\cdots\!08\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(45\!\cdots\!80\)\( \beta_{5}) q^{68} +(-\)\(11\!\cdots\!12\)\( - \)\(30\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!86\)\( \beta_{2} - \)\(27\!\cdots\!08\)\( \beta_{3} + \)\(16\!\cdots\!26\)\( \beta_{4} - \)\(67\!\cdots\!52\)\( \beta_{5}) q^{69} +(-\)\(29\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(45\!\cdots\!20\)\( \beta_{2} + \)\(36\!\cdots\!40\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4} + \)\(23\!\cdots\!40\)\( \beta_{5}) q^{70} +(\)\(42\!\cdots\!12\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!74\)\( \beta_{2} - \)\(12\!\cdots\!12\)\( \beta_{3} - \)\(42\!\cdots\!06\)\( \beta_{4} + \)\(59\!\cdots\!16\)\( \beta_{5}) q^{71} +(\)\(89\!\cdots\!28\)\( - \)\(13\!\cdots\!95\)\( \beta_{1} - \)\(78\!\cdots\!25\)\( \beta_{2} - \)\(66\!\cdots\!45\)\( \beta_{3} - \)\(33\!\cdots\!90\)\( \beta_{4} - \)\(43\!\cdots\!89\)\( \beta_{5}) q^{72} +(\)\(27\!\cdots\!54\)\( + \)\(39\!\cdots\!56\)\( \beta_{1} - \)\(20\!\cdots\!90\)\( \beta_{2} - \)\(54\!\cdots\!02\)\( \beta_{3} - \)\(11\!\cdots\!26\)\( \beta_{4} - \)\(18\!\cdots\!96\)\( \beta_{5}) q^{73} +(\)\(16\!\cdots\!78\)\( + \)\(16\!\cdots\!02\)\( \beta_{1} + \)\(24\!\cdots\!56\)\( \beta_{2} + \)\(74\!\cdots\!56\)\( \beta_{3} + \)\(33\!\cdots\!68\)\( \beta_{4} - \)\(45\!\cdots\!76\)\( \beta_{5}) q^{74} +(-\)\(11\!\cdots\!25\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(66\!\cdots\!50\)\( \beta_{2} - \)\(71\!\cdots\!50\)\( \beta_{3} + \)\(91\!\cdots\!50\)\( \beta_{4} + \)\(27\!\cdots\!00\)\( \beta_{5}) q^{75} +(-\)\(86\!\cdots\!52\)\( + \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(42\!\cdots\!88\)\( \beta_{2} - \)\(13\!\cdots\!84\)\( \beta_{3} - \)\(21\!\cdots\!52\)\( \beta_{4} + \)\(21\!\cdots\!40\)\( \beta_{5}) q^{76} +(\)\(58\!\cdots\!24\)\( + \)\(44\!\cdots\!16\)\( \beta_{1} + \)\(35\!\cdots\!84\)\( \beta_{2} - \)\(15\!\cdots\!52\)\( \beta_{3} - \)\(59\!\cdots\!76\)\( \beta_{4} - \)\(17\!\cdots\!32\)\( \beta_{5}) q^{77} +(-\)\(13\!\cdots\!86\)\( - \)\(53\!\cdots\!86\)\( \beta_{1} - \)\(74\!\cdots\!88\)\( \beta_{2} + \)\(56\!\cdots\!04\)\( \beta_{3} - \)\(25\!\cdots\!08\)\( \beta_{4} - \)\(22\!\cdots\!28\)\( \beta_{5}) q^{78} +(-\)\(79\!\cdots\!40\)\( - \)\(25\!\cdots\!46\)\( \beta_{1} - \)\(13\!\cdots\!51\)\( \beta_{2} - \)\(30\!\cdots\!38\)\( \beta_{3} + \)\(15\!\cdots\!71\)\( \beta_{4} + \)\(86\!\cdots\!22\)\( \beta_{5}) q^{79} +(\)\(36\!\cdots\!00\)\( - \)\(17\!\cdots\!64\)\( \beta_{1} + \)\(64\!\cdots\!68\)\( \beta_{2} + \)\(91\!\cdots\!76\)\( \beta_{3} + \)\(19\!\cdots\!80\)\( \beta_{4} - \)\(34\!\cdots\!20\)\( \beta_{5}) q^{80} +\)\(95\!\cdots\!41\)\( q^{81} +(\)\(10\!\cdots\!14\)\( - \)\(45\!\cdots\!02\)\( \beta_{1} + \)\(44\!\cdots\!36\)\( \beta_{2} - \)\(25\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!88\)\( \beta_{4} + \)\(42\!\cdots\!24\)\( \beta_{5}) q^{82} +(\)\(61\!\cdots\!72\)\( - \)\(38\!\cdots\!08\)\( \beta_{1} - \)\(61\!\cdots\!20\)\( \beta_{2} - \)\(17\!\cdots\!82\)\( \beta_{3} + \)\(39\!\cdots\!44\)\( \beta_{4} + \)\(10\!\cdots\!04\)\( \beta_{5}) q^{83} +(\)\(13\!\cdots\!52\)\( - \)\(36\!\cdots\!08\)\( \beta_{1} + \)\(10\!\cdots\!16\)\( \beta_{2} - \)\(72\!\cdots\!40\)\( \beta_{3} - \)\(46\!\cdots\!40\)\( \beta_{4} - \)\(31\!\cdots\!60\)\( \beta_{5}) q^{84} +(\)\(41\!\cdots\!00\)\( - \)\(24\!\cdots\!64\)\( \beta_{1} + \)\(12\!\cdots\!08\)\( \beta_{2} + \)\(29\!\cdots\!46\)\( \beta_{3} - \)\(44\!\cdots\!00\)\( \beta_{4} - \)\(35\!\cdots\!00\)\( \beta_{5}) q^{85} +(\)\(18\!\cdots\!84\)\( - \)\(50\!\cdots\!68\)\( \beta_{1} - \)\(25\!\cdots\!48\)\( \beta_{2} + \)\(12\!\cdots\!24\)\( \beta_{3} + \)\(37\!\cdots\!32\)\( \beta_{4} + \)\(17\!\cdots\!68\)\( \beta_{5}) q^{86} +(\)\(17\!\cdots\!62\)\( + \)\(67\!\cdots\!90\)\( \beta_{1} + \)\(11\!\cdots\!54\)\( \beta_{2} + \)\(35\!\cdots\!91\)\( \beta_{3} + \)\(42\!\cdots\!58\)\( \beta_{4} + \)\(16\!\cdots\!24\)\( \beta_{5}) q^{87} +(\)\(49\!\cdots\!32\)\( + \)\(70\!\cdots\!32\)\( \beta_{1} + \)\(12\!\cdots\!64\)\( \beta_{2} - \)\(40\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!20\)\( \beta_{4} + \)\(45\!\cdots\!16\)\( \beta_{5}) q^{88} +(-\)\(41\!\cdots\!82\)\( + \)\(57\!\cdots\!72\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} - \)\(37\!\cdots\!68\)\( \beta_{3} - \)\(65\!\cdots\!24\)\( \beta_{4} - \)\(20\!\cdots\!32\)\( \beta_{5}) q^{89} +(\)\(37\!\cdots\!50\)\( - \)\(67\!\cdots\!32\)\( \beta_{1} + \)\(34\!\cdots\!74\)\( \beta_{2} - \)\(35\!\cdots\!42\)\( \beta_{3} - \)\(88\!\cdots\!40\)\( \beta_{4} + \)\(50\!\cdots\!10\)\( \beta_{5}) q^{90} +(\)\(27\!\cdots\!96\)\( + \)\(15\!\cdots\!08\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3} + \)\(13\!\cdots\!86\)\( \beta_{4} + \)\(24\!\cdots\!88\)\( \beta_{5}) q^{91} +(\)\(69\!\cdots\!12\)\( + \)\(13\!\cdots\!64\)\( \beta_{1} + \)\(35\!\cdots\!28\)\( \beta_{2} - \)\(16\!\cdots\!28\)\( \beta_{3} + \)\(29\!\cdots\!16\)\( \beta_{4} - \)\(25\!\cdots\!28\)\( \beta_{5}) q^{92} +(\)\(20\!\cdots\!32\)\( + \)\(77\!\cdots\!14\)\( \beta_{1} - \)\(16\!\cdots\!31\)\( \beta_{2} - \)\(42\!\cdots\!02\)\( \beta_{3} - \)\(40\!\cdots\!21\)\( \beta_{4} + \)\(20\!\cdots\!14\)\( \beta_{5}) q^{93} +(\)\(17\!\cdots\!52\)\( + \)\(64\!\cdots\!84\)\( \beta_{1} + \)\(73\!\cdots\!12\)\( \beta_{2} + \)\(21\!\cdots\!32\)\( \beta_{3} - \)\(23\!\cdots\!84\)\( \beta_{4} + \)\(17\!\cdots\!52\)\( \beta_{5}) q^{94} +(\)\(14\!\cdots\!00\)\( - \)\(52\!\cdots\!16\)\( \beta_{1} - \)\(57\!\cdots\!28\)\( \beta_{2} + \)\(65\!\cdots\!84\)\( \beta_{3} - \)\(40\!\cdots\!40\)\( \beta_{4} - \)\(14\!\cdots\!40\)\( \beta_{5}) q^{95} +(\)\(60\!\cdots\!84\)\( - \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(21\!\cdots\!48\)\( \beta_{2} + \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(57\!\cdots\!88\)\( \beta_{4} - \)\(57\!\cdots\!24\)\( \beta_{5}) q^{96} +(\)\(64\!\cdots\!94\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(19\!\cdots\!04\)\( \beta_{2} - \)\(85\!\cdots\!60\)\( \beta_{3} + \)\(43\!\cdots\!60\)\( \beta_{4} + \)\(68\!\cdots\!68\)\( \beta_{5}) q^{97} +(\)\(43\!\cdots\!29\)\( - \)\(16\!\cdots\!01\)\( \beta_{1} + \)\(12\!\cdots\!04\)\( \beta_{2} - \)\(57\!\cdots\!88\)\( \beta_{3} - \)\(27\!\cdots\!44\)\( \beta_{4} + \)\(18\!\cdots\!56\)\( \beta_{5}) q^{98} +(\)\(64\!\cdots\!16\)\( - \)\(32\!\cdots\!52\)\( \beta_{1} - \)\(73\!\cdots\!72\)\( \beta_{2} + \)\(31\!\cdots\!30\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4} - \)\(87\!\cdots\!04\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 13735355166q^{2} - 33354363399333138q^{3} + \)\(46\!\cdots\!52\)\(q^{4} - \)\(18\!\cdots\!00\)\(q^{5} - \)\(76\!\cdots\!18\)\(q^{6} - \)\(28\!\cdots\!08\)\(q^{7} + \)\(17\!\cdots\!92\)\(q^{8} + \)\(18\!\cdots\!74\)\(q^{9} + O(q^{10}) \) \( 6q + 13735355166q^{2} - 33354363399333138q^{3} + \)\(46\!\cdots\!52\)\(q^{4} - \)\(18\!\cdots\!00\)\(q^{5} - \)\(76\!\cdots\!18\)\(q^{6} - \)\(28\!\cdots\!08\)\(q^{7} + \)\(17\!\cdots\!92\)\(q^{8} + \)\(18\!\cdots\!74\)\(q^{9} + \)\(73\!\cdots\!00\)\(q^{10} + \)\(12\!\cdots\!24\)\(q^{11} - \)\(25\!\cdots\!96\)\(q^{12} - \)\(10\!\cdots\!64\)\(q^{13} - \)\(44\!\cdots\!52\)\(q^{14} + \)\(10\!\cdots\!00\)\(q^{15} - \)\(10\!\cdots\!80\)\(q^{16} - \)\(22\!\cdots\!96\)\(q^{17} + \)\(42\!\cdots\!14\)\(q^{18} - \)\(67\!\cdots\!52\)\(q^{19} + \)\(46\!\cdots\!00\)\(q^{20} + \)\(15\!\cdots\!84\)\(q^{21} + \)\(14\!\cdots\!68\)\(q^{22} + \)\(12\!\cdots\!64\)\(q^{23} - \)\(96\!\cdots\!16\)\(q^{24} + \)\(12\!\cdots\!50\)\(q^{25} + \)\(14\!\cdots\!92\)\(q^{26} - \)\(10\!\cdots\!02\)\(q^{27} - \)\(14\!\cdots\!44\)\(q^{28} - \)\(18\!\cdots\!64\)\(q^{29} - \)\(40\!\cdots\!00\)\(q^{30} - \)\(22\!\cdots\!04\)\(q^{31} - \)\(64\!\cdots\!48\)\(q^{32} - \)\(69\!\cdots\!52\)\(q^{33} + \)\(69\!\cdots\!04\)\(q^{34} - \)\(64\!\cdots\!00\)\(q^{35} + \)\(14\!\cdots\!08\)\(q^{36} + \)\(27\!\cdots\!52\)\(q^{37} + \)\(12\!\cdots\!04\)\(q^{38} + \)\(57\!\cdots\!72\)\(q^{39} + \)\(11\!\cdots\!00\)\(q^{40} + \)\(26\!\cdots\!96\)\(q^{41} + \)\(24\!\cdots\!96\)\(q^{42} - \)\(11\!\cdots\!00\)\(q^{43} - \)\(34\!\cdots\!04\)\(q^{44} - \)\(55\!\cdots\!00\)\(q^{45} - \)\(45\!\cdots\!24\)\(q^{46} + \)\(58\!\cdots\!60\)\(q^{47} + \)\(59\!\cdots\!40\)\(q^{48} + \)\(98\!\cdots\!70\)\(q^{49} + \)\(47\!\cdots\!50\)\(q^{50} + \)\(12\!\cdots\!08\)\(q^{51} - \)\(77\!\cdots\!76\)\(q^{52} - \)\(13\!\cdots\!36\)\(q^{53} - \)\(23\!\cdots\!22\)\(q^{54} - \)\(51\!\cdots\!00\)\(q^{55} - \)\(53\!\cdots\!40\)\(q^{56} + \)\(37\!\cdots\!96\)\(q^{57} - \)\(27\!\cdots\!12\)\(q^{58} + \)\(43\!\cdots\!12\)\(q^{59} - \)\(25\!\cdots\!00\)\(q^{60} - \)\(30\!\cdots\!40\)\(q^{61} + \)\(13\!\cdots\!84\)\(q^{62} - \)\(86\!\cdots\!32\)\(q^{63} - \)\(33\!\cdots\!12\)\(q^{64} + \)\(91\!\cdots\!00\)\(q^{65} - \)\(78\!\cdots\!64\)\(q^{66} - \)\(36\!\cdots\!96\)\(q^{67} + \)\(62\!\cdots\!88\)\(q^{68} - \)\(68\!\cdots\!72\)\(q^{69} - \)\(17\!\cdots\!00\)\(q^{70} + \)\(25\!\cdots\!72\)\(q^{71} + \)\(53\!\cdots\!68\)\(q^{72} + \)\(16\!\cdots\!24\)\(q^{73} + \)\(10\!\cdots\!68\)\(q^{74} - \)\(69\!\cdots\!50\)\(q^{75} - \)\(51\!\cdots\!12\)\(q^{76} + \)\(35\!\cdots\!44\)\(q^{77} - \)\(82\!\cdots\!16\)\(q^{78} - \)\(47\!\cdots\!40\)\(q^{79} + \)\(22\!\cdots\!00\)\(q^{80} + \)\(57\!\cdots\!46\)\(q^{81} + \)\(64\!\cdots\!84\)\(q^{82} + \)\(36\!\cdots\!32\)\(q^{83} + \)\(81\!\cdots\!12\)\(q^{84} + \)\(25\!\cdots\!00\)\(q^{85} + \)\(10\!\cdots\!04\)\(q^{86} + \)\(10\!\cdots\!72\)\(q^{87} + \)\(29\!\cdots\!92\)\(q^{88} - \)\(24\!\cdots\!92\)\(q^{89} + \)\(22\!\cdots\!00\)\(q^{90} + \)\(16\!\cdots\!76\)\(q^{91} + \)\(41\!\cdots\!72\)\(q^{92} + \)\(12\!\cdots\!92\)\(q^{93} + \)\(10\!\cdots\!12\)\(q^{94} + \)\(88\!\cdots\!00\)\(q^{95} + \)\(36\!\cdots\!04\)\(q^{96} + \)\(38\!\cdots\!64\)\(q^{97} + \)\(26\!\cdots\!74\)\(q^{98} + \)\(38\!\cdots\!96\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 18265801580559590892 x^{4} - 7523970115455178716247853536 x^{3} + 78501261395396048023472482629898922880 x^{2} + 29605717530431234535889524616359650095893868800 x - 80637078962263575025979556823830075867035962985404800000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 3 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} - 22243446972 \nu - 219189618955593367272 \)
\(\beta_{3}\)\(=\)\((\)\(547656442731 \nu^{5} - 1948427421673813232157 \nu^{4} - 6610099066817607565987436485344 \nu^{3} + 22045821613742858828448779691123530008800 \nu^{2} + 15511387419718206739455612128355924712055616441600 \nu - 43849158917089635084556338654244679976800281181975924394240\)\()/ \)\(50\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(152285563749136221 \nu^{5} - 44198762943573650977329627 \nu^{4} - 2091774541352086674869707906974556704 \nu^{3} - 940502006414093841225016418875952277260352480 \nu^{2} + 3792145608704657568185822853913921837818945092995541760 \nu + 2514372192078802698287077358613077227357378667670079244064272640\)\()/ \)\(35\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-35543898440648379 \nu^{5} + 48817395431103207716732493 \nu^{4} + 576416861397703537014097471318617312 \nu^{3} - 496748380676434680942250393566449513324067552 \nu^{2} - 1863568152173038225032573567221321416281990886847102208 \nu + 1175497631740102854763090187082864397921472619598164033538144512\)\()/ \)\(70\!\cdots\!16\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3707241162 \beta_{1} + 219189618966715090758\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(141 \beta_{5} + 110 \beta_{4} + 2166905 \beta_{3} + 9413817317 \beta_{2} + 348133172179229341527 \beta_{1} + 812588778387279013456075631868\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(215683758605 \beta_{5} + 626554370414 \beta_{4} - 14890455560985607 \beta_{3} + 270176227379470727469 \beta_{2} + 1593848457539098681444566001255 \beta_{1} + 38153588681100594335049668773620993718764\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(17618615270372932220961 \beta_{5} + 18636474309890062085734 \beta_{4} + 254913665554799621734702653 \beta_{3} + 1732506467044683769447592070401 \beta_{2} + 37593243180169074975655494272242500516387 \beta_{1} + 174677518108355491808139991369890052791801708585372\)\()/1944\)

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
3.84072e9
2.02873e9
9.72648e8
−1.58092e9
−2.19503e9
−3.06614e9
−2.07551e10 −5.55906e15 2.83199e20 −8.85856e22 1.15379e26 −1.53665e28 −2.81491e30 3.09032e31 1.83860e33
1.2 −9.88316e9 −5.55906e15 −4.98972e19 −3.20918e23 5.49411e25 2.08311e28 1.95164e30 3.09032e31 3.17168e33
1.3 −3.54666e9 −5.55906e15 −1.34995e20 1.66367e23 1.97161e25 −1.26991e28 1.00218e30 3.09032e31 −5.90048e32
1.4 1.17747e10 −5.55906e15 −8.92992e18 8.66624e22 −6.54564e25 3.38994e28 −1.84279e30 3.09032e31 1.02043e33
1.5 1.54594e10 −5.55906e15 9.14202e19 −4.50800e23 −8.59399e25 −2.44076e28 −8.68106e29 3.09032e31 −6.96912e33
1.6 2.06861e10 −5.55906e15 2.80340e20 4.26985e23 −1.14995e26 −3.03250e28 2.74642e30 3.09032e31 8.83265e33
\(n\): 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 3.68.a.b 6
3.b odd 2 1 9.68.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.68.a.b 6 1.a even 1 1 trivial
9.68.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 \( T_{2}^{6} - 13735355166 T_{2}^{5} - \)\(57\!\cdots\!32\)\( T_{2}^{4} + \)\(74\!\cdots\!48\)\( T_{2}^{3} + \)\(70\!\cdots\!96\)\( T_{2}^{2} - \)\(63\!\cdots\!44\)\( T_{2} - \)\(27\!\cdots\!52\)\( \) acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 13735355166 T + \)\(30\!\cdots\!36\)\( T^{2} - \)\(27\!\cdots\!92\)\( T^{3} + \)\(55\!\cdots\!72\)\( T^{4} - \)\(35\!\cdots\!52\)\( T^{5} + \)\(66\!\cdots\!36\)\( T^{6} - \)\(51\!\cdots\!56\)\( T^{7} + \)\(12\!\cdots\!48\)\( T^{8} - \)\(87\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!16\)\( T^{10} - \)\(96\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!04\)\( T^{12} \)
$3$ \( ( 1 + 5559060566555523 T )^{6} \)
$5$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(15\!\cdots\!50\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!75\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(90\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(61\!\cdots\!75\)\( T^{8} + \)\(87\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!50\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{11} + \)\(96\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 + \)\(28\!\cdots\!08\)\( T + \)\(11\!\cdots\!26\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(52\!\cdots\!11\)\( T^{4} + \)\(65\!\cdots\!84\)\( T^{5} + \)\(28\!\cdots\!12\)\( T^{6} + \)\(27\!\cdots\!12\)\( T^{7} + \)\(92\!\cdots\!39\)\( T^{8} + \)\(97\!\cdots\!52\)\( T^{9} + \)\(35\!\cdots\!26\)\( T^{10} + \)\(35\!\cdots\!44\)\( T^{11} + \)\(53\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 - \)\(12\!\cdots\!24\)\( T + \)\(26\!\cdots\!34\)\( T^{2} + \)\(77\!\cdots\!12\)\( T^{3} + \)\(32\!\cdots\!63\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} + \)\(88\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!83\)\( T^{8} + \)\(16\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!54\)\( T^{10} - \)\(92\!\cdots\!24\)\( T^{11} + \)\(43\!\cdots\!21\)\( T^{12} \)
$13$ \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(60\!\cdots\!10\)\( T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!15\)\( T^{4} - \)\(86\!\cdots\!96\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} - \)\(37\!\cdots\!32\)\( T^{7} + \)\(50\!\cdots\!35\)\( T^{8} - \)\(92\!\cdots\!56\)\( T^{9} + \)\(20\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!48\)\( T^{11} + \)\(63\!\cdots\!69\)\( T^{12} \)
$17$ \( 1 + \)\(22\!\cdots\!96\)\( T + \)\(75\!\cdots\!78\)\( T^{2} - \)\(32\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!55\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{5} + \)\(69\!\cdots\!56\)\( T^{6} - \)\(81\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} - \)\(67\!\cdots\!80\)\( T^{9} + \)\(43\!\cdots\!98\)\( T^{10} + \)\(35\!\cdots\!28\)\( T^{11} + \)\(43\!\cdots\!89\)\( T^{12} \)
$19$ \( 1 + \)\(67\!\cdots\!52\)\( T + \)\(82\!\cdots\!42\)\( T^{2} + \)\(39\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{5} - \)\(70\!\cdots\!92\)\( T^{6} - \)\(64\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!63\)\( T^{8} + \)\(42\!\cdots\!16\)\( T^{9} + \)\(42\!\cdots\!22\)\( T^{10} + \)\(16\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!61\)\( T^{12} \)
$23$ \( 1 - \)\(12\!\cdots\!64\)\( T + \)\(91\!\cdots\!54\)\( T^{2} - \)\(39\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} + \)\(43\!\cdots\!48\)\( T^{5} - \)\(92\!\cdots\!12\)\( T^{6} + \)\(75\!\cdots\!56\)\( T^{7} + \)\(30\!\cdots\!79\)\( T^{8} - \)\(20\!\cdots\!96\)\( T^{9} + \)\(80\!\cdots\!74\)\( T^{10} - \)\(18\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} \)
$29$ \( 1 + \)\(18\!\cdots\!64\)\( T + \)\(49\!\cdots\!66\)\( T^{2} + \)\(61\!\cdots\!32\)\( T^{3} + \)\(95\!\cdots\!67\)\( T^{4} + \)\(90\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + \)\(86\!\cdots\!96\)\( T^{7} + \)\(87\!\cdots\!27\)\( T^{8} + \)\(53\!\cdots\!28\)\( T^{9} + \)\(41\!\cdots\!26\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{11} + \)\(76\!\cdots\!41\)\( T^{12} \)
$31$ \( 1 + \)\(22\!\cdots\!04\)\( T + \)\(65\!\cdots\!38\)\( T^{2} + \)\(94\!\cdots\!52\)\( T^{3} + \)\(15\!\cdots\!83\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!56\)\( T^{7} + \)\(10\!\cdots\!43\)\( T^{8} + \)\(54\!\cdots\!12\)\( T^{9} + \)\(31\!\cdots\!58\)\( T^{10} + \)\(90\!\cdots\!04\)\( T^{11} + \)\(33\!\cdots\!61\)\( T^{12} \)
$37$ \( 1 - \)\(27\!\cdots\!52\)\( T + \)\(27\!\cdots\!26\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!91\)\( T^{4} + \)\(79\!\cdots\!04\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} + \)\(93\!\cdots\!32\)\( T^{7} + \)\(39\!\cdots\!99\)\( T^{8} - \)\(16\!\cdots\!68\)\( T^{9} + \)\(52\!\cdots\!46\)\( T^{10} - \)\(61\!\cdots\!36\)\( T^{11} + \)\(26\!\cdots\!69\)\( T^{12} \)
$41$ \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(71\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!43\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!72\)\( T^{6} - \)\(30\!\cdots\!04\)\( T^{7} + \)\(26\!\cdots\!23\)\( T^{8} - \)\(18\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!18\)\( T^{10} - \)\(50\!\cdots\!96\)\( T^{11} + \)\(21\!\cdots\!81\)\( T^{12} \)
$43$ \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(98\!\cdots\!10\)\( T^{2} + \)\(79\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!47\)\( T^{4} + \)\(36\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(99\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!03\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!00\)\( T^{11} + \)\(45\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(58\!\cdots\!60\)\( T + \)\(21\!\cdots\!70\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!07\)\( T^{4} + \)\(34\!\cdots\!20\)\( T^{5} + \)\(37\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{7} + \)\(38\!\cdots\!83\)\( T^{8} - \)\(30\!\cdots\!60\)\( T^{9} + \)\(29\!\cdots\!70\)\( T^{10} - \)\(82\!\cdots\!80\)\( T^{11} + \)\(15\!\cdots\!09\)\( T^{12} \)
$53$ \( 1 + \)\(13\!\cdots\!36\)\( T + \)\(21\!\cdots\!74\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!11\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(76\!\cdots\!08\)\( T^{6} + \)\(38\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!59\)\( T^{8} + \)\(72\!\cdots\!04\)\( T^{9} + \)\(27\!\cdots\!14\)\( T^{10} + \)\(58\!\cdots\!52\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \)
$59$ \( 1 - \)\(43\!\cdots\!12\)\( T + \)\(22\!\cdots\!62\)\( T^{2} - \)\(61\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!63\)\( T^{4} - \)\(44\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(19\!\cdots\!92\)\( T^{7} + \)\(40\!\cdots\!43\)\( T^{8} - \)\(53\!\cdots\!36\)\( T^{9} + \)\(87\!\cdots\!02\)\( T^{10} - \)\(74\!\cdots\!88\)\( T^{11} + \)\(76\!\cdots\!81\)\( T^{12} \)
$61$ \( 1 + \)\(30\!\cdots\!40\)\( T + \)\(11\!\cdots\!74\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(11\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!94\)\( T^{10} + \)\(36\!\cdots\!40\)\( T^{11} + \)\(50\!\cdots\!21\)\( T^{12} \)
$67$ \( 1 + \)\(36\!\cdots\!96\)\( T + \)\(83\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(32\!\cdots\!55\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} + \)\(84\!\cdots\!56\)\( T^{6} - \)\(62\!\cdots\!52\)\( T^{7} + \)\(16\!\cdots\!95\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(20\!\cdots\!98\)\( T^{10} + \)\(19\!\cdots\!28\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$71$ \( 1 - \)\(25\!\cdots\!72\)\( T + \)\(73\!\cdots\!06\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!95\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{5} + \)\(29\!\cdots\!44\)\( T^{6} - \)\(25\!\cdots\!72\)\( T^{7} + \)\(23\!\cdots\!95\)\( T^{8} - \)\(14\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!66\)\( T^{10} - \)\(37\!\cdots\!72\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} \)
$73$ \( 1 - \)\(16\!\cdots\!24\)\( T + \)\(21\!\cdots\!14\)\( T^{2} - \)\(26\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!11\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} - \)\(20\!\cdots\!24\)\( T^{7} + \)\(13\!\cdots\!99\)\( T^{8} - \)\(89\!\cdots\!16\)\( T^{9} + \)\(50\!\cdots\!34\)\( T^{10} - \)\(27\!\cdots\!68\)\( T^{11} + \)\(11\!\cdots\!29\)\( T^{12} \)
$79$ \( 1 + \)\(47\!\cdots\!40\)\( T + \)\(66\!\cdots\!54\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} + \)\(55\!\cdots\!00\)\( T^{5} + \)\(35\!\cdots\!80\)\( T^{6} + \)\(77\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!15\)\( T^{8} + \)\(61\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!94\)\( T^{10} + \)\(24\!\cdots\!60\)\( T^{11} + \)\(70\!\cdots\!41\)\( T^{12} \)
$83$ \( 1 - \)\(36\!\cdots\!32\)\( T + \)\(21\!\cdots\!74\)\( T^{2} - \)\(63\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} - \)\(45\!\cdots\!64\)\( T^{5} + \)\(96\!\cdots\!44\)\( T^{6} - \)\(17\!\cdots\!28\)\( T^{7} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(34\!\cdots\!48\)\( T^{9} + \)\(43\!\cdots\!34\)\( T^{10} - \)\(28\!\cdots\!24\)\( T^{11} + \)\(29\!\cdots\!89\)\( T^{12} \)
$89$ \( 1 + \)\(24\!\cdots\!92\)\( T + \)\(88\!\cdots\!02\)\( T^{2} - \)\(17\!\cdots\!96\)\( T^{3} + \)\(51\!\cdots\!03\)\( T^{4} + \)\(14\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!88\)\( T^{6} + \)\(60\!\cdots\!52\)\( T^{7} + \)\(85\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{9} + \)\(24\!\cdots\!62\)\( T^{10} + \)\(27\!\cdots\!08\)\( T^{11} + \)\(45\!\cdots\!21\)\( T^{12} \)
$97$ \( 1 - \)\(38\!\cdots\!64\)\( T + \)\(36\!\cdots\!18\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!55\)\( T^{4} - \)\(40\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!96\)\( T^{6} - \)\(52\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} - \)\(43\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!98\)\( T^{10} - \)\(14\!\cdots\!52\)\( T^{11} + \)\(48\!\cdots\!09\)\( T^{12} \)
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