Properties

Label 1.116.a.a
Level 1
Weight 116
Character orbit 1.a
Self dual yes
Analytic conductor 83.750
Analytic rank 0
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 116 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(83.7504016273\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{142}\cdot 3^{52}\cdot 5^{17}\cdot 7^{8}\cdot 11^{3}\cdot 13^{3}\cdot 17\cdot 19^{3}\cdot 23^{3}\cdot 29^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-12908974454383949 - \beta_{1}) q^{2} +(\)\(38\!\cdots\!92\)\( - 548079911 \beta_{1} + \beta_{2}) q^{3} +(\)\(11\!\cdots\!47\)\( + 7744231697620209 \beta_{1} - 484553 \beta_{2} + \beta_{3}) q^{4} +(-\)\(10\!\cdots\!70\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + 586675963404 \beta_{2} + 4036 \beta_{3} + \beta_{4}) q^{5} +(\)\(28\!\cdots\!05\)\( + \)\(52\!\cdots\!39\)\( \beta_{1} - 40505871977025988 \beta_{2} - 107907690 \beta_{3} + 797 \beta_{4} - \beta_{5}) q^{6} +(\)\(20\!\cdots\!38\)\( + \)\(17\!\cdots\!73\)\( \beta_{1} + \)\(30\!\cdots\!57\)\( \beta_{2} + 7240853888179 \beta_{3} - 25994649 \beta_{4} + 606 \beta_{5} - \beta_{6}) q^{7} +(-\)\(24\!\cdots\!22\)\( - \)\(51\!\cdots\!28\)\( \beta_{1} - \)\(95\!\cdots\!00\)\( \beta_{2} - 619136513711257 \beta_{3} - 189393293777 \beta_{4} - 664275 \beta_{5} - 413 \beta_{6} - \beta_{7}) q^{8} +(\)\(28\!\cdots\!37\)\( + \)\(53\!\cdots\!97\)\( \beta_{1} - \)\(26\!\cdots\!54\)\( \beta_{2} - 39934242747451754594 \beta_{3} - 133277535964188 \beta_{4} - 459502997 \beta_{5} + 313389 \beta_{6} - 182 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\) \( q +(-12908974454383949 - \beta_{1}) q^{2} +(\)\(38\!\cdots\!92\)\( - 548079911 \beta_{1} + \beta_{2}) q^{3} +(\)\(11\!\cdots\!47\)\( + 7744231697620209 \beta_{1} - 484553 \beta_{2} + \beta_{3}) q^{4} +(-\)\(10\!\cdots\!70\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + 586675963404 \beta_{2} + 4036 \beta_{3} + \beta_{4}) q^{5} +(\)\(28\!\cdots\!05\)\( + \)\(52\!\cdots\!39\)\( \beta_{1} - 40505871977025988 \beta_{2} - 107907690 \beta_{3} + 797 \beta_{4} - \beta_{5}) q^{6} +(\)\(20\!\cdots\!38\)\( + \)\(17\!\cdots\!73\)\( \beta_{1} + \)\(30\!\cdots\!57\)\( \beta_{2} + 7240853888179 \beta_{3} - 25994649 \beta_{4} + 606 \beta_{5} - \beta_{6}) q^{7} +(-\)\(24\!\cdots\!22\)\( - \)\(51\!\cdots\!28\)\( \beta_{1} - \)\(95\!\cdots\!00\)\( \beta_{2} - 619136513711257 \beta_{3} - 189393293777 \beta_{4} - 664275 \beta_{5} - 413 \beta_{6} - \beta_{7}) q^{8} +(\)\(28\!\cdots\!37\)\( + \)\(53\!\cdots\!97\)\( \beta_{1} - \)\(26\!\cdots\!54\)\( \beta_{2} - 39934242747451754594 \beta_{3} - 133277535964188 \beta_{4} - 459502997 \beta_{5} + 313389 \beta_{6} - 182 \beta_{7} + \beta_{8}) q^{9} +(\)\(76\!\cdots\!22\)\( + \)\(88\!\cdots\!90\)\( \beta_{1} - \)\(12\!\cdots\!16\)\( \beta_{2} - \)\(10\!\cdots\!20\)\( \beta_{3} - 56147047023408348 \beta_{4} - 184957797900 \beta_{5} + 21531704 \beta_{6} - 156480 \beta_{7} + 264 \beta_{8}) q^{10} +(-\)\(49\!\cdots\!08\)\( - \)\(43\!\cdots\!11\)\( \beta_{1} + \)\(29\!\cdots\!57\)\( \beta_{2} + \)\(93\!\cdots\!02\)\( \beta_{3} - 12632865744118755930 \beta_{4} - 28420377803664 \beta_{5} - 69136732774 \beta_{6} - 72116648 \beta_{7} - 18756 \beta_{8}) q^{11} +(-\)\(47\!\cdots\!56\)\( - \)\(55\!\cdots\!68\)\( \beta_{1} + \)\(46\!\cdots\!96\)\( \beta_{2} + \)\(23\!\cdots\!88\)\( \beta_{3} - \)\(83\!\cdots\!92\)\( \beta_{4} + 636374606564640 \beta_{5} - 19813923913248 \beta_{6} - 3599055456 \beta_{7} - 2892480 \beta_{8}) q^{12} +(\)\(21\!\cdots\!42\)\( + \)\(11\!\cdots\!71\)\( \beta_{1} + \)\(64\!\cdots\!72\)\( \beta_{2} - \)\(30\!\cdots\!32\)\( \beta_{3} + \)\(44\!\cdots\!21\)\( \beta_{4} - 2414876052438376326 \beta_{5} - 1880087776576106 \beta_{6} + 807139399116 \beta_{7} + 515975790 \beta_{8}) q^{13} +(-\)\(97\!\cdots\!94\)\( - \)\(41\!\cdots\!18\)\( \beta_{1} - \)\(39\!\cdots\!64\)\( \beta_{2} + \)\(14\!\cdots\!36\)\( \beta_{3} - \)\(47\!\cdots\!22\)\( \beta_{4} - \)\(87\!\cdots\!62\)\( \beta_{5} - 20784814855981408 \beta_{6} - 34340071544576 \beta_{7} - 40630437792 \beta_{8}) q^{14} +(\)\(63\!\cdots\!34\)\( + \)\(78\!\cdots\!55\)\( \beta_{1} - \)\(46\!\cdots\!77\)\( \beta_{2} - \)\(13\!\cdots\!15\)\( \beta_{3} + \)\(27\!\cdots\!69\)\( \beta_{4} - \)\(70\!\cdots\!50\)\( \beta_{5} + 7522015980496972113 \beta_{6} - 45823238274960 \beta_{7} + 2123970677208 \beta_{8}) q^{15} +(-\)\(21\!\cdots\!96\)\( - \)\(31\!\cdots\!40\)\( \beta_{1} + \)\(47\!\cdots\!20\)\( \beta_{2} + \)\(48\!\cdots\!08\)\( \beta_{3} + \)\(51\!\cdots\!56\)\( \beta_{4} - \)\(75\!\cdots\!72\)\( \beta_{5} - \)\(19\!\cdots\!84\)\( \beta_{6} + 63376353383092152 \beta_{7} - 83303711279616 \beta_{8}) q^{16} +(\)\(37\!\cdots\!94\)\( - \)\(34\!\cdots\!71\)\( \beta_{1} + \)\(41\!\cdots\!02\)\( \beta_{2} - \)\(62\!\cdots\!54\)\( \beta_{3} + \)\(14\!\cdots\!84\)\( \beta_{4} + \)\(59\!\cdots\!59\)\( \beta_{5} - \)\(21\!\cdots\!59\)\( \beta_{6} - 3268486108013674110 \beta_{7} + 2603010144162285 \beta_{8}) q^{17} +(-\)\(32\!\cdots\!41\)\( - \)\(14\!\cdots\!53\)\( \beta_{1} + \)\(26\!\cdots\!32\)\( \beta_{2} + \)\(32\!\cdots\!64\)\( \beta_{3} + \)\(11\!\cdots\!56\)\( \beta_{4} + \)\(64\!\cdots\!96\)\( \beta_{5} + \)\(20\!\cdots\!44\)\( \beta_{6} + 99000356531144415360 \beta_{7} - 67158677161258320 \beta_{8}) q^{18} +(\)\(62\!\cdots\!36\)\( - \)\(54\!\cdots\!57\)\( \beta_{1} + \)\(17\!\cdots\!79\)\( \beta_{2} + \)\(76\!\cdots\!30\)\( \beta_{3} + \)\(37\!\cdots\!90\)\( \beta_{4} - \)\(35\!\cdots\!32\)\( \beta_{5} - \)\(45\!\cdots\!22\)\( \beta_{6} - \)\(21\!\cdots\!44\)\( \beta_{7} + 1464583823069070132 \beta_{8}) q^{19} +(-\)\(13\!\cdots\!10\)\( + \)\(19\!\cdots\!62\)\( \beta_{1} - \)\(15\!\cdots\!98\)\( \beta_{2} + \)\(57\!\cdots\!18\)\( \beta_{3} - \)\(59\!\cdots\!12\)\( \beta_{4} + \)\(74\!\cdots\!00\)\( \beta_{5} + \)\(33\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!00\)\( \beta_{7} - 27434369381284396800 \beta_{8}) q^{20} +(\)\(31\!\cdots\!60\)\( + \)\(68\!\cdots\!02\)\( \beta_{1} - \)\(13\!\cdots\!24\)\( \beta_{2} + \)\(13\!\cdots\!44\)\( \beta_{3} - \)\(53\!\cdots\!24\)\( \beta_{4} + \)\(89\!\cdots\!66\)\( \beta_{5} + \)\(56\!\cdots\!34\)\( \beta_{6} - \)\(34\!\cdots\!52\)\( \beta_{7} + \)\(44\!\cdots\!66\)\( \beta_{8}) q^{21} +(\)\(23\!\cdots\!87\)\( + \)\(14\!\cdots\!41\)\( \beta_{1} + \)\(59\!\cdots\!32\)\( \beta_{2} + \)\(34\!\cdots\!98\)\( \beta_{3} + \)\(17\!\cdots\!31\)\( \beta_{4} - \)\(97\!\cdots\!91\)\( \beta_{5} - \)\(19\!\cdots\!16\)\( \beta_{6} + \)\(18\!\cdots\!76\)\( \beta_{7} - \)\(63\!\cdots\!40\)\( \beta_{8}) q^{22} +(\)\(56\!\cdots\!02\)\( - \)\(57\!\cdots\!53\)\( \beta_{1} - \)\(55\!\cdots\!61\)\( \beta_{2} + \)\(33\!\cdots\!05\)\( \beta_{3} + \)\(78\!\cdots\!05\)\( \beta_{4} - \)\(71\!\cdots\!10\)\( \beta_{5} + \)\(26\!\cdots\!45\)\( \beta_{6} + \)\(19\!\cdots\!40\)\( \beta_{7} + \)\(79\!\cdots\!40\)\( \beta_{8}) q^{23} +(-\)\(83\!\cdots\!20\)\( - \)\(53\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2} + \)\(21\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!68\)\( \beta_{4} + \)\(21\!\cdots\!40\)\( \beta_{5} - \)\(19\!\cdots\!44\)\( \beta_{6} - \)\(64\!\cdots\!28\)\( \beta_{7} - \)\(87\!\cdots\!96\)\( \beta_{8}) q^{24} +(\)\(13\!\cdots\!35\)\( + \)\(22\!\cdots\!70\)\( \beta_{1} + \)\(26\!\cdots\!40\)\( \beta_{2} - \)\(24\!\cdots\!20\)\( \beta_{3} - \)\(76\!\cdots\!60\)\( \beta_{4} - \)\(11\!\cdots\!50\)\( \beta_{5} + \)\(72\!\cdots\!70\)\( \beta_{6} + \)\(98\!\cdots\!00\)\( \beta_{7} + \)\(83\!\cdots\!70\)\( \beta_{8}) q^{25} +(-\)\(65\!\cdots\!26\)\( - \)\(18\!\cdots\!38\)\( \beta_{1} + \)\(88\!\cdots\!56\)\( \beta_{2} - \)\(35\!\cdots\!52\)\( \beta_{3} + \)\(77\!\cdots\!24\)\( \beta_{4} - \)\(10\!\cdots\!40\)\( \beta_{5} + \)\(14\!\cdots\!72\)\( \beta_{6} - \)\(10\!\cdots\!36\)\( \beta_{7} - \)\(69\!\cdots\!52\)\( \beta_{8}) q^{26} +(-\)\(60\!\cdots\!48\)\( - \)\(15\!\cdots\!12\)\( \beta_{1} + \)\(47\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!38\)\( \beta_{3} + \)\(69\!\cdots\!82\)\( \beta_{4} + \)\(18\!\cdots\!60\)\( \beta_{5} - \)\(17\!\cdots\!42\)\( \beta_{6} + \)\(81\!\cdots\!16\)\( \beta_{7} + \)\(48\!\cdots\!60\)\( \beta_{8}) q^{27} +(\)\(14\!\cdots\!20\)\( - \)\(33\!\cdots\!96\)\( \beta_{1} + \)\(27\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!52\)\( \beta_{3} - \)\(21\!\cdots\!48\)\( \beta_{4} - \)\(74\!\cdots\!40\)\( \beta_{5} + \)\(81\!\cdots\!88\)\( \beta_{6} - \)\(43\!\cdots\!44\)\( \beta_{7} - \)\(25\!\cdots\!80\)\( \beta_{8}) q^{28} +(\)\(27\!\cdots\!58\)\( + \)\(51\!\cdots\!93\)\( \beta_{1} + \)\(30\!\cdots\!84\)\( \beta_{2} + \)\(15\!\cdots\!92\)\( \beta_{3} - \)\(33\!\cdots\!11\)\( \beta_{4} - \)\(56\!\cdots\!80\)\( \beta_{5} + \)\(21\!\cdots\!92\)\( \beta_{6} + \)\(73\!\cdots\!84\)\( \beta_{7} + \)\(74\!\cdots\!48\)\( \beta_{8}) q^{29} +(-\)\(12\!\cdots\!90\)\( - \)\(19\!\cdots\!82\)\( \beta_{1} + \)\(33\!\cdots\!28\)\( \beta_{2} - \)\(58\!\cdots\!48\)\( \beta_{3} + \)\(15\!\cdots\!82\)\( \beta_{4} + \)\(74\!\cdots\!50\)\( \beta_{5} - \)\(65\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!00\)\( \beta_{7} + \)\(39\!\cdots\!00\)\( \beta_{8}) q^{30} +(-\)\(12\!\cdots\!64\)\( - \)\(57\!\cdots\!28\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(75\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!24\)\( \beta_{4} - \)\(21\!\cdots\!68\)\( \beta_{5} + \)\(48\!\cdots\!44\)\( \beta_{6} - \)\(19\!\cdots\!12\)\( \beta_{7} - \)\(86\!\cdots\!64\)\( \beta_{8}) q^{31} +(\)\(20\!\cdots\!04\)\( + \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(35\!\cdots\!44\)\( \beta_{2} - \)\(23\!\cdots\!64\)\( \beta_{3} - \)\(69\!\cdots\!76\)\( \beta_{4} - \)\(16\!\cdots\!96\)\( \beta_{5} - \)\(12\!\cdots\!24\)\( \beta_{6} + \)\(14\!\cdots\!60\)\( \beta_{7} + \)\(85\!\cdots\!80\)\( \beta_{8}) q^{32} +(\)\(41\!\cdots\!88\)\( - \)\(13\!\cdots\!93\)\( \beta_{1} - \)\(12\!\cdots\!14\)\( \beta_{2} + \)\(22\!\cdots\!06\)\( \beta_{3} - \)\(41\!\cdots\!32\)\( \beta_{4} + \)\(15\!\cdots\!61\)\( \beta_{5} - \)\(93\!\cdots\!53\)\( \beta_{6} - \)\(74\!\cdots\!34\)\( \beta_{7} - \)\(60\!\cdots\!05\)\( \beta_{8}) q^{33} +(\)\(17\!\cdots\!34\)\( - \)\(13\!\cdots\!10\)\( \beta_{1} - \)\(27\!\cdots\!40\)\( \beta_{2} + \)\(10\!\cdots\!28\)\( \beta_{3} + \)\(35\!\cdots\!00\)\( \beta_{4} - \)\(25\!\cdots\!92\)\( \beta_{5} + \)\(12\!\cdots\!88\)\( \beta_{6} + \)\(19\!\cdots\!56\)\( \beta_{7} + \)\(33\!\cdots\!92\)\( \beta_{8}) q^{34} +(-\)\(58\!\cdots\!48\)\( + \)\(65\!\cdots\!40\)\( \beta_{1} - \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(17\!\cdots\!20\)\( \beta_{3} + \)\(53\!\cdots\!32\)\( \beta_{4} - \)\(31\!\cdots\!00\)\( \beta_{5} - \)\(61\!\cdots\!36\)\( \beta_{6} + \)\(54\!\cdots\!20\)\( \beta_{7} - \)\(13\!\cdots\!76\)\( \beta_{8}) q^{35} +(-\)\(38\!\cdots\!45\)\( - \)\(20\!\cdots\!35\)\( \beta_{1} - \)\(27\!\cdots\!25\)\( \beta_{2} - \)\(39\!\cdots\!71\)\( \beta_{3} - \)\(10\!\cdots\!08\)\( \beta_{4} + \)\(19\!\cdots\!04\)\( \beta_{5} + \)\(85\!\cdots\!80\)\( \beta_{6} - \)\(10\!\cdots\!60\)\( \beta_{7} + \)\(21\!\cdots\!40\)\( \beta_{8}) q^{36} +(\)\(31\!\cdots\!10\)\( - \)\(31\!\cdots\!77\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2} - \)\(24\!\cdots\!36\)\( \beta_{3} + \)\(14\!\cdots\!69\)\( \beta_{4} + \)\(24\!\cdots\!70\)\( \beta_{5} + \)\(10\!\cdots\!86\)\( \beta_{6} + \)\(67\!\cdots\!12\)\( \beta_{7} + \)\(21\!\cdots\!50\)\( \beta_{8}) q^{37} +(\)\(20\!\cdots\!97\)\( - \)\(89\!\cdots\!53\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2} + \)\(93\!\cdots\!46\)\( \beta_{3} + \)\(15\!\cdots\!65\)\( \beta_{4} - \)\(43\!\cdots\!73\)\( \beta_{5} - \)\(87\!\cdots\!80\)\( \beta_{6} - \)\(27\!\cdots\!36\)\( \beta_{7} - \)\(26\!\cdots\!20\)\( \beta_{8}) q^{38} +(\)\(63\!\cdots\!06\)\( - \)\(19\!\cdots\!09\)\( \beta_{1} + \)\(47\!\cdots\!03\)\( \beta_{2} + \)\(75\!\cdots\!05\)\( \beta_{3} - \)\(29\!\cdots\!35\)\( \beta_{4} + \)\(14\!\cdots\!86\)\( \beta_{5} + \)\(31\!\cdots\!01\)\( \beta_{6} + \)\(58\!\cdots\!32\)\( \beta_{7} + \)\(18\!\cdots\!64\)\( \beta_{8}) q^{39} +(-\)\(41\!\cdots\!60\)\( - \)\(22\!\cdots\!00\)\( \beta_{1} - \)\(28\!\cdots\!20\)\( \beta_{2} - \)\(82\!\cdots\!50\)\( \beta_{3} - \)\(21\!\cdots\!10\)\( \beta_{4} + \)\(36\!\cdots\!50\)\( \beta_{5} - \)\(17\!\cdots\!70\)\( \beta_{6} + \)\(84\!\cdots\!50\)\( \beta_{7} - \)\(95\!\cdots\!20\)\( \beta_{8}) q^{40} +(\)\(50\!\cdots\!30\)\( - \)\(10\!\cdots\!18\)\( \beta_{1} - \)\(93\!\cdots\!24\)\( \beta_{2} - \)\(59\!\cdots\!96\)\( \beta_{3} + \)\(98\!\cdots\!08\)\( \beta_{4} - \)\(37\!\cdots\!54\)\( \beta_{5} - \)\(37\!\cdots\!30\)\( \beta_{6} - \)\(11\!\cdots\!40\)\( \beta_{7} + \)\(38\!\cdots\!10\)\( \beta_{8}) q^{41} +(-\)\(40\!\cdots\!16\)\( - \)\(76\!\cdots\!00\)\( \beta_{1} - \)\(41\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(67\!\cdots\!76\)\( \beta_{4} + \)\(52\!\cdots\!36\)\( \beta_{5} + \)\(17\!\cdots\!24\)\( \beta_{6} + \)\(28\!\cdots\!80\)\( \beta_{7} - \)\(11\!\cdots\!60\)\( \beta_{8}) q^{42} +(\)\(12\!\cdots\!36\)\( - \)\(25\!\cdots\!97\)\( \beta_{1} - \)\(12\!\cdots\!33\)\( \beta_{2} + \)\(83\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} + \)\(44\!\cdots\!88\)\( \beta_{5} - \)\(91\!\cdots\!76\)\( \beta_{6} + \)\(16\!\cdots\!44\)\( \beta_{7} + \)\(17\!\cdots\!00\)\( \beta_{8}) q^{43} +(\)\(96\!\cdots\!68\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(38\!\cdots\!20\)\( \beta_{2} - \)\(51\!\cdots\!28\)\( \beta_{3} - \)\(98\!\cdots\!16\)\( \beta_{4} - \)\(20\!\cdots\!28\)\( \beta_{5} - \)\(21\!\cdots\!56\)\( \beta_{6} - \)\(19\!\cdots\!52\)\( \beta_{7} + \)\(51\!\cdots\!76\)\( \beta_{8}) q^{44} +(-\)\(39\!\cdots\!90\)\( - \)\(53\!\cdots\!97\)\( \beta_{1} + \)\(22\!\cdots\!88\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} + \)\(52\!\cdots\!97\)\( \beta_{4} - \)\(35\!\cdots\!50\)\( \beta_{5} + \)\(88\!\cdots\!50\)\( \beta_{6} + \)\(10\!\cdots\!00\)\( \beta_{7} - \)\(55\!\cdots\!50\)\( \beta_{8}) q^{45} +(\)\(23\!\cdots\!50\)\( - \)\(17\!\cdots\!98\)\( \beta_{1} + \)\(31\!\cdots\!96\)\( \beta_{2} - \)\(23\!\cdots\!96\)\( \beta_{3} - \)\(44\!\cdots\!82\)\( \beta_{4} + \)\(22\!\cdots\!66\)\( \beta_{5} + \)\(27\!\cdots\!00\)\( \beta_{6} - \)\(38\!\cdots\!20\)\( \beta_{7} + \)\(25\!\cdots\!20\)\( \beta_{8}) q^{46} +(-\)\(35\!\cdots\!36\)\( - \)\(31\!\cdots\!46\)\( \beta_{1} - \)\(15\!\cdots\!06\)\( \beta_{2} + \)\(17\!\cdots\!30\)\( \beta_{3} - \)\(14\!\cdots\!98\)\( \beta_{4} - \)\(26\!\cdots\!44\)\( \beta_{5} - \)\(16\!\cdots\!82\)\( \beta_{6} + \)\(80\!\cdots\!28\)\( \beta_{7} - \)\(73\!\cdots\!20\)\( \beta_{8}) q^{47} +(\)\(49\!\cdots\!80\)\( - \)\(63\!\cdots\!76\)\( \beta_{1} - \)\(12\!\cdots\!92\)\( \beta_{2} + \)\(22\!\cdots\!56\)\( \beta_{3} - \)\(33\!\cdots\!16\)\( \beta_{4} - \)\(28\!\cdots\!56\)\( \beta_{5} + \)\(80\!\cdots\!16\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(97\!\cdots\!00\)\( \beta_{8}) q^{48} +(-\)\(60\!\cdots\!59\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(33\!\cdots\!76\)\( \beta_{2} - \)\(94\!\cdots\!16\)\( \beta_{3} + \)\(16\!\cdots\!08\)\( \beta_{4} + \)\(10\!\cdots\!56\)\( \beta_{5} - \)\(18\!\cdots\!40\)\( \beta_{6} - \)\(68\!\cdots\!00\)\( \beta_{7} + \)\(30\!\cdots\!60\)\( \beta_{8}) q^{49} +(-\)\(11\!\cdots\!15\)\( + \)\(75\!\cdots\!45\)\( \beta_{1} + \)\(53\!\cdots\!40\)\( \beta_{2} - \)\(37\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(29\!\cdots\!00\)\( \beta_{7} - \)\(25\!\cdots\!80\)\( \beta_{8}) q^{50} +(\)\(43\!\cdots\!76\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} + \)\(22\!\cdots\!56\)\( \beta_{2} - \)\(51\!\cdots\!90\)\( \beta_{3} - \)\(59\!\cdots\!50\)\( \beta_{4} - \)\(85\!\cdots\!48\)\( \beta_{5} - \)\(35\!\cdots\!78\)\( \beta_{6} - \)\(65\!\cdots\!36\)\( \beta_{7} + \)\(88\!\cdots\!48\)\( \beta_{8}) q^{51} +(\)\(20\!\cdots\!34\)\( + \)\(14\!\cdots\!26\)\( \beta_{1} + \)\(21\!\cdots\!66\)\( \beta_{2} + \)\(50\!\cdots\!50\)\( \beta_{3} + \)\(18\!\cdots\!92\)\( \beta_{4} + \)\(42\!\cdots\!36\)\( \beta_{5} + \)\(42\!\cdots\!28\)\( \beta_{6} + \)\(57\!\cdots\!68\)\( \beta_{7} - \)\(16\!\cdots\!00\)\( \beta_{8}) q^{52} +(\)\(37\!\cdots\!50\)\( + \)\(36\!\cdots\!83\)\( \beta_{1} + \)\(72\!\cdots\!68\)\( \beta_{2} + \)\(54\!\cdots\!88\)\( \beta_{3} + \)\(29\!\cdots\!53\)\( \beta_{4} + \)\(12\!\cdots\!10\)\( \beta_{5} - \)\(23\!\cdots\!18\)\( \beta_{6} + \)\(81\!\cdots\!24\)\( \beta_{7} - \)\(77\!\cdots\!70\)\( \beta_{8}) q^{53} +(\)\(90\!\cdots\!30\)\( + \)\(51\!\cdots\!98\)\( \beta_{1} - \)\(10\!\cdots\!36\)\( \beta_{2} - \)\(48\!\cdots\!96\)\( \beta_{3} + \)\(95\!\cdots\!14\)\( \beta_{4} - \)\(38\!\cdots\!78\)\( \beta_{5} + \)\(63\!\cdots\!04\)\( \beta_{6} - \)\(78\!\cdots\!92\)\( \beta_{7} + \)\(17\!\cdots\!76\)\( \beta_{8}) q^{54} +(-\)\(26\!\cdots\!90\)\( - \)\(69\!\cdots\!27\)\( \beta_{1} - \)\(10\!\cdots\!67\)\( \beta_{2} - \)\(48\!\cdots\!53\)\( \beta_{3} - \)\(24\!\cdots\!73\)\( \beta_{4} + \)\(18\!\cdots\!50\)\( \beta_{5} - \)\(49\!\cdots\!25\)\( \beta_{6} - \)\(12\!\cdots\!00\)\( \beta_{7} - \)\(65\!\cdots\!00\)\( \beta_{8}) q^{55} +(\)\(56\!\cdots\!12\)\( - \)\(54\!\cdots\!92\)\( \beta_{1} + \)\(36\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!56\)\( \beta_{3} - \)\(25\!\cdots\!52\)\( \beta_{4} + \)\(41\!\cdots\!24\)\( \beta_{5} - \)\(24\!\cdots\!52\)\( \beta_{6} + \)\(24\!\cdots\!96\)\( \beta_{7} + \)\(12\!\cdots\!12\)\( \beta_{8}) q^{56} +(\)\(21\!\cdots\!04\)\( - \)\(30\!\cdots\!63\)\( \beta_{1} + \)\(16\!\cdots\!74\)\( \beta_{2} + \)\(77\!\cdots\!90\)\( \beta_{3} + \)\(27\!\cdots\!36\)\( \beta_{4} + \)\(57\!\cdots\!83\)\( \beta_{5} + \)\(92\!\cdots\!49\)\( \beta_{6} + \)\(16\!\cdots\!54\)\( \beta_{7} + \)\(39\!\cdots\!65\)\( \beta_{8}) q^{57} +(-\)\(30\!\cdots\!50\)\( - \)\(77\!\cdots\!46\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(49\!\cdots\!96\)\( \beta_{3} - \)\(81\!\cdots\!88\)\( \beta_{4} - \)\(33\!\cdots\!92\)\( \beta_{5} - \)\(14\!\cdots\!32\)\( \beta_{6} - \)\(11\!\cdots\!48\)\( \beta_{7} - \)\(10\!\cdots\!40\)\( \beta_{8}) q^{58} +(\)\(29\!\cdots\!52\)\( - \)\(46\!\cdots\!97\)\( \beta_{1} - \)\(25\!\cdots\!21\)\( \beta_{2} - \)\(77\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!96\)\( \beta_{4} + \)\(65\!\cdots\!44\)\( \beta_{5} + \)\(27\!\cdots\!36\)\( \beta_{6} + \)\(31\!\cdots\!32\)\( \beta_{7} + \)\(32\!\cdots\!24\)\( \beta_{8}) q^{59} +(-\)\(16\!\cdots\!08\)\( + \)\(60\!\cdots\!40\)\( \beta_{1} - \)\(27\!\cdots\!76\)\( \beta_{2} - \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!56\)\( \beta_{6} - \)\(17\!\cdots\!80\)\( \beta_{7} - \)\(45\!\cdots\!96\)\( \beta_{8}) q^{60} +(\)\(51\!\cdots\!90\)\( + \)\(47\!\cdots\!35\)\( \beta_{1} - \)\(52\!\cdots\!00\)\( \beta_{2} + \)\(33\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!73\)\( \beta_{4} + \)\(11\!\cdots\!18\)\( \beta_{5} + \)\(49\!\cdots\!02\)\( \beta_{6} - \)\(18\!\cdots\!16\)\( \beta_{7} - \)\(50\!\cdots\!42\)\( \beta_{8}) q^{61} +(\)\(32\!\cdots\!64\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} + \)\(52\!\cdots\!36\)\( \beta_{2} + \)\(75\!\cdots\!44\)\( \beta_{3} + \)\(78\!\cdots\!96\)\( \beta_{4} - \)\(11\!\cdots\!24\)\( \beta_{5} - \)\(46\!\cdots\!96\)\( \beta_{6} + \)\(73\!\cdots\!40\)\( \beta_{7} + \)\(41\!\cdots\!80\)\( \beta_{8}) q^{62} +(-\)\(29\!\cdots\!82\)\( + \)\(63\!\cdots\!89\)\( \beta_{1} + \)\(60\!\cdots\!77\)\( \beta_{2} - \)\(55\!\cdots\!29\)\( \beta_{3} - \)\(59\!\cdots\!69\)\( \beta_{4} + \)\(29\!\cdots\!70\)\( \beta_{5} - \)\(24\!\cdots\!61\)\( \beta_{6} - \)\(11\!\cdots\!72\)\( \beta_{7} - \)\(95\!\cdots\!80\)\( \beta_{8}) q^{63} +(-\)\(51\!\cdots\!28\)\( + \)\(79\!\cdots\!52\)\( \beta_{1} + \)\(40\!\cdots\!76\)\( \beta_{2} - \)\(76\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!04\)\( \beta_{4} - \)\(59\!\cdots\!48\)\( \beta_{5} + \)\(96\!\cdots\!00\)\( \beta_{6} - \)\(77\!\cdots\!40\)\( \beta_{7} + \)\(59\!\cdots\!40\)\( \beta_{8}) q^{64} +(\)\(34\!\cdots\!04\)\( - \)\(80\!\cdots\!70\)\( \beta_{1} - \)\(34\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!40\)\( \beta_{3} + \)\(66\!\cdots\!64\)\( \beta_{4} - \)\(80\!\cdots\!50\)\( \beta_{5} - \)\(11\!\cdots\!22\)\( \beta_{6} + \)\(67\!\cdots\!40\)\( \beta_{7} + \)\(27\!\cdots\!98\)\( \beta_{8}) q^{65} +(\)\(65\!\cdots\!96\)\( - \)\(84\!\cdots\!24\)\( \beta_{1} - \)\(67\!\cdots\!12\)\( \beta_{2} + \)\(38\!\cdots\!20\)\( \beta_{3} - \)\(33\!\cdots\!40\)\( \beta_{4} + \)\(18\!\cdots\!36\)\( \beta_{5} - \)\(80\!\cdots\!24\)\( \beta_{6} - \)\(11\!\cdots\!88\)\( \beta_{7} - \)\(84\!\cdots\!16\)\( \beta_{8}) q^{66} +(\)\(58\!\cdots\!48\)\( - \)\(24\!\cdots\!73\)\( \beta_{1} - \)\(10\!\cdots\!93\)\( \beta_{2} + \)\(59\!\cdots\!66\)\( \beta_{3} - \)\(14\!\cdots\!10\)\( \beta_{4} + \)\(19\!\cdots\!32\)\( \beta_{5} + \)\(50\!\cdots\!70\)\( \beta_{6} + \)\(22\!\cdots\!44\)\( \beta_{7} + \)\(74\!\cdots\!20\)\( \beta_{8}) q^{67} +(-\)\(10\!\cdots\!70\)\( - \)\(41\!\cdots\!30\)\( \beta_{1} + \)\(47\!\cdots\!10\)\( \beta_{2} - \)\(11\!\cdots\!78\)\( \beta_{3} - \)\(79\!\cdots\!08\)\( \beta_{4} + \)\(12\!\cdots\!40\)\( \beta_{5} + \)\(15\!\cdots\!48\)\( \beta_{6} + \)\(10\!\cdots\!96\)\( \beta_{7} + \)\(10\!\cdots\!40\)\( \beta_{8}) q^{68} +(-\)\(52\!\cdots\!24\)\( - \)\(68\!\cdots\!86\)\( \beta_{1} + \)\(78\!\cdots\!52\)\( \beta_{2} - \)\(46\!\cdots\!76\)\( \beta_{3} + \)\(23\!\cdots\!36\)\( \beta_{4} - \)\(62\!\cdots\!62\)\( \beta_{5} - \)\(25\!\cdots\!34\)\( \beta_{6} + \)\(80\!\cdots\!52\)\( \beta_{7} - \)\(90\!\cdots\!66\)\( \beta_{8}) q^{69} +(\)\(72\!\cdots\!80\)\( + \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} + \)\(22\!\cdots\!56\)\( \beta_{3} - \)\(18\!\cdots\!04\)\( \beta_{4} + \)\(82\!\cdots\!00\)\( \beta_{5} - \)\(93\!\cdots\!00\)\( \beta_{6} - \)\(27\!\cdots\!00\)\( \beta_{7} - \)\(11\!\cdots\!00\)\( \beta_{8}) q^{70} +(\)\(35\!\cdots\!66\)\( + \)\(49\!\cdots\!45\)\( \beta_{1} - \)\(14\!\cdots\!75\)\( \beta_{2} + \)\(18\!\cdots\!79\)\( \beta_{3} + \)\(45\!\cdots\!79\)\( \beta_{4} + \)\(14\!\cdots\!74\)\( \beta_{5} + \)\(42\!\cdots\!71\)\( \beta_{6} - \)\(18\!\cdots\!88\)\( \beta_{7} + \)\(27\!\cdots\!04\)\( \beta_{8}) q^{71} +(\)\(14\!\cdots\!34\)\( + \)\(23\!\cdots\!92\)\( \beta_{1} - \)\(83\!\cdots\!12\)\( \beta_{2} + \)\(42\!\cdots\!23\)\( \beta_{3} - \)\(26\!\cdots\!13\)\( \beta_{4} + \)\(95\!\cdots\!17\)\( \beta_{5} - \)\(43\!\cdots\!37\)\( \beta_{6} + \)\(29\!\cdots\!75\)\( \beta_{7} + \)\(56\!\cdots\!20\)\( \beta_{8}) q^{72} +(-\)\(12\!\cdots\!86\)\( + \)\(11\!\cdots\!97\)\( \beta_{1} - \)\(17\!\cdots\!42\)\( \beta_{2} - \)\(49\!\cdots\!74\)\( \beta_{3} + \)\(38\!\cdots\!60\)\( \beta_{4} - \)\(75\!\cdots\!53\)\( \beta_{5} - \)\(85\!\cdots\!75\)\( \beta_{6} - \)\(69\!\cdots\!86\)\( \beta_{7} - \)\(41\!\cdots\!95\)\( \beta_{8}) q^{73} +(\)\(16\!\cdots\!10\)\( + \)\(64\!\cdots\!78\)\( \beta_{1} - \)\(64\!\cdots\!56\)\( \beta_{2} - \)\(28\!\cdots\!92\)\( \beta_{3} - \)\(95\!\cdots\!56\)\( \beta_{4} - \)\(26\!\cdots\!24\)\( \beta_{5} + \)\(20\!\cdots\!48\)\( \beta_{6} + \)\(88\!\cdots\!56\)\( \beta_{7} + \)\(79\!\cdots\!52\)\( \beta_{8}) q^{74} +(\)\(21\!\cdots\!20\)\( - \)\(99\!\cdots\!85\)\( \beta_{1} + \)\(11\!\cdots\!55\)\( \beta_{2} - \)\(18\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!80\)\( \beta_{4} + \)\(65\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!40\)\( \beta_{6} + \)\(30\!\cdots\!00\)\( \beta_{7} + \)\(53\!\cdots\!40\)\( \beta_{8}) q^{75} +(\)\(21\!\cdots\!84\)\( - \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} + \)\(15\!\cdots\!72\)\( \beta_{3} - \)\(57\!\cdots\!84\)\( \beta_{4} - \)\(11\!\cdots\!32\)\( \beta_{5} - \)\(64\!\cdots\!44\)\( \beta_{6} - \)\(66\!\cdots\!28\)\( \beta_{7} - \)\(59\!\cdots\!96\)\( \beta_{8}) q^{76} +(\)\(94\!\cdots\!20\)\( - \)\(53\!\cdots\!38\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2} + \)\(10\!\cdots\!16\)\( \beta_{3} - \)\(25\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!54\)\( \beta_{5} - \)\(19\!\cdots\!14\)\( \beta_{6} + \)\(17\!\cdots\!40\)\( \beta_{7} + \)\(11\!\cdots\!50\)\( \beta_{8}) q^{77} +(\)\(93\!\cdots\!78\)\( - \)\(86\!\cdots\!90\)\( \beta_{1} - \)\(86\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(16\!\cdots\!50\)\( \beta_{4} - \)\(12\!\cdots\!06\)\( \beta_{5} + \)\(84\!\cdots\!20\)\( \beta_{6} + \)\(13\!\cdots\!88\)\( \beta_{7} + \)\(18\!\cdots\!00\)\( \beta_{8}) q^{78} +(-\)\(29\!\cdots\!92\)\( + \)\(14\!\cdots\!86\)\( \beta_{1} - \)\(13\!\cdots\!22\)\( \beta_{2} - \)\(12\!\cdots\!94\)\( \beta_{3} + \)\(89\!\cdots\!54\)\( \beta_{4} + \)\(94\!\cdots\!92\)\( \beta_{5} - \)\(53\!\cdots\!26\)\( \beta_{6} - \)\(86\!\cdots\!92\)\( \beta_{7} - \)\(55\!\cdots\!04\)\( \beta_{8}) q^{79} +(\)\(13\!\cdots\!80\)\( + \)\(61\!\cdots\!64\)\( \beta_{1} - \)\(95\!\cdots\!56\)\( \beta_{2} + \)\(99\!\cdots\!96\)\( \beta_{3} - \)\(35\!\cdots\!64\)\( \beta_{4} - \)\(45\!\cdots\!00\)\( \beta_{5} - \)\(24\!\cdots\!00\)\( \beta_{6} - \)\(35\!\cdots\!00\)\( \beta_{7} + \)\(99\!\cdots\!00\)\( \beta_{8}) q^{80} +(\)\(27\!\cdots\!41\)\( + \)\(14\!\cdots\!91\)\( \beta_{1} + \)\(30\!\cdots\!98\)\( \beta_{2} + \)\(49\!\cdots\!58\)\( \beta_{3} - \)\(17\!\cdots\!56\)\( \beta_{4} - \)\(46\!\cdots\!91\)\( \beta_{5} + \)\(45\!\cdots\!91\)\( \beta_{6} - \)\(27\!\cdots\!18\)\( \beta_{7} + \)\(34\!\cdots\!79\)\( \beta_{8}) q^{81} +(\)\(52\!\cdots\!06\)\( + \)\(14\!\cdots\!10\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!44\)\( \beta_{3} + \)\(46\!\cdots\!64\)\( \beta_{4} + \)\(79\!\cdots\!80\)\( \beta_{5} + \)\(58\!\cdots\!16\)\( \beta_{6} + \)\(45\!\cdots\!12\)\( \beta_{7} - \)\(39\!\cdots\!60\)\( \beta_{8}) q^{82} +(\)\(10\!\cdots\!72\)\( + \)\(51\!\cdots\!01\)\( \beta_{1} + \)\(16\!\cdots\!93\)\( \beta_{2} - \)\(17\!\cdots\!80\)\( \beta_{3} - \)\(21\!\cdots\!20\)\( \beta_{4} - \)\(44\!\cdots\!00\)\( \beta_{5} - \)\(18\!\cdots\!80\)\( \beta_{6} - \)\(81\!\cdots\!00\)\( \beta_{7} + \)\(33\!\cdots\!20\)\( \beta_{8}) q^{83} +(\)\(28\!\cdots\!76\)\( - \)\(49\!\cdots\!52\)\( \beta_{1} - \)\(53\!\cdots\!36\)\( \beta_{2} - \)\(64\!\cdots\!56\)\( \beta_{3} + \)\(40\!\cdots\!16\)\( \beta_{4} + \)\(12\!\cdots\!52\)\( \beta_{5} - \)\(34\!\cdots\!80\)\( \beta_{6} - \)\(73\!\cdots\!80\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8}) q^{84} +(\)\(39\!\cdots\!12\)\( - \)\(13\!\cdots\!60\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(19\!\cdots\!58\)\( \beta_{4} - \)\(70\!\cdots\!50\)\( \beta_{5} + \)\(49\!\cdots\!34\)\( \beta_{6} + \)\(47\!\cdots\!20\)\( \beta_{7} - \)\(25\!\cdots\!06\)\( \beta_{8}) q^{85} +(\)\(13\!\cdots\!23\)\( - \)\(43\!\cdots\!95\)\( \beta_{1} - \)\(21\!\cdots\!80\)\( \beta_{2} + \)\(22\!\cdots\!86\)\( \beta_{3} + \)\(57\!\cdots\!87\)\( \beta_{4} + \)\(28\!\cdots\!61\)\( \beta_{5} + \)\(11\!\cdots\!12\)\( \beta_{6} - \)\(51\!\cdots\!36\)\( \beta_{7} - \)\(34\!\cdots\!12\)\( \beta_{8}) q^{86} +(\)\(30\!\cdots\!06\)\( - \)\(32\!\cdots\!13\)\( \beta_{1} + \)\(32\!\cdots\!27\)\( \beta_{2} - \)\(85\!\cdots\!91\)\( \beta_{3} + \)\(74\!\cdots\!37\)\( \beta_{4} + \)\(25\!\cdots\!14\)\( \beta_{5} - \)\(35\!\cdots\!27\)\( \beta_{6} - \)\(48\!\cdots\!36\)\( \beta_{7} + \)\(20\!\cdots\!60\)\( \beta_{8}) q^{87} +(\)\(57\!\cdots\!68\)\( - \)\(84\!\cdots\!12\)\( \beta_{1} + \)\(65\!\cdots\!80\)\( \beta_{2} + \)\(85\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!24\)\( \beta_{4} - \)\(28\!\cdots\!16\)\( \beta_{5} + \)\(12\!\cdots\!76\)\( \beta_{6} + \)\(91\!\cdots\!20\)\( \beta_{7} - \)\(18\!\cdots\!20\)\( \beta_{8}) q^{88} +(\)\(66\!\cdots\!82\)\( + \)\(13\!\cdots\!29\)\( \beta_{1} + \)\(14\!\cdots\!22\)\( \beta_{2} - \)\(18\!\cdots\!26\)\( \beta_{3} + \)\(88\!\cdots\!00\)\( \beta_{4} - \)\(93\!\cdots\!17\)\( \beta_{5} + \)\(25\!\cdots\!13\)\( \beta_{6} + \)\(20\!\cdots\!06\)\( \beta_{7} - \)\(64\!\cdots\!83\)\( \beta_{8}) q^{89} +(\)\(28\!\cdots\!34\)\( + \)\(75\!\cdots\!30\)\( \beta_{1} - \)\(77\!\cdots\!52\)\( \beta_{2} - \)\(20\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} - \)\(11\!\cdots\!00\)\( \beta_{5} + \)\(12\!\cdots\!88\)\( \beta_{6} - \)\(12\!\cdots\!60\)\( \beta_{7} + \)\(20\!\cdots\!08\)\( \beta_{8}) q^{90} +(\)\(27\!\cdots\!84\)\( + \)\(46\!\cdots\!88\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(34\!\cdots\!44\)\( \beta_{3} + \)\(60\!\cdots\!08\)\( \beta_{4} + \)\(39\!\cdots\!04\)\( \beta_{5} - \)\(29\!\cdots\!32\)\( \beta_{6} - \)\(23\!\cdots\!44\)\( \beta_{7} - \)\(13\!\cdots\!28\)\( \beta_{8}) q^{91} +(\)\(68\!\cdots\!60\)\( + \)\(75\!\cdots\!24\)\( \beta_{1} - \)\(97\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!04\)\( \beta_{4} + \)\(46\!\cdots\!00\)\( \beta_{5} - \)\(63\!\cdots\!24\)\( \beta_{6} + \)\(55\!\cdots\!52\)\( \beta_{7} - \)\(52\!\cdots\!00\)\( \beta_{8}) q^{92} +(\)\(13\!\cdots\!20\)\( - \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!12\)\( \beta_{3} + \)\(32\!\cdots\!40\)\( \beta_{4} - \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(26\!\cdots\!40\)\( \beta_{6} + \)\(16\!\cdots\!72\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{93} +(\)\(16\!\cdots\!16\)\( - \)\(59\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} + \)\(14\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!96\)\( \beta_{4} + \)\(21\!\cdots\!92\)\( \beta_{5} - \)\(20\!\cdots\!36\)\( \beta_{6} - \)\(27\!\cdots\!32\)\( \beta_{7} - \)\(64\!\cdots\!24\)\( \beta_{8}) q^{94} +(\)\(88\!\cdots\!50\)\( - \)\(10\!\cdots\!05\)\( \beta_{1} + \)\(93\!\cdots\!95\)\( \beta_{2} - \)\(59\!\cdots\!95\)\( \beta_{3} + \)\(33\!\cdots\!05\)\( \beta_{4} + \)\(88\!\cdots\!50\)\( \beta_{5} - \)\(35\!\cdots\!75\)\( \beta_{6} + \)\(37\!\cdots\!00\)\( \beta_{7} - \)\(24\!\cdots\!00\)\( \beta_{8}) q^{95} +(\)\(36\!\cdots\!24\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(19\!\cdots\!28\)\( \beta_{3} + \)\(11\!\cdots\!96\)\( \beta_{4} + \)\(69\!\cdots\!12\)\( \beta_{5} + \)\(57\!\cdots\!20\)\( \beta_{6} + \)\(28\!\cdots\!20\)\( \beta_{7} + \)\(43\!\cdots\!00\)\( \beta_{8}) q^{96} +(\)\(28\!\cdots\!38\)\( - \)\(10\!\cdots\!31\)\( \beta_{1} - \)\(10\!\cdots\!78\)\( \beta_{2} + \)\(55\!\cdots\!82\)\( \beta_{3} - \)\(37\!\cdots\!68\)\( \beta_{4} - \)\(14\!\cdots\!45\)\( \beta_{5} - \)\(86\!\cdots\!67\)\( \beta_{6} - \)\(13\!\cdots\!54\)\( \beta_{7} - \)\(56\!\cdots\!35\)\( \beta_{8}) q^{97} +(\)\(65\!\cdots\!27\)\( + \)\(39\!\cdots\!99\)\( \beta_{1} - \)\(46\!\cdots\!84\)\( \beta_{2} + \)\(23\!\cdots\!24\)\( \beta_{3} + \)\(70\!\cdots\!44\)\( \beta_{4} - \)\(68\!\cdots\!40\)\( \beta_{5} + \)\(15\!\cdots\!36\)\( \beta_{6} + \)\(86\!\cdots\!52\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{98} +(-\)\(90\!\cdots\!48\)\( + \)\(16\!\cdots\!27\)\( \beta_{1} - \)\(44\!\cdots\!09\)\( \beta_{2} + \)\(34\!\cdots\!44\)\( \beta_{3} - \)\(16\!\cdots\!28\)\( \beta_{4} + \)\(25\!\cdots\!36\)\( \beta_{5} - \)\(53\!\cdots\!48\)\( \beta_{6} + \)\(21\!\cdots\!64\)\( \beta_{7} - \)\(13\!\cdots\!72\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 116180770089455544q^{2} + \)\(34\!\cdots\!92\)\(q^{3} + \)\(10\!\cdots\!12\)\(q^{4} - \)\(94\!\cdots\!90\)\(q^{5} + \)\(25\!\cdots\!68\)\(q^{6} + \)\(18\!\cdots\!56\)\(q^{7} - \)\(21\!\cdots\!00\)\(q^{8} + \)\(26\!\cdots\!13\)\(q^{9} + O(q^{10}) \) \( 9q - 116180770089455544q^{2} + \)\(34\!\cdots\!92\)\(q^{3} + \)\(10\!\cdots\!12\)\(q^{4} - \)\(94\!\cdots\!90\)\(q^{5} + \)\(25\!\cdots\!68\)\(q^{6} + \)\(18\!\cdots\!56\)\(q^{7} - \)\(21\!\cdots\!00\)\(q^{8} + \)\(26\!\cdots\!13\)\(q^{9} + \)\(68\!\cdots\!60\)\(q^{10} - \)\(44\!\cdots\!32\)\(q^{11} - \)\(42\!\cdots\!44\)\(q^{12} + \)\(19\!\cdots\!42\)\(q^{13} - \)\(87\!\cdots\!76\)\(q^{14} + \)\(57\!\cdots\!20\)\(q^{15} - \)\(18\!\cdots\!16\)\(q^{16} + \)\(33\!\cdots\!06\)\(q^{17} - \)\(28\!\cdots\!08\)\(q^{18} + \)\(56\!\cdots\!00\)\(q^{19} - \)\(12\!\cdots\!20\)\(q^{20} + \)\(28\!\cdots\!68\)\(q^{21} + \)\(21\!\cdots\!12\)\(q^{22} + \)\(50\!\cdots\!92\)\(q^{23} - \)\(74\!\cdots\!00\)\(q^{24} + \)\(12\!\cdots\!75\)\(q^{25} - \)\(59\!\cdots\!32\)\(q^{26} - \)\(54\!\cdots\!00\)\(q^{27} + \)\(13\!\cdots\!08\)\(q^{28} + \)\(24\!\cdots\!50\)\(q^{29} - \)\(11\!\cdots\!80\)\(q^{30} - \)\(11\!\cdots\!32\)\(q^{31} + \)\(18\!\cdots\!56\)\(q^{32} + \)\(36\!\cdots\!84\)\(q^{33} + \)\(16\!\cdots\!24\)\(q^{34} - \)\(52\!\cdots\!40\)\(q^{35} - \)\(34\!\cdots\!16\)\(q^{36} + \)\(28\!\cdots\!06\)\(q^{37} + \)\(18\!\cdots\!00\)\(q^{38} + \)\(57\!\cdots\!76\)\(q^{39} - \)\(37\!\cdots\!00\)\(q^{40} + \)\(45\!\cdots\!18\)\(q^{41} - \)\(36\!\cdots\!88\)\(q^{42} + \)\(11\!\cdots\!92\)\(q^{43} + \)\(86\!\cdots\!24\)\(q^{44} - \)\(35\!\cdots\!30\)\(q^{45} + \)\(21\!\cdots\!68\)\(q^{46} - \)\(31\!\cdots\!44\)\(q^{47} + \)\(44\!\cdots\!92\)\(q^{48} - \)\(54\!\cdots\!63\)\(q^{49} - \)\(10\!\cdots\!00\)\(q^{50} + \)\(39\!\cdots\!68\)\(q^{51} + \)\(18\!\cdots\!56\)\(q^{52} + \)\(33\!\cdots\!42\)\(q^{53} + \)\(81\!\cdots\!00\)\(q^{54} - \)\(23\!\cdots\!80\)\(q^{55} + \)\(50\!\cdots\!00\)\(q^{56} + \)\(19\!\cdots\!00\)\(q^{57} - \)\(27\!\cdots\!00\)\(q^{58} + \)\(26\!\cdots\!00\)\(q^{59} - \)\(14\!\cdots\!40\)\(q^{60} + \)\(46\!\cdots\!18\)\(q^{61} + \)\(28\!\cdots\!12\)\(q^{62} - \)\(26\!\cdots\!08\)\(q^{63} - \)\(46\!\cdots\!88\)\(q^{64} + \)\(30\!\cdots\!20\)\(q^{65} + \)\(58\!\cdots\!36\)\(q^{66} + \)\(52\!\cdots\!56\)\(q^{67} - \)\(96\!\cdots\!92\)\(q^{68} - \)\(47\!\cdots\!24\)\(q^{69} + \)\(64\!\cdots\!60\)\(q^{70} + \)\(32\!\cdots\!68\)\(q^{71} + \)\(13\!\cdots\!00\)\(q^{72} - \)\(11\!\cdots\!58\)\(q^{73} + \)\(14\!\cdots\!24\)\(q^{74} + \)\(19\!\cdots\!00\)\(q^{75} + \)\(18\!\cdots\!00\)\(q^{76} + \)\(85\!\cdots\!12\)\(q^{77} + \)\(84\!\cdots\!84\)\(q^{78} - \)\(26\!\cdots\!00\)\(q^{79} + \)\(11\!\cdots\!60\)\(q^{80} + \)\(24\!\cdots\!09\)\(q^{81} + \)\(47\!\cdots\!12\)\(q^{82} + \)\(97\!\cdots\!92\)\(q^{83} + \)\(25\!\cdots\!24\)\(q^{84} + \)\(35\!\cdots\!60\)\(q^{85} + \)\(11\!\cdots\!68\)\(q^{86} + \)\(27\!\cdots\!00\)\(q^{87} + \)\(52\!\cdots\!00\)\(q^{88} + \)\(59\!\cdots\!50\)\(q^{89} + \)\(25\!\cdots\!20\)\(q^{90} + \)\(24\!\cdots\!68\)\(q^{91} + \)\(61\!\cdots\!56\)\(q^{92} + \)\(12\!\cdots\!84\)\(q^{93} + \)\(14\!\cdots\!24\)\(q^{94} + \)\(79\!\cdots\!00\)\(q^{95} + \)\(32\!\cdots\!68\)\(q^{96} + \)\(25\!\cdots\!06\)\(q^{97} + \)\(58\!\cdots\!08\)\(q^{98} - \)\(81\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 2 x^{8} - \)\(41\!\cdots\!36\)\( x^{7} + \)\(20\!\cdots\!12\)\( x^{6} + \)\(53\!\cdots\!86\)\( x^{5} - \)\(39\!\cdots\!00\)\( x^{4} - \)\(25\!\cdots\!00\)\( x^{3} + \)\(49\!\cdots\!00\)\( x^{2} + \)\(38\!\cdots\!25\)\( x + \)\(17\!\cdots\!50\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(18\!\cdots\!49\)\( \nu^{8} - \)\(65\!\cdots\!75\)\( \nu^{7} + \)\(47\!\cdots\!73\)\( \nu^{6} + \)\(22\!\cdots\!89\)\( \nu^{5} - \)\(21\!\cdots\!69\)\( \nu^{4} - \)\(20\!\cdots\!93\)\( \nu^{3} + \)\(22\!\cdots\!83\)\( \nu^{2} + \)\(42\!\cdots\!35\)\( \nu + \)\(90\!\cdots\!30\)\(\)\()/ \)\(40\!\cdots\!84\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(90\!\cdots\!97\)\( \nu^{8} - \)\(31\!\cdots\!75\)\( \nu^{7} + \)\(23\!\cdots\!69\)\( \nu^{6} + \)\(10\!\cdots\!17\)\( \nu^{5} - \)\(10\!\cdots\!57\)\( \nu^{4} - \)\(97\!\cdots\!29\)\( \nu^{3} + \)\(24\!\cdots\!83\)\( \nu^{2} + \)\(22\!\cdots\!19\)\( \nu - \)\(20\!\cdots\!66\)\(\)\()/ \)\(40\!\cdots\!84\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(28\!\cdots\!31\)\( \nu^{8} - \)\(17\!\cdots\!27\)\( \nu^{7} - \)\(10\!\cdots\!11\)\( \nu^{6} + \)\(71\!\cdots\!37\)\( \nu^{5} + \)\(11\!\cdots\!11\)\( \nu^{4} - \)\(81\!\cdots\!65\)\( \nu^{3} - \)\(31\!\cdots\!25\)\( \nu^{2} + \)\(19\!\cdots\!75\)\( \nu + \)\(15\!\cdots\!50\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(23\!\cdots\!11\)\( \nu^{8} - \)\(84\!\cdots\!13\)\( \nu^{7} + \)\(81\!\cdots\!91\)\( \nu^{6} + \)\(20\!\cdots\!03\)\( \nu^{5} - \)\(75\!\cdots\!91\)\( \nu^{4} - \)\(94\!\cdots\!35\)\( \nu^{3} + \)\(15\!\cdots\!25\)\( \nu^{2} + \)\(14\!\cdots\!25\)\( \nu + \)\(13\!\cdots\!50\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(38\!\cdots\!21\)\( \nu^{8} + \)\(46\!\cdots\!43\)\( \nu^{7} - \)\(12\!\cdots\!01\)\( \nu^{6} + \)\(53\!\cdots\!67\)\( \nu^{5} + \)\(97\!\cdots\!01\)\( \nu^{4} - \)\(32\!\cdots\!15\)\( \nu^{3} + \)\(21\!\cdots\!25\)\( \nu^{2} + \)\(11\!\cdots\!25\)\( \nu - \)\(42\!\cdots\!50\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(20\!\cdots\!19\)\( \nu^{8} + \)\(15\!\cdots\!23\)\( \nu^{7} + \)\(80\!\cdots\!39\)\( \nu^{6} - \)\(64\!\cdots\!13\)\( \nu^{5} - \)\(86\!\cdots\!39\)\( \nu^{4} + \)\(73\!\cdots\!85\)\( \nu^{3} + \)\(22\!\cdots\!25\)\( \nu^{2} - \)\(19\!\cdots\!75\)\( \nu - \)\(98\!\cdots\!50\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(37\!\cdots\!91\)\( \nu^{8} - \)\(14\!\cdots\!47\)\( \nu^{7} - \)\(14\!\cdots\!71\)\( \nu^{6} + \)\(64\!\cdots\!57\)\( \nu^{5} + \)\(15\!\cdots\!71\)\( \nu^{4} - \)\(75\!\cdots\!65\)\( \nu^{3} - \)\(39\!\cdots\!25\)\( \nu^{2} + \)\(17\!\cdots\!75\)\( \nu + \)\(82\!\cdots\!50\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 484553 \beta_{2} - 18073717211147679 \beta_{1} + 53001739878028145233323179512730439\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 413 \beta_{6} + 664275 \beta_{5} + 189393293777 \beta_{4} - 38107786849440575 \beta_{3} + 114056587572313116197696 \beta_{2} + 88405175090489855863641579900017084 \beta_{1} - 957938458254282665056357244169716144561633024779486\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(-10412963909952 \beta_{8} + 1467556945694547 \beta_{7} - 27009661388808667609 \beta_{6} - 98749583594328355423959 \beta_{5} + 5183614690002396536607437763 \beta_{4} + 16303251739383817902272811577456587 \beta_{3} - 2273068339929917024625036724765707901880 \beta_{2} - 487678643604962842851760126691162171713004223700820 \beta_{1} + 585703511752209383105760939369323734653716598776592595231776182697422\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(-3063728300247593546060314440 \beta_{8} + 145445726412599068359363126821843 \beta_{7} + 92220362597007090321083244188580655 \beta_{6} + 313331271415733645970061805205391868289 \beta_{5} + 34592650781384667306935247707708693752285955 \beta_{4} - 5779427120278069279131207652491160242292929801858 \beta_{3} - 18631557476300263724627194944056472259557411442379445491 \beta_{2} + 9108238802121355524826978257971989991069092802549671776960558125515 \beta_{1} - 201936067284529692989337825142643067881337601859473172714700829894858794207478375641\)\()/7776\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(17\!\cdots\!20\)\( \beta_{8} + \)\(84\!\cdots\!67\)\( \beta_{7} - \)\(40\!\cdots\!25\)\( \beta_{6} - \)\(20\!\cdots\!47\)\( \beta_{5} + \)\(70\!\cdots\!71\)\( \beta_{4} + \)\(17\!\cdots\!59\)\( \beta_{3} + \)\(85\!\cdots\!84\)\( \beta_{2} - \)\(63\!\cdots\!28\)\( \beta_{1} + \)\(53\!\cdots\!14\)\(\)\()/20736\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(22\!\cdots\!40\)\( \beta_{8} + \)\(17\!\cdots\!13\)\( \beta_{7} + \)\(13\!\cdots\!41\)\( \beta_{6} + \)\(57\!\cdots\!15\)\( \beta_{5} + \)\(45\!\cdots\!65\)\( \beta_{4} - \)\(78\!\cdots\!83\)\( \beta_{3} - \)\(54\!\cdots\!24\)\( \beta_{2} + \)\(95\!\cdots\!32\)\( \beta_{1} - \)\(27\!\cdots\!58\)\(\)\()/41472\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(26\!\cdots\!44\)\( \beta_{8} - \)\(13\!\cdots\!37\)\( \beta_{7} - \)\(62\!\cdots\!77\)\( \beta_{6} - \)\(34\!\cdots\!63\)\( \beta_{5} + \)\(91\!\cdots\!47\)\( \beta_{4} + \)\(21\!\cdots\!31\)\( \beta_{3} + \)\(22\!\cdots\!12\)\( \beta_{2} - \)\(90\!\cdots\!56\)\( \beta_{1} + \)\(63\!\cdots\!02\)\(\)\()/124416\)

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
1.42762e16
9.41344e15
9.00047e15
5.63699e15
−4.70183e14
−5.44635e15
−5.76359e15
−1.17725e16
−1.48745e16
−3.55538e17 −3.54906e26 8.48689e34 1.00650e40 1.26183e44 6.58681e47 −1.54056e52 −7.26915e54 −3.57850e57
1.2 −2.38832e17 4.64741e27 1.55022e34 −1.89028e40 −1.10995e45 5.62740e48 6.21827e51 1.42033e55 4.51459e57
1.3 −2.28920e17 −5.05419e27 1.08661e34 −2.40937e40 1.15700e45 −2.38510e48 7.02151e51 1.81497e55 5.51552e57
1.4 −1.48197e17 7.29391e26 −1.95761e34 6.05511e39 −1.08093e44 −4.10385e48 9.05697e51 −6.86309e54 −8.97348e56
1.5 −1.62458e15 −2.61982e27 −4.15357e34 1.47212e40 4.25609e42 6.19961e48 1.34960e50 −5.31663e53 −2.39156e55
1.6 1.17803e17 5.60840e26 −2.76608e34 −2.33935e40 6.60688e43 −1.66757e47 −8.15189e51 −7.08056e54 −2.75583e57
1.7 1.25417e17 4.55926e27 −2.58089e34 2.05215e40 5.71809e44 −2.84362e48 −8.44651e51 1.33917e55 2.57375e57
1.8 2.69631e17 −3.76423e27 3.11624e34 5.22690e39 −1.01495e45 −4.51498e48 −2.79767e51 6.77433e54 1.40933e57
1.9 3.44079e17 1.64109e27 7.68520e34 3.84023e38 5.64664e44 3.35173e48 1.21507e52 −4.70193e54 1.32134e56
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 1.116.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.116.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{116}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 116180770089455544 T + \)\(14\!\cdots\!68\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!96\)\( T^{4} + \)\(10\!\cdots\!24\)\( T^{5} + \)\(65\!\cdots\!28\)\( T^{6} + \)\(58\!\cdots\!00\)\( T^{7} + \)\(31\!\cdots\!56\)\( T^{8} + \)\(27\!\cdots\!64\)\( T^{9} + \)\(13\!\cdots\!08\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(47\!\cdots\!96\)\( T^{12} + \)\(30\!\cdots\!24\)\( T^{13} + \)\(14\!\cdots\!28\)\( T^{14} + \)\(73\!\cdots\!00\)\( T^{15} + \)\(30\!\cdots\!76\)\( T^{16} + \)\(10\!\cdots\!44\)\( T^{17} + \)\(36\!\cdots\!68\)\( T^{18} \)
$3$ \( 1 - \)\(34\!\cdots\!92\)\( T + \)\(20\!\cdots\!07\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!96\)\( T^{4} - \)\(28\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} + \)\(75\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!06\)\( T^{8} + \)\(12\!\cdots\!48\)\( T^{9} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(65\!\cdots\!96\)\( T^{12} - \)\(84\!\cdots\!32\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} - \)\(70\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!01\)\( T^{16} - \)\(30\!\cdots\!92\)\( T^{17} + \)\(66\!\cdots\!07\)\( T^{18} \)
$5$ \( 1 + \)\(94\!\cdots\!90\)\( T + \)\(10\!\cdots\!25\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!00\)\( T^{4} + \)\(85\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!00\)\( T^{7} + \)\(89\!\cdots\!50\)\( T^{8} + \)\(95\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!50\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{11} + \)\(39\!\cdots\!00\)\( T^{12} + \)\(28\!\cdots\!00\)\( T^{13} + \)\(52\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!00\)\( T^{15} + \)\(49\!\cdots\!25\)\( T^{16} + \)\(10\!\cdots\!50\)\( T^{17} + \)\(27\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 - \)\(18\!\cdots\!56\)\( T + \)\(73\!\cdots\!43\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!96\)\( T^{4} - \)\(81\!\cdots\!76\)\( T^{5} + \)\(81\!\cdots\!28\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(38\!\cdots\!36\)\( T^{9} + \)\(25\!\cdots\!58\)\( T^{10} - \)\(49\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!96\)\( T^{12} - \)\(45\!\cdots\!76\)\( T^{13} + \)\(25\!\cdots\!28\)\( T^{14} - \)\(24\!\cdots\!00\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} - \)\(56\!\cdots\!56\)\( T^{17} + \)\(47\!\cdots\!43\)\( T^{18} \)
$11$ \( 1 + \)\(44\!\cdots\!32\)\( T + \)\(31\!\cdots\!03\)\( T^{2} + \)\(12\!\cdots\!84\)\( T^{3} + \)\(48\!\cdots\!60\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(49\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!76\)\( T^{7} + \)\(36\!\cdots\!58\)\( T^{8} + \)\(92\!\cdots\!80\)\( T^{9} + \)\(21\!\cdots\!58\)\( T^{10} + \)\(47\!\cdots\!76\)\( T^{11} + \)\(93\!\cdots\!64\)\( T^{12} + \)\(17\!\cdots\!32\)\( T^{13} + \)\(30\!\cdots\!60\)\( T^{14} + \)\(44\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!53\)\( T^{16} + \)\(53\!\cdots\!32\)\( T^{17} + \)\(69\!\cdots\!51\)\( T^{18} \)
$13$ \( 1 - \)\(19\!\cdots\!42\)\( T + \)\(73\!\cdots\!57\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(66\!\cdots\!32\)\( T^{5} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} - \)\(42\!\cdots\!94\)\( T^{8} + \)\(45\!\cdots\!48\)\( T^{9} - \)\(53\!\cdots\!58\)\( T^{10} + \)\(29\!\cdots\!00\)\( T^{11} + \)\(31\!\cdots\!96\)\( T^{12} - \)\(17\!\cdots\!32\)\( T^{13} + \)\(59\!\cdots\!72\)\( T^{14} - \)\(33\!\cdots\!00\)\( T^{15} + \)\(39\!\cdots\!01\)\( T^{16} - \)\(12\!\cdots\!42\)\( T^{17} + \)\(85\!\cdots\!57\)\( T^{18} \)
$17$ \( 1 - \)\(33\!\cdots\!06\)\( T + \)\(12\!\cdots\!93\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!96\)\( T^{4} - \)\(28\!\cdots\!76\)\( T^{5} + \)\(30\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} + \)\(14\!\cdots\!64\)\( T^{9} + \)\(33\!\cdots\!58\)\( T^{10} + \)\(24\!\cdots\!00\)\( T^{11} + \)\(97\!\cdots\!96\)\( T^{12} - \)\(29\!\cdots\!76\)\( T^{13} + \)\(23\!\cdots\!28\)\( T^{14} - \)\(22\!\cdots\!00\)\( T^{15} + \)\(40\!\cdots\!01\)\( T^{16} - \)\(34\!\cdots\!06\)\( T^{17} + \)\(32\!\cdots\!93\)\( T^{18} \)
$19$ \( 1 - \)\(56\!\cdots\!00\)\( T + \)\(55\!\cdots\!91\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!36\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{5} + \)\(30\!\cdots\!16\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(50\!\cdots\!74\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!84\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(30\!\cdots\!64\)\( T^{14} - \)\(54\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!09\)\( T^{16} - \)\(15\!\cdots\!00\)\( T^{17} + \)\(32\!\cdots\!99\)\( T^{18} \)
$23$ \( 1 - \)\(50\!\cdots\!92\)\( T + \)\(23\!\cdots\!07\)\( T^{2} - \)\(75\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!96\)\( T^{4} - \)\(63\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} - \)\(37\!\cdots\!00\)\( T^{7} + \)\(82\!\cdots\!06\)\( T^{8} - \)\(16\!\cdots\!52\)\( T^{9} + \)\(32\!\cdots\!42\)\( T^{10} - \)\(59\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!32\)\( T^{13} + \)\(23\!\cdots\!72\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(36\!\cdots\!01\)\( T^{16} - \)\(31\!\cdots\!92\)\( T^{17} + \)\(24\!\cdots\!07\)\( T^{18} \)
$29$ \( 1 - \)\(24\!\cdots\!50\)\( T + \)\(79\!\cdots\!41\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!36\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!16\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(53\!\cdots\!26\)\( T^{8} - \)\(41\!\cdots\!00\)\( T^{9} + \)\(80\!\cdots\!74\)\( T^{10} - \)\(68\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!84\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!64\)\( T^{14} - \)\(13\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!09\)\( T^{16} - \)\(62\!\cdots\!50\)\( T^{17} + \)\(38\!\cdots\!49\)\( T^{18} \)
$31$ \( 1 + \)\(11\!\cdots\!32\)\( T + \)\(16\!\cdots\!03\)\( T^{2} + \)\(85\!\cdots\!84\)\( T^{3} + \)\(58\!\cdots\!60\)\( T^{4} + \)\(20\!\cdots\!32\)\( T^{5} + \)\(31\!\cdots\!64\)\( T^{6} - \)\(69\!\cdots\!24\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{9} + \)\(37\!\cdots\!58\)\( T^{10} - \)\(71\!\cdots\!24\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} + \)\(21\!\cdots\!32\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!84\)\( T^{15} + \)\(56\!\cdots\!53\)\( T^{16} + \)\(12\!\cdots\!32\)\( T^{17} + \)\(36\!\cdots\!51\)\( T^{18} \)
$37$ \( 1 - \)\(28\!\cdots\!06\)\( T + \)\(16\!\cdots\!93\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!96\)\( T^{4} - \)\(48\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!28\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!06\)\( T^{8} - \)\(62\!\cdots\!36\)\( T^{9} + \)\(40\!\cdots\!58\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!76\)\( T^{13} + \)\(65\!\cdots\!28\)\( T^{14} - \)\(67\!\cdots\!00\)\( T^{15} + \)\(41\!\cdots\!01\)\( T^{16} - \)\(15\!\cdots\!06\)\( T^{17} + \)\(12\!\cdots\!93\)\( T^{18} \)
$41$ \( 1 - \)\(45\!\cdots\!18\)\( T + \)\(83\!\cdots\!53\)\( T^{2} - \)\(41\!\cdots\!16\)\( T^{3} + \)\(38\!\cdots\!60\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!64\)\( T^{6} - \)\(59\!\cdots\!24\)\( T^{7} + \)\(34\!\cdots\!58\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} - \)\(51\!\cdots\!24\)\( T^{11} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{13} + \)\(87\!\cdots\!60\)\( T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(16\!\cdots\!53\)\( T^{16} - \)\(25\!\cdots\!18\)\( T^{17} + \)\(17\!\cdots\!01\)\( T^{18} \)
$43$ \( 1 - \)\(11\!\cdots\!92\)\( T + \)\(39\!\cdots\!07\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!96\)\( T^{4} - \)\(89\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(87\!\cdots\!52\)\( T^{9} + \)\(71\!\cdots\!42\)\( T^{10} - \)\(52\!\cdots\!00\)\( T^{11} + \)\(37\!\cdots\!96\)\( T^{12} - \)\(22\!\cdots\!32\)\( T^{13} + \)\(13\!\cdots\!72\)\( T^{14} - \)\(58\!\cdots\!00\)\( T^{15} + \)\(34\!\cdots\!01\)\( T^{16} - \)\(69\!\cdots\!92\)\( T^{17} + \)\(43\!\cdots\!07\)\( T^{18} \)
$47$ \( 1 + \)\(31\!\cdots\!44\)\( T + \)\(58\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} - \)\(83\!\cdots\!76\)\( T^{5} + \)\(59\!\cdots\!28\)\( T^{6} - \)\(87\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!06\)\( T^{8} - \)\(24\!\cdots\!36\)\( T^{9} + \)\(26\!\cdots\!58\)\( T^{10} - \)\(33\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!96\)\( T^{12} - \)\(12\!\cdots\!76\)\( T^{13} + \)\(61\!\cdots\!28\)\( T^{14} + \)\(74\!\cdots\!00\)\( T^{15} + \)\(63\!\cdots\!01\)\( T^{16} + \)\(67\!\cdots\!44\)\( T^{17} + \)\(41\!\cdots\!43\)\( T^{18} \)
$53$ \( 1 - \)\(33\!\cdots\!42\)\( T + \)\(67\!\cdots\!57\)\( T^{2} - \)\(73\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!96\)\( T^{4} - \)\(32\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(98\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!06\)\( T^{8} - \)\(24\!\cdots\!52\)\( T^{9} + \)\(50\!\cdots\!42\)\( T^{10} - \)\(37\!\cdots\!00\)\( T^{11} + \)\(80\!\cdots\!96\)\( T^{12} - \)\(46\!\cdots\!32\)\( T^{13} + \)\(85\!\cdots\!72\)\( T^{14} - \)\(41\!\cdots\!00\)\( T^{15} + \)\(73\!\cdots\!01\)\( T^{16} - \)\(72\!\cdots\!42\)\( T^{17} + \)\(42\!\cdots\!57\)\( T^{18} \)
$59$ \( 1 - \)\(26\!\cdots\!00\)\( T + \)\(16\!\cdots\!91\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!16\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!26\)\( T^{8} + \)\(83\!\cdots\!00\)\( T^{9} + \)\(14\!\cdots\!74\)\( T^{10} + \)\(30\!\cdots\!00\)\( T^{11} + \)\(64\!\cdots\!84\)\( T^{12} + \)\(73\!\cdots\!00\)\( T^{13} + \)\(23\!\cdots\!64\)\( T^{14} + \)\(88\!\cdots\!00\)\( T^{15} + \)\(56\!\cdots\!09\)\( T^{16} - \)\(40\!\cdots\!00\)\( T^{17} + \)\(67\!\cdots\!99\)\( T^{18} \)
$61$ \( 1 - \)\(46\!\cdots\!18\)\( T + \)\(11\!\cdots\!53\)\( T^{2} - \)\(56\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!60\)\( T^{4} - \)\(25\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!64\)\( T^{6} - \)\(69\!\cdots\!24\)\( T^{7} + \)\(70\!\cdots\!58\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(29\!\cdots\!24\)\( T^{11} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(45\!\cdots\!68\)\( T^{13} + \)\(24\!\cdots\!60\)\( T^{14} - \)\(42\!\cdots\!16\)\( T^{15} + \)\(18\!\cdots\!53\)\( T^{16} - \)\(14\!\cdots\!18\)\( T^{17} + \)\(65\!\cdots\!01\)\( T^{18} \)
$67$ \( 1 - \)\(52\!\cdots\!56\)\( T + \)\(67\!\cdots\!43\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{5} + \)\(42\!\cdots\!28\)\( T^{6} - \)\(29\!\cdots\!00\)\( T^{7} + \)\(59\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!36\)\( T^{9} + \)\(59\!\cdots\!58\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(42\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!76\)\( T^{13} + \)\(21\!\cdots\!28\)\( T^{14} - \)\(43\!\cdots\!00\)\( T^{15} + \)\(65\!\cdots\!01\)\( T^{16} - \)\(50\!\cdots\!56\)\( T^{17} + \)\(97\!\cdots\!43\)\( T^{18} \)
$71$ \( 1 - \)\(32\!\cdots\!68\)\( T + \)\(52\!\cdots\!03\)\( T^{2} - \)\(14\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!60\)\( T^{4} - \)\(32\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!64\)\( T^{6} - \)\(44\!\cdots\!24\)\( T^{7} + \)\(22\!\cdots\!58\)\( T^{8} - \)\(42\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!58\)\( T^{10} - \)\(27\!\cdots\!24\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!68\)\( T^{13} + \)\(39\!\cdots\!60\)\( T^{14} - \)\(34\!\cdots\!16\)\( T^{15} + \)\(95\!\cdots\!53\)\( T^{16} - \)\(46\!\cdots\!68\)\( T^{17} + \)\(11\!\cdots\!51\)\( T^{18} \)
$73$ \( 1 + \)\(11\!\cdots\!58\)\( T + \)\(10\!\cdots\!57\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!96\)\( T^{4} + \)\(48\!\cdots\!68\)\( T^{5} + \)\(19\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(50\!\cdots\!06\)\( T^{8} + \)\(34\!\cdots\!48\)\( T^{9} + \)\(97\!\cdots\!42\)\( T^{10} + \)\(56\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!96\)\( T^{12} + \)\(65\!\cdots\!68\)\( T^{13} + \)\(14\!\cdots\!72\)\( T^{14} + \)\(51\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!01\)\( T^{16} + \)\(20\!\cdots\!58\)\( T^{17} + \)\(34\!\cdots\!57\)\( T^{18} \)
$79$ \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(97\!\cdots\!91\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(34\!\cdots\!26\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!74\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{11} + \)\(72\!\cdots\!84\)\( T^{12} + \)\(54\!\cdots\!00\)\( T^{13} + \)\(65\!\cdots\!64\)\( T^{14} + \)\(44\!\cdots\!00\)\( T^{15} + \)\(38\!\cdots\!09\)\( T^{16} + \)\(17\!\cdots\!00\)\( T^{17} + \)\(11\!\cdots\!99\)\( T^{18} \)
$83$ \( 1 - \)\(97\!\cdots\!92\)\( T + \)\(80\!\cdots\!07\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} - \)\(81\!\cdots\!32\)\( T^{5} + \)\(28\!\cdots\!72\)\( T^{6} - \)\(83\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!06\)\( T^{8} - \)\(51\!\cdots\!52\)\( T^{9} + \)\(11\!\cdots\!42\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!96\)\( T^{12} - \)\(48\!\cdots\!32\)\( T^{13} + \)\(62\!\cdots\!72\)\( T^{14} - \)\(64\!\cdots\!00\)\( T^{15} + \)\(57\!\cdots\!01\)\( T^{16} - \)\(34\!\cdots\!92\)\( T^{17} + \)\(17\!\cdots\!07\)\( T^{18} \)
$89$ \( 1 - \)\(59\!\cdots\!50\)\( T + \)\(23\!\cdots\!41\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!36\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(59\!\cdots\!16\)\( T^{6} - \)\(95\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!26\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{9} + \)\(20\!\cdots\!74\)\( T^{10} - \)\(21\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!84\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} + \)\(12\!\cdots\!64\)\( T^{14} - \)\(80\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!09\)\( T^{16} - \)\(16\!\cdots\!50\)\( T^{17} + \)\(41\!\cdots\!49\)\( T^{18} \)
$97$ \( 1 - \)\(25\!\cdots\!06\)\( T + \)\(18\!\cdots\!93\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(41\!\cdots\!76\)\( T^{5} + \)\(11\!\cdots\!28\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!06\)\( T^{8} - \)\(79\!\cdots\!36\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!96\)\( T^{12} - \)\(33\!\cdots\!76\)\( T^{13} + \)\(44\!\cdots\!28\)\( T^{14} - \)\(35\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!01\)\( T^{16} - \)\(17\!\cdots\!06\)\( T^{17} + \)\(20\!\cdots\!93\)\( T^{18} \)
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