Properties

Label 1.122.a.a
Level 1
Weight 122
Character orbit 1.a
Self dual yes
Analytic conductor 92.717
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 122 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.7173263878\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{145}\cdot 3^{53}\cdot 5^{20}\cdot 7^{8}\cdot 11^{6}\cdot 13^{2}\cdot 17^{2} \)
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.

\(f(q)\) \(=\) \( q +(-260556173583835525 + \beta_{1}) q^{2} +(-\)\(50\!\cdots\!47\)\( - 5694448773 \beta_{1} + \beta_{2}) q^{3} +(\)\(94\!\cdots\!52\)\( - 331283849028211141 \beta_{1} + 3414351 \beta_{2} + \beta_{3}) q^{4} +(-\)\(21\!\cdots\!17\)\( - \)\(69\!\cdots\!36\)\( \beta_{1} + 1315393166598 \beta_{2} + 60136 \beta_{3} + \beta_{4}) q^{5} +(-\)\(18\!\cdots\!28\)\( + \)\(13\!\cdots\!86\)\( \beta_{1} - 286415241388034362 \beta_{2} - 10945374801 \beta_{3} + 10106 \beta_{4} - \beta_{5}) q^{6} +(\)\(24\!\cdots\!97\)\( - \)\(18\!\cdots\!82\)\( \beta_{1} + \)\(22\!\cdots\!26\)\( \beta_{2} + 117761204996058 \beta_{3} + 27640033 \beta_{4} - 884 \beta_{5} + \beta_{6}) q^{7} +(-\)\(72\!\cdots\!85\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} - \)\(10\!\cdots\!81\)\( \beta_{2} - 520035242032518678 \beta_{3} + 611104015289 \beta_{4} - 8716729 \beta_{5} - 100 \beta_{6} + \beta_{7}) q^{8} +(\)\(88\!\cdots\!41\)\( - \)\(15\!\cdots\!83\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(29\!\cdots\!47\)\( \beta_{3} + 281352546038066 \beta_{4} + 136881610 \beta_{5} + 89884 \beta_{6} - 111 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\) \( q +(-260556173583835525 + \beta_{1}) q^{2} +(-\)\(50\!\cdots\!47\)\( - 5694448773 \beta_{1} + \beta_{2}) q^{3} +(\)\(94\!\cdots\!52\)\( - 331283849028211141 \beta_{1} + 3414351 \beta_{2} + \beta_{3}) q^{4} +(-\)\(21\!\cdots\!17\)\( - \)\(69\!\cdots\!36\)\( \beta_{1} + 1315393166598 \beta_{2} + 60136 \beta_{3} + \beta_{4}) q^{5} +(-\)\(18\!\cdots\!28\)\( + \)\(13\!\cdots\!86\)\( \beta_{1} - 286415241388034362 \beta_{2} - 10945374801 \beta_{3} + 10106 \beta_{4} - \beta_{5}) q^{6} +(\)\(24\!\cdots\!97\)\( - \)\(18\!\cdots\!82\)\( \beta_{1} + \)\(22\!\cdots\!26\)\( \beta_{2} + 117761204996058 \beta_{3} + 27640033 \beta_{4} - 884 \beta_{5} + \beta_{6}) q^{7} +(-\)\(72\!\cdots\!85\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} - \)\(10\!\cdots\!81\)\( \beta_{2} - 520035242032518678 \beta_{3} + 611104015289 \beta_{4} - 8716729 \beta_{5} - 100 \beta_{6} + \beta_{7}) q^{8} +(\)\(88\!\cdots\!41\)\( - \)\(15\!\cdots\!83\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(29\!\cdots\!47\)\( \beta_{3} + 281352546038066 \beta_{4} + 136881610 \beta_{5} + 89884 \beta_{6} - 111 \beta_{7} + \beta_{8}) q^{9} +(-\)\(19\!\cdots\!34\)\( - \)\(28\!\cdots\!02\)\( \beta_{1} + \)\(38\!\cdots\!76\)\( \beta_{2} + \)\(86\!\cdots\!52\)\( \beta_{3} - 85411994472303248 \beta_{4} + 223590377740 \beta_{5} - 21046880 \beta_{6} - 2520 \beta_{7} - 240 \beta_{8}) q^{10} +(-\)\(10\!\cdots\!35\)\( - \)\(67\!\cdots\!59\)\( \beta_{1} + \)\(31\!\cdots\!51\)\( \beta_{2} + \)\(82\!\cdots\!84\)\( \beta_{3} - 86913908567829979798 \beta_{4} + 547339293703264 \beta_{5} - 20065513526 \beta_{6} - 45440956 \beta_{7} - 168444 \beta_{8}) q^{11} +(\)\(22\!\cdots\!92\)\( - \)\(36\!\cdots\!40\)\( \beta_{1} + \)\(33\!\cdots\!80\)\( \beta_{2} + \)\(15\!\cdots\!88\)\( \beta_{3} - \)\(21\!\cdots\!04\)\( \beta_{4} + 321442233029511384 \beta_{5} - 13036671918240 \beta_{6} - 34941366936 \beta_{7} + 2046720 \beta_{8}) q^{12} +(\)\(22\!\cdots\!79\)\( - \)\(19\!\cdots\!46\)\( \beta_{1} - \)\(72\!\cdots\!74\)\( \beta_{2} + \)\(13\!\cdots\!46\)\( \beta_{3} - \)\(15\!\cdots\!51\)\( \beta_{4} + 39150026746055710940 \beta_{5} - 7810320313768024 \beta_{6} - 3364521724746 \beta_{7} + 2252737110 \beta_{8}) q^{13} +(-\)\(72\!\cdots\!52\)\( + \)\(58\!\cdots\!32\)\( \beta_{1} + \)\(88\!\cdots\!68\)\( \beta_{2} - \)\(21\!\cdots\!02\)\( \beta_{3} + \)\(47\!\cdots\!12\)\( \beta_{4} + \)\(74\!\cdots\!86\)\( \beta_{5} - 1424711924174411648 \beta_{6} + 319490020035872 \beta_{7} - 250410186432 \beta_{8}) q^{14} +(\)\(10\!\cdots\!83\)\( + \)\(78\!\cdots\!74\)\( \beta_{1} - \)\(91\!\cdots\!62\)\( \beta_{2} - \)\(46\!\cdots\!74\)\( \beta_{3} - \)\(21\!\cdots\!49\)\( \beta_{4} + \)\(49\!\cdots\!20\)\( \beta_{5} - 42708662191974162465 \beta_{6} - 1283199651234360 \beta_{7} + 14385279460680 \beta_{8}) q^{15} +(\)\(16\!\cdots\!84\)\( - \)\(14\!\cdots\!20\)\( \beta_{1} + \)\(29\!\cdots\!44\)\( \beta_{2} + \)\(13\!\cdots\!40\)\( \beta_{3} - \)\(11\!\cdots\!60\)\( \beta_{4} + \)\(49\!\cdots\!76\)\( \beta_{5} + \)\(30\!\cdots\!64\)\( \beta_{6} - 674307616144702896 \beta_{7} - 539902754586624 \beta_{8}) q^{16} +(\)\(32\!\cdots\!46\)\( + \)\(31\!\cdots\!33\)\( \beta_{1} - \)\(10\!\cdots\!90\)\( \beta_{2} - \)\(67\!\cdots\!23\)\( \beta_{3} - \)\(11\!\cdots\!58\)\( \beta_{4} + \)\(13\!\cdots\!74\)\( \beta_{5} + \)\(34\!\cdots\!84\)\( \beta_{6} + 33600631746098946165 \beta_{7} + 13988525269125285 \beta_{8}) q^{17} +(-\)\(28\!\cdots\!37\)\( - \)\(40\!\cdots\!99\)\( \beta_{1} + \)\(73\!\cdots\!64\)\( \beta_{2} - \)\(12\!\cdots\!88\)\( \beta_{3} - \)\(23\!\cdots\!88\)\( \beta_{4} + \)\(61\!\cdots\!84\)\( \beta_{5} - \)\(37\!\cdots\!96\)\( \beta_{6} - \)\(76\!\cdots\!60\)\( \beta_{7} - 233921939232398880 \beta_{8}) q^{18} +(\)\(33\!\cdots\!51\)\( + \)\(42\!\cdots\!03\)\( \beta_{1} - \)\(54\!\cdots\!59\)\( \beta_{2} - \)\(16\!\cdots\!88\)\( \beta_{3} - \)\(67\!\cdots\!06\)\( \beta_{4} - \)\(13\!\cdots\!72\)\( \beta_{5} + \)\(53\!\cdots\!58\)\( \beta_{6} + \)\(56\!\cdots\!48\)\( \beta_{7} + 1194095236891187052 \beta_{8}) q^{19} +(\)\(50\!\cdots\!36\)\( + \)\(26\!\cdots\!38\)\( \beta_{1} - \)\(63\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!38\)\( \beta_{3} - \)\(20\!\cdots\!08\)\( \beta_{4} - \)\(29\!\cdots\!00\)\( \beta_{5} + \)\(81\!\cdots\!00\)\( \beta_{6} + \)\(20\!\cdots\!00\)\( \beta_{7} + 72483187728374092800 \beta_{8}) q^{20} +(\)\(39\!\cdots\!04\)\( - \)\(49\!\cdots\!22\)\( \beta_{1} + \)\(22\!\cdots\!92\)\( \beta_{2} + \)\(77\!\cdots\!42\)\( \beta_{3} - \)\(33\!\cdots\!84\)\( \beta_{4} + \)\(59\!\cdots\!08\)\( \beta_{5} - \)\(36\!\cdots\!24\)\( \beta_{6} - \)\(83\!\cdots\!14\)\( \beta_{7} - \)\(30\!\cdots\!66\)\( \beta_{8}) q^{21} +(-\)\(21\!\cdots\!84\)\( + \)\(14\!\cdots\!26\)\( \beta_{1} - \)\(37\!\cdots\!06\)\( \beta_{2} - \)\(23\!\cdots\!51\)\( \beta_{3} + \)\(47\!\cdots\!66\)\( \beta_{4} - \)\(42\!\cdots\!95\)\( \beta_{5} + \)\(45\!\cdots\!64\)\( \beta_{6} + \)\(16\!\cdots\!96\)\( \beta_{7} + \)\(76\!\cdots\!60\)\( \beta_{8}) q^{22} +(-\)\(73\!\cdots\!73\)\( - \)\(15\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!14\)\( \beta_{2} - \)\(35\!\cdots\!70\)\( \beta_{3} + \)\(70\!\cdots\!35\)\( \beta_{4} - \)\(74\!\cdots\!80\)\( \beta_{5} + \)\(48\!\cdots\!95\)\( \beta_{6} - \)\(21\!\cdots\!40\)\( \beta_{7} - \)\(14\!\cdots\!40\)\( \beta_{8}) q^{23} +(-\)\(83\!\cdots\!68\)\( + \)\(67\!\cdots\!80\)\( \beta_{1} - \)\(57\!\cdots\!16\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3} + \)\(64\!\cdots\!32\)\( \beta_{4} + \)\(95\!\cdots\!32\)\( \beta_{5} - \)\(90\!\cdots\!84\)\( \beta_{6} + \)\(16\!\cdots\!36\)\( \beta_{7} + \)\(22\!\cdots\!24\)\( \beta_{8}) q^{24} +(-\)\(67\!\cdots\!25\)\( + \)\(19\!\cdots\!50\)\( \beta_{1} + \)\(85\!\cdots\!00\)\( \beta_{2} - \)\(92\!\cdots\!50\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(57\!\cdots\!00\)\( \beta_{5} + \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!50\)\( \beta_{7} - \)\(29\!\cdots\!50\)\( \beta_{8}) q^{25} +(-\)\(76\!\cdots\!82\)\( + \)\(24\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(11\!\cdots\!08\)\( \beta_{3} - \)\(21\!\cdots\!64\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(61\!\cdots\!92\)\( \beta_{6} - \)\(21\!\cdots\!32\)\( \beta_{7} + \)\(32\!\cdots\!12\)\( \beta_{8}) q^{26} +(-\)\(66\!\cdots\!80\)\( + \)\(37\!\cdots\!22\)\( \beta_{1} + \)\(10\!\cdots\!14\)\( \beta_{2} + \)\(60\!\cdots\!92\)\( \beta_{3} + \)\(11\!\cdots\!54\)\( \beta_{4} + \)\(19\!\cdots\!36\)\( \beta_{5} - \)\(52\!\cdots\!30\)\( \beta_{6} + \)\(36\!\cdots\!56\)\( \beta_{7} - \)\(30\!\cdots\!40\)\( \beta_{8}) q^{27} +(\)\(16\!\cdots\!72\)\( - \)\(86\!\cdots\!04\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!76\)\( \beta_{4} + \)\(13\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} - \)\(37\!\cdots\!76\)\( \beta_{7} + \)\(24\!\cdots\!80\)\( \beta_{8}) q^{28} +(\)\(10\!\cdots\!39\)\( + \)\(33\!\cdots\!08\)\( \beta_{1} - \)\(11\!\cdots\!30\)\( \beta_{2} + \)\(30\!\cdots\!52\)\( \beta_{3} - \)\(55\!\cdots\!71\)\( \beta_{4} - \)\(11\!\cdots\!96\)\( \beta_{5} - \)\(88\!\cdots\!48\)\( \beta_{6} + \)\(25\!\cdots\!12\)\( \beta_{7} - \)\(15\!\cdots\!12\)\( \beta_{8}) q^{29} +(\)\(24\!\cdots\!76\)\( - \)\(35\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} + \)\(17\!\cdots\!42\)\( \beta_{3} - \)\(26\!\cdots\!28\)\( \beta_{4} - \)\(76\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6} - \)\(91\!\cdots\!00\)\( \beta_{7} + \)\(66\!\cdots\!00\)\( \beta_{8}) q^{30} +(-\)\(11\!\cdots\!36\)\( + \)\(52\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} - \)\(18\!\cdots\!48\)\( \beta_{3} + \)\(32\!\cdots\!80\)\( \beta_{4} + \)\(62\!\cdots\!24\)\( \beta_{5} + \)\(34\!\cdots\!16\)\( \beta_{6} - \)\(32\!\cdots\!04\)\( \beta_{7} - \)\(33\!\cdots\!96\)\( \beta_{8}) q^{31} +(-\)\(36\!\cdots\!20\)\( + \)\(28\!\cdots\!00\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} - \)\(28\!\cdots\!68\)\( \beta_{3} + \)\(15\!\cdots\!32\)\( \beta_{4} - \)\(28\!\cdots\!96\)\( \beta_{5} - \)\(36\!\cdots\!36\)\( \beta_{6} + \)\(81\!\cdots\!60\)\( \beta_{7} - \)\(29\!\cdots\!20\)\( \beta_{8}) q^{32} +(\)\(21\!\cdots\!60\)\( - \)\(54\!\cdots\!57\)\( \beta_{1} - \)\(89\!\cdots\!86\)\( \beta_{2} + \)\(59\!\cdots\!95\)\( \beta_{3} - \)\(18\!\cdots\!22\)\( \beta_{4} - \)\(10\!\cdots\!22\)\( \beta_{5} + \)\(15\!\cdots\!84\)\( \beta_{6} - \)\(70\!\cdots\!41\)\( \beta_{7} + \)\(37\!\cdots\!55\)\( \beta_{8}) q^{33} +(\)\(10\!\cdots\!70\)\( + \)\(27\!\cdots\!30\)\( \beta_{1} - \)\(37\!\cdots\!16\)\( \beta_{2} + \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(32\!\cdots\!76\)\( \beta_{4} + \)\(12\!\cdots\!28\)\( \beta_{5} + \)\(26\!\cdots\!28\)\( \beta_{6} + \)\(37\!\cdots\!88\)\( \beta_{7} - \)\(29\!\cdots\!08\)\( \beta_{8}) q^{34} +(\)\(14\!\cdots\!56\)\( + \)\(18\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!84\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3} + \)\(61\!\cdots\!32\)\( \beta_{4} - \)\(45\!\cdots\!60\)\( \beta_{5} - \)\(69\!\cdots\!80\)\( \beta_{6} - \)\(93\!\cdots\!20\)\( \beta_{7} + \)\(17\!\cdots\!60\)\( \beta_{8}) q^{35} +(-\)\(24\!\cdots\!64\)\( - \)\(55\!\cdots\!65\)\( \beta_{1} + \)\(90\!\cdots\!99\)\( \beta_{2} - \)\(16\!\cdots\!15\)\( \beta_{3} + \)\(56\!\cdots\!56\)\( \beta_{4} - \)\(42\!\cdots\!76\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6} - \)\(36\!\cdots\!20\)\( \beta_{7} - \)\(78\!\cdots\!60\)\( \beta_{8}) q^{36} +(-\)\(98\!\cdots\!89\)\( + \)\(90\!\cdots\!38\)\( \beta_{1} + \)\(39\!\cdots\!10\)\( \beta_{2} - \)\(15\!\cdots\!06\)\( \beta_{3} - \)\(15\!\cdots\!07\)\( \beta_{4} + \)\(16\!\cdots\!32\)\( \beta_{5} - \)\(27\!\cdots\!80\)\( \beta_{6} + \)\(55\!\cdots\!22\)\( \beta_{7} + \)\(18\!\cdots\!50\)\( \beta_{8}) q^{37} +(\)\(14\!\cdots\!12\)\( - \)\(47\!\cdots\!14\)\( \beta_{1} + \)\(37\!\cdots\!54\)\( \beta_{2} + \)\(24\!\cdots\!31\)\( \beta_{3} - \)\(26\!\cdots\!02\)\( \beta_{4} + \)\(72\!\cdots\!99\)\( \beta_{5} - \)\(78\!\cdots\!12\)\( \beta_{6} - \)\(32\!\cdots\!84\)\( \beta_{7} + \)\(61\!\cdots\!20\)\( \beta_{8}) q^{38} +(\)\(24\!\cdots\!63\)\( - \)\(21\!\cdots\!54\)\( \beta_{1} + \)\(12\!\cdots\!62\)\( \beta_{2} + \)\(34\!\cdots\!34\)\( \beta_{3} + \)\(51\!\cdots\!23\)\( \beta_{4} - \)\(74\!\cdots\!84\)\( \beta_{5} + \)\(48\!\cdots\!71\)\( \beta_{6} + \)\(10\!\cdots\!96\)\( \beta_{7} - \)\(11\!\cdots\!16\)\( \beta_{8}) q^{39} +(\)\(12\!\cdots\!90\)\( - \)\(16\!\cdots\!80\)\( \beta_{1} - \)\(74\!\cdots\!10\)\( \beta_{2} + \)\(28\!\cdots\!80\)\( \beta_{3} - \)\(36\!\cdots\!70\)\( \beta_{4} + \)\(11\!\cdots\!50\)\( \beta_{5} - \)\(70\!\cdots\!00\)\( \beta_{6} - \)\(68\!\cdots\!50\)\( \beta_{7} + \)\(83\!\cdots\!00\)\( \beta_{8}) q^{40} +(-\)\(40\!\cdots\!38\)\( + \)\(71\!\cdots\!98\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} - \)\(32\!\cdots\!98\)\( \beta_{3} - \)\(12\!\cdots\!96\)\( \beta_{4} + \)\(11\!\cdots\!16\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!30\)\( \beta_{7} - \)\(44\!\cdots\!90\)\( \beta_{8}) q^{41} +(-\)\(27\!\cdots\!92\)\( + \)\(74\!\cdots\!52\)\( \beta_{1} - \)\(18\!\cdots\!72\)\( \beta_{2} - \)\(33\!\cdots\!52\)\( \beta_{3} + \)\(24\!\cdots\!48\)\( \beta_{4} - \)\(59\!\cdots\!84\)\( \beta_{5} + \)\(45\!\cdots\!36\)\( \beta_{6} + \)\(88\!\cdots\!80\)\( \beta_{7} + \)\(18\!\cdots\!40\)\( \beta_{8}) q^{42} +(-\)\(25\!\cdots\!53\)\( - \)\(83\!\cdots\!87\)\( \beta_{1} - \)\(16\!\cdots\!73\)\( \beta_{2} + \)\(65\!\cdots\!68\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} - \)\(25\!\cdots\!32\)\( \beta_{5} - \)\(84\!\cdots\!48\)\( \beta_{6} - \)\(88\!\cdots\!84\)\( \beta_{7} - \)\(62\!\cdots\!00\)\( \beta_{8}) q^{43} +(\)\(86\!\cdots\!92\)\( - \)\(72\!\cdots\!80\)\( \beta_{1} + \)\(91\!\cdots\!44\)\( \beta_{2} + \)\(53\!\cdots\!00\)\( \beta_{3} - \)\(30\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!96\)\( \beta_{5} - \)\(37\!\cdots\!76\)\( \beta_{6} - \)\(15\!\cdots\!16\)\( \beta_{7} + \)\(14\!\cdots\!76\)\( \beta_{8}) q^{44} +(\)\(36\!\cdots\!39\)\( - \)\(19\!\cdots\!38\)\( \beta_{1} + \)\(58\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!62\)\( \beta_{3} - \)\(12\!\cdots\!67\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(25\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!50\)\( \beta_{7} - \)\(19\!\cdots\!50\)\( \beta_{8}) q^{45} +(-\)\(35\!\cdots\!68\)\( - \)\(17\!\cdots\!92\)\( \beta_{1} + \)\(26\!\cdots\!68\)\( \beta_{2} - \)\(47\!\cdots\!58\)\( \beta_{3} + \)\(50\!\cdots\!84\)\( \beta_{4} - \)\(50\!\cdots\!94\)\( \beta_{5} - \)\(31\!\cdots\!20\)\( \beta_{6} - \)\(52\!\cdots\!40\)\( \beta_{7} - \)\(20\!\cdots\!40\)\( \beta_{8}) q^{46} +(-\)\(14\!\cdots\!54\)\( - \)\(34\!\cdots\!64\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!36\)\( \beta_{3} + \)\(32\!\cdots\!90\)\( \beta_{4} + \)\(27\!\cdots\!24\)\( \beta_{5} - \)\(29\!\cdots\!34\)\( \beta_{6} + \)\(13\!\cdots\!88\)\( \beta_{7} + \)\(14\!\cdots\!80\)\( \beta_{8}) q^{47} +(\)\(19\!\cdots\!52\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(21\!\cdots\!24\)\( \beta_{2} + \)\(85\!\cdots\!88\)\( \beta_{3} - \)\(57\!\cdots\!72\)\( \beta_{4} + \)\(63\!\cdots\!16\)\( \beta_{5} + \)\(16\!\cdots\!56\)\( \beta_{6} - \)\(22\!\cdots\!80\)\( \beta_{7} - \)\(82\!\cdots\!00\)\( \beta_{8}) q^{48} +(\)\(44\!\cdots\!17\)\( - \)\(83\!\cdots\!68\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} + \)\(15\!\cdots\!68\)\( \beta_{3} - \)\(27\!\cdots\!84\)\( \beta_{4} - \)\(57\!\cdots\!16\)\( \beta_{5} - \)\(25\!\cdots\!20\)\( \beta_{6} + \)\(35\!\cdots\!00\)\( \beta_{7} - \)\(13\!\cdots\!20\)\( \beta_{8}) q^{49} +(\)\(85\!\cdots\!25\)\( - \)\(32\!\cdots\!25\)\( \beta_{1} - \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3} + \)\(29\!\cdots\!00\)\( \beta_{4} + \)\(59\!\cdots\!00\)\( \beta_{5} - \)\(60\!\cdots\!00\)\( \beta_{6} - \)\(29\!\cdots\!00\)\( \beta_{7} + \)\(22\!\cdots\!00\)\( \beta_{8}) q^{50} +(-\)\(14\!\cdots\!12\)\( - \)\(69\!\cdots\!02\)\( \beta_{1} + \)\(53\!\cdots\!26\)\( \beta_{2} - \)\(33\!\cdots\!08\)\( \beta_{3} + \)\(46\!\cdots\!14\)\( \beta_{4} + \)\(36\!\cdots\!08\)\( \beta_{5} + \)\(21\!\cdots\!58\)\( \beta_{6} + \)\(21\!\cdots\!68\)\( \beta_{7} + \)\(35\!\cdots\!12\)\( \beta_{8}) q^{51} +(\)\(46\!\cdots\!00\)\( - \)\(34\!\cdots\!02\)\( \beta_{1} + \)\(53\!\cdots\!54\)\( \beta_{2} + \)\(55\!\cdots\!74\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(38\!\cdots\!76\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{6} - \)\(86\!\cdots\!12\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{52} +(\)\(12\!\cdots\!19\)\( - \)\(51\!\cdots\!58\)\( \beta_{1} + \)\(39\!\cdots\!62\)\( \beta_{2} - \)\(70\!\cdots\!58\)\( \beta_{3} - \)\(21\!\cdots\!31\)\( \beta_{4} - \)\(55\!\cdots\!44\)\( \beta_{5} - \)\(93\!\cdots\!40\)\( \beta_{6} + \)\(17\!\cdots\!46\)\( \beta_{7} + \)\(12\!\cdots\!70\)\( \beta_{8}) q^{53} +(\)\(14\!\cdots\!48\)\( - \)\(51\!\cdots\!32\)\( \beta_{1} + \)\(44\!\cdots\!96\)\( \beta_{2} + \)\(14\!\cdots\!22\)\( \beta_{3} + \)\(35\!\cdots\!80\)\( \beta_{4} + \)\(14\!\cdots\!26\)\( \beta_{5} + \)\(28\!\cdots\!24\)\( \beta_{6} + \)\(98\!\cdots\!44\)\( \beta_{7} - \)\(25\!\cdots\!44\)\( \beta_{8}) q^{54} +(-\)\(14\!\cdots\!79\)\( + \)\(34\!\cdots\!18\)\( \beta_{1} - \)\(92\!\cdots\!74\)\( \beta_{2} + \)\(21\!\cdots\!82\)\( \beta_{3} + \)\(20\!\cdots\!37\)\( \beta_{4} + \)\(36\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!75\)\( \beta_{6} - \)\(20\!\cdots\!00\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{55} +(-\)\(15\!\cdots\!56\)\( + \)\(33\!\cdots\!12\)\( \beta_{1} - \)\(51\!\cdots\!40\)\( \beta_{2} - \)\(15\!\cdots\!72\)\( \beta_{3} + \)\(36\!\cdots\!80\)\( \beta_{4} - \)\(21\!\cdots\!12\)\( \beta_{5} + \)\(31\!\cdots\!72\)\( \beta_{6} + \)\(72\!\cdots\!32\)\( \beta_{7} + \)\(39\!\cdots\!68\)\( \beta_{8}) q^{56} +(-\)\(34\!\cdots\!88\)\( + \)\(65\!\cdots\!37\)\( \beta_{1} + \)\(32\!\cdots\!86\)\( \beta_{2} - \)\(35\!\cdots\!43\)\( \beta_{3} - \)\(44\!\cdots\!30\)\( \beta_{4} + \)\(25\!\cdots\!62\)\( \beta_{5} - \)\(36\!\cdots\!92\)\( \beta_{6} - \)\(11\!\cdots\!31\)\( \beta_{7} - \)\(16\!\cdots\!35\)\( \beta_{8}) q^{57} +(\)\(11\!\cdots\!02\)\( + \)\(87\!\cdots\!54\)\( \beta_{1} + \)\(30\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!72\)\( \beta_{3} + \)\(20\!\cdots\!28\)\( \beta_{4} + \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(23\!\cdots\!52\)\( \beta_{6} - \)\(24\!\cdots\!52\)\( \beta_{7} + \)\(31\!\cdots\!40\)\( \beta_{8}) q^{58} +(-\)\(23\!\cdots\!33\)\( + \)\(32\!\cdots\!73\)\( \beta_{1} + \)\(72\!\cdots\!27\)\( \beta_{2} + \)\(48\!\cdots\!72\)\( \beta_{3} + \)\(29\!\cdots\!76\)\( \beta_{4} - \)\(27\!\cdots\!92\)\( \beta_{5} - \)\(62\!\cdots\!84\)\( \beta_{6} + \)\(34\!\cdots\!36\)\( \beta_{7} + \)\(13\!\cdots\!24\)\( \beta_{8}) q^{59} +(-\)\(46\!\cdots\!64\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(64\!\cdots\!96\)\( \beta_{2} - \)\(12\!\cdots\!08\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(43\!\cdots\!60\)\( \beta_{5} + \)\(41\!\cdots\!20\)\( \beta_{6} + \)\(43\!\cdots\!80\)\( \beta_{7} - \)\(32\!\cdots\!40\)\( \beta_{8}) q^{60} +(\)\(13\!\cdots\!03\)\( - \)\(17\!\cdots\!30\)\( \beta_{1} - \)\(69\!\cdots\!10\)\( \beta_{2} - \)\(16\!\cdots\!70\)\( \beta_{3} - \)\(17\!\cdots\!87\)\( \beta_{4} - \)\(30\!\cdots\!16\)\( \beta_{5} + \)\(20\!\cdots\!28\)\( \beta_{6} - \)\(42\!\cdots\!02\)\( \beta_{7} + \)\(11\!\cdots\!22\)\( \beta_{8}) q^{61} +(\)\(21\!\cdots\!60\)\( - \)\(71\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!28\)\( \beta_{2} + \)\(57\!\cdots\!48\)\( \beta_{3} - \)\(18\!\cdots\!92\)\( \beta_{4} + \)\(13\!\cdots\!96\)\( \beta_{5} - \)\(46\!\cdots\!04\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7} - \)\(17\!\cdots\!20\)\( \beta_{8}) q^{62} +(\)\(50\!\cdots\!81\)\( - \)\(21\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!30\)\( \beta_{2} + \)\(35\!\cdots\!34\)\( \beta_{3} + \)\(19\!\cdots\!33\)\( \beta_{4} - \)\(41\!\cdots\!48\)\( \beta_{5} - \)\(28\!\cdots\!15\)\( \beta_{6} + \)\(60\!\cdots\!32\)\( \beta_{7} - \)\(18\!\cdots\!20\)\( \beta_{8}) q^{63} +(\)\(65\!\cdots\!72\)\( - \)\(89\!\cdots\!28\)\( \beta_{1} + \)\(80\!\cdots\!36\)\( \beta_{2} + \)\(30\!\cdots\!48\)\( \beta_{3} - \)\(39\!\cdots\!08\)\( \beta_{4} + \)\(49\!\cdots\!28\)\( \beta_{5} + \)\(30\!\cdots\!40\)\( \beta_{6} - \)\(24\!\cdots\!20\)\( \beta_{7} + \)\(17\!\cdots\!80\)\( \beta_{8}) q^{64} +(-\)\(45\!\cdots\!92\)\( - \)\(40\!\cdots\!26\)\( \beta_{1} + \)\(29\!\cdots\!88\)\( \beta_{2} - \)\(25\!\cdots\!74\)\( \beta_{3} - \)\(78\!\cdots\!24\)\( \beta_{4} - \)\(88\!\cdots\!80\)\( \beta_{5} + \)\(13\!\cdots\!60\)\( \beta_{6} + \)\(36\!\cdots\!90\)\( \beta_{7} - \)\(45\!\cdots\!70\)\( \beta_{8}) q^{65} +(-\)\(24\!\cdots\!68\)\( + \)\(32\!\cdots\!84\)\( \beta_{1} + \)\(35\!\cdots\!68\)\( \beta_{2} - \)\(22\!\cdots\!64\)\( \beta_{3} - \)\(82\!\cdots\!68\)\( \beta_{4} + \)\(57\!\cdots\!64\)\( \beta_{5} - \)\(56\!\cdots\!56\)\( \beta_{6} + \)\(56\!\cdots\!24\)\( \beta_{7} + \)\(37\!\cdots\!16\)\( \beta_{8}) q^{66} +(-\)\(15\!\cdots\!45\)\( + \)\(32\!\cdots\!31\)\( \beta_{1} - \)\(10\!\cdots\!55\)\( \beta_{2} - \)\(46\!\cdots\!56\)\( \beta_{3} + \)\(42\!\cdots\!62\)\( \beta_{4} - \)\(15\!\cdots\!44\)\( \beta_{5} + \)\(73\!\cdots\!22\)\( \beta_{6} - \)\(37\!\cdots\!56\)\( \beta_{7} + \)\(13\!\cdots\!20\)\( \beta_{8}) q^{67} +(-\)\(15\!\cdots\!40\)\( + \)\(28\!\cdots\!10\)\( \beta_{1} - \)\(32\!\cdots\!74\)\( \beta_{2} + \)\(10\!\cdots\!78\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4} + \)\(10\!\cdots\!84\)\( \beta_{5} + \)\(92\!\cdots\!20\)\( \beta_{6} + \)\(70\!\cdots\!44\)\( \beta_{7} - \)\(52\!\cdots\!40\)\( \beta_{8}) q^{68} +(\)\(14\!\cdots\!28\)\( + \)\(57\!\cdots\!94\)\( \beta_{1} - \)\(81\!\cdots\!48\)\( \beta_{2} + \)\(12\!\cdots\!46\)\( \beta_{3} - \)\(33\!\cdots\!80\)\( \beta_{4} + \)\(72\!\cdots\!44\)\( \beta_{5} - \)\(35\!\cdots\!44\)\( \beta_{6} - \)\(11\!\cdots\!34\)\( \beta_{7} + \)\(72\!\cdots\!54\)\( \beta_{8}) q^{69} +(\)\(27\!\cdots\!32\)\( + \)\(42\!\cdots\!56\)\( \beta_{1} + \)\(74\!\cdots\!92\)\( \beta_{2} - \)\(38\!\cdots\!56\)\( \beta_{3} - \)\(48\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(59\!\cdots\!00\)\( \beta_{6} - \)\(16\!\cdots\!00\)\( \beta_{7} + \)\(32\!\cdots\!00\)\( \beta_{8}) q^{70} +(-\)\(23\!\cdots\!95\)\( - \)\(41\!\cdots\!50\)\( \beta_{1} + \)\(41\!\cdots\!70\)\( \beta_{2} - \)\(17\!\cdots\!50\)\( \beta_{3} + \)\(54\!\cdots\!09\)\( \beta_{4} - \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(10\!\cdots\!69\)\( \beta_{6} + \)\(92\!\cdots\!84\)\( \beta_{7} - \)\(24\!\cdots\!04\)\( \beta_{8}) q^{71} +(-\)\(57\!\cdots\!93\)\( - \)\(68\!\cdots\!82\)\( \beta_{1} + \)\(11\!\cdots\!63\)\( \beta_{2} - \)\(50\!\cdots\!74\)\( \beta_{3} + \)\(67\!\cdots\!21\)\( \beta_{4} - \)\(19\!\cdots\!93\)\( \beta_{5} + \)\(60\!\cdots\!72\)\( \beta_{6} + \)\(50\!\cdots\!25\)\( \beta_{7} + \)\(87\!\cdots\!20\)\( \beta_{8}) q^{72} +(\)\(10\!\cdots\!98\)\( - \)\(13\!\cdots\!63\)\( \beta_{1} + \)\(54\!\cdots\!26\)\( \beta_{2} + \)\(41\!\cdots\!81\)\( \beta_{3} - \)\(11\!\cdots\!82\)\( \beta_{4} + \)\(22\!\cdots\!94\)\( \beta_{5} - \)\(22\!\cdots\!52\)\( \beta_{6} + \)\(10\!\cdots\!01\)\( \beta_{7} + \)\(75\!\cdots\!45\)\( \beta_{8}) q^{73} +(\)\(34\!\cdots\!10\)\( - \)\(51\!\cdots\!82\)\( \beta_{1} - \)\(47\!\cdots\!72\)\( \beta_{2} + \)\(19\!\cdots\!32\)\( \beta_{3} - \)\(78\!\cdots\!48\)\( \beta_{4} - \)\(16\!\cdots\!80\)\( \beta_{5} + \)\(18\!\cdots\!48\)\( \beta_{6} - \)\(11\!\cdots\!72\)\( \beta_{7} + \)\(82\!\cdots\!32\)\( \beta_{8}) q^{74} +(\)\(47\!\cdots\!75\)\( - \)\(61\!\cdots\!75\)\( \beta_{1} - \)\(16\!\cdots\!25\)\( \beta_{2} - \)\(94\!\cdots\!00\)\( \beta_{3} - \)\(32\!\cdots\!00\)\( \beta_{4} - \)\(31\!\cdots\!00\)\( \beta_{5} + \)\(59\!\cdots\!00\)\( \beta_{6} + \)\(26\!\cdots\!00\)\( \beta_{7} - \)\(13\!\cdots\!00\)\( \beta_{8}) q^{75} +(-\)\(18\!\cdots\!52\)\( + \)\(96\!\cdots\!64\)\( \beta_{1} - \)\(20\!\cdots\!40\)\( \beta_{2} - \)\(12\!\cdots\!84\)\( \beta_{3} + \)\(24\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} - \)\(14\!\cdots\!16\)\( \beta_{6} + \)\(38\!\cdots\!64\)\( \beta_{7} + \)\(33\!\cdots\!76\)\( \beta_{8}) q^{76} +(\)\(65\!\cdots\!04\)\( + \)\(15\!\cdots\!38\)\( \beta_{1} + \)\(76\!\cdots\!36\)\( \beta_{2} + \)\(39\!\cdots\!62\)\( \beta_{3} - \)\(35\!\cdots\!08\)\( \beta_{4} + \)\(83\!\cdots\!04\)\( \beta_{5} + \)\(30\!\cdots\!04\)\( \beta_{6} - \)\(14\!\cdots\!10\)\( \beta_{7} + \)\(57\!\cdots\!50\)\( \beta_{8}) q^{77} +(-\)\(76\!\cdots\!16\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} + \)\(16\!\cdots\!28\)\( \beta_{2} + \)\(75\!\cdots\!82\)\( \beta_{3} + \)\(34\!\cdots\!16\)\( \beta_{4} - \)\(83\!\cdots\!82\)\( \beta_{5} - \)\(14\!\cdots\!64\)\( \beta_{6} + \)\(30\!\cdots\!52\)\( \beta_{7} - \)\(23\!\cdots\!00\)\( \beta_{8}) q^{78} +(\)\(13\!\cdots\!38\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(82\!\cdots\!88\)\( \beta_{2} + \)\(71\!\cdots\!64\)\( \beta_{3} - \)\(18\!\cdots\!10\)\( \beta_{4} - \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(19\!\cdots\!34\)\( \beta_{6} - \)\(98\!\cdots\!56\)\( \beta_{7} + \)\(58\!\cdots\!16\)\( \beta_{8}) q^{79} +(-\)\(22\!\cdots\!92\)\( + \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(14\!\cdots\!52\)\( \beta_{2} + \)\(37\!\cdots\!36\)\( \beta_{3} + \)\(39\!\cdots\!76\)\( \beta_{4} + \)\(71\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!00\)\( \beta_{6} - \)\(53\!\cdots\!00\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{80} +(-\)\(45\!\cdots\!51\)\( + \)\(25\!\cdots\!79\)\( \beta_{1} - \)\(12\!\cdots\!14\)\( \beta_{2} - \)\(23\!\cdots\!69\)\( \beta_{3} - \)\(25\!\cdots\!46\)\( \beta_{4} + \)\(49\!\cdots\!22\)\( \beta_{5} - \)\(30\!\cdots\!76\)\( \beta_{6} + \)\(38\!\cdots\!19\)\( \beta_{7} - \)\(22\!\cdots\!69\)\( \beta_{8}) q^{81} +(\)\(35\!\cdots\!02\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} - \)\(32\!\cdots\!28\)\( \beta_{2} - \)\(56\!\cdots\!36\)\( \beta_{3} + \)\(99\!\cdots\!48\)\( \beta_{4} + \)\(96\!\cdots\!52\)\( \beta_{5} + \)\(21\!\cdots\!20\)\( \beta_{6} + \)\(68\!\cdots\!32\)\( \beta_{7} + \)\(57\!\cdots\!40\)\( \beta_{8}) q^{82} +(-\)\(10\!\cdots\!31\)\( - \)\(27\!\cdots\!65\)\( \beta_{1} - \)\(15\!\cdots\!67\)\( \beta_{2} - \)\(73\!\cdots\!60\)\( \beta_{3} - \)\(33\!\cdots\!80\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} - \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(46\!\cdots\!00\)\( \beta_{7} - \)\(28\!\cdots\!20\)\( \beta_{8}) q^{83} +(\)\(16\!\cdots\!64\)\( - \)\(10\!\cdots\!12\)\( \beta_{1} + \)\(19\!\cdots\!40\)\( \beta_{2} + \)\(52\!\cdots\!72\)\( \beta_{3} - \)\(41\!\cdots\!88\)\( \beta_{4} + \)\(75\!\cdots\!08\)\( \beta_{5} - \)\(73\!\cdots\!60\)\( \beta_{6} - \)\(24\!\cdots\!40\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{84} +(-\)\(50\!\cdots\!14\)\( - \)\(30\!\cdots\!42\)\( \beta_{1} + \)\(22\!\cdots\!96\)\( \beta_{2} + \)\(27\!\cdots\!42\)\( \beta_{3} + \)\(77\!\cdots\!42\)\( \beta_{4} + \)\(22\!\cdots\!40\)\( \beta_{5} - \)\(66\!\cdots\!80\)\( \beta_{6} + \)\(29\!\cdots\!30\)\( \beta_{7} + \)\(27\!\cdots\!10\)\( \beta_{8}) q^{85} +(-\)\(28\!\cdots\!12\)\( + \)\(15\!\cdots\!50\)\( \beta_{1} + \)\(88\!\cdots\!38\)\( \beta_{2} - \)\(84\!\cdots\!35\)\( \beta_{3} + \)\(13\!\cdots\!90\)\( \beta_{4} + \)\(15\!\cdots\!17\)\( \beta_{5} + \)\(28\!\cdots\!28\)\( \beta_{6} + \)\(60\!\cdots\!08\)\( \beta_{7} + \)\(98\!\cdots\!52\)\( \beta_{8}) q^{86} +(-\)\(76\!\cdots\!49\)\( + \)\(40\!\cdots\!10\)\( \beta_{1} - \)\(67\!\cdots\!14\)\( \beta_{2} - \)\(12\!\cdots\!30\)\( \beta_{3} - \)\(25\!\cdots\!17\)\( \beta_{4} - \)\(40\!\cdots\!52\)\( \beta_{5} + \)\(21\!\cdots\!59\)\( \beta_{6} - \)\(22\!\cdots\!16\)\( \beta_{7} - \)\(82\!\cdots\!40\)\( \beta_{8}) q^{87} +(-\)\(22\!\cdots\!32\)\( + \)\(20\!\cdots\!20\)\( \beta_{1} - \)\(29\!\cdots\!72\)\( \beta_{2} - \)\(78\!\cdots\!32\)\( \beta_{3} - \)\(92\!\cdots\!52\)\( \beta_{4} - \)\(80\!\cdots\!24\)\( \beta_{5} - \)\(15\!\cdots\!24\)\( \beta_{6} + \)\(17\!\cdots\!80\)\( \beta_{7} - \)\(41\!\cdots\!80\)\( \beta_{8}) q^{88} +(-\)\(67\!\cdots\!14\)\( + \)\(11\!\cdots\!49\)\( \beta_{1} - \)\(18\!\cdots\!10\)\( \beta_{2} + \)\(18\!\cdots\!81\)\( \beta_{3} + \)\(15\!\cdots\!14\)\( \beta_{4} + \)\(28\!\cdots\!58\)\( \beta_{5} + \)\(22\!\cdots\!08\)\( \beta_{6} + \)\(18\!\cdots\!93\)\( \beta_{7} + \)\(57\!\cdots\!37\)\( \beta_{8}) q^{89} +(-\)\(70\!\cdots\!22\)\( + \)\(16\!\cdots\!34\)\( \beta_{1} + \)\(45\!\cdots\!08\)\( \beta_{2} + \)\(96\!\cdots\!16\)\( \beta_{3} + \)\(48\!\cdots\!16\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5} + \)\(44\!\cdots\!60\)\( \beta_{6} + \)\(35\!\cdots\!40\)\( \beta_{7} - \)\(76\!\cdots\!20\)\( \beta_{8}) q^{90} +(-\)\(13\!\cdots\!84\)\( - \)\(84\!\cdots\!88\)\( \beta_{1} + \)\(22\!\cdots\!40\)\( \beta_{2} + \)\(45\!\cdots\!28\)\( \beta_{3} + \)\(55\!\cdots\!60\)\( \beta_{4} + \)\(24\!\cdots\!48\)\( \beta_{5} + \)\(30\!\cdots\!32\)\( \beta_{6} - \)\(12\!\cdots\!88\)\( \beta_{7} - \)\(16\!\cdots\!32\)\( \beta_{8}) q^{91} +(-\)\(40\!\cdots\!72\)\( - \)\(99\!\cdots\!36\)\( \beta_{1} + \)\(17\!\cdots\!04\)\( \beta_{2} - \)\(21\!\cdots\!56\)\( \beta_{3} - \)\(16\!\cdots\!72\)\( \beta_{4} - \)\(22\!\cdots\!08\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(13\!\cdots\!48\)\( \beta_{7} + \)\(71\!\cdots\!00\)\( \beta_{8}) q^{92} +(-\)\(73\!\cdots\!76\)\( - \)\(20\!\cdots\!72\)\( \beta_{1} + \)\(77\!\cdots\!08\)\( \beta_{2} + \)\(14\!\cdots\!28\)\( \beta_{3} + \)\(54\!\cdots\!84\)\( \beta_{4} + \)\(22\!\cdots\!52\)\( \beta_{5} + \)\(16\!\cdots\!44\)\( \beta_{6} + \)\(14\!\cdots\!08\)\( \beta_{7} - \)\(86\!\cdots\!80\)\( \beta_{8}) q^{93} +(-\)\(11\!\cdots\!68\)\( - \)\(70\!\cdots\!72\)\( \beta_{1} - \)\(55\!\cdots\!64\)\( \beta_{2} + \)\(83\!\cdots\!12\)\( \beta_{3} - \)\(23\!\cdots\!80\)\( \beta_{4} + \)\(59\!\cdots\!96\)\( \beta_{5} + \)\(83\!\cdots\!64\)\( \beta_{6} - \)\(22\!\cdots\!56\)\( \beta_{7} - \)\(85\!\cdots\!04\)\( \beta_{8}) q^{94} +(-\)\(21\!\cdots\!45\)\( - \)\(23\!\cdots\!10\)\( \beta_{1} - \)\(25\!\cdots\!70\)\( \beta_{2} + \)\(90\!\cdots\!10\)\( \beta_{3} + \)\(10\!\cdots\!35\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!25\)\( \beta_{6} - \)\(15\!\cdots\!00\)\( \beta_{7} + \)\(47\!\cdots\!00\)\( \beta_{8}) q^{95} +(-\)\(88\!\cdots\!48\)\( + \)\(26\!\cdots\!80\)\( \beta_{1} - \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(90\!\cdots\!20\)\( \beta_{3} + \)\(36\!\cdots\!48\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5} - \)\(46\!\cdots\!40\)\( \beta_{6} + \)\(93\!\cdots\!40\)\( \beta_{7} - \)\(78\!\cdots\!20\)\( \beta_{8}) q^{96} +(-\)\(11\!\cdots\!26\)\( + \)\(24\!\cdots\!85\)\( \beta_{1} + \)\(39\!\cdots\!22\)\( \beta_{2} + \)\(14\!\cdots\!37\)\( \beta_{3} - \)\(34\!\cdots\!26\)\( \beta_{4} - \)\(20\!\cdots\!34\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} - \)\(98\!\cdots\!59\)\( \beta_{7} + \)\(40\!\cdots\!65\)\( \beta_{8}) q^{97} +(-\)\(30\!\cdots\!57\)\( + \)\(86\!\cdots\!17\)\( \beta_{1} + \)\(14\!\cdots\!48\)\( \beta_{2} - \)\(69\!\cdots\!44\)\( \beta_{3} - \)\(61\!\cdots\!08\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(41\!\cdots\!20\)\( \beta_{6} - \)\(58\!\cdots\!32\)\( \beta_{7} + \)\(14\!\cdots\!80\)\( \beta_{8}) q^{98} +(-\)\(48\!\cdots\!05\)\( + \)\(43\!\cdots\!57\)\( \beta_{1} + \)\(27\!\cdots\!15\)\( \beta_{2} + \)\(69\!\cdots\!08\)\( \beta_{3} - \)\(27\!\cdots\!60\)\( \beta_{4} - \)\(66\!\cdots\!32\)\( \beta_{5} + \)\(50\!\cdots\!32\)\( \beta_{6} - \)\(33\!\cdots\!68\)\( \beta_{7} - \)\(53\!\cdots\!72\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 2345005562254519728q^{2} - \)\(45\!\cdots\!04\)\(q^{3} + \)\(84\!\cdots\!88\)\(q^{4} - \)\(18\!\cdots\!50\)\(q^{5} - \)\(16\!\cdots\!92\)\(q^{6} + \)\(21\!\cdots\!92\)\(q^{7} - \)\(64\!\cdots\!40\)\(q^{8} + \)\(79\!\cdots\!17\)\(q^{9} + O(q^{10}) \) \( 9q - 2345005562254519728q^{2} - \)\(45\!\cdots\!04\)\(q^{3} + \)\(84\!\cdots\!88\)\(q^{4} - \)\(18\!\cdots\!50\)\(q^{5} - \)\(16\!\cdots\!92\)\(q^{6} + \)\(21\!\cdots\!92\)\(q^{7} - \)\(64\!\cdots\!40\)\(q^{8} + \)\(79\!\cdots\!17\)\(q^{9} - \)\(17\!\cdots\!00\)\(q^{10} - \)\(94\!\cdots\!92\)\(q^{11} + \)\(20\!\cdots\!72\)\(q^{12} + \)\(20\!\cdots\!86\)\(q^{13} - \)\(65\!\cdots\!84\)\(q^{14} + \)\(94\!\cdots\!00\)\(q^{15} + \)\(15\!\cdots\!04\)\(q^{16} + \)\(29\!\cdots\!82\)\(q^{17} - \)\(25\!\cdots\!64\)\(q^{18} + \)\(30\!\cdots\!80\)\(q^{19} + \)\(45\!\cdots\!00\)\(q^{20} + \)\(35\!\cdots\!88\)\(q^{21} - \)\(19\!\cdots\!36\)\(q^{22} - \)\(66\!\cdots\!24\)\(q^{23} - \)\(75\!\cdots\!60\)\(q^{24} - \)\(60\!\cdots\!25\)\(q^{25} - \)\(68\!\cdots\!72\)\(q^{26} - \)\(59\!\cdots\!40\)\(q^{27} + \)\(14\!\cdots\!44\)\(q^{28} + \)\(90\!\cdots\!70\)\(q^{29} + \)\(22\!\cdots\!00\)\(q^{30} - \)\(10\!\cdots\!12\)\(q^{31} - \)\(32\!\cdots\!88\)\(q^{32} + \)\(18\!\cdots\!52\)\(q^{33} + \)\(93\!\cdots\!36\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(21\!\cdots\!56\)\(q^{36} - \)\(88\!\cdots\!38\)\(q^{37} + \)\(12\!\cdots\!40\)\(q^{38} + \)\(21\!\cdots\!04\)\(q^{39} + \)\(11\!\cdots\!00\)\(q^{40} - \)\(36\!\cdots\!22\)\(q^{41} - \)\(24\!\cdots\!96\)\(q^{42} - \)\(23\!\cdots\!44\)\(q^{43} + \)\(77\!\cdots\!56\)\(q^{44} + \)\(32\!\cdots\!50\)\(q^{45} - \)\(32\!\cdots\!52\)\(q^{46} - \)\(13\!\cdots\!48\)\(q^{47} + \)\(17\!\cdots\!76\)\(q^{48} + \)\(39\!\cdots\!13\)\(q^{49} + \)\(76\!\cdots\!00\)\(q^{50} - \)\(12\!\cdots\!52\)\(q^{51} + \)\(41\!\cdots\!52\)\(q^{52} + \)\(10\!\cdots\!46\)\(q^{53} + \)\(13\!\cdots\!80\)\(q^{54} - \)\(13\!\cdots\!00\)\(q^{55} - \)\(14\!\cdots\!20\)\(q^{56} - \)\(31\!\cdots\!80\)\(q^{57} + \)\(10\!\cdots\!60\)\(q^{58} - \)\(21\!\cdots\!60\)\(q^{59} - \)\(42\!\cdots\!00\)\(q^{60} + \)\(11\!\cdots\!58\)\(q^{61} + \)\(19\!\cdots\!04\)\(q^{62} + \)\(45\!\cdots\!96\)\(q^{63} + \)\(58\!\cdots\!68\)\(q^{64} - \)\(41\!\cdots\!00\)\(q^{65} - \)\(22\!\cdots\!04\)\(q^{66} - \)\(13\!\cdots\!68\)\(q^{67} - \)\(14\!\cdots\!76\)\(q^{68} + \)\(13\!\cdots\!64\)\(q^{69} + \)\(25\!\cdots\!00\)\(q^{70} - \)\(21\!\cdots\!52\)\(q^{71} - \)\(51\!\cdots\!20\)\(q^{72} + \)\(91\!\cdots\!26\)\(q^{73} + \)\(31\!\cdots\!76\)\(q^{74} + \)\(43\!\cdots\!00\)\(q^{75} - \)\(16\!\cdots\!40\)\(q^{76} + \)\(59\!\cdots\!04\)\(q^{77} - \)\(68\!\cdots\!68\)\(q^{78} + \)\(12\!\cdots\!20\)\(q^{79} - \)\(20\!\cdots\!00\)\(q^{80} - \)\(40\!\cdots\!11\)\(q^{81} + \)\(32\!\cdots\!24\)\(q^{82} - \)\(92\!\cdots\!84\)\(q^{83} + \)\(14\!\cdots\!16\)\(q^{84} - \)\(45\!\cdots\!00\)\(q^{85} - \)\(25\!\cdots\!12\)\(q^{86} - \)\(68\!\cdots\!20\)\(q^{87} - \)\(20\!\cdots\!80\)\(q^{88} - \)\(60\!\cdots\!90\)\(q^{89} - \)\(63\!\cdots\!00\)\(q^{90} - \)\(12\!\cdots\!92\)\(q^{91} - \)\(36\!\cdots\!68\)\(q^{92} - \)\(66\!\cdots\!28\)\(q^{93} - \)\(10\!\cdots\!04\)\(q^{94} - \)\(19\!\cdots\!00\)\(q^{95} - \)\(79\!\cdots\!32\)\(q^{96} - \)\(10\!\cdots\!98\)\(q^{97} - \)\(27\!\cdots\!96\)\(q^{98} - \)\(44\!\cdots\!96\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 2 x^{8} - \)\(68\!\cdots\!36\)\( x^{7} - \)\(18\!\cdots\!28\)\( x^{6} + \)\(14\!\cdots\!06\)\( x^{5} + \)\(61\!\cdots\!64\)\( x^{4} - \)\(98\!\cdots\!68\)\( x^{3} - \)\(28\!\cdots\!44\)\( x^{2} + \)\(18\!\cdots\!33\)\( x + \)\(32\!\cdots\!74\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 11 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(69\!\cdots\!11\)\( \nu^{8} + \)\(15\!\cdots\!53\)\( \nu^{7} + \)\(40\!\cdots\!19\)\( \nu^{6} - \)\(80\!\cdots\!47\)\( \nu^{5} - \)\(59\!\cdots\!71\)\( \nu^{4} + \)\(10\!\cdots\!27\)\( \nu^{3} + \)\(16\!\cdots\!17\)\( \nu^{2} - \)\(44\!\cdots\!81\)\( \nu + \)\(15\!\cdots\!02\)\(\)\()/ \)\(44\!\cdots\!52\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(79\!\cdots\!87\)\( \nu^{8} - \)\(17\!\cdots\!01\)\( \nu^{7} - \)\(46\!\cdots\!23\)\( \nu^{6} + \)\(91\!\cdots\!99\)\( \nu^{5} + \)\(67\!\cdots\!07\)\( \nu^{4} - \)\(11\!\cdots\!59\)\( \nu^{3} + \)\(14\!\cdots\!47\)\( \nu^{2} + \)\(36\!\cdots\!85\)\( \nu - \)\(53\!\cdots\!90\)\(\)\()/ \)\(14\!\cdots\!84\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(14\!\cdots\!31\)\( \nu^{8} - \)\(35\!\cdots\!19\)\( \nu^{7} + \)\(10\!\cdots\!47\)\( \nu^{6} + \)\(23\!\cdots\!65\)\( \nu^{5} - \)\(21\!\cdots\!71\)\( \nu^{4} - \)\(38\!\cdots\!05\)\( \nu^{3} + \)\(13\!\cdots\!53\)\( \nu^{2} + \)\(12\!\cdots\!67\)\( \nu - \)\(15\!\cdots\!06\)\(\)\()/ \)\(46\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(55\!\cdots\!91\)\( \nu^{8} + \)\(19\!\cdots\!41\)\( \nu^{7} + \)\(34\!\cdots\!67\)\( \nu^{6} - \)\(10\!\cdots\!35\)\( \nu^{5} - \)\(57\!\cdots\!31\)\( \nu^{4} + \)\(13\!\cdots\!95\)\( \nu^{3} + \)\(27\!\cdots\!33\)\( \nu^{2} - \)\(34\!\cdots\!13\)\( \nu - \)\(47\!\cdots\!66\)\(\)\()/ \)\(27\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(33\!\cdots\!11\)\( \nu^{8} - \)\(11\!\cdots\!61\)\( \nu^{7} - \)\(18\!\cdots\!07\)\( \nu^{6} + \)\(60\!\cdots\!35\)\( \nu^{5} + \)\(27\!\cdots\!51\)\( \nu^{4} - \)\(70\!\cdots\!95\)\( \nu^{3} - \)\(11\!\cdots\!93\)\( \nu^{2} + \)\(17\!\cdots\!73\)\( \nu + \)\(11\!\cdots\!86\)\(\)\()/ \)\(69\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(95\!\cdots\!97\)\( \nu^{8} + \)\(14\!\cdots\!47\)\( \nu^{7} + \)\(31\!\cdots\!89\)\( \nu^{6} - \)\(85\!\cdots\!45\)\( \nu^{5} + \)\(32\!\cdots\!23\)\( \nu^{4} + \)\(13\!\cdots\!65\)\( \nu^{3} - \)\(10\!\cdots\!89\)\( \nu^{2} - \)\(76\!\cdots\!71\)\( \nu + \)\(31\!\cdots\!78\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(50\!\cdots\!01\)\( \nu^{8} - \)\(85\!\cdots\!51\)\( \nu^{7} - \)\(27\!\cdots\!37\)\( \nu^{6} + \)\(28\!\cdots\!85\)\( \nu^{5} + \)\(33\!\cdots\!41\)\( \nu^{4} + \)\(82\!\cdots\!55\)\( \nu^{3} - \)\(48\!\cdots\!63\)\( \nu^{2} - \)\(24\!\cdots\!57\)\( \nu - \)\(26\!\cdots\!74\)\(\)\()/ \)\(55\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 11\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3414351 \beta_{2} + 189828498139459931 \beta_{1} + 3530888858815588277459233691606826100\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 100 \beta_{6} - 8716729 \beta_{5} + 611104015289 \beta_{4} + 261633278718987930 \beta_{3} - 7852379881357415610550873 \beta_{2} + 6392297614796539847227904912944140426 \beta_{1} + 670263329160507439760026724614221985787578575603698103\)\()/110592\)
\(\nu^{4}\)\(=\)\((\)\(-33743922161664 \beta_{8} + 22994817386914953 \beta_{7} + 185098793423871777404 \beta_{6} + 2504204149487045150693375 \beta_{5} - 30109893022878964364258792959 \beta_{4} + 576961848958890314932551097193358490 \beta_{3} + 2924045494290224782521215472492163016111407 \beta_{2} + 160690031202447687101840435956556935214181300547721706 \beta_{1} + 1410655776895744825634505644465995221491836208865564342313094813099015599\)\()/331776\)
\(\nu^{5}\)\(=\)\((\)\(-977710028302938496794410972160 \beta_{8} + 10982027054486582087150658654706277 \beta_{7} - 804239144813557581512932253680820084 \beta_{6} - 61749678315799720816719286887561336919037 \beta_{5} + 6885741500484871576889697797771796121426834941 \beta_{4} + 3515690829166565982410567204916235296549117532215482 \beta_{3} - 178763837174342394896597444434978400630110164948157560128405 \beta_{2} + 48077367563605355957548557806232488176704933143418647772251942064726234 \beta_{1} + 8865291263913725629783995190494740789504015754670068598075343284757366059693509637005139\)\()/248832\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(86\!\cdots\!80\)\( \beta_{8} + \)\(83\!\cdots\!21\)\( \beta_{7} + \)\(49\!\cdots\!88\)\( \beta_{6} + \)\(75\!\cdots\!75\)\( \beta_{5} - \)\(47\!\cdots\!83\)\( \beta_{4} + \)\(11\!\cdots\!94\)\( \beta_{3} + \)\(47\!\cdots\!79\)\( \beta_{2} + \)\(37\!\cdots\!66\)\( \beta_{1} + \)\(23\!\cdots\!35\)\(\)\()/165888\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(78\!\cdots\!60\)\( \beta_{8} + \)\(61\!\cdots\!01\)\( \beta_{7} - \)\(13\!\cdots\!28\)\( \beta_{6} - \)\(16\!\cdots\!49\)\( \beta_{5} + \)\(37\!\cdots\!97\)\( \beta_{4} + \)\(23\!\cdots\!10\)\( \beta_{3} - \)\(12\!\cdots\!21\)\( \beta_{2} + \)\(23\!\cdots\!82\)\( \beta_{1} + \)\(54\!\cdots\!07\)\(\)\()/331776\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(75\!\cdots\!56\)\( \beta_{8} + \)\(90\!\cdots\!07\)\( \beta_{7} + \)\(42\!\cdots\!40\)\( \beta_{6} + \)\(69\!\cdots\!21\)\( \beta_{5} - \)\(25\!\cdots\!41\)\( \beta_{4} + \)\(84\!\cdots\!38\)\( \beta_{3} + \)\(25\!\cdots\!05\)\( \beta_{2} + \)\(33\!\cdots\!26\)\( \beta_{1} + \)\(17\!\cdots\!21\)\(\)\()/331776\)

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
−6.03020e16
−3.77468e16
−3.52210e16
−1.87667e16
−1.78332e14
1.54621e16
2.68569e16
4.65892e16
6.33066e16
−3.15505e18 6.48457e28 7.29589e36 −2.23954e41 −2.04592e47 1.98659e51 −1.46313e55 −1.18606e57 7.06586e59
1.2 −2.07240e18 −2.64580e28 1.63640e36 −5.16214e41 5.48316e46 −2.39892e51 2.11812e54 −4.69101e57 1.06980e60
1.3 −1.95116e18 −1.11766e29 1.14858e36 9.20370e41 2.18074e47 2.38466e51 2.94601e54 7.10063e57 −1.79579e60
1.4 −1.16136e18 1.09371e29 −1.30970e36 1.47325e42 −1.27019e47 −1.67990e50 4.60846e54 6.57089e57 −1.71098e60
1.5 −2.69116e17 3.14043e28 −2.58603e36 −3.76295e42 −8.45141e45 1.39208e51 1.41138e54 −4.40480e57 1.01267e60
1.6 4.81626e17 −2.96062e28 −2.42649e36 2.39673e42 −1.42591e46 2.39069e50 −2.44905e54 −4.51451e57 1.15433e60
1.7 1.02857e18 −1.22395e29 −1.60049e36 −2.11004e42 −1.25893e47 −1.42065e51 −4.38064e54 9.58953e57 −2.17033e60
1.8 1.97572e18 8.85385e28 1.24503e36 −2.55522e41 1.74928e47 −5.36128e50 −2.79253e54 2.44803e57 −5.04842e59
1.9 2.77816e18 −4.92324e28 5.05972e36 1.85701e41 −1.36775e47 7.17028e50 6.67109e54 −2.96721e57 5.15907e59
\(n\): 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 1.122.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.122.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{122}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2345005562254519728 T + \)\(10\!\cdots\!32\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!76\)\( T^{4} + \)\(95\!\cdots\!48\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} + \)\(33\!\cdots\!80\)\( T^{7} + \)\(57\!\cdots\!16\)\( T^{8} + \)\(96\!\cdots\!88\)\( T^{9} + \)\(15\!\cdots\!32\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(36\!\cdots\!36\)\( T^{12} + \)\(47\!\cdots\!68\)\( T^{13} + \)\(72\!\cdots\!32\)\( T^{14} + \)\(71\!\cdots\!60\)\( T^{15} + \)\(98\!\cdots\!96\)\( T^{16} + \)\(58\!\cdots\!68\)\( T^{17} + \)\(66\!\cdots\!12\)\( T^{18} \)
$3$ \( 1 + \)\(45\!\cdots\!04\)\( T + \)\(21\!\cdots\!63\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!16\)\( T^{4} + \)\(13\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!46\)\( T^{8} + \)\(65\!\cdots\!44\)\( T^{9} + \)\(81\!\cdots\!38\)\( T^{10} + \)\(31\!\cdots\!20\)\( T^{11} + \)\(34\!\cdots\!56\)\( T^{12} + \)\(11\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!88\)\( T^{14} + \)\(25\!\cdots\!60\)\( T^{15} + \)\(28\!\cdots\!81\)\( T^{16} + \)\(32\!\cdots\!44\)\( T^{17} + \)\(38\!\cdots\!83\)\( T^{18} \)
$5$ \( 1 + \)\(18\!\cdots\!50\)\( T + \)\(21\!\cdots\!25\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!00\)\( T^{4} + \)\(52\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{7} + \)\(75\!\cdots\!50\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(28\!\cdots\!50\)\( T^{10} + \)\(50\!\cdots\!00\)\( T^{11} + \)\(81\!\cdots\!00\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!00\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} + \)\(75\!\cdots\!50\)\( T^{17} + \)\(15\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 - \)\(21\!\cdots\!92\)\( T + \)\(85\!\cdots\!07\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!96\)\( T^{4} - \)\(38\!\cdots\!32\)\( T^{5} + \)\(68\!\cdots\!72\)\( T^{6} - \)\(69\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(21\!\cdots\!42\)\( T^{10} - \)\(22\!\cdots\!00\)\( T^{11} + \)\(40\!\cdots\!96\)\( T^{12} - \)\(41\!\cdots\!32\)\( T^{13} + \)\(59\!\cdots\!72\)\( T^{14} - \)\(48\!\cdots\!00\)\( T^{15} + \)\(53\!\cdots\!01\)\( T^{16} - \)\(24\!\cdots\!92\)\( T^{17} + \)\(20\!\cdots\!07\)\( T^{18} \)
$11$ \( 1 + \)\(94\!\cdots\!92\)\( T + \)\(52\!\cdots\!83\)\( T^{2} + \)\(54\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!60\)\( T^{4} + \)\(14\!\cdots\!32\)\( T^{5} + \)\(30\!\cdots\!84\)\( T^{6} + \)\(25\!\cdots\!36\)\( T^{7} + \)\(41\!\cdots\!78\)\( T^{8} + \)\(30\!\cdots\!80\)\( T^{9} + \)\(42\!\cdots\!58\)\( T^{10} + \)\(26\!\cdots\!56\)\( T^{11} + \)\(31\!\cdots\!04\)\( T^{12} + \)\(15\!\cdots\!12\)\( T^{13} + \)\(16\!\cdots\!60\)\( T^{14} + \)\(61\!\cdots\!84\)\( T^{15} + \)\(60\!\cdots\!93\)\( T^{16} + \)\(11\!\cdots\!52\)\( T^{17} + \)\(11\!\cdots\!91\)\( T^{18} \)
$13$ \( 1 - \)\(20\!\cdots\!86\)\( T + \)\(30\!\cdots\!33\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!36\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(48\!\cdots\!28\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} - \)\(98\!\cdots\!76\)\( T^{9} + \)\(22\!\cdots\!18\)\( T^{10} - \)\(50\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!16\)\( T^{12} - \)\(17\!\cdots\!96\)\( T^{13} + \)\(40\!\cdots\!48\)\( T^{14} - \)\(39\!\cdots\!80\)\( T^{15} + \)\(99\!\cdots\!61\)\( T^{16} - \)\(40\!\cdots\!06\)\( T^{17} + \)\(12\!\cdots\!73\)\( T^{18} \)
$17$ \( 1 - \)\(29\!\cdots\!82\)\( T + \)\(40\!\cdots\!57\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(83\!\cdots\!36\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!32\)\( T^{6} - \)\(30\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!32\)\( T^{9} + \)\(91\!\cdots\!62\)\( T^{10} - \)\(17\!\cdots\!40\)\( T^{11} + \)\(52\!\cdots\!16\)\( T^{12} - \)\(84\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!52\)\( T^{14} - \)\(25\!\cdots\!20\)\( T^{15} + \)\(62\!\cdots\!61\)\( T^{16} - \)\(34\!\cdots\!62\)\( T^{17} + \)\(90\!\cdots\!97\)\( T^{18} \)
$19$ \( 1 - \)\(30\!\cdots\!80\)\( T + \)\(22\!\cdots\!71\)\( T^{2} - \)\(28\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!96\)\( T^{4} - \)\(46\!\cdots\!40\)\( T^{5} + \)\(26\!\cdots\!56\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!46\)\( T^{8} - \)\(26\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!04\)\( T^{12} - \)\(38\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!04\)\( T^{14} - \)\(66\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!69\)\( T^{16} - \)\(20\!\cdots\!80\)\( T^{17} + \)\(36\!\cdots\!79\)\( T^{18} \)
$23$ \( 1 + \)\(66\!\cdots\!24\)\( T + \)\(47\!\cdots\!03\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!56\)\( T^{4} + \)\(23\!\cdots\!84\)\( T^{5} + \)\(67\!\cdots\!28\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!26\)\( T^{8} + \)\(93\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!98\)\( T^{10} + \)\(56\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!76\)\( T^{12} + \)\(28\!\cdots\!44\)\( T^{13} + \)\(55\!\cdots\!08\)\( T^{14} + \)\(81\!\cdots\!80\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} + \)\(94\!\cdots\!44\)\( T^{17} + \)\(83\!\cdots\!63\)\( T^{18} \)
$29$ \( 1 - \)\(90\!\cdots\!70\)\( T + \)\(40\!\cdots\!61\)\( T^{2} - \)\(58\!\cdots\!40\)\( T^{3} + \)\(82\!\cdots\!76\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!76\)\( T^{6} - \)\(27\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!06\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} - \)\(21\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!60\)\( T^{13} + \)\(46\!\cdots\!24\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} + \)\(18\!\cdots\!49\)\( T^{16} - \)\(36\!\cdots\!70\)\( T^{17} + \)\(35\!\cdots\!69\)\( T^{18} \)
$31$ \( 1 + \)\(10\!\cdots\!12\)\( T + \)\(15\!\cdots\!43\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!60\)\( T^{4} + \)\(11\!\cdots\!32\)\( T^{5} + \)\(51\!\cdots\!24\)\( T^{6} + \)\(55\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!18\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(52\!\cdots\!58\)\( T^{10} + \)\(44\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!84\)\( T^{12} + \)\(75\!\cdots\!72\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} + \)\(82\!\cdots\!84\)\( T^{15} + \)\(22\!\cdots\!73\)\( T^{16} + \)\(44\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!71\)\( T^{18} \)
$37$ \( 1 + \)\(88\!\cdots\!38\)\( T + \)\(28\!\cdots\!57\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!16\)\( T^{4} + \)\(16\!\cdots\!88\)\( T^{5} + \)\(32\!\cdots\!52\)\( T^{6} + \)\(13\!\cdots\!20\)\( T^{7} + \)\(23\!\cdots\!46\)\( T^{8} + \)\(88\!\cdots\!08\)\( T^{9} + \)\(13\!\cdots\!02\)\( T^{10} + \)\(41\!\cdots\!80\)\( T^{11} + \)\(58\!\cdots\!56\)\( T^{12} + \)\(16\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!12\)\( T^{14} + \)\(55\!\cdots\!40\)\( T^{15} + \)\(53\!\cdots\!81\)\( T^{16} + \)\(92\!\cdots\!98\)\( T^{17} + \)\(59\!\cdots\!77\)\( T^{18} \)
$41$ \( 1 + \)\(36\!\cdots\!22\)\( T + \)\(60\!\cdots\!73\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!60\)\( T^{4} + \)\(81\!\cdots\!32\)\( T^{5} + \)\(56\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!38\)\( T^{8} + \)\(30\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} + \)\(36\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!24\)\( T^{12} + \)\(31\!\cdots\!52\)\( T^{13} + \)\(12\!\cdots\!60\)\( T^{14} + \)\(18\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!13\)\( T^{16} + \)\(54\!\cdots\!62\)\( T^{17} + \)\(20\!\cdots\!61\)\( T^{18} \)
$43$ \( 1 + \)\(23\!\cdots\!44\)\( T + \)\(29\!\cdots\!43\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!96\)\( T^{4} + \)\(73\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!28\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!06\)\( T^{8} + \)\(30\!\cdots\!64\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{11} + \)\(33\!\cdots\!96\)\( T^{12} + \)\(29\!\cdots\!24\)\( T^{13} + \)\(75\!\cdots\!28\)\( T^{14} + \)\(46\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!01\)\( T^{16} + \)\(36\!\cdots\!44\)\( T^{17} + \)\(70\!\cdots\!43\)\( T^{18} \)
$47$ \( 1 + \)\(13\!\cdots\!48\)\( T + \)\(10\!\cdots\!07\)\( T^{2} + \)\(80\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!56\)\( T^{4} + \)\(98\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!12\)\( T^{6} - \)\(41\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!26\)\( T^{8} - \)\(16\!\cdots\!72\)\( T^{9} + \)\(51\!\cdots\!22\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} + \)\(19\!\cdots\!68\)\( T^{13} + \)\(19\!\cdots\!92\)\( T^{14} + \)\(70\!\cdots\!20\)\( T^{15} + \)\(19\!\cdots\!41\)\( T^{16} + \)\(51\!\cdots\!28\)\( T^{17} + \)\(82\!\cdots\!67\)\( T^{18} \)
$53$ \( 1 - \)\(10\!\cdots\!46\)\( T + \)\(22\!\cdots\!13\)\( T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!16\)\( T^{4} - \)\(39\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(34\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!46\)\( T^{8} - \)\(19\!\cdots\!56\)\( T^{9} + \)\(38\!\cdots\!38\)\( T^{10} - \)\(64\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!56\)\( T^{12} - \)\(13\!\cdots\!36\)\( T^{13} + \)\(37\!\cdots\!88\)\( T^{14} - \)\(19\!\cdots\!40\)\( T^{15} + \)\(65\!\cdots\!81\)\( T^{16} - \)\(13\!\cdots\!06\)\( T^{17} + \)\(54\!\cdots\!33\)\( T^{18} \)
$59$ \( 1 + \)\(21\!\cdots\!60\)\( T + \)\(10\!\cdots\!31\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!16\)\( T^{4} + \)\(70\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!36\)\( T^{6} + \)\(18\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} + \)\(38\!\cdots\!00\)\( T^{9} + \)\(69\!\cdots\!74\)\( T^{10} + \)\(65\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!44\)\( T^{12} + \)\(86\!\cdots\!80\)\( T^{13} + \)\(11\!\cdots\!84\)\( T^{14} + \)\(75\!\cdots\!20\)\( T^{15} + \)\(84\!\cdots\!89\)\( T^{16} + \)\(32\!\cdots\!60\)\( T^{17} + \)\(28\!\cdots\!39\)\( T^{18} \)
$61$ \( 1 - \)\(11\!\cdots\!58\)\( T + \)\(45\!\cdots\!33\)\( T^{2} - \)\(66\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} - \)\(23\!\cdots\!64\)\( T^{7} + \)\(28\!\cdots\!78\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!58\)\( T^{10} - \)\(26\!\cdots\!44\)\( T^{11} + \)\(24\!\cdots\!04\)\( T^{12} - \)\(20\!\cdots\!88\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(94\!\cdots\!16\)\( T^{15} + \)\(68\!\cdots\!93\)\( T^{16} - \)\(18\!\cdots\!98\)\( T^{17} + \)\(16\!\cdots\!41\)\( T^{18} \)
$67$ \( 1 + \)\(13\!\cdots\!68\)\( T + \)\(32\!\cdots\!07\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!36\)\( T^{4} - \)\(36\!\cdots\!92\)\( T^{5} + \)\(75\!\cdots\!32\)\( T^{6} - \)\(52\!\cdots\!60\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!32\)\( T^{9} + \)\(80\!\cdots\!62\)\( T^{10} - \)\(43\!\cdots\!40\)\( T^{11} + \)\(55\!\cdots\!16\)\( T^{12} - \)\(24\!\cdots\!32\)\( T^{13} + \)\(32\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!80\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} + \)\(59\!\cdots\!88\)\( T^{17} + \)\(39\!\cdots\!47\)\( T^{18} \)
$71$ \( 1 + \)\(21\!\cdots\!52\)\( T + \)\(81\!\cdots\!63\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(30\!\cdots\!60\)\( T^{4} + \)\(44\!\cdots\!32\)\( T^{5} + \)\(69\!\cdots\!04\)\( T^{6} + \)\(82\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!98\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} + \)\(83\!\cdots\!36\)\( T^{11} + \)\(70\!\cdots\!44\)\( T^{12} + \)\(45\!\cdots\!92\)\( T^{13} + \)\(31\!\cdots\!60\)\( T^{14} + \)\(14\!\cdots\!84\)\( T^{15} + \)\(84\!\cdots\!33\)\( T^{16} + \)\(22\!\cdots\!72\)\( T^{17} + \)\(10\!\cdots\!31\)\( T^{18} \)
$73$ \( 1 - \)\(91\!\cdots\!26\)\( T + \)\(16\!\cdots\!53\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(93\!\cdots\!16\)\( T^{5} + \)\(76\!\cdots\!28\)\( T^{6} - \)\(45\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!96\)\( T^{9} + \)\(88\!\cdots\!98\)\( T^{10} - \)\(37\!\cdots\!40\)\( T^{11} + \)\(18\!\cdots\!76\)\( T^{12} - \)\(66\!\cdots\!56\)\( T^{13} + \)\(28\!\cdots\!08\)\( T^{14} - \)\(75\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!41\)\( T^{16} - \)\(45\!\cdots\!06\)\( T^{17} + \)\(14\!\cdots\!13\)\( T^{18} \)
$79$ \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(26\!\cdots\!11\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!76\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!76\)\( T^{6} - \)\(10\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(51\!\cdots\!00\)\( T^{9} + \)\(44\!\cdots\!74\)\( T^{10} - \)\(18\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!64\)\( T^{12} - \)\(49\!\cdots\!60\)\( T^{13} + \)\(33\!\cdots\!24\)\( T^{14} - \)\(10\!\cdots\!40\)\( T^{15} + \)\(52\!\cdots\!49\)\( T^{16} - \)\(99\!\cdots\!20\)\( T^{17} + \)\(32\!\cdots\!19\)\( T^{18} \)
$83$ \( 1 + \)\(92\!\cdots\!84\)\( T + \)\(79\!\cdots\!23\)\( T^{2} + \)\(51\!\cdots\!60\)\( T^{3} + \)\(32\!\cdots\!76\)\( T^{4} + \)\(15\!\cdots\!04\)\( T^{5} + \)\(87\!\cdots\!28\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!66\)\( T^{8} + \)\(53\!\cdots\!84\)\( T^{9} + \)\(28\!\cdots\!78\)\( T^{10} + \)\(81\!\cdots\!80\)\( T^{11} + \)\(37\!\cdots\!36\)\( T^{12} + \)\(10\!\cdots\!84\)\( T^{13} + \)\(35\!\cdots\!68\)\( T^{14} + \)\(91\!\cdots\!40\)\( T^{15} + \)\(22\!\cdots\!21\)\( T^{16} + \)\(43\!\cdots\!44\)\( T^{17} + \)\(75\!\cdots\!03\)\( T^{18} \)
$89$ \( 1 + \)\(60\!\cdots\!90\)\( T + \)\(32\!\cdots\!01\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!56\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!96\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(84\!\cdots\!66\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(63\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!24\)\( T^{12} + \)\(48\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!44\)\( T^{14} + \)\(23\!\cdots\!80\)\( T^{15} + \)\(44\!\cdots\!29\)\( T^{16} + \)\(61\!\cdots\!90\)\( T^{17} + \)\(76\!\cdots\!09\)\( T^{18} \)
$97$ \( 1 + \)\(10\!\cdots\!98\)\( T + \)\(63\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(29\!\cdots\!28\)\( T^{5} + \)\(74\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!26\)\( T^{8} + \)\(51\!\cdots\!28\)\( T^{9} + \)\(77\!\cdots\!22\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} + \)\(11\!\cdots\!68\)\( T^{13} + \)\(10\!\cdots\!92\)\( T^{14} + \)\(70\!\cdots\!20\)\( T^{15} + \)\(39\!\cdots\!41\)\( T^{16} + \)\(16\!\cdots\!78\)\( T^{17} + \)\(39\!\cdots\!17\)\( T^{18} \)
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