Properties

Label 1.108.a.a
Level 1
Weight 108
Character orbit 1.a
Self dual yes
Analytic conductor 72.504
Analytic rank 0
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 108 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.5037502298\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{143}\cdot 3^{48}\cdot 5^{18}\cdot 7^{8}\cdot 11^{2}\cdot 13^{2} \)
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.

\(f(q)\) \(=\) \( q +(607308845937277 - \beta_{1}) q^{2} +(\)\(17\!\cdots\!33\)\( + 62113396 \beta_{1} + \beta_{2}) q^{3} +(\)\(71\!\cdots\!70\)\( + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4} +(\)\(27\!\cdots\!09\)\( + \)\(50\!\cdots\!29\)\( \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5} +(-\)\(13\!\cdots\!31\)\( + \)\(25\!\cdots\!77\)\( \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6} +(-\)\(10\!\cdots\!24\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(-\)\(11\!\cdots\!77\)\( - \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(51\!\cdots\!47\)\( \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8} +(\)\(41\!\cdots\!23\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(17\!\cdots\!19\)\( \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\) \( q +(607308845937277 - \beta_{1}) q^{2} +(\)\(17\!\cdots\!33\)\( + 62113396 \beta_{1} + \beta_{2}) q^{3} +(\)\(71\!\cdots\!70\)\( + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4} +(\)\(27\!\cdots\!09\)\( + \)\(50\!\cdots\!29\)\( \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5} +(-\)\(13\!\cdots\!31\)\( + \)\(25\!\cdots\!77\)\( \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6} +(-\)\(10\!\cdots\!24\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(-\)\(11\!\cdots\!77\)\( - \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(51\!\cdots\!47\)\( \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8} +(\)\(41\!\cdots\!23\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(17\!\cdots\!19\)\( \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9} +(-\)\(11\!\cdots\!14\)\( - \)\(85\!\cdots\!98\)\( \beta_{1} + \)\(34\!\cdots\!28\)\( \beta_{2} - \)\(42\!\cdots\!40\)\( \beta_{3} + 3035644428618332 \beta_{4} + 137787538404 \beta_{5} + 147169648 \beta_{6} - 120312 \beta_{7} - 216 \beta_{8}) q^{10} +(\)\(83\!\cdots\!59\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(26\!\cdots\!77\)\( \beta_{2} - \)\(30\!\cdots\!24\)\( \beta_{3} + 39655237059879084 \beta_{4} + 18943495258882 \beta_{5} + 20412816178 \beta_{6} + 12038260 \beta_{7} + 23004 \beta_{8}) q^{11} +(-\)\(88\!\cdots\!96\)\( + \)\(84\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!32\)\( \beta_{3} + 17417687351454383040 \beta_{4} + 8277620636103552 \beta_{5} + 1126938004992 \beta_{6} - 91849536 \beta_{7} - 1609920 \beta_{8}) q^{12} +(\)\(42\!\cdots\!13\)\( - \)\(29\!\cdots\!41\)\( \beta_{1} + \)\(84\!\cdots\!80\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3} - 1292007267852162005 \beta_{4} + 688583074305208092 \beta_{5} - 165811929566128 \beta_{6} - 46020832614 \beta_{7} + 83255310 \beta_{8}) q^{13} +(-\)\(82\!\cdots\!78\)\( + \)\(43\!\cdots\!78\)\( \beta_{1} + \)\(78\!\cdots\!18\)\( \beta_{2} + \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(34\!\cdots\!22\)\( \beta_{4} + 17161982422390714858 \beta_{5} + 4494593484494656 \beta_{6} + 3819461803360 \beta_{7} - 3391893792 \beta_{8}) q^{14} +(\)\(12\!\cdots\!72\)\( - \)\(49\!\cdots\!31\)\( \beta_{1} - \)\(43\!\cdots\!19\)\( \beta_{2} + \)\(11\!\cdots\!20\)\( \beta_{3} + \)\(28\!\cdots\!84\)\( \beta_{4} - \)\(14\!\cdots\!57\)\( \beta_{5} + 71321094534085191 \beta_{6} - 177279761520504 \beta_{7} + 113342680728 \beta_{8}) q^{15} +(\)\(32\!\cdots\!36\)\( + \)\(77\!\cdots\!76\)\( \beta_{1} - \)\(67\!\cdots\!24\)\( \beta_{2} + \)\(88\!\cdots\!92\)\( \beta_{3} + \)\(73\!\cdots\!32\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5} - 7893052997101223272 \beta_{6} + 5829377304694680 \beta_{7} - 3193371357696 \beta_{8}) q^{16} +(\)\(61\!\cdots\!04\)\( - \)\(51\!\cdots\!51\)\( \beta_{1} + \)\(30\!\cdots\!87\)\( \beta_{2} - \)\(44\!\cdots\!23\)\( \beta_{3} + \)\(45\!\cdots\!85\)\( \beta_{4} + \)\(86\!\cdots\!14\)\( \beta_{5} + \)\(25\!\cdots\!84\)\( \beta_{6} - 147577032261572145 \beta_{7} + 77391112194765 \beta_{8}) q^{17} +(\)\(26\!\cdots\!33\)\( - \)\(54\!\cdots\!45\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(56\!\cdots\!08\)\( \beta_{3} - \)\(44\!\cdots\!20\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(45\!\cdots\!64\)\( \beta_{6} + 2995387893492815280 \beta_{7} - 1637786543560080 \beta_{8}) q^{18} +(\)\(95\!\cdots\!49\)\( - \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(75\!\cdots\!37\)\( \beta_{2} + \)\(95\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!68\)\( \beta_{4} - \)\(67\!\cdots\!94\)\( \beta_{5} + \)\(45\!\cdots\!14\)\( \beta_{6} - 49760167465115702820 \beta_{7} + 30620433423116052 \beta_{8}) q^{19} +(\)\(14\!\cdots\!52\)\( + \)\(10\!\cdots\!32\)\( \beta_{1} + \)\(59\!\cdots\!86\)\( \beta_{2} + \)\(26\!\cdots\!70\)\( \beta_{3} + \)\(90\!\cdots\!88\)\( \beta_{4} + \)\(51\!\cdots\!80\)\( \beta_{5} + \)\(96\!\cdots\!60\)\( \beta_{6} + \)\(68\!\cdots\!60\)\( \beta_{7} - 510445792877080320 \beta_{8}) q^{20} +(\)\(10\!\cdots\!08\)\( - \)\(50\!\cdots\!94\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} + \)\(67\!\cdots\!70\)\( \beta_{3} + \)\(16\!\cdots\!94\)\( \beta_{4} - \)\(69\!\cdots\!64\)\( \beta_{5} - \)\(89\!\cdots\!68\)\( \beta_{6} - \)\(76\!\cdots\!30\)\( \beta_{7} + 7643011086349644726 \beta_{8}) q^{21} +(-\)\(23\!\cdots\!17\)\( - \)\(35\!\cdots\!21\)\( \beta_{1} + \)\(45\!\cdots\!75\)\( \beta_{2} - \)\(25\!\cdots\!70\)\( \beta_{3} - \)\(13\!\cdots\!85\)\( \beta_{4} + \)\(23\!\cdots\!47\)\( \beta_{5} + \)\(17\!\cdots\!32\)\( \beta_{6} + \)\(66\!\cdots\!56\)\( \beta_{7} - \)\(10\!\cdots\!60\)\( \beta_{8}) q^{22} +(-\)\(11\!\cdots\!80\)\( + \)\(57\!\cdots\!65\)\( \beta_{1} + \)\(88\!\cdots\!29\)\( \beta_{2} - \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(44\!\cdots\!95\)\( \beta_{5} - \)\(20\!\cdots\!45\)\( \beta_{6} - \)\(39\!\cdots\!00\)\( \beta_{7} + \)\(12\!\cdots\!60\)\( \beta_{8}) q^{23} +(-\)\(32\!\cdots\!60\)\( + \)\(54\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} + \)\(46\!\cdots\!44\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4} - \)\(63\!\cdots\!52\)\( \beta_{5} + \)\(16\!\cdots\!08\)\( \beta_{6} + \)\(18\!\cdots\!20\)\( \beta_{7} - \)\(14\!\cdots\!56\)\( \beta_{8}) q^{24} +(\)\(23\!\cdots\!75\)\( + \)\(76\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3} - \)\(96\!\cdots\!50\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!50\)\( \beta_{7} + \)\(14\!\cdots\!50\)\( \beta_{8}) q^{25} +(\)\(33\!\cdots\!98\)\( - \)\(58\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(65\!\cdots\!56\)\( \beta_{3} - \)\(55\!\cdots\!60\)\( \beta_{4} + \)\(29\!\cdots\!96\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} - \)\(53\!\cdots\!60\)\( \beta_{7} - \)\(13\!\cdots\!72\)\( \beta_{8}) q^{26} +(\)\(13\!\cdots\!82\)\( + \)\(74\!\cdots\!46\)\( \beta_{1} + \)\(91\!\cdots\!08\)\( \beta_{2} - \)\(84\!\cdots\!72\)\( \beta_{3} + \)\(70\!\cdots\!40\)\( \beta_{4} - \)\(42\!\cdots\!42\)\( \beta_{5} + \)\(48\!\cdots\!38\)\( \beta_{6} + \)\(58\!\cdots\!36\)\( \beta_{7} + \)\(11\!\cdots\!40\)\( \beta_{8}) q^{27} +(-\)\(85\!\cdots\!36\)\( - \)\(32\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!92\)\( \beta_{2} - \)\(63\!\cdots\!32\)\( \beta_{3} + \)\(65\!\cdots\!00\)\( \beta_{4} + \)\(21\!\cdots\!48\)\( \beta_{5} - \)\(38\!\cdots\!32\)\( \beta_{6} - \)\(48\!\cdots\!24\)\( \beta_{7} - \)\(91\!\cdots\!20\)\( \beta_{8}) q^{28} +(\)\(39\!\cdots\!69\)\( + \)\(80\!\cdots\!49\)\( \beta_{1} - \)\(17\!\cdots\!98\)\( \beta_{2} + \)\(19\!\cdots\!10\)\( \beta_{3} - \)\(36\!\cdots\!15\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{6} + \)\(33\!\cdots\!20\)\( \beta_{7} + \)\(66\!\cdots\!48\)\( \beta_{8}) q^{29} +(\)\(19\!\cdots\!58\)\( - \)\(14\!\cdots\!22\)\( \beta_{1} - \)\(25\!\cdots\!06\)\( \beta_{2} + \)\(33\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!02\)\( \beta_{4} - \)\(79\!\cdots\!30\)\( \beta_{5} + \)\(90\!\cdots\!40\)\( \beta_{6} - \)\(18\!\cdots\!60\)\( \beta_{7} - \)\(43\!\cdots\!80\)\( \beta_{8}) q^{30} +(\)\(88\!\cdots\!68\)\( + \)\(88\!\cdots\!24\)\( \beta_{1} - \)\(76\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(56\!\cdots\!48\)\( \beta_{4} + \)\(36\!\cdots\!20\)\( \beta_{5} - \)\(14\!\cdots\!28\)\( \beta_{6} + \)\(75\!\cdots\!40\)\( \beta_{7} + \)\(26\!\cdots\!96\)\( \beta_{8}) q^{31} +(-\)\(16\!\cdots\!28\)\( - \)\(69\!\cdots\!40\)\( \beta_{1} + \)\(84\!\cdots\!44\)\( \beta_{2} - \)\(13\!\cdots\!08\)\( \beta_{3} - \)\(36\!\cdots\!20\)\( \beta_{4} + \)\(96\!\cdots\!04\)\( \beta_{5} + \)\(10\!\cdots\!04\)\( \beta_{6} - \)\(19\!\cdots\!20\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{32} +(\)\(43\!\cdots\!10\)\( - \)\(26\!\cdots\!29\)\( \beta_{1} + \)\(21\!\cdots\!69\)\( \beta_{2} + \)\(86\!\cdots\!91\)\( \beta_{3} - \)\(15\!\cdots\!25\)\( \beta_{4} - \)\(19\!\cdots\!86\)\( \beta_{5} - \)\(49\!\cdots\!96\)\( \beta_{6} - \)\(20\!\cdots\!79\)\( \beta_{7} + \)\(69\!\cdots\!55\)\( \beta_{8}) q^{33} +(\)\(12\!\cdots\!46\)\( - \)\(37\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + \)\(36\!\cdots\!80\)\( \beta_{3} + \)\(16\!\cdots\!92\)\( \beta_{4} + \)\(10\!\cdots\!96\)\( \beta_{5} + \)\(11\!\cdots\!44\)\( \beta_{6} + \)\(52\!\cdots\!60\)\( \beta_{7} - \)\(29\!\cdots\!08\)\( \beta_{8}) q^{34} +(\)\(54\!\cdots\!76\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} + \)\(11\!\cdots\!48\)\( \beta_{2} - \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(11\!\cdots\!68\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(30\!\cdots\!88\)\( \beta_{6} - \)\(27\!\cdots\!72\)\( \beta_{7} + \)\(10\!\cdots\!04\)\( \beta_{8}) q^{35} +(-\)\(52\!\cdots\!14\)\( - \)\(10\!\cdots\!58\)\( \beta_{1} - \)\(29\!\cdots\!21\)\( \beta_{2} - \)\(51\!\cdots\!11\)\( \beta_{3} - \)\(32\!\cdots\!04\)\( \beta_{4} - \)\(13\!\cdots\!44\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} + \)\(41\!\cdots\!80\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{36} +(-\)\(23\!\cdots\!87\)\( - \)\(34\!\cdots\!45\)\( \beta_{1} - \)\(84\!\cdots\!56\)\( \beta_{2} - \)\(51\!\cdots\!04\)\( \beta_{3} + \)\(94\!\cdots\!35\)\( \beta_{4} + \)\(75\!\cdots\!56\)\( \beta_{5} + \)\(17\!\cdots\!36\)\( \beta_{6} + \)\(57\!\cdots\!82\)\( \beta_{7} + \)\(39\!\cdots\!50\)\( \beta_{8}) q^{37} +(\)\(91\!\cdots\!89\)\( - \)\(23\!\cdots\!71\)\( \beta_{1} - \)\(13\!\cdots\!27\)\( \beta_{2} + \)\(81\!\cdots\!98\)\( \beta_{3} + \)\(38\!\cdots\!25\)\( \beta_{4} - \)\(78\!\cdots\!31\)\( \beta_{5} - \)\(26\!\cdots\!36\)\( \beta_{6} - \)\(63\!\cdots\!76\)\( \beta_{7} + \)\(18\!\cdots\!20\)\( \beta_{8}) q^{38} +(\)\(13\!\cdots\!32\)\( - \)\(79\!\cdots\!55\)\( \beta_{1} + \)\(11\!\cdots\!41\)\( \beta_{2} + \)\(29\!\cdots\!84\)\( \beta_{3} - \)\(24\!\cdots\!36\)\( \beta_{4} - \)\(81\!\cdots\!53\)\( \beta_{5} - \)\(12\!\cdots\!37\)\( \beta_{6} + \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(18\!\cdots\!16\)\( \beta_{8}) q^{39} +(-\)\(51\!\cdots\!50\)\( - \)\(42\!\cdots\!30\)\( \beta_{1} + \)\(96\!\cdots\!50\)\( \beta_{2} - \)\(22\!\cdots\!50\)\( \beta_{3} - \)\(20\!\cdots\!40\)\( \beta_{4} + \)\(33\!\cdots\!80\)\( \beta_{5} + \)\(86\!\cdots\!10\)\( \beta_{6} - \)\(16\!\cdots\!90\)\( \beta_{7} + \)\(88\!\cdots\!80\)\( \beta_{8}) q^{40} +(-\)\(16\!\cdots\!62\)\( - \)\(41\!\cdots\!22\)\( \beta_{1} + \)\(51\!\cdots\!10\)\( \beta_{2} + \)\(35\!\cdots\!34\)\( \beta_{3} + \)\(27\!\cdots\!74\)\( \beta_{4} + \)\(35\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(51\!\cdots\!70\)\( \beta_{7} - \)\(29\!\cdots\!50\)\( \beta_{8}) q^{41} +(\)\(12\!\cdots\!80\)\( - \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{3} - \)\(22\!\cdots\!60\)\( \beta_{4} - \)\(65\!\cdots\!84\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} - \)\(81\!\cdots\!60\)\( \beta_{7} + \)\(58\!\cdots\!60\)\( \beta_{8}) q^{42} +(\)\(65\!\cdots\!03\)\( + \)\(28\!\cdots\!76\)\( \beta_{1} - \)\(27\!\cdots\!41\)\( \beta_{2} + \)\(89\!\cdots\!76\)\( \beta_{3} - \)\(26\!\cdots\!80\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5} + \)\(11\!\cdots\!60\)\( \beta_{6} - \)\(18\!\cdots\!76\)\( \beta_{7} + \)\(34\!\cdots\!00\)\( \beta_{8}) q^{43} +(-\)\(68\!\cdots\!44\)\( + \)\(47\!\cdots\!44\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} + \)\(78\!\cdots\!08\)\( \beta_{3} + \)\(29\!\cdots\!68\)\( \beta_{4} + \)\(37\!\cdots\!60\)\( \beta_{5} - \)\(44\!\cdots\!08\)\( \beta_{6} + \)\(19\!\cdots\!00\)\( \beta_{7} - \)\(90\!\cdots\!44\)\( \beta_{8}) q^{44} +(-\)\(35\!\cdots\!67\)\( + \)\(11\!\cdots\!63\)\( \beta_{1} + \)\(46\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} + \)\(54\!\cdots\!07\)\( \beta_{4} - \)\(33\!\cdots\!40\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} - \)\(81\!\cdots\!30\)\( \beta_{7} + \)\(44\!\cdots\!10\)\( \beta_{8}) q^{45} +(-\)\(20\!\cdots\!62\)\( + \)\(53\!\cdots\!18\)\( \beta_{1} + \)\(92\!\cdots\!90\)\( \beta_{2} + \)\(14\!\cdots\!44\)\( \beta_{3} - \)\(13\!\cdots\!66\)\( \beta_{4} + \)\(78\!\cdots\!34\)\( \beta_{5} - \)\(11\!\cdots\!40\)\( \beta_{6} + \)\(19\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!20\)\( \beta_{8}) q^{46} +(\)\(45\!\cdots\!20\)\( + \)\(64\!\cdots\!90\)\( \beta_{1} - \)\(18\!\cdots\!22\)\( \beta_{2} + \)\(60\!\cdots\!72\)\( \beta_{3} - \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!50\)\( \beta_{5} - \)\(12\!\cdots\!90\)\( \beta_{6} - \)\(12\!\cdots\!32\)\( \beta_{7} + \)\(15\!\cdots\!20\)\( \beta_{8}) q^{47} +(\)\(15\!\cdots\!00\)\( - \)\(36\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} - \)\(25\!\cdots\!28\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!76\)\( \beta_{5} + \)\(34\!\cdots\!04\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(56\!\cdots\!00\)\( \beta_{8}) q^{48} +(-\)\(55\!\cdots\!51\)\( - \)\(45\!\cdots\!52\)\( \beta_{1} - \)\(21\!\cdots\!20\)\( \beta_{2} - \)\(37\!\cdots\!16\)\( \beta_{3} + \)\(16\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!56\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6} + \)\(59\!\cdots\!80\)\( \beta_{7} - \)\(42\!\cdots\!80\)\( \beta_{8}) q^{49} +(-\)\(36\!\cdots\!25\)\( - \)\(48\!\cdots\!75\)\( \beta_{1} + \)\(70\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} - \)\(51\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{50} +(\)\(47\!\cdots\!38\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(63\!\cdots\!88\)\( \beta_{3} - \)\(39\!\cdots\!72\)\( \beta_{4} - \)\(32\!\cdots\!86\)\( \beta_{5} - \)\(22\!\cdots\!54\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7} - \)\(22\!\cdots\!72\)\( \beta_{8}) q^{51} +(\)\(65\!\cdots\!08\)\( - \)\(43\!\cdots\!24\)\( \beta_{1} + \)\(62\!\cdots\!50\)\( \beta_{2} + \)\(10\!\cdots\!42\)\( \beta_{3} + \)\(98\!\cdots\!40\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(52\!\cdots\!08\)\( \beta_{7} - \)\(32\!\cdots\!00\)\( \beta_{8}) q^{52} +(-\)\(59\!\cdots\!91\)\( + \)\(22\!\cdots\!71\)\( \beta_{1} - \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(31\!\cdots\!68\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} - \)\(90\!\cdots\!88\)\( \beta_{5} - \)\(66\!\cdots\!08\)\( \beta_{6} - \)\(22\!\cdots\!46\)\( \beta_{7} + \)\(33\!\cdots\!70\)\( \beta_{8}) q^{53} +(-\)\(17\!\cdots\!66\)\( + \)\(14\!\cdots\!14\)\( \beta_{1} - \)\(52\!\cdots\!30\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(89\!\cdots\!22\)\( \beta_{4} + \)\(55\!\cdots\!10\)\( \beta_{5} + \)\(81\!\cdots\!32\)\( \beta_{6} + \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(10\!\cdots\!24\)\( \beta_{8}) q^{54} +(\)\(16\!\cdots\!48\)\( + \)\(59\!\cdots\!63\)\( \beta_{1} - \)\(77\!\cdots\!61\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!72\)\( \beta_{4} - \)\(18\!\cdots\!75\)\( \beta_{5} + \)\(18\!\cdots\!25\)\( \beta_{6} + \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(15\!\cdots\!00\)\( \beta_{8}) q^{55} +(\)\(82\!\cdots\!64\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(25\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!48\)\( \beta_{3} + \)\(12\!\cdots\!36\)\( \beta_{4} + \)\(14\!\cdots\!60\)\( \beta_{5} - \)\(10\!\cdots\!76\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(26\!\cdots\!32\)\( \beta_{8}) q^{56} +(-\)\(10\!\cdots\!98\)\( + \)\(96\!\cdots\!49\)\( \beta_{1} + \)\(13\!\cdots\!91\)\( \beta_{2} - \)\(11\!\cdots\!99\)\( \beta_{3} + \)\(83\!\cdots\!85\)\( \beta_{4} + \)\(80\!\cdots\!50\)\( \beta_{5} + \)\(20\!\cdots\!80\)\( \beta_{6} - \)\(13\!\cdots\!81\)\( \beta_{7} - \)\(22\!\cdots\!15\)\( \beta_{8}) q^{57} +(-\)\(16\!\cdots\!70\)\( - \)\(78\!\cdots\!94\)\( \beta_{1} + \)\(96\!\cdots\!08\)\( \beta_{2} - \)\(77\!\cdots\!60\)\( \beta_{3} - \)\(60\!\cdots\!80\)\( \beta_{4} - \)\(25\!\cdots\!16\)\( \beta_{5} - \)\(14\!\cdots\!36\)\( \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{7} + \)\(61\!\cdots\!40\)\( \beta_{8}) q^{58} +(-\)\(19\!\cdots\!21\)\( - \)\(18\!\cdots\!36\)\( \beta_{1} - \)\(46\!\cdots\!17\)\( \beta_{2} + \)\(54\!\cdots\!20\)\( \beta_{3} + \)\(98\!\cdots\!56\)\( \beta_{4} + \)\(18\!\cdots\!64\)\( \beta_{5} - \)\(52\!\cdots\!32\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7} - \)\(58\!\cdots\!76\)\( \beta_{8}) q^{59} +(\)\(13\!\cdots\!16\)\( - \)\(61\!\cdots\!28\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(29\!\cdots\!60\)\( \beta_{3} - \)\(12\!\cdots\!28\)\( \beta_{4} + \)\(63\!\cdots\!64\)\( \beta_{5} - \)\(41\!\cdots\!32\)\( \beta_{6} - \)\(10\!\cdots\!92\)\( \beta_{7} - \)\(23\!\cdots\!56\)\( \beta_{8}) q^{60} +(\)\(11\!\cdots\!97\)\( + \)\(15\!\cdots\!03\)\( \beta_{1} + \)\(77\!\cdots\!84\)\( \beta_{2} + \)\(66\!\cdots\!56\)\( \beta_{3} + \)\(35\!\cdots\!07\)\( \beta_{4} + \)\(22\!\cdots\!28\)\( \beta_{5} + \)\(38\!\cdots\!16\)\( \beta_{6} + \)\(50\!\cdots\!50\)\( \beta_{7} + \)\(11\!\cdots\!38\)\( \beta_{8}) q^{61} +(-\)\(20\!\cdots\!56\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(52\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!92\)\( \beta_{3} - \)\(21\!\cdots\!20\)\( \beta_{4} - \)\(95\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!96\)\( \beta_{6} - \)\(83\!\cdots\!40\)\( \beta_{7} - \)\(24\!\cdots\!80\)\( \beta_{8}) q^{62} +(-\)\(58\!\cdots\!92\)\( + \)\(79\!\cdots\!67\)\( \beta_{1} + \)\(24\!\cdots\!71\)\( \beta_{2} - \)\(22\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!60\)\( \beta_{4} + \)\(80\!\cdots\!09\)\( \beta_{5} - \)\(54\!\cdots\!91\)\( \beta_{6} - \)\(11\!\cdots\!12\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{63} +(\)\(98\!\cdots\!04\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!76\)\( \beta_{2} + \)\(20\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!32\)\( \beta_{4} + \)\(29\!\cdots\!68\)\( \beta_{5} + \)\(79\!\cdots\!20\)\( \beta_{6} + \)\(94\!\cdots\!60\)\( \beta_{7} + \)\(14\!\cdots\!60\)\( \beta_{8}) q^{64} +(\)\(37\!\cdots\!12\)\( + \)\(10\!\cdots\!34\)\( \beta_{1} - \)\(14\!\cdots\!74\)\( \beta_{2} + \)\(19\!\cdots\!70\)\( \beta_{3} + \)\(15\!\cdots\!94\)\( \beta_{4} - \)\(61\!\cdots\!32\)\( \beta_{5} + \)\(12\!\cdots\!16\)\( \beta_{6} - \)\(20\!\cdots\!54\)\( \beta_{7} - \)\(45\!\cdots\!22\)\( \beta_{8}) q^{65} +(\)\(64\!\cdots\!12\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(31\!\cdots\!44\)\( \beta_{2} + \)\(15\!\cdots\!04\)\( \beta_{3} - \)\(62\!\cdots\!36\)\( \beta_{4} - \)\(37\!\cdots\!28\)\( \beta_{5} - \)\(66\!\cdots\!12\)\( \beta_{6} + \)\(38\!\cdots\!20\)\( \beta_{7} + \)\(51\!\cdots\!84\)\( \beta_{8}) q^{66} +(\)\(74\!\cdots\!65\)\( - \)\(59\!\cdots\!54\)\( \beta_{1} + \)\(26\!\cdots\!79\)\( \beta_{2} - \)\(10\!\cdots\!92\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4} - \)\(33\!\cdots\!66\)\( \beta_{5} + \)\(71\!\cdots\!14\)\( \beta_{6} + \)\(97\!\cdots\!24\)\( \beta_{7} + \)\(10\!\cdots\!80\)\( \beta_{8}) q^{67} +(-\)\(58\!\cdots\!72\)\( - \)\(52\!\cdots\!72\)\( \beta_{1} + \)\(15\!\cdots\!18\)\( \beta_{2} - \)\(94\!\cdots\!02\)\( \beta_{3} + \)\(18\!\cdots\!20\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(14\!\cdots\!68\)\( \beta_{6} - \)\(26\!\cdots\!24\)\( \beta_{7} - \)\(57\!\cdots\!40\)\( \beta_{8}) q^{68} +(\)\(13\!\cdots\!76\)\( - \)\(60\!\cdots\!94\)\( \beta_{1} + \)\(17\!\cdots\!34\)\( \beta_{2} - \)\(43\!\cdots\!02\)\( \beta_{3} - \)\(15\!\cdots\!98\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(35\!\cdots\!52\)\( \beta_{6} + \)\(21\!\cdots\!30\)\( \beta_{7} + \)\(10\!\cdots\!14\)\( \beta_{8}) q^{69} +(\)\(33\!\cdots\!64\)\( + \)\(21\!\cdots\!44\)\( \beta_{1} - \)\(27\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!40\)\( \beta_{3} - \)\(39\!\cdots\!24\)\( \beta_{4} - \)\(67\!\cdots\!60\)\( \beta_{5} - \)\(57\!\cdots\!20\)\( \beta_{6} + \)\(53\!\cdots\!80\)\( \beta_{7} + \)\(89\!\cdots\!40\)\( \beta_{8}) q^{70} +(\)\(61\!\cdots\!12\)\( + \)\(59\!\cdots\!79\)\( \beta_{1} - \)\(10\!\cdots\!93\)\( \beta_{2} - \)\(22\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!84\)\( \beta_{4} + \)\(14\!\cdots\!89\)\( \beta_{5} + \)\(28\!\cdots\!33\)\( \beta_{6} - \)\(17\!\cdots\!20\)\( \beta_{7} - \)\(52\!\cdots\!56\)\( \beta_{8}) q^{71} +(\)\(18\!\cdots\!19\)\( + \)\(13\!\cdots\!31\)\( \beta_{1} - \)\(16\!\cdots\!17\)\( \beta_{2} + \)\(35\!\cdots\!41\)\( \beta_{3} + \)\(57\!\cdots\!20\)\( \beta_{4} - \)\(17\!\cdots\!38\)\( \beta_{5} - \)\(81\!\cdots\!93\)\( \beta_{6} + \)\(18\!\cdots\!55\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{72} +(\)\(14\!\cdots\!56\)\( + \)\(12\!\cdots\!33\)\( \beta_{1} + \)\(26\!\cdots\!15\)\( \beta_{2} - \)\(14\!\cdots\!07\)\( \beta_{3} - \)\(55\!\cdots\!75\)\( \beta_{4} - \)\(52\!\cdots\!06\)\( \beta_{5} - \)\(14\!\cdots\!36\)\( \beta_{6} + \)\(42\!\cdots\!79\)\( \beta_{7} - \)\(86\!\cdots\!55\)\( \beta_{8}) q^{73} +(\)\(80\!\cdots\!30\)\( + \)\(93\!\cdots\!34\)\( \beta_{1} + \)\(88\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!44\)\( \beta_{4} - \)\(27\!\cdots\!52\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(15\!\cdots\!40\)\( \beta_{7} - \)\(34\!\cdots\!68\)\( \beta_{8}) q^{74} +(\)\(20\!\cdots\!75\)\( - \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(25\!\cdots\!75\)\( \beta_{2} - \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!00\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5} - \)\(35\!\cdots\!00\)\( \beta_{6} - \)\(24\!\cdots\!00\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8}) q^{75} +(\)\(41\!\cdots\!68\)\( - \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3} + \)\(59\!\cdots\!52\)\( \beta_{4} - \)\(57\!\cdots\!40\)\( \beta_{5} - \)\(72\!\cdots\!92\)\( \beta_{6} - \)\(27\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!56\)\( \beta_{8}) q^{76} +(\)\(43\!\cdots\!92\)\( - \)\(19\!\cdots\!50\)\( \beta_{1} - \)\(70\!\cdots\!54\)\( \beta_{2} - \)\(31\!\cdots\!18\)\( \beta_{3} + \)\(96\!\cdots\!50\)\( \beta_{4} - \)\(65\!\cdots\!36\)\( \beta_{5} + \)\(78\!\cdots\!44\)\( \beta_{6} + \)\(13\!\cdots\!90\)\( \beta_{7} - \)\(14\!\cdots\!50\)\( \beta_{8}) q^{77} +(\)\(18\!\cdots\!66\)\( - \)\(46\!\cdots\!86\)\( \beta_{1} - \)\(16\!\cdots\!74\)\( \beta_{2} - \)\(37\!\cdots\!64\)\( \beta_{3} - \)\(25\!\cdots\!10\)\( \beta_{4} + \)\(65\!\cdots\!58\)\( \beta_{5} - \)\(29\!\cdots\!92\)\( \beta_{6} - \)\(18\!\cdots\!92\)\( \beta_{7} + \)\(75\!\cdots\!00\)\( \beta_{8}) q^{78} +(\)\(19\!\cdots\!28\)\( + \)\(83\!\cdots\!14\)\( \beta_{1} + \)\(61\!\cdots\!46\)\( \beta_{2} - \)\(58\!\cdots\!68\)\( \beta_{3} + \)\(46\!\cdots\!68\)\( \beta_{4} + \)\(89\!\cdots\!30\)\( \beta_{5} + \)\(50\!\cdots\!62\)\( \beta_{6} - \)\(34\!\cdots\!00\)\( \beta_{7} - \)\(10\!\cdots\!84\)\( \beta_{8}) q^{79} +(\)\(74\!\cdots\!44\)\( + \)\(29\!\cdots\!24\)\( \beta_{1} + \)\(37\!\cdots\!92\)\( \beta_{2} + \)\(37\!\cdots\!40\)\( \beta_{3} - \)\(77\!\cdots\!04\)\( \beta_{4} + \)\(25\!\cdots\!40\)\( \beta_{5} - \)\(15\!\cdots\!20\)\( \beta_{6} + \)\(16\!\cdots\!80\)\( \beta_{7} - \)\(17\!\cdots\!60\)\( \beta_{8}) q^{80} +(\)\(95\!\cdots\!79\)\( + \)\(50\!\cdots\!19\)\( \beta_{1} + \)\(15\!\cdots\!29\)\( \beta_{2} - \)\(23\!\cdots\!13\)\( \beta_{3} + \)\(32\!\cdots\!71\)\( \beta_{4} - \)\(15\!\cdots\!06\)\( \beta_{5} - \)\(19\!\cdots\!92\)\( \beta_{6} - \)\(15\!\cdots\!15\)\( \beta_{7} + \)\(31\!\cdots\!19\)\( \beta_{8}) q^{81} +(\)\(95\!\cdots\!02\)\( + \)\(17\!\cdots\!38\)\( \beta_{1} + \)\(40\!\cdots\!32\)\( \beta_{2} - \)\(38\!\cdots\!04\)\( \beta_{3} - \)\(13\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!36\)\( \beta_{5} + \)\(86\!\cdots\!76\)\( \beta_{6} - \)\(39\!\cdots\!88\)\( \beta_{7} - \)\(50\!\cdots\!40\)\( \beta_{8}) q^{82} +(\)\(11\!\cdots\!81\)\( + \)\(49\!\cdots\!52\)\( \beta_{1} - \)\(32\!\cdots\!47\)\( \beta_{2} - \)\(79\!\cdots\!20\)\( \beta_{3} - \)\(58\!\cdots\!80\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} - \)\(14\!\cdots\!40\)\( \beta_{6} + \)\(11\!\cdots\!80\)\( \beta_{7} + \)\(25\!\cdots\!80\)\( \beta_{8}) q^{83} +(\)\(32\!\cdots\!20\)\( - \)\(26\!\cdots\!24\)\( \beta_{1} - \)\(13\!\cdots\!12\)\( \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(50\!\cdots\!72\)\( \beta_{4} - \)\(67\!\cdots\!72\)\( \beta_{5} - \)\(48\!\cdots\!80\)\( \beta_{6} + \)\(70\!\cdots\!80\)\( \beta_{7} + \)\(19\!\cdots\!60\)\( \beta_{8}) q^{84} +(\)\(31\!\cdots\!06\)\( - \)\(78\!\cdots\!88\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} + \)\(16\!\cdots\!10\)\( \beta_{3} + \)\(21\!\cdots\!32\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5} + \)\(32\!\cdots\!68\)\( \beta_{6} - \)\(42\!\cdots\!42\)\( \beta_{7} - \)\(31\!\cdots\!06\)\( \beta_{8}) q^{85} +(-\)\(62\!\cdots\!33\)\( - \)\(32\!\cdots\!33\)\( \beta_{1} + \)\(23\!\cdots\!07\)\( \beta_{2} + \)\(36\!\cdots\!54\)\( \beta_{3} + \)\(45\!\cdots\!39\)\( \beta_{4} - \)\(97\!\cdots\!45\)\( \beta_{5} + \)\(50\!\cdots\!16\)\( \beta_{6} + \)\(44\!\cdots\!60\)\( \beta_{7} + \)\(19\!\cdots\!88\)\( \beta_{8}) q^{86} +(-\)\(26\!\cdots\!60\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + \)\(76\!\cdots\!49\)\( \beta_{2} - \)\(79\!\cdots\!16\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4} + \)\(64\!\cdots\!31\)\( \beta_{5} - \)\(64\!\cdots\!89\)\( \beta_{6} + \)\(12\!\cdots\!44\)\( \beta_{7} - \)\(93\!\cdots\!60\)\( \beta_{8}) q^{87} +(-\)\(72\!\cdots\!36\)\( - \)\(31\!\cdots\!56\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} - \)\(47\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(37\!\cdots\!76\)\( \beta_{5} - \)\(18\!\cdots\!76\)\( \beta_{6} - \)\(34\!\cdots\!60\)\( \beta_{7} + \)\(17\!\cdots\!20\)\( \beta_{8}) q^{88} +(-\)\(98\!\cdots\!68\)\( + \)\(59\!\cdots\!65\)\( \beta_{1} - \)\(10\!\cdots\!05\)\( \beta_{2} + \)\(23\!\cdots\!41\)\( \beta_{3} - \)\(41\!\cdots\!43\)\( \beta_{4} - \)\(18\!\cdots\!34\)\( \beta_{5} + \)\(74\!\cdots\!24\)\( \beta_{6} + \)\(42\!\cdots\!35\)\( \beta_{7} + \)\(63\!\cdots\!57\)\( \beta_{8}) q^{89} +(-\)\(26\!\cdots\!18\)\( + \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(45\!\cdots\!64\)\( \beta_{2} + \)\(89\!\cdots\!20\)\( \beta_{3} + \)\(58\!\cdots\!04\)\( \beta_{4} + \)\(30\!\cdots\!08\)\( \beta_{5} - \)\(38\!\cdots\!04\)\( \beta_{6} + \)\(11\!\cdots\!76\)\( \beta_{7} - \)\(86\!\cdots\!32\)\( \beta_{8}) q^{90} +(-\)\(22\!\cdots\!24\)\( - \)\(26\!\cdots\!72\)\( \beta_{1} - \)\(38\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!48\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} - \)\(20\!\cdots\!36\)\( \beta_{6} - \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!52\)\( \beta_{8}) q^{91} +(-\)\(10\!\cdots\!88\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} + \)\(53\!\cdots\!60\)\( \beta_{2} - \)\(52\!\cdots\!04\)\( \beta_{3} - \)\(12\!\cdots\!40\)\( \beta_{4} + \)\(67\!\cdots\!96\)\( \beta_{5} + \)\(43\!\cdots\!76\)\( \beta_{6} - \)\(53\!\cdots\!48\)\( \beta_{7} + \)\(12\!\cdots\!00\)\( \beta_{8}) q^{92} +(-\)\(11\!\cdots\!56\)\( - \)\(35\!\cdots\!64\)\( \beta_{1} + \)\(29\!\cdots\!80\)\( \beta_{2} + \)\(17\!\cdots\!84\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4} - \)\(12\!\cdots\!48\)\( \beta_{5} - \)\(20\!\cdots\!28\)\( \beta_{6} + \)\(46\!\cdots\!32\)\( \beta_{7} - \)\(40\!\cdots\!80\)\( \beta_{8}) q^{93} +(-\)\(14\!\cdots\!80\)\( - \)\(15\!\cdots\!96\)\( \beta_{1} + \)\(38\!\cdots\!60\)\( \beta_{2} - \)\(77\!\cdots\!40\)\( \beta_{3} - \)\(40\!\cdots\!72\)\( \beta_{4} - \)\(19\!\cdots\!60\)\( \beta_{5} - \)\(34\!\cdots\!88\)\( \beta_{6} - \)\(39\!\cdots\!20\)\( \beta_{7} + \)\(52\!\cdots\!16\)\( \beta_{8}) q^{94} +(\)\(21\!\cdots\!00\)\( - \)\(46\!\cdots\!95\)\( \beta_{1} + \)\(54\!\cdots\!25\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!40\)\( \beta_{4} + \)\(68\!\cdots\!95\)\( \beta_{5} + \)\(89\!\cdots\!15\)\( \beta_{6} - \)\(13\!\cdots\!60\)\( \beta_{7} + \)\(12\!\cdots\!20\)\( \beta_{8}) q^{95} +(\)\(90\!\cdots\!12\)\( - \)\(11\!\cdots\!84\)\( \beta_{1} - \)\(25\!\cdots\!28\)\( \beta_{2} + \)\(18\!\cdots\!72\)\( \beta_{3} - \)\(16\!\cdots\!92\)\( \beta_{4} - \)\(21\!\cdots\!92\)\( \beta_{5} - \)\(20\!\cdots\!80\)\( \beta_{6} - \)\(52\!\cdots\!20\)\( \beta_{7} - \)\(49\!\cdots\!40\)\( \beta_{8}) q^{96} +(\)\(20\!\cdots\!52\)\( + \)\(52\!\cdots\!41\)\( \beta_{1} + \)\(93\!\cdots\!15\)\( \beta_{2} - \)\(15\!\cdots\!47\)\( \beta_{3} - \)\(10\!\cdots\!55\)\( \beta_{4} - \)\(49\!\cdots\!82\)\( \beta_{5} + \)\(22\!\cdots\!68\)\( \beta_{6} + \)\(21\!\cdots\!71\)\( \beta_{7} - \)\(10\!\cdots\!15\)\( \beta_{8}) q^{97} +(\)\(10\!\cdots\!89\)\( + \)\(10\!\cdots\!67\)\( \beta_{1} - \)\(38\!\cdots\!28\)\( \beta_{2} - \)\(36\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(27\!\cdots\!84\)\( \beta_{5} + \)\(50\!\cdots\!76\)\( \beta_{6} + \)\(18\!\cdots\!92\)\( \beta_{7} + \)\(11\!\cdots\!80\)\( \beta_{8}) q^{98} +(\)\(23\!\cdots\!75\)\( + \)\(97\!\cdots\!92\)\( \beta_{1} + \)\(12\!\cdots\!91\)\( \beta_{2} - \)\(20\!\cdots\!88\)\( \beta_{3} + \)\(46\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!60\)\( \beta_{5} - \)\(13\!\cdots\!84\)\( \beta_{6} - \)\(15\!\cdots\!20\)\( \beta_{7} + \)\(85\!\cdots\!88\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 5465779613435496q^{2} + \)\(15\!\cdots\!12\)\(q^{3} + \)\(64\!\cdots\!92\)\(q^{4} + \)\(24\!\cdots\!50\)\(q^{5} - \)\(12\!\cdots\!32\)\(q^{6} - \)\(93\!\cdots\!44\)\(q^{7} - \)\(10\!\cdots\!40\)\(q^{8} + \)\(36\!\cdots\!13\)\(q^{9} + O(q^{10}) \) \( 9q + 5465779613435496q^{2} + \)\(15\!\cdots\!12\)\(q^{3} + \)\(64\!\cdots\!92\)\(q^{4} + \)\(24\!\cdots\!50\)\(q^{5} - \)\(12\!\cdots\!32\)\(q^{6} - \)\(93\!\cdots\!44\)\(q^{7} - \)\(10\!\cdots\!40\)\(q^{8} + \)\(36\!\cdots\!13\)\(q^{9} - \)\(10\!\cdots\!00\)\(q^{10} + \)\(75\!\cdots\!48\)\(q^{11} - \)\(79\!\cdots\!44\)\(q^{12} + \)\(38\!\cdots\!82\)\(q^{13} - \)\(74\!\cdots\!16\)\(q^{14} + \)\(11\!\cdots\!00\)\(q^{15} + \)\(29\!\cdots\!64\)\(q^{16} + \)\(55\!\cdots\!26\)\(q^{17} + \)\(23\!\cdots\!72\)\(q^{18} + \)\(85\!\cdots\!60\)\(q^{19} + \)\(13\!\cdots\!00\)\(q^{20} + \)\(98\!\cdots\!48\)\(q^{21} - \)\(21\!\cdots\!88\)\(q^{22} - \)\(10\!\cdots\!48\)\(q^{23} - \)\(29\!\cdots\!20\)\(q^{24} + \)\(20\!\cdots\!75\)\(q^{25} + \)\(29\!\cdots\!48\)\(q^{26} + \)\(12\!\cdots\!40\)\(q^{27} - \)\(77\!\cdots\!72\)\(q^{28} + \)\(35\!\cdots\!90\)\(q^{29} + \)\(17\!\cdots\!00\)\(q^{30} + \)\(80\!\cdots\!08\)\(q^{31} - \)\(14\!\cdots\!64\)\(q^{32} + \)\(39\!\cdots\!64\)\(q^{33} + \)\(11\!\cdots\!64\)\(q^{34} + \)\(49\!\cdots\!00\)\(q^{35} - \)\(47\!\cdots\!56\)\(q^{36} - \)\(21\!\cdots\!34\)\(q^{37} + \)\(82\!\cdots\!40\)\(q^{38} + \)\(12\!\cdots\!56\)\(q^{39} - \)\(46\!\cdots\!00\)\(q^{40} - \)\(15\!\cdots\!62\)\(q^{41} + \)\(11\!\cdots\!12\)\(q^{42} + \)\(58\!\cdots\!92\)\(q^{43} - \)\(61\!\cdots\!76\)\(q^{44} - \)\(31\!\cdots\!50\)\(q^{45} - \)\(18\!\cdots\!72\)\(q^{46} + \)\(41\!\cdots\!36\)\(q^{47} + \)\(14\!\cdots\!52\)\(q^{48} - \)\(49\!\cdots\!63\)\(q^{49} - \)\(32\!\cdots\!00\)\(q^{50} + \)\(42\!\cdots\!08\)\(q^{51} + \)\(59\!\cdots\!16\)\(q^{52} - \)\(53\!\cdots\!38\)\(q^{53} - \)\(15\!\cdots\!40\)\(q^{54} + \)\(14\!\cdots\!00\)\(q^{55} + \)\(74\!\cdots\!40\)\(q^{56} - \)\(93\!\cdots\!20\)\(q^{57} - \)\(14\!\cdots\!40\)\(q^{58} - \)\(17\!\cdots\!20\)\(q^{59} + \)\(11\!\cdots\!00\)\(q^{60} + \)\(99\!\cdots\!98\)\(q^{61} - \)\(18\!\cdots\!48\)\(q^{62} - \)\(52\!\cdots\!08\)\(q^{63} + \)\(88\!\cdots\!12\)\(q^{64} + \)\(33\!\cdots\!00\)\(q^{65} + \)\(57\!\cdots\!96\)\(q^{66} + \)\(67\!\cdots\!76\)\(q^{67} - \)\(52\!\cdots\!12\)\(q^{68} + \)\(12\!\cdots\!16\)\(q^{69} + \)\(30\!\cdots\!00\)\(q^{70} + \)\(55\!\cdots\!28\)\(q^{71} + \)\(16\!\cdots\!20\)\(q^{72} + \)\(13\!\cdots\!02\)\(q^{73} + \)\(72\!\cdots\!24\)\(q^{74} + \)\(18\!\cdots\!00\)\(q^{75} + \)\(36\!\cdots\!80\)\(q^{76} + \)\(38\!\cdots\!32\)\(q^{77} + \)\(16\!\cdots\!64\)\(q^{78} + \)\(17\!\cdots\!40\)\(q^{79} + \)\(67\!\cdots\!00\)\(q^{80} + \)\(85\!\cdots\!89\)\(q^{81} + \)\(86\!\cdots\!72\)\(q^{82} + \)\(10\!\cdots\!72\)\(q^{83} + \)\(29\!\cdots\!24\)\(q^{84} + \)\(28\!\cdots\!00\)\(q^{85} - \)\(56\!\cdots\!12\)\(q^{86} - \)\(23\!\cdots\!80\)\(q^{87} - \)\(64\!\cdots\!80\)\(q^{88} - \)\(88\!\cdots\!30\)\(q^{89} - \)\(23\!\cdots\!00\)\(q^{90} - \)\(20\!\cdots\!72\)\(q^{91} - \)\(96\!\cdots\!24\)\(q^{92} - \)\(10\!\cdots\!56\)\(q^{93} - \)\(13\!\cdots\!96\)\(q^{94} + \)\(19\!\cdots\!00\)\(q^{95} + \)\(81\!\cdots\!88\)\(q^{96} + \)\(18\!\cdots\!86\)\(q^{97} + \)\(91\!\cdots\!28\)\(q^{98} + \)\(21\!\cdots\!36\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - \)\(18\!\cdots\!64\)\( x^{7} - \)\(73\!\cdots\!04\)\( x^{6} + \)\(10\!\cdots\!46\)\( x^{5} + \)\(37\!\cdots\!92\)\( x^{4} - \)\(21\!\cdots\!48\)\( x^{3} - \)\(45\!\cdots\!88\)\( x^{2} + \)\(13\!\cdots\!17\)\( x + \)\(27\!\cdots\!52\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 11 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(12\!\cdots\!79\)\( \nu^{8} - \)\(17\!\cdots\!29\)\( \nu^{7} + \)\(22\!\cdots\!45\)\( \nu^{6} + \)\(27\!\cdots\!23\)\( \nu^{5} - \)\(10\!\cdots\!49\)\( \nu^{4} - \)\(11\!\cdots\!03\)\( \nu^{3} + \)\(11\!\cdots\!11\)\( \nu^{2} + \)\(11\!\cdots\!21\)\( \nu + \)\(21\!\cdots\!68\)\(\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(47\!\cdots\!19\)\( \nu^{8} - \)\(66\!\cdots\!69\)\( \nu^{7} + \)\(86\!\cdots\!45\)\( \nu^{6} + \)\(10\!\cdots\!03\)\( \nu^{5} - \)\(40\!\cdots\!89\)\( \nu^{4} - \)\(44\!\cdots\!83\)\( \nu^{3} + \)\(10\!\cdots\!43\)\( \nu^{2} + \)\(42\!\cdots\!73\)\( \nu - \)\(24\!\cdots\!20\)\(\)\()/ \)\(10\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(32\!\cdots\!89\)\( \nu^{8} + \)\(90\!\cdots\!09\)\( \nu^{7} + \)\(58\!\cdots\!03\)\( \nu^{6} - \)\(12\!\cdots\!75\)\( \nu^{5} - \)\(29\!\cdots\!19\)\( \nu^{4} + \)\(41\!\cdots\!75\)\( \nu^{3} + \)\(28\!\cdots\!97\)\( \nu^{2} - \)\(53\!\cdots\!37\)\( \nu + \)\(16\!\cdots\!36\)\(\)\()/ \)\(96\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(59\!\cdots\!89\)\( \nu^{8} + \)\(35\!\cdots\!91\)\( \nu^{7} - \)\(91\!\cdots\!03\)\( \nu^{6} - \)\(13\!\cdots\!25\)\( \nu^{5} + \)\(38\!\cdots\!19\)\( \nu^{4} + \)\(73\!\cdots\!25\)\( \nu^{3} - \)\(40\!\cdots\!97\)\( \nu^{2} - \)\(91\!\cdots\!63\)\( \nu - \)\(93\!\cdots\!36\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(50\!\cdots\!53\)\( \nu^{8} + \)\(54\!\cdots\!93\)\( \nu^{7} + \)\(89\!\cdots\!31\)\( \nu^{6} - \)\(76\!\cdots\!75\)\( \nu^{5} - \)\(47\!\cdots\!63\)\( \nu^{4} + \)\(50\!\cdots\!75\)\( \nu^{3} + \)\(84\!\cdots\!69\)\( \nu^{2} - \)\(88\!\cdots\!49\)\( \nu - \)\(31\!\cdots\!28\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(26\!\cdots\!07\)\( \nu^{8} + \)\(47\!\cdots\!67\)\( \nu^{7} + \)\(44\!\cdots\!89\)\( \nu^{6} - \)\(57\!\cdots\!25\)\( \nu^{5} - \)\(21\!\cdots\!97\)\( \nu^{4} + \)\(41\!\cdots\!25\)\( \nu^{3} + \)\(28\!\cdots\!11\)\( \nu^{2} - \)\(13\!\cdots\!31\)\( \nu + \)\(53\!\cdots\!68\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(45\!\cdots\!37\)\( \nu^{8} + \)\(14\!\cdots\!97\)\( \nu^{7} + \)\(10\!\cdots\!99\)\( \nu^{6} + \)\(31\!\cdots\!25\)\( \nu^{5} - \)\(90\!\cdots\!27\)\( \nu^{4} - \)\(18\!\cdots\!25\)\( \nu^{3} + \)\(26\!\cdots\!01\)\( \nu^{2} + \)\(43\!\cdots\!79\)\( \nu - \)\(14\!\cdots\!12\)\(\)\()/ \)\(76\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 11\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 664037 \beta_{2} + 1454496237795986 \beta_{1} + 233205805914080463879155578823890\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 287 \beta_{6} + 326098 \beta_{5} - 11536093720 \beta_{4} + 6080152289344487 \beta_{3} - 1729190174695774540615 \beta_{2} + 389982933498588803529772202388069 \beta_{1} + 339196967390022887798427489622681198989944874045\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(-399171419712 \beta_{8} + 1032326586055479 \beta_{7} - 1073780444029653737 \beta_{6} + 398942241277981412382 \beta_{5} + 88256816768613430614080024 \beta_{4} + 73419319872788947466478796229417 \beta_{3} - 40830610558608177706493963427534012393 \beta_{2} + 229589003907921368619341708370163101878428065411 \beta_{1} + 11368285537427622943708687074161312670632101143477642887442636587\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(1040276766472715976193601280 \beta_{8} + 5386861631393577261729528541249 \beta_{7} - 2503564908944438757999586187538911 \beta_{6} + 965686401281995211115857145364280914 \beta_{5} - 13058856798979542643874920464718010944280 \beta_{4} + 45718502092925457823634876203711469669762023863 \beta_{3} - 16942881902956622105520393094251233220884436358727511 \beta_{2} + 1438922850832020154332150699622892850112530279258047393493043109 \beta_{1} + 3346343043049345609121544223011877028625201201516347548008693867606683570380605\)\()/62208\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(69\!\cdots\!00\)\( \beta_{8} + \)\(88\!\cdots\!81\)\( \beta_{7} - \)\(73\!\cdots\!19\)\( \beta_{6} + \)\(58\!\cdots\!94\)\( \beta_{5} + \)\(61\!\cdots\!88\)\( \beta_{4} + \)\(35\!\cdots\!43\)\( \beta_{3} - \)\(16\!\cdots\!35\)\( \beta_{2} + \)\(17\!\cdots\!85\)\( \beta_{1} + \)\(46\!\cdots\!93\)\(\)\()/20736\)
\(\nu^{7}\)\(=\)\((\)\(\)\(10\!\cdots\!80\)\( \beta_{8} + \)\(37\!\cdots\!81\)\( \beta_{7} - \)\(21\!\cdots\!15\)\( \beta_{6} + \)\(41\!\cdots\!14\)\( \beta_{5} + \)\(20\!\cdots\!24\)\( \beta_{4} + \)\(39\!\cdots\!51\)\( \beta_{3} - \)\(20\!\cdots\!27\)\( \beta_{2} + \)\(86\!\cdots\!33\)\( \beta_{1} + \)\(33\!\cdots\!45\)\(\)\()/41472\)
\(\nu^{8}\)\(=\)\((\)\(\)\(42\!\cdots\!96\)\( \beta_{8} + \)\(69\!\cdots\!03\)\( \beta_{7} - \)\(51\!\cdots\!17\)\( \beta_{6} + \)\(51\!\cdots\!14\)\( \beta_{5} + \)\(40\!\cdots\!76\)\( \beta_{4} + \)\(21\!\cdots\!81\)\( \beta_{3} - \)\(85\!\cdots\!13\)\( \beta_{2} + \)\(13\!\cdots\!47\)\( \beta_{1} + \)\(25\!\cdots\!07\)\(\)\()/124416\)

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
1.03210e15
6.73733e14
4.82720e14
3.47594e14
−2.01415e13
−3.22655e14
−5.32911e14
−7.86617e14
−8.73826e14
−2.41632e16 −3.68177e24 4.21601e32 3.92322e37 8.89632e40 −1.24740e45 −6.26652e48 −1.11358e51 −9.47976e53
1.2 −1.55623e16 1.35366e25 7.99254e31 −3.05298e37 −2.10660e41 6.30256e44 1.28130e48 −9.43892e50 4.75113e53
1.3 −1.09780e16 −5.69800e25 −4.17435e31 2.81503e36 6.25524e41 3.42403e44 2.23954e48 2.11959e51 −3.09033e52
1.4 −7.73495e15 6.57221e25 −1.02430e32 3.43777e37 −5.08357e41 9.22106e44 2.04736e48 3.19227e51 −2.65909e53
1.5 1.09070e15 6.73736e23 −1.61070e32 1.05761e37 7.34847e38 −2.03867e45 −3.52656e47 −1.12668e51 1.15354e52
1.6 8.35103e15 −1.73817e25 −9.25196e31 3.17576e36 −1.45155e41 2.61780e45 −2.12767e48 −8.25006e50 2.65209e52
1.7 1.33972e16 4.78035e25 1.72251e31 −4.75138e37 6.40432e41 −7.19244e44 −1.94305e48 1.15805e51 −6.36552e53
1.8 1.94861e16 −5.50693e25 2.17450e32 −2.12470e37 −1.07309e42 −2.04209e45 1.07545e48 1.90550e51 −4.14022e53
1.9 2.15791e16 2.13607e25 3.03400e32 3.38441e37 4.60946e41 5.98905e44 3.04570e48 −6.70851e50 7.30327e53
\(n\): 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 1.108.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.108.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{108}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5465779613435496 T + \)\(42\!\cdots\!88\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!16\)\( T^{4} - \)\(47\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} + \)\(78\!\cdots\!80\)\( T^{7} + \)\(24\!\cdots\!96\)\( T^{8} + \)\(19\!\cdots\!44\)\( T^{9} + \)\(40\!\cdots\!88\)\( T^{10} + \)\(20\!\cdots\!20\)\( T^{11} + \)\(73\!\cdots\!56\)\( T^{12} - \)\(32\!\cdots\!36\)\( T^{13} + \)\(11\!\cdots\!88\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!56\)\( T^{16} - \)\(26\!\cdots\!56\)\( T^{17} + \)\(77\!\cdots\!08\)\( T^{18} \)
$3$ \( 1 - \)\(15\!\cdots\!12\)\( T + \)\(33\!\cdots\!07\)\( T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!16\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(47\!\cdots\!52\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(69\!\cdots\!46\)\( T^{8} - \)\(30\!\cdots\!92\)\( T^{9} + \)\(77\!\cdots\!02\)\( T^{10} - \)\(26\!\cdots\!20\)\( T^{11} + \)\(68\!\cdots\!56\)\( T^{12} - \)\(16\!\cdots\!32\)\( T^{13} + \)\(81\!\cdots\!12\)\( T^{14} - \)\(10\!\cdots\!60\)\( T^{15} + \)\(77\!\cdots\!81\)\( T^{16} - \)\(41\!\cdots\!52\)\( T^{17} + \)\(29\!\cdots\!27\)\( T^{18} \)
$5$ \( 1 - \)\(24\!\cdots\!50\)\( T + \)\(20\!\cdots\!25\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!00\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!50\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!50\)\( T^{10} + \)\(38\!\cdots\!00\)\( T^{11} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(93\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!00\)\( T^{14} - \)\(15\!\cdots\!00\)\( T^{15} + \)\(68\!\cdots\!25\)\( T^{16} - \)\(51\!\cdots\!50\)\( T^{17} + \)\(12\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 + \)\(93\!\cdots\!44\)\( T + \)\(14\!\cdots\!43\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(57\!\cdots\!24\)\( T^{5} + \)\(48\!\cdots\!28\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!06\)\( T^{8} + \)\(45\!\cdots\!64\)\( T^{9} + \)\(42\!\cdots\!58\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(91\!\cdots\!96\)\( T^{12} + \)\(28\!\cdots\!24\)\( T^{13} + \)\(14\!\cdots\!28\)\( T^{14} + \)\(40\!\cdots\!00\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} + \)\(23\!\cdots\!44\)\( T^{17} + \)\(67\!\cdots\!43\)\( T^{18} \)
$11$ \( 1 - \)\(75\!\cdots\!48\)\( T + \)\(13\!\cdots\!63\)\( T^{2} - \)\(79\!\cdots\!96\)\( T^{3} + \)\(90\!\cdots\!60\)\( T^{4} - \)\(49\!\cdots\!68\)\( T^{5} + \)\(42\!\cdots\!04\)\( T^{6} - \)\(20\!\cdots\!04\)\( T^{7} + \)\(14\!\cdots\!98\)\( T^{8} - \)\(62\!\cdots\!20\)\( T^{9} + \)\(39\!\cdots\!58\)\( T^{10} - \)\(14\!\cdots\!64\)\( T^{11} + \)\(82\!\cdots\!44\)\( T^{12} - \)\(25\!\cdots\!08\)\( T^{13} + \)\(12\!\cdots\!60\)\( T^{14} - \)\(29\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!33\)\( T^{16} - \)\(20\!\cdots\!28\)\( T^{17} + \)\(72\!\cdots\!31\)\( T^{18} \)
$13$ \( 1 - \)\(38\!\cdots\!82\)\( T + \)\(55\!\cdots\!57\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!36\)\( T^{4} - \)\(55\!\cdots\!92\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!86\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!62\)\( T^{10} - \)\(25\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!16\)\( T^{12} - \)\(32\!\cdots\!32\)\( T^{13} + \)\(15\!\cdots\!52\)\( T^{14} - \)\(28\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!61\)\( T^{16} - \)\(13\!\cdots\!62\)\( T^{17} + \)\(53\!\cdots\!97\)\( T^{18} \)
$17$ \( 1 - \)\(55\!\cdots\!26\)\( T + \)\(20\!\cdots\!53\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!56\)\( T^{4} - \)\(16\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(62\!\cdots\!96\)\( T^{9} + \)\(47\!\cdots\!98\)\( T^{10} - \)\(24\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!76\)\( T^{12} - \)\(70\!\cdots\!56\)\( T^{13} + \)\(42\!\cdots\!08\)\( T^{14} - \)\(12\!\cdots\!20\)\( T^{15} + \)\(83\!\cdots\!41\)\( T^{16} - \)\(10\!\cdots\!06\)\( T^{17} + \)\(83\!\cdots\!13\)\( T^{18} \)
$19$ \( 1 - \)\(85\!\cdots\!60\)\( T + \)\(67\!\cdots\!51\)\( T^{2} - \)\(35\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!56\)\( T^{4} - \)\(63\!\cdots\!80\)\( T^{5} + \)\(22\!\cdots\!96\)\( T^{6} - \)\(70\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!66\)\( T^{8} - \)\(54\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!74\)\( T^{10} - \)\(31\!\cdots\!40\)\( T^{11} + \)\(68\!\cdots\!24\)\( T^{12} - \)\(12\!\cdots\!80\)\( T^{13} + \)\(22\!\cdots\!44\)\( T^{14} - \)\(31\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!29\)\( T^{16} - \)\(35\!\cdots\!60\)\( T^{17} + \)\(27\!\cdots\!59\)\( T^{18} \)
$23$ \( 1 + \)\(10\!\cdots\!48\)\( T + \)\(26\!\cdots\!07\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!56\)\( T^{4} + \)\(27\!\cdots\!28\)\( T^{5} + \)\(35\!\cdots\!12\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!26\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!22\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{11} + \)\(46\!\cdots\!76\)\( T^{12} + \)\(18\!\cdots\!68\)\( T^{13} + \)\(12\!\cdots\!92\)\( T^{14} + \)\(37\!\cdots\!20\)\( T^{15} + \)\(23\!\cdots\!41\)\( T^{16} + \)\(43\!\cdots\!28\)\( T^{17} + \)\(22\!\cdots\!67\)\( T^{18} \)
$29$ \( 1 - \)\(35\!\cdots\!90\)\( T + \)\(18\!\cdots\!81\)\( T^{2} - \)\(47\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!16\)\( T^{4} - \)\(34\!\cdots\!20\)\( T^{5} + \)\(91\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} - \)\(57\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!44\)\( T^{12} - \)\(27\!\cdots\!20\)\( T^{13} + \)\(39\!\cdots\!84\)\( T^{14} - \)\(34\!\cdots\!80\)\( T^{15} + \)\(40\!\cdots\!89\)\( T^{16} - \)\(23\!\cdots\!90\)\( T^{17} + \)\(19\!\cdots\!89\)\( T^{18} \)
$31$ \( 1 - \)\(80\!\cdots\!08\)\( T + \)\(20\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!60\)\( T^{4} - \)\(95\!\cdots\!68\)\( T^{5} + \)\(84\!\cdots\!84\)\( T^{6} - \)\(39\!\cdots\!64\)\( T^{7} + \)\(30\!\cdots\!78\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!58\)\( T^{10} - \)\(56\!\cdots\!44\)\( T^{11} + \)\(45\!\cdots\!04\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{13} + \)\(13\!\cdots\!60\)\( T^{14} - \)\(38\!\cdots\!16\)\( T^{15} + \)\(21\!\cdots\!93\)\( T^{16} - \)\(32\!\cdots\!48\)\( T^{17} + \)\(15\!\cdots\!91\)\( T^{18} \)
$37$ \( 1 + \)\(21\!\cdots\!34\)\( T + \)\(25\!\cdots\!73\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!76\)\( T^{4} + \)\(97\!\cdots\!04\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} + \)\(99\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!66\)\( T^{8} + \)\(73\!\cdots\!84\)\( T^{9} + \)\(17\!\cdots\!78\)\( T^{10} + \)\(39\!\cdots\!80\)\( T^{11} + \)\(87\!\cdots\!36\)\( T^{12} + \)\(15\!\cdots\!84\)\( T^{13} + \)\(34\!\cdots\!68\)\( T^{14} + \)\(38\!\cdots\!40\)\( T^{15} + \)\(97\!\cdots\!21\)\( T^{16} + \)\(50\!\cdots\!94\)\( T^{17} + \)\(15\!\cdots\!53\)\( T^{18} \)
$41$ \( 1 + \)\(15\!\cdots\!62\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(34\!\cdots\!64\)\( T^{3} + \)\(23\!\cdots\!60\)\( T^{4} + \)\(36\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{7} + \)\(85\!\cdots\!18\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(31\!\cdots\!58\)\( T^{10} + \)\(31\!\cdots\!16\)\( T^{11} + \)\(85\!\cdots\!84\)\( T^{12} + \)\(67\!\cdots\!72\)\( T^{13} + \)\(16\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!84\)\( T^{15} + \)\(20\!\cdots\!73\)\( T^{16} + \)\(53\!\cdots\!42\)\( T^{17} + \)\(12\!\cdots\!21\)\( T^{18} \)
$43$ \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(33\!\cdots\!07\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!32\)\( T^{5} + \)\(57\!\cdots\!72\)\( T^{6} - \)\(81\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(54\!\cdots\!52\)\( T^{9} + \)\(27\!\cdots\!42\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} - \)\(14\!\cdots\!32\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} - \)\(56\!\cdots\!00\)\( T^{15} + \)\(97\!\cdots\!01\)\( T^{16} - \)\(10\!\cdots\!92\)\( T^{17} + \)\(10\!\cdots\!07\)\( T^{18} \)
$47$ \( 1 - \)\(41\!\cdots\!36\)\( T + \)\(65\!\cdots\!83\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!36\)\( T^{4} - \)\(60\!\cdots\!36\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} - \)\(93\!\cdots\!40\)\( T^{7} + \)\(43\!\cdots\!86\)\( T^{8} - \)\(93\!\cdots\!76\)\( T^{9} + \)\(35\!\cdots\!18\)\( T^{10} - \)\(62\!\cdots\!60\)\( T^{11} + \)\(19\!\cdots\!16\)\( T^{12} - \)\(27\!\cdots\!96\)\( T^{13} + \)\(74\!\cdots\!48\)\( T^{14} - \)\(71\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!61\)\( T^{16} - \)\(85\!\cdots\!56\)\( T^{17} + \)\(16\!\cdots\!23\)\( T^{18} \)
$53$ \( 1 + \)\(53\!\cdots\!38\)\( T + \)\(18\!\cdots\!57\)\( T^{2} + \)\(66\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!16\)\( T^{4} + \)\(47\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} + \)\(22\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!46\)\( T^{8} + \)\(81\!\cdots\!08\)\( T^{9} + \)\(13\!\cdots\!02\)\( T^{10} + \)\(22\!\cdots\!80\)\( T^{11} + \)\(31\!\cdots\!56\)\( T^{12} + \)\(46\!\cdots\!68\)\( T^{13} + \)\(51\!\cdots\!12\)\( T^{14} + \)\(64\!\cdots\!40\)\( T^{15} + \)\(55\!\cdots\!81\)\( T^{16} + \)\(50\!\cdots\!98\)\( T^{17} + \)\(30\!\cdots\!77\)\( T^{18} \)
$59$ \( 1 + \)\(17\!\cdots\!20\)\( T + \)\(18\!\cdots\!71\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!96\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(92\!\cdots\!56\)\( T^{6} + \)\(96\!\cdots\!80\)\( T^{7} + \)\(37\!\cdots\!46\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} + \)\(88\!\cdots\!80\)\( T^{11} + \)\(25\!\cdots\!04\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(41\!\cdots\!04\)\( T^{14} + \)\(21\!\cdots\!40\)\( T^{15} + \)\(42\!\cdots\!69\)\( T^{16} + \)\(12\!\cdots\!20\)\( T^{17} + \)\(21\!\cdots\!79\)\( T^{18} \)
$61$ \( 1 - \)\(99\!\cdots\!98\)\( T + \)\(10\!\cdots\!13\)\( T^{2} - \)\(67\!\cdots\!96\)\( T^{3} + \)\(41\!\cdots\!60\)\( T^{4} - \)\(20\!\cdots\!68\)\( T^{5} + \)\(94\!\cdots\!04\)\( T^{6} - \)\(37\!\cdots\!04\)\( T^{7} + \)\(14\!\cdots\!98\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{9} + \)\(15\!\cdots\!58\)\( T^{10} - \)\(43\!\cdots\!64\)\( T^{11} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(27\!\cdots\!08\)\( T^{13} + \)\(59\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(17\!\cdots\!33\)\( T^{16} - \)\(17\!\cdots\!78\)\( T^{17} + \)\(18\!\cdots\!81\)\( T^{18} \)
$67$ \( 1 - \)\(67\!\cdots\!76\)\( T + \)\(17\!\cdots\!03\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(80\!\cdots\!16\)\( T^{5} + \)\(72\!\cdots\!28\)\( T^{6} - \)\(36\!\cdots\!60\)\( T^{7} + \)\(25\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} + \)\(61\!\cdots\!98\)\( T^{10} - \)\(21\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!76\)\( T^{12} - \)\(29\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!08\)\( T^{14} - \)\(23\!\cdots\!20\)\( T^{15} + \)\(93\!\cdots\!41\)\( T^{16} - \)\(88\!\cdots\!56\)\( T^{17} + \)\(32\!\cdots\!63\)\( T^{18} \)
$71$ \( 1 - \)\(55\!\cdots\!28\)\( T + \)\(30\!\cdots\!23\)\( T^{2} - \)\(98\!\cdots\!76\)\( T^{3} + \)\(54\!\cdots\!60\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} + \)\(90\!\cdots\!44\)\( T^{6} - \)\(25\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!38\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(37\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!24\)\( T^{12} - \)\(49\!\cdots\!48\)\( T^{13} + \)\(14\!\cdots\!60\)\( T^{14} - \)\(31\!\cdots\!16\)\( T^{15} + \)\(11\!\cdots\!13\)\( T^{16} - \)\(26\!\cdots\!88\)\( T^{17} + \)\(57\!\cdots\!11\)\( T^{18} \)
$73$ \( 1 - \)\(13\!\cdots\!02\)\( T + \)\(21\!\cdots\!57\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!56\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{5} + \)\(89\!\cdots\!12\)\( T^{6} - \)\(52\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!26\)\( T^{8} - \)\(14\!\cdots\!72\)\( T^{9} + \)\(69\!\cdots\!22\)\( T^{10} - \)\(29\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} - \)\(41\!\cdots\!32\)\( T^{13} + \)\(13\!\cdots\!92\)\( T^{14} - \)\(35\!\cdots\!80\)\( T^{15} + \)\(90\!\cdots\!41\)\( T^{16} - \)\(13\!\cdots\!22\)\( T^{17} + \)\(23\!\cdots\!17\)\( T^{18} \)
$79$ \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(21\!\cdots\!31\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} - \)\(82\!\cdots\!20\)\( T^{5} + \)\(42\!\cdots\!36\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(77\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!74\)\( T^{10} - \)\(23\!\cdots\!60\)\( T^{11} + \)\(58\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{13} + \)\(23\!\cdots\!84\)\( T^{14} - \)\(35\!\cdots\!80\)\( T^{15} + \)\(45\!\cdots\!89\)\( T^{16} - \)\(41\!\cdots\!40\)\( T^{17} + \)\(25\!\cdots\!39\)\( T^{18} \)
$83$ \( 1 - \)\(10\!\cdots\!72\)\( T + \)\(14\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(92\!\cdots\!76\)\( T^{4} - \)\(56\!\cdots\!52\)\( T^{5} + \)\(38\!\cdots\!92\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{9} + \)\(24\!\cdots\!82\)\( T^{10} - \)\(95\!\cdots\!80\)\( T^{11} + \)\(40\!\cdots\!36\)\( T^{12} - \)\(13\!\cdots\!32\)\( T^{13} + \)\(46\!\cdots\!32\)\( T^{14} - \)\(11\!\cdots\!40\)\( T^{15} + \)\(35\!\cdots\!21\)\( T^{16} - \)\(56\!\cdots\!32\)\( T^{17} + \)\(11\!\cdots\!87\)\( T^{18} \)
$89$ \( 1 + \)\(88\!\cdots\!30\)\( T + \)\(55\!\cdots\!61\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!76\)\( T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(59\!\cdots\!06\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!74\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(56\!\cdots\!64\)\( T^{12} + \)\(74\!\cdots\!40\)\( T^{13} + \)\(86\!\cdots\!24\)\( T^{14} + \)\(84\!\cdots\!60\)\( T^{15} + \)\(68\!\cdots\!49\)\( T^{16} + \)\(42\!\cdots\!30\)\( T^{17} + \)\(18\!\cdots\!69\)\( T^{18} \)
$97$ \( 1 - \)\(18\!\cdots\!86\)\( T + \)\(30\!\cdots\!33\)\( T^{2} - \)\(47\!\cdots\!20\)\( T^{3} + \)\(42\!\cdots\!36\)\( T^{4} - \)\(55\!\cdots\!36\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} - \)\(39\!\cdots\!40\)\( T^{7} + \)\(19\!\cdots\!86\)\( T^{8} - \)\(18\!\cdots\!76\)\( T^{9} + \)\(76\!\cdots\!18\)\( T^{10} - \)\(57\!\cdots\!60\)\( T^{11} + \)\(20\!\cdots\!16\)\( T^{12} - \)\(12\!\cdots\!96\)\( T^{13} + \)\(35\!\cdots\!48\)\( T^{14} - \)\(15\!\cdots\!80\)\( T^{15} + \)\(38\!\cdots\!61\)\( T^{16} - \)\(86\!\cdots\!06\)\( T^{17} + \)\(18\!\cdots\!73\)\( T^{18} \)
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