Properties

Label 1.114.a.a
Level 1
Weight 114
Character orbit 1.a
Self dual yes
Analytic conductor 80.863
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 114 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(80.8627478904\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{144}\cdot 3^{48}\cdot 5^{19}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 19^{3}\cdot 23 \)
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.

\(f(q)\) \(=\) \( q +(552636039763781 + \beta_{1}) q^{2} +(-\)\(11\!\cdots\!03\)\( - 53786200 \beta_{1} + \beta_{2}) q^{3} +(\)\(63\!\cdots\!76\)\( - 7521194705788522 \beta_{1} + 233291 \beta_{2} + \beta_{3}) q^{4} +(-\)\(34\!\cdots\!43\)\( - \)\(10\!\cdots\!25\)\( \beta_{1} - 149841320469 \beta_{2} - 84572 \beta_{3} - \beta_{4}) q^{5} +(-\)\(96\!\cdots\!48\)\( - \)\(10\!\cdots\!51\)\( \beta_{1} + 3078932092137945 \beta_{2} + 522089023 \beta_{3} - 13268 \beta_{4} + \beta_{5}) q^{6} +(-\)\(19\!\cdots\!47\)\( + \)\(15\!\cdots\!31\)\( \beta_{1} - \)\(26\!\cdots\!43\)\( \beta_{2} - 8203997863513 \beta_{3} - 19535798 \beta_{4} - 1095 \beta_{5} - \beta_{6}) q^{7} +(-\)\(12\!\cdots\!17\)\( + \)\(46\!\cdots\!51\)\( \beta_{1} + \)\(94\!\cdots\!50\)\( \beta_{2} - 15716006786227561 \beta_{3} - 49231023484 \beta_{4} + 1827020 \beta_{5} + 54 \beta_{6} + \beta_{7}) q^{8} +(\)\(61\!\cdots\!67\)\( + \)\(14\!\cdots\!41\)\( \beta_{1} - \)\(26\!\cdots\!17\)\( \beta_{2} - 111141217798568264 \beta_{3} - 1279755571357 \beta_{4} + 352669740 \beta_{5} - 209767 \beta_{6} + 93 \beta_{7} - \beta_{8}) q^{9} +O(q^{10})\) \( q +(552636039763781 + \beta_{1}) q^{2} +(-\)\(11\!\cdots\!03\)\( - 53786200 \beta_{1} + \beta_{2}) q^{3} +(\)\(63\!\cdots\!76\)\( - 7521194705788522 \beta_{1} + 233291 \beta_{2} + \beta_{3}) q^{4} +(-\)\(34\!\cdots\!43\)\( - \)\(10\!\cdots\!25\)\( \beta_{1} - 149841320469 \beta_{2} - 84572 \beta_{3} - \beta_{4}) q^{5} +(-\)\(96\!\cdots\!48\)\( - \)\(10\!\cdots\!51\)\( \beta_{1} + 3078932092137945 \beta_{2} + 522089023 \beta_{3} - 13268 \beta_{4} + \beta_{5}) q^{6} +(-\)\(19\!\cdots\!47\)\( + \)\(15\!\cdots\!31\)\( \beta_{1} - \)\(26\!\cdots\!43\)\( \beta_{2} - 8203997863513 \beta_{3} - 19535798 \beta_{4} - 1095 \beta_{5} - \beta_{6}) q^{7} +(-\)\(12\!\cdots\!17\)\( + \)\(46\!\cdots\!51\)\( \beta_{1} + \)\(94\!\cdots\!50\)\( \beta_{2} - 15716006786227561 \beta_{3} - 49231023484 \beta_{4} + 1827020 \beta_{5} + 54 \beta_{6} + \beta_{7}) q^{8} +(\)\(61\!\cdots\!67\)\( + \)\(14\!\cdots\!41\)\( \beta_{1} - \)\(26\!\cdots\!17\)\( \beta_{2} - 111141217798568264 \beta_{3} - 1279755571357 \beta_{4} + 352669740 \beta_{5} - 209767 \beta_{6} + 93 \beta_{7} - \beta_{8}) q^{9} +(-\)\(17\!\cdots\!46\)\( - \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(35\!\cdots\!56\)\( \beta_{3} + 11752642414485088 \beta_{4} - 598794841860 \beta_{5} + 172191680 \beta_{6} - 273240 \beta_{7} + 720 \beta_{8}) q^{10} +(\)\(13\!\cdots\!29\)\( - \)\(20\!\cdots\!34\)\( \beta_{1} - \)\(25\!\cdots\!95\)\( \beta_{2} + \)\(10\!\cdots\!66\)\( \beta_{3} + 3016148434268673000 \beta_{4} - 13973036603174 \beta_{5} + 10347763338 \beta_{6} - 36310252 \beta_{7} - 115236 \beta_{8}) q^{11} +(-\)\(52\!\cdots\!16\)\( + \)\(49\!\cdots\!08\)\( \beta_{1} + \)\(62\!\cdots\!04\)\( \beta_{2} + \)\(81\!\cdots\!56\)\( \beta_{3} + \)\(32\!\cdots\!44\)\( \beta_{4} + 4905906456797280 \beta_{5} + 15568162228656 \beta_{6} + 700932744 \beta_{7} + 9734400 \beta_{8}) q^{12} +(\)\(13\!\cdots\!93\)\( + \)\(18\!\cdots\!65\)\( \beta_{1} + \)\(11\!\cdots\!41\)\( \beta_{2} + \)\(61\!\cdots\!48\)\( \beta_{3} + \)\(49\!\cdots\!41\)\( \beta_{4} - 1705651472282584440 \beta_{5} + 804733821555910 \beta_{6} + 374426482014 \beta_{7} - 538082550 \beta_{8}) q^{13} +(\)\(25\!\cdots\!48\)\( - \)\(94\!\cdots\!42\)\( \beta_{1} - \)\(13\!\cdots\!58\)\( \beta_{2} - \)\(11\!\cdots\!82\)\( \beta_{3} + \)\(80\!\cdots\!52\)\( \beta_{4} - \)\(61\!\cdots\!46\)\( \beta_{5} - 52162133780342016 \beta_{6} - 34620392506336 \beta_{7} + 21291826752 \beta_{8}) q^{14} +(-\)\(89\!\cdots\!53\)\( + \)\(95\!\cdots\!85\)\( \beta_{1} - \)\(29\!\cdots\!69\)\( \beta_{2} - \)\(26\!\cdots\!67\)\( \beta_{3} + \)\(37\!\cdots\!34\)\( \beta_{4} - \)\(19\!\cdots\!05\)\( \beta_{5} - 928742678667824535 \beta_{6} + 1551521939483880 \beta_{7} - 624217733640 \beta_{8}) q^{15} +(\)\(12\!\cdots\!24\)\( - \)\(22\!\cdots\!60\)\( \beta_{1} + \)\(32\!\cdots\!64\)\( \beta_{2} + \)\(31\!\cdots\!64\)\( \beta_{3} + \)\(18\!\cdots\!36\)\( \beta_{4} + \)\(34\!\cdots\!04\)\( \beta_{5} + \)\(11\!\cdots\!28\)\( \beta_{6} - 42031723946208912 \beta_{7} + 13320235229184 \beta_{8}) q^{16} +(\)\(11\!\cdots\!28\)\( + \)\(21\!\cdots\!13\)\( \beta_{1} - \)\(51\!\cdots\!21\)\( \beta_{2} + \)\(46\!\cdots\!48\)\( \beta_{3} + \)\(52\!\cdots\!03\)\( \beta_{4} + \)\(19\!\cdots\!20\)\( \beta_{5} - \)\(27\!\cdots\!19\)\( \beta_{6} + 644440847452975185 \beta_{7} - 176948799905925 \beta_{8}) q^{17} +(\)\(24\!\cdots\!41\)\( + \)\(55\!\cdots\!93\)\( \beta_{1} + \)\(44\!\cdots\!28\)\( \beta_{2} + \)\(78\!\cdots\!88\)\( \beta_{3} + \)\(23\!\cdots\!08\)\( \beta_{4} - \)\(31\!\cdots\!80\)\( \beta_{5} + \)\(10\!\cdots\!56\)\( \beta_{6} - 184252463709552720 \beta_{7} - 212945718813600 \beta_{8}) q^{18} +(-\)\(22\!\cdots\!65\)\( - \)\(53\!\cdots\!46\)\( \beta_{1} + \)\(22\!\cdots\!79\)\( \beta_{2} - \)\(37\!\cdots\!74\)\( \beta_{3} + \)\(77\!\cdots\!20\)\( \beta_{4} - \)\(63\!\cdots\!58\)\( \beta_{5} + \)\(10\!\cdots\!66\)\( \beta_{6} - \)\(29\!\cdots\!64\)\( \beta_{7} + 93757583067879348 \beta_{8}) q^{19} +(-\)\(14\!\cdots\!76\)\( + \)\(35\!\cdots\!00\)\( \beta_{1} - \)\(84\!\cdots\!58\)\( \beta_{2} - \)\(24\!\cdots\!54\)\( \beta_{3} - \)\(10\!\cdots\!32\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5} - \)\(29\!\cdots\!00\)\( \beta_{6} + \)\(99\!\cdots\!00\)\( \beta_{7} - 3314985135660057600 \beta_{8}) q^{20} +(-\)\(22\!\cdots\!04\)\( - \)\(69\!\cdots\!82\)\( \beta_{1} + \)\(88\!\cdots\!54\)\( \beta_{2} - \)\(35\!\cdots\!68\)\( \beta_{3} - \)\(57\!\cdots\!94\)\( \beta_{4} - \)\(35\!\cdots\!68\)\( \beta_{5} + \)\(39\!\cdots\!62\)\( \beta_{6} - \)\(20\!\cdots\!98\)\( \beta_{7} + 79592770128386681586 \beta_{8}) q^{21} +(-\)\(34\!\cdots\!48\)\( + \)\(24\!\cdots\!95\)\( \beta_{1} - \)\(79\!\cdots\!61\)\( \beta_{2} - \)\(54\!\cdots\!03\)\( \beta_{3} + \)\(15\!\cdots\!44\)\( \beta_{4} - \)\(29\!\cdots\!85\)\( \beta_{5} - \)\(22\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!56\)\( \beta_{7} - \)\(15\!\cdots\!00\)\( \beta_{8}) q^{22} +(-\)\(18\!\cdots\!01\)\( - \)\(18\!\cdots\!79\)\( \beta_{1} - \)\(57\!\cdots\!21\)\( \beta_{2} - \)\(26\!\cdots\!55\)\( \beta_{3} + \)\(11\!\cdots\!30\)\( \beta_{4} + \)\(31\!\cdots\!75\)\( \beta_{5} - \)\(21\!\cdots\!55\)\( \beta_{6} - \)\(39\!\cdots\!20\)\( \beta_{7} + \)\(24\!\cdots\!00\)\( \beta_{8}) q^{23} +(\)\(93\!\cdots\!00\)\( + \)\(12\!\cdots\!24\)\( \beta_{1} + \)\(44\!\cdots\!68\)\( \beta_{2} + \)\(94\!\cdots\!24\)\( \beta_{3} - \)\(17\!\cdots\!72\)\( \beta_{4} + \)\(15\!\cdots\!28\)\( \beta_{5} + \)\(63\!\cdots\!12\)\( \beta_{6} + \)\(39\!\cdots\!52\)\( \beta_{7} - \)\(35\!\cdots\!64\)\( \beta_{8}) q^{24} +(\)\(40\!\cdots\!75\)\( - \)\(11\!\cdots\!50\)\( \beta_{1} - \)\(32\!\cdots\!50\)\( \beta_{2} + \)\(54\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!50\)\( \beta_{4} - \)\(59\!\cdots\!00\)\( \beta_{5} - \)\(68\!\cdots\!50\)\( \beta_{6} - \)\(30\!\cdots\!50\)\( \beta_{7} + \)\(44\!\cdots\!50\)\( \beta_{8}) q^{25} +(\)\(30\!\cdots\!70\)\( + \)\(51\!\cdots\!98\)\( \beta_{1} - \)\(32\!\cdots\!84\)\( \beta_{2} + \)\(46\!\cdots\!24\)\( \beta_{3} + \)\(10\!\cdots\!04\)\( \beta_{4} + \)\(48\!\cdots\!84\)\( \beta_{5} + \)\(34\!\cdots\!56\)\( \beta_{6} + \)\(15\!\cdots\!76\)\( \beta_{7} - \)\(50\!\cdots\!32\)\( \beta_{8}) q^{26} +(-\)\(14\!\cdots\!72\)\( + \)\(26\!\cdots\!46\)\( \beta_{1} - \)\(34\!\cdots\!80\)\( \beta_{2} - \)\(26\!\cdots\!66\)\( \beta_{3} - \)\(57\!\cdots\!04\)\( \beta_{4} - \)\(99\!\cdots\!30\)\( \beta_{5} + \)\(12\!\cdots\!74\)\( \beta_{6} - \)\(21\!\cdots\!44\)\( \beta_{7} + \)\(51\!\cdots\!00\)\( \beta_{8}) q^{27} +(-\)\(13\!\cdots\!20\)\( + \)\(59\!\cdots\!84\)\( \beta_{1} - \)\(75\!\cdots\!36\)\( \beta_{2} - \)\(42\!\cdots\!64\)\( \beta_{3} - \)\(61\!\cdots\!76\)\( \beta_{4} - \)\(28\!\cdots\!20\)\( \beta_{5} - \)\(41\!\cdots\!84\)\( \beta_{6} - \)\(61\!\cdots\!56\)\( \beta_{7} - \)\(47\!\cdots\!00\)\( \beta_{8}) q^{28} +(\)\(18\!\cdots\!13\)\( + \)\(30\!\cdots\!99\)\( \beta_{1} - \)\(19\!\cdots\!17\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} + \)\(56\!\cdots\!91\)\( \beta_{4} + \)\(22\!\cdots\!16\)\( \beta_{5} + \)\(41\!\cdots\!64\)\( \beta_{6} + \)\(81\!\cdots\!44\)\( \beta_{7} + \)\(39\!\cdots\!92\)\( \beta_{8}) q^{29} +(\)\(16\!\cdots\!24\)\( - \)\(40\!\cdots\!50\)\( \beta_{1} - \)\(34\!\cdots\!58\)\( \beta_{2} + \)\(18\!\cdots\!46\)\( \beta_{3} + \)\(39\!\cdots\!68\)\( \beta_{4} - \)\(22\!\cdots\!50\)\( \beta_{5} - \)\(23\!\cdots\!00\)\( \beta_{6} - \)\(59\!\cdots\!00\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{30} +(\)\(14\!\cdots\!72\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(41\!\cdots\!08\)\( \beta_{2} + \)\(25\!\cdots\!48\)\( \beta_{3} - \)\(95\!\cdots\!36\)\( \beta_{4} - \)\(73\!\cdots\!84\)\( \beta_{5} + \)\(62\!\cdots\!32\)\( \beta_{6} + \)\(24\!\cdots\!72\)\( \beta_{7} + \)\(20\!\cdots\!96\)\( \beta_{8}) q^{31} +(-\)\(23\!\cdots\!60\)\( - \)\(68\!\cdots\!96\)\( \beta_{1} + \)\(75\!\cdots\!92\)\( \beta_{2} - \)\(42\!\cdots\!72\)\( \beta_{3} - \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(51\!\cdots\!20\)\( \beta_{5} + \)\(21\!\cdots\!76\)\( \beta_{6} + \)\(21\!\cdots\!20\)\( \beta_{7} - \)\(12\!\cdots\!00\)\( \beta_{8}) q^{32} +(-\)\(23\!\cdots\!38\)\( - \)\(17\!\cdots\!57\)\( \beta_{1} + \)\(50\!\cdots\!77\)\( \beta_{2} - \)\(22\!\cdots\!56\)\( \beta_{3} + \)\(74\!\cdots\!17\)\( \beta_{4} - \)\(66\!\cdots\!00\)\( \beta_{5} - \)\(29\!\cdots\!93\)\( \beta_{6} - \)\(13\!\cdots\!81\)\( \beta_{7} + \)\(72\!\cdots\!25\)\( \beta_{8}) q^{33} +(\)\(35\!\cdots\!06\)\( + \)\(49\!\cdots\!38\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(69\!\cdots\!64\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4} - \)\(91\!\cdots\!08\)\( \beta_{5} + \)\(10\!\cdots\!36\)\( \beta_{6} + \)\(13\!\cdots\!56\)\( \beta_{7} - \)\(37\!\cdots\!92\)\( \beta_{8}) q^{34} +(\)\(38\!\cdots\!24\)\( - \)\(57\!\cdots\!80\)\( \beta_{1} + \)\(71\!\cdots\!52\)\( \beta_{2} + \)\(76\!\cdots\!36\)\( \beta_{3} - \)\(29\!\cdots\!72\)\( \beta_{4} + \)\(42\!\cdots\!40\)\( \beta_{5} + \)\(29\!\cdots\!80\)\( \beta_{6} - \)\(82\!\cdots\!40\)\( \beta_{7} + \)\(16\!\cdots\!20\)\( \beta_{8}) q^{35} +(\)\(30\!\cdots\!20\)\( + \)\(72\!\cdots\!02\)\( \beta_{1} - \)\(60\!\cdots\!29\)\( \beta_{2} - \)\(82\!\cdots\!11\)\( \beta_{3} - \)\(36\!\cdots\!88\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} - \)\(56\!\cdots\!20\)\( \beta_{6} + \)\(36\!\cdots\!80\)\( \beta_{7} - \)\(66\!\cdots\!60\)\( \beta_{8}) q^{36} +(-\)\(66\!\cdots\!95\)\( - \)\(62\!\cdots\!43\)\( \beta_{1} - \)\(14\!\cdots\!23\)\( \beta_{2} - \)\(11\!\cdots\!72\)\( \beta_{3} + \)\(88\!\cdots\!37\)\( \beta_{4} - \)\(16\!\cdots\!60\)\( \beta_{5} + \)\(32\!\cdots\!98\)\( \beta_{6} - \)\(90\!\cdots\!58\)\( \beta_{7} + \)\(22\!\cdots\!50\)\( \beta_{8}) q^{37} +(-\)\(91\!\cdots\!48\)\( - \)\(54\!\cdots\!15\)\( \beta_{1} - \)\(10\!\cdots\!67\)\( \beta_{2} - \)\(36\!\cdots\!85\)\( \beta_{3} - \)\(15\!\cdots\!08\)\( \beta_{4} + \)\(40\!\cdots\!85\)\( \beta_{5} - \)\(82\!\cdots\!04\)\( \beta_{6} - \)\(13\!\cdots\!84\)\( \beta_{7} - \)\(58\!\cdots\!00\)\( \beta_{8}) q^{38} +(\)\(96\!\cdots\!51\)\( - \)\(19\!\cdots\!07\)\( \beta_{1} + \)\(16\!\cdots\!63\)\( \beta_{2} + \)\(17\!\cdots\!17\)\( \beta_{3} - \)\(13\!\cdots\!90\)\( \beta_{4} + \)\(18\!\cdots\!99\)\( \beta_{5} - \)\(11\!\cdots\!03\)\( \beta_{6} + \)\(29\!\cdots\!12\)\( \beta_{7} + \)\(91\!\cdots\!16\)\( \beta_{8}) q^{39} +(\)\(75\!\cdots\!10\)\( - \)\(24\!\cdots\!50\)\( \beta_{1} + \)\(23\!\cdots\!80\)\( \beta_{2} + \)\(15\!\cdots\!90\)\( \beta_{3} + \)\(58\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(19\!\cdots\!50\)\( \beta_{7} + \)\(10\!\cdots\!00\)\( \beta_{8}) q^{40} +(-\)\(38\!\cdots\!94\)\( - \)\(45\!\cdots\!86\)\( \beta_{1} + \)\(51\!\cdots\!66\)\( \beta_{2} - \)\(44\!\cdots\!92\)\( \beta_{3} + \)\(52\!\cdots\!98\)\( \beta_{4} + \)\(28\!\cdots\!04\)\( \beta_{5} - \)\(61\!\cdots\!30\)\( \beta_{6} + \)\(82\!\cdots\!70\)\( \beta_{7} - \)\(13\!\cdots\!90\)\( \beta_{8}) q^{41} +(-\)\(11\!\cdots\!28\)\( - \)\(52\!\cdots\!84\)\( \beta_{1} - \)\(72\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!28\)\( \beta_{3} - \)\(64\!\cdots\!68\)\( \beta_{4} + \)\(10\!\cdots\!80\)\( \beta_{5} - \)\(60\!\cdots\!96\)\( \beta_{6} - \)\(25\!\cdots\!40\)\( \beta_{7} + \)\(57\!\cdots\!00\)\( \beta_{8}) q^{42} +(\)\(23\!\cdots\!19\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} - \)\(70\!\cdots\!85\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3} - \)\(38\!\cdots\!20\)\( \beta_{4} - \)\(48\!\cdots\!20\)\( \beta_{5} + \)\(14\!\cdots\!92\)\( \beta_{6} + \)\(51\!\cdots\!36\)\( \beta_{7} - \)\(14\!\cdots\!00\)\( \beta_{8}) q^{43} +(\)\(25\!\cdots\!32\)\( - \)\(62\!\cdots\!20\)\( \beta_{1} - \)\(49\!\cdots\!96\)\( \beta_{2} + \)\(33\!\cdots\!24\)\( \beta_{3} + \)\(80\!\cdots\!36\)\( \beta_{4} - \)\(10\!\cdots\!76\)\( \beta_{5} - \)\(63\!\cdots\!12\)\( \beta_{6} - \)\(63\!\cdots\!52\)\( \beta_{7} + \)\(19\!\cdots\!64\)\( \beta_{8}) q^{44} +(\)\(28\!\cdots\!21\)\( - \)\(18\!\cdots\!75\)\( \beta_{1} + \)\(46\!\cdots\!93\)\( \beta_{2} + \)\(24\!\cdots\!84\)\( \beta_{3} - \)\(10\!\cdots\!03\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!50\)\( \beta_{6} + \)\(11\!\cdots\!50\)\( \beta_{7} - \)\(26\!\cdots\!50\)\( \beta_{8}) q^{45} +(-\)\(31\!\cdots\!24\)\( - \)\(39\!\cdots\!26\)\( \beta_{1} + \)\(39\!\cdots\!66\)\( \beta_{2} - \)\(46\!\cdots\!62\)\( \beta_{3} - \)\(76\!\cdots\!92\)\( \beta_{4} - \)\(25\!\cdots\!26\)\( \beta_{5} - \)\(29\!\cdots\!60\)\( \beta_{6} - \)\(11\!\cdots\!60\)\( \beta_{7} + \)\(29\!\cdots\!20\)\( \beta_{8}) q^{46} +(-\)\(54\!\cdots\!66\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + \)\(13\!\cdots\!10\)\( \beta_{2} - \)\(64\!\cdots\!86\)\( \beta_{3} - \)\(53\!\cdots\!60\)\( \beta_{4} - \)\(47\!\cdots\!10\)\( \beta_{5} - \)\(20\!\cdots\!54\)\( \beta_{6} + \)\(71\!\cdots\!68\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{47} +(\)\(27\!\cdots\!20\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(45\!\cdots\!72\)\( \beta_{3} + \)\(81\!\cdots\!52\)\( \beta_{4} + \)\(23\!\cdots\!80\)\( \beta_{5} - \)\(52\!\cdots\!36\)\( \beta_{6} - \)\(20\!\cdots\!80\)\( \beta_{7} + \)\(94\!\cdots\!00\)\( \beta_{8}) q^{48} +(\)\(83\!\cdots\!33\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} - \)\(49\!\cdots\!76\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3} + \)\(37\!\cdots\!92\)\( \beta_{4} + \)\(63\!\cdots\!16\)\( \beta_{5} + \)\(37\!\cdots\!80\)\( \beta_{6} - \)\(36\!\cdots\!20\)\( \beta_{7} - \)\(16\!\cdots\!60\)\( \beta_{8}) q^{49} +(-\)\(19\!\cdots\!25\)\( + \)\(87\!\cdots\!75\)\( \beta_{1} - \)\(83\!\cdots\!00\)\( \beta_{2} - \)\(52\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} - \)\(56\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(33\!\cdots\!00\)\( \beta_{7} - \)\(55\!\cdots\!00\)\( \beta_{8}) q^{50} +(-\)\(46\!\cdots\!48\)\( + \)\(91\!\cdots\!34\)\( \beta_{1} + \)\(64\!\cdots\!24\)\( \beta_{2} - \)\(55\!\cdots\!94\)\( \beta_{3} + \)\(43\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!82\)\( \beta_{5} + \)\(24\!\cdots\!06\)\( \beta_{6} - \)\(17\!\cdots\!24\)\( \beta_{7} + \)\(56\!\cdots\!68\)\( \beta_{8}) q^{51} +(-\)\(33\!\cdots\!84\)\( + \)\(51\!\cdots\!56\)\( \beta_{1} + \)\(40\!\cdots\!58\)\( \beta_{2} + \)\(99\!\cdots\!94\)\( \beta_{3} + \)\(79\!\cdots\!40\)\( \beta_{4} + \)\(13\!\cdots\!40\)\( \beta_{5} - \)\(56\!\cdots\!84\)\( \beta_{6} + \)\(45\!\cdots\!28\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{52} +(\)\(50\!\cdots\!53\)\( - \)\(22\!\cdots\!15\)\( \beta_{1} + \)\(11\!\cdots\!73\)\( \beta_{2} + \)\(97\!\cdots\!04\)\( \beta_{3} - \)\(41\!\cdots\!19\)\( \beta_{4} - \)\(38\!\cdots\!80\)\( \beta_{5} - \)\(61\!\cdots\!66\)\( \beta_{6} - \)\(31\!\cdots\!74\)\( \beta_{7} + \)\(45\!\cdots\!50\)\( \beta_{8}) q^{53} +(\)\(42\!\cdots\!56\)\( - \)\(37\!\cdots\!54\)\( \beta_{1} - \)\(38\!\cdots\!94\)\( \beta_{2} + \)\(13\!\cdots\!86\)\( \beta_{3} - \)\(39\!\cdots\!44\)\( \beta_{4} - \)\(35\!\cdots\!06\)\( \beta_{5} - \)\(17\!\cdots\!32\)\( \beta_{6} - \)\(27\!\cdots\!72\)\( \beta_{7} + \)\(40\!\cdots\!04\)\( \beta_{8}) q^{54} +(-\)\(48\!\cdots\!51\)\( - \)\(70\!\cdots\!25\)\( \beta_{1} - \)\(80\!\cdots\!83\)\( \beta_{2} - \)\(76\!\cdots\!29\)\( \beta_{3} - \)\(17\!\cdots\!82\)\( \beta_{4} + \)\(19\!\cdots\!25\)\( \beta_{5} + \)\(53\!\cdots\!75\)\( \beta_{6} + \)\(13\!\cdots\!00\)\( \beta_{7} - \)\(67\!\cdots\!00\)\( \beta_{8}) q^{55} +(-\)\(17\!\cdots\!04\)\( - \)\(41\!\cdots\!56\)\( \beta_{1} - \)\(21\!\cdots\!00\)\( \beta_{2} - \)\(60\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(37\!\cdots\!08\)\( \beta_{5} + \)\(63\!\cdots\!44\)\( \beta_{6} - \)\(29\!\cdots\!76\)\( \beta_{7} + \)\(25\!\cdots\!32\)\( \beta_{8}) q^{56} +(\)\(22\!\cdots\!46\)\( - \)\(55\!\cdots\!67\)\( \beta_{1} - \)\(44\!\cdots\!17\)\( \beta_{2} + \)\(24\!\cdots\!92\)\( \beta_{3} - \)\(10\!\cdots\!05\)\( \beta_{4} + \)\(97\!\cdots\!20\)\( \beta_{5} - \)\(13\!\cdots\!87\)\( \beta_{6} + \)\(13\!\cdots\!29\)\( \beta_{7} - \)\(46\!\cdots\!25\)\( \beta_{8}) q^{57} +(\)\(51\!\cdots\!62\)\( + \)\(42\!\cdots\!10\)\( \beta_{1} + \)\(31\!\cdots\!40\)\( \beta_{2} + \)\(11\!\cdots\!76\)\( \beta_{3} - \)\(39\!\cdots\!88\)\( \beta_{4} - \)\(18\!\cdots\!80\)\( \beta_{5} + \)\(49\!\cdots\!80\)\( \beta_{6} + \)\(13\!\cdots\!28\)\( \beta_{7} - \)\(45\!\cdots\!00\)\( \beta_{8}) q^{58} +(-\)\(88\!\cdots\!01\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(44\!\cdots\!59\)\( \beta_{2} + \)\(21\!\cdots\!32\)\( \beta_{3} - \)\(25\!\cdots\!04\)\( \beta_{4} + \)\(59\!\cdots\!12\)\( \beta_{5} - \)\(63\!\cdots\!08\)\( \beta_{6} - \)\(48\!\cdots\!68\)\( \beta_{7} + \)\(59\!\cdots\!76\)\( \beta_{8}) q^{59} +(-\)\(57\!\cdots\!96\)\( + \)\(21\!\cdots\!20\)\( \beta_{1} - \)\(70\!\cdots\!08\)\( \beta_{2} - \)\(11\!\cdots\!44\)\( \beta_{3} + \)\(46\!\cdots\!88\)\( \beta_{4} - \)\(19\!\cdots\!60\)\( \beta_{5} - \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(73\!\cdots\!60\)\( \beta_{7} - \)\(19\!\cdots\!80\)\( \beta_{8}) q^{60} +(-\)\(22\!\cdots\!95\)\( + \)\(37\!\cdots\!01\)\( \beta_{1} - \)\(11\!\cdots\!39\)\( \beta_{2} - \)\(33\!\cdots\!40\)\( \beta_{3} + \)\(52\!\cdots\!53\)\( \beta_{4} + \)\(16\!\cdots\!36\)\( \beta_{5} + \)\(55\!\cdots\!66\)\( \beta_{6} - \)\(41\!\cdots\!14\)\( \beta_{7} + \)\(24\!\cdots\!98\)\( \beta_{8}) q^{61} +(\)\(32\!\cdots\!28\)\( + \)\(34\!\cdots\!84\)\( \beta_{1} - \)\(83\!\cdots\!12\)\( \beta_{2} + \)\(32\!\cdots\!12\)\( \beta_{3} - \)\(37\!\cdots\!48\)\( \beta_{4} + \)\(29\!\cdots\!80\)\( \beta_{5} - \)\(49\!\cdots\!76\)\( \beta_{6} - \)\(17\!\cdots\!00\)\( \beta_{7} + \)\(58\!\cdots\!00\)\( \beta_{8}) q^{62} +(\)\(12\!\cdots\!17\)\( - \)\(39\!\cdots\!41\)\( \beta_{1} - \)\(10\!\cdots\!63\)\( \beta_{2} + \)\(16\!\cdots\!43\)\( \beta_{3} + \)\(41\!\cdots\!42\)\( \beta_{4} + \)\(13\!\cdots\!65\)\( \beta_{5} - \)\(10\!\cdots\!77\)\( \beta_{6} + \)\(20\!\cdots\!12\)\( \beta_{7} - \)\(39\!\cdots\!00\)\( \beta_{8}) q^{63} +(-\)\(24\!\cdots\!88\)\( - \)\(36\!\cdots\!12\)\( \beta_{1} + \)\(64\!\cdots\!40\)\( \beta_{2} - \)\(20\!\cdots\!64\)\( \beta_{3} + \)\(62\!\cdots\!64\)\( \beta_{4} - \)\(29\!\cdots\!08\)\( \beta_{5} - \)\(27\!\cdots\!80\)\( \beta_{6} - \)\(32\!\cdots\!80\)\( \beta_{7} + \)\(94\!\cdots\!60\)\( \beta_{8}) q^{64} +(-\)\(67\!\cdots\!88\)\( - \)\(11\!\cdots\!90\)\( \beta_{1} + \)\(15\!\cdots\!26\)\( \beta_{2} - \)\(69\!\cdots\!32\)\( \beta_{3} + \)\(31\!\cdots\!14\)\( \beta_{4} - \)\(14\!\cdots\!80\)\( \beta_{5} + \)\(18\!\cdots\!90\)\( \beta_{6} + \)\(39\!\cdots\!30\)\( \beta_{7} - \)\(46\!\cdots\!90\)\( \beta_{8}) q^{65} +(-\)\(31\!\cdots\!16\)\( - \)\(41\!\cdots\!08\)\( \beta_{1} - \)\(23\!\cdots\!28\)\( \beta_{2} - \)\(12\!\cdots\!72\)\( \beta_{3} - \)\(20\!\cdots\!20\)\( \beta_{4} + \)\(36\!\cdots\!96\)\( \beta_{5} - \)\(50\!\cdots\!72\)\( \beta_{6} - \)\(16\!\cdots\!12\)\( \beta_{7} - \)\(47\!\cdots\!16\)\( \beta_{8}) q^{66} +(\)\(27\!\cdots\!15\)\( + \)\(74\!\cdots\!74\)\( \beta_{1} - \)\(14\!\cdots\!69\)\( \beta_{2} + \)\(54\!\cdots\!30\)\( \beta_{3} + \)\(26\!\cdots\!28\)\( \beta_{4} - \)\(83\!\cdots\!10\)\( \beta_{5} + \)\(79\!\cdots\!94\)\( \beta_{6} + \)\(23\!\cdots\!84\)\( \beta_{7} + \)\(18\!\cdots\!00\)\( \beta_{8}) q^{67} +(\)\(70\!\cdots\!04\)\( + \)\(72\!\cdots\!16\)\( \beta_{1} - \)\(12\!\cdots\!90\)\( \beta_{2} + \)\(88\!\cdots\!66\)\( \beta_{3} - \)\(44\!\cdots\!96\)\( \beta_{4} + \)\(61\!\cdots\!80\)\( \beta_{5} + \)\(27\!\cdots\!76\)\( \beta_{6} + \)\(55\!\cdots\!44\)\( \beta_{7} - \)\(27\!\cdots\!00\)\( \beta_{8}) q^{68} +(-\)\(26\!\cdots\!08\)\( + \)\(23\!\cdots\!18\)\( \beta_{1} - \)\(28\!\cdots\!58\)\( \beta_{2} - \)\(40\!\cdots\!68\)\( \beta_{3} + \)\(13\!\cdots\!74\)\( \beta_{4} - \)\(11\!\cdots\!84\)\( \beta_{5} - \)\(63\!\cdots\!58\)\( \beta_{6} - \)\(30\!\cdots\!18\)\( \beta_{7} - \)\(26\!\cdots\!74\)\( \beta_{8}) q^{69} +(-\)\(95\!\cdots\!92\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!64\)\( \beta_{2} - \)\(10\!\cdots\!68\)\( \beta_{3} - \)\(54\!\cdots\!44\)\( \beta_{4} + \)\(65\!\cdots\!00\)\( \beta_{5} + \)\(49\!\cdots\!00\)\( \beta_{6} + \)\(47\!\cdots\!00\)\( \beta_{7} + \)\(23\!\cdots\!00\)\( \beta_{8}) q^{70} +(\)\(10\!\cdots\!21\)\( + \)\(13\!\cdots\!83\)\( \beta_{1} + \)\(57\!\cdots\!13\)\( \beta_{2} + \)\(46\!\cdots\!55\)\( \beta_{3} + \)\(46\!\cdots\!74\)\( \beta_{4} - \)\(57\!\cdots\!87\)\( \beta_{5} - \)\(26\!\cdots\!97\)\( \beta_{6} + \)\(42\!\cdots\!88\)\( \beta_{7} - \)\(51\!\cdots\!16\)\( \beta_{8}) q^{71} +(\)\(97\!\cdots\!83\)\( - \)\(11\!\cdots\!93\)\( \beta_{1} + \)\(15\!\cdots\!50\)\( \beta_{2} + \)\(28\!\cdots\!99\)\( \beta_{3} + \)\(49\!\cdots\!84\)\( \beta_{4} - \)\(38\!\cdots\!40\)\( \beta_{5} + \)\(19\!\cdots\!38\)\( \beta_{6} - \)\(27\!\cdots\!35\)\( \beta_{7} + \)\(22\!\cdots\!00\)\( \beta_{8}) q^{72} +(\)\(60\!\cdots\!88\)\( - \)\(53\!\cdots\!63\)\( \beta_{1} - \)\(53\!\cdots\!85\)\( \beta_{2} + \)\(70\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!87\)\( \beta_{4} + \)\(10\!\cdots\!60\)\( \beta_{5} + \)\(39\!\cdots\!21\)\( \beta_{6} + \)\(24\!\cdots\!21\)\( \beta_{7} + \)\(20\!\cdots\!75\)\( \beta_{8}) q^{73} +(-\)\(10\!\cdots\!34\)\( - \)\(16\!\cdots\!70\)\( \beta_{1} - \)\(41\!\cdots\!88\)\( \beta_{2} - \)\(18\!\cdots\!92\)\( \beta_{3} - \)\(54\!\cdots\!64\)\( \beta_{4} - \)\(56\!\cdots\!40\)\( \beta_{5} - \)\(33\!\cdots\!44\)\( \beta_{6} + \)\(80\!\cdots\!76\)\( \beta_{7} - \)\(64\!\cdots\!32\)\( \beta_{8}) q^{74} +(-\)\(32\!\cdots\!25\)\( - \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(27\!\cdots\!75\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!00\)\( \beta_{4} + \)\(34\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!00\)\( \beta_{7} + \)\(71\!\cdots\!00\)\( \beta_{8}) q^{75} +(-\)\(67\!\cdots\!88\)\( - \)\(31\!\cdots\!92\)\( \beta_{1} + \)\(54\!\cdots\!80\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(21\!\cdots\!96\)\( \beta_{4} - \)\(80\!\cdots\!96\)\( \beta_{5} - \)\(65\!\cdots\!12\)\( \beta_{6} - \)\(32\!\cdots\!52\)\( \beta_{7} + \)\(10\!\cdots\!64\)\( \beta_{8}) q^{76} +(\)\(11\!\cdots\!48\)\( + \)\(98\!\cdots\!46\)\( \beta_{1} - \)\(55\!\cdots\!90\)\( \beta_{2} + \)\(17\!\cdots\!88\)\( \beta_{3} - \)\(88\!\cdots\!22\)\( \beta_{4} + \)\(16\!\cdots\!20\)\( \beta_{5} - \)\(28\!\cdots\!34\)\( \beta_{6} + \)\(16\!\cdots\!90\)\( \beta_{7} - \)\(57\!\cdots\!50\)\( \beta_{8}) q^{77} +(-\)\(31\!\cdots\!40\)\( + \)\(26\!\cdots\!26\)\( \beta_{1} + \)\(54\!\cdots\!46\)\( \beta_{2} + \)\(12\!\cdots\!70\)\( \beta_{3} + \)\(27\!\cdots\!24\)\( \beta_{4} + \)\(19\!\cdots\!70\)\( \beta_{5} + \)\(67\!\cdots\!72\)\( \beta_{6} - \)\(12\!\cdots\!68\)\( \beta_{7} + \)\(92\!\cdots\!00\)\( \beta_{8}) q^{78} +(-\)\(66\!\cdots\!26\)\( + \)\(49\!\cdots\!82\)\( \beta_{1} - \)\(52\!\cdots\!22\)\( \beta_{2} - \)\(16\!\cdots\!22\)\( \beta_{3} - \)\(18\!\cdots\!24\)\( \beta_{4} - \)\(10\!\cdots\!26\)\( \beta_{5} - \)\(14\!\cdots\!22\)\( \beta_{6} - \)\(68\!\cdots\!12\)\( \beta_{7} + \)\(66\!\cdots\!84\)\( \beta_{8}) q^{79} +(-\)\(26\!\cdots\!88\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!04\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} + \)\(17\!\cdots\!84\)\( \beta_{4} + \)\(97\!\cdots\!00\)\( \beta_{5} - \)\(72\!\cdots\!00\)\( \beta_{6} + \)\(20\!\cdots\!00\)\( \beta_{7} - \)\(35\!\cdots\!00\)\( \beta_{8}) q^{80} +(-\)\(33\!\cdots\!61\)\( - \)\(31\!\cdots\!69\)\( \beta_{1} - \)\(58\!\cdots\!91\)\( \beta_{2} + \)\(12\!\cdots\!56\)\( \beta_{3} - \)\(58\!\cdots\!31\)\( \beta_{4} - \)\(59\!\cdots\!12\)\( \beta_{5} - \)\(19\!\cdots\!77\)\( \beta_{6} - \)\(10\!\cdots\!17\)\( \beta_{7} + \)\(77\!\cdots\!69\)\( \beta_{8}) q^{81} +(-\)\(77\!\cdots\!66\)\( - \)\(72\!\cdots\!38\)\( \beta_{1} + \)\(26\!\cdots\!04\)\( \beta_{2} + \)\(29\!\cdots\!08\)\( \beta_{3} + \)\(15\!\cdots\!12\)\( \beta_{4} + \)\(48\!\cdots\!40\)\( \beta_{5} + \)\(65\!\cdots\!68\)\( \beta_{6} - \)\(56\!\cdots\!48\)\( \beta_{7} - \)\(51\!\cdots\!00\)\( \beta_{8}) q^{82} +(-\)\(12\!\cdots\!75\)\( - \)\(70\!\cdots\!96\)\( \beta_{1} + \)\(10\!\cdots\!93\)\( \beta_{2} + \)\(11\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!00\)\( \beta_{4} - \)\(77\!\cdots\!00\)\( \beta_{5} + \)\(49\!\cdots\!20\)\( \beta_{6} + \)\(12\!\cdots\!60\)\( \beta_{7} - \)\(13\!\cdots\!00\)\( \beta_{8}) q^{83} +(-\)\(64\!\cdots\!20\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} - \)\(31\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!56\)\( \beta_{4} - \)\(16\!\cdots\!08\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} - \)\(39\!\cdots\!00\)\( \beta_{7} + \)\(44\!\cdots\!00\)\( \beta_{8}) q^{84} +(-\)\(71\!\cdots\!86\)\( - \)\(33\!\cdots\!80\)\( \beta_{1} - \)\(92\!\cdots\!28\)\( \beta_{2} + \)\(23\!\cdots\!96\)\( \beta_{3} + \)\(28\!\cdots\!08\)\( \beta_{4} + \)\(50\!\cdots\!40\)\( \beta_{5} + \)\(35\!\cdots\!30\)\( \beta_{6} - \)\(21\!\cdots\!90\)\( \beta_{7} - \)\(57\!\cdots\!30\)\( \beta_{8}) q^{85} +(-\)\(23\!\cdots\!92\)\( + \)\(48\!\cdots\!55\)\( \beta_{1} - \)\(64\!\cdots\!17\)\( \beta_{2} - \)\(10\!\cdots\!87\)\( \beta_{3} + \)\(12\!\cdots\!32\)\( \beta_{4} - \)\(11\!\cdots\!97\)\( \beta_{5} + \)\(42\!\cdots\!76\)\( \beta_{6} + \)\(14\!\cdots\!96\)\( \beta_{7} + \)\(45\!\cdots\!28\)\( \beta_{8}) q^{86} +(-\)\(17\!\cdots\!81\)\( + \)\(15\!\cdots\!41\)\( \beta_{1} - \)\(93\!\cdots\!41\)\( \beta_{2} + \)\(24\!\cdots\!29\)\( \beta_{3} + \)\(17\!\cdots\!02\)\( \beta_{4} + \)\(74\!\cdots\!75\)\( \beta_{5} + \)\(16\!\cdots\!77\)\( \beta_{6} + \)\(46\!\cdots\!44\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{87} +(-\)\(67\!\cdots\!40\)\( + \)\(34\!\cdots\!64\)\( \beta_{1} - \)\(21\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!08\)\( \beta_{3} - \)\(10\!\cdots\!28\)\( \beta_{4} - \)\(33\!\cdots\!20\)\( \beta_{5} - \)\(34\!\cdots\!96\)\( \beta_{6} + \)\(12\!\cdots\!20\)\( \beta_{7} - \)\(36\!\cdots\!00\)\( \beta_{8}) q^{88} +(-\)\(72\!\cdots\!84\)\( + \)\(12\!\cdots\!21\)\( \beta_{1} + \)\(54\!\cdots\!23\)\( \beta_{2} + \)\(73\!\cdots\!40\)\( \beta_{3} + \)\(44\!\cdots\!15\)\( \beta_{4} + \)\(42\!\cdots\!52\)\( \beta_{5} - \)\(29\!\cdots\!59\)\( \beta_{6} - \)\(42\!\cdots\!39\)\( \beta_{7} + \)\(46\!\cdots\!23\)\( \beta_{8}) q^{89} +(-\)\(30\!\cdots\!38\)\( + \)\(39\!\cdots\!10\)\( \beta_{1} + \)\(17\!\cdots\!76\)\( \beta_{2} - \)\(15\!\cdots\!32\)\( \beta_{3} - \)\(65\!\cdots\!36\)\( \beta_{4} + \)\(10\!\cdots\!20\)\( \beta_{5} + \)\(22\!\cdots\!40\)\( \beta_{6} + \)\(78\!\cdots\!80\)\( \beta_{7} - \)\(26\!\cdots\!40\)\( \beta_{8}) q^{90} +(-\)\(25\!\cdots\!92\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} - \)\(60\!\cdots\!20\)\( \beta_{2} + \)\(27\!\cdots\!60\)\( \beta_{3} + \)\(80\!\cdots\!88\)\( \beta_{4} - \)\(20\!\cdots\!68\)\( \beta_{5} - \)\(35\!\cdots\!76\)\( \beta_{6} + \)\(61\!\cdots\!04\)\( \beta_{7} - \)\(13\!\cdots\!28\)\( \beta_{8}) q^{91} +(-\)\(47\!\cdots\!04\)\( - \)\(48\!\cdots\!80\)\( \beta_{1} - \)\(19\!\cdots\!24\)\( \beta_{2} - \)\(16\!\cdots\!72\)\( \beta_{3} - \)\(71\!\cdots\!68\)\( \beta_{4} - \)\(85\!\cdots\!60\)\( \beta_{5} - \)\(10\!\cdots\!92\)\( \beta_{6} - \)\(23\!\cdots\!48\)\( \beta_{7} + \)\(31\!\cdots\!00\)\( \beta_{8}) q^{92} +(\)\(17\!\cdots\!76\)\( - \)\(45\!\cdots\!92\)\( \beta_{1} - \)\(61\!\cdots\!20\)\( \beta_{2} + \)\(57\!\cdots\!60\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!20\)\( \beta_{5} + \)\(57\!\cdots\!28\)\( \beta_{6} + \)\(51\!\cdots\!08\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{93} +(\)\(20\!\cdots\!80\)\( - \)\(10\!\cdots\!84\)\( \beta_{1} - \)\(66\!\cdots\!44\)\( \beta_{2} + \)\(78\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!04\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(89\!\cdots\!12\)\( \beta_{6} - \)\(93\!\cdots\!52\)\( \beta_{7} - \)\(12\!\cdots\!36\)\( \beta_{8}) q^{94} +(-\)\(45\!\cdots\!05\)\( + \)\(15\!\cdots\!25\)\( \beta_{1} + \)\(11\!\cdots\!35\)\( \beta_{2} - \)\(11\!\cdots\!95\)\( \beta_{3} + \)\(30\!\cdots\!90\)\( \beta_{4} - \)\(96\!\cdots\!25\)\( \beta_{5} + \)\(15\!\cdots\!25\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7} + \)\(71\!\cdots\!00\)\( \beta_{8}) q^{95} +(\)\(93\!\cdots\!64\)\( + \)\(52\!\cdots\!16\)\( \beta_{1} + \)\(21\!\cdots\!88\)\( \beta_{2} - \)\(20\!\cdots\!08\)\( \beta_{3} + \)\(14\!\cdots\!56\)\( \beta_{4} - \)\(37\!\cdots\!92\)\( \beta_{5} - \)\(84\!\cdots\!00\)\( \beta_{6} + \)\(20\!\cdots\!00\)\( \beta_{7} - \)\(12\!\cdots\!00\)\( \beta_{8}) q^{96} +(\)\(40\!\cdots\!16\)\( + \)\(98\!\cdots\!73\)\( \beta_{1} + \)\(56\!\cdots\!63\)\( \beta_{2} - \)\(50\!\cdots\!16\)\( \beta_{3} - \)\(26\!\cdots\!69\)\( \beta_{4} + \)\(13\!\cdots\!20\)\( \beta_{5} + \)\(71\!\cdots\!29\)\( \beta_{6} - \)\(13\!\cdots\!39\)\( \beta_{7} + \)\(28\!\cdots\!75\)\( \beta_{8}) q^{97} +(\)\(25\!\cdots\!05\)\( + \)\(17\!\cdots\!77\)\( \beta_{1} + \)\(95\!\cdots\!16\)\( \beta_{2} + \)\(14\!\cdots\!72\)\( \beta_{3} - \)\(11\!\cdots\!92\)\( \beta_{4} + \)\(97\!\cdots\!60\)\( \beta_{5} + \)\(23\!\cdots\!12\)\( \beta_{6} + \)\(22\!\cdots\!68\)\( \beta_{7} - \)\(53\!\cdots\!00\)\( \beta_{8}) q^{98} +(\)\(35\!\cdots\!87\)\( + \)\(49\!\cdots\!24\)\( \beta_{1} - \)\(35\!\cdots\!45\)\( \beta_{2} - \)\(52\!\cdots\!40\)\( \beta_{3} - \)\(45\!\cdots\!92\)\( \beta_{4} + \)\(13\!\cdots\!12\)\( \beta_{5} - \)\(46\!\cdots\!16\)\( \beta_{6} + \)\(91\!\cdots\!64\)\( \beta_{7} - \)\(74\!\cdots\!48\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 4973724357874032q^{2} - \)\(10\!\cdots\!24\)\(q^{3} + \)\(57\!\cdots\!88\)\(q^{4} - \)\(31\!\cdots\!50\)\(q^{5} - \)\(86\!\cdots\!12\)\(q^{6} - \)\(17\!\cdots\!08\)\(q^{7} - \)\(11\!\cdots\!20\)\(q^{8} + \)\(55\!\cdots\!77\)\(q^{9} + O(q^{10}) \) \( 9q + 4973724357874032q^{2} - \)\(10\!\cdots\!24\)\(q^{3} + \)\(57\!\cdots\!88\)\(q^{4} - \)\(31\!\cdots\!50\)\(q^{5} - \)\(86\!\cdots\!12\)\(q^{6} - \)\(17\!\cdots\!08\)\(q^{7} - \)\(11\!\cdots\!20\)\(q^{8} + \)\(55\!\cdots\!77\)\(q^{9} - \)\(15\!\cdots\!00\)\(q^{10} + \)\(12\!\cdots\!88\)\(q^{11} - \)\(47\!\cdots\!68\)\(q^{12} + \)\(11\!\cdots\!46\)\(q^{13} + \)\(23\!\cdots\!96\)\(q^{14} - \)\(80\!\cdots\!00\)\(q^{15} + \)\(10\!\cdots\!44\)\(q^{16} + \)\(10\!\cdots\!62\)\(q^{17} + \)\(21\!\cdots\!96\)\(q^{18} - \)\(20\!\cdots\!60\)\(q^{19} - \)\(12\!\cdots\!00\)\(q^{20} - \)\(20\!\cdots\!72\)\(q^{21} - \)\(31\!\cdots\!76\)\(q^{22} - \)\(16\!\cdots\!84\)\(q^{23} + \)\(83\!\cdots\!20\)\(q^{24} + \)\(36\!\cdots\!75\)\(q^{25} + \)\(27\!\cdots\!48\)\(q^{26} - \)\(12\!\cdots\!20\)\(q^{27} - \)\(12\!\cdots\!56\)\(q^{28} + \)\(16\!\cdots\!10\)\(q^{29} + \)\(14\!\cdots\!00\)\(q^{30} + \)\(13\!\cdots\!28\)\(q^{31} - \)\(21\!\cdots\!48\)\(q^{32} - \)\(21\!\cdots\!68\)\(q^{33} + \)\(31\!\cdots\!56\)\(q^{34} + \)\(34\!\cdots\!00\)\(q^{35} + \)\(27\!\cdots\!64\)\(q^{36} - \)\(59\!\cdots\!98\)\(q^{37} - \)\(82\!\cdots\!80\)\(q^{38} + \)\(87\!\cdots\!64\)\(q^{39} + \)\(68\!\cdots\!00\)\(q^{40} - \)\(34\!\cdots\!02\)\(q^{41} - \)\(10\!\cdots\!56\)\(q^{42} + \)\(20\!\cdots\!56\)\(q^{43} + \)\(23\!\cdots\!16\)\(q^{44} + \)\(25\!\cdots\!50\)\(q^{45} - \)\(28\!\cdots\!92\)\(q^{46} - \)\(49\!\cdots\!28\)\(q^{47} + \)\(24\!\cdots\!16\)\(q^{48} + \)\(75\!\cdots\!13\)\(q^{49} - \)\(17\!\cdots\!00\)\(q^{50} - \)\(41\!\cdots\!92\)\(q^{51} - \)\(29\!\cdots\!28\)\(q^{52} + \)\(45\!\cdots\!26\)\(q^{53} + \)\(38\!\cdots\!40\)\(q^{54} - \)\(43\!\cdots\!00\)\(q^{55} - \)\(15\!\cdots\!60\)\(q^{56} + \)\(19\!\cdots\!60\)\(q^{57} + \)\(46\!\cdots\!80\)\(q^{58} - \)\(79\!\cdots\!80\)\(q^{59} - \)\(51\!\cdots\!00\)\(q^{60} - \)\(20\!\cdots\!62\)\(q^{61} + \)\(29\!\cdots\!44\)\(q^{62} + \)\(10\!\cdots\!76\)\(q^{63} - \)\(21\!\cdots\!72\)\(q^{64} - \)\(60\!\cdots\!00\)\(q^{65} - \)\(28\!\cdots\!84\)\(q^{66} + \)\(24\!\cdots\!12\)\(q^{67} + \)\(63\!\cdots\!84\)\(q^{68} - \)\(24\!\cdots\!56\)\(q^{69} - \)\(86\!\cdots\!00\)\(q^{70} + \)\(91\!\cdots\!08\)\(q^{71} + \)\(87\!\cdots\!40\)\(q^{72} + \)\(54\!\cdots\!66\)\(q^{73} - \)\(93\!\cdots\!24\)\(q^{74} - \)\(29\!\cdots\!00\)\(q^{75} - \)\(60\!\cdots\!20\)\(q^{76} + \)\(10\!\cdots\!44\)\(q^{77} - \)\(28\!\cdots\!28\)\(q^{78} - \)\(59\!\cdots\!40\)\(q^{79} - \)\(24\!\cdots\!00\)\(q^{80} - \)\(30\!\cdots\!51\)\(q^{81} - \)\(69\!\cdots\!96\)\(q^{82} - \)\(11\!\cdots\!64\)\(q^{83} - \)\(57\!\cdots\!04\)\(q^{84} - \)\(64\!\cdots\!00\)\(q^{85} - \)\(21\!\cdots\!72\)\(q^{86} - \)\(15\!\cdots\!60\)\(q^{87} - \)\(60\!\cdots\!40\)\(q^{88} - \)\(65\!\cdots\!70\)\(q^{89} - \)\(27\!\cdots\!00\)\(q^{90} - \)\(23\!\cdots\!12\)\(q^{91} - \)\(42\!\cdots\!88\)\(q^{92} + \)\(15\!\cdots\!92\)\(q^{93} + \)\(18\!\cdots\!36\)\(q^{94} - \)\(40\!\cdots\!00\)\(q^{95} + \)\(84\!\cdots\!68\)\(q^{96} + \)\(36\!\cdots\!22\)\(q^{97} + \)\(22\!\cdots\!24\)\(q^{98} + \)\(32\!\cdots\!64\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - \)\(32\!\cdots\!04\)\( x^{7} + \)\(39\!\cdots\!64\)\( x^{6} + \)\(35\!\cdots\!16\)\( x^{5} - \)\(67\!\cdots\!92\)\( x^{4} - \)\(14\!\cdots\!88\)\( x^{3} + \)\(31\!\cdots\!68\)\( x^{2} + \)\(14\!\cdots\!72\)\( x - \)\(66\!\cdots\!92\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(83\!\cdots\!59\)\( \nu^{8} - \)\(16\!\cdots\!81\)\( \nu^{7} + \)\(36\!\cdots\!00\)\( \nu^{6} + \)\(56\!\cdots\!12\)\( \nu^{5} - \)\(51\!\cdots\!24\)\( \nu^{4} - \)\(58\!\cdots\!72\)\( \nu^{3} + \)\(23\!\cdots\!16\)\( \nu^{2} + \)\(17\!\cdots\!08\)\( \nu - \)\(21\!\cdots\!40\)\(\)\()/ \)\(77\!\cdots\!28\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(11\!\cdots\!57\)\( \nu^{8} + \)\(22\!\cdots\!63\)\( \nu^{7} - \)\(50\!\cdots\!00\)\( \nu^{6} - \)\(77\!\cdots\!76\)\( \nu^{5} + \)\(70\!\cdots\!52\)\( \nu^{4} + \)\(80\!\cdots\!56\)\( \nu^{3} + \)\(72\!\cdots\!68\)\( \nu^{2} - \)\(58\!\cdots\!16\)\( \nu - \)\(73\!\cdots\!48\)\(\)\()/ \)\(45\!\cdots\!84\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(22\!\cdots\!33\)\( \nu^{8} - \)\(36\!\cdots\!43\)\( \nu^{7} + \)\(64\!\cdots\!36\)\( \nu^{6} + \)\(76\!\cdots\!80\)\( \nu^{5} - \)\(54\!\cdots\!68\)\( \nu^{4} - \)\(40\!\cdots\!60\)\( \nu^{3} + \)\(13\!\cdots\!84\)\( \nu^{2} + \)\(40\!\cdots\!04\)\( \nu - \)\(49\!\cdots\!88\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(61\!\cdots\!93\)\( \nu^{8} + \)\(88\!\cdots\!97\)\( \nu^{7} + \)\(19\!\cdots\!56\)\( \nu^{6} - \)\(20\!\cdots\!20\)\( \nu^{5} - \)\(18\!\cdots\!28\)\( \nu^{4} + \)\(12\!\cdots\!40\)\( \nu^{3} + \)\(51\!\cdots\!64\)\( \nu^{2} - \)\(88\!\cdots\!16\)\( \nu - \)\(12\!\cdots\!48\)\(\)\()/ \)\(24\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(13\!\cdots\!91\)\( \nu^{8} + \)\(74\!\cdots\!61\)\( \nu^{7} - \)\(43\!\cdots\!72\)\( \nu^{6} - \)\(16\!\cdots\!60\)\( \nu^{5} + \)\(44\!\cdots\!36\)\( \nu^{4} + \)\(14\!\cdots\!20\)\( \nu^{3} - \)\(16\!\cdots\!68\)\( \nu^{2} - \)\(52\!\cdots\!08\)\( \nu + \)\(12\!\cdots\!76\)\(\)\()/ \)\(48\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(32\!\cdots\!73\)\( \nu^{8} - \)\(28\!\cdots\!83\)\( \nu^{7} + \)\(36\!\cdots\!16\)\( \nu^{6} + \)\(57\!\cdots\!80\)\( \nu^{5} + \)\(29\!\cdots\!92\)\( \nu^{4} - \)\(50\!\cdots\!60\)\( \nu^{3} - \)\(24\!\cdots\!96\)\( \nu^{2} - \)\(23\!\cdots\!76\)\( \nu - \)\(85\!\cdots\!28\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(50\!\cdots\!73\)\( \nu^{8} + \)\(24\!\cdots\!83\)\( \nu^{7} - \)\(12\!\cdots\!16\)\( \nu^{6} - \)\(65\!\cdots\!80\)\( \nu^{5} + \)\(89\!\cdots\!08\)\( \nu^{4} + \)\(52\!\cdots\!60\)\( \nu^{3} - \)\(25\!\cdots\!04\)\( \nu^{2} - \)\(10\!\cdots\!24\)\( \nu + \)\(44\!\cdots\!28\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 233291 \beta_{2} - 8626466785316074 \beta_{1} + 16733190786889893764973572264708832\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 54 \beta_{6} + 1827020 \beta_{5} - 49231023484 \beta_{4} - 17373914905518889 \beta_{3} + 94357237491658779056302 \beta_{2} + 25450765450947347315780708407499879 \beta_{1} - 144348314535107460480424942609861832895932180528325\)\()/110592\)
\(\nu^{4}\)\(=\)\((\)\(832514701824 \beta_{8} - 2765141756579001 \beta_{7} + 6946973400136530682 \beta_{6} + 21141823508226342818964 \beta_{5} + 1139173923285625331184168092 \beta_{4} + 2147164449978826711610930479658465 \beta_{3} + 2477893574206486168252478874096488413186 \beta_{2} - 32054464969139057704892063162013178160964878990063 \beta_{1} + 26617032122701961034418401648430041724999585885114115987802567375965\)\()/331776\)
\(\nu^{5}\)\(=\)\((\)\(-50427023822868726326442078720 \beta_{8} + 163551734811077361753504119093093 \beta_{7} + 15838759907024084470952284692122958 \beta_{6} + 495340693749304401081259955541927054460 \beta_{5} - 14190224035519964910538321026303280867605484 \beta_{4} - 4585280352109805668271447470351863471821186041965 \beta_{3} + 17892103407323365203630525268140012620340148590969230822 \beta_{2} + 2842237229076009581805829080353737246520310347603912133999408914467 \beta_{1} - 33523342368531939213043617462018388045570540823503359218626770883062512783505091257\)\()/995328\)
\(\nu^{6}\)\(=\)\((\)\(\)\(45\!\cdots\!40\)\( \beta_{8} - \)\(63\!\cdots\!77\)\( \beta_{7} + \)\(15\!\cdots\!18\)\( \beta_{6} + \)\(38\!\cdots\!08\)\( \beta_{5} + \)\(27\!\cdots\!32\)\( \beta_{4} + \)\(30\!\cdots\!13\)\( \beta_{3} + \)\(69\!\cdots\!06\)\( \beta_{2} - \)\(67\!\cdots\!43\)\( \beta_{1} + \)\(33\!\cdots\!81\)\(\)\()/331776\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(40\!\cdots\!20\)\( \beta_{8} + \)\(85\!\cdots\!91\)\( \beta_{7} + \)\(60\!\cdots\!42\)\( \beta_{6} + \)\(33\!\cdots\!68\)\( \beta_{5} - \)\(10\!\cdots\!44\)\( \beta_{4} - \)\(30\!\cdots\!67\)\( \beta_{3} + \)\(66\!\cdots\!18\)\( \beta_{2} + \)\(12\!\cdots\!53\)\( \beta_{1} - \)\(23\!\cdots\!35\)\(\)\()/331776\)
\(\nu^{8}\)\(=\)\((\)\(\)\(35\!\cdots\!88\)\( \beta_{8} - \)\(36\!\cdots\!47\)\( \beta_{7} + \)\(82\!\cdots\!66\)\( \beta_{6} + \)\(15\!\cdots\!92\)\( \beta_{5} + \)\(15\!\cdots\!88\)\( \beta_{4} + \)\(12\!\cdots\!95\)\( \beta_{3} + \)\(42\!\cdots\!90\)\( \beta_{2} - \)\(37\!\cdots\!09\)\( \beta_{1} + \)\(13\!\cdots\!55\)\(\)\()/995328\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.94089e15
−3.29839e15
−2.30101e15
−1.42245e15
5.24199e14
9.66339e14
2.64772e15
3.15688e15
3.66758e15
−1.88610e17 4.27041e26 2.51891e34 −5.60099e39 −8.05442e43 −6.14078e47 −2.79228e51 −6.39314e53 1.05640e57
1.2 −1.57770e17 −1.34800e27 1.45067e34 3.80102e39 2.12674e44 5.08173e47 −6.50350e50 9.95431e53 −5.99686e56
1.3 −1.09896e17 1.02220e27 1.69249e33 3.22293e39 −1.12336e44 −9.90814e46 9.55225e50 2.23221e53 −3.54186e56
1.4 −6.77248e16 −4.98640e26 −5.79795e33 −2.60538e39 3.37703e43 3.08065e47 1.09596e51 −5.73036e53 1.76449e56
1.5 2.57142e16 −9.84309e26 −9.72337e33 1.11425e39 −2.53107e43 −8.63240e47 −5.17060e50 1.47185e53 2.86521e55
1.6 4.69369e16 1.07652e27 −8.18152e33 −1.53406e39 5.05287e43 2.72254e47 −8.71436e50 3.37224e53 −7.20041e55
1.7 1.27643e17 −2.19803e26 5.90823e33 5.13288e39 −2.80563e43 6.06875e47 −5.71378e50 −7.73365e53 6.55179e56
1.8 1.52083e17 −1.34317e27 1.27446e34 −5.29218e39 −2.04274e44 7.47667e47 3.58920e50 9.82432e53 −8.04850e56
1.9 1.76597e17 8.20335e26 2.08018e34 −1.35461e39 1.44868e44 −1.04223e48 1.83964e51 −1.48729e53 −2.39220e56
\(n\): 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 1.114.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.114.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{114}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4973724357874032 T + \)\(18\!\cdots\!32\)\( T^{2} + \)\(31\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!36\)\( T^{4} + \)\(77\!\cdots\!08\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} + \)\(14\!\cdots\!40\)\( T^{7} + \)\(43\!\cdots\!36\)\( T^{8} + \)\(12\!\cdots\!68\)\( T^{9} + \)\(45\!\cdots\!12\)\( T^{10} + \)\(15\!\cdots\!60\)\( T^{11} + \)\(42\!\cdots\!16\)\( T^{12} + \)\(90\!\cdots\!68\)\( T^{13} + \)\(34\!\cdots\!52\)\( T^{14} + \)\(39\!\cdots\!80\)\( T^{15} + \)\(23\!\cdots\!36\)\( T^{16} - \)\(67\!\cdots\!12\)\( T^{17} + \)\(14\!\cdots\!72\)\( T^{18} \)
$3$ \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(39\!\cdots\!03\)\( T^{2} + \)\(39\!\cdots\!20\)\( T^{3} + \)\(88\!\cdots\!56\)\( T^{4} + \)\(78\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!04\)\( T^{9} + \)\(11\!\cdots\!98\)\( T^{10} + \)\(71\!\cdots\!60\)\( T^{11} + \)\(73\!\cdots\!76\)\( T^{12} + \)\(35\!\cdots\!44\)\( T^{13} + \)\(32\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!80\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} + \)\(21\!\cdots\!44\)\( T^{17} + \)\(17\!\cdots\!63\)\( T^{18} \)
$5$ \( 1 + \)\(31\!\cdots\!50\)\( T + \)\(30\!\cdots\!25\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(77\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!50\)\( T^{8} + \)\(22\!\cdots\!00\)\( T^{9} + \)\(86\!\cdots\!50\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(68\!\cdots\!00\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(44\!\cdots\!00\)\( T^{14} + \)\(75\!\cdots\!00\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} + \)\(23\!\cdots\!50\)\( T^{17} + \)\(71\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 + \)\(17\!\cdots\!08\)\( T + \)\(10\!\cdots\!07\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!96\)\( T^{4} + \)\(39\!\cdots\!68\)\( T^{5} + \)\(36\!\cdots\!72\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!06\)\( T^{8} + \)\(72\!\cdots\!48\)\( T^{9} + \)\(46\!\cdots\!42\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!96\)\( T^{12} + \)\(38\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!72\)\( T^{14} + \)\(49\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!01\)\( T^{16} + \)\(16\!\cdots\!08\)\( T^{17} + \)\(29\!\cdots\!07\)\( T^{18} \)
$11$ \( 1 - \)\(12\!\cdots\!88\)\( T + \)\(17\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(60\!\cdots\!68\)\( T^{5} + \)\(53\!\cdots\!24\)\( T^{6} - \)\(17\!\cdots\!44\)\( T^{7} + \)\(16\!\cdots\!18\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{9} + \)\(80\!\cdots\!58\)\( T^{10} - \)\(40\!\cdots\!84\)\( T^{11} + \)\(58\!\cdots\!84\)\( T^{12} - \)\(31\!\cdots\!28\)\( T^{13} + \)\(28\!\cdots\!60\)\( T^{14} - \)\(13\!\cdots\!16\)\( T^{15} + \)\(97\!\cdots\!73\)\( T^{16} - \)\(32\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!71\)\( T^{18} \)
$13$ \( 1 - \)\(11\!\cdots\!46\)\( T + \)\(41\!\cdots\!13\)\( T^{2} - \)\(49\!\cdots\!60\)\( T^{3} + \)\(89\!\cdots\!16\)\( T^{4} - \)\(90\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!46\)\( T^{8} - \)\(97\!\cdots\!56\)\( T^{9} + \)\(96\!\cdots\!38\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} + \)\(53\!\cdots\!56\)\( T^{12} - \)\(28\!\cdots\!36\)\( T^{13} + \)\(21\!\cdots\!88\)\( T^{14} - \)\(88\!\cdots\!40\)\( T^{15} + \)\(55\!\cdots\!81\)\( T^{16} - \)\(11\!\cdots\!06\)\( T^{17} + \)\(75\!\cdots\!33\)\( T^{18} \)
$17$ \( 1 - \)\(10\!\cdots\!62\)\( T + \)\(46\!\cdots\!57\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(17\!\cdots\!88\)\( T^{5} + \)\(23\!\cdots\!52\)\( T^{6} + \)\(50\!\cdots\!20\)\( T^{7} + \)\(32\!\cdots\!46\)\( T^{8} + \)\(82\!\cdots\!08\)\( T^{9} + \)\(35\!\cdots\!02\)\( T^{10} + \)\(60\!\cdots\!80\)\( T^{11} + \)\(30\!\cdots\!56\)\( T^{12} + \)\(25\!\cdots\!68\)\( T^{13} + \)\(19\!\cdots\!12\)\( T^{14} - \)\(52\!\cdots\!60\)\( T^{15} + \)\(90\!\cdots\!81\)\( T^{16} - \)\(21\!\cdots\!02\)\( T^{17} + \)\(23\!\cdots\!77\)\( T^{18} \)
$19$ \( 1 + \)\(20\!\cdots\!60\)\( T + \)\(24\!\cdots\!31\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!16\)\( T^{4} + \)\(44\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!40\)\( T^{7} + \)\(86\!\cdots\!86\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!74\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!44\)\( T^{12} + \)\(44\!\cdots\!80\)\( T^{13} + \)\(87\!\cdots\!84\)\( T^{14} + \)\(44\!\cdots\!20\)\( T^{15} + \)\(77\!\cdots\!89\)\( T^{16} + \)\(20\!\cdots\!60\)\( T^{17} + \)\(31\!\cdots\!39\)\( T^{18} \)
$23$ \( 1 + \)\(16\!\cdots\!84\)\( T + \)\(57\!\cdots\!23\)\( T^{2} + \)\(79\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(24\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!66\)\( T^{8} + \)\(21\!\cdots\!84\)\( T^{9} + \)\(20\!\cdots\!78\)\( T^{10} + \)\(13\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!36\)\( T^{12} + \)\(56\!\cdots\!84\)\( T^{13} + \)\(36\!\cdots\!68\)\( T^{14} + \)\(14\!\cdots\!40\)\( T^{15} + \)\(76\!\cdots\!21\)\( T^{16} + \)\(16\!\cdots\!44\)\( T^{17} + \)\(75\!\cdots\!03\)\( T^{18} \)
$29$ \( 1 - \)\(16\!\cdots\!10\)\( T + \)\(70\!\cdots\!01\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!56\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(78\!\cdots\!96\)\( T^{6} - \)\(41\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!66\)\( T^{8} - \)\(85\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!74\)\( T^{10} - \)\(13\!\cdots\!40\)\( T^{11} + \)\(44\!\cdots\!24\)\( T^{12} - \)\(17\!\cdots\!80\)\( T^{13} + \)\(49\!\cdots\!44\)\( T^{14} - \)\(24\!\cdots\!20\)\( T^{15} + \)\(40\!\cdots\!29\)\( T^{16} - \)\(16\!\cdots\!10\)\( T^{17} + \)\(18\!\cdots\!09\)\( T^{18} \)
$31$ \( 1 - \)\(13\!\cdots\!28\)\( T + \)\(20\!\cdots\!23\)\( T^{2} - \)\(19\!\cdots\!76\)\( T^{3} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(14\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} - \)\(70\!\cdots\!84\)\( T^{7} + \)\(44\!\cdots\!38\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(78\!\cdots\!04\)\( T^{11} + \)\(39\!\cdots\!24\)\( T^{12} - \)\(17\!\cdots\!48\)\( T^{13} + \)\(75\!\cdots\!60\)\( T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(93\!\cdots\!13\)\( T^{16} - \)\(20\!\cdots\!88\)\( T^{17} + \)\(51\!\cdots\!11\)\( T^{18} \)
$37$ \( 1 + \)\(59\!\cdots\!98\)\( T + \)\(57\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!56\)\( T^{4} + \)\(89\!\cdots\!28\)\( T^{5} + \)\(46\!\cdots\!12\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(92\!\cdots\!26\)\( T^{8} + \)\(33\!\cdots\!28\)\( T^{9} + \)\(14\!\cdots\!22\)\( T^{10} + \)\(48\!\cdots\!40\)\( T^{11} + \)\(19\!\cdots\!76\)\( T^{12} + \)\(60\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!20\)\( T^{15} + \)\(16\!\cdots\!41\)\( T^{16} + \)\(27\!\cdots\!78\)\( T^{17} + \)\(72\!\cdots\!17\)\( T^{18} \)
$41$ \( 1 + \)\(34\!\cdots\!02\)\( T + \)\(12\!\cdots\!13\)\( T^{2} + \)\(27\!\cdots\!04\)\( T^{3} + \)\(60\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!04\)\( T^{6} + \)\(28\!\cdots\!96\)\( T^{7} + \)\(43\!\cdots\!98\)\( T^{8} + \)\(57\!\cdots\!80\)\( T^{9} + \)\(76\!\cdots\!58\)\( T^{10} + \)\(87\!\cdots\!36\)\( T^{11} + \)\(10\!\cdots\!44\)\( T^{12} + \)\(99\!\cdots\!92\)\( T^{13} + \)\(10\!\cdots\!60\)\( T^{14} + \)\(79\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!33\)\( T^{16} + \)\(31\!\cdots\!22\)\( T^{17} + \)\(15\!\cdots\!81\)\( T^{18} \)
$43$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(13\!\cdots\!43\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(99\!\cdots\!96\)\( T^{4} - \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!28\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!06\)\( T^{8} - \)\(53\!\cdots\!36\)\( T^{9} + \)\(77\!\cdots\!58\)\( T^{10} - \)\(30\!\cdots\!00\)\( T^{11} + \)\(27\!\cdots\!96\)\( T^{12} - \)\(17\!\cdots\!76\)\( T^{13} + \)\(80\!\cdots\!28\)\( T^{14} - \)\(52\!\cdots\!00\)\( T^{15} + \)\(16\!\cdots\!01\)\( T^{16} - \)\(93\!\cdots\!56\)\( T^{17} + \)\(17\!\cdots\!43\)\( T^{18} \)
$47$ \( 1 + \)\(49\!\cdots\!28\)\( T + \)\(76\!\cdots\!07\)\( T^{2} + \)\(29\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!76\)\( T^{4} + \)\(80\!\cdots\!48\)\( T^{5} + \)\(48\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!66\)\( T^{8} + \)\(14\!\cdots\!88\)\( T^{9} + \)\(55\!\cdots\!82\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{11} + \)\(33\!\cdots\!36\)\( T^{12} + \)\(49\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!32\)\( T^{14} + \)\(14\!\cdots\!60\)\( T^{15} + \)\(32\!\cdots\!21\)\( T^{16} + \)\(18\!\cdots\!68\)\( T^{17} + \)\(33\!\cdots\!87\)\( T^{18} \)
$53$ \( 1 - \)\(45\!\cdots\!26\)\( T + \)\(58\!\cdots\!53\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!56\)\( T^{4} - \)\(46\!\cdots\!16\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} - \)\(59\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!26\)\( T^{8} - \)\(50\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!98\)\( T^{10} - \)\(29\!\cdots\!40\)\( T^{11} + \)\(77\!\cdots\!76\)\( T^{12} - \)\(10\!\cdots\!56\)\( T^{13} + \)\(24\!\cdots\!08\)\( T^{14} - \)\(24\!\cdots\!20\)\( T^{15} + \)\(46\!\cdots\!41\)\( T^{16} - \)\(25\!\cdots\!06\)\( T^{17} + \)\(38\!\cdots\!13\)\( T^{18} \)
$59$ \( 1 + \)\(79\!\cdots\!80\)\( T + \)\(66\!\cdots\!11\)\( T^{2} + \)\(55\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!76\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(50\!\cdots\!76\)\( T^{6} + \)\(34\!\cdots\!20\)\( T^{7} + \)\(84\!\cdots\!06\)\( T^{8} + \)\(50\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!74\)\( T^{10} + \)\(56\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} + \)\(46\!\cdots\!40\)\( T^{13} + \)\(76\!\cdots\!24\)\( T^{14} + \)\(24\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!49\)\( T^{16} + \)\(56\!\cdots\!80\)\( T^{17} + \)\(90\!\cdots\!19\)\( T^{18} \)
$61$ \( 1 + \)\(20\!\cdots\!62\)\( T + \)\(32\!\cdots\!93\)\( T^{2} + \)\(41\!\cdots\!64\)\( T^{3} + \)\(51\!\cdots\!60\)\( T^{4} + \)\(42\!\cdots\!32\)\( T^{5} + \)\(52\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!56\)\( T^{7} + \)\(39\!\cdots\!18\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(21\!\cdots\!58\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{11} + \)\(88\!\cdots\!84\)\( T^{12} + \)\(39\!\cdots\!72\)\( T^{13} + \)\(26\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(50\!\cdots\!73\)\( T^{16} + \)\(17\!\cdots\!42\)\( T^{17} + \)\(47\!\cdots\!21\)\( T^{18} \)
$67$ \( 1 - \)\(24\!\cdots\!12\)\( T + \)\(82\!\cdots\!07\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!16\)\( T^{4} - \)\(62\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!52\)\( T^{6} - \)\(19\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!46\)\( T^{8} - \)\(50\!\cdots\!92\)\( T^{9} + \)\(69\!\cdots\!02\)\( T^{10} - \)\(96\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(15\!\cdots\!32\)\( T^{13} + \)\(18\!\cdots\!12\)\( T^{14} - \)\(20\!\cdots\!60\)\( T^{15} + \)\(21\!\cdots\!81\)\( T^{16} - \)\(14\!\cdots\!52\)\( T^{17} + \)\(13\!\cdots\!27\)\( T^{18} \)
$71$ \( 1 - \)\(91\!\cdots\!08\)\( T + \)\(72\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(25\!\cdots\!60\)\( T^{4} - \)\(73\!\cdots\!68\)\( T^{5} + \)\(60\!\cdots\!84\)\( T^{6} - \)\(22\!\cdots\!64\)\( T^{7} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(43\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!58\)\( T^{10} - \)\(55\!\cdots\!44\)\( T^{11} + \)\(22\!\cdots\!04\)\( T^{12} - \)\(43\!\cdots\!88\)\( T^{13} + \)\(23\!\cdots\!60\)\( T^{14} - \)\(19\!\cdots\!16\)\( T^{15} + \)\(16\!\cdots\!93\)\( T^{16} - \)\(31\!\cdots\!48\)\( T^{17} + \)\(53\!\cdots\!91\)\( T^{18} \)
$73$ \( 1 - \)\(54\!\cdots\!66\)\( T + \)\(14\!\cdots\!73\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!76\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{5} + \)\(76\!\cdots\!28\)\( T^{6} - \)\(78\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!66\)\( T^{8} - \)\(32\!\cdots\!16\)\( T^{9} + \)\(12\!\cdots\!78\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(35\!\cdots\!36\)\( T^{12} - \)\(21\!\cdots\!16\)\( T^{13} + \)\(75\!\cdots\!68\)\( T^{14} - \)\(29\!\cdots\!60\)\( T^{15} + \)\(10\!\cdots\!21\)\( T^{16} - \)\(15\!\cdots\!06\)\( T^{17} + \)\(99\!\cdots\!53\)\( T^{18} \)
$79$ \( 1 + \)\(59\!\cdots\!40\)\( T + \)\(26\!\cdots\!51\)\( T^{2} + \)\(81\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(78\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!66\)\( T^{8} + \)\(30\!\cdots\!00\)\( T^{9} + \)\(52\!\cdots\!74\)\( T^{10} + \)\(91\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!24\)\( T^{12} + \)\(23\!\cdots\!20\)\( T^{13} + \)\(30\!\cdots\!44\)\( T^{14} + \)\(31\!\cdots\!80\)\( T^{15} + \)\(28\!\cdots\!29\)\( T^{16} + \)\(16\!\cdots\!40\)\( T^{17} + \)\(77\!\cdots\!59\)\( T^{18} \)
$83$ \( 1 + \)\(11\!\cdots\!64\)\( T + \)\(81\!\cdots\!83\)\( T^{2} + \)\(42\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!36\)\( T^{4} + \)\(56\!\cdots\!64\)\( T^{5} + \)\(15\!\cdots\!28\)\( T^{6} + \)\(34\!\cdots\!60\)\( T^{7} + \)\(70\!\cdots\!86\)\( T^{8} + \)\(15\!\cdots\!24\)\( T^{9} + \)\(50\!\cdots\!18\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(56\!\cdots\!16\)\( T^{12} + \)\(15\!\cdots\!04\)\( T^{13} + \)\(33\!\cdots\!48\)\( T^{14} + \)\(57\!\cdots\!20\)\( T^{15} + \)\(80\!\cdots\!61\)\( T^{16} + \)\(77\!\cdots\!44\)\( T^{17} + \)\(50\!\cdots\!23\)\( T^{18} \)
$89$ \( 1 + \)\(65\!\cdots\!70\)\( T + \)\(29\!\cdots\!21\)\( T^{2} + \)\(93\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!96\)\( T^{4} + \)\(56\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(32\!\cdots\!46\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{9} + \)\(61\!\cdots\!74\)\( T^{10} + \)\(73\!\cdots\!80\)\( T^{11} + \)\(78\!\cdots\!04\)\( T^{12} + \)\(75\!\cdots\!60\)\( T^{13} + \)\(63\!\cdots\!04\)\( T^{14} + \)\(45\!\cdots\!40\)\( T^{15} + \)\(26\!\cdots\!69\)\( T^{16} + \)\(11\!\cdots\!70\)\( T^{17} + \)\(33\!\cdots\!29\)\( T^{18} \)
$97$ \( 1 - \)\(36\!\cdots\!22\)\( T + \)\(17\!\cdots\!57\)\( T^{2} - \)\(47\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!76\)\( T^{4} - \)\(28\!\cdots\!52\)\( T^{5} + \)\(64\!\cdots\!92\)\( T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(24\!\cdots\!66\)\( T^{8} - \)\(41\!\cdots\!12\)\( T^{9} + \)\(77\!\cdots\!82\)\( T^{10} - \)\(12\!\cdots\!80\)\( T^{11} + \)\(21\!\cdots\!36\)\( T^{12} - \)\(29\!\cdots\!32\)\( T^{13} + \)\(44\!\cdots\!32\)\( T^{14} - \)\(50\!\cdots\!40\)\( T^{15} + \)\(59\!\cdots\!21\)\( T^{16} - \)\(40\!\cdots\!82\)\( T^{17} + \)\(35\!\cdots\!37\)\( T^{18} \)
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