Properties

Label 1.110.a.a
Level 1
Weight 110
Character orbit 1.a
Self dual yes
Analytic conductor 75.239
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 110 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.2394221917\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{118}\cdot 3^{40}\cdot 5^{14}\cdot 7^{6}\cdot 11^{3}\cdot 13 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(286049075864400 - \beta_{1}) q^{2} +(-\)\(99\!\cdots\!00\)\( - 852079581 \beta_{1} + \beta_{2}) q^{3} +(\)\(32\!\cdots\!32\)\( + 1401168417189145 \beta_{1} - 733523 \beta_{2} + \beta_{3}) q^{4} +(-\)\(26\!\cdots\!50\)\( - \)\(24\!\cdots\!63\)\( \beta_{1} + 256532420749 \beta_{2} - 43853 \beta_{3} - \beta_{4}) q^{5} +(\)\(82\!\cdots\!92\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} - 4952444410002432 \beta_{2} + 1320331506 \beta_{3} + 1134 \beta_{4} + \beta_{5}) q^{6} +(\)\(29\!\cdots\!00\)\( + \)\(65\!\cdots\!95\)\( \beta_{1} - 29921845411816752196 \beta_{2} + 5353680223264 \beta_{3} + 7148829 \beta_{4} + 67 \beta_{5} - \beta_{6}) q^{7} +(-\)\(14\!\cdots\!00\)\( - \)\(36\!\cdots\!36\)\( \beta_{1} + \)\(36\!\cdots\!21\)\( \beta_{2} - 526031820577024 \beta_{3} + 23678718391 \beta_{4} - 2011782 \beta_{5} - 79 \beta_{6} - \beta_{7}) q^{8} +(\)\(30\!\cdots\!73\)\( + \)\(76\!\cdots\!54\)\( \beta_{1} - \)\(71\!\cdots\!02\)\( \beta_{2} - 15308494200942822 \beta_{3} - 5285573276346 \beta_{4} + 431210340 \beta_{5} + 40716 \beta_{6} + 216 \beta_{7}) q^{9} +O(q^{10})\) \( q +(286049075864400 - \beta_{1}) q^{2} +(-\)\(99\!\cdots\!00\)\( - 852079581 \beta_{1} + \beta_{2}) q^{3} +(\)\(32\!\cdots\!32\)\( + 1401168417189145 \beta_{1} - 733523 \beta_{2} + \beta_{3}) q^{4} +(-\)\(26\!\cdots\!50\)\( - \)\(24\!\cdots\!63\)\( \beta_{1} + 256532420749 \beta_{2} - 43853 \beta_{3} - \beta_{4}) q^{5} +(\)\(82\!\cdots\!92\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} - 4952444410002432 \beta_{2} + 1320331506 \beta_{3} + 1134 \beta_{4} + \beta_{5}) q^{6} +(\)\(29\!\cdots\!00\)\( + \)\(65\!\cdots\!95\)\( \beta_{1} - 29921845411816752196 \beta_{2} + 5353680223264 \beta_{3} + 7148829 \beta_{4} + 67 \beta_{5} - \beta_{6}) q^{7} +(-\)\(14\!\cdots\!00\)\( - \)\(36\!\cdots\!36\)\( \beta_{1} + \)\(36\!\cdots\!21\)\( \beta_{2} - 526031820577024 \beta_{3} + 23678718391 \beta_{4} - 2011782 \beta_{5} - 79 \beta_{6} - \beta_{7}) q^{8} +(\)\(30\!\cdots\!73\)\( + \)\(76\!\cdots\!54\)\( \beta_{1} - \)\(71\!\cdots\!02\)\( \beta_{2} - 15308494200942822 \beta_{3} - 5285573276346 \beta_{4} + 431210340 \beta_{5} + 40716 \beta_{6} + 216 \beta_{7}) q^{9} +(\)\(22\!\cdots\!00\)\( + \)\(58\!\cdots\!34\)\( \beta_{1} - \)\(44\!\cdots\!32\)\( \beta_{2} - \)\(24\!\cdots\!96\)\( \beta_{3} - 4561081552650912 \beta_{4} + 335860266940 \beta_{5} + 74398280 \beta_{6} + 173880 \beta_{7}) q^{10} +(-\)\(20\!\cdots\!08\)\( - \)\(19\!\cdots\!97\)\( \beta_{1} + \)\(10\!\cdots\!35\)\( \beta_{2} - \)\(21\!\cdots\!64\)\( \beta_{3} - 322715467190875022 \beta_{4} - 14356077251154 \beta_{5} + 36376798246 \beta_{6} + 2070496 \beta_{7}) q^{11} +(-\)\(11\!\cdots\!00\)\( - \)\(12\!\cdots\!52\)\( \beta_{1} + \)\(24\!\cdots\!88\)\( \beta_{2} - \)\(33\!\cdots\!76\)\( \beta_{3} + 18868717616355676584 \beta_{4} - 13421783109153168 \beta_{5} + 3844922465304 \beta_{6} - 2246540184 \beta_{7}) q^{12} +(-\)\(89\!\cdots\!50\)\( + \)\(55\!\cdots\!57\)\( \beta_{1} - \)\(61\!\cdots\!83\)\( \beta_{2} - \)\(76\!\cdots\!77\)\( \beta_{3} - \)\(27\!\cdots\!77\)\( \beta_{4} - 1174394252859531496 \beta_{5} - 194701425963512 \beta_{6} + 196226359056 \beta_{7}) q^{13} +(-\)\(63\!\cdots\!96\)\( - \)\(42\!\cdots\!72\)\( \beta_{1} + \)\(41\!\cdots\!12\)\( \beta_{2} - \)\(56\!\cdots\!64\)\( \beta_{3} + \)\(67\!\cdots\!28\)\( \beta_{4} - 97742305884835341726 \beta_{5} - 6693323875281632 \beta_{6} - 9107509497632 \beta_{7}) q^{14} +(\)\(29\!\cdots\!00\)\( + \)\(97\!\cdots\!97\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} + \)\(11\!\cdots\!32\)\( \beta_{3} + \)\(48\!\cdots\!79\)\( \beta_{4} + \)\(20\!\cdots\!45\)\( \beta_{5} + 657840936324634065 \beta_{6} + 268557631842240 \beta_{7}) q^{15} +(\)\(14\!\cdots\!96\)\( + \)\(17\!\cdots\!20\)\( \beta_{1} - \)\(15\!\cdots\!88\)\( \beta_{2} + \)\(32\!\cdots\!68\)\( \beta_{3} + \)\(33\!\cdots\!60\)\( \beta_{4} + \)\(81\!\cdots\!44\)\( \beta_{5} - 18752777077785451904 \beta_{6} - 4948338306443904 \beta_{7}) q^{16} +(-\)\(10\!\cdots\!50\)\( - \)\(30\!\cdots\!30\)\( \beta_{1} + \)\(26\!\cdots\!62\)\( \beta_{2} - \)\(37\!\cdots\!34\)\( \beta_{3} + \)\(48\!\cdots\!26\)\( \beta_{4} - \)\(20\!\cdots\!52\)\( \beta_{5} + \)\(17\!\cdots\!56\)\( \beta_{6} + 33117702287146680 \beta_{7}) q^{17} +(-\)\(74\!\cdots\!00\)\( - \)\(47\!\cdots\!09\)\( \beta_{1} + \)\(42\!\cdots\!76\)\( \beta_{2} - \)\(20\!\cdots\!44\)\( \beta_{3} - \)\(73\!\cdots\!84\)\( \beta_{4} - \)\(15\!\cdots\!32\)\( \beta_{5} + \)\(42\!\cdots\!96\)\( \beta_{6} + 1288165981990582800 \beta_{7}) q^{18} +(-\)\(14\!\cdots\!20\)\( - \)\(19\!\cdots\!99\)\( \beta_{1} + \)\(13\!\cdots\!33\)\( \beta_{2} - \)\(36\!\cdots\!64\)\( \beta_{3} - \)\(21\!\cdots\!54\)\( \beta_{4} + \)\(96\!\cdots\!02\)\( \beta_{5} - \)\(18\!\cdots\!58\)\( \beta_{6} - 58545559046077715808 \beta_{7}) q^{19} +(-\)\(39\!\cdots\!00\)\( - \)\(33\!\cdots\!26\)\( \beta_{1} + \)\(15\!\cdots\!98\)\( \beta_{2} - \)\(13\!\cdots\!06\)\( \beta_{3} + \)\(30\!\cdots\!48\)\( \beta_{4} - \)\(80\!\cdots\!00\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!00\)\( \beta_{7}) q^{20} +(-\)\(34\!\cdots\!28\)\( - \)\(85\!\cdots\!76\)\( \beta_{1} - \)\(58\!\cdots\!68\)\( \beta_{2} - \)\(92\!\cdots\!24\)\( \beta_{3} + \)\(61\!\cdots\!64\)\( \beta_{4} - \)\(13\!\cdots\!48\)\( \beta_{5} - \)\(41\!\cdots\!56\)\( \beta_{6} - \)\(25\!\cdots\!56\)\( \beta_{7}) q^{21} +(\)\(18\!\cdots\!00\)\( + \)\(17\!\cdots\!88\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} + \)\(58\!\cdots\!62\)\( \beta_{3} - \)\(31\!\cdots\!38\)\( \beta_{4} + \)\(35\!\cdots\!51\)\( \beta_{5} + \)\(21\!\cdots\!72\)\( \beta_{6} + \)\(37\!\cdots\!64\)\( \beta_{7}) q^{22} +(\)\(67\!\cdots\!00\)\( - \)\(18\!\cdots\!35\)\( \beta_{1} + \)\(11\!\cdots\!44\)\( \beta_{2} + \)\(76\!\cdots\!20\)\( \beta_{3} - \)\(58\!\cdots\!05\)\( \beta_{4} - \)\(32\!\cdots\!15\)\( \beta_{5} + \)\(15\!\cdots\!45\)\( \beta_{6} - \)\(44\!\cdots\!40\)\( \beta_{7}) q^{23} +(\)\(66\!\cdots\!40\)\( + \)\(26\!\cdots\!68\)\( \beta_{1} - \)\(10\!\cdots\!12\)\( \beta_{2} + \)\(18\!\cdots\!36\)\( \beta_{3} + \)\(38\!\cdots\!68\)\( \beta_{4} + \)\(41\!\cdots\!08\)\( \beta_{5} - \)\(48\!\cdots\!56\)\( \beta_{6} + \)\(44\!\cdots\!44\)\( \beta_{7}) q^{24} +(\)\(22\!\cdots\!75\)\( - \)\(40\!\cdots\!00\)\( \beta_{1} - \)\(77\!\cdots\!00\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} + \)\(49\!\cdots\!00\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(57\!\cdots\!00\)\( \beta_{6} - \)\(35\!\cdots\!00\)\( \beta_{7}) q^{25} +(-\)\(54\!\cdots\!88\)\( + \)\(12\!\cdots\!54\)\( \beta_{1} - \)\(91\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3} - \)\(52\!\cdots\!36\)\( \beta_{4} - \)\(15\!\cdots\!76\)\( \beta_{5} - \)\(39\!\cdots\!28\)\( \beta_{6} + \)\(22\!\cdots\!72\)\( \beta_{7}) q^{26} +(-\)\(34\!\cdots\!00\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} - \)\(78\!\cdots\!34\)\( \beta_{2} - \)\(37\!\cdots\!64\)\( \beta_{3} - \)\(97\!\cdots\!74\)\( \beta_{4} - \)\(57\!\cdots\!02\)\( \beta_{5} + \)\(11\!\cdots\!06\)\( \beta_{6} - \)\(89\!\cdots\!56\)\( \beta_{7}) q^{27} +(\)\(39\!\cdots\!00\)\( + \)\(69\!\cdots\!12\)\( \beta_{1} - \)\(65\!\cdots\!72\)\( \beta_{2} + \)\(78\!\cdots\!84\)\( \beta_{3} + \)\(30\!\cdots\!44\)\( \beta_{4} + \)\(10\!\cdots\!12\)\( \beta_{5} + \)\(93\!\cdots\!64\)\( \beta_{6} - \)\(84\!\cdots\!04\)\( \beta_{7}) q^{28} +(\)\(37\!\cdots\!70\)\( + \)\(29\!\cdots\!01\)\( \beta_{1} - \)\(16\!\cdots\!55\)\( \beta_{2} + \)\(24\!\cdots\!99\)\( \beta_{3} - \)\(50\!\cdots\!89\)\( \beta_{4} - \)\(71\!\cdots\!24\)\( \beta_{5} - \)\(15\!\cdots\!72\)\( \beta_{6} + \)\(55\!\cdots\!28\)\( \beta_{7}) q^{29} +(-\)\(93\!\cdots\!00\)\( - \)\(39\!\cdots\!56\)\( \beta_{1} + \)\(18\!\cdots\!88\)\( \beta_{2} - \)\(26\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!12\)\( \beta_{4} + \)\(95\!\cdots\!50\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} - \)\(61\!\cdots\!00\)\( \beta_{7}) q^{30} +(-\)\(18\!\cdots\!28\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} + \)\(82\!\cdots\!12\)\( \beta_{2} - \)\(62\!\cdots\!76\)\( \beta_{3} + \)\(40\!\cdots\!80\)\( \beta_{4} + \)\(31\!\cdots\!36\)\( \beta_{5} - \)\(39\!\cdots\!16\)\( \beta_{6} + \)\(44\!\cdots\!84\)\( \beta_{7}) q^{31} +(-\)\(74\!\cdots\!00\)\( - \)\(14\!\cdots\!24\)\( \beta_{1} + \)\(66\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3} + \)\(28\!\cdots\!56\)\( \beta_{4} - \)\(22\!\cdots\!12\)\( \beta_{5} - \)\(37\!\cdots\!64\)\( \beta_{6} - \)\(23\!\cdots\!00\)\( \beta_{7}) q^{32} +(\)\(12\!\cdots\!00\)\( + \)\(42\!\cdots\!22\)\( \beta_{1} + \)\(36\!\cdots\!94\)\( \beta_{2} + \)\(32\!\cdots\!70\)\( \beta_{3} - \)\(22\!\cdots\!10\)\( \beta_{4} + \)\(48\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6} + \)\(82\!\cdots\!76\)\( \beta_{7}) q^{33} +(\)\(28\!\cdots\!24\)\( + \)\(36\!\cdots\!06\)\( \beta_{1} - \)\(17\!\cdots\!16\)\( \beta_{2} + \)\(64\!\cdots\!44\)\( \beta_{3} - \)\(80\!\cdots\!84\)\( \beta_{4} + \)\(29\!\cdots\!52\)\( \beta_{5} - \)\(10\!\cdots\!28\)\( \beta_{6} - \)\(39\!\cdots\!28\)\( \beta_{7}) q^{34} +(-\)\(36\!\cdots\!00\)\( + \)\(48\!\cdots\!84\)\( \beta_{1} - \)\(57\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!96\)\( \beta_{3} + \)\(50\!\cdots\!88\)\( \beta_{4} - \)\(26\!\cdots\!60\)\( \beta_{5} + \)\(33\!\cdots\!80\)\( \beta_{6} - \)\(20\!\cdots\!20\)\( \beta_{7}) q^{35} +(\)\(24\!\cdots\!36\)\( + \)\(17\!\cdots\!09\)\( \beta_{1} - \)\(17\!\cdots\!15\)\( \beta_{2} - \)\(50\!\cdots\!03\)\( \beta_{3} - \)\(10\!\cdots\!96\)\( \beta_{4} + \)\(74\!\cdots\!76\)\( \beta_{5} + \)\(17\!\cdots\!80\)\( \beta_{6} + \)\(19\!\cdots\!80\)\( \beta_{7}) q^{36} +(\)\(41\!\cdots\!50\)\( + \)\(17\!\cdots\!41\)\( \beta_{1} + \)\(12\!\cdots\!29\)\( \beta_{2} + \)\(24\!\cdots\!07\)\( \beta_{3} - \)\(56\!\cdots\!13\)\( \beta_{4} + \)\(89\!\cdots\!76\)\( \beta_{5} - \)\(78\!\cdots\!28\)\( \beta_{6} - \)\(11\!\cdots\!52\)\( \beta_{7}) q^{37} +(\)\(14\!\cdots\!00\)\( + \)\(39\!\cdots\!96\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2} + \)\(61\!\cdots\!18\)\( \beta_{3} + \)\(10\!\cdots\!78\)\( \beta_{4} - \)\(93\!\cdots\!31\)\( \beta_{5} + \)\(40\!\cdots\!68\)\( \beta_{6} + \)\(49\!\cdots\!84\)\( \beta_{7}) q^{38} +(-\)\(96\!\cdots\!84\)\( + \)\(96\!\cdots\!17\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(22\!\cdots\!48\)\( \beta_{3} + \)\(16\!\cdots\!07\)\( \beta_{4} - \)\(15\!\cdots\!31\)\( \beta_{5} - \)\(71\!\cdots\!71\)\( \beta_{6} - \)\(16\!\cdots\!96\)\( \beta_{7}) q^{39} +(-\)\(12\!\cdots\!00\)\( + \)\(95\!\cdots\!60\)\( \beta_{1} - \)\(86\!\cdots\!30\)\( \beta_{2} - \)\(59\!\cdots\!40\)\( \beta_{3} - \)\(65\!\cdots\!30\)\( \beta_{4} + \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!50\)\( \beta_{6} + \)\(41\!\cdots\!50\)\( \beta_{7}) q^{40} +(\)\(32\!\cdots\!62\)\( + \)\(23\!\cdots\!16\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} + \)\(30\!\cdots\!80\)\( \beta_{3} - \)\(10\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!36\)\( \beta_{5} + \)\(34\!\cdots\!20\)\( \beta_{6} - \)\(60\!\cdots\!80\)\( \beta_{7}) q^{41} +(\)\(81\!\cdots\!00\)\( + \)\(98\!\cdots\!08\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} + \)\(27\!\cdots\!84\)\( \beta_{3} + \)\(10\!\cdots\!24\)\( \beta_{4} - \)\(22\!\cdots\!48\)\( \beta_{5} - \)\(13\!\cdots\!56\)\( \beta_{6} + \)\(84\!\cdots\!00\)\( \beta_{7}) q^{42} +(-\)\(39\!\cdots\!00\)\( + \)\(15\!\cdots\!65\)\( \beta_{1} + \)\(39\!\cdots\!15\)\( \beta_{2} - \)\(32\!\cdots\!76\)\( \beta_{3} - \)\(33\!\cdots\!56\)\( \beta_{4} + \)\(15\!\cdots\!12\)\( \beta_{5} + \)\(29\!\cdots\!64\)\( \beta_{6} + \)\(20\!\cdots\!04\)\( \beta_{7}) q^{43} +(-\)\(15\!\cdots\!56\)\( - \)\(14\!\cdots\!40\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2} - \)\(33\!\cdots\!32\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(40\!\cdots\!16\)\( \beta_{5} - \)\(24\!\cdots\!84\)\( \beta_{6} + \)\(14\!\cdots\!16\)\( \beta_{7}) q^{44} +(\)\(35\!\cdots\!50\)\( - \)\(20\!\cdots\!19\)\( \beta_{1} + \)\(20\!\cdots\!37\)\( \beta_{2} + \)\(16\!\cdots\!11\)\( \beta_{3} + \)\(12\!\cdots\!87\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} + \)\(74\!\cdots\!00\)\( \beta_{6} - \)\(84\!\cdots\!00\)\( \beta_{7}) q^{45} +(\)\(18\!\cdots\!32\)\( - \)\(55\!\cdots\!64\)\( \beta_{1} - \)\(26\!\cdots\!36\)\( \beta_{2} + \)\(46\!\cdots\!40\)\( \beta_{3} + \)\(13\!\cdots\!76\)\( \beta_{4} + \)\(50\!\cdots\!14\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7}) q^{46} +(\)\(20\!\cdots\!00\)\( - \)\(31\!\cdots\!82\)\( \beta_{1} - \)\(38\!\cdots\!00\)\( \beta_{2} - \)\(47\!\cdots\!48\)\( \beta_{3} - \)\(30\!\cdots\!38\)\( \beta_{4} + \)\(21\!\cdots\!26\)\( \beta_{5} + \)\(78\!\cdots\!22\)\( \beta_{6} + \)\(11\!\cdots\!92\)\( \beta_{7}) q^{47} +(-\)\(17\!\cdots\!00\)\( - \)\(12\!\cdots\!12\)\( \beta_{1} + \)\(69\!\cdots\!44\)\( \beta_{2} - \)\(29\!\cdots\!36\)\( \beta_{3} - \)\(76\!\cdots\!96\)\( \beta_{4} - \)\(92\!\cdots\!08\)\( \beta_{5} - \)\(26\!\cdots\!76\)\( \beta_{6} - \)\(88\!\cdots\!20\)\( \beta_{7}) q^{48} +(\)\(19\!\cdots\!57\)\( - \)\(15\!\cdots\!56\)\( \beta_{1} + \)\(31\!\cdots\!56\)\( \beta_{2} - \)\(59\!\cdots\!60\)\( \beta_{3} + \)\(21\!\cdots\!04\)\( \beta_{4} + \)\(33\!\cdots\!16\)\( \beta_{5} + \)\(43\!\cdots\!40\)\( \beta_{6} + \)\(33\!\cdots\!40\)\( \beta_{7}) q^{49} +(\)\(40\!\cdots\!00\)\( - \)\(15\!\cdots\!75\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4} - \)\(41\!\cdots\!00\)\( \beta_{5} + \)\(44\!\cdots\!00\)\( \beta_{6} - \)\(58\!\cdots\!00\)\( \beta_{7}) q^{50} +(\)\(61\!\cdots\!32\)\( + \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(33\!\cdots\!82\)\( \beta_{2} + \)\(56\!\cdots\!36\)\( \beta_{3} - \)\(22\!\cdots\!34\)\( \beta_{4} - \)\(23\!\cdots\!98\)\( \beta_{5} - \)\(44\!\cdots\!78\)\( \beta_{6} - \)\(88\!\cdots\!28\)\( \beta_{7}) q^{51} +(-\)\(67\!\cdots\!00\)\( + \)\(11\!\cdots\!54\)\( \beta_{1} - \)\(97\!\cdots\!82\)\( \beta_{2} - \)\(26\!\cdots\!38\)\( \beta_{3} - \)\(94\!\cdots\!28\)\( \beta_{4} - \)\(16\!\cdots\!44\)\( \beta_{5} + \)\(10\!\cdots\!32\)\( \beta_{6} + \)\(98\!\cdots\!52\)\( \beta_{7}) q^{52} +(\)\(11\!\cdots\!50\)\( + \)\(14\!\cdots\!17\)\( \beta_{1} - \)\(43\!\cdots\!11\)\( \beta_{2} - \)\(22\!\cdots\!69\)\( \beta_{3} + \)\(40\!\cdots\!71\)\( \beta_{4} + \)\(19\!\cdots\!08\)\( \beta_{5} - \)\(93\!\cdots\!24\)\( \beta_{6} - \)\(33\!\cdots\!16\)\( \beta_{7}) q^{53} +(-\)\(30\!\cdots\!20\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} + \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(30\!\cdots\!40\)\( \beta_{3} - \)\(82\!\cdots\!40\)\( \beta_{4} - \)\(62\!\cdots\!46\)\( \beta_{5} + \)\(68\!\cdots\!36\)\( \beta_{6} + \)\(50\!\cdots\!36\)\( \beta_{7}) q^{54} +(\)\(14\!\cdots\!00\)\( - \)\(31\!\cdots\!21\)\( \beta_{1} + \)\(19\!\cdots\!08\)\( \beta_{2} + \)\(54\!\cdots\!24\)\( \beta_{3} - \)\(15\!\cdots\!67\)\( \beta_{4} + \)\(90\!\cdots\!75\)\( \beta_{5} - \)\(12\!\cdots\!25\)\( \beta_{6} + \)\(75\!\cdots\!00\)\( \beta_{7}) q^{55} +(-\)\(25\!\cdots\!20\)\( - \)\(71\!\cdots\!20\)\( \beta_{1} + \)\(11\!\cdots\!12\)\( \beta_{2} - \)\(49\!\cdots\!48\)\( \beta_{3} - \)\(24\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(61\!\cdots\!28\)\( \beta_{6} - \)\(64\!\cdots\!72\)\( \beta_{7}) q^{56} +(\)\(28\!\cdots\!00\)\( - \)\(28\!\cdots\!22\)\( \beta_{1} - \)\(18\!\cdots\!02\)\( \beta_{2} - \)\(90\!\cdots\!54\)\( \beta_{3} + \)\(17\!\cdots\!26\)\( \beta_{4} + \)\(79\!\cdots\!48\)\( \beta_{5} - \)\(81\!\cdots\!44\)\( \beta_{6} + \)\(17\!\cdots\!16\)\( \beta_{7}) q^{57} +(-\)\(28\!\cdots\!00\)\( - \)\(22\!\cdots\!74\)\( \beta_{1} - \)\(41\!\cdots\!48\)\( \beta_{2} - \)\(53\!\cdots\!44\)\( \beta_{3} - \)\(57\!\cdots\!44\)\( \beta_{4} - \)\(25\!\cdots\!12\)\( \beta_{5} - \)\(31\!\cdots\!64\)\( \beta_{6} - \)\(14\!\cdots\!68\)\( \beta_{7}) q^{58} +(\)\(52\!\cdots\!40\)\( - \)\(18\!\cdots\!11\)\( \beta_{1} - \)\(44\!\cdots\!73\)\( \beta_{2} + \)\(69\!\cdots\!36\)\( \beta_{3} - \)\(29\!\cdots\!96\)\( \beta_{4} + \)\(64\!\cdots\!72\)\( \beta_{5} + \)\(17\!\cdots\!84\)\( \beta_{6} - \)\(53\!\cdots\!16\)\( \beta_{7}) q^{59} +(\)\(19\!\cdots\!00\)\( + \)\(23\!\cdots\!44\)\( \beta_{1} - \)\(12\!\cdots\!12\)\( \beta_{2} + \)\(64\!\cdots\!64\)\( \beta_{3} - \)\(26\!\cdots\!92\)\( \beta_{4} + \)\(18\!\cdots\!40\)\( \beta_{5} - \)\(31\!\cdots\!20\)\( \beta_{6} + \)\(22\!\cdots\!80\)\( \beta_{7}) q^{60} +(-\)\(37\!\cdots\!58\)\( + \)\(15\!\cdots\!77\)\( \beta_{1} + \)\(93\!\cdots\!89\)\( \beta_{2} - \)\(57\!\cdots\!93\)\( \beta_{3} + \)\(11\!\cdots\!67\)\( \beta_{4} - \)\(48\!\cdots\!04\)\( \beta_{5} - \)\(21\!\cdots\!08\)\( \beta_{6} - \)\(35\!\cdots\!08\)\( \beta_{7}) q^{61} +(-\)\(12\!\cdots\!00\)\( + \)\(64\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2} - \)\(36\!\cdots\!36\)\( \beta_{3} - \)\(11\!\cdots\!96\)\( \beta_{4} + \)\(28\!\cdots\!92\)\( \beta_{5} + \)\(11\!\cdots\!24\)\( \beta_{6} - \)\(92\!\cdots\!80\)\( \beta_{7}) q^{62} +(\)\(18\!\cdots\!00\)\( + \)\(15\!\cdots\!59\)\( \beta_{1} - \)\(26\!\cdots\!16\)\( \beta_{2} + \)\(10\!\cdots\!92\)\( \beta_{3} + \)\(47\!\cdots\!97\)\( \beta_{4} + \)\(21\!\cdots\!31\)\( \beta_{5} - \)\(14\!\cdots\!93\)\( \beta_{6} + \)\(13\!\cdots\!48\)\( \beta_{7}) q^{63} +(\)\(43\!\cdots\!12\)\( - \)\(91\!\cdots\!12\)\( \beta_{1} - \)\(10\!\cdots\!04\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3} - \)\(11\!\cdots\!32\)\( \beta_{4} + \)\(32\!\cdots\!52\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!00\)\( \beta_{7}) q^{64} +(\)\(59\!\cdots\!00\)\( - \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(23\!\cdots\!36\)\( \beta_{2} + \)\(25\!\cdots\!92\)\( \beta_{3} + \)\(38\!\cdots\!24\)\( \beta_{4} - \)\(12\!\cdots\!80\)\( \beta_{5} - \)\(41\!\cdots\!60\)\( \beta_{6} - \)\(19\!\cdots\!60\)\( \beta_{7}) q^{65} +(-\)\(40\!\cdots\!36\)\( - \)\(35\!\cdots\!72\)\( \beta_{1} + \)\(48\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!32\)\( \beta_{3} - \)\(34\!\cdots\!12\)\( \beta_{4} + \)\(28\!\cdots\!76\)\( \beta_{5} + \)\(48\!\cdots\!56\)\( \beta_{6} - \)\(76\!\cdots\!44\)\( \beta_{7}) q^{66} +(-\)\(30\!\cdots\!00\)\( - \)\(18\!\cdots\!79\)\( \beta_{1} + \)\(65\!\cdots\!09\)\( \beta_{2} - \)\(14\!\cdots\!28\)\( \beta_{3} + \)\(17\!\cdots\!62\)\( \beta_{4} + \)\(67\!\cdots\!26\)\( \beta_{5} - \)\(10\!\cdots\!78\)\( \beta_{6} + \)\(44\!\cdots\!96\)\( \beta_{7}) q^{67} +(-\)\(28\!\cdots\!00\)\( - \)\(72\!\cdots\!06\)\( \beta_{1} + \)\(34\!\cdots\!54\)\( \beta_{2} - \)\(21\!\cdots\!26\)\( \beta_{3} - \)\(25\!\cdots\!16\)\( \beta_{4} - \)\(12\!\cdots\!68\)\( \beta_{5} - \)\(59\!\cdots\!96\)\( \beta_{6} - \)\(12\!\cdots\!44\)\( \beta_{7}) q^{68} +(\)\(26\!\cdots\!76\)\( + \)\(23\!\cdots\!40\)\( \beta_{1} - \)\(27\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(58\!\cdots\!80\)\( \beta_{4} - \)\(73\!\cdots\!04\)\( \beta_{5} + \)\(74\!\cdots\!24\)\( \beta_{6} - \)\(13\!\cdots\!76\)\( \beta_{7}) q^{69} +(-\)\(48\!\cdots\!00\)\( + \)\(12\!\cdots\!68\)\( \beta_{1} - \)\(22\!\cdots\!64\)\( \beta_{2} + \)\(83\!\cdots\!08\)\( \beta_{3} + \)\(28\!\cdots\!36\)\( \beta_{4} + \)\(38\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(92\!\cdots\!00\)\( \beta_{7}) q^{70} +(-\)\(22\!\cdots\!68\)\( + \)\(65\!\cdots\!91\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} - \)\(10\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!39\)\( \beta_{4} + \)\(93\!\cdots\!43\)\( \beta_{5} - \)\(13\!\cdots\!89\)\( \beta_{6} - \)\(24\!\cdots\!64\)\( \beta_{7}) q^{71} +(-\)\(11\!\cdots\!00\)\( - \)\(68\!\cdots\!56\)\( \beta_{1} + \)\(49\!\cdots\!89\)\( \beta_{2} - \)\(16\!\cdots\!92\)\( \beta_{3} - \)\(21\!\cdots\!37\)\( \beta_{4} - \)\(18\!\cdots\!26\)\( \beta_{5} + \)\(27\!\cdots\!53\)\( \beta_{6} + \)\(11\!\cdots\!15\)\( \beta_{7}) q^{72} +(-\)\(12\!\cdots\!50\)\( - \)\(21\!\cdots\!18\)\( \beta_{1} + \)\(12\!\cdots\!82\)\( \beta_{2} + \)\(12\!\cdots\!58\)\( \beta_{3} - \)\(42\!\cdots\!82\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5} + \)\(10\!\cdots\!08\)\( \beta_{6} + \)\(89\!\cdots\!44\)\( \beta_{7}) q^{73} +(-\)\(16\!\cdots\!36\)\( - \)\(20\!\cdots\!62\)\( \beta_{1} + \)\(99\!\cdots\!28\)\( \beta_{2} + \)\(49\!\cdots\!92\)\( \beta_{3} + \)\(44\!\cdots\!68\)\( \beta_{4} + \)\(80\!\cdots\!00\)\( \beta_{5} - \)\(86\!\cdots\!48\)\( \beta_{6} - \)\(23\!\cdots\!48\)\( \beta_{7}) q^{74} +(-\)\(73\!\cdots\!00\)\( - \)\(19\!\cdots\!75\)\( \beta_{1} - \)\(20\!\cdots\!25\)\( \beta_{2} + \)\(74\!\cdots\!00\)\( \beta_{3} - \)\(59\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!00\)\( \beta_{5} - \)\(27\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7}) q^{75} +(-\)\(28\!\cdots\!40\)\( - \)\(65\!\cdots\!20\)\( \beta_{1} - \)\(78\!\cdots\!36\)\( \beta_{2} - \)\(38\!\cdots\!16\)\( \beta_{3} - \)\(24\!\cdots\!00\)\( \beta_{4} + \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(14\!\cdots\!56\)\( \beta_{6} + \)\(54\!\cdots\!56\)\( \beta_{7}) q^{76} +(-\)\(55\!\cdots\!00\)\( + \)\(45\!\cdots\!96\)\( \beta_{1} - \)\(11\!\cdots\!12\)\( \beta_{2} - \)\(31\!\cdots\!64\)\( \beta_{3} - \)\(90\!\cdots\!04\)\( \beta_{4} - \)\(11\!\cdots\!92\)\( \beta_{5} - \)\(14\!\cdots\!24\)\( \beta_{6} - \)\(10\!\cdots\!40\)\( \beta_{7}) q^{77} +(-\)\(93\!\cdots\!00\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(58\!\cdots\!16\)\( \beta_{4} + \)\(18\!\cdots\!18\)\( \beta_{5} - \)\(34\!\cdots\!04\)\( \beta_{6} - \)\(50\!\cdots\!72\)\( \beta_{7}) q^{78} +(-\)\(11\!\cdots\!80\)\( + \)\(52\!\cdots\!10\)\( \beta_{1} + \)\(81\!\cdots\!20\)\( \beta_{2} + \)\(29\!\cdots\!92\)\( \beta_{3} - \)\(98\!\cdots\!30\)\( \beta_{4} + \)\(25\!\cdots\!14\)\( \beta_{5} + \)\(10\!\cdots\!66\)\( \beta_{6} + \)\(30\!\cdots\!16\)\( \beta_{7}) q^{79} +(-\)\(67\!\cdots\!00\)\( + \)\(33\!\cdots\!32\)\( \beta_{1} + \)\(24\!\cdots\!64\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3} - \)\(18\!\cdots\!36\)\( \beta_{4} - \)\(85\!\cdots\!00\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} + \)\(30\!\cdots\!00\)\( \beta_{7}) q^{80} +(-\)\(80\!\cdots\!19\)\( - \)\(81\!\cdots\!54\)\( \beta_{1} - \)\(27\!\cdots\!86\)\( \beta_{2} + \)\(35\!\cdots\!42\)\( \beta_{3} + \)\(98\!\cdots\!46\)\( \beta_{4} - \)\(54\!\cdots\!32\)\( \beta_{5} - \)\(97\!\cdots\!24\)\( \beta_{6} - \)\(24\!\cdots\!24\)\( \beta_{7}) q^{81} +(-\)\(23\!\cdots\!00\)\( - \)\(11\!\cdots\!06\)\( \beta_{1} - \)\(70\!\cdots\!24\)\( \beta_{2} - \)\(20\!\cdots\!08\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4} + \)\(68\!\cdots\!56\)\( \beta_{5} - \)\(24\!\cdots\!68\)\( \beta_{6} + \)\(27\!\cdots\!88\)\( \beta_{7}) q^{82} +(-\)\(10\!\cdots\!00\)\( - \)\(20\!\cdots\!37\)\( \beta_{1} - \)\(22\!\cdots\!27\)\( \beta_{2} + \)\(37\!\cdots\!60\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4} + \)\(67\!\cdots\!80\)\( \beta_{5} + \)\(93\!\cdots\!60\)\( \beta_{6} + \)\(71\!\cdots\!60\)\( \beta_{7}) q^{83} +(-\)\(73\!\cdots\!96\)\( - \)\(23\!\cdots\!52\)\( \beta_{1} + \)\(25\!\cdots\!44\)\( \beta_{2} - \)\(33\!\cdots\!84\)\( \beta_{3} - \)\(23\!\cdots\!52\)\( \beta_{4} - \)\(16\!\cdots\!08\)\( \beta_{5} + \)\(79\!\cdots\!80\)\( \beta_{6} - \)\(27\!\cdots\!20\)\( \beta_{7}) q^{84} +(-\)\(58\!\cdots\!00\)\( + \)\(67\!\cdots\!34\)\( \beta_{1} + \)\(10\!\cdots\!18\)\( \beta_{2} + \)\(12\!\cdots\!54\)\( \beta_{3} - \)\(11\!\cdots\!62\)\( \beta_{4} - \)\(46\!\cdots\!60\)\( \beta_{5} - \)\(67\!\cdots\!20\)\( \beta_{6} + \)\(31\!\cdots\!80\)\( \beta_{7}) q^{85} +(-\)\(14\!\cdots\!28\)\( + \)\(26\!\cdots\!80\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!54\)\( \beta_{3} - \)\(22\!\cdots\!90\)\( \beta_{4} + \)\(50\!\cdots\!63\)\( \beta_{5} + \)\(83\!\cdots\!52\)\( \beta_{6} + \)\(28\!\cdots\!52\)\( \beta_{7}) q^{86} +(-\)\(18\!\cdots\!00\)\( + \)\(19\!\cdots\!37\)\( \beta_{1} + \)\(88\!\cdots\!16\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!55\)\( \beta_{4} - \)\(46\!\cdots\!35\)\( \beta_{5} + \)\(57\!\cdots\!05\)\( \beta_{6} - \)\(15\!\cdots\!24\)\( \beta_{7}) q^{87} +(\)\(19\!\cdots\!00\)\( + \)\(28\!\cdots\!40\)\( \beta_{1} - \)\(40\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!44\)\( \beta_{3} + \)\(78\!\cdots\!84\)\( \beta_{4} + \)\(61\!\cdots\!32\)\( \beta_{5} - \)\(19\!\cdots\!96\)\( \beta_{6} + \)\(28\!\cdots\!60\)\( \beta_{7}) q^{88} +(\)\(67\!\cdots\!10\)\( + \)\(39\!\cdots\!66\)\( \beta_{1} - \)\(34\!\cdots\!54\)\( \beta_{2} - \)\(75\!\cdots\!70\)\( \beta_{3} - \)\(68\!\cdots\!94\)\( \beta_{4} - \)\(31\!\cdots\!68\)\( \beta_{5} + \)\(80\!\cdots\!52\)\( \beta_{6} - \)\(25\!\cdots\!48\)\( \beta_{7}) q^{89} +(\)\(20\!\cdots\!00\)\( - \)\(96\!\cdots\!58\)\( \beta_{1} - \)\(69\!\cdots\!16\)\( \beta_{2} + \)\(28\!\cdots\!52\)\( \beta_{3} + \)\(34\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!20\)\( \beta_{5} - \)\(39\!\cdots\!60\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7}) q^{90} +(\)\(24\!\cdots\!92\)\( - \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(48\!\cdots\!32\)\( \beta_{2} - \)\(30\!\cdots\!88\)\( \beta_{3} - \)\(84\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!72\)\( \beta_{5} + \)\(17\!\cdots\!08\)\( \beta_{6} + \)\(19\!\cdots\!08\)\( \beta_{7}) q^{91} +(\)\(49\!\cdots\!00\)\( - \)\(38\!\cdots\!56\)\( \beta_{1} + \)\(58\!\cdots\!48\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} + \)\(53\!\cdots\!32\)\( \beta_{4} - \)\(23\!\cdots\!64\)\( \beta_{5} - \)\(92\!\cdots\!08\)\( \beta_{6} - \)\(41\!\cdots\!52\)\( \beta_{7}) q^{92} +(\)\(71\!\cdots\!00\)\( - \)\(16\!\cdots\!96\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} - \)\(12\!\cdots\!76\)\( \beta_{3} - \)\(37\!\cdots\!96\)\( \beta_{4} + \)\(23\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!76\)\( \beta_{6} + \)\(17\!\cdots\!92\)\( \beta_{7}) q^{93} +(\)\(30\!\cdots\!84\)\( + \)\(32\!\cdots\!00\)\( \beta_{1} - \)\(98\!\cdots\!44\)\( \beta_{2} + \)\(20\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!04\)\( \beta_{5} + \)\(90\!\cdots\!96\)\( \beta_{6} + \)\(11\!\cdots\!96\)\( \beta_{7}) q^{94} +(\)\(46\!\cdots\!00\)\( + \)\(13\!\cdots\!45\)\( \beta_{1} - \)\(22\!\cdots\!60\)\( \beta_{2} + \)\(58\!\cdots\!20\)\( \beta_{3} + \)\(55\!\cdots\!15\)\( \beta_{4} + \)\(57\!\cdots\!25\)\( \beta_{5} + \)\(12\!\cdots\!25\)\( \beta_{6} - \)\(24\!\cdots\!00\)\( \beta_{7}) q^{95} +(\)\(76\!\cdots\!52\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(63\!\cdots\!36\)\( \beta_{3} + \)\(87\!\cdots\!32\)\( \beta_{4} - \)\(47\!\cdots\!72\)\( \beta_{5} - \)\(21\!\cdots\!80\)\( \beta_{6} + \)\(68\!\cdots\!20\)\( \beta_{7}) q^{96} +(\)\(38\!\cdots\!50\)\( + \)\(43\!\cdots\!14\)\( \beta_{1} - \)\(63\!\cdots\!86\)\( \beta_{2} - \)\(10\!\cdots\!74\)\( \beta_{3} + \)\(62\!\cdots\!66\)\( \beta_{4} - \)\(19\!\cdots\!32\)\( \beta_{5} - \)\(67\!\cdots\!04\)\( \beta_{6} + \)\(38\!\cdots\!04\)\( \beta_{7}) q^{97} +(\)\(15\!\cdots\!00\)\( - \)\(16\!\cdots\!93\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} - \)\(12\!\cdots\!32\)\( \beta_{4} + \)\(22\!\cdots\!64\)\( \beta_{5} + \)\(17\!\cdots\!08\)\( \beta_{6} - \)\(29\!\cdots\!68\)\( \beta_{7}) q^{98} +(\)\(89\!\cdots\!16\)\( - \)\(62\!\cdots\!35\)\( \beta_{1} + \)\(91\!\cdots\!07\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} - \)\(14\!\cdots\!00\)\( \beta_{4} - \)\(10\!\cdots\!48\)\( \beta_{5} + \)\(44\!\cdots\!48\)\( \beta_{6} - \)\(58\!\cdots\!52\)\( \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2288392606915200q^{2} - \)\(79\!\cdots\!00\)\(q^{3} + \)\(25\!\cdots\!56\)\(q^{4} - \)\(21\!\cdots\!00\)\(q^{5} + \)\(65\!\cdots\!36\)\(q^{6} + \)\(23\!\cdots\!00\)\(q^{7} - \)\(11\!\cdots\!00\)\(q^{8} + \)\(24\!\cdots\!84\)\(q^{9} + O(q^{10}) \) \( 8q + 2288392606915200q^{2} - \)\(79\!\cdots\!00\)\(q^{3} + \)\(25\!\cdots\!56\)\(q^{4} - \)\(21\!\cdots\!00\)\(q^{5} + \)\(65\!\cdots\!36\)\(q^{6} + \)\(23\!\cdots\!00\)\(q^{7} - \)\(11\!\cdots\!00\)\(q^{8} + \)\(24\!\cdots\!84\)\(q^{9} + \)\(18\!\cdots\!00\)\(q^{10} - \)\(16\!\cdots\!64\)\(q^{11} - \)\(91\!\cdots\!00\)\(q^{12} - \)\(71\!\cdots\!00\)\(q^{13} - \)\(50\!\cdots\!68\)\(q^{14} + \)\(23\!\cdots\!00\)\(q^{15} + \)\(11\!\cdots\!68\)\(q^{16} - \)\(86\!\cdots\!00\)\(q^{17} - \)\(59\!\cdots\!00\)\(q^{18} - \)\(11\!\cdots\!60\)\(q^{19} - \)\(31\!\cdots\!00\)\(q^{20} - \)\(27\!\cdots\!24\)\(q^{21} + \)\(14\!\cdots\!00\)\(q^{22} + \)\(53\!\cdots\!00\)\(q^{23} + \)\(53\!\cdots\!20\)\(q^{24} + \)\(18\!\cdots\!00\)\(q^{25} - \)\(43\!\cdots\!04\)\(q^{26} - \)\(27\!\cdots\!00\)\(q^{27} + \)\(31\!\cdots\!00\)\(q^{28} + \)\(29\!\cdots\!60\)\(q^{29} - \)\(75\!\cdots\!00\)\(q^{30} - \)\(14\!\cdots\!24\)\(q^{31} - \)\(59\!\cdots\!00\)\(q^{32} + \)\(98\!\cdots\!00\)\(q^{33} + \)\(23\!\cdots\!92\)\(q^{34} - \)\(28\!\cdots\!00\)\(q^{35} + \)\(19\!\cdots\!88\)\(q^{36} + \)\(33\!\cdots\!00\)\(q^{37} + \)\(11\!\cdots\!00\)\(q^{38} - \)\(77\!\cdots\!72\)\(q^{39} - \)\(10\!\cdots\!00\)\(q^{40} + \)\(26\!\cdots\!96\)\(q^{41} + \)\(65\!\cdots\!00\)\(q^{42} - \)\(31\!\cdots\!00\)\(q^{43} - \)\(12\!\cdots\!48\)\(q^{44} + \)\(28\!\cdots\!00\)\(q^{45} + \)\(14\!\cdots\!56\)\(q^{46} + \)\(16\!\cdots\!00\)\(q^{47} - \)\(14\!\cdots\!00\)\(q^{48} + \)\(15\!\cdots\!56\)\(q^{49} + \)\(32\!\cdots\!00\)\(q^{50} + \)\(48\!\cdots\!56\)\(q^{51} - \)\(54\!\cdots\!00\)\(q^{52} + \)\(90\!\cdots\!00\)\(q^{53} - \)\(24\!\cdots\!60\)\(q^{54} + \)\(11\!\cdots\!00\)\(q^{55} - \)\(20\!\cdots\!60\)\(q^{56} + \)\(22\!\cdots\!00\)\(q^{57} - \)\(22\!\cdots\!00\)\(q^{58} + \)\(42\!\cdots\!20\)\(q^{59} + \)\(15\!\cdots\!00\)\(q^{60} - \)\(30\!\cdots\!64\)\(q^{61} - \)\(10\!\cdots\!00\)\(q^{62} + \)\(14\!\cdots\!00\)\(q^{63} + \)\(35\!\cdots\!96\)\(q^{64} + \)\(47\!\cdots\!00\)\(q^{65} - \)\(32\!\cdots\!88\)\(q^{66} - \)\(24\!\cdots\!00\)\(q^{67} - \)\(22\!\cdots\!00\)\(q^{68} + \)\(20\!\cdots\!08\)\(q^{69} - \)\(38\!\cdots\!00\)\(q^{70} - \)\(18\!\cdots\!44\)\(q^{71} - \)\(93\!\cdots\!00\)\(q^{72} - \)\(99\!\cdots\!00\)\(q^{73} - \)\(13\!\cdots\!88\)\(q^{74} - \)\(58\!\cdots\!00\)\(q^{75} - \)\(23\!\cdots\!20\)\(q^{76} - \)\(44\!\cdots\!00\)\(q^{77} - \)\(74\!\cdots\!00\)\(q^{78} - \)\(90\!\cdots\!40\)\(q^{79} - \)\(53\!\cdots\!00\)\(q^{80} - \)\(64\!\cdots\!52\)\(q^{81} - \)\(18\!\cdots\!00\)\(q^{82} - \)\(86\!\cdots\!00\)\(q^{83} - \)\(58\!\cdots\!68\)\(q^{84} - \)\(46\!\cdots\!00\)\(q^{85} - \)\(11\!\cdots\!24\)\(q^{86} - \)\(14\!\cdots\!00\)\(q^{87} + \)\(15\!\cdots\!00\)\(q^{88} + \)\(53\!\cdots\!80\)\(q^{89} + \)\(16\!\cdots\!00\)\(q^{90} + \)\(19\!\cdots\!36\)\(q^{91} + \)\(39\!\cdots\!00\)\(q^{92} + \)\(56\!\cdots\!00\)\(q^{93} + \)\(24\!\cdots\!72\)\(q^{94} + \)\(37\!\cdots\!00\)\(q^{95} + \)\(61\!\cdots\!16\)\(q^{96} + \)\(31\!\cdots\!00\)\(q^{97} + \)\(12\!\cdots\!00\)\(q^{98} + \)\(71\!\cdots\!28\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - \)\(10\!\cdots\!01\)\( x^{6} - \)\(72\!\cdots\!28\)\( x^{5} + \)\(31\!\cdots\!56\)\( x^{4} + \)\(37\!\cdots\!80\)\( x^{3} - \)\(26\!\cdots\!00\)\( x^{2} - \)\(71\!\cdots\!00\)\( x + \)\(46\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 192 \nu - 48 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(18\!\cdots\!09\)\( \nu^{7} + \)\(28\!\cdots\!72\)\( \nu^{6} + \)\(15\!\cdots\!61\)\( \nu^{5} - \)\(22\!\cdots\!90\)\( \nu^{4} - \)\(26\!\cdots\!64\)\( \nu^{3} + \)\(33\!\cdots\!04\)\( \nu^{2} + \)\(52\!\cdots\!64\)\( \nu - \)\(59\!\cdots\!64\)\(\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!01\)\( \nu^{7} + \)\(29\!\cdots\!08\)\( \nu^{6} + \)\(16\!\cdots\!29\)\( \nu^{5} - \)\(23\!\cdots\!10\)\( \nu^{4} - \)\(27\!\cdots\!96\)\( \nu^{3} + \)\(35\!\cdots\!08\)\( \nu^{2} + \)\(49\!\cdots\!00\)\( \nu - \)\(20\!\cdots\!36\)\(\)\()/ \)\(15\!\cdots\!68\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(25\!\cdots\!07\)\( \nu^{7} + \)\(86\!\cdots\!88\)\( \nu^{6} + \)\(24\!\cdots\!91\)\( \nu^{5} - \)\(34\!\cdots\!66\)\( \nu^{4} - \)\(58\!\cdots\!80\)\( \nu^{3} - \)\(91\!\cdots\!80\)\( \nu^{2} + \)\(31\!\cdots\!00\)\( \nu + \)\(99\!\cdots\!60\)\(\)\()/ \)\(64\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(37\!\cdots\!77\)\( \nu^{7} + \)\(56\!\cdots\!08\)\( \nu^{6} + \)\(31\!\cdots\!21\)\( \nu^{5} - \)\(44\!\cdots\!06\)\( \nu^{4} - \)\(54\!\cdots\!20\)\( \nu^{3} + \)\(66\!\cdots\!20\)\( \nu^{2} + \)\(23\!\cdots\!20\)\( \nu - \)\(11\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(29\!\cdots\!71\)\( \nu^{7} + \)\(42\!\cdots\!44\)\( \nu^{6} + \)\(24\!\cdots\!63\)\( \nu^{5} - \)\(33\!\cdots\!98\)\( \nu^{4} - \)\(44\!\cdots\!60\)\( \nu^{3} + \)\(49\!\cdots\!80\)\( \nu^{2} + \)\(45\!\cdots\!40\)\( \nu - \)\(92\!\cdots\!20\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(21\!\cdots\!61\)\( \nu^{7} - \)\(32\!\cdots\!28\)\( \nu^{6} - \)\(18\!\cdots\!81\)\( \nu^{5} + \)\(25\!\cdots\!70\)\( \nu^{4} + \)\(32\!\cdots\!48\)\( \nu^{3} - \)\(38\!\cdots\!80\)\( \nu^{2} - \)\(31\!\cdots\!16\)\( \nu + \)\(76\!\cdots\!00\)\(\)\()/ \)\(33\!\cdots\!76\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 48\)\()/192\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 733523 \beta_{2} + 1973266568918041 \beta_{1} + 969131934460330611034670049512448\)\()/36864\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 79 \beta_{6} + 2011782 \beta_{5} - 23678718391 \beta_{4} + 1384179048170368 \beta_{3} - 37016620395754701897533 \beta_{2} + 1662515232343756217142566943276552 \beta_{1} + 1912355646808829026711061052601232496903554531328\)\()/7077888\)
\(\nu^{4}\)\(=\)\((\)\(-29719859398329 \beta_{7} - 145799887264158487 \beta_{6} + 652495846394054466346 \beta_{5} + 25721593923614443863307071 \beta_{4} + 17723063282997046737413637201312 \beta_{3} - 23663368907972630582218452884002380875 \beta_{2} + 50079391174913080622776913061670979014869966936 \beta_{1} + 12587473462299087857831108687417268263598920316167838249896411136\)\()/10616832\)
\(\nu^{5}\)\(=\)\((\)\(690081898468688194372948118093 \beta_{7} + 52712705502696217851643644636611 \beta_{6} + 1851396046364780165157952552942886478 \beta_{5} - 20736003028845115535796221761437035534859 \beta_{4} + 686874930443596549204656381482858041942526976 \beta_{3} - 40384008968747526252706688812928724407833709582813913 \beta_{2} + 780750341310787037771810044164864831223409858253746164631892712 \beta_{1} + 1516673038600256798883004165784241181727374958343599126469274830406359768956928\)\()/63700992\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(33\!\cdots\!81\)\( \beta_{7} - \)\(16\!\cdots\!47\)\( \beta_{6} + \)\(81\!\cdots\!26\)\( \beta_{5} + \)\(25\!\cdots\!31\)\( \beta_{4} + \)\(13\!\cdots\!76\)\( \beta_{3} - \)\(24\!\cdots\!31\)\( \beta_{2} + \)\(41\!\cdots\!32\)\( \beta_{1} + \)\(82\!\cdots\!88\)\(\)\()/1327104\)
\(\nu^{7}\)\(=\)\((\)\(\)\(14\!\cdots\!23\)\( \beta_{7} + \)\(62\!\cdots\!45\)\( \beta_{6} + \)\(46\!\cdots\!46\)\( \beta_{5} - \)\(47\!\cdots\!81\)\( \beta_{4} + \)\(15\!\cdots\!28\)\( \beta_{3} - \)\(10\!\cdots\!07\)\( \beta_{2} + \)\(14\!\cdots\!44\)\( \beta_{1} + \)\(33\!\cdots\!64\)\(\)\()/21233664\)

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
2.47185e14
1.86979e14
1.06740e14
1.20500e13
−1.45972e13
−1.48022e14
−1.48210e14
−2.42126e14
−4.71735e16 −1.38575e26 1.57631e33 −1.39304e38 6.53705e42 1.59904e46 −4.37425e49 9.05875e51 6.57147e54
1.2 −3.56139e16 1.00458e26 6.19316e32 −8.07110e36 −3.57771e42 −7.18940e45 1.05848e48 −5.23267e49 2.87444e53
1.3 −2.02081e16 −7.88466e25 −2.40671e32 1.60701e38 1.59334e42 1.73591e45 1.79793e49 −3.92739e51 −3.24746e54
1.4 −2.02754e15 −9.96254e25 −6.44926e32 −2.14431e38 2.01995e41 −1.47165e46 2.62357e48 −2.18949e50 4.34769e53
1.5 3.08871e15 1.03355e26 −6.39497e32 −4.34627e37 3.19235e41 1.35697e46 −3.97991e48 5.38156e50 −1.34244e53
1.6 2.87062e16 1.16403e26 1.75010e32 1.23460e38 3.34150e42 −1.99654e46 −1.36075e49 3.40559e51 3.54406e54
1.7 2.87423e16 −1.13913e26 1.77081e32 7.39260e37 −3.27412e42 4.38791e45 −1.35651e49 2.83200e51 2.12480e54
1.8 4.67743e16 3.08304e25 1.53880e33 −1.66073e38 1.44207e42 8.52831e45 4.16178e49 −9.19366e51 −7.76792e54
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 1.110.a.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.110.a.a 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{110}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2288392606915200 T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(98\!\cdots\!76\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(61\!\cdots\!40\)\( T^{6} + \)\(37\!\cdots\!00\)\( T^{7} + \)\(33\!\cdots\!16\)\( T^{8} + \)\(24\!\cdots\!00\)\( T^{9} + \)\(25\!\cdots\!60\)\( T^{10} + \)\(86\!\cdots\!00\)\( T^{11} + \)\(17\!\cdots\!36\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(98\!\cdots\!80\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!96\)\( T^{16} \)
$3$ \( 1 + \)\(79\!\cdots\!00\)\( T + \)\(42\!\cdots\!40\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(98\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!26\)\( T^{8} + \)\(99\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!20\)\( T^{10} + \)\(70\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} + \)\(32\!\cdots\!00\)\( T^{13} + \)\(46\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} \)
$5$ \( 1 + \)\(21\!\cdots\!00\)\( T + \)\(75\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(42\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!00\)\( T^{12} + \)\(11\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} + \)\(43\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 - \)\(23\!\cdots\!00\)\( T + \)\(44\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(38\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!00\)\( T^{6} + \)\(80\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!06\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!00\)\( T^{10} + \)\(85\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!96\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(15\!\cdots\!00\)\( T^{15} + \)\(84\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + \)\(16\!\cdots\!64\)\( T + \)\(17\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!20\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(87\!\cdots\!68\)\( T^{6} + \)\(57\!\cdots\!60\)\( T^{7} + \)\(33\!\cdots\!70\)\( T^{8} + \)\(18\!\cdots\!60\)\( T^{9} + \)\(92\!\cdots\!08\)\( T^{10} + \)\(44\!\cdots\!72\)\( T^{11} + \)\(17\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!40\)\( T^{13} + \)\(21\!\cdots\!20\)\( T^{14} + \)\(63\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!21\)\( T^{16} \)
$13$ \( 1 + \)\(71\!\cdots\!00\)\( T + \)\(10\!\cdots\!80\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!16\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} + \)\(64\!\cdots\!00\)\( T^{7} + \)\(47\!\cdots\!46\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!40\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(25\!\cdots\!56\)\( T^{12} + \)\(70\!\cdots\!00\)\( T^{13} + \)\(33\!\cdots\!20\)\( T^{14} + \)\(62\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 + \)\(86\!\cdots\!00\)\( T + \)\(74\!\cdots\!60\)\( T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!36\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(58\!\cdots\!20\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(91\!\cdots\!86\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!80\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(79\!\cdots\!16\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!40\)\( T^{14} + \)\(58\!\cdots\!00\)\( T^{15} + \)\(89\!\cdots\!61\)\( T^{16} \)
$19$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(13\!\cdots\!32\)\( T^{2} + \)\(86\!\cdots\!80\)\( T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!84\)\( T^{6} + \)\(78\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!70\)\( T^{8} + \)\(19\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!44\)\( T^{10} + \)\(43\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!88\)\( T^{12} + \)\(72\!\cdots\!20\)\( T^{13} + \)\(27\!\cdots\!72\)\( T^{14} + \)\(54\!\cdots\!40\)\( T^{15} + \)\(11\!\cdots\!61\)\( T^{16} \)
$23$ \( 1 - \)\(53\!\cdots\!00\)\( T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!76\)\( T^{4} - \)\(29\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(78\!\cdots\!66\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!60\)\( T^{10} - \)\(57\!\cdots\!00\)\( T^{11} + \)\(31\!\cdots\!36\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!80\)\( T^{14} - \)\(53\!\cdots\!00\)\( T^{15} + \)\(26\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(14\!\cdots\!52\)\( T^{2} - \)\(39\!\cdots\!80\)\( T^{3} + \)\(98\!\cdots\!08\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(43\!\cdots\!04\)\( T^{6} - \)\(93\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!44\)\( T^{10} - \)\(39\!\cdots\!40\)\( T^{11} + \)\(39\!\cdots\!68\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{13} + \)\(36\!\cdots\!12\)\( T^{14} - \)\(19\!\cdots\!40\)\( T^{15} + \)\(16\!\cdots\!41\)\( T^{16} \)
$31$ \( 1 + \)\(14\!\cdots\!24\)\( T + \)\(17\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!20\)\( T^{4} + \)\(63\!\cdots\!32\)\( T^{5} + \)\(72\!\cdots\!88\)\( T^{6} + \)\(93\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!70\)\( T^{8} + \)\(33\!\cdots\!60\)\( T^{9} + \)\(94\!\cdots\!08\)\( T^{10} + \)\(29\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!40\)\( T^{13} + \)\(40\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(29\!\cdots\!61\)\( T^{16} \)
$37$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(27\!\cdots\!80\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!16\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!60\)\( T^{6} - \)\(78\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!46\)\( T^{8} - \)\(67\!\cdots\!00\)\( T^{9} + \)\(28\!\cdots\!40\)\( T^{10} - \)\(41\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!56\)\( T^{12} - \)\(28\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!20\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(29\!\cdots\!81\)\( T^{16} \)
$41$ \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(14\!\cdots\!20\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(26\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(67\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{9} + \)\(41\!\cdots\!08\)\( T^{10} - \)\(63\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!20\)\( T^{12} - \)\(99\!\cdots\!60\)\( T^{13} + \)\(82\!\cdots\!20\)\( T^{14} - \)\(94\!\cdots\!16\)\( T^{15} + \)\(22\!\cdots\!81\)\( T^{16} \)
$43$ \( 1 + \)\(31\!\cdots\!00\)\( T + \)\(60\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!96\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!06\)\( T^{8} + \)\(22\!\cdots\!00\)\( T^{9} + \)\(42\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(27\!\cdots\!96\)\( T^{12} + \)\(58\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} + \)\(68\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - \)\(16\!\cdots\!00\)\( T + \)\(57\!\cdots\!40\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(54\!\cdots\!80\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!20\)\( T^{10} - \)\(67\!\cdots\!00\)\( T^{11} + \)\(22\!\cdots\!76\)\( T^{12} - \)\(48\!\cdots\!00\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - \)\(90\!\cdots\!00\)\( T + \)\(20\!\cdots\!40\)\( T^{2} - \)\(95\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!56\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!26\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!20\)\( T^{10} - \)\(66\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!76\)\( T^{12} - \)\(51\!\cdots\!00\)\( T^{13} + \)\(95\!\cdots\!60\)\( T^{14} - \)\(38\!\cdots\!00\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - \)\(42\!\cdots\!20\)\( T + \)\(70\!\cdots\!12\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!88\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{5} + \)\(35\!\cdots\!64\)\( T^{6} - \)\(58\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!70\)\( T^{8} - \)\(61\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!44\)\( T^{10} - \)\(51\!\cdots\!80\)\( T^{11} + \)\(25\!\cdots\!08\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{13} + \)\(96\!\cdots\!32\)\( T^{14} - \)\(60\!\cdots\!80\)\( T^{15} + \)\(15\!\cdots\!81\)\( T^{16} \)
$61$ \( 1 + \)\(30\!\cdots\!64\)\( T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(84\!\cdots\!20\)\( T^{4} + \)\(12\!\cdots\!32\)\( T^{5} + \)\(30\!\cdots\!68\)\( T^{6} + \)\(33\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} + \)\(13\!\cdots\!60\)\( T^{9} + \)\(47\!\cdots\!08\)\( T^{10} + \)\(78\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!20\)\( T^{12} + \)\(28\!\cdots\!40\)\( T^{13} + \)\(55\!\cdots\!20\)\( T^{14} + \)\(48\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!21\)\( T^{16} \)
$67$ \( 1 + \)\(24\!\cdots\!00\)\( T + \)\(64\!\cdots\!60\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!36\)\( T^{4} + \)\(33\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!20\)\( T^{6} + \)\(53\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!86\)\( T^{8} + \)\(59\!\cdots\!00\)\( T^{9} + \)\(45\!\cdots\!80\)\( T^{10} + \)\(44\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!16\)\( T^{12} + \)\(21\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!40\)\( T^{14} + \)\(48\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!61\)\( T^{16} \)
$71$ \( 1 + \)\(18\!\cdots\!44\)\( T + \)\(48\!\cdots\!20\)\( T^{2} + \)\(57\!\cdots\!40\)\( T^{3} + \)\(85\!\cdots\!20\)\( T^{4} + \)\(71\!\cdots\!32\)\( T^{5} + \)\(80\!\cdots\!28\)\( T^{6} + \)\(53\!\cdots\!60\)\( T^{7} + \)\(53\!\cdots\!70\)\( T^{8} + \)\(32\!\cdots\!60\)\( T^{9} + \)\(30\!\cdots\!08\)\( T^{10} + \)\(16\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!20\)\( T^{12} + \)\(49\!\cdots\!40\)\( T^{13} + \)\(25\!\cdots\!20\)\( T^{14} + \)\(59\!\cdots\!84\)\( T^{15} + \)\(19\!\cdots\!41\)\( T^{16} \)
$73$ \( 1 + \)\(99\!\cdots\!00\)\( T + \)\(11\!\cdots\!20\)\( T^{2} + \)\(82\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!76\)\( T^{4} + \)\(29\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!66\)\( T^{8} + \)\(78\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!60\)\( T^{10} + \)\(60\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!36\)\( T^{12} + \)\(26\!\cdots\!00\)\( T^{13} + \)\(48\!\cdots\!80\)\( T^{14} + \)\(51\!\cdots\!00\)\( T^{15} + \)\(65\!\cdots\!21\)\( T^{16} \)
$79$ \( 1 + \)\(90\!\cdots\!40\)\( T + \)\(47\!\cdots\!52\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!08\)\( T^{4} + \)\(14\!\cdots\!40\)\( T^{5} + \)\(32\!\cdots\!04\)\( T^{6} + \)\(69\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!70\)\( T^{8} + \)\(48\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!44\)\( T^{10} + \)\(47\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!68\)\( T^{12} + \)\(28\!\cdots\!80\)\( T^{13} + \)\(53\!\cdots\!12\)\( T^{14} + \)\(70\!\cdots\!60\)\( T^{15} + \)\(53\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + \)\(86\!\cdots\!00\)\( T + \)\(55\!\cdots\!60\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!36\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} - \)\(84\!\cdots\!14\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(78\!\cdots\!80\)\( T^{10} + \)\(80\!\cdots\!00\)\( T^{11} + \)\(53\!\cdots\!16\)\( T^{12} + \)\(21\!\cdots\!00\)\( T^{13} + \)\(65\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - \)\(53\!\cdots\!80\)\( T + \)\(17\!\cdots\!72\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!68\)\( T^{4} - \)\(90\!\cdots\!80\)\( T^{5} + \)\(77\!\cdots\!24\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(71\!\cdots\!44\)\( T^{10} - \)\(25\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!60\)\( T^{13} + \)\(13\!\cdots\!52\)\( T^{14} - \)\(13\!\cdots\!20\)\( T^{15} + \)\(73\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(13\!\cdots\!40\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!56\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!26\)\( T^{8} - \)\(92\!\cdots\!00\)\( T^{9} + \)\(28\!\cdots\!20\)\( T^{10} - \)\(58\!\cdots\!00\)\( T^{11} + \)\(72\!\cdots\!76\)\( T^{12} - \)\(85\!\cdots\!00\)\( T^{13} + \)\(29\!\cdots\!60\)\( T^{14} - \)\(25\!\cdots\!00\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} \)
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