Properties

Label 1.102.a.a
Level 1
Weight 102
Character orbit 1.a
Self dual yes
Analytic conductor 64.601
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 102 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.6006978936\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{119}\cdot 3^{37}\cdot 5^{14}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-54373636474380 - \beta_{1}) q^{2} +(-\)\(15\!\cdots\!20\)\( - 82394580 \beta_{1} - \beta_{2}) q^{3} +(\)\(11\!\cdots\!12\)\( + 229422224535561 \beta_{1} + 310415 \beta_{2} + \beta_{3}) q^{4} +(\)\(47\!\cdots\!50\)\( + 12781604295053357539 \beta_{1} + 15054417655 \beta_{2} + 30556 \beta_{3} + \beta_{4}) q^{5} +(\)\(30\!\cdots\!92\)\( + \)\(34\!\cdots\!32\)\( \beta_{1} + 389788608094608 \beta_{2} + 189420581 \beta_{3} + 1682 \beta_{4} - \beta_{5}) q^{6} +(-\)\(72\!\cdots\!00\)\( + \)\(17\!\cdots\!75\)\( \beta_{1} - 207531242386384435 \beta_{2} + 311630903937 \beta_{3} - 4702714 \beta_{4} - 80 \beta_{5} + \beta_{6}) q^{7} +(-\)\(76\!\cdots\!40\)\( - \)\(40\!\cdots\!76\)\( \beta_{1} - \)\(49\!\cdots\!83\)\( \beta_{2} + 124289729660408 \beta_{3} - 1912817011 \beta_{4} + 181393 \beta_{5} - 183 \beta_{6} - \beta_{7}) q^{8} +(\)\(65\!\cdots\!73\)\( + \)\(23\!\cdots\!86\)\( \beta_{1} + \)\(37\!\cdots\!38\)\( \beta_{2} - 150777539087639100 \beta_{3} + 2172379267758 \beta_{4} - 83581560 \beta_{5} - 61884 \beta_{6} + 696 \beta_{7}) q^{9} +O(q^{10})\) \( q +(-54373636474380 - \beta_{1}) q^{2} +(-\)\(15\!\cdots\!20\)\( - 82394580 \beta_{1} - \beta_{2}) q^{3} +(\)\(11\!\cdots\!12\)\( + 229422224535561 \beta_{1} + 310415 \beta_{2} + \beta_{3}) q^{4} +(\)\(47\!\cdots\!50\)\( + 12781604295053357539 \beta_{1} + 15054417655 \beta_{2} + 30556 \beta_{3} + \beta_{4}) q^{5} +(\)\(30\!\cdots\!92\)\( + \)\(34\!\cdots\!32\)\( \beta_{1} + 389788608094608 \beta_{2} + 189420581 \beta_{3} + 1682 \beta_{4} - \beta_{5}) q^{6} +(-\)\(72\!\cdots\!00\)\( + \)\(17\!\cdots\!75\)\( \beta_{1} - 207531242386384435 \beta_{2} + 311630903937 \beta_{3} - 4702714 \beta_{4} - 80 \beta_{5} + \beta_{6}) q^{7} +(-\)\(76\!\cdots\!40\)\( - \)\(40\!\cdots\!76\)\( \beta_{1} - \)\(49\!\cdots\!83\)\( \beta_{2} + 124289729660408 \beta_{3} - 1912817011 \beta_{4} + 181393 \beta_{5} - 183 \beta_{6} - \beta_{7}) q^{8} +(\)\(65\!\cdots\!73\)\( + \)\(23\!\cdots\!86\)\( \beta_{1} + \)\(37\!\cdots\!38\)\( \beta_{2} - 150777539087639100 \beta_{3} + 2172379267758 \beta_{4} - 83581560 \beta_{5} - 61884 \beta_{6} + 696 \beta_{7}) q^{9} +(-\)\(47\!\cdots\!00\)\( - \)\(62\!\cdots\!42\)\( \beta_{1} - \)\(51\!\cdots\!00\)\( \beta_{2} - 11902232918987832068 \beta_{3} - 2436035283808 \beta_{4} + 58736901340 \beta_{5} - 72318680 \beta_{6} - 98280 \beta_{7}) q^{10} +(\)\(57\!\cdots\!12\)\( - \)\(21\!\cdots\!90\)\( \beta_{1} - \)\(22\!\cdots\!29\)\( \beta_{2} - \)\(37\!\cdots\!62\)\( \beta_{3} - 22303986140125060 \beta_{4} + 12998547407744 \beta_{5} + 1900924986 \beta_{6} + 7198816 \beta_{7}) q^{11} +(-\)\(90\!\cdots\!60\)\( - \)\(55\!\cdots\!68\)\( \beta_{1} - \)\(77\!\cdots\!36\)\( \beta_{2} - \)\(20\!\cdots\!08\)\( \beta_{3} - 8004291691235921064 \beta_{4} + 1277464914935352 \beta_{5} + 276758020728 \beta_{6} - 339515064 \beta_{7}) q^{12} +(\)\(31\!\cdots\!10\)\( + \)\(28\!\cdots\!71\)\( \beta_{1} + \)\(13\!\cdots\!99\)\( \beta_{2} - \)\(22\!\cdots\!76\)\( \beta_{3} - \)\(39\!\cdots\!43\)\( \beta_{4} - 29465342186438608 \beta_{5} - 22175840367016 \beta_{6} + 11181327696 \beta_{7}) q^{13} +(-\)\(60\!\cdots\!36\)\( + \)\(19\!\cdots\!04\)\( \beta_{1} - \)\(15\!\cdots\!04\)\( \beta_{2} - \)\(61\!\cdots\!98\)\( \beta_{3} - \)\(35\!\cdots\!08\)\( \beta_{4} - 1858053851670474274 \beta_{5} + 765440512740768 \beta_{6} - 259232530592 \beta_{7}) q^{14} +(-\)\(37\!\cdots\!00\)\( - \)\(37\!\cdots\!71\)\( \beta_{1} - \)\(36\!\cdots\!25\)\( \beta_{2} - \)\(53\!\cdots\!09\)\( \beta_{3} - \)\(43\!\cdots\!54\)\( \beta_{4} + 75175579037504959120 \beta_{5} - 13679001364346865 \beta_{6} + 3745840800960 \beta_{7}) q^{15} +(-\)\(13\!\cdots\!84\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(78\!\cdots\!80\)\( \beta_{2} - \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!76\)\( \beta_{4} - \)\(53\!\cdots\!44\)\( \beta_{5} + 36199878363452576 \beta_{6} - 3581416918944 \beta_{7}) q^{16} +(-\)\(49\!\cdots\!90\)\( + \)\(41\!\cdots\!06\)\( \beta_{1} - \)\(33\!\cdots\!86\)\( \beta_{2} - \)\(34\!\cdots\!92\)\( \beta_{3} + \)\(21\!\cdots\!74\)\( \beta_{4} - \)\(23\!\cdots\!00\)\( \beta_{5} + 5303318752936720404 \beta_{6} - 1680372172971240 \beta_{7}) q^{17} +(-\)\(90\!\cdots\!20\)\( - \)\(51\!\cdots\!69\)\( \beta_{1} - \)\(48\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!72\)\( \beta_{3} - \)\(75\!\cdots\!16\)\( \beta_{4} + \)\(68\!\cdots\!00\)\( \beta_{5} - \)\(17\!\cdots\!36\)\( \beta_{6} + 61158459347994960 \beta_{7}) q^{18} +(-\)\(26\!\cdots\!60\)\( + \)\(24\!\cdots\!74\)\( \beta_{1} + \)\(37\!\cdots\!97\)\( \beta_{2} + \)\(10\!\cdots\!50\)\( \beta_{3} - \)\(17\!\cdots\!00\)\( \beta_{4} - \)\(75\!\cdots\!12\)\( \beta_{5} + \)\(34\!\cdots\!62\)\( \beta_{6} - 1448443710124252128 \beta_{7}) q^{19} +(\)\(21\!\cdots\!00\)\( + \)\(55\!\cdots\!38\)\( \beta_{1} + \)\(17\!\cdots\!10\)\( \beta_{2} + \)\(14\!\cdots\!02\)\( \beta_{3} + \)\(17\!\cdots\!92\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5} - \)\(47\!\cdots\!00\)\( \beta_{6} + 26979311695698549600 \beta_{7}) q^{20} +(\)\(50\!\cdots\!92\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!48\)\( \beta_{2} + \)\(89\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!84\)\( \beta_{4} + \)\(10\!\cdots\!28\)\( \beta_{5} + \)\(47\!\cdots\!84\)\( \beta_{6} - \)\(42\!\cdots\!96\)\( \beta_{7}) q^{21} +(\)\(76\!\cdots\!40\)\( + \)\(65\!\cdots\!24\)\( \beta_{1} + \)\(41\!\cdots\!76\)\( \beta_{2} - \)\(10\!\cdots\!09\)\( \beta_{3} - \)\(55\!\cdots\!62\)\( \beta_{4} - \)\(16\!\cdots\!67\)\( \beta_{5} - \)\(31\!\cdots\!64\)\( \beta_{6} + \)\(56\!\cdots\!04\)\( \beta_{7}) q^{22} +(-\)\(17\!\cdots\!60\)\( - \)\(34\!\cdots\!51\)\( \beta_{1} + \)\(32\!\cdots\!31\)\( \beta_{2} - \)\(91\!\cdots\!25\)\( \beta_{3} + \)\(10\!\cdots\!50\)\( \beta_{4} + \)\(10\!\cdots\!80\)\( \beta_{5} + \)\(49\!\cdots\!55\)\( \beta_{6} - \)\(66\!\cdots\!60\)\( \beta_{7}) q^{23} +(\)\(13\!\cdots\!20\)\( + \)\(61\!\cdots\!56\)\( \beta_{1} + \)\(35\!\cdots\!16\)\( \beta_{2} + \)\(24\!\cdots\!24\)\( \beta_{3} + \)\(73\!\cdots\!48\)\( \beta_{4} - \)\(79\!\cdots\!48\)\( \beta_{5} + \)\(19\!\cdots\!84\)\( \beta_{6} + \)\(69\!\cdots\!04\)\( \beta_{7}) q^{24} +(\)\(96\!\cdots\!75\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2} + \)\(75\!\cdots\!00\)\( \beta_{3} - \)\(84\!\cdots\!00\)\( \beta_{4} - \)\(56\!\cdots\!00\)\( \beta_{5} - \)\(32\!\cdots\!00\)\( \beta_{6} - \)\(64\!\cdots\!00\)\( \beta_{7}) q^{25} +(-\)\(12\!\cdots\!68\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(72\!\cdots\!52\)\( \beta_{2} - \)\(51\!\cdots\!76\)\( \beta_{3} + \)\(39\!\cdots\!64\)\( \beta_{4} + \)\(54\!\cdots\!56\)\( \beta_{5} + \)\(31\!\cdots\!92\)\( \beta_{6} + \)\(54\!\cdots\!52\)\( \beta_{7}) q^{26} +(-\)\(74\!\cdots\!40\)\( - \)\(22\!\cdots\!26\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} - \)\(30\!\cdots\!82\)\( \beta_{3} + \)\(22\!\cdots\!44\)\( \beta_{4} - \)\(17\!\cdots\!92\)\( \beta_{5} - \)\(21\!\cdots\!38\)\( \beta_{6} - \)\(40\!\cdots\!56\)\( \beta_{7}) q^{27} +(\)\(11\!\cdots\!60\)\( + \)\(12\!\cdots\!36\)\( \beta_{1} - \)\(35\!\cdots\!96\)\( \beta_{2} - \)\(79\!\cdots\!28\)\( \beta_{3} - \)\(25\!\cdots\!24\)\( \beta_{4} - \)\(11\!\cdots\!68\)\( \beta_{5} + \)\(11\!\cdots\!48\)\( \beta_{6} + \)\(27\!\cdots\!76\)\( \beta_{7}) q^{28} +(\)\(19\!\cdots\!10\)\( - \)\(10\!\cdots\!01\)\( \beta_{1} - \)\(14\!\cdots\!53\)\( \beta_{2} + \)\(15\!\cdots\!20\)\( \beta_{3} + \)\(51\!\cdots\!21\)\( \beta_{4} + \)\(16\!\cdots\!44\)\( \beta_{5} - \)\(35\!\cdots\!72\)\( \beta_{6} - \)\(16\!\cdots\!32\)\( \beta_{7}) q^{29} +(\)\(13\!\cdots\!00\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(27\!\cdots\!60\)\( \beta_{2} + \)\(18\!\cdots\!82\)\( \beta_{3} + \)\(10\!\cdots\!72\)\( \beta_{4} - \)\(93\!\cdots\!50\)\( \beta_{5} - \)\(19\!\cdots\!00\)\( \beta_{6} + \)\(88\!\cdots\!00\)\( \beta_{7}) q^{30} +(-\)\(82\!\cdots\!68\)\( + \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(25\!\cdots\!00\)\( \beta_{2} - \)\(45\!\cdots\!48\)\( \beta_{3} - \)\(59\!\cdots\!76\)\( \beta_{4} + \)\(17\!\cdots\!44\)\( \beta_{5} + \)\(85\!\cdots\!24\)\( \beta_{6} - \)\(42\!\cdots\!56\)\( \beta_{7}) q^{31} +(\)\(15\!\cdots\!20\)\( + \)\(34\!\cdots\!20\)\( \beta_{1} - \)\(86\!\cdots\!24\)\( \beta_{2} - \)\(85\!\cdots\!12\)\( \beta_{3} + \)\(19\!\cdots\!64\)\( \beta_{4} + \)\(10\!\cdots\!60\)\( \beta_{5} - \)\(56\!\cdots\!96\)\( \beta_{6} + \)\(17\!\cdots\!40\)\( \beta_{7}) q^{32} +(\)\(54\!\cdots\!60\)\( + \)\(25\!\cdots\!46\)\( \beta_{1} - \)\(17\!\cdots\!26\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(99\!\cdots\!70\)\( \beta_{4} - \)\(93\!\cdots\!08\)\( \beta_{5} + \)\(15\!\cdots\!32\)\( \beta_{6} - \)\(61\!\cdots\!84\)\( \beta_{7}) q^{33} +(\)\(11\!\cdots\!64\)\( + \)\(10\!\cdots\!22\)\( \beta_{1} - \)\(64\!\cdots\!28\)\( \beta_{2} - \)\(13\!\cdots\!92\)\( \beta_{3} - \)\(26\!\cdots\!20\)\( \beta_{4} + \)\(23\!\cdots\!88\)\( \beta_{5} + \)\(45\!\cdots\!72\)\( \beta_{6} + \)\(17\!\cdots\!32\)\( \beta_{7}) q^{34} +(-\)\(14\!\cdots\!00\)\( + \)\(44\!\cdots\!28\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2} - \)\(25\!\cdots\!88\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} + \)\(77\!\cdots\!40\)\( \beta_{5} - \)\(78\!\cdots\!80\)\( \beta_{6} - \)\(32\!\cdots\!80\)\( \beta_{7}) q^{35} +(\)\(24\!\cdots\!76\)\( + \)\(25\!\cdots\!53\)\( \beta_{1} + \)\(20\!\cdots\!83\)\( \beta_{2} + \)\(50\!\cdots\!97\)\( \beta_{3} + \)\(67\!\cdots\!92\)\( \beta_{4} - \)\(78\!\cdots\!76\)\( \beta_{5} + \)\(49\!\cdots\!20\)\( \beta_{6} + \)\(48\!\cdots\!20\)\( \beta_{7}) q^{36} +(\)\(49\!\cdots\!30\)\( + \)\(48\!\cdots\!23\)\( \beta_{1} + \)\(29\!\cdots\!27\)\( \beta_{2} + \)\(19\!\cdots\!16\)\( \beta_{3} + \)\(10\!\cdots\!53\)\( \beta_{4} + \)\(21\!\cdots\!96\)\( \beta_{5} - \)\(20\!\cdots\!56\)\( \beta_{6} + \)\(25\!\cdots\!28\)\( \beta_{7}) q^{37} +(-\)\(88\!\cdots\!60\)\( - \)\(22\!\cdots\!92\)\( \beta_{1} - \)\(24\!\cdots\!40\)\( \beta_{2} - \)\(16\!\cdots\!31\)\( \beta_{3} - \)\(16\!\cdots\!78\)\( \beta_{4} + \)\(39\!\cdots\!23\)\( \beta_{5} + \)\(59\!\cdots\!40\)\( \beta_{6} - \)\(11\!\cdots\!16\)\( \beta_{7}) q^{38} +(-\)\(33\!\cdots\!04\)\( - \)\(42\!\cdots\!87\)\( \beta_{1} - \)\(72\!\cdots\!61\)\( \beta_{2} - \)\(28\!\cdots\!25\)\( \beta_{3} + \)\(34\!\cdots\!30\)\( \beta_{4} - \)\(50\!\cdots\!64\)\( \beta_{5} - \)\(11\!\cdots\!01\)\( \beta_{6} + \)\(23\!\cdots\!44\)\( \beta_{7}) q^{39} +(-\)\(95\!\cdots\!00\)\( - \)\(35\!\cdots\!80\)\( \beta_{1} - \)\(49\!\cdots\!50\)\( \beta_{2} - \)\(71\!\cdots\!20\)\( \beta_{3} + \)\(92\!\cdots\!30\)\( \beta_{4} + \)\(16\!\cdots\!50\)\( \beta_{5} + \)\(14\!\cdots\!50\)\( \beta_{6} - \)\(18\!\cdots\!50\)\( \beta_{7}) q^{40} +(\)\(70\!\cdots\!42\)\( - \)\(10\!\cdots\!16\)\( \beta_{1} + \)\(11\!\cdots\!52\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} - \)\(43\!\cdots\!72\)\( \beta_{4} - \)\(97\!\cdots\!84\)\( \beta_{5} - \)\(64\!\cdots\!20\)\( \beta_{6} + \)\(51\!\cdots\!80\)\( \beta_{7}) q^{41} +(\)\(37\!\cdots\!60\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!88\)\( \beta_{2} + \)\(48\!\cdots\!92\)\( \beta_{3} - \)\(81\!\cdots\!24\)\( \beta_{4} - \)\(70\!\cdots\!40\)\( \beta_{5} + \)\(52\!\cdots\!56\)\( \beta_{6} - \)\(10\!\cdots\!80\)\( \beta_{7}) q^{42} +(-\)\(35\!\cdots\!00\)\( - \)\(22\!\cdots\!64\)\( \beta_{1} - \)\(41\!\cdots\!19\)\( \beta_{2} - \)\(40\!\cdots\!48\)\( \beta_{3} + \)\(75\!\cdots\!96\)\( \beta_{4} - \)\(72\!\cdots\!72\)\( \beta_{5} - \)\(24\!\cdots\!76\)\( \beta_{6} + \)\(78\!\cdots\!84\)\( \beta_{7}) q^{43} +(-\)\(25\!\cdots\!56\)\( + \)\(74\!\cdots\!60\)\( \beta_{1} - \)\(53\!\cdots\!20\)\( \beta_{2} - \)\(83\!\cdots\!80\)\( \beta_{3} - \)\(82\!\cdots\!04\)\( \beta_{4} + \)\(25\!\cdots\!76\)\( \beta_{5} + \)\(57\!\cdots\!96\)\( \beta_{6} - \)\(30\!\cdots\!24\)\( \beta_{7}) q^{44} +(\)\(89\!\cdots\!50\)\( + \)\(15\!\cdots\!87\)\( \beta_{1} + \)\(69\!\cdots\!15\)\( \beta_{2} + \)\(52\!\cdots\!48\)\( \beta_{3} - \)\(44\!\cdots\!67\)\( \beta_{4} - \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(46\!\cdots\!00\)\( \beta_{6} + \)\(50\!\cdots\!00\)\( \beta_{7}) q^{45} +(\)\(13\!\cdots\!72\)\( + \)\(27\!\cdots\!64\)\( \beta_{1} + \)\(20\!\cdots\!12\)\( \beta_{2} + \)\(20\!\cdots\!70\)\( \beta_{3} + \)\(11\!\cdots\!28\)\( \beta_{4} + \)\(29\!\cdots\!06\)\( \beta_{5} - \)\(93\!\cdots\!60\)\( \beta_{6} + \)\(16\!\cdots\!40\)\( \beta_{7}) q^{46} +(-\)\(57\!\cdots\!60\)\( + \)\(60\!\cdots\!26\)\( \beta_{1} + \)\(87\!\cdots\!98\)\( \beta_{2} - \)\(93\!\cdots\!14\)\( \beta_{3} + \)\(14\!\cdots\!28\)\( \beta_{4} - \)\(28\!\cdots\!96\)\( \beta_{5} + \)\(37\!\cdots\!82\)\( \beta_{6} - \)\(15\!\cdots\!88\)\( \beta_{7}) q^{47} +(-\)\(72\!\cdots\!60\)\( - \)\(10\!\cdots\!92\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} - \)\(20\!\cdots\!04\)\( \beta_{4} + \)\(14\!\cdots\!80\)\( \beta_{5} - \)\(73\!\cdots\!04\)\( \beta_{6} + \)\(56\!\cdots\!80\)\( \beta_{7}) q^{48} +(\)\(15\!\cdots\!57\)\( - \)\(69\!\cdots\!04\)\( \beta_{1} - \)\(67\!\cdots\!12\)\( \beta_{2} + \)\(21\!\cdots\!40\)\( \beta_{3} - \)\(39\!\cdots\!88\)\( \beta_{4} - \)\(29\!\cdots\!16\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(94\!\cdots\!00\)\( \beta_{7}) q^{49} +(-\)\(50\!\cdots\!00\)\( - \)\(24\!\cdots\!75\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4} + \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(57\!\cdots\!00\)\( \beta_{6} - \)\(16\!\cdots\!00\)\( \beta_{7}) q^{50} +(\)\(14\!\cdots\!92\)\( - \)\(29\!\cdots\!86\)\( \beta_{1} + \)\(32\!\cdots\!72\)\( \beta_{2} - \)\(23\!\cdots\!10\)\( \beta_{3} + \)\(34\!\cdots\!80\)\( \beta_{4} - \)\(22\!\cdots\!32\)\( \beta_{5} - \)\(16\!\cdots\!58\)\( \beta_{6} + \)\(16\!\cdots\!52\)\( \beta_{7}) q^{51} +(-\)\(92\!\cdots\!00\)\( + \)\(26\!\cdots\!30\)\( \beta_{1} + \)\(14\!\cdots\!90\)\( \beta_{2} - \)\(40\!\cdots\!14\)\( \beta_{3} - \)\(22\!\cdots\!72\)\( \beta_{4} - \)\(66\!\cdots\!96\)\( \beta_{5} + \)\(17\!\cdots\!32\)\( \beta_{6} - \)\(52\!\cdots\!88\)\( \beta_{7}) q^{52} +(\)\(16\!\cdots\!30\)\( + \)\(22\!\cdots\!55\)\( \beta_{1} + \)\(85\!\cdots\!59\)\( \beta_{2} - \)\(78\!\cdots\!52\)\( \beta_{3} - \)\(30\!\cdots\!91\)\( \beta_{4} + \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(28\!\cdots\!72\)\( \beta_{6} + \)\(68\!\cdots\!04\)\( \beta_{7}) q^{53} +(\)\(85\!\cdots\!40\)\( + \)\(15\!\cdots\!12\)\( \beta_{1} + \)\(35\!\cdots\!40\)\( \beta_{2} + \)\(88\!\cdots\!42\)\( \beta_{3} + \)\(14\!\cdots\!56\)\( \beta_{4} - \)\(64\!\cdots\!94\)\( \beta_{5} - \)\(84\!\cdots\!64\)\( \beta_{6} + \)\(18\!\cdots\!16\)\( \beta_{7}) q^{54} +(-\)\(18\!\cdots\!00\)\( + \)\(77\!\cdots\!43\)\( \beta_{1} - \)\(36\!\cdots\!15\)\( \beta_{2} - \)\(43\!\cdots\!03\)\( \beta_{3} - \)\(22\!\cdots\!38\)\( \beta_{4} + \)\(27\!\cdots\!00\)\( \beta_{5} - \)\(11\!\cdots\!75\)\( \beta_{6} - \)\(12\!\cdots\!00\)\( \beta_{7}) q^{55} +(-\)\(29\!\cdots\!60\)\( + \)\(16\!\cdots\!88\)\( \beta_{1} - \)\(95\!\cdots\!80\)\( \beta_{2} - \)\(33\!\cdots\!32\)\( \beta_{3} + \)\(13\!\cdots\!48\)\( \beta_{4} - \)\(57\!\cdots\!72\)\( \beta_{5} + \)\(68\!\cdots\!08\)\( \beta_{6} + \)\(30\!\cdots\!48\)\( \beta_{7}) q^{56} +(-\)\(83\!\cdots\!80\)\( - \)\(14\!\cdots\!70\)\( \beta_{1} + \)\(81\!\cdots\!86\)\( \beta_{2} - \)\(38\!\cdots\!32\)\( \beta_{3} - \)\(95\!\cdots\!86\)\( \beta_{4} + \)\(67\!\cdots\!52\)\( \beta_{5} + \)\(64\!\cdots\!16\)\( \beta_{6} - \)\(16\!\cdots\!44\)\( \beta_{7}) q^{57} +(\)\(36\!\cdots\!60\)\( - \)\(28\!\cdots\!38\)\( \beta_{1} + \)\(45\!\cdots\!60\)\( \beta_{2} + \)\(27\!\cdots\!28\)\( \beta_{3} + \)\(43\!\cdots\!04\)\( \beta_{4} - \)\(11\!\cdots\!96\)\( \beta_{5} - \)\(10\!\cdots\!72\)\( \beta_{6} - \)\(15\!\cdots\!48\)\( \beta_{7}) q^{58} +(\)\(26\!\cdots\!20\)\( - \)\(21\!\cdots\!60\)\( \beta_{1} + \)\(37\!\cdots\!73\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(45\!\cdots\!84\)\( \beta_{4} - \)\(43\!\cdots\!72\)\( \beta_{5} + \)\(28\!\cdots\!84\)\( \beta_{6} + \)\(62\!\cdots\!04\)\( \beta_{7}) q^{59} +(-\)\(43\!\cdots\!00\)\( - \)\(15\!\cdots\!32\)\( \beta_{1} - \)\(36\!\cdots\!00\)\( \beta_{2} - \)\(13\!\cdots\!28\)\( \beta_{3} - \)\(11\!\cdots\!68\)\( \beta_{4} + \)\(21\!\cdots\!40\)\( \beta_{5} - \)\(95\!\cdots\!80\)\( \beta_{6} - \)\(98\!\cdots\!80\)\( \beta_{7}) q^{60} +(-\)\(42\!\cdots\!38\)\( + \)\(34\!\cdots\!79\)\( \beta_{1} - \)\(27\!\cdots\!21\)\( \beta_{2} - \)\(36\!\cdots\!84\)\( \beta_{3} + \)\(20\!\cdots\!33\)\( \beta_{4} - \)\(29\!\cdots\!56\)\( \beta_{5} - \)\(15\!\cdots\!88\)\( \beta_{6} - \)\(79\!\cdots\!28\)\( \beta_{7}) q^{61} +(-\)\(73\!\cdots\!60\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} + \)\(70\!\cdots\!24\)\( \beta_{2} + \)\(91\!\cdots\!12\)\( \beta_{3} + \)\(35\!\cdots\!36\)\( \beta_{4} - \)\(25\!\cdots\!60\)\( \beta_{5} + \)\(48\!\cdots\!96\)\( \beta_{6} + \)\(81\!\cdots\!60\)\( \beta_{7}) q^{62} +(-\)\(25\!\cdots\!80\)\( + \)\(12\!\cdots\!31\)\( \beta_{1} + \)\(18\!\cdots\!37\)\( \beta_{2} + \)\(19\!\cdots\!41\)\( \beta_{3} - \)\(19\!\cdots\!22\)\( \beta_{4} + \)\(97\!\cdots\!76\)\( \beta_{5} - \)\(45\!\cdots\!51\)\( \beta_{6} - \)\(19\!\cdots\!32\)\( \beta_{7}) q^{63} +(-\)\(93\!\cdots\!88\)\( - \)\(84\!\cdots\!96\)\( \beta_{1} + \)\(24\!\cdots\!36\)\( \beta_{2} - \)\(40\!\cdots\!88\)\( \beta_{3} - \)\(45\!\cdots\!36\)\( \beta_{4} + \)\(68\!\cdots\!28\)\( \beta_{5} - \)\(96\!\cdots\!80\)\( \beta_{6} + \)\(10\!\cdots\!20\)\( \beta_{7}) q^{64} +(-\)\(20\!\cdots\!00\)\( - \)\(33\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!00\)\( \beta_{2} + \)\(22\!\cdots\!16\)\( \beta_{3} + \)\(97\!\cdots\!96\)\( \beta_{4} - \)\(31\!\cdots\!80\)\( \beta_{5} + \)\(36\!\cdots\!60\)\( \beta_{6} + \)\(71\!\cdots\!60\)\( \beta_{7}) q^{65} +(-\)\(93\!\cdots\!96\)\( - \)\(34\!\cdots\!28\)\( \beta_{1} - \)\(37\!\cdots\!64\)\( \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(29\!\cdots\!00\)\( \beta_{4} - \)\(95\!\cdots\!56\)\( \beta_{5} - \)\(43\!\cdots\!44\)\( \beta_{6} - \)\(24\!\cdots\!64\)\( \beta_{7}) q^{66} +(-\)\(77\!\cdots\!40\)\( - \)\(33\!\cdots\!42\)\( \beta_{1} - \)\(14\!\cdots\!79\)\( \beta_{2} + \)\(34\!\cdots\!26\)\( \beta_{3} + \)\(81\!\cdots\!88\)\( \beta_{4} + \)\(45\!\cdots\!32\)\( \beta_{5} + \)\(11\!\cdots\!50\)\( \beta_{6} + \)\(28\!\cdots\!56\)\( \beta_{7}) q^{67} +(-\)\(27\!\cdots\!20\)\( + \)\(13\!\cdots\!34\)\( \beta_{1} + \)\(79\!\cdots\!18\)\( \beta_{2} - \)\(45\!\cdots\!78\)\( \beta_{3} - \)\(64\!\cdots\!24\)\( \beta_{4} - \)\(40\!\cdots\!08\)\( \beta_{5} - \)\(26\!\cdots\!92\)\( \beta_{6} + \)\(36\!\cdots\!56\)\( \beta_{7}) q^{68} +(-\)\(67\!\cdots\!84\)\( + \)\(15\!\cdots\!36\)\( \beta_{1} + \)\(20\!\cdots\!40\)\( \beta_{2} + \)\(16\!\cdots\!56\)\( \beta_{3} + \)\(68\!\cdots\!24\)\( \beta_{4} - \)\(76\!\cdots\!36\)\( \beta_{5} + \)\(82\!\cdots\!04\)\( \beta_{6} - \)\(21\!\cdots\!76\)\( \beta_{7}) q^{69} +(-\)\(16\!\cdots\!00\)\( + \)\(47\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(45\!\cdots\!76\)\( \beta_{3} + \)\(37\!\cdots\!04\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(59\!\cdots\!00\)\( \beta_{6} + \)\(35\!\cdots\!00\)\( \beta_{7}) q^{70} +(-\)\(19\!\cdots\!28\)\( + \)\(59\!\cdots\!07\)\( \beta_{1} - \)\(51\!\cdots\!43\)\( \beta_{2} + \)\(23\!\cdots\!53\)\( \beta_{3} - \)\(36\!\cdots\!86\)\( \beta_{4} + \)\(93\!\cdots\!52\)\( \beta_{5} - \)\(31\!\cdots\!79\)\( \beta_{6} - \)\(57\!\cdots\!24\)\( \beta_{7}) q^{71} +(-\)\(68\!\cdots\!20\)\( - \)\(62\!\cdots\!00\)\( \beta_{1} - \)\(30\!\cdots\!31\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(74\!\cdots\!03\)\( \beta_{4} + \)\(47\!\cdots\!65\)\( \beta_{5} + \)\(47\!\cdots\!77\)\( \beta_{6} - \)\(12\!\cdots\!25\)\( \beta_{7}) q^{72} +(-\)\(25\!\cdots\!10\)\( - \)\(29\!\cdots\!94\)\( \beta_{1} + \)\(22\!\cdots\!58\)\( \beta_{2} + \)\(23\!\cdots\!44\)\( \beta_{3} - \)\(64\!\cdots\!78\)\( \beta_{4} - \)\(34\!\cdots\!72\)\( \beta_{5} + \)\(49\!\cdots\!20\)\( \beta_{6} + \)\(28\!\cdots\!24\)\( \beta_{7}) q^{73} +(-\)\(17\!\cdots\!36\)\( - \)\(96\!\cdots\!82\)\( \beta_{1} + \)\(27\!\cdots\!16\)\( \beta_{2} - \)\(37\!\cdots\!64\)\( \beta_{3} + \)\(76\!\cdots\!16\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} - \)\(29\!\cdots\!68\)\( \beta_{6} - \)\(20\!\cdots\!08\)\( \beta_{7}) q^{74} +(-\)\(25\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} + \)\(92\!\cdots\!25\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} - \)\(73\!\cdots\!00\)\( \beta_{4} + \)\(73\!\cdots\!00\)\( \beta_{5} + \)\(33\!\cdots\!00\)\( \beta_{6} - \)\(43\!\cdots\!00\)\( \beta_{7}) q^{75} +(\)\(80\!\cdots\!80\)\( + \)\(99\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!40\)\( \beta_{2} - \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(86\!\cdots\!04\)\( \beta_{4} - \)\(24\!\cdots\!44\)\( \beta_{5} + \)\(37\!\cdots\!16\)\( \beta_{6} + \)\(14\!\cdots\!96\)\( \beta_{7}) q^{76} +(\)\(31\!\cdots\!00\)\( + \)\(24\!\cdots\!08\)\( \beta_{1} - \)\(38\!\cdots\!32\)\( \beta_{2} + \)\(20\!\cdots\!68\)\( \beta_{3} + \)\(61\!\cdots\!04\)\( \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5} - \)\(11\!\cdots\!16\)\( \beta_{6} - \)\(20\!\cdots\!40\)\( \beta_{7}) q^{77} +(\)\(15\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(46\!\cdots\!98\)\( \beta_{3} + \)\(18\!\cdots\!24\)\( \beta_{4} + \)\(62\!\cdots\!06\)\( \beta_{5} - \)\(23\!\cdots\!80\)\( \beta_{6} + \)\(58\!\cdots\!48\)\( \beta_{7}) q^{78} +(\)\(18\!\cdots\!60\)\( + \)\(12\!\cdots\!54\)\( \beta_{1} - \)\(36\!\cdots\!70\)\( \beta_{2} - \)\(10\!\cdots\!06\)\( \beta_{3} + \)\(12\!\cdots\!96\)\( \beta_{4} + \)\(87\!\cdots\!36\)\( \beta_{5} + \)\(33\!\cdots\!86\)\( \beta_{6} + \)\(42\!\cdots\!16\)\( \beta_{7}) q^{79} +(\)\(75\!\cdots\!00\)\( + \)\(38\!\cdots\!64\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2} - \)\(52\!\cdots\!44\)\( \beta_{3} - \)\(35\!\cdots\!24\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(12\!\cdots\!00\)\( \beta_{7}) q^{80} +(\)\(18\!\cdots\!01\)\( - \)\(22\!\cdots\!42\)\( \beta_{1} + \)\(11\!\cdots\!42\)\( \beta_{2} - \)\(28\!\cdots\!96\)\( \beta_{3} + \)\(16\!\cdots\!34\)\( \beta_{4} - \)\(32\!\cdots\!48\)\( \beta_{5} - \)\(14\!\cdots\!64\)\( \beta_{6} + \)\(17\!\cdots\!16\)\( \beta_{7}) q^{81} +(\)\(37\!\cdots\!40\)\( - \)\(13\!\cdots\!42\)\( \beta_{1} - \)\(36\!\cdots\!40\)\( \beta_{2} + \)\(11\!\cdots\!16\)\( \beta_{3} - \)\(13\!\cdots\!72\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(23\!\cdots\!24\)\( \beta_{6} - \)\(29\!\cdots\!32\)\( \beta_{7}) q^{82} +(\)\(42\!\cdots\!20\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(45\!\cdots\!33\)\( \beta_{2} + \)\(49\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4} - \)\(50\!\cdots\!00\)\( \beta_{5} - \)\(73\!\cdots\!00\)\( \beta_{6} - \)\(44\!\cdots\!00\)\( \beta_{7}) q^{83} +(\)\(72\!\cdots\!04\)\( - \)\(10\!\cdots\!72\)\( \beta_{1} - \)\(64\!\cdots\!84\)\( \beta_{2} - \)\(86\!\cdots\!44\)\( \beta_{3} + \)\(59\!\cdots\!64\)\( \beta_{4} - \)\(59\!\cdots\!92\)\( \beta_{5} - \)\(19\!\cdots\!60\)\( \beta_{6} + \)\(96\!\cdots\!40\)\( \beta_{7}) q^{84} +(\)\(21\!\cdots\!00\)\( + \)\(28\!\cdots\!18\)\( \beta_{1} + \)\(97\!\cdots\!50\)\( \beta_{2} - \)\(63\!\cdots\!28\)\( \beta_{3} - \)\(78\!\cdots\!18\)\( \beta_{4} - \)\(27\!\cdots\!60\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} - \)\(70\!\cdots\!80\)\( \beta_{7}) q^{85} +(\)\(84\!\cdots\!52\)\( + \)\(11\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!60\)\( \beta_{2} + \)\(11\!\cdots\!55\)\( \beta_{3} + \)\(87\!\cdots\!22\)\( \beta_{4} + \)\(41\!\cdots\!77\)\( \beta_{5} - \)\(52\!\cdots\!48\)\( \beta_{6} - \)\(34\!\cdots\!88\)\( \beta_{7}) q^{86} +(\)\(32\!\cdots\!80\)\( + \)\(24\!\cdots\!93\)\( \beta_{1} + \)\(45\!\cdots\!47\)\( \beta_{2} + \)\(13\!\cdots\!35\)\( \beta_{3} - \)\(45\!\cdots\!10\)\( \beta_{4} + \)\(18\!\cdots\!92\)\( \beta_{5} + \)\(91\!\cdots\!27\)\( \beta_{6} + \)\(59\!\cdots\!16\)\( \beta_{7}) q^{87} +(-\)\(45\!\cdots\!80\)\( + \)\(32\!\cdots\!52\)\( \beta_{1} - \)\(74\!\cdots\!52\)\( \beta_{2} + \)\(20\!\cdots\!12\)\( \beta_{3} + \)\(26\!\cdots\!36\)\( \beta_{4} - \)\(45\!\cdots\!40\)\( \beta_{5} + \)\(70\!\cdots\!16\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7}) q^{88} +(-\)\(77\!\cdots\!70\)\( - \)\(36\!\cdots\!82\)\( \beta_{1} + \)\(54\!\cdots\!82\)\( \beta_{2} - \)\(26\!\cdots\!96\)\( \beta_{3} - \)\(17\!\cdots\!70\)\( \beta_{4} + \)\(12\!\cdots\!68\)\( \beta_{5} - \)\(56\!\cdots\!08\)\( \beta_{6} + \)\(10\!\cdots\!52\)\( \beta_{7}) q^{89} +(-\)\(58\!\cdots\!00\)\( - \)\(20\!\cdots\!86\)\( \beta_{1} - \)\(41\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!44\)\( \beta_{3} - \)\(12\!\cdots\!64\)\( \beta_{4} + \)\(75\!\cdots\!20\)\( \beta_{5} + \)\(74\!\cdots\!60\)\( \beta_{6} - \)\(27\!\cdots\!40\)\( \beta_{7}) q^{90} +(-\)\(46\!\cdots\!68\)\( - \)\(15\!\cdots\!72\)\( \beta_{1} + \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(14\!\cdots\!48\)\( \beta_{3} + \)\(12\!\cdots\!88\)\( \beta_{4} + \)\(13\!\cdots\!88\)\( \beta_{5} + \)\(56\!\cdots\!28\)\( \beta_{6} + \)\(19\!\cdots\!68\)\( \beta_{7}) q^{91} +(-\)\(57\!\cdots\!40\)\( - \)\(16\!\cdots\!40\)\( \beta_{1} - \)\(17\!\cdots\!32\)\( \beta_{2} - \)\(65\!\cdots\!84\)\( \beta_{3} + \)\(20\!\cdots\!28\)\( \beta_{4} - \)\(12\!\cdots\!64\)\( \beta_{5} - \)\(25\!\cdots\!16\)\( \beta_{6} + \)\(11\!\cdots\!48\)\( \beta_{7}) q^{92} +(-\)\(49\!\cdots\!40\)\( + \)\(51\!\cdots\!36\)\( \beta_{1} + \)\(18\!\cdots\!64\)\( \beta_{2} + \)\(70\!\cdots\!52\)\( \beta_{3} + \)\(17\!\cdots\!76\)\( \beta_{4} - \)\(23\!\cdots\!76\)\( \beta_{5} + \)\(16\!\cdots\!60\)\( \beta_{6} - \)\(16\!\cdots\!08\)\( \beta_{7}) q^{93} +(-\)\(21\!\cdots\!36\)\( + \)\(46\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!20\)\( \beta_{2} - \)\(75\!\cdots\!28\)\( \beta_{3} - \)\(14\!\cdots\!84\)\( \beta_{4} + \)\(63\!\cdots\!76\)\( \beta_{5} + \)\(23\!\cdots\!36\)\( \beta_{6} - \)\(12\!\cdots\!84\)\( \beta_{7}) q^{94} +(-\)\(33\!\cdots\!00\)\( + \)\(22\!\cdots\!65\)\( \beta_{1} + \)\(21\!\cdots\!75\)\( \beta_{2} - \)\(93\!\cdots\!65\)\( \beta_{3} + \)\(45\!\cdots\!10\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!25\)\( \beta_{6} + \)\(55\!\cdots\!00\)\( \beta_{7}) q^{95} +(\)\(58\!\cdots\!32\)\( + \)\(98\!\cdots\!24\)\( \beta_{1} - \)\(78\!\cdots\!56\)\( \beta_{2} - \)\(33\!\cdots\!84\)\( \beta_{3} + \)\(19\!\cdots\!56\)\( \beta_{4} + \)\(14\!\cdots\!32\)\( \beta_{5} - \)\(56\!\cdots\!40\)\( \beta_{6} - \)\(71\!\cdots\!40\)\( \beta_{7}) q^{96} +(\)\(80\!\cdots\!90\)\( + \)\(36\!\cdots\!22\)\( \beta_{1} + \)\(18\!\cdots\!34\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(10\!\cdots\!54\)\( \beta_{4} - \)\(34\!\cdots\!12\)\( \beta_{5} - \)\(37\!\cdots\!48\)\( \beta_{6} - \)\(15\!\cdots\!16\)\( \beta_{7}) q^{97} +(\)\(25\!\cdots\!40\)\( - \)\(27\!\cdots\!37\)\( \beta_{1} - \)\(33\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!24\)\( \beta_{3} + \)\(31\!\cdots\!92\)\( \beta_{4} - \)\(62\!\cdots\!76\)\( \beta_{5} + \)\(34\!\cdots\!96\)\( \beta_{6} + \)\(38\!\cdots\!32\)\( \beta_{7}) q^{98} +(\)\(28\!\cdots\!76\)\( - \)\(25\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!05\)\( \beta_{2} - \)\(17\!\cdots\!92\)\( \beta_{3} - \)\(18\!\cdots\!12\)\( \beta_{4} + \)\(29\!\cdots\!08\)\( \beta_{5} - \)\(25\!\cdots\!92\)\( \beta_{6} + \)\(54\!\cdots\!48\)\( \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 434989091795040q^{2} - \)\(12\!\cdots\!60\)\(q^{3} + \)\(90\!\cdots\!96\)\(q^{4} + \)\(38\!\cdots\!00\)\(q^{5} + \)\(24\!\cdots\!36\)\(q^{6} - \)\(57\!\cdots\!00\)\(q^{7} - \)\(61\!\cdots\!20\)\(q^{8} + \)\(52\!\cdots\!84\)\(q^{9} + O(q^{10}) \) \( 8q - 434989091795040q^{2} - \)\(12\!\cdots\!60\)\(q^{3} + \)\(90\!\cdots\!96\)\(q^{4} + \)\(38\!\cdots\!00\)\(q^{5} + \)\(24\!\cdots\!36\)\(q^{6} - \)\(57\!\cdots\!00\)\(q^{7} - \)\(61\!\cdots\!20\)\(q^{8} + \)\(52\!\cdots\!84\)\(q^{9} - \)\(37\!\cdots\!00\)\(q^{10} + \)\(46\!\cdots\!96\)\(q^{11} - \)\(72\!\cdots\!80\)\(q^{12} + \)\(25\!\cdots\!80\)\(q^{13} - \)\(48\!\cdots\!88\)\(q^{14} - \)\(29\!\cdots\!00\)\(q^{15} - \)\(10\!\cdots\!72\)\(q^{16} - \)\(39\!\cdots\!20\)\(q^{17} - \)\(72\!\cdots\!60\)\(q^{18} - \)\(21\!\cdots\!80\)\(q^{19} + \)\(17\!\cdots\!00\)\(q^{20} + \)\(40\!\cdots\!36\)\(q^{21} + \)\(61\!\cdots\!20\)\(q^{22} - \)\(13\!\cdots\!80\)\(q^{23} + \)\(10\!\cdots\!60\)\(q^{24} + \)\(77\!\cdots\!00\)\(q^{25} - \)\(97\!\cdots\!44\)\(q^{26} - \)\(59\!\cdots\!20\)\(q^{27} + \)\(92\!\cdots\!80\)\(q^{28} + \)\(15\!\cdots\!80\)\(q^{29} + \)\(11\!\cdots\!00\)\(q^{30} - \)\(65\!\cdots\!44\)\(q^{31} + \)\(12\!\cdots\!60\)\(q^{32} + \)\(43\!\cdots\!80\)\(q^{33} + \)\(95\!\cdots\!12\)\(q^{34} - \)\(11\!\cdots\!00\)\(q^{35} + \)\(19\!\cdots\!08\)\(q^{36} + \)\(39\!\cdots\!40\)\(q^{37} - \)\(70\!\cdots\!80\)\(q^{38} - \)\(26\!\cdots\!32\)\(q^{39} - \)\(76\!\cdots\!00\)\(q^{40} + \)\(56\!\cdots\!36\)\(q^{41} + \)\(30\!\cdots\!80\)\(q^{42} - \)\(28\!\cdots\!00\)\(q^{43} - \)\(20\!\cdots\!48\)\(q^{44} + \)\(71\!\cdots\!00\)\(q^{45} + \)\(10\!\cdots\!76\)\(q^{46} - \)\(45\!\cdots\!80\)\(q^{47} - \)\(58\!\cdots\!80\)\(q^{48} + \)\(12\!\cdots\!56\)\(q^{49} - \)\(40\!\cdots\!00\)\(q^{50} + \)\(11\!\cdots\!36\)\(q^{51} - \)\(73\!\cdots\!00\)\(q^{52} + \)\(13\!\cdots\!40\)\(q^{53} + \)\(68\!\cdots\!20\)\(q^{54} - \)\(14\!\cdots\!00\)\(q^{55} - \)\(23\!\cdots\!80\)\(q^{56} - \)\(66\!\cdots\!40\)\(q^{57} + \)\(29\!\cdots\!80\)\(q^{58} + \)\(21\!\cdots\!60\)\(q^{59} - \)\(34\!\cdots\!00\)\(q^{60} - \)\(33\!\cdots\!04\)\(q^{61} - \)\(58\!\cdots\!80\)\(q^{62} - \)\(20\!\cdots\!40\)\(q^{63} - \)\(74\!\cdots\!04\)\(q^{64} - \)\(16\!\cdots\!00\)\(q^{65} - \)\(74\!\cdots\!68\)\(q^{66} - \)\(61\!\cdots\!20\)\(q^{67} - \)\(21\!\cdots\!60\)\(q^{68} - \)\(53\!\cdots\!72\)\(q^{69} - \)\(12\!\cdots\!00\)\(q^{70} - \)\(15\!\cdots\!24\)\(q^{71} - \)\(55\!\cdots\!60\)\(q^{72} - \)\(20\!\cdots\!80\)\(q^{73} - \)\(14\!\cdots\!88\)\(q^{74} - \)\(20\!\cdots\!00\)\(q^{75} + \)\(64\!\cdots\!40\)\(q^{76} + \)\(25\!\cdots\!00\)\(q^{77} + \)\(12\!\cdots\!00\)\(q^{78} + \)\(14\!\cdots\!80\)\(q^{79} + \)\(60\!\cdots\!00\)\(q^{80} + \)\(14\!\cdots\!08\)\(q^{81} + \)\(30\!\cdots\!20\)\(q^{82} + \)\(33\!\cdots\!60\)\(q^{83} + \)\(57\!\cdots\!32\)\(q^{84} + \)\(17\!\cdots\!00\)\(q^{85} + \)\(67\!\cdots\!16\)\(q^{86} + \)\(25\!\cdots\!40\)\(q^{87} - \)\(36\!\cdots\!40\)\(q^{88} - \)\(62\!\cdots\!60\)\(q^{89} - \)\(47\!\cdots\!00\)\(q^{90} - \)\(36\!\cdots\!44\)\(q^{91} - \)\(46\!\cdots\!20\)\(q^{92} - \)\(39\!\cdots\!20\)\(q^{93} - \)\(17\!\cdots\!88\)\(q^{94} - \)\(26\!\cdots\!00\)\(q^{95} + \)\(46\!\cdots\!56\)\(q^{96} + \)\(64\!\cdots\!20\)\(q^{97} + \)\(20\!\cdots\!20\)\(q^{98} + \)\(22\!\cdots\!08\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - \)\(15\!\cdots\!64\)\( x^{6} - \)\(13\!\cdots\!76\)\( x^{5} + \)\(79\!\cdots\!56\)\( x^{4} + \)\(16\!\cdots\!20\)\( x^{3} - \)\(12\!\cdots\!00\)\( x^{2} - \)\(46\!\cdots\!00\)\( x + \)\(14\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 96 \nu - 12 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(27\!\cdots\!29\)\( \nu^{7} + \)\(70\!\cdots\!21\)\( \nu^{6} + \)\(45\!\cdots\!04\)\( \nu^{5} - \)\(80\!\cdots\!20\)\( \nu^{4} - \)\(22\!\cdots\!44\)\( \nu^{3} + \)\(23\!\cdots\!12\)\( \nu^{2} + \)\(29\!\cdots\!28\)\( \nu - \)\(62\!\cdots\!32\)\(\)\()/ \)\(50\!\cdots\!92\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(24\!\cdots\!45\)\( \nu^{7} - \)\(63\!\cdots\!05\)\( \nu^{6} - \)\(40\!\cdots\!20\)\( \nu^{5} + \)\(73\!\cdots\!00\)\( \nu^{4} + \)\(20\!\cdots\!20\)\( \nu^{3} - \)\(73\!\cdots\!56\)\( \nu^{2} - \)\(28\!\cdots\!40\)\( \nu - \)\(47\!\cdots\!92\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(10\!\cdots\!11\)\( \nu^{7} - \)\(52\!\cdots\!09\)\( \nu^{6} + \)\(11\!\cdots\!24\)\( \nu^{5} + \)\(18\!\cdots\!16\)\( \nu^{4} - \)\(28\!\cdots\!96\)\( \nu^{3} - \)\(11\!\cdots\!20\)\( \nu^{2} - \)\(52\!\cdots\!20\)\( \nu + \)\(85\!\cdots\!80\)\(\)\()/ \)\(35\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(55\!\cdots\!57\)\( \nu^{7} + \)\(44\!\cdots\!17\)\( \nu^{6} + \)\(68\!\cdots\!28\)\( \nu^{5} - \)\(81\!\cdots\!88\)\( \nu^{4} - \)\(23\!\cdots\!52\)\( \nu^{3} + \)\(56\!\cdots\!20\)\( \nu^{2} + \)\(19\!\cdots\!00\)\( \nu - \)\(39\!\cdots\!40\)\(\)\()/ \)\(35\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(18\!\cdots\!09\)\( \nu^{7} - \)\(29\!\cdots\!91\)\( \nu^{6} + \)\(29\!\cdots\!56\)\( \nu^{5} + \)\(32\!\cdots\!44\)\( \nu^{4} - \)\(13\!\cdots\!64\)\( \nu^{3} - \)\(88\!\cdots\!20\)\( \nu^{2} + \)\(15\!\cdots\!20\)\( \nu + \)\(43\!\cdots\!60\)\(\)\()/ \)\(12\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(93\!\cdots\!41\)\( \nu^{7} - \)\(78\!\cdots\!61\)\( \nu^{6} - \)\(17\!\cdots\!64\)\( \nu^{5} + \)\(67\!\cdots\!64\)\( \nu^{4} + \)\(11\!\cdots\!76\)\( \nu^{3} - \)\(76\!\cdots\!20\)\( \nu^{2} - \)\(27\!\cdots\!20\)\( \nu - \)\(59\!\cdots\!00\)\(\)\()/ \)\(27\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 12\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 310415 \beta_{2} + 120674951586825 \beta_{1} + 3658465598508525014815957089408\)\()/9216\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 183 \beta_{6} - 181393 \beta_{5} + 1912817011 \beta_{4} - 287410639083512 \beta_{3} + 445211688083035631123 \beta_{2} + 5446795118464140208309951300408 \beta_{1} + 441485158567510392680440580527129275362783232\)\()/884736\)
\(\nu^{4}\)\(=\)\((\)\(-6908623838013 \beta_{7} - 112550735493275 \beta_{6} - 15409778375279271539 \beta_{5} + 638680884319226457526937 \beta_{4} + 218388789881400125335150732568 \beta_{3} + 73042206322836907577922932150840697 \beta_{2} + 21463000735052476685530295597495512985440936 \beta_{1} + 622716017575955242197588480382533197831239774225040133525504\)\()/2654208\)
\(\nu^{5}\)\(=\)\((\)\(8218298097895220964852696257 \beta_{7} + 2358026885524086501532101010935 \beta_{6} - 2722541821488810891732549582337361 \beta_{5} + 11544797948364774240846859265081059251 \beta_{4} - 2246920756066341539294995600654021605022968 \beta_{3} + 5118314875838883981914228975238504442757409571667 \beta_{2} + 31676186311443576444645909909326000637448508622607582972472 \beta_{1} + 2453801554137476819559674420827512093324808121982095371414910936356466688\)\()/7962624\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(59\!\cdots\!37\)\( \beta_{7} - \)\(43\!\cdots\!55\)\( \beta_{6} - \)\(17\!\cdots\!75\)\( \beta_{5} + \)\(72\!\cdots\!21\)\( \beta_{4} + \)\(15\!\cdots\!32\)\( \beta_{3} + \)\(66\!\cdots\!21\)\( \beta_{2} + \)\(11\!\cdots\!44\)\( \beta_{1} + \)\(40\!\cdots\!04\)\(\)\()/2654208\)
\(\nu^{7}\)\(=\)\((\)\(\)\(21\!\cdots\!63\)\( \beta_{7} + \)\(74\!\cdots\!61\)\( \beta_{6} - \)\(10\!\cdots\!23\)\( \beta_{5} + \)\(14\!\cdots\!73\)\( \beta_{4} - \)\(57\!\cdots\!68\)\( \beta_{3} + \)\(15\!\cdots\!25\)\( \beta_{2} + \)\(71\!\cdots\!72\)\( \beta_{1} + \)\(44\!\cdots\!96\)\(\)\()/2654208\)

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
2.71282e13
2.42719e13
1.78363e13
2.10926e12
−6.74954e12
−1.40661e13
−2.32579e13
−2.72721e13
−2.65868e15 −2.23967e24 4.53328e30 2.85912e35 5.95456e39 3.30273e41 −5.31200e45 3.46997e48 −7.60148e50
1.2 −2.38447e15 1.43714e24 3.15040e30 7.99157e34 −3.42682e39 3.14769e42 −1.46670e45 5.19237e47 −1.90557e50
1.3 −1.76666e15 −3.81746e23 5.85773e29 −3.44135e35 6.74413e38 −5.35995e42 3.44415e45 −1.40040e48 6.07967e50
1.4 −2.56863e14 −3.45553e23 −2.46932e30 1.16158e35 8.87596e37 2.79663e42 1.28550e45 −1.42673e48 −2.98366e49
1.5 5.93582e14 2.15060e24 −2.18296e30 2.07725e34 1.27656e39 −6.15512e42 −2.80068e45 3.07894e48 1.23302e49
1.6 1.29597e15 −2.26314e24 −8.55769e29 −1.52987e35 −2.93295e39 −2.30272e42 −4.39472e45 3.57566e48 −1.98266e50
1.7 2.17839e15 6.93716e23 2.21006e30 −2.56275e35 1.51118e39 8.01618e42 −7.08493e44 −1.06489e48 −5.58265e50
1.8 2.56375e15 −2.60728e23 4.03750e30 2.88877e35 −6.68439e38 −6.25978e42 3.85125e45 −1.47815e48 7.40607e50
\(n\): 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 1.102.a.a 8
3.b odd 2 1 9.102.a.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.102.a.a 8 1.a even 1 1 trivial
9.102.a.b 8 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{102}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 434989091795040 T + \)\(57\!\cdots\!60\)\( T^{2} + \)\(41\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(90\!\cdots\!20\)\( T^{6} + \)\(52\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!96\)\( T^{8} + \)\(13\!\cdots\!20\)\( T^{9} + \)\(58\!\cdots\!80\)\( T^{10} + \)\(23\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!56\)\( T^{12} + \)\(43\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!40\)\( T^{14} + \)\(29\!\cdots\!20\)\( T^{15} + \)\(17\!\cdots\!56\)\( T^{16} \)
$3$ \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(42\!\cdots\!20\)\( T^{2} + \)\(59\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!36\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(32\!\cdots\!86\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{9} + \)\(43\!\cdots\!60\)\( T^{10} + \)\(48\!\cdots\!40\)\( T^{11} + \)\(51\!\cdots\!16\)\( T^{12} + \)\(52\!\cdots\!40\)\( T^{13} + \)\(58\!\cdots\!80\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(32\!\cdots\!61\)\( T^{16} \)
$5$ \( 1 - \)\(38\!\cdots\!00\)\( T + \)\(11\!\cdots\!00\)\( T^{2} + \)\(81\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!00\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(43\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!50\)\( T^{8} + \)\(69\!\cdots\!00\)\( T^{9} + \)\(67\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!00\)\( T^{12} + \)\(77\!\cdots\!00\)\( T^{13} + \)\(45\!\cdots\!00\)\( T^{14} - \)\(56\!\cdots\!00\)\( T^{15} + \)\(58\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + \)\(57\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!96\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!06\)\( T^{8} + \)\(66\!\cdots\!00\)\( T^{9} + \)\(70\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(24\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 - \)\(46\!\cdots\!96\)\( T + \)\(53\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(68\!\cdots\!68\)\( T^{5} + \)\(32\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!40\)\( T^{7} + \)\(64\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!40\)\( T^{9} + \)\(75\!\cdots\!08\)\( T^{10} - \)\(23\!\cdots\!08\)\( T^{11} + \)\(71\!\cdots\!20\)\( T^{12} - \)\(13\!\cdots\!60\)\( T^{13} + \)\(64\!\cdots\!20\)\( T^{14} - \)\(85\!\cdots\!16\)\( T^{15} + \)\(27\!\cdots\!81\)\( T^{16} \)
$13$ \( 1 - \)\(25\!\cdots\!80\)\( T + \)\(17\!\cdots\!40\)\( T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} - \)\(30\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!66\)\( T^{8} - \)\(47\!\cdots\!40\)\( T^{9} + \)\(86\!\cdots\!20\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!36\)\( T^{12} - \)\(13\!\cdots\!20\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} - \)\(90\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \)
$17$ \( 1 + \)\(39\!\cdots\!20\)\( T + \)\(16\!\cdots\!80\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!60\)\( T^{6} + \)\(53\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!40\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(97\!\cdots\!80\)\( T^{13} + \)\(74\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!41\)\( T^{16} \)
$19$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(87\!\cdots\!52\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(36\!\cdots\!08\)\( T^{4} + \)\(48\!\cdots\!80\)\( T^{5} + \)\(92\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!70\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!44\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!68\)\( T^{12} + \)\(85\!\cdots\!60\)\( T^{13} + \)\(73\!\cdots\!12\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} \)
$23$ \( 1 + \)\(13\!\cdots\!80\)\( T + \)\(17\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} + \)\(96\!\cdots\!60\)\( T^{5} + \)\(71\!\cdots\!20\)\( T^{6} + \)\(45\!\cdots\!80\)\( T^{7} + \)\(28\!\cdots\!46\)\( T^{8} + \)\(15\!\cdots\!40\)\( T^{9} + \)\(83\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!56\)\( T^{12} + \)\(71\!\cdots\!20\)\( T^{13} + \)\(28\!\cdots\!40\)\( T^{14} + \)\(75\!\cdots\!60\)\( T^{15} + \)\(18\!\cdots\!81\)\( T^{16} \)
$29$ \( 1 - \)\(15\!\cdots\!80\)\( T + \)\(17\!\cdots\!32\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!48\)\( T^{4} - \)\(97\!\cdots\!80\)\( T^{5} + \)\(74\!\cdots\!84\)\( T^{6} - \)\(75\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!70\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!44\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(89\!\cdots\!88\)\( T^{12} - \)\(23\!\cdots\!60\)\( T^{13} + \)\(27\!\cdots\!72\)\( T^{14} - \)\(12\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!61\)\( T^{16} \)
$31$ \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(33\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!20\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} + \)\(62\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!70\)\( T^{8} + \)\(26\!\cdots\!60\)\( T^{9} + \)\(49\!\cdots\!08\)\( T^{10} + \)\(81\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!20\)\( T^{12} + \)\(16\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!20\)\( T^{14} + \)\(16\!\cdots\!84\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} \)
$37$ \( 1 - \)\(39\!\cdots\!40\)\( T + \)\(16\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!76\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} - \)\(31\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!66\)\( T^{8} - \)\(76\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!20\)\( T^{10} - \)\(19\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!36\)\( T^{12} - \)\(31\!\cdots\!60\)\( T^{13} + \)\(34\!\cdots\!60\)\( T^{14} - \)\(20\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!21\)\( T^{16} \)
$41$ \( 1 - \)\(56\!\cdots\!36\)\( T + \)\(41\!\cdots\!20\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!20\)\( T^{4} - \)\(28\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} - \)\(30\!\cdots\!40\)\( T^{7} + \)\(96\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!40\)\( T^{9} + \)\(64\!\cdots\!08\)\( T^{10} - \)\(13\!\cdots\!28\)\( T^{11} + \)\(30\!\cdots\!20\)\( T^{12} - \)\(49\!\cdots\!60\)\( T^{13} + \)\(91\!\cdots\!20\)\( T^{14} - \)\(97\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$43$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(58\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!96\)\( T^{4} + \)\(26\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(44\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + \)\(45\!\cdots\!80\)\( T + \)\(42\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(80\!\cdots\!36\)\( T^{4} + \)\(22\!\cdots\!60\)\( T^{5} + \)\(99\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} + \)\(18\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!60\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(26\!\cdots\!16\)\( T^{12} + \)\(36\!\cdots\!20\)\( T^{13} + \)\(82\!\cdots\!80\)\( T^{14} + \)\(68\!\cdots\!40\)\( T^{15} + \)\(11\!\cdots\!61\)\( T^{16} \)
$53$ \( 1 - \)\(13\!\cdots\!40\)\( T + \)\(52\!\cdots\!20\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!36\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!40\)\( T^{7} + \)\(58\!\cdots\!86\)\( T^{8} - \)\(31\!\cdots\!20\)\( T^{9} + \)\(71\!\cdots\!60\)\( T^{10} - \)\(30\!\cdots\!60\)\( T^{11} + \)\(63\!\cdots\!16\)\( T^{12} - \)\(17\!\cdots\!60\)\( T^{13} + \)\(42\!\cdots\!80\)\( T^{14} - \)\(15\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!61\)\( T^{16} \)
$59$ \( 1 - \)\(21\!\cdots\!60\)\( T + \)\(28\!\cdots\!72\)\( T^{2} - \)\(86\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!68\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{5} + \)\(50\!\cdots\!24\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(26\!\cdots\!44\)\( T^{10} - \)\(57\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{13} + \)\(39\!\cdots\!52\)\( T^{14} - \)\(20\!\cdots\!40\)\( T^{15} + \)\(70\!\cdots\!21\)\( T^{16} \)
$61$ \( 1 + \)\(33\!\cdots\!04\)\( T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!20\)\( T^{4} + \)\(18\!\cdots\!32\)\( T^{5} + \)\(34\!\cdots\!48\)\( T^{6} + \)\(56\!\cdots\!60\)\( T^{7} + \)\(86\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} + \)\(16\!\cdots\!92\)\( T^{11} + \)\(16\!\cdots\!20\)\( T^{12} + \)\(13\!\cdots\!40\)\( T^{13} + \)\(11\!\cdots\!20\)\( T^{14} + \)\(57\!\cdots\!84\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} \)
$67$ \( 1 + \)\(61\!\cdots\!20\)\( T + \)\(25\!\cdots\!80\)\( T^{2} + \)\(85\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!56\)\( T^{4} + \)\(57\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!80\)\( T^{7} + \)\(41\!\cdots\!26\)\( T^{8} + \)\(64\!\cdots\!60\)\( T^{9} + \)\(90\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(67\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} \)
$71$ \( 1 + \)\(15\!\cdots\!24\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!20\)\( T^{4} + \)\(37\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!88\)\( T^{6} + \)\(61\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} + \)\(58\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} + \)\(32\!\cdots\!52\)\( T^{11} + \)\(61\!\cdots\!20\)\( T^{12} + \)\(95\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!20\)\( T^{14} + \)\(10\!\cdots\!84\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$73$ \( 1 + \)\(20\!\cdots\!80\)\( T + \)\(67\!\cdots\!60\)\( T^{2} + \)\(75\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!46\)\( T^{8} + \)\(16\!\cdots\!40\)\( T^{9} + \)\(52\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(92\!\cdots\!56\)\( T^{12} + \)\(72\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(47\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!81\)\( T^{16} \)
$79$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(34\!\cdots\!32\)\( T^{2} - \)\(37\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!48\)\( T^{4} - \)\(44\!\cdots\!80\)\( T^{5} + \)\(44\!\cdots\!84\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(93\!\cdots\!44\)\( T^{10} - \)\(42\!\cdots\!20\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} - \)\(74\!\cdots\!60\)\( T^{13} + \)\(31\!\cdots\!72\)\( T^{14} - \)\(61\!\cdots\!20\)\( T^{15} + \)\(19\!\cdots\!61\)\( T^{16} \)
$83$ \( 1 - \)\(33\!\cdots\!60\)\( T + \)\(94\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!56\)\( T^{4} - \)\(38\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} - \)\(44\!\cdots\!60\)\( T^{7} + \)\(39\!\cdots\!26\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!76\)\( T^{12} - \)\(24\!\cdots\!40\)\( T^{13} + \)\(86\!\cdots\!20\)\( T^{14} - \)\(20\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!41\)\( T^{16} \)
$89$ \( 1 + \)\(62\!\cdots\!60\)\( T + \)\(43\!\cdots\!12\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!88\)\( T^{4} + \)\(37\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} + \)\(32\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!44\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(35\!\cdots\!08\)\( T^{12} + \)\(57\!\cdots\!20\)\( T^{13} + \)\(93\!\cdots\!32\)\( T^{14} + \)\(10\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$97$ \( 1 - \)\(64\!\cdots\!20\)\( T + \)\(50\!\cdots\!20\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(94\!\cdots\!36\)\( T^{4} - \)\(29\!\cdots\!40\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!80\)\( T^{7} + \)\(55\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(20\!\cdots\!60\)\( T^{10} - \)\(28\!\cdots\!20\)\( T^{11} + \)\(42\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!80\)\( T^{13} + \)\(48\!\cdots\!80\)\( T^{14} - \)\(28\!\cdots\!60\)\( T^{15} + \)\(20\!\cdots\!61\)\( T^{16} \)
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