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 \( 2^{119}\cdot 3^{37}\cdot 5^{14}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 17^{2} \)
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.

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

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

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

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.

Hecke kernels

There are no other newforms in \(S_{102}^{\mathrm{new}}(\Gamma_0(1))\).