Properties

Label 1.106.a.a
Level 1
Weight 106
Character orbit 1.a
Self dual Yes
Analytic conductor 69.819
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 106 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(69.8187388595\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{111}\cdot 3^{44}\cdot 5^{13}\cdot 7^{7}\cdot 11\cdot 13^{3}\cdot 17 \)
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.

\(f(q)\) \(=\) \( q\) \(+(-1146879061146990 + \beta_{1}) q^{2}\) \(+(-\)\(44\!\cdots\!60\)\( - 314342399 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(15\!\cdots\!72\)\( - 2165061586903924 \beta_{1} + 1007775 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(93\!\cdots\!50\)\( + \)\(10\!\cdots\!87\)\( \beta_{1} - 55401614498 \beta_{2} - 19997 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(16\!\cdots\!08\)\( + \)\(28\!\cdots\!12\)\( \beta_{1} - 2707251382104775 \beta_{2} - 419725070 \beta_{3} - 243 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(88\!\cdots\!00\)\( - \)\(18\!\cdots\!72\)\( \beta_{1} + 2699450686949966629 \beta_{2} - 1080220275289 \beta_{3} - 1300315 \beta_{4} - 83 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(88\!\cdots\!20\)\( + \)\(11\!\cdots\!13\)\( \beta_{1} - \)\(57\!\cdots\!87\)\( \beta_{2} - 351950150081493 \beta_{3} + 3050104560 \beta_{4} + 920971 \beta_{5} + 253 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(39\!\cdots\!73\)\( - \)\(52\!\cdots\!50\)\( \beta_{1} + \)\(24\!\cdots\!40\)\( \beta_{2} + 247697597880192966 \beta_{3} - 1288526907606 \beta_{4} - 421417308 \beta_{5} - 193020 \beta_{6} - 456 \beta_{7}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-1146879061146990 + \beta_{1}) q^{2}\) \(+(-\)\(44\!\cdots\!60\)\( - 314342399 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(15\!\cdots\!72\)\( - 2165061586903924 \beta_{1} + 1007775 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(93\!\cdots\!50\)\( + \)\(10\!\cdots\!87\)\( \beta_{1} - 55401614498 \beta_{2} - 19997 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(16\!\cdots\!08\)\( + \)\(28\!\cdots\!12\)\( \beta_{1} - 2707251382104775 \beta_{2} - 419725070 \beta_{3} - 243 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(88\!\cdots\!00\)\( - \)\(18\!\cdots\!72\)\( \beta_{1} + 2699450686949966629 \beta_{2} - 1080220275289 \beta_{3} - 1300315 \beta_{4} - 83 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(88\!\cdots\!20\)\( + \)\(11\!\cdots\!13\)\( \beta_{1} - \)\(57\!\cdots\!87\)\( \beta_{2} - 351950150081493 \beta_{3} + 3050104560 \beta_{4} + 920971 \beta_{5} + 253 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(39\!\cdots\!73\)\( - \)\(52\!\cdots\!50\)\( \beta_{1} + \)\(24\!\cdots\!40\)\( \beta_{2} + 247697597880192966 \beta_{3} - 1288526907606 \beta_{4} - 421417308 \beta_{5} - 193020 \beta_{6} - 456 \beta_{7}) q^{9}\) \(+(\)\(55\!\cdots\!00\)\( - \)\(81\!\cdots\!26\)\( \beta_{1} + \)\(21\!\cdots\!04\)\( \beta_{2} + \)\(12\!\cdots\!56\)\( \beta_{3} - 92058652701748 \beta_{4} - 77306877500 \beta_{5} - 100613000 \beta_{6} - 66600 \beta_{7}) q^{10}\) \(+(-\)\(11\!\cdots\!48\)\( + \)\(11\!\cdots\!59\)\( \beta_{1} - \)\(77\!\cdots\!35\)\( \beta_{2} + \)\(20\!\cdots\!18\)\( \beta_{3} + 64914103221863026 \beta_{4} - 4193024729182 \beta_{5} - 12671658070 \beta_{6} + 14891744 \beta_{7}) q^{11}\) \(+(\)\(19\!\cdots\!20\)\( - \)\(28\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!16\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} - 7958310877059154560 \beta_{4} - 6059025539409096 \beta_{5} + 344959819272 \beta_{6} - 1180575576 \beta_{7}) q^{12}\) \(+(\)\(50\!\cdots\!30\)\( - \)\(39\!\cdots\!61\)\( \beta_{1} + \)\(72\!\cdots\!70\)\( \beta_{2} + \)\(89\!\cdots\!79\)\( \beta_{3} - \)\(12\!\cdots\!75\)\( \beta_{4} - 263724497697627304 \beta_{5} + 58723631479448 \beta_{6} + 55401123024 \beta_{7}) q^{13}\) \(+(-\)\(20\!\cdots\!16\)\( - \)\(17\!\cdots\!84\)\( \beta_{1} - \)\(97\!\cdots\!30\)\( \beta_{2} + \)\(50\!\cdots\!16\)\( \beta_{3} - \)\(18\!\cdots\!70\)\( \beta_{4} + 5754584147636521490 \beta_{5} - 4288029853862560 \beta_{6} - 1737608742688 \beta_{7}) q^{14}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(62\!\cdots\!52\)\( \beta_{1} - \)\(46\!\cdots\!33\)\( \beta_{2} - \)\(18\!\cdots\!87\)\( \beta_{3} - \)\(10\!\cdots\!29\)\( \beta_{4} + \)\(41\!\cdots\!75\)\( \beta_{5} + 131167256677905375 \beta_{6} + 36476646667200 \beta_{7}) q^{15}\) \(+(\)\(11\!\cdots\!56\)\( - \)\(40\!\cdots\!36\)\( \beta_{1} + \)\(33\!\cdots\!40\)\( \beta_{2} + \)\(31\!\cdots\!92\)\( \beta_{3} - \)\(14\!\cdots\!68\)\( \beta_{4} - \)\(13\!\cdots\!24\)\( \beta_{5} - 1816885189362163120 \beta_{6} - 404880132971376 \beta_{7}) q^{16}\) \(+(-\)\(59\!\cdots\!70\)\( + \)\(13\!\cdots\!02\)\( \beta_{1} - \)\(67\!\cdots\!28\)\( \beta_{2} - \)\(50\!\cdots\!06\)\( \beta_{3} + \)\(39\!\cdots\!90\)\( \beta_{4} + \)\(18\!\cdots\!28\)\( \beta_{5} - 11574678955880903916 \beta_{6} - 3995118584607720 \beta_{7}) q^{17}\) \(+(-\)\(32\!\cdots\!10\)\( + \)\(60\!\cdots\!61\)\( \beta_{1} - \)\(33\!\cdots\!24\)\( \beta_{2} - \)\(20\!\cdots\!76\)\( \beta_{3} + \)\(79\!\cdots\!40\)\( \beta_{4} + \)\(55\!\cdots\!68\)\( \beta_{5} + \)\(11\!\cdots\!04\)\( \beta_{6} + 328918256323171920 \beta_{7}) q^{18}\) \(+(-\)\(23\!\cdots\!40\)\( + \)\(13\!\cdots\!17\)\( \beta_{1} - \)\(53\!\cdots\!45\)\( \beta_{2} - \)\(19\!\cdots\!30\)\( \beta_{3} - \)\(11\!\cdots\!58\)\( \beta_{4} - \)\(10\!\cdots\!94\)\( \beta_{5} - \)\(29\!\cdots\!90\)\( \beta_{6} - 9634352666571803232 \beta_{7}) q^{19}\) \(+(-\)\(54\!\cdots\!00\)\( + \)\(67\!\cdots\!64\)\( \beta_{1} - \)\(33\!\cdots\!06\)\( \beta_{2} - \)\(25\!\cdots\!34\)\( \beta_{3} - \)\(15\!\cdots\!28\)\( \beta_{4} + \)\(52\!\cdots\!00\)\( \beta_{5} + \)\(47\!\cdots\!00\)\( \beta_{6} + \)\(19\!\cdots\!00\)\( \beta_{7}) q^{20}\) \(+(\)\(42\!\cdots\!32\)\( - \)\(13\!\cdots\!84\)\( \beta_{1} - \)\(48\!\cdots\!40\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3} + \)\(26\!\cdots\!60\)\( \beta_{4} + \)\(87\!\cdots\!80\)\( \beta_{5} - \)\(52\!\cdots\!80\)\( \beta_{6} - \)\(33\!\cdots\!24\)\( \beta_{7}) q^{21}\) \(+(\)\(77\!\cdots\!20\)\( - \)\(83\!\cdots\!44\)\( \beta_{1} + \)\(55\!\cdots\!35\)\( \beta_{2} + \)\(52\!\cdots\!46\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} - \)\(17\!\cdots\!41\)\( \beta_{5} + \)\(37\!\cdots\!92\)\( \beta_{6} + \)\(45\!\cdots\!16\)\( \beta_{7}) q^{22}\) \(+(\)\(44\!\cdots\!20\)\( - \)\(35\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!21\)\( \beta_{2} + \)\(13\!\cdots\!65\)\( \beta_{3} - \)\(36\!\cdots\!25\)\( \beta_{4} + \)\(89\!\cdots\!55\)\( \beta_{5} - \)\(87\!\cdots\!85\)\( \beta_{6} - \)\(53\!\cdots\!00\)\( \beta_{7}) q^{23}\) \(+(-\)\(10\!\cdots\!20\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(29\!\cdots\!40\)\( \beta_{2} - \)\(41\!\cdots\!60\)\( \beta_{3} + \)\(98\!\cdots\!72\)\( \beta_{4} + \)\(87\!\cdots\!96\)\( \beta_{5} - \)\(18\!\cdots\!80\)\( \beta_{6} + \)\(54\!\cdots\!76\)\( \beta_{7}) q^{24}\) \(+(\)\(45\!\cdots\!75\)\( + \)\(38\!\cdots\!00\)\( \beta_{1} - \)\(15\!\cdots\!00\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4} - \)\(19\!\cdots\!00\)\( \beta_{5} + \)\(30\!\cdots\!00\)\( \beta_{6} - \)\(47\!\cdots\!00\)\( \beta_{7}) q^{25}\) \(+(-\)\(22\!\cdots\!28\)\( + \)\(58\!\cdots\!38\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!32\)\( \beta_{3} - \)\(13\!\cdots\!24\)\( \beta_{4} + \)\(15\!\cdots\!68\)\( \beta_{5} - \)\(25\!\cdots\!40\)\( \beta_{6} + \)\(35\!\cdots\!88\)\( \beta_{7}) q^{26}\) \(+(\)\(51\!\cdots\!80\)\( - \)\(65\!\cdots\!42\)\( \beta_{1} + \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(17\!\cdots\!82\)\( \beta_{3} - \)\(65\!\cdots\!90\)\( \beta_{4} - \)\(47\!\cdots\!94\)\( \beta_{5} + \)\(12\!\cdots\!58\)\( \beta_{6} - \)\(22\!\cdots\!44\)\( \beta_{7}) q^{27}\) \(+(-\)\(12\!\cdots\!20\)\( + \)\(13\!\cdots\!96\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2} - \)\(60\!\cdots\!72\)\( \beta_{3} + \)\(97\!\cdots\!40\)\( \beta_{4} - \)\(24\!\cdots\!96\)\( \beta_{5} - \)\(17\!\cdots\!28\)\( \beta_{6} + \)\(10\!\cdots\!64\)\( \beta_{7}) q^{28}\) \(+(-\)\(16\!\cdots\!10\)\( + \)\(84\!\cdots\!31\)\( \beta_{1} + \)\(19\!\cdots\!70\)\( \beta_{2} - \)\(82\!\cdots\!53\)\( \beta_{3} - \)\(12\!\cdots\!11\)\( \beta_{4} + \)\(34\!\cdots\!52\)\( \beta_{5} - \)\(54\!\cdots\!60\)\( \beta_{6} - \)\(30\!\cdots\!48\)\( \beta_{7}) q^{29}\) \(+(\)\(46\!\cdots\!00\)\( - \)\(18\!\cdots\!96\)\( \beta_{1} + \)\(24\!\cdots\!34\)\( \beta_{2} + \)\(17\!\cdots\!76\)\( \beta_{3} - \)\(30\!\cdots\!58\)\( \beta_{4} - \)\(14\!\cdots\!50\)\( \beta_{5} + \)\(53\!\cdots\!00\)\( \beta_{6} - \)\(95\!\cdots\!00\)\( \beta_{7}) q^{30}\) \(+(\)\(26\!\cdots\!52\)\( - \)\(51\!\cdots\!72\)\( \beta_{1} - \)\(62\!\cdots\!40\)\( \beta_{2} + \)\(26\!\cdots\!60\)\( \beta_{3} + \)\(14\!\cdots\!08\)\( \beta_{4} - \)\(12\!\cdots\!56\)\( \beta_{5} - \)\(28\!\cdots\!80\)\( \beta_{6} + \)\(21\!\cdots\!36\)\( \beta_{7}) q^{31}\) \(+(\)\(12\!\cdots\!60\)\( - \)\(11\!\cdots\!44\)\( \beta_{1} - \)\(57\!\cdots\!40\)\( \beta_{2} - \)\(51\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4} + \)\(40\!\cdots\!88\)\( \beta_{5} + \)\(62\!\cdots\!64\)\( \beta_{6} - \)\(19\!\cdots\!20\)\( \beta_{7}) q^{32}\) \(+(-\)\(13\!\cdots\!20\)\( + \)\(69\!\cdots\!98\)\( \beta_{1} - \)\(29\!\cdots\!88\)\( \beta_{2} - \)\(50\!\cdots\!98\)\( \beta_{3} - \)\(42\!\cdots\!70\)\( \beta_{4} - \)\(12\!\cdots\!88\)\( \beta_{5} + \)\(35\!\cdots\!96\)\( \beta_{6} + \)\(12\!\cdots\!04\)\( \beta_{7}) q^{33}\) \(+(\)\(78\!\cdots\!44\)\( - \)\(25\!\cdots\!70\)\( \beta_{1} + \)\(34\!\cdots\!80\)\( \beta_{2} + \)\(42\!\cdots\!28\)\( \beta_{3} + \)\(80\!\cdots\!72\)\( \beta_{4} - \)\(74\!\cdots\!04\)\( \beta_{5} - \)\(45\!\cdots\!40\)\( \beta_{6} - \)\(68\!\cdots\!52\)\( \beta_{7}) q^{34}\) \(+(-\)\(22\!\cdots\!00\)\( - \)\(97\!\cdots\!96\)\( \beta_{1} + \)\(33\!\cdots\!84\)\( \beta_{2} + \)\(64\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!92\)\( \beta_{4} + \)\(85\!\cdots\!00\)\( \beta_{5} + \)\(25\!\cdots\!00\)\( \beta_{6} + \)\(29\!\cdots\!00\)\( \beta_{7}) q^{35}\) \(+(\)\(20\!\cdots\!56\)\( - \)\(92\!\cdots\!92\)\( \beta_{1} + \)\(35\!\cdots\!75\)\( \beta_{2} + \)\(35\!\cdots\!01\)\( \beta_{3} - \)\(28\!\cdots\!68\)\( \beta_{4} - \)\(33\!\cdots\!24\)\( \beta_{5} - \)\(75\!\cdots\!00\)\( \beta_{6} - \)\(10\!\cdots\!00\)\( \beta_{7}) q^{36}\) \(+(-\)\(28\!\cdots\!10\)\( - \)\(21\!\cdots\!37\)\( \beta_{1} - \)\(45\!\cdots\!62\)\( \beta_{2} - \)\(22\!\cdots\!21\)\( \beta_{3} - \)\(10\!\cdots\!55\)\( \beta_{4} + \)\(24\!\cdots\!92\)\( \beta_{5} - \)\(10\!\cdots\!44\)\( \beta_{6} + \)\(28\!\cdots\!12\)\( \beta_{7}) q^{37}\) \(+(\)\(10\!\cdots\!20\)\( - \)\(87\!\cdots\!00\)\( \beta_{1} - \)\(50\!\cdots\!63\)\( \beta_{2} - \)\(77\!\cdots\!90\)\( \beta_{3} + \)\(63\!\cdots\!65\)\( \beta_{4} + \)\(35\!\cdots\!17\)\( \beta_{5} + \)\(14\!\cdots\!16\)\( \beta_{6} - \)\(48\!\cdots\!84\)\( \beta_{7}) q^{38}\) \(+(\)\(12\!\cdots\!56\)\( - \)\(46\!\cdots\!16\)\( \beta_{1} - \)\(73\!\cdots\!65\)\( \beta_{2} + \)\(50\!\cdots\!45\)\( \beta_{3} + \)\(67\!\cdots\!79\)\( \beta_{4} - \)\(14\!\cdots\!53\)\( \beta_{5} - \)\(88\!\cdots\!05\)\( \beta_{6} + \)\(34\!\cdots\!76\)\( \beta_{7}) q^{39}\) \(+(\)\(20\!\cdots\!00\)\( - \)\(11\!\cdots\!90\)\( \beta_{1} + \)\(37\!\cdots\!10\)\( \beta_{2} + \)\(58\!\cdots\!90\)\( \beta_{3} - \)\(11\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!50\)\( \beta_{5} + \)\(30\!\cdots\!50\)\( \beta_{6} + \)\(27\!\cdots\!50\)\( \beta_{7}) q^{40}\) \(+(-\)\(11\!\cdots\!98\)\( + \)\(69\!\cdots\!84\)\( \beta_{1} + \)\(79\!\cdots\!00\)\( \beta_{2} - \)\(51\!\cdots\!84\)\( \beta_{3} + \)\(10\!\cdots\!68\)\( \beta_{4} + \)\(26\!\cdots\!24\)\( \beta_{5} - \)\(63\!\cdots\!00\)\( \beta_{6} - \)\(68\!\cdots\!00\)\( \beta_{7}) q^{41}\) \(+(-\)\(12\!\cdots\!20\)\( + \)\(85\!\cdots\!60\)\( \beta_{1} + \)\(53\!\cdots\!16\)\( \beta_{2} - \)\(10\!\cdots\!84\)\( \beta_{3} + \)\(12\!\cdots\!60\)\( \beta_{4} - \)\(83\!\cdots\!68\)\( \beta_{5} + \)\(73\!\cdots\!96\)\( \beta_{6} - \)\(68\!\cdots\!60\)\( \beta_{7}) q^{42}\) \(+(\)\(37\!\cdots\!00\)\( + \)\(46\!\cdots\!23\)\( \beta_{1} - \)\(63\!\cdots\!33\)\( \beta_{2} + \)\(56\!\cdots\!64\)\( \beta_{3} - \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(15\!\cdots\!60\)\( \beta_{5} - \)\(26\!\cdots\!80\)\( \beta_{6} + \)\(56\!\cdots\!56\)\( \beta_{7}) q^{43}\) \(+(-\)\(76\!\cdots\!56\)\( + \)\(20\!\cdots\!04\)\( \beta_{1} - \)\(75\!\cdots\!60\)\( \beta_{2} + \)\(11\!\cdots\!32\)\( \beta_{3} - \)\(10\!\cdots\!68\)\( \beta_{4} + \)\(61\!\cdots\!76\)\( \beta_{5} + \)\(28\!\cdots\!80\)\( \beta_{6} + \)\(24\!\cdots\!04\)\( \beta_{7}) q^{44}\) \(+(-\)\(91\!\cdots\!50\)\( + \)\(62\!\cdots\!51\)\( \beta_{1} - \)\(19\!\cdots\!54\)\( \beta_{2} - \)\(31\!\cdots\!81\)\( \beta_{3} + \)\(42\!\cdots\!73\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5} - \)\(15\!\cdots\!00\)\( \beta_{6} - \)\(35\!\cdots\!00\)\( \beta_{7}) q^{45}\) \(+(-\)\(24\!\cdots\!48\)\( + \)\(53\!\cdots\!04\)\( \beta_{1} + \)\(72\!\cdots\!50\)\( \beta_{2} - \)\(19\!\cdots\!04\)\( \beta_{3} + \)\(66\!\cdots\!58\)\( \beta_{4} - \)\(12\!\cdots\!06\)\( \beta_{5} + \)\(49\!\cdots\!00\)\( \beta_{6} + \)\(18\!\cdots\!80\)\( \beta_{7}) q^{46}\) \(+(-\)\(24\!\cdots\!80\)\( - \)\(33\!\cdots\!24\)\( \beta_{1} + \)\(16\!\cdots\!06\)\( \beta_{2} + \)\(18\!\cdots\!62\)\( \beta_{3} - \)\(51\!\cdots\!10\)\( \beta_{4} + \)\(54\!\cdots\!30\)\( \beta_{5} - \)\(54\!\cdots\!90\)\( \beta_{6} - \)\(50\!\cdots\!52\)\( \beta_{7}) q^{47}\) \(+(\)\(59\!\cdots\!20\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} + \)\(54\!\cdots\!04\)\( \beta_{2} + \)\(26\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!60\)\( \beta_{4} - \)\(10\!\cdots\!88\)\( \beta_{5} - \)\(26\!\cdots\!64\)\( \beta_{6} + \)\(24\!\cdots\!20\)\( \beta_{7}) q^{48}\) \(+(\)\(11\!\cdots\!57\)\( - \)\(31\!\cdots\!24\)\( \beta_{1} - \)\(22\!\cdots\!00\)\( \beta_{2} - \)\(39\!\cdots\!16\)\( \beta_{3} + \)\(37\!\cdots\!92\)\( \beta_{4} + \)\(11\!\cdots\!56\)\( \beta_{5} + \)\(16\!\cdots\!00\)\( \beta_{6} + \)\(49\!\cdots\!80\)\( \beta_{7}) q^{49}\) \(+(\)\(15\!\cdots\!50\)\( - \)\(81\!\cdots\!25\)\( \beta_{1} - \)\(58\!\cdots\!00\)\( \beta_{2} - \)\(97\!\cdots\!00\)\( \beta_{3} - \)\(37\!\cdots\!00\)\( \beta_{4} - \)\(60\!\cdots\!00\)\( \beta_{5} - \)\(42\!\cdots\!00\)\( \beta_{6} - \)\(26\!\cdots\!00\)\( \beta_{7}) q^{50}\) \(+(-\)\(12\!\cdots\!88\)\( + \)\(14\!\cdots\!02\)\( \beta_{1} - \)\(98\!\cdots\!20\)\( \beta_{2} - \)\(16\!\cdots\!10\)\( \beta_{3} - \)\(39\!\cdots\!18\)\( \beta_{4} + \)\(41\!\cdots\!26\)\( \beta_{5} + \)\(44\!\cdots\!10\)\( \beta_{6} + \)\(67\!\cdots\!88\)\( \beta_{7}) q^{51}\) \(+(\)\(32\!\cdots\!00\)\( + \)\(25\!\cdots\!80\)\( \beta_{1} + \)\(94\!\cdots\!34\)\( \beta_{2} + \)\(15\!\cdots\!42\)\( \beta_{3} + \)\(63\!\cdots\!40\)\( \beta_{4} - \)\(12\!\cdots\!20\)\( \beta_{5} + \)\(81\!\cdots\!60\)\( \beta_{6} - \)\(34\!\cdots\!32\)\( \beta_{7}) q^{52}\) \(+(-\)\(63\!\cdots\!10\)\( + \)\(24\!\cdots\!15\)\( \beta_{1} + \)\(85\!\cdots\!46\)\( \beta_{2} + \)\(84\!\cdots\!67\)\( \beta_{3} + \)\(10\!\cdots\!85\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} - \)\(37\!\cdots\!32\)\( \beta_{6} - \)\(46\!\cdots\!44\)\( \beta_{7}) q^{53}\) \(+(-\)\(41\!\cdots\!40\)\( + \)\(11\!\cdots\!76\)\( \beta_{1} - \)\(36\!\cdots\!10\)\( \beta_{2} - \)\(93\!\cdots\!56\)\( \beta_{3} + \)\(68\!\cdots\!42\)\( \beta_{4} + \)\(42\!\cdots\!06\)\( \beta_{5} + \)\(53\!\cdots\!80\)\( \beta_{6} + \)\(21\!\cdots\!24\)\( \beta_{7}) q^{54}\) \(+(\)\(23\!\cdots\!00\)\( + \)\(17\!\cdots\!24\)\( \beta_{1} - \)\(44\!\cdots\!71\)\( \beta_{2} - \)\(51\!\cdots\!69\)\( \beta_{3} - \)\(27\!\cdots\!23\)\( \beta_{4} - \)\(12\!\cdots\!75\)\( \beta_{5} - \)\(66\!\cdots\!75\)\( \beta_{6} - \)\(42\!\cdots\!00\)\( \beta_{7}) q^{55}\) \(+(\)\(97\!\cdots\!60\)\( - \)\(20\!\cdots\!40\)\( \beta_{1} - \)\(12\!\cdots\!80\)\( \beta_{2} + \)\(18\!\cdots\!64\)\( \beta_{3} + \)\(64\!\cdots\!76\)\( \beta_{4} - \)\(17\!\cdots\!32\)\( \beta_{5} + \)\(47\!\cdots\!40\)\( \beta_{6} - \)\(53\!\cdots\!88\)\( \beta_{7}) q^{56}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(12\!\cdots\!06\)\( \beta_{1} + \)\(13\!\cdots\!76\)\( \beta_{2} + \)\(15\!\cdots\!66\)\( \beta_{3} - \)\(12\!\cdots\!30\)\( \beta_{4} + \)\(43\!\cdots\!40\)\( \beta_{5} - \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(31\!\cdots\!64\)\( \beta_{7}) q^{57}\) \(+(\)\(65\!\cdots\!80\)\( - \)\(37\!\cdots\!74\)\( \beta_{1} + \)\(15\!\cdots\!08\)\( \beta_{2} - \)\(25\!\cdots\!72\)\( \beta_{3} + \)\(70\!\cdots\!00\)\( \beta_{4} + \)\(89\!\cdots\!52\)\( \beta_{5} + \)\(31\!\cdots\!76\)\( \beta_{6} - \)\(10\!\cdots\!92\)\( \beta_{7}) q^{58}\) \(+(-\)\(10\!\cdots\!20\)\( - \)\(47\!\cdots\!49\)\( \beta_{1} + \)\(32\!\cdots\!35\)\( \beta_{2} - \)\(32\!\cdots\!44\)\( \beta_{3} - \)\(12\!\cdots\!40\)\( \beta_{4} - \)\(76\!\cdots\!20\)\( \beta_{5} + \)\(79\!\cdots\!20\)\( \beta_{6} + \)\(12\!\cdots\!56\)\( \beta_{7}) q^{59}\) \(+(-\)\(61\!\cdots\!00\)\( + \)\(16\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!76\)\( \beta_{2} - \)\(10\!\cdots\!64\)\( \beta_{3} + \)\(33\!\cdots\!12\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5} - \)\(47\!\cdots\!00\)\( \beta_{6} + \)\(22\!\cdots\!00\)\( \beta_{7}) q^{60}\) \(+(\)\(11\!\cdots\!02\)\( + \)\(17\!\cdots\!87\)\( \beta_{1} - \)\(61\!\cdots\!70\)\( \beta_{2} + \)\(45\!\cdots\!07\)\( \beta_{3} - \)\(26\!\cdots\!75\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(84\!\cdots\!60\)\( \beta_{6} - \)\(13\!\cdots\!32\)\( \beta_{7}) q^{61}\) \(+(-\)\(31\!\cdots\!80\)\( + \)\(11\!\cdots\!32\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2} + \)\(53\!\cdots\!16\)\( \beta_{3} + \)\(40\!\cdots\!60\)\( \beta_{4} - \)\(25\!\cdots\!28\)\( \beta_{5} + \)\(20\!\cdots\!16\)\( \beta_{6} + \)\(25\!\cdots\!60\)\( \beta_{7}) q^{62}\) \(+(-\)\(87\!\cdots\!40\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(73\!\cdots\!93\)\( \beta_{2} - \)\(53\!\cdots\!61\)\( \beta_{3} - \)\(32\!\cdots\!55\)\( \beta_{4} - \)\(31\!\cdots\!03\)\( \beta_{5} - \)\(34\!\cdots\!79\)\( \beta_{6} - \)\(14\!\cdots\!08\)\( \beta_{7}) q^{63}\) \(+(-\)\(12\!\cdots\!88\)\( - \)\(32\!\cdots\!76\)\( \beta_{1} + \)\(20\!\cdots\!00\)\( \beta_{2} - \)\(83\!\cdots\!00\)\( \beta_{3} + \)\(60\!\cdots\!24\)\( \beta_{4} - \)\(74\!\cdots\!68\)\( \beta_{5} + \)\(34\!\cdots\!00\)\( \beta_{6} - \)\(10\!\cdots\!60\)\( \beta_{7}) q^{64}\) \(+(-\)\(45\!\cdots\!00\)\( - \)\(58\!\cdots\!68\)\( \beta_{1} - \)\(18\!\cdots\!28\)\( \beta_{2} + \)\(41\!\cdots\!08\)\( \beta_{3} + \)\(49\!\cdots\!36\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(71\!\cdots\!00\)\( \beta_{6} + \)\(24\!\cdots\!00\)\( \beta_{7}) q^{65}\) \(+(\)\(19\!\cdots\!84\)\( - \)\(38\!\cdots\!64\)\( \beta_{1} + \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(34\!\cdots\!80\)\( \beta_{3} - \)\(17\!\cdots\!84\)\( \beta_{4} - \)\(28\!\cdots\!12\)\( \beta_{5} - \)\(19\!\cdots\!20\)\( \beta_{6} - \)\(10\!\cdots\!96\)\( \beta_{7}) q^{66}\) \(+(-\)\(13\!\cdots\!20\)\( - \)\(45\!\cdots\!87\)\( \beta_{1} - \)\(24\!\cdots\!93\)\( \beta_{2} + \)\(48\!\cdots\!70\)\( \beta_{3} - \)\(18\!\cdots\!90\)\( \beta_{4} + \)\(13\!\cdots\!98\)\( \beta_{5} - \)\(10\!\cdots\!46\)\( \beta_{6} - \)\(54\!\cdots\!76\)\( \beta_{7}) q^{67}\) \(+(-\)\(12\!\cdots\!60\)\( + \)\(74\!\cdots\!04\)\( \beta_{1} - \)\(46\!\cdots\!78\)\( \beta_{2} - \)\(28\!\cdots\!02\)\( \beta_{3} - \)\(11\!\cdots\!60\)\( \beta_{4} + \)\(79\!\cdots\!04\)\( \beta_{5} + \)\(25\!\cdots\!72\)\( \beta_{6} + \)\(80\!\cdots\!44\)\( \beta_{7}) q^{68}\) \(+(-\)\(18\!\cdots\!04\)\( + \)\(15\!\cdots\!72\)\( \beta_{1} + \)\(11\!\cdots\!60\)\( \beta_{2} - \)\(74\!\cdots\!76\)\( \beta_{3} + \)\(42\!\cdots\!88\)\( \beta_{4} - \)\(11\!\cdots\!16\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} + \)\(14\!\cdots\!76\)\( \beta_{7}) q^{69}\) \(+(-\)\(52\!\cdots\!00\)\( + \)\(38\!\cdots\!08\)\( \beta_{1} + \)\(26\!\cdots\!68\)\( \beta_{2} + \)\(22\!\cdots\!52\)\( \beta_{3} - \)\(77\!\cdots\!16\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} - \)\(16\!\cdots\!00\)\( \beta_{6} - \)\(38\!\cdots\!00\)\( \beta_{7}) q^{70}\) \(+(-\)\(63\!\cdots\!48\)\( + \)\(17\!\cdots\!96\)\( \beta_{1} + \)\(49\!\cdots\!65\)\( \beta_{2} + \)\(24\!\cdots\!31\)\( \beta_{3} - \)\(12\!\cdots\!75\)\( \beta_{4} - \)\(38\!\cdots\!75\)\( \beta_{5} + \)\(28\!\cdots\!05\)\( \beta_{6} - \)\(10\!\cdots\!56\)\( \beta_{7}) q^{71}\) \(+(-\)\(39\!\cdots\!60\)\( + \)\(27\!\cdots\!97\)\( \beta_{1} - \)\(22\!\cdots\!11\)\( \beta_{2} - \)\(44\!\cdots\!53\)\( \beta_{3} - \)\(63\!\cdots\!80\)\( \beta_{4} + \)\(27\!\cdots\!19\)\( \beta_{5} + \)\(14\!\cdots\!57\)\( \beta_{6} + \)\(57\!\cdots\!05\)\( \beta_{7}) q^{72}\) \(+(-\)\(38\!\cdots\!30\)\( - \)\(33\!\cdots\!26\)\( \beta_{1} - \)\(13\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!90\)\( \beta_{3} + \)\(62\!\cdots\!90\)\( \beta_{4} - \)\(37\!\cdots\!48\)\( \beta_{5} - \)\(10\!\cdots\!04\)\( \beta_{6} - \)\(73\!\cdots\!84\)\( \beta_{7}) q^{73}\) \(+(-\)\(11\!\cdots\!36\)\( - \)\(90\!\cdots\!34\)\( \beta_{1} + \)\(49\!\cdots\!80\)\( \beta_{2} - \)\(17\!\cdots\!12\)\( \beta_{3} - \)\(30\!\cdots\!52\)\( \beta_{4} - \)\(63\!\cdots\!36\)\( \beta_{5} + \)\(37\!\cdots\!60\)\( \beta_{6} - \)\(16\!\cdots\!72\)\( \beta_{7}) q^{74}\) \(+(-\)\(25\!\cdots\!00\)\( - \)\(19\!\cdots\!25\)\( \beta_{1} + \)\(59\!\cdots\!75\)\( \beta_{2} + \)\(43\!\cdots\!00\)\( \beta_{3} - \)\(22\!\cdots\!00\)\( \beta_{4} + \)\(23\!\cdots\!00\)\( \beta_{5} + \)\(28\!\cdots\!00\)\( \beta_{6} + \)\(79\!\cdots\!00\)\( \beta_{7}) q^{75}\) \(+(-\)\(39\!\cdots\!80\)\( - \)\(26\!\cdots\!80\)\( \beta_{1} + \)\(23\!\cdots\!40\)\( \beta_{2} - \)\(18\!\cdots\!52\)\( \beta_{3} + \)\(21\!\cdots\!32\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} - \)\(14\!\cdots\!20\)\( \beta_{6} - \)\(11\!\cdots\!76\)\( \beta_{7}) q^{76}\) \(+(-\)\(74\!\cdots\!00\)\( + \)\(54\!\cdots\!96\)\( \beta_{1} + \)\(88\!\cdots\!88\)\( \beta_{2} + \)\(11\!\cdots\!04\)\( \beta_{3} + \)\(97\!\cdots\!40\)\( \beta_{4} - \)\(57\!\cdots\!32\)\( \beta_{5} - \)\(21\!\cdots\!96\)\( \beta_{6} - \)\(90\!\cdots\!60\)\( \beta_{7}) q^{77}\) \(+(-\)\(26\!\cdots\!00\)\( + \)\(30\!\cdots\!76\)\( \beta_{1} - \)\(11\!\cdots\!86\)\( \beta_{2} - \)\(41\!\cdots\!80\)\( \beta_{3} + \)\(18\!\cdots\!30\)\( \beta_{4} - \)\(54\!\cdots\!06\)\( \beta_{5} + \)\(36\!\cdots\!12\)\( \beta_{6} + \)\(65\!\cdots\!12\)\( \beta_{7}) q^{78}\) \(+(-\)\(19\!\cdots\!60\)\( + \)\(17\!\cdots\!88\)\( \beta_{1} - \)\(72\!\cdots\!10\)\( \beta_{2} + \)\(14\!\cdots\!66\)\( \beta_{3} - \)\(54\!\cdots\!18\)\( \beta_{4} + \)\(10\!\cdots\!26\)\( \beta_{5} + \)\(37\!\cdots\!30\)\( \beta_{6} - \)\(11\!\cdots\!16\)\( \beta_{7}) q^{79}\) \(+(-\)\(45\!\cdots\!00\)\( + \)\(34\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!88\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} + \)\(73\!\cdots\!56\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5} - \)\(20\!\cdots\!00\)\( \beta_{6} + \)\(16\!\cdots\!00\)\( \beta_{7}) q^{80}\) \(+(-\)\(21\!\cdots\!59\)\( - \)\(68\!\cdots\!58\)\( \beta_{1} + \)\(66\!\cdots\!40\)\( \beta_{2} + \)\(17\!\cdots\!42\)\( \beta_{3} + \)\(70\!\cdots\!10\)\( \beta_{4} - \)\(76\!\cdots\!20\)\( \beta_{5} + \)\(20\!\cdots\!80\)\( \beta_{6} + \)\(30\!\cdots\!04\)\( \beta_{7}) q^{81}\) \(+(\)\(50\!\cdots\!20\)\( - \)\(26\!\cdots\!50\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} + \)\(82\!\cdots\!20\)\( \beta_{4} - \)\(21\!\cdots\!88\)\( \beta_{5} + \)\(15\!\cdots\!16\)\( \beta_{6} - \)\(71\!\cdots\!28\)\( \beta_{7}) q^{82}\) \(+(-\)\(34\!\cdots\!40\)\( - \)\(17\!\cdots\!83\)\( \beta_{1} - \)\(11\!\cdots\!07\)\( \beta_{2} - \)\(56\!\cdots\!20\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(34\!\cdots\!00\)\( \beta_{6} + \)\(74\!\cdots\!20\)\( \beta_{7}) q^{83}\) \(+(\)\(30\!\cdots\!04\)\( - \)\(33\!\cdots\!32\)\( \beta_{1} - \)\(28\!\cdots\!00\)\( \beta_{2} + \)\(22\!\cdots\!84\)\( \beta_{3} - \)\(11\!\cdots\!16\)\( \beta_{4} + \)\(46\!\cdots\!12\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(27\!\cdots\!40\)\( \beta_{7}) q^{84}\) \(+(\)\(13\!\cdots\!00\)\( + \)\(44\!\cdots\!14\)\( \beta_{1} - \)\(50\!\cdots\!56\)\( \beta_{2} + \)\(53\!\cdots\!66\)\( \beta_{3} + \)\(14\!\cdots\!22\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(11\!\cdots\!00\)\( \beta_{6} - \)\(30\!\cdots\!00\)\( \beta_{7}) q^{85}\) \(+(\)\(25\!\cdots\!12\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} + \)\(56\!\cdots\!55\)\( \beta_{2} + \)\(55\!\cdots\!14\)\( \beta_{3} + \)\(64\!\cdots\!19\)\( \beta_{4} - \)\(71\!\cdots\!33\)\( \beta_{5} + \)\(52\!\cdots\!60\)\( \beta_{6} + \)\(71\!\cdots\!68\)\( \beta_{7}) q^{86}\) \(+(\)\(30\!\cdots\!40\)\( + \)\(63\!\cdots\!28\)\( \beta_{1} - \)\(78\!\cdots\!61\)\( \beta_{2} - \)\(73\!\cdots\!83\)\( \beta_{3} + \)\(27\!\cdots\!55\)\( \beta_{4} + \)\(36\!\cdots\!67\)\( \beta_{5} - \)\(37\!\cdots\!89\)\( \beta_{6} - \)\(69\!\cdots\!96\)\( \beta_{7}) q^{87}\) \(+(\)\(82\!\cdots\!60\)\( + \)\(19\!\cdots\!88\)\( \beta_{1} + \)\(68\!\cdots\!16\)\( \beta_{2} + \)\(43\!\cdots\!76\)\( \beta_{3} - \)\(95\!\cdots\!40\)\( \beta_{4} - \)\(19\!\cdots\!08\)\( \beta_{5} - \)\(11\!\cdots\!24\)\( \beta_{6} - \)\(87\!\cdots\!40\)\( \beta_{7}) q^{88}\) \(+(\)\(57\!\cdots\!70\)\( + \)\(11\!\cdots\!26\)\( \beta_{1} - \)\(46\!\cdots\!20\)\( \beta_{2} - \)\(29\!\cdots\!26\)\( \beta_{3} - \)\(79\!\cdots\!78\)\( \beta_{4} + \)\(28\!\cdots\!96\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6} + \)\(38\!\cdots\!48\)\( \beta_{7}) q^{89}\) \(+(\)\(34\!\cdots\!00\)\( - \)\(31\!\cdots\!98\)\( \beta_{1} + \)\(10\!\cdots\!92\)\( \beta_{2} + \)\(37\!\cdots\!88\)\( \beta_{3} + \)\(31\!\cdots\!96\)\( \beta_{4} - \)\(22\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(41\!\cdots\!00\)\( \beta_{7}) q^{90}\) \(+(\)\(34\!\cdots\!12\)\( - \)\(38\!\cdots\!00\)\( \beta_{1} - \)\(38\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!24\)\( \beta_{3} - \)\(25\!\cdots\!44\)\( \beta_{4} + \)\(18\!\cdots\!08\)\( \beta_{5} + \)\(40\!\cdots\!40\)\( \beta_{6} - \)\(20\!\cdots\!88\)\( \beta_{7}) q^{91}\) \(+(\)\(14\!\cdots\!80\)\( - \)\(54\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!88\)\( \beta_{2} + \)\(58\!\cdots\!64\)\( \beta_{3} + \)\(40\!\cdots\!20\)\( \beta_{4} + \)\(81\!\cdots\!92\)\( \beta_{5} - \)\(34\!\cdots\!44\)\( \beta_{6} + \)\(68\!\cdots\!52\)\( \beta_{7}) q^{92}\) \(+(-\)\(22\!\cdots\!20\)\( - \)\(74\!\cdots\!16\)\( \beta_{1} + \)\(18\!\cdots\!12\)\( \beta_{2} - \)\(44\!\cdots\!20\)\( \beta_{3} - \)\(24\!\cdots\!80\)\( \beta_{4} - \)\(15\!\cdots\!04\)\( \beta_{5} + \)\(38\!\cdots\!08\)\( \beta_{6} + \)\(27\!\cdots\!08\)\( \beta_{7}) q^{93}\) \(+(-\)\(18\!\cdots\!76\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2} - \)\(56\!\cdots\!16\)\( \beta_{3} + \)\(10\!\cdots\!92\)\( \beta_{4} + \)\(50\!\cdots\!56\)\( \beta_{5} + \)\(10\!\cdots\!80\)\( \beta_{6} - \)\(50\!\cdots\!56\)\( \beta_{7}) q^{94}\) \(+(-\)\(24\!\cdots\!00\)\( + \)\(31\!\cdots\!20\)\( \beta_{1} - \)\(78\!\cdots\!05\)\( \beta_{2} + \)\(64\!\cdots\!05\)\( \beta_{3} + \)\(35\!\cdots\!35\)\( \beta_{4} - \)\(16\!\cdots\!25\)\( \beta_{5} - \)\(32\!\cdots\!25\)\( \beta_{6} - \)\(75\!\cdots\!00\)\( \beta_{7}) q^{95}\) \(+(-\)\(90\!\cdots\!08\)\( + \)\(88\!\cdots\!84\)\( \beta_{1} + \)\(22\!\cdots\!00\)\( \beta_{2} - \)\(31\!\cdots\!12\)\( \beta_{3} + \)\(21\!\cdots\!96\)\( \beta_{4} + \)\(44\!\cdots\!28\)\( \beta_{5} + \)\(19\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!60\)\( \beta_{7}) q^{96}\) \(+(-\)\(95\!\cdots\!30\)\( + \)\(69\!\cdots\!78\)\( \beta_{1} - \)\(10\!\cdots\!44\)\( \beta_{2} + \)\(21\!\cdots\!42\)\( \beta_{3} - \)\(26\!\cdots\!90\)\( \beta_{4} + \)\(28\!\cdots\!76\)\( \beta_{5} + \)\(52\!\cdots\!68\)\( \beta_{6} + \)\(13\!\cdots\!56\)\( \beta_{7}) q^{97}\) \(+(-\)\(17\!\cdots\!30\)\( - \)\(14\!\cdots\!91\)\( \beta_{1} + \)\(86\!\cdots\!80\)\( \beta_{2} - \)\(73\!\cdots\!44\)\( \beta_{3} - \)\(35\!\cdots\!20\)\( \beta_{4} - \)\(25\!\cdots\!12\)\( \beta_{5} - \)\(11\!\cdots\!16\)\( \beta_{6} - \)\(73\!\cdots\!32\)\( \beta_{7}) q^{98}\) \(+(-\)\(31\!\cdots\!04\)\( - \)\(88\!\cdots\!65\)\( \beta_{1} - \)\(31\!\cdots\!05\)\( \beta_{2} - \)\(28\!\cdots\!16\)\( \beta_{3} + \)\(42\!\cdots\!56\)\( \beta_{4} + \)\(93\!\cdots\!08\)\( \beta_{5} + \)\(67\!\cdots\!40\)\( \beta_{6} + \)\(47\!\cdots\!52\)\( \beta_{7}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 9175032489175920q^{2} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!80\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!76\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!64\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!84\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 9175032489175920q^{2} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!80\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!76\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!64\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!84\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!84\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!60\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!40\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!28\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!48\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!60\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!80\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!56\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!60\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!60\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!60\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!24\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!40\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!80\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!16\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!52\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!48\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!60\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!48\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!84\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!60\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!48\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!84\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!40\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!60\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!56\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!04\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!80\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!80\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!40\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!60\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!16\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!40\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!20\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!04\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!80\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!32\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!84\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!80\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!40\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!88\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!40\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!80\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!72\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!60\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!32\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!96\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!80\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!60\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!96\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!60\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!08\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!64\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!40\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!40\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!32\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!04\) \(x^{6}\mathstrut -\mathstrut \) \(62\!\cdots\!96\) \(x^{5}\mathstrut +\mathstrut \) \(32\!\cdots\!36\) \(x^{4}\mathstrut -\mathstrut \) \(88\!\cdots\!20\) \(x^{3}\mathstrut -\mathstrut \) \(32\!\cdots\!00\) \(x^{2}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(x\mathstrut +\mathstrut \) \(48\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 144 \nu - 18 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(27\!\cdots\!11\) \(\nu^{7}\mathstrut -\mathstrut \) \(69\!\cdots\!21\) \(\nu^{6}\mathstrut +\mathstrut \) \(16\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(76\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(74\!\cdots\!44\) \(\nu^{3}\mathstrut -\mathstrut \) \(14\!\cdots\!52\) \(\nu^{2}\mathstrut -\mathstrut \) \(40\!\cdots\!76\) \(\nu\mathstrut +\mathstrut \) \(29\!\cdots\!68\)\()/\)\(78\!\cdots\!52\)
\(\beta_{3}\)\(=\)\((\)\(27\!\cdots\!25\) \(\nu^{7}\mathstrut +\mathstrut \) \(70\!\cdots\!75\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(77\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(74\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(30\!\cdots\!72\) \(\nu^{2}\mathstrut +\mathstrut \) \(39\!\cdots\!04\) \(\nu\mathstrut -\mathstrut \) \(45\!\cdots\!44\)\()/\)\(78\!\cdots\!52\)
\(\beta_{4}\)\(=\)\((\)\(33\!\cdots\!37\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!03\) \(\nu^{6}\mathstrut -\mathstrut \) \(33\!\cdots\!88\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!92\) \(\nu^{4}\mathstrut +\mathstrut \) \(91\!\cdots\!52\) \(\nu^{3}\mathstrut +\mathstrut \) \(33\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(73\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(11\!\cdots\!80\)\()/\)\(89\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(71\!\cdots\!67\) \(\nu^{7}\mathstrut -\mathstrut \) \(99\!\cdots\!53\) \(\nu^{6}\mathstrut +\mathstrut \) \(68\!\cdots\!68\) \(\nu^{5}\mathstrut +\mathstrut \) \(76\!\cdots\!72\) \(\nu^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(18\!\cdots\!60\) \(\nu\mathstrut +\mathstrut \) \(89\!\cdots\!20\)\()/\)\(44\!\cdots\!40\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(11\!\cdots\!43\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!37\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!92\) \(\nu^{5}\mathstrut +\mathstrut \) \(28\!\cdots\!88\) \(\nu^{4}\mathstrut -\mathstrut \) \(26\!\cdots\!88\) \(\nu^{3}\mathstrut -\mathstrut \) \(29\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(35\!\cdots\!80\)\()/\)\(47\!\cdots\!20\)
\(\beta_{7}\)\(=\)\((\)\(83\!\cdots\!23\) \(\nu^{7}\mathstrut +\mathstrut \) \(18\!\cdots\!25\) \(\nu^{6}\mathstrut -\mathstrut \) \(88\!\cdots\!08\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!84\) \(\nu^{4}\mathstrut +\mathstrut \) \(29\!\cdots\!16\) \(\nu^{3}\mathstrut -\mathstrut \) \(54\!\cdots\!04\) \(\nu^{2}\mathstrut -\mathstrut \) \(38\!\cdots\!44\) \(\nu\mathstrut +\mathstrut \) \(36\!\cdots\!80\)\()/\)\(17\!\cdots\!96\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(18\)\()/144\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(1007775\) \(\beta_{2}\mathstrut +\mathstrut \) \(128696535390092\) \(\beta_{1}\mathstrut +\mathstrut \) \(54299611946833427723289037367328\)\()/20736\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(253\) \(\beta_{6}\mathstrut +\mathstrut \) \(920971\) \(\beta_{5}\mathstrut +\mathstrut \) \(3050104560\) \(\beta_{4}\mathstrut +\mathstrut \) \(3088687033359531\) \(\beta_{3}\mathstrut -\mathstrut \) \(2263599774724471097887\) \(\beta_{2}\mathstrut +\mathstrut \) \(89051647372276658322091865838593\) \(\beta_{1}\mathstrut +\mathstrut \) \(6988171918589951075120044186801148188176450048\)\()/2985984\)
\(\nu^{4}\)\(=\)\((\)\(261414756976041\) \(\beta_{7}\mathstrut -\mathstrut \) \(41015223717586939\) \(\beta_{6}\mathstrut -\mathstrut \) \(552670782673499559997\) \(\beta_{5}\mathstrut -\mathstrut \) \(7999591704992749065968528\) \(\beta_{4}\mathstrut +\mathstrut \) \(8192065749979863001066188876131\) \(\beta_{3}\mathstrut +\mathstrut \) \(8597496544545301247185630515625394441\) \(\beta_{2}\mathstrut +\mathstrut \) \(6864505788744185250647448452831809459841072041\) \(\beta_{1}\mathstrut +\mathstrut \) \(302216868472988346430007805211295563972859169357674792401371648\)\()/26873856\)
\(\nu^{5}\)\(=\)\((\)\(599029354963908732587800047593\) \(\beta_{7}\mathstrut +\mathstrut \) \(157163612661037350333679064701253\) \(\beta_{6}\mathstrut +\mathstrut \) \(496475835179650333701935015246469251\) \(\beta_{5}\mathstrut +\mathstrut \) \(254864816742689250761195560126735679600\) \(\beta_{4}\mathstrut +\mathstrut \) \(2411159489065473859791029310325348545240967715\) \(\beta_{3}\mathstrut -\mathstrut \) \(2623336585702135787435427177465182492512558183277751\) \(\beta_{2}\mathstrut +\mathstrut \) \(36791662786884071520165540138754948459883234888758553087149545\) \(\beta_{1}\mathstrut +\mathstrut \) \(23296250027079726429270104958638534168404286880351223849641929719349022786048\)\()/\)\(241864704\)
\(\nu^{6}\)\(=\)\((\)\(73\!\cdots\!07\) \(\beta_{7}\mathstrut -\mathstrut \) \(32\!\cdots\!53\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\!\cdots\!55\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\!\cdots\!28\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\!\cdots\!13\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\!\cdots\!91\) \(\beta_{2}\mathstrut +\mathstrut \) \(22\!\cdots\!59\) \(\beta_{1}\mathstrut +\mathstrut \) \(46\!\cdots\!24\)\()/80621568\)
\(\nu^{7}\)\(=\)\((\)\(38\!\cdots\!61\) \(\beta_{7}\mathstrut +\mathstrut \) \(89\!\cdots\!49\) \(\beta_{6}\mathstrut +\mathstrut \) \(28\!\cdots\!95\) \(\beta_{5}\mathstrut -\mathstrut \) \(49\!\cdots\!36\) \(\beta_{4}\mathstrut +\mathstrut \) \(18\!\cdots\!07\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\!\cdots\!95\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\!\cdots\!33\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\!\cdots\!08\)\()/\)\(241864704\)

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
−7.21371e13
−5.83850e13
−4.32050e13
−9.82818e12
1.99379e13
3.47548e13
4.91848e13
7.96777e13
−1.15346e16 2.00325e25 9.24825e31 −4.02660e36 −2.31067e41 −1.49411e43 −5.98851e47 2.76064e50 4.64453e52
1.2 −9.55432e15 −1.37857e25 5.07201e31 −2.80039e36 1.31713e41 −2.78207e44 −9.70270e46 6.48087e49 2.67558e52
1.3 −7.36840e15 −8.33754e23 1.37285e31 6.83377e36 6.14343e39 4.41963e44 1.97741e47 −1.24542e50 −5.03539e52
1.4 −2.56214e15 7.24197e24 −3.40003e31 −1.17115e36 −1.85549e40 −1.82598e44 1.91046e47 −7.27906e49 3.00065e51
1.5 1.72418e15 −1.28608e25 −3.75920e31 −7.91382e36 −2.21743e40 2.76433e44 −1.34757e47 4.01631e49 −1.36449e52
1.6 3.85781e15 −1.61266e25 −2.56821e31 9.75211e36 −6.22135e40 −2.29314e44 −2.55568e47 1.34832e50 3.76218e52
1.7 5.93573e15 1.55541e25 −5.33194e30 1.46253e36 9.23248e40 8.76445e43 −2.72431e47 1.16692e50 8.68120e51
1.8 1.03267e16 −2.77402e24 6.60762e31 −1.38808e36 −2.86465e40 −3.04428e43 2.63449e47 −1.17542e50 −1.43343e52
\(n\): 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 \(S_{106}^{\mathrm{new}}(\Gamma_0(1))\).