Properties

Label 1.116.a.a
Level 1
Weight 116
Character orbit 1.a
Self dual Yes
Analytic conductor 83.750
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 116 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(83.7504016273\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{142}\cdot 3^{52}\cdot 5^{17}\cdot 7^{8}\cdot 11^{3}\cdot 13^{3}\cdot 17\cdot 19^{3}\cdot 23^{3}\cdot 29^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(-12908974454383949 - \beta_{1}) q^{2}\) \(+(\)\(38\!\cdots\!92\)\( - 548079911 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(11\!\cdots\!47\)\( + 7744231697620209 \beta_{1} - 484553 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(10\!\cdots\!70\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + 586675963404 \beta_{2} + 4036 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(28\!\cdots\!05\)\( + \)\(52\!\cdots\!39\)\( \beta_{1} - 40505871977025988 \beta_{2} - 107907690 \beta_{3} + 797 \beta_{4} - \beta_{5}) q^{6}\) \(+(\)\(20\!\cdots\!38\)\( + \)\(17\!\cdots\!73\)\( \beta_{1} + \)\(30\!\cdots\!57\)\( \beta_{2} + 7240853888179 \beta_{3} - 25994649 \beta_{4} + 606 \beta_{5} - \beta_{6}) q^{7}\) \(+(-\)\(24\!\cdots\!22\)\( - \)\(51\!\cdots\!28\)\( \beta_{1} - \)\(95\!\cdots\!00\)\( \beta_{2} - 619136513711257 \beta_{3} - 189393293777 \beta_{4} - 664275 \beta_{5} - 413 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(28\!\cdots\!37\)\( + \)\(53\!\cdots\!97\)\( \beta_{1} - \)\(26\!\cdots\!54\)\( \beta_{2} - 39934242747451754594 \beta_{3} - 133277535964188 \beta_{4} - 459502997 \beta_{5} + 313389 \beta_{6} - 182 \beta_{7} + \beta_{8}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-12908974454383949 - \beta_{1}) q^{2}\) \(+(\)\(38\!\cdots\!92\)\( - 548079911 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(11\!\cdots\!47\)\( + 7744231697620209 \beta_{1} - 484553 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(10\!\cdots\!70\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + 586675963404 \beta_{2} + 4036 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(28\!\cdots\!05\)\( + \)\(52\!\cdots\!39\)\( \beta_{1} - 40505871977025988 \beta_{2} - 107907690 \beta_{3} + 797 \beta_{4} - \beta_{5}) q^{6}\) \(+(\)\(20\!\cdots\!38\)\( + \)\(17\!\cdots\!73\)\( \beta_{1} + \)\(30\!\cdots\!57\)\( \beta_{2} + 7240853888179 \beta_{3} - 25994649 \beta_{4} + 606 \beta_{5} - \beta_{6}) q^{7}\) \(+(-\)\(24\!\cdots\!22\)\( - \)\(51\!\cdots\!28\)\( \beta_{1} - \)\(95\!\cdots\!00\)\( \beta_{2} - 619136513711257 \beta_{3} - 189393293777 \beta_{4} - 664275 \beta_{5} - 413 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(28\!\cdots\!37\)\( + \)\(53\!\cdots\!97\)\( \beta_{1} - \)\(26\!\cdots\!54\)\( \beta_{2} - 39934242747451754594 \beta_{3} - 133277535964188 \beta_{4} - 459502997 \beta_{5} + 313389 \beta_{6} - 182 \beta_{7} + \beta_{8}) q^{9}\) \(+(\)\(76\!\cdots\!22\)\( + \)\(88\!\cdots\!90\)\( \beta_{1} - \)\(12\!\cdots\!16\)\( \beta_{2} - \)\(10\!\cdots\!20\)\( \beta_{3} - 56147047023408348 \beta_{4} - 184957797900 \beta_{5} + 21531704 \beta_{6} - 156480 \beta_{7} + 264 \beta_{8}) q^{10}\) \(+(-\)\(49\!\cdots\!08\)\( - \)\(43\!\cdots\!11\)\( \beta_{1} + \)\(29\!\cdots\!57\)\( \beta_{2} + \)\(93\!\cdots\!02\)\( \beta_{3} - 12632865744118755930 \beta_{4} - 28420377803664 \beta_{5} - 69136732774 \beta_{6} - 72116648 \beta_{7} - 18756 \beta_{8}) q^{11}\) \(+(-\)\(47\!\cdots\!56\)\( - \)\(55\!\cdots\!68\)\( \beta_{1} + \)\(46\!\cdots\!96\)\( \beta_{2} + \)\(23\!\cdots\!88\)\( \beta_{3} - \)\(83\!\cdots\!92\)\( \beta_{4} + 636374606564640 \beta_{5} - 19813923913248 \beta_{6} - 3599055456 \beta_{7} - 2892480 \beta_{8}) q^{12}\) \(+(\)\(21\!\cdots\!42\)\( + \)\(11\!\cdots\!71\)\( \beta_{1} + \)\(64\!\cdots\!72\)\( \beta_{2} - \)\(30\!\cdots\!32\)\( \beta_{3} + \)\(44\!\cdots\!21\)\( \beta_{4} - 2414876052438376326 \beta_{5} - 1880087776576106 \beta_{6} + 807139399116 \beta_{7} + 515975790 \beta_{8}) q^{13}\) \(+(-\)\(97\!\cdots\!94\)\( - \)\(41\!\cdots\!18\)\( \beta_{1} - \)\(39\!\cdots\!64\)\( \beta_{2} + \)\(14\!\cdots\!36\)\( \beta_{3} - \)\(47\!\cdots\!22\)\( \beta_{4} - \)\(87\!\cdots\!62\)\( \beta_{5} - 20784814855981408 \beta_{6} - 34340071544576 \beta_{7} - 40630437792 \beta_{8}) q^{14}\) \(+(\)\(63\!\cdots\!34\)\( + \)\(78\!\cdots\!55\)\( \beta_{1} - \)\(46\!\cdots\!77\)\( \beta_{2} - \)\(13\!\cdots\!15\)\( \beta_{3} + \)\(27\!\cdots\!69\)\( \beta_{4} - \)\(70\!\cdots\!50\)\( \beta_{5} + 7522015980496972113 \beta_{6} - 45823238274960 \beta_{7} + 2123970677208 \beta_{8}) q^{15}\) \(+(-\)\(21\!\cdots\!96\)\( - \)\(31\!\cdots\!40\)\( \beta_{1} + \)\(47\!\cdots\!20\)\( \beta_{2} + \)\(48\!\cdots\!08\)\( \beta_{3} + \)\(51\!\cdots\!56\)\( \beta_{4} - \)\(75\!\cdots\!72\)\( \beta_{5} - \)\(19\!\cdots\!84\)\( \beta_{6} + 63376353383092152 \beta_{7} - 83303711279616 \beta_{8}) q^{16}\) \(+(\)\(37\!\cdots\!94\)\( - \)\(34\!\cdots\!71\)\( \beta_{1} + \)\(41\!\cdots\!02\)\( \beta_{2} - \)\(62\!\cdots\!54\)\( \beta_{3} + \)\(14\!\cdots\!84\)\( \beta_{4} + \)\(59\!\cdots\!59\)\( \beta_{5} - \)\(21\!\cdots\!59\)\( \beta_{6} - 3268486108013674110 \beta_{7} + 2603010144162285 \beta_{8}) q^{17}\) \(+(-\)\(32\!\cdots\!41\)\( - \)\(14\!\cdots\!53\)\( \beta_{1} + \)\(26\!\cdots\!32\)\( \beta_{2} + \)\(32\!\cdots\!64\)\( \beta_{3} + \)\(11\!\cdots\!56\)\( \beta_{4} + \)\(64\!\cdots\!96\)\( \beta_{5} + \)\(20\!\cdots\!44\)\( \beta_{6} + 99000356531144415360 \beta_{7} - 67158677161258320 \beta_{8}) q^{18}\) \(+(\)\(62\!\cdots\!36\)\( - \)\(54\!\cdots\!57\)\( \beta_{1} + \)\(17\!\cdots\!79\)\( \beta_{2} + \)\(76\!\cdots\!30\)\( \beta_{3} + \)\(37\!\cdots\!90\)\( \beta_{4} - \)\(35\!\cdots\!32\)\( \beta_{5} - \)\(45\!\cdots\!22\)\( \beta_{6} - \)\(21\!\cdots\!44\)\( \beta_{7} + 1464583823069070132 \beta_{8}) q^{19}\) \(+(-\)\(13\!\cdots\!10\)\( + \)\(19\!\cdots\!62\)\( \beta_{1} - \)\(15\!\cdots\!98\)\( \beta_{2} + \)\(57\!\cdots\!18\)\( \beta_{3} - \)\(59\!\cdots\!12\)\( \beta_{4} + \)\(74\!\cdots\!00\)\( \beta_{5} + \)\(33\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!00\)\( \beta_{7} - 27434369381284396800 \beta_{8}) q^{20}\) \(+(\)\(31\!\cdots\!60\)\( + \)\(68\!\cdots\!02\)\( \beta_{1} - \)\(13\!\cdots\!24\)\( \beta_{2} + \)\(13\!\cdots\!44\)\( \beta_{3} - \)\(53\!\cdots\!24\)\( \beta_{4} + \)\(89\!\cdots\!66\)\( \beta_{5} + \)\(56\!\cdots\!34\)\( \beta_{6} - \)\(34\!\cdots\!52\)\( \beta_{7} + \)\(44\!\cdots\!66\)\( \beta_{8}) q^{21}\) \(+(\)\(23\!\cdots\!87\)\( + \)\(14\!\cdots\!41\)\( \beta_{1} + \)\(59\!\cdots\!32\)\( \beta_{2} + \)\(34\!\cdots\!98\)\( \beta_{3} + \)\(17\!\cdots\!31\)\( \beta_{4} - \)\(97\!\cdots\!91\)\( \beta_{5} - \)\(19\!\cdots\!16\)\( \beta_{6} + \)\(18\!\cdots\!76\)\( \beta_{7} - \)\(63\!\cdots\!40\)\( \beta_{8}) q^{22}\) \(+(\)\(56\!\cdots\!02\)\( - \)\(57\!\cdots\!53\)\( \beta_{1} - \)\(55\!\cdots\!61\)\( \beta_{2} + \)\(33\!\cdots\!05\)\( \beta_{3} + \)\(78\!\cdots\!05\)\( \beta_{4} - \)\(71\!\cdots\!10\)\( \beta_{5} + \)\(26\!\cdots\!45\)\( \beta_{6} + \)\(19\!\cdots\!40\)\( \beta_{7} + \)\(79\!\cdots\!40\)\( \beta_{8}) q^{23}\) \(+(-\)\(83\!\cdots\!20\)\( - \)\(53\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2} + \)\(21\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!68\)\( \beta_{4} + \)\(21\!\cdots\!40\)\( \beta_{5} - \)\(19\!\cdots\!44\)\( \beta_{6} - \)\(64\!\cdots\!28\)\( \beta_{7} - \)\(87\!\cdots\!96\)\( \beta_{8}) q^{24}\) \(+(\)\(13\!\cdots\!35\)\( + \)\(22\!\cdots\!70\)\( \beta_{1} + \)\(26\!\cdots\!40\)\( \beta_{2} - \)\(24\!\cdots\!20\)\( \beta_{3} - \)\(76\!\cdots\!60\)\( \beta_{4} - \)\(11\!\cdots\!50\)\( \beta_{5} + \)\(72\!\cdots\!70\)\( \beta_{6} + \)\(98\!\cdots\!00\)\( \beta_{7} + \)\(83\!\cdots\!70\)\( \beta_{8}) q^{25}\) \(+(-\)\(65\!\cdots\!26\)\( - \)\(18\!\cdots\!38\)\( \beta_{1} + \)\(88\!\cdots\!56\)\( \beta_{2} - \)\(35\!\cdots\!52\)\( \beta_{3} + \)\(77\!\cdots\!24\)\( \beta_{4} - \)\(10\!\cdots\!40\)\( \beta_{5} + \)\(14\!\cdots\!72\)\( \beta_{6} - \)\(10\!\cdots\!36\)\( \beta_{7} - \)\(69\!\cdots\!52\)\( \beta_{8}) q^{26}\) \(+(-\)\(60\!\cdots\!48\)\( - \)\(15\!\cdots\!12\)\( \beta_{1} + \)\(47\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!38\)\( \beta_{3} + \)\(69\!\cdots\!82\)\( \beta_{4} + \)\(18\!\cdots\!60\)\( \beta_{5} - \)\(17\!\cdots\!42\)\( \beta_{6} + \)\(81\!\cdots\!16\)\( \beta_{7} + \)\(48\!\cdots\!60\)\( \beta_{8}) q^{27}\) \(+(\)\(14\!\cdots\!20\)\( - \)\(33\!\cdots\!96\)\( \beta_{1} + \)\(27\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!52\)\( \beta_{3} - \)\(21\!\cdots\!48\)\( \beta_{4} - \)\(74\!\cdots\!40\)\( \beta_{5} + \)\(81\!\cdots\!88\)\( \beta_{6} - \)\(43\!\cdots\!44\)\( \beta_{7} - \)\(25\!\cdots\!80\)\( \beta_{8}) q^{28}\) \(+(\)\(27\!\cdots\!58\)\( + \)\(51\!\cdots\!93\)\( \beta_{1} + \)\(30\!\cdots\!84\)\( \beta_{2} + \)\(15\!\cdots\!92\)\( \beta_{3} - \)\(33\!\cdots\!11\)\( \beta_{4} - \)\(56\!\cdots\!80\)\( \beta_{5} + \)\(21\!\cdots\!92\)\( \beta_{6} + \)\(73\!\cdots\!84\)\( \beta_{7} + \)\(74\!\cdots\!48\)\( \beta_{8}) q^{29}\) \(+(-\)\(12\!\cdots\!90\)\( - \)\(19\!\cdots\!82\)\( \beta_{1} + \)\(33\!\cdots\!28\)\( \beta_{2} - \)\(58\!\cdots\!48\)\( \beta_{3} + \)\(15\!\cdots\!82\)\( \beta_{4} + \)\(74\!\cdots\!50\)\( \beta_{5} - \)\(65\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!00\)\( \beta_{7} + \)\(39\!\cdots\!00\)\( \beta_{8}) q^{30}\) \(+(-\)\(12\!\cdots\!64\)\( - \)\(57\!\cdots\!28\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(75\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!24\)\( \beta_{4} - \)\(21\!\cdots\!68\)\( \beta_{5} + \)\(48\!\cdots\!44\)\( \beta_{6} - \)\(19\!\cdots\!12\)\( \beta_{7} - \)\(86\!\cdots\!64\)\( \beta_{8}) q^{31}\) \(+(\)\(20\!\cdots\!04\)\( + \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(35\!\cdots\!44\)\( \beta_{2} - \)\(23\!\cdots\!64\)\( \beta_{3} - \)\(69\!\cdots\!76\)\( \beta_{4} - \)\(16\!\cdots\!96\)\( \beta_{5} - \)\(12\!\cdots\!24\)\( \beta_{6} + \)\(14\!\cdots\!60\)\( \beta_{7} + \)\(85\!\cdots\!80\)\( \beta_{8}) q^{32}\) \(+(\)\(41\!\cdots\!88\)\( - \)\(13\!\cdots\!93\)\( \beta_{1} - \)\(12\!\cdots\!14\)\( \beta_{2} + \)\(22\!\cdots\!06\)\( \beta_{3} - \)\(41\!\cdots\!32\)\( \beta_{4} + \)\(15\!\cdots\!61\)\( \beta_{5} - \)\(93\!\cdots\!53\)\( \beta_{6} - \)\(74\!\cdots\!34\)\( \beta_{7} - \)\(60\!\cdots\!05\)\( \beta_{8}) q^{33}\) \(+(\)\(17\!\cdots\!34\)\( - \)\(13\!\cdots\!10\)\( \beta_{1} - \)\(27\!\cdots\!40\)\( \beta_{2} + \)\(10\!\cdots\!28\)\( \beta_{3} + \)\(35\!\cdots\!00\)\( \beta_{4} - \)\(25\!\cdots\!92\)\( \beta_{5} + \)\(12\!\cdots\!88\)\( \beta_{6} + \)\(19\!\cdots\!56\)\( \beta_{7} + \)\(33\!\cdots\!92\)\( \beta_{8}) q^{34}\) \(+(-\)\(58\!\cdots\!48\)\( + \)\(65\!\cdots\!40\)\( \beta_{1} - \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(17\!\cdots\!20\)\( \beta_{3} + \)\(53\!\cdots\!32\)\( \beta_{4} - \)\(31\!\cdots\!00\)\( \beta_{5} - \)\(61\!\cdots\!36\)\( \beta_{6} + \)\(54\!\cdots\!20\)\( \beta_{7} - \)\(13\!\cdots\!76\)\( \beta_{8}) q^{35}\) \(+(-\)\(38\!\cdots\!45\)\( - \)\(20\!\cdots\!35\)\( \beta_{1} - \)\(27\!\cdots\!25\)\( \beta_{2} - \)\(39\!\cdots\!71\)\( \beta_{3} - \)\(10\!\cdots\!08\)\( \beta_{4} + \)\(19\!\cdots\!04\)\( \beta_{5} + \)\(85\!\cdots\!80\)\( \beta_{6} - \)\(10\!\cdots\!60\)\( \beta_{7} + \)\(21\!\cdots\!40\)\( \beta_{8}) q^{36}\) \(+(\)\(31\!\cdots\!10\)\( - \)\(31\!\cdots\!77\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2} - \)\(24\!\cdots\!36\)\( \beta_{3} + \)\(14\!\cdots\!69\)\( \beta_{4} + \)\(24\!\cdots\!70\)\( \beta_{5} + \)\(10\!\cdots\!86\)\( \beta_{6} + \)\(67\!\cdots\!12\)\( \beta_{7} + \)\(21\!\cdots\!50\)\( \beta_{8}) q^{37}\) \(+(\)\(20\!\cdots\!97\)\( - \)\(89\!\cdots\!53\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2} + \)\(93\!\cdots\!46\)\( \beta_{3} + \)\(15\!\cdots\!65\)\( \beta_{4} - \)\(43\!\cdots\!73\)\( \beta_{5} - \)\(87\!\cdots\!80\)\( \beta_{6} - \)\(27\!\cdots\!36\)\( \beta_{7} - \)\(26\!\cdots\!20\)\( \beta_{8}) q^{38}\) \(+(\)\(63\!\cdots\!06\)\( - \)\(19\!\cdots\!09\)\( \beta_{1} + \)\(47\!\cdots\!03\)\( \beta_{2} + \)\(75\!\cdots\!05\)\( \beta_{3} - \)\(29\!\cdots\!35\)\( \beta_{4} + \)\(14\!\cdots\!86\)\( \beta_{5} + \)\(31\!\cdots\!01\)\( \beta_{6} + \)\(58\!\cdots\!32\)\( \beta_{7} + \)\(18\!\cdots\!64\)\( \beta_{8}) q^{39}\) \(+(-\)\(41\!\cdots\!60\)\( - \)\(22\!\cdots\!00\)\( \beta_{1} - \)\(28\!\cdots\!20\)\( \beta_{2} - \)\(82\!\cdots\!50\)\( \beta_{3} - \)\(21\!\cdots\!10\)\( \beta_{4} + \)\(36\!\cdots\!50\)\( \beta_{5} - \)\(17\!\cdots\!70\)\( \beta_{6} + \)\(84\!\cdots\!50\)\( \beta_{7} - \)\(95\!\cdots\!20\)\( \beta_{8}) q^{40}\) \(+(\)\(50\!\cdots\!30\)\( - \)\(10\!\cdots\!18\)\( \beta_{1} - \)\(93\!\cdots\!24\)\( \beta_{2} - \)\(59\!\cdots\!96\)\( \beta_{3} + \)\(98\!\cdots\!08\)\( \beta_{4} - \)\(37\!\cdots\!54\)\( \beta_{5} - \)\(37\!\cdots\!30\)\( \beta_{6} - \)\(11\!\cdots\!40\)\( \beta_{7} + \)\(38\!\cdots\!10\)\( \beta_{8}) q^{41}\) \(+(-\)\(40\!\cdots\!16\)\( - \)\(76\!\cdots\!00\)\( \beta_{1} - \)\(41\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(67\!\cdots\!76\)\( \beta_{4} + \)\(52\!\cdots\!36\)\( \beta_{5} + \)\(17\!\cdots\!24\)\( \beta_{6} + \)\(28\!\cdots\!80\)\( \beta_{7} - \)\(11\!\cdots\!60\)\( \beta_{8}) q^{42}\) \(+(\)\(12\!\cdots\!36\)\( - \)\(25\!\cdots\!97\)\( \beta_{1} - \)\(12\!\cdots\!33\)\( \beta_{2} + \)\(83\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} + \)\(44\!\cdots\!88\)\( \beta_{5} - \)\(91\!\cdots\!76\)\( \beta_{6} + \)\(16\!\cdots\!44\)\( \beta_{7} + \)\(17\!\cdots\!00\)\( \beta_{8}) q^{43}\) \(+(\)\(96\!\cdots\!68\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(38\!\cdots\!20\)\( \beta_{2} - \)\(51\!\cdots\!28\)\( \beta_{3} - \)\(98\!\cdots\!16\)\( \beta_{4} - \)\(20\!\cdots\!28\)\( \beta_{5} - \)\(21\!\cdots\!56\)\( \beta_{6} - \)\(19\!\cdots\!52\)\( \beta_{7} + \)\(51\!\cdots\!76\)\( \beta_{8}) q^{44}\) \(+(-\)\(39\!\cdots\!90\)\( - \)\(53\!\cdots\!97\)\( \beta_{1} + \)\(22\!\cdots\!88\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} + \)\(52\!\cdots\!97\)\( \beta_{4} - \)\(35\!\cdots\!50\)\( \beta_{5} + \)\(88\!\cdots\!50\)\( \beta_{6} + \)\(10\!\cdots\!00\)\( \beta_{7} - \)\(55\!\cdots\!50\)\( \beta_{8}) q^{45}\) \(+(\)\(23\!\cdots\!50\)\( - \)\(17\!\cdots\!98\)\( \beta_{1} + \)\(31\!\cdots\!96\)\( \beta_{2} - \)\(23\!\cdots\!96\)\( \beta_{3} - \)\(44\!\cdots\!82\)\( \beta_{4} + \)\(22\!\cdots\!66\)\( \beta_{5} + \)\(27\!\cdots\!00\)\( \beta_{6} - \)\(38\!\cdots\!20\)\( \beta_{7} + \)\(25\!\cdots\!20\)\( \beta_{8}) q^{46}\) \(+(-\)\(35\!\cdots\!36\)\( - \)\(31\!\cdots\!46\)\( \beta_{1} - \)\(15\!\cdots\!06\)\( \beta_{2} + \)\(17\!\cdots\!30\)\( \beta_{3} - \)\(14\!\cdots\!98\)\( \beta_{4} - \)\(26\!\cdots\!44\)\( \beta_{5} - \)\(16\!\cdots\!82\)\( \beta_{6} + \)\(80\!\cdots\!28\)\( \beta_{7} - \)\(73\!\cdots\!20\)\( \beta_{8}) q^{47}\) \(+(\)\(49\!\cdots\!80\)\( - \)\(63\!\cdots\!76\)\( \beta_{1} - \)\(12\!\cdots\!92\)\( \beta_{2} + \)\(22\!\cdots\!56\)\( \beta_{3} - \)\(33\!\cdots\!16\)\( \beta_{4} - \)\(28\!\cdots\!56\)\( \beta_{5} + \)\(80\!\cdots\!16\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(97\!\cdots\!00\)\( \beta_{8}) q^{48}\) \(+(-\)\(60\!\cdots\!59\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(33\!\cdots\!76\)\( \beta_{2} - \)\(94\!\cdots\!16\)\( \beta_{3} + \)\(16\!\cdots\!08\)\( \beta_{4} + \)\(10\!\cdots\!56\)\( \beta_{5} - \)\(18\!\cdots\!40\)\( \beta_{6} - \)\(68\!\cdots\!00\)\( \beta_{7} + \)\(30\!\cdots\!60\)\( \beta_{8}) q^{49}\) \(+(-\)\(11\!\cdots\!15\)\( + \)\(75\!\cdots\!45\)\( \beta_{1} + \)\(53\!\cdots\!40\)\( \beta_{2} - \)\(37\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(29\!\cdots\!00\)\( \beta_{7} - \)\(25\!\cdots\!80\)\( \beta_{8}) q^{50}\) \(+(\)\(43\!\cdots\!76\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} + \)\(22\!\cdots\!56\)\( \beta_{2} - \)\(51\!\cdots\!90\)\( \beta_{3} - \)\(59\!\cdots\!50\)\( \beta_{4} - \)\(85\!\cdots\!48\)\( \beta_{5} - \)\(35\!\cdots\!78\)\( \beta_{6} - \)\(65\!\cdots\!36\)\( \beta_{7} + \)\(88\!\cdots\!48\)\( \beta_{8}) q^{51}\) \(+(\)\(20\!\cdots\!34\)\( + \)\(14\!\cdots\!26\)\( \beta_{1} + \)\(21\!\cdots\!66\)\( \beta_{2} + \)\(50\!\cdots\!50\)\( \beta_{3} + \)\(18\!\cdots\!92\)\( \beta_{4} + \)\(42\!\cdots\!36\)\( \beta_{5} + \)\(42\!\cdots\!28\)\( \beta_{6} + \)\(57\!\cdots\!68\)\( \beta_{7} - \)\(16\!\cdots\!00\)\( \beta_{8}) q^{52}\) \(+(\)\(37\!\cdots\!50\)\( + \)\(36\!\cdots\!83\)\( \beta_{1} + \)\(72\!\cdots\!68\)\( \beta_{2} + \)\(54\!\cdots\!88\)\( \beta_{3} + \)\(29\!\cdots\!53\)\( \beta_{4} + \)\(12\!\cdots\!10\)\( \beta_{5} - \)\(23\!\cdots\!18\)\( \beta_{6} + \)\(81\!\cdots\!24\)\( \beta_{7} - \)\(77\!\cdots\!70\)\( \beta_{8}) q^{53}\) \(+(\)\(90\!\cdots\!30\)\( + \)\(51\!\cdots\!98\)\( \beta_{1} - \)\(10\!\cdots\!36\)\( \beta_{2} - \)\(48\!\cdots\!96\)\( \beta_{3} + \)\(95\!\cdots\!14\)\( \beta_{4} - \)\(38\!\cdots\!78\)\( \beta_{5} + \)\(63\!\cdots\!04\)\( \beta_{6} - \)\(78\!\cdots\!92\)\( \beta_{7} + \)\(17\!\cdots\!76\)\( \beta_{8}) q^{54}\) \(+(-\)\(26\!\cdots\!90\)\( - \)\(69\!\cdots\!27\)\( \beta_{1} - \)\(10\!\cdots\!67\)\( \beta_{2} - \)\(48\!\cdots\!53\)\( \beta_{3} - \)\(24\!\cdots\!73\)\( \beta_{4} + \)\(18\!\cdots\!50\)\( \beta_{5} - \)\(49\!\cdots\!25\)\( \beta_{6} - \)\(12\!\cdots\!00\)\( \beta_{7} - \)\(65\!\cdots\!00\)\( \beta_{8}) q^{55}\) \(+(\)\(56\!\cdots\!12\)\( - \)\(54\!\cdots\!92\)\( \beta_{1} + \)\(36\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!56\)\( \beta_{3} - \)\(25\!\cdots\!52\)\( \beta_{4} + \)\(41\!\cdots\!24\)\( \beta_{5} - \)\(24\!\cdots\!52\)\( \beta_{6} + \)\(24\!\cdots\!96\)\( \beta_{7} + \)\(12\!\cdots\!12\)\( \beta_{8}) q^{56}\) \(+(\)\(21\!\cdots\!04\)\( - \)\(30\!\cdots\!63\)\( \beta_{1} + \)\(16\!\cdots\!74\)\( \beta_{2} + \)\(77\!\cdots\!90\)\( \beta_{3} + \)\(27\!\cdots\!36\)\( \beta_{4} + \)\(57\!\cdots\!83\)\( \beta_{5} + \)\(92\!\cdots\!49\)\( \beta_{6} + \)\(16\!\cdots\!54\)\( \beta_{7} + \)\(39\!\cdots\!65\)\( \beta_{8}) q^{57}\) \(+(-\)\(30\!\cdots\!50\)\( - \)\(77\!\cdots\!46\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(49\!\cdots\!96\)\( \beta_{3} - \)\(81\!\cdots\!88\)\( \beta_{4} - \)\(33\!\cdots\!92\)\( \beta_{5} - \)\(14\!\cdots\!32\)\( \beta_{6} - \)\(11\!\cdots\!48\)\( \beta_{7} - \)\(10\!\cdots\!40\)\( \beta_{8}) q^{58}\) \(+(\)\(29\!\cdots\!52\)\( - \)\(46\!\cdots\!97\)\( \beta_{1} - \)\(25\!\cdots\!21\)\( \beta_{2} - \)\(77\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!96\)\( \beta_{4} + \)\(65\!\cdots\!44\)\( \beta_{5} + \)\(27\!\cdots\!36\)\( \beta_{6} + \)\(31\!\cdots\!32\)\( \beta_{7} + \)\(32\!\cdots\!24\)\( \beta_{8}) q^{59}\) \(+(-\)\(16\!\cdots\!08\)\( + \)\(60\!\cdots\!40\)\( \beta_{1} - \)\(27\!\cdots\!76\)\( \beta_{2} - \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!56\)\( \beta_{6} - \)\(17\!\cdots\!80\)\( \beta_{7} - \)\(45\!\cdots\!96\)\( \beta_{8}) q^{60}\) \(+(\)\(51\!\cdots\!90\)\( + \)\(47\!\cdots\!35\)\( \beta_{1} - \)\(52\!\cdots\!00\)\( \beta_{2} + \)\(33\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!73\)\( \beta_{4} + \)\(11\!\cdots\!18\)\( \beta_{5} + \)\(49\!\cdots\!02\)\( \beta_{6} - \)\(18\!\cdots\!16\)\( \beta_{7} - \)\(50\!\cdots\!42\)\( \beta_{8}) q^{61}\) \(+(\)\(32\!\cdots\!64\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} + \)\(52\!\cdots\!36\)\( \beta_{2} + \)\(75\!\cdots\!44\)\( \beta_{3} + \)\(78\!\cdots\!96\)\( \beta_{4} - \)\(11\!\cdots\!24\)\( \beta_{5} - \)\(46\!\cdots\!96\)\( \beta_{6} + \)\(73\!\cdots\!40\)\( \beta_{7} + \)\(41\!\cdots\!80\)\( \beta_{8}) q^{62}\) \(+(-\)\(29\!\cdots\!82\)\( + \)\(63\!\cdots\!89\)\( \beta_{1} + \)\(60\!\cdots\!77\)\( \beta_{2} - \)\(55\!\cdots\!29\)\( \beta_{3} - \)\(59\!\cdots\!69\)\( \beta_{4} + \)\(29\!\cdots\!70\)\( \beta_{5} - \)\(24\!\cdots\!61\)\( \beta_{6} - \)\(11\!\cdots\!72\)\( \beta_{7} - \)\(95\!\cdots\!80\)\( \beta_{8}) q^{63}\) \(+(-\)\(51\!\cdots\!28\)\( + \)\(79\!\cdots\!52\)\( \beta_{1} + \)\(40\!\cdots\!76\)\( \beta_{2} - \)\(76\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!04\)\( \beta_{4} - \)\(59\!\cdots\!48\)\( \beta_{5} + \)\(96\!\cdots\!00\)\( \beta_{6} - \)\(77\!\cdots\!40\)\( \beta_{7} + \)\(59\!\cdots\!40\)\( \beta_{8}) q^{64}\) \(+(\)\(34\!\cdots\!04\)\( - \)\(80\!\cdots\!70\)\( \beta_{1} - \)\(34\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!40\)\( \beta_{3} + \)\(66\!\cdots\!64\)\( \beta_{4} - \)\(80\!\cdots\!50\)\( \beta_{5} - \)\(11\!\cdots\!22\)\( \beta_{6} + \)\(67\!\cdots\!40\)\( \beta_{7} + \)\(27\!\cdots\!98\)\( \beta_{8}) q^{65}\) \(+(\)\(65\!\cdots\!96\)\( - \)\(84\!\cdots\!24\)\( \beta_{1} - \)\(67\!\cdots\!12\)\( \beta_{2} + \)\(38\!\cdots\!20\)\( \beta_{3} - \)\(33\!\cdots\!40\)\( \beta_{4} + \)\(18\!\cdots\!36\)\( \beta_{5} - \)\(80\!\cdots\!24\)\( \beta_{6} - \)\(11\!\cdots\!88\)\( \beta_{7} - \)\(84\!\cdots\!16\)\( \beta_{8}) q^{66}\) \(+(\)\(58\!\cdots\!48\)\( - \)\(24\!\cdots\!73\)\( \beta_{1} - \)\(10\!\cdots\!93\)\( \beta_{2} + \)\(59\!\cdots\!66\)\( \beta_{3} - \)\(14\!\cdots\!10\)\( \beta_{4} + \)\(19\!\cdots\!32\)\( \beta_{5} + \)\(50\!\cdots\!70\)\( \beta_{6} + \)\(22\!\cdots\!44\)\( \beta_{7} + \)\(74\!\cdots\!20\)\( \beta_{8}) q^{67}\) \(+(-\)\(10\!\cdots\!70\)\( - \)\(41\!\cdots\!30\)\( \beta_{1} + \)\(47\!\cdots\!10\)\( \beta_{2} - \)\(11\!\cdots\!78\)\( \beta_{3} - \)\(79\!\cdots\!08\)\( \beta_{4} + \)\(12\!\cdots\!40\)\( \beta_{5} + \)\(15\!\cdots\!48\)\( \beta_{6} + \)\(10\!\cdots\!96\)\( \beta_{7} + \)\(10\!\cdots\!40\)\( \beta_{8}) q^{68}\) \(+(-\)\(52\!\cdots\!24\)\( - \)\(68\!\cdots\!86\)\( \beta_{1} + \)\(78\!\cdots\!52\)\( \beta_{2} - \)\(46\!\cdots\!76\)\( \beta_{3} + \)\(23\!\cdots\!36\)\( \beta_{4} - \)\(62\!\cdots\!62\)\( \beta_{5} - \)\(25\!\cdots\!34\)\( \beta_{6} + \)\(80\!\cdots\!52\)\( \beta_{7} - \)\(90\!\cdots\!66\)\( \beta_{8}) q^{69}\) \(+(\)\(72\!\cdots\!80\)\( + \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} + \)\(22\!\cdots\!56\)\( \beta_{3} - \)\(18\!\cdots\!04\)\( \beta_{4} + \)\(82\!\cdots\!00\)\( \beta_{5} - \)\(93\!\cdots\!00\)\( \beta_{6} - \)\(27\!\cdots\!00\)\( \beta_{7} - \)\(11\!\cdots\!00\)\( \beta_{8}) q^{70}\) \(+(\)\(35\!\cdots\!66\)\( + \)\(49\!\cdots\!45\)\( \beta_{1} - \)\(14\!\cdots\!75\)\( \beta_{2} + \)\(18\!\cdots\!79\)\( \beta_{3} + \)\(45\!\cdots\!79\)\( \beta_{4} + \)\(14\!\cdots\!74\)\( \beta_{5} + \)\(42\!\cdots\!71\)\( \beta_{6} - \)\(18\!\cdots\!88\)\( \beta_{7} + \)\(27\!\cdots\!04\)\( \beta_{8}) q^{71}\) \(+(\)\(14\!\cdots\!34\)\( + \)\(23\!\cdots\!92\)\( \beta_{1} - \)\(83\!\cdots\!12\)\( \beta_{2} + \)\(42\!\cdots\!23\)\( \beta_{3} - \)\(26\!\cdots\!13\)\( \beta_{4} + \)\(95\!\cdots\!17\)\( \beta_{5} - \)\(43\!\cdots\!37\)\( \beta_{6} + \)\(29\!\cdots\!75\)\( \beta_{7} + \)\(56\!\cdots\!20\)\( \beta_{8}) q^{72}\) \(+(-\)\(12\!\cdots\!86\)\( + \)\(11\!\cdots\!97\)\( \beta_{1} - \)\(17\!\cdots\!42\)\( \beta_{2} - \)\(49\!\cdots\!74\)\( \beta_{3} + \)\(38\!\cdots\!60\)\( \beta_{4} - \)\(75\!\cdots\!53\)\( \beta_{5} - \)\(85\!\cdots\!75\)\( \beta_{6} - \)\(69\!\cdots\!86\)\( \beta_{7} - \)\(41\!\cdots\!95\)\( \beta_{8}) q^{73}\) \(+(\)\(16\!\cdots\!10\)\( + \)\(64\!\cdots\!78\)\( \beta_{1} - \)\(64\!\cdots\!56\)\( \beta_{2} - \)\(28\!\cdots\!92\)\( \beta_{3} - \)\(95\!\cdots\!56\)\( \beta_{4} - \)\(26\!\cdots\!24\)\( \beta_{5} + \)\(20\!\cdots\!48\)\( \beta_{6} + \)\(88\!\cdots\!56\)\( \beta_{7} + \)\(79\!\cdots\!52\)\( \beta_{8}) q^{74}\) \(+(\)\(21\!\cdots\!20\)\( - \)\(99\!\cdots\!85\)\( \beta_{1} + \)\(11\!\cdots\!55\)\( \beta_{2} - \)\(18\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!80\)\( \beta_{4} + \)\(65\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!40\)\( \beta_{6} + \)\(30\!\cdots\!00\)\( \beta_{7} + \)\(53\!\cdots\!40\)\( \beta_{8}) q^{75}\) \(+(\)\(21\!\cdots\!84\)\( - \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} + \)\(15\!\cdots\!72\)\( \beta_{3} - \)\(57\!\cdots\!84\)\( \beta_{4} - \)\(11\!\cdots\!32\)\( \beta_{5} - \)\(64\!\cdots\!44\)\( \beta_{6} - \)\(66\!\cdots\!28\)\( \beta_{7} - \)\(59\!\cdots\!96\)\( \beta_{8}) q^{76}\) \(+(\)\(94\!\cdots\!20\)\( - \)\(53\!\cdots\!38\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2} + \)\(10\!\cdots\!16\)\( \beta_{3} - \)\(25\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!54\)\( \beta_{5} - \)\(19\!\cdots\!14\)\( \beta_{6} + \)\(17\!\cdots\!40\)\( \beta_{7} + \)\(11\!\cdots\!50\)\( \beta_{8}) q^{77}\) \(+(\)\(93\!\cdots\!78\)\( - \)\(86\!\cdots\!90\)\( \beta_{1} - \)\(86\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(16\!\cdots\!50\)\( \beta_{4} - \)\(12\!\cdots\!06\)\( \beta_{5} + \)\(84\!\cdots\!20\)\( \beta_{6} + \)\(13\!\cdots\!88\)\( \beta_{7} + \)\(18\!\cdots\!00\)\( \beta_{8}) q^{78}\) \(+(-\)\(29\!\cdots\!92\)\( + \)\(14\!\cdots\!86\)\( \beta_{1} - \)\(13\!\cdots\!22\)\( \beta_{2} - \)\(12\!\cdots\!94\)\( \beta_{3} + \)\(89\!\cdots\!54\)\( \beta_{4} + \)\(94\!\cdots\!92\)\( \beta_{5} - \)\(53\!\cdots\!26\)\( \beta_{6} - \)\(86\!\cdots\!92\)\( \beta_{7} - \)\(55\!\cdots\!04\)\( \beta_{8}) q^{79}\) \(+(\)\(13\!\cdots\!80\)\( + \)\(61\!\cdots\!64\)\( \beta_{1} - \)\(95\!\cdots\!56\)\( \beta_{2} + \)\(99\!\cdots\!96\)\( \beta_{3} - \)\(35\!\cdots\!64\)\( \beta_{4} - \)\(45\!\cdots\!00\)\( \beta_{5} - \)\(24\!\cdots\!00\)\( \beta_{6} - \)\(35\!\cdots\!00\)\( \beta_{7} + \)\(99\!\cdots\!00\)\( \beta_{8}) q^{80}\) \(+(\)\(27\!\cdots\!41\)\( + \)\(14\!\cdots\!91\)\( \beta_{1} + \)\(30\!\cdots\!98\)\( \beta_{2} + \)\(49\!\cdots\!58\)\( \beta_{3} - \)\(17\!\cdots\!56\)\( \beta_{4} - \)\(46\!\cdots\!91\)\( \beta_{5} + \)\(45\!\cdots\!91\)\( \beta_{6} - \)\(27\!\cdots\!18\)\( \beta_{7} + \)\(34\!\cdots\!79\)\( \beta_{8}) q^{81}\) \(+(\)\(52\!\cdots\!06\)\( + \)\(14\!\cdots\!10\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!44\)\( \beta_{3} + \)\(46\!\cdots\!64\)\( \beta_{4} + \)\(79\!\cdots\!80\)\( \beta_{5} + \)\(58\!\cdots\!16\)\( \beta_{6} + \)\(45\!\cdots\!12\)\( \beta_{7} - \)\(39\!\cdots\!60\)\( \beta_{8}) q^{82}\) \(+(\)\(10\!\cdots\!72\)\( + \)\(51\!\cdots\!01\)\( \beta_{1} + \)\(16\!\cdots\!93\)\( \beta_{2} - \)\(17\!\cdots\!80\)\( \beta_{3} - \)\(21\!\cdots\!20\)\( \beta_{4} - \)\(44\!\cdots\!00\)\( \beta_{5} - \)\(18\!\cdots\!80\)\( \beta_{6} - \)\(81\!\cdots\!00\)\( \beta_{7} + \)\(33\!\cdots\!20\)\( \beta_{8}) q^{83}\) \(+(\)\(28\!\cdots\!76\)\( - \)\(49\!\cdots\!52\)\( \beta_{1} - \)\(53\!\cdots\!36\)\( \beta_{2} - \)\(64\!\cdots\!56\)\( \beta_{3} + \)\(40\!\cdots\!16\)\( \beta_{4} + \)\(12\!\cdots\!52\)\( \beta_{5} - \)\(34\!\cdots\!80\)\( \beta_{6} - \)\(73\!\cdots\!80\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8}) q^{84}\) \(+(\)\(39\!\cdots\!12\)\( - \)\(13\!\cdots\!60\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(19\!\cdots\!58\)\( \beta_{4} - \)\(70\!\cdots\!50\)\( \beta_{5} + \)\(49\!\cdots\!34\)\( \beta_{6} + \)\(47\!\cdots\!20\)\( \beta_{7} - \)\(25\!\cdots\!06\)\( \beta_{8}) q^{85}\) \(+(\)\(13\!\cdots\!23\)\( - \)\(43\!\cdots\!95\)\( \beta_{1} - \)\(21\!\cdots\!80\)\( \beta_{2} + \)\(22\!\cdots\!86\)\( \beta_{3} + \)\(57\!\cdots\!87\)\( \beta_{4} + \)\(28\!\cdots\!61\)\( \beta_{5} + \)\(11\!\cdots\!12\)\( \beta_{6} - \)\(51\!\cdots\!36\)\( \beta_{7} - \)\(34\!\cdots\!12\)\( \beta_{8}) q^{86}\) \(+(\)\(30\!\cdots\!06\)\( - \)\(32\!\cdots\!13\)\( \beta_{1} + \)\(32\!\cdots\!27\)\( \beta_{2} - \)\(85\!\cdots\!91\)\( \beta_{3} + \)\(74\!\cdots\!37\)\( \beta_{4} + \)\(25\!\cdots\!14\)\( \beta_{5} - \)\(35\!\cdots\!27\)\( \beta_{6} - \)\(48\!\cdots\!36\)\( \beta_{7} + \)\(20\!\cdots\!60\)\( \beta_{8}) q^{87}\) \(+(\)\(57\!\cdots\!68\)\( - \)\(84\!\cdots\!12\)\( \beta_{1} + \)\(65\!\cdots\!80\)\( \beta_{2} + \)\(85\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!24\)\( \beta_{4} - \)\(28\!\cdots\!16\)\( \beta_{5} + \)\(12\!\cdots\!76\)\( \beta_{6} + \)\(91\!\cdots\!20\)\( \beta_{7} - \)\(18\!\cdots\!20\)\( \beta_{8}) q^{88}\) \(+(\)\(66\!\cdots\!82\)\( + \)\(13\!\cdots\!29\)\( \beta_{1} + \)\(14\!\cdots\!22\)\( \beta_{2} - \)\(18\!\cdots\!26\)\( \beta_{3} + \)\(88\!\cdots\!00\)\( \beta_{4} - \)\(93\!\cdots\!17\)\( \beta_{5} + \)\(25\!\cdots\!13\)\( \beta_{6} + \)\(20\!\cdots\!06\)\( \beta_{7} - \)\(64\!\cdots\!83\)\( \beta_{8}) q^{89}\) \(+(\)\(28\!\cdots\!34\)\( + \)\(75\!\cdots\!30\)\( \beta_{1} - \)\(77\!\cdots\!52\)\( \beta_{2} - \)\(20\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} - \)\(11\!\cdots\!00\)\( \beta_{5} + \)\(12\!\cdots\!88\)\( \beta_{6} - \)\(12\!\cdots\!60\)\( \beta_{7} + \)\(20\!\cdots\!08\)\( \beta_{8}) q^{90}\) \(+(\)\(27\!\cdots\!84\)\( + \)\(46\!\cdots\!88\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(34\!\cdots\!44\)\( \beta_{3} + \)\(60\!\cdots\!08\)\( \beta_{4} + \)\(39\!\cdots\!04\)\( \beta_{5} - \)\(29\!\cdots\!32\)\( \beta_{6} - \)\(23\!\cdots\!44\)\( \beta_{7} - \)\(13\!\cdots\!28\)\( \beta_{8}) q^{91}\) \(+(\)\(68\!\cdots\!60\)\( + \)\(75\!\cdots\!24\)\( \beta_{1} - \)\(97\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!04\)\( \beta_{4} + \)\(46\!\cdots\!00\)\( \beta_{5} - \)\(63\!\cdots\!24\)\( \beta_{6} + \)\(55\!\cdots\!52\)\( \beta_{7} - \)\(52\!\cdots\!00\)\( \beta_{8}) q^{92}\) \(+(\)\(13\!\cdots\!20\)\( - \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!12\)\( \beta_{3} + \)\(32\!\cdots\!40\)\( \beta_{4} - \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(26\!\cdots\!40\)\( \beta_{6} + \)\(16\!\cdots\!72\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{93}\) \(+(\)\(16\!\cdots\!16\)\( - \)\(59\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} + \)\(14\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!96\)\( \beta_{4} + \)\(21\!\cdots\!92\)\( \beta_{5} - \)\(20\!\cdots\!36\)\( \beta_{6} - \)\(27\!\cdots\!32\)\( \beta_{7} - \)\(64\!\cdots\!24\)\( \beta_{8}) q^{94}\) \(+(\)\(88\!\cdots\!50\)\( - \)\(10\!\cdots\!05\)\( \beta_{1} + \)\(93\!\cdots\!95\)\( \beta_{2} - \)\(59\!\cdots\!95\)\( \beta_{3} + \)\(33\!\cdots\!05\)\( \beta_{4} + \)\(88\!\cdots\!50\)\( \beta_{5} - \)\(35\!\cdots\!75\)\( \beta_{6} + \)\(37\!\cdots\!00\)\( \beta_{7} - \)\(24\!\cdots\!00\)\( \beta_{8}) q^{95}\) \(+(\)\(36\!\cdots\!24\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(19\!\cdots\!28\)\( \beta_{3} + \)\(11\!\cdots\!96\)\( \beta_{4} + \)\(69\!\cdots\!12\)\( \beta_{5} + \)\(57\!\cdots\!20\)\( \beta_{6} + \)\(28\!\cdots\!20\)\( \beta_{7} + \)\(43\!\cdots\!00\)\( \beta_{8}) q^{96}\) \(+(\)\(28\!\cdots\!38\)\( - \)\(10\!\cdots\!31\)\( \beta_{1} - \)\(10\!\cdots\!78\)\( \beta_{2} + \)\(55\!\cdots\!82\)\( \beta_{3} - \)\(37\!\cdots\!68\)\( \beta_{4} - \)\(14\!\cdots\!45\)\( \beta_{5} - \)\(86\!\cdots\!67\)\( \beta_{6} - \)\(13\!\cdots\!54\)\( \beta_{7} - \)\(56\!\cdots\!35\)\( \beta_{8}) q^{97}\) \(+(\)\(65\!\cdots\!27\)\( + \)\(39\!\cdots\!99\)\( \beta_{1} - \)\(46\!\cdots\!84\)\( \beta_{2} + \)\(23\!\cdots\!24\)\( \beta_{3} + \)\(70\!\cdots\!44\)\( \beta_{4} - \)\(68\!\cdots\!40\)\( \beta_{5} + \)\(15\!\cdots\!36\)\( \beta_{6} + \)\(86\!\cdots\!52\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{98}\) \(+(-\)\(90\!\cdots\!48\)\( + \)\(16\!\cdots\!27\)\( \beta_{1} - \)\(44\!\cdots\!09\)\( \beta_{2} + \)\(34\!\cdots\!44\)\( \beta_{3} - \)\(16\!\cdots\!28\)\( \beta_{4} + \)\(25\!\cdots\!36\)\( \beta_{5} - \)\(53\!\cdots\!48\)\( \beta_{6} + \)\(21\!\cdots\!64\)\( \beta_{7} - \)\(13\!\cdots\!72\)\( \beta_{8}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut -\mathstrut 116180770089455544q^{2} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!92\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!12\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!90\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!68\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!13\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 116180770089455544q^{2} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!92\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!12\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!90\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!68\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!13\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!60\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!32\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!44\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!42\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!76\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!20\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!16\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!06\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!08\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!68\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!12\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!92\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!32\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!08\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!32\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!84\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!24\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!40\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!16\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!06\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!76\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!18\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!88\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!92\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!24\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!30\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!68\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!44\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!92\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!63\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!68\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!42\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!18\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!12\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!08\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!88\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!20\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!36\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!56\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!92\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!24\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!60\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!68\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!58\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!24\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!12\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!84\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!09\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!12\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!92\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!24\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!60\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!50\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!68\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!56\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!84\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!24\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!68\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!06\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!08\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!24\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(2\) \(x^{8}\mathstrut -\mathstrut \) \(41\!\cdots\!36\) \(x^{7}\mathstrut +\mathstrut \) \(20\!\cdots\!12\) \(x^{6}\mathstrut +\mathstrut \) \(53\!\cdots\!86\) \(x^{5}\mathstrut -\mathstrut \) \(39\!\cdots\!00\) \(x^{4}\mathstrut -\mathstrut \) \(25\!\cdots\!00\) \(x^{3}\mathstrut +\mathstrut \) \(49\!\cdots\!00\) \(x^{2}\mathstrut +\mathstrut \) \(38\!\cdots\!25\) \(x\mathstrut +\mathstrut \) \(17\!\cdots\!50\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(18\!\cdots\!49\) \(\nu^{8}\mathstrut -\mathstrut \) \(65\!\cdots\!75\) \(\nu^{7}\mathstrut +\mathstrut \) \(47\!\cdots\!73\) \(\nu^{6}\mathstrut +\mathstrut \) \(22\!\cdots\!89\) \(\nu^{5}\mathstrut -\mathstrut \) \(21\!\cdots\!69\) \(\nu^{4}\mathstrut -\mathstrut \) \(20\!\cdots\!93\) \(\nu^{3}\mathstrut +\mathstrut \) \(22\!\cdots\!83\) \(\nu^{2}\mathstrut +\mathstrut \) \(42\!\cdots\!35\) \(\nu\mathstrut +\mathstrut \) \(90\!\cdots\!30\)\()/\)\(40\!\cdots\!84\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(90\!\cdots\!97\) \(\nu^{8}\mathstrut -\mathstrut \) \(31\!\cdots\!75\) \(\nu^{7}\mathstrut +\mathstrut \) \(23\!\cdots\!69\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!17\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!57\) \(\nu^{4}\mathstrut -\mathstrut \) \(97\!\cdots\!29\) \(\nu^{3}\mathstrut +\mathstrut \) \(24\!\cdots\!83\) \(\nu^{2}\mathstrut +\mathstrut \) \(22\!\cdots\!19\) \(\nu\mathstrut -\mathstrut \) \(20\!\cdots\!66\)\()/\)\(40\!\cdots\!84\)
\(\beta_{4}\)\(=\)\((\)\(28\!\cdots\!31\) \(\nu^{8}\mathstrut -\mathstrut \) \(17\!\cdots\!27\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!11\) \(\nu^{6}\mathstrut +\mathstrut \) \(71\!\cdots\!37\) \(\nu^{5}\mathstrut +\mathstrut \) \(11\!\cdots\!11\) \(\nu^{4}\mathstrut -\mathstrut \) \(81\!\cdots\!65\) \(\nu^{3}\mathstrut -\mathstrut \) \(31\!\cdots\!25\) \(\nu^{2}\mathstrut +\mathstrut \) \(19\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!50\)\()/\)\(31\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(23\!\cdots\!11\) \(\nu^{8}\mathstrut -\mathstrut \) \(84\!\cdots\!13\) \(\nu^{7}\mathstrut +\mathstrut \) \(81\!\cdots\!91\) \(\nu^{6}\mathstrut +\mathstrut \) \(20\!\cdots\!03\) \(\nu^{5}\mathstrut -\mathstrut \) \(75\!\cdots\!91\) \(\nu^{4}\mathstrut -\mathstrut \) \(94\!\cdots\!35\) \(\nu^{3}\mathstrut +\mathstrut \) \(15\!\cdots\!25\) \(\nu^{2}\mathstrut +\mathstrut \) \(14\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!50\)\()/\)\(62\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(38\!\cdots\!21\) \(\nu^{8}\mathstrut +\mathstrut \) \(46\!\cdots\!43\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!01\) \(\nu^{6}\mathstrut +\mathstrut \) \(53\!\cdots\!67\) \(\nu^{5}\mathstrut +\mathstrut \) \(97\!\cdots\!01\) \(\nu^{4}\mathstrut -\mathstrut \) \(32\!\cdots\!15\) \(\nu^{3}\mathstrut +\mathstrut \) \(21\!\cdots\!25\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!25\) \(\nu\mathstrut -\mathstrut \) \(42\!\cdots\!50\)\()/\)\(62\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(20\!\cdots\!19\) \(\nu^{8}\mathstrut +\mathstrut \) \(15\!\cdots\!23\) \(\nu^{7}\mathstrut +\mathstrut \) \(80\!\cdots\!39\) \(\nu^{6}\mathstrut -\mathstrut \) \(64\!\cdots\!13\) \(\nu^{5}\mathstrut -\mathstrut \) \(86\!\cdots\!39\) \(\nu^{4}\mathstrut +\mathstrut \) \(73\!\cdots\!85\) \(\nu^{3}\mathstrut +\mathstrut \) \(22\!\cdots\!25\) \(\nu^{2}\mathstrut -\mathstrut \) \(19\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(98\!\cdots\!50\)\()/\)\(12\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(37\!\cdots\!91\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!47\) \(\nu^{7}\mathstrut -\mathstrut \) \(14\!\cdots\!71\) \(\nu^{6}\mathstrut +\mathstrut \) \(64\!\cdots\!57\) \(\nu^{5}\mathstrut +\mathstrut \) \(15\!\cdots\!71\) \(\nu^{4}\mathstrut -\mathstrut \) \(75\!\cdots\!65\) \(\nu^{3}\mathstrut -\mathstrut \) \(39\!\cdots\!25\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(82\!\cdots\!50\)\()/\)\(62\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(484553\) \(\beta_{2}\mathstrut -\mathstrut \) \(18073717211147679\) \(\beta_{1}\mathstrut +\mathstrut \) \(53001739878028145233323179512730439\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(413\) \(\beta_{6}\mathstrut +\mathstrut \) \(664275\) \(\beta_{5}\mathstrut +\mathstrut \) \(189393293777\) \(\beta_{4}\mathstrut -\mathstrut \) \(38107786849440575\) \(\beta_{3}\mathstrut +\mathstrut \) \(114056587572313116197696\) \(\beta_{2}\mathstrut +\mathstrut \) \(88405175090489855863641579900017084\) \(\beta_{1}\mathstrut -\mathstrut \) \(957938458254282665056357244169716144561633024779486\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(10412963909952\) \(\beta_{8}\mathstrut +\mathstrut \) \(1467556945694547\) \(\beta_{7}\mathstrut -\mathstrut \) \(27009661388808667609\) \(\beta_{6}\mathstrut -\mathstrut \) \(98749583594328355423959\) \(\beta_{5}\mathstrut +\mathstrut \) \(5183614690002396536607437763\) \(\beta_{4}\mathstrut +\mathstrut \) \(16303251739383817902272811577456587\) \(\beta_{3}\mathstrut -\mathstrut \) \(2273068339929917024625036724765707901880\) \(\beta_{2}\mathstrut -\mathstrut \) \(487678643604962842851760126691162171713004223700820\) \(\beta_{1}\mathstrut +\mathstrut \) \(585703511752209383105760939369323734653716598776592595231776182697422\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(3063728300247593546060314440\) \(\beta_{8}\mathstrut +\mathstrut \) \(145445726412599068359363126821843\) \(\beta_{7}\mathstrut +\mathstrut \) \(92220362597007090321083244188580655\) \(\beta_{6}\mathstrut +\mathstrut \) \(313331271415733645970061805205391868289\) \(\beta_{5}\mathstrut +\mathstrut \) \(34592650781384667306935247707708693752285955\) \(\beta_{4}\mathstrut -\mathstrut \) \(5779427120278069279131207652491160242292929801858\) \(\beta_{3}\mathstrut -\mathstrut \) \(18631557476300263724627194944056472259557411442379445491\) \(\beta_{2}\mathstrut +\mathstrut \) \(9108238802121355524826978257971989991069092802549671776960558125515\) \(\beta_{1}\mathstrut -\mathstrut \) \(201936067284529692989337825142643067881337601859473172714700829894858794207478375641\)\()/7776\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(17\!\cdots\!20\) \(\beta_{8}\mathstrut +\mathstrut \) \(84\!\cdots\!67\) \(\beta_{7}\mathstrut -\mathstrut \) \(40\!\cdots\!25\) \(\beta_{6}\mathstrut -\mathstrut \) \(20\!\cdots\!47\) \(\beta_{5}\mathstrut +\mathstrut \) \(70\!\cdots\!71\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\!\cdots\!59\) \(\beta_{3}\mathstrut +\mathstrut \) \(85\!\cdots\!84\) \(\beta_{2}\mathstrut -\mathstrut \) \(63\!\cdots\!28\) \(\beta_{1}\mathstrut +\mathstrut \) \(53\!\cdots\!14\)\()/20736\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(22\!\cdots\!40\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\!\cdots\!13\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\!\cdots\!41\) \(\beta_{6}\mathstrut +\mathstrut \) \(57\!\cdots\!15\) \(\beta_{5}\mathstrut +\mathstrut \) \(45\!\cdots\!65\) \(\beta_{4}\mathstrut -\mathstrut \) \(78\!\cdots\!83\) \(\beta_{3}\mathstrut -\mathstrut \) \(54\!\cdots\!24\) \(\beta_{2}\mathstrut +\mathstrut \) \(95\!\cdots\!32\) \(\beta_{1}\mathstrut -\mathstrut \) \(27\!\cdots\!58\)\()/41472\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(26\!\cdots\!44\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\!\cdots\!37\) \(\beta_{7}\mathstrut -\mathstrut \) \(62\!\cdots\!77\) \(\beta_{6}\mathstrut -\mathstrut \) \(34\!\cdots\!63\) \(\beta_{5}\mathstrut +\mathstrut \) \(91\!\cdots\!47\) \(\beta_{4}\mathstrut +\mathstrut \) \(21\!\cdots\!31\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\!\cdots\!12\) \(\beta_{2}\mathstrut -\mathstrut \) \(90\!\cdots\!56\) \(\beta_{1}\mathstrut +\mathstrut \) \(63\!\cdots\!02\)\()/124416\)

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
1.42762e16
9.41344e15
9.00047e15
5.63699e15
−4.70183e14
−5.44635e15
−5.76359e15
−1.17725e16
−1.48745e16
−3.55538e17 −3.54906e26 8.48689e34 1.00650e40 1.26183e44 6.58681e47 −1.54056e52 −7.26915e54 −3.57850e57
1.2 −2.38832e17 4.64741e27 1.55022e34 −1.89028e40 −1.10995e45 5.62740e48 6.21827e51 1.42033e55 4.51459e57
1.3 −2.28920e17 −5.05419e27 1.08661e34 −2.40937e40 1.15700e45 −2.38510e48 7.02151e51 1.81497e55 5.51552e57
1.4 −1.48197e17 7.29391e26 −1.95761e34 6.05511e39 −1.08093e44 −4.10385e48 9.05697e51 −6.86309e54 −8.97348e56
1.5 −1.62458e15 −2.61982e27 −4.15357e34 1.47212e40 4.25609e42 6.19961e48 1.34960e50 −5.31663e53 −2.39156e55
1.6 1.17803e17 5.60840e26 −2.76608e34 −2.33935e40 6.60688e43 −1.66757e47 −8.15189e51 −7.08056e54 −2.75583e57
1.7 1.25417e17 4.55926e27 −2.58089e34 2.05215e40 5.71809e44 −2.84362e48 −8.44651e51 1.33917e55 2.57375e57
1.8 2.69631e17 −3.76423e27 3.11624e34 5.22690e39 −1.01495e45 −4.51498e48 −2.79767e51 6.77433e54 1.40933e57
1.9 3.44079e17 1.64109e27 7.68520e34 3.84023e38 5.64664e44 3.35173e48 1.21507e52 −4.70193e54 1.32134e56
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{116}^{\mathrm{new}}(\Gamma_0(1))\).