Properties

Label 1.96.a.a
Level 1
Weight 96
Character orbit 1.a
Self dual Yes
Analytic conductor 57.154
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 96 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(57.1535908815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{38}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
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.

\(f(q)\) \(=\) \( q\) \(+(-729457392285 + \beta_{1}) q^{2}\) \(+(-\)\(11\!\cdots\!60\)\( + 20242501 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(26\!\cdots\!48\)\( - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(24\!\cdots\!70\)\( - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(39\!\cdots\!00\)\( + \)\(16\!\cdots\!42\)\( \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(18\!\cdots\!20\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(11\!\cdots\!17\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(34\!\cdots\!86\)\( \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-729457392285 + \beta_{1}) q^{2}\) \(+(-\)\(11\!\cdots\!60\)\( + 20242501 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(26\!\cdots\!48\)\( - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(24\!\cdots\!70\)\( - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(39\!\cdots\!00\)\( + \)\(16\!\cdots\!42\)\( \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(18\!\cdots\!20\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(11\!\cdots\!17\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(34\!\cdots\!86\)\( \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9}\) \(+(-\)\(44\!\cdots\!30\)\( + \)\(40\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!56\)\( \beta_{2} + 1823205551639504480 \beta_{3} + 31201730362844 \beta_{4} + 32408570756 \beta_{5} - 8596448 \beta_{6} + 18144 \beta_{7}) q^{10}\) \(+(\)\(66\!\cdots\!52\)\( - \)\(38\!\cdots\!81\)\( \beta_{1} + \)\(40\!\cdots\!65\)\( \beta_{2} + 1971098222936635454 \beta_{3} - 2281046769584042 \beta_{4} + 1373746304736 \beta_{5} - 186517774 \beta_{6} - 1124608 \beta_{7}) q^{11}\) \(+(\)\(13\!\cdots\!80\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!36\)\( \beta_{2} + \)\(34\!\cdots\!28\)\( \beta_{3} - 1437017081453628624 \beta_{4} - 130671188070048 \beta_{5} + 46052813520 \beta_{6} + 51404976 \beta_{7}) q^{12}\) \(+(\)\(14\!\cdots\!30\)\( - \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(76\!\cdots\!69\)\( \beta_{2} + \)\(43\!\cdots\!38\)\( \beta_{3} - 22836391586997725369 \beta_{4} + 1277203065211264 \beta_{5} - 2768620169896 \beta_{6} - 1847131776 \beta_{7}) q^{13}\) \(+(\)\(11\!\cdots\!76\)\( + \)\(29\!\cdots\!16\)\( \beta_{1} + \)\(68\!\cdots\!26\)\( \beta_{2} + \)\(36\!\cdots\!08\)\( \beta_{3} + \)\(67\!\cdots\!10\)\( \beta_{4} + 113268416831067546 \beta_{5} + 101942184647552 \beta_{6} + 54313261184 \beta_{7}) q^{14}\) \(+(\)\(23\!\cdots\!40\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} - \)\(15\!\cdots\!68\)\( \beta_{2} + \)\(13\!\cdots\!85\)\( \beta_{3} + \)\(21\!\cdots\!93\)\( \beta_{4} - 4760217497661248368 \beta_{5} - 2699459604617481 \beta_{6} - 1343169464832 \beta_{7}) q^{15}\) \(+(\)\(11\!\cdots\!16\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} + \)\(35\!\cdots\!16\)\( \beta_{3} - \)\(51\!\cdots\!44\)\( \beta_{4} + 86123790987775929904 \beta_{5} + 54813613926115496 \beta_{6} + 28494357719832 \beta_{7}) q^{16}\) \(+(-\)\(17\!\cdots\!30\)\( - \)\(31\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!74\)\( \beta_{2} - \)\(76\!\cdots\!44\)\( \beta_{3} - \)\(11\!\cdots\!38\)\( \beta_{4} - \)\(63\!\cdots\!64\)\( \beta_{5} - 878472313378704156 \beta_{6} - 526281973855680 \beta_{7}) q^{17}\) \(+(-\)\(22\!\cdots\!85\)\( + \)\(79\!\cdots\!33\)\( \beta_{1} - \)\(21\!\cdots\!60\)\( \beta_{2} - \)\(75\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!28\)\( \beta_{4} - \)\(62\!\cdots\!56\)\( \beta_{5} + 11177988501349285056 \beta_{6} + 8559395719398720 \beta_{7}) q^{18}\) \(+(-\)\(65\!\cdots\!80\)\( - \)\(31\!\cdots\!75\)\( \beta_{1} - \)\(57\!\cdots\!09\)\( \beta_{2} - \)\(39\!\cdots\!34\)\( \beta_{3} - \)\(25\!\cdots\!86\)\( \beta_{4} + \)\(23\!\cdots\!48\)\( \beta_{5} - \)\(11\!\cdots\!22\)\( \beta_{6} - 123672945181895424 \beta_{7}) q^{19}\) \(+(\)\(17\!\cdots\!60\)\( + \)\(47\!\cdots\!28\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} + \)\(88\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!88\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} + \)\(76\!\cdots\!00\)\( \beta_{6} + 1598525547088209600 \beta_{7}) q^{20}\) \(+(\)\(16\!\cdots\!92\)\( - \)\(29\!\cdots\!24\)\( \beta_{1} - \)\(93\!\cdots\!80\)\( \beta_{2} + \)\(48\!\cdots\!08\)\( \beta_{3} + \)\(30\!\cdots\!80\)\( \beta_{4} + \)\(24\!\cdots\!52\)\( \beta_{5} - \)\(20\!\cdots\!96\)\( \beta_{6} - 18582876201795196032 \beta_{7}) q^{21}\) \(+(-\)\(25\!\cdots\!20\)\( + \)\(76\!\cdots\!86\)\( \beta_{1} - \)\(63\!\cdots\!89\)\( \beta_{2} - \)\(60\!\cdots\!52\)\( \beta_{3} - \)\(13\!\cdots\!99\)\( \beta_{4} - \)\(42\!\cdots\!61\)\( \beta_{5} - \)\(37\!\cdots\!76\)\( \beta_{6} + \)\(19\!\cdots\!84\)\( \beta_{7}) q^{22}\) \(+(-\)\(89\!\cdots\!80\)\( - \)\(36\!\cdots\!22\)\( \beta_{1} + \)\(49\!\cdots\!56\)\( \beta_{2} - \)\(50\!\cdots\!35\)\( \beta_{3} - \)\(86\!\cdots\!95\)\( \beta_{4} - \)\(13\!\cdots\!60\)\( \beta_{5} + \)\(75\!\cdots\!35\)\( \beta_{6} - \)\(18\!\cdots\!00\)\( \beta_{7}) q^{23}\) \(+(\)\(36\!\cdots\!60\)\( + \)\(25\!\cdots\!36\)\( \beta_{1} + \)\(77\!\cdots\!24\)\( \beta_{2} + \)\(55\!\cdots\!04\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4} + \)\(18\!\cdots\!32\)\( \beta_{5} - \)\(83\!\cdots\!04\)\( \beta_{6} + \)\(16\!\cdots\!32\)\( \beta_{7}) q^{24}\) \(+(\)\(10\!\cdots\!75\)\( - \)\(26\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(63\!\cdots\!00\)\( \beta_{3} - \)\(54\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!20\)\( \beta_{5} + \)\(68\!\cdots\!60\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7}) q^{25}\) \(+(-\)\(86\!\cdots\!38\)\( + \)\(38\!\cdots\!38\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(24\!\cdots\!92\)\( \beta_{4} + \)\(54\!\cdots\!04\)\( \beta_{5} - \)\(43\!\cdots\!48\)\( \beta_{6} + \)\(89\!\cdots\!84\)\( \beta_{7}) q^{26}\) \(+(\)\(80\!\cdots\!20\)\( + \)\(84\!\cdots\!74\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} - \)\(77\!\cdots\!58\)\( \beta_{3} + \)\(35\!\cdots\!14\)\( \beta_{4} + \)\(11\!\cdots\!48\)\( \beta_{5} + \)\(21\!\cdots\!70\)\( \beta_{6} - \)\(57\!\cdots\!56\)\( \beta_{7}) q^{27}\) \(+(\)\(37\!\cdots\!80\)\( + \)\(20\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!44\)\( \beta_{2} + \)\(85\!\cdots\!72\)\( \beta_{3} - \)\(13\!\cdots\!76\)\( \beta_{4} - \)\(16\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!00\)\( \beta_{6} + \)\(33\!\cdots\!64\)\( \beta_{7}) q^{28}\) \(+(\)\(96\!\cdots\!30\)\( + \)\(45\!\cdots\!36\)\( \beta_{1} + \)\(22\!\cdots\!31\)\( \beta_{2} - \)\(10\!\cdots\!70\)\( \beta_{3} - \)\(36\!\cdots\!37\)\( \beta_{4} + \)\(11\!\cdots\!04\)\( \beta_{5} - \)\(82\!\cdots\!08\)\( \beta_{6} - \)\(17\!\cdots\!36\)\( \beta_{7}) q^{29}\) \(+(-\)\(78\!\cdots\!60\)\( + \)\(92\!\cdots\!92\)\( \beta_{1} + \)\(20\!\cdots\!94\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(18\!\cdots\!18\)\( \beta_{4} - \)\(26\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6} + \)\(77\!\cdots\!00\)\( \beta_{7}) q^{30}\) \(+(\)\(60\!\cdots\!52\)\( + \)\(20\!\cdots\!72\)\( \beta_{1} + \)\(75\!\cdots\!92\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!64\)\( \beta_{4} - \)\(12\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!56\)\( \beta_{6} - \)\(29\!\cdots\!52\)\( \beta_{7}) q^{31}\) \(+(\)\(11\!\cdots\!40\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - \)\(31\!\cdots\!56\)\( \beta_{2} + \)\(47\!\cdots\!76\)\( \beta_{3} - \)\(65\!\cdots\!48\)\( \beta_{4} + \)\(14\!\cdots\!56\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(91\!\cdots\!20\)\( \beta_{7}) q^{32}\) \(+(-\)\(19\!\cdots\!20\)\( + \)\(33\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(68\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!42\)\( \beta_{4} - \)\(63\!\cdots\!48\)\( \beta_{5} + \)\(67\!\cdots\!96\)\( \beta_{6} - \)\(17\!\cdots\!96\)\( \beta_{7}) q^{33}\) \(+(-\)\(20\!\cdots\!54\)\( - \)\(87\!\cdots\!54\)\( \beta_{1} + \)\(43\!\cdots\!88\)\( \beta_{2} - \)\(61\!\cdots\!76\)\( \beta_{3} + \)\(79\!\cdots\!24\)\( \beta_{4} + \)\(10\!\cdots\!48\)\( \beta_{5} - \)\(14\!\cdots\!92\)\( \beta_{6} - \)\(23\!\cdots\!64\)\( \beta_{7}) q^{34}\) \(+(\)\(73\!\cdots\!20\)\( - \)\(29\!\cdots\!84\)\( \beta_{1} + \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} - \)\(77\!\cdots\!96\)\( \beta_{4} + \)\(35\!\cdots\!96\)\( \beta_{5} + \)\(10\!\cdots\!32\)\( \beta_{6} + \)\(40\!\cdots\!04\)\( \beta_{7}) q^{35}\) \(+(\)\(64\!\cdots\!16\)\( - \)\(45\!\cdots\!56\)\( \beta_{1} + \)\(35\!\cdots\!13\)\( \beta_{2} + \)\(76\!\cdots\!01\)\( \beta_{3} + \)\(20\!\cdots\!56\)\( \beta_{4} - \)\(23\!\cdots\!24\)\( \beta_{5} - \)\(46\!\cdots\!00\)\( \beta_{6} - \)\(21\!\cdots\!00\)\( \beta_{7}) q^{36}\) \(+(-\)\(22\!\cdots\!90\)\( - \)\(39\!\cdots\!24\)\( \beta_{1} - \)\(29\!\cdots\!93\)\( \beta_{2} - \)\(15\!\cdots\!06\)\( \beta_{3} + \)\(74\!\cdots\!23\)\( \beta_{4} + \)\(90\!\cdots\!56\)\( \beta_{5} + \)\(14\!\cdots\!80\)\( \beta_{6} + \)\(76\!\cdots\!88\)\( \beta_{7}) q^{37}\) \(+(-\)\(20\!\cdots\!80\)\( - \)\(22\!\cdots\!54\)\( \beta_{1} - \)\(67\!\cdots\!87\)\( \beta_{2} - \)\(95\!\cdots\!64\)\( \beta_{3} - \)\(85\!\cdots\!33\)\( \beta_{4} + \)\(50\!\cdots\!45\)\( \beta_{5} - \)\(25\!\cdots\!48\)\( \beta_{6} - \)\(16\!\cdots\!84\)\( \beta_{7}) q^{38}\) \(+(\)\(58\!\cdots\!04\)\( + \)\(11\!\cdots\!30\)\( \beta_{1} - \)\(90\!\cdots\!28\)\( \beta_{2} + \)\(15\!\cdots\!57\)\( \beta_{3} - \)\(58\!\cdots\!07\)\( \beta_{4} - \)\(30\!\cdots\!44\)\( \beta_{5} - \)\(37\!\cdots\!29\)\( \beta_{6} - \)\(15\!\cdots\!68\)\( \beta_{7}) q^{39}\) \(+(\)\(48\!\cdots\!00\)\( + \)\(45\!\cdots\!20\)\( \beta_{1} + \)\(20\!\cdots\!20\)\( \beta_{2} + \)\(13\!\cdots\!50\)\( \beta_{3} + \)\(41\!\cdots\!30\)\( \beta_{4} + \)\(98\!\cdots\!20\)\( \beta_{5} + \)\(46\!\cdots\!90\)\( \beta_{6} + \)\(20\!\cdots\!30\)\( \beta_{7}) q^{40}\) \(+(-\)\(10\!\cdots\!98\)\( + \)\(61\!\cdots\!32\)\( \beta_{1} + \)\(62\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!68\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4} - \)\(17\!\cdots\!76\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(10\!\cdots\!00\)\( \beta_{7}) q^{41}\) \(+(-\)\(19\!\cdots\!80\)\( + \)\(43\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!28\)\( \beta_{2} - \)\(99\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!72\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5} + \)\(41\!\cdots\!76\)\( \beta_{6} + \)\(29\!\cdots\!60\)\( \beta_{7}) q^{42}\) \(+(\)\(45\!\cdots\!00\)\( - \)\(29\!\cdots\!45\)\( \beta_{1} - \)\(34\!\cdots\!31\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(55\!\cdots\!20\)\( \beta_{4} - \)\(44\!\cdots\!56\)\( \beta_{5} - \)\(18\!\cdots\!32\)\( \beta_{6} - \)\(34\!\cdots\!44\)\( \beta_{7}) q^{43}\) \(+(\)\(24\!\cdots\!96\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} + \)\(73\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(33\!\cdots\!44\)\( \beta_{4} + \)\(41\!\cdots\!36\)\( \beta_{5} - \)\(24\!\cdots\!16\)\( \beta_{6} - \)\(13\!\cdots\!72\)\( \beta_{7}) q^{44}\) \(+(-\)\(20\!\cdots\!10\)\( - \)\(66\!\cdots\!68\)\( \beta_{1} + \)\(10\!\cdots\!99\)\( \beta_{2} - \)\(16\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!97\)\( \beta_{4} - \)\(14\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(92\!\cdots\!00\)\( \beta_{7}) q^{45}\) \(+(-\)\(24\!\cdots\!48\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(83\!\cdots\!74\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(38\!\cdots\!14\)\( \beta_{4} + \)\(91\!\cdots\!54\)\( \beta_{5} - \)\(19\!\cdots\!80\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7}) q^{46}\) \(+(\)\(48\!\cdots\!80\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(17\!\cdots\!50\)\( \beta_{3} + \)\(75\!\cdots\!90\)\( \beta_{4} + \)\(12\!\cdots\!48\)\( \beta_{5} + \)\(49\!\cdots\!06\)\( \beta_{6} + \)\(36\!\cdots\!52\)\( \beta_{7}) q^{47}\) \(+(\)\(11\!\cdots\!20\)\( + \)\(20\!\cdots\!96\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!76\)\( \beta_{3} - \)\(69\!\cdots\!48\)\( \beta_{4} - \)\(54\!\cdots\!44\)\( \beta_{5} + \)\(76\!\cdots\!24\)\( \beta_{6} + \)\(92\!\cdots\!20\)\( \beta_{7}) q^{48}\) \(+(\)\(33\!\cdots\!93\)\( + \)\(27\!\cdots\!12\)\( \beta_{1} + \)\(25\!\cdots\!04\)\( \beta_{2} - \)\(85\!\cdots\!48\)\( \beta_{3} - \)\(47\!\cdots\!36\)\( \beta_{4} + \)\(90\!\cdots\!84\)\( \beta_{5} - \)\(17\!\cdots\!20\)\( \beta_{6} - \)\(67\!\cdots\!40\)\( \beta_{7}) q^{49}\) \(+(-\)\(17\!\cdots\!75\)\( + \)\(13\!\cdots\!35\)\( \beta_{1} + \)\(72\!\cdots\!40\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!40\)\( \beta_{4} + \)\(26\!\cdots\!80\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6} + \)\(18\!\cdots\!20\)\( \beta_{7}) q^{50}\) \(+(\)\(30\!\cdots\!32\)\( - \)\(21\!\cdots\!10\)\( \beta_{1} + \)\(29\!\cdots\!74\)\( \beta_{2} + \)\(49\!\cdots\!34\)\( \beta_{3} - \)\(30\!\cdots\!94\)\( \beta_{4} - \)\(33\!\cdots\!88\)\( \beta_{5} + \)\(32\!\cdots\!02\)\( \beta_{6} - \)\(17\!\cdots\!16\)\( \beta_{7}) q^{51}\) \(+(\)\(19\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(20\!\cdots\!74\)\( \beta_{2} + \)\(36\!\cdots\!50\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!68\)\( \beta_{5} + \)\(20\!\cdots\!96\)\( \beta_{6} - \)\(74\!\cdots\!68\)\( \beta_{7}) q^{52}\) \(+(-\)\(36\!\cdots\!10\)\( - \)\(85\!\cdots\!60\)\( \beta_{1} - \)\(62\!\cdots\!33\)\( \beta_{2} - \)\(59\!\cdots\!82\)\( \beta_{3} - \)\(10\!\cdots\!69\)\( \beta_{4} + \)\(14\!\cdots\!12\)\( \beta_{5} - \)\(13\!\cdots\!80\)\( \beta_{6} + \)\(38\!\cdots\!56\)\( \beta_{7}) q^{53}\) \(+(\)\(55\!\cdots\!20\)\( - \)\(26\!\cdots\!20\)\( \beta_{1} + \)\(42\!\cdots\!38\)\( \beta_{2} - \)\(17\!\cdots\!24\)\( \beta_{3} + \)\(65\!\cdots\!14\)\( \beta_{4} + \)\(12\!\cdots\!66\)\( \beta_{5} + \)\(25\!\cdots\!04\)\( \beta_{6} - \)\(81\!\cdots\!32\)\( \beta_{7}) q^{54}\) \(+(\)\(97\!\cdots\!40\)\( + \)\(68\!\cdots\!62\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} - \)\(33\!\cdots\!45\)\( \beta_{3} - \)\(71\!\cdots\!77\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!25\)\( \beta_{6} + \)\(11\!\cdots\!00\)\( \beta_{7}) q^{55}\) \(+(\)\(90\!\cdots\!80\)\( + \)\(27\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!92\)\( \beta_{4} + \)\(67\!\cdots\!32\)\( \beta_{5} - \)\(33\!\cdots\!52\)\( \beta_{6} + \)\(48\!\cdots\!16\)\( \beta_{7}) q^{56}\) \(+(\)\(18\!\cdots\!40\)\( + \)\(29\!\cdots\!56\)\( \beta_{1} - \)\(45\!\cdots\!58\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(88\!\cdots\!70\)\( \beta_{4} - \)\(23\!\cdots\!36\)\( \beta_{5} + \)\(76\!\cdots\!08\)\( \beta_{6} - \)\(15\!\cdots\!64\)\( \beta_{7}) q^{57}\) \(+(\)\(29\!\cdots\!30\)\( + \)\(76\!\cdots\!30\)\( \beta_{1} - \)\(72\!\cdots\!76\)\( \beta_{2} + \)\(48\!\cdots\!56\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(28\!\cdots\!52\)\( \beta_{5} - \)\(14\!\cdots\!92\)\( \beta_{6} + \)\(19\!\cdots\!08\)\( \beta_{7}) q^{58}\) \(+(\)\(30\!\cdots\!60\)\( - \)\(99\!\cdots\!21\)\( \beta_{1} + \)\(18\!\cdots\!25\)\( \beta_{2} - \)\(16\!\cdots\!68\)\( \beta_{3} + \)\(23\!\cdots\!20\)\( \beta_{4} + \)\(68\!\cdots\!88\)\( \beta_{5} - \)\(38\!\cdots\!24\)\( \beta_{6} + \)\(33\!\cdots\!92\)\( \beta_{7}) q^{59}\) \(+(\)\(51\!\cdots\!20\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(42\!\cdots\!84\)\( \beta_{2} + \)\(41\!\cdots\!80\)\( \beta_{3} - \)\(85\!\cdots\!16\)\( \beta_{4} - \)\(45\!\cdots\!84\)\( \beta_{5} + \)\(10\!\cdots\!72\)\( \beta_{6} - \)\(20\!\cdots\!16\)\( \beta_{7}) q^{60}\) \(+(\)\(32\!\cdots\!02\)\( - \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(75\!\cdots\!19\)\( \beta_{2} - \)\(71\!\cdots\!90\)\( \beta_{3} + \)\(97\!\cdots\!75\)\( \beta_{4} - \)\(52\!\cdots\!04\)\( \beta_{5} - \)\(77\!\cdots\!28\)\( \beta_{6} + \)\(39\!\cdots\!24\)\( \beta_{7}) q^{61}\) \(+(\)\(13\!\cdots\!80\)\( - \)\(46\!\cdots\!36\)\( \beta_{1} + \)\(38\!\cdots\!96\)\( \beta_{2} + \)\(90\!\cdots\!44\)\( \beta_{3} + \)\(21\!\cdots\!88\)\( \beta_{4} - \)\(22\!\cdots\!96\)\( \beta_{5} - \)\(22\!\cdots\!64\)\( \beta_{6} - \)\(22\!\cdots\!60\)\( \beta_{7}) q^{62}\) \(+(\)\(21\!\cdots\!60\)\( + \)\(29\!\cdots\!22\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} - \)\(66\!\cdots\!29\)\( \beta_{3} - \)\(51\!\cdots\!93\)\( \beta_{4} + \)\(43\!\cdots\!04\)\( \beta_{5} + \)\(67\!\cdots\!45\)\( \beta_{6} - \)\(18\!\cdots\!08\)\( \beta_{7}) q^{63}\) \(+(\)\(50\!\cdots\!28\)\( + \)\(27\!\cdots\!68\)\( \beta_{1} - \)\(30\!\cdots\!64\)\( \beta_{2} + \)\(98\!\cdots\!76\)\( \beta_{3} - \)\(14\!\cdots\!92\)\( \beta_{4} + \)\(19\!\cdots\!88\)\( \beta_{5} - \)\(53\!\cdots\!60\)\( \beta_{6} + \)\(49\!\cdots\!80\)\( \beta_{7}) q^{64}\) \(+(\)\(88\!\cdots\!40\)\( + \)\(49\!\cdots\!32\)\( \beta_{1} - \)\(69\!\cdots\!08\)\( \beta_{2} - \)\(23\!\cdots\!40\)\( \beta_{3} - \)\(13\!\cdots\!92\)\( \beta_{4} - \)\(88\!\cdots\!08\)\( \beta_{5} - \)\(25\!\cdots\!36\)\( \beta_{6} - \)\(37\!\cdots\!92\)\( \beta_{7}) q^{65}\) \(+(\)\(21\!\cdots\!44\)\( - \)\(55\!\cdots\!20\)\( \beta_{1} + \)\(35\!\cdots\!12\)\( \beta_{2} + \)\(16\!\cdots\!72\)\( \beta_{3} + \)\(12\!\cdots\!28\)\( \beta_{4} + \)\(95\!\cdots\!76\)\( \beta_{5} - \)\(92\!\cdots\!84\)\( \beta_{6} - \)\(12\!\cdots\!28\)\( \beta_{7}) q^{66}\) \(+(-\)\(80\!\cdots\!80\)\( + \)\(21\!\cdots\!01\)\( \beta_{1} + \)\(33\!\cdots\!87\)\( \beta_{2} - \)\(34\!\cdots\!34\)\( \beta_{3} - \)\(21\!\cdots\!98\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5} + \)\(44\!\cdots\!02\)\( \beta_{6} + \)\(43\!\cdots\!76\)\( \beta_{7}) q^{67}\) \(+(\)\(11\!\cdots\!40\)\( - \)\(37\!\cdots\!28\)\( \beta_{1} - \)\(54\!\cdots\!50\)\( \beta_{2} - \)\(97\!\cdots\!78\)\( \beta_{3} - \)\(11\!\cdots\!76\)\( \beta_{4} - \)\(48\!\cdots\!72\)\( \beta_{5} + \)\(79\!\cdots\!40\)\( \beta_{6} - \)\(50\!\cdots\!56\)\( \beta_{7}) q^{68}\) \(+(-\)\(20\!\cdots\!16\)\( - \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!56\)\( \beta_{2} + \)\(15\!\cdots\!12\)\( \beta_{3} + \)\(71\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5} - \)\(64\!\cdots\!04\)\( \beta_{6} - \)\(50\!\cdots\!68\)\( \beta_{7}) q^{69}\) \(+(-\)\(19\!\cdots\!80\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} - \)\(13\!\cdots\!68\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(56\!\cdots\!96\)\( \beta_{4} + \)\(43\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(27\!\cdots\!00\)\( \beta_{7}) q^{70}\) \(+(-\)\(18\!\cdots\!48\)\( + \)\(85\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!55\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} + \)\(22\!\cdots\!68\)\( \beta_{5} + \)\(12\!\cdots\!01\)\( \beta_{6} - \)\(38\!\cdots\!08\)\( \beta_{7}) q^{71}\) \(+(-\)\(21\!\cdots\!40\)\( + \)\(21\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(29\!\cdots\!37\)\( \beta_{3} - \)\(55\!\cdots\!99\)\( \beta_{4} - \)\(66\!\cdots\!82\)\( \beta_{5} - \)\(69\!\cdots\!33\)\( \beta_{6} - \)\(60\!\cdots\!55\)\( \beta_{7}) q^{72}\) \(+(-\)\(20\!\cdots\!30\)\( + \)\(66\!\cdots\!76\)\( \beta_{1} + \)\(65\!\cdots\!26\)\( \beta_{2} + \)\(21\!\cdots\!16\)\( \beta_{3} + \)\(55\!\cdots\!02\)\( \beta_{4} - \)\(27\!\cdots\!00\)\( \beta_{5} + \)\(65\!\cdots\!72\)\( \beta_{6} + \)\(10\!\cdots\!16\)\( \beta_{7}) q^{73}\) \(+(-\)\(26\!\cdots\!14\)\( - \)\(74\!\cdots\!62\)\( \beta_{1} + \)\(17\!\cdots\!12\)\( \beta_{2} - \)\(68\!\cdots\!72\)\( \beta_{3} + \)\(12\!\cdots\!16\)\( \beta_{4} + \)\(31\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!88\)\( \beta_{6} - \)\(15\!\cdots\!04\)\( \beta_{7}) q^{74}\) \(+(-\)\(39\!\cdots\!00\)\( - \)\(32\!\cdots\!05\)\( \beta_{1} - \)\(18\!\cdots\!95\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(16\!\cdots\!20\)\( \beta_{4} - \)\(29\!\cdots\!40\)\( \beta_{5} - \)\(40\!\cdots\!80\)\( \beta_{6} + \)\(72\!\cdots\!40\)\( \beta_{7}) q^{75}\) \(+(-\)\(11\!\cdots\!40\)\( - \)\(44\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3} - \)\(20\!\cdots\!44\)\( \beta_{4} - \)\(32\!\cdots\!96\)\( \beta_{5} - \)\(16\!\cdots\!04\)\( \beta_{6} - \)\(66\!\cdots\!68\)\( \beta_{7}) q^{76}\) \(+(-\)\(44\!\cdots\!00\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(32\!\cdots\!64\)\( \beta_{3} - \)\(67\!\cdots\!28\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} + \)\(17\!\cdots\!84\)\( \beta_{6} + \)\(12\!\cdots\!60\)\( \beta_{7}) q^{77}\) \(+(\)\(73\!\cdots\!00\)\( + \)\(79\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!86\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3} + \)\(34\!\cdots\!94\)\( \beta_{4} + \)\(28\!\cdots\!90\)\( \beta_{5} - \)\(11\!\cdots\!36\)\( \beta_{6} + \)\(51\!\cdots\!12\)\( \beta_{7}) q^{78}\) \(+(\)\(58\!\cdots\!80\)\( + \)\(80\!\cdots\!64\)\( \beta_{1} - \)\(92\!\cdots\!44\)\( \beta_{2} + \)\(43\!\cdots\!18\)\( \beta_{3} - \)\(10\!\cdots\!06\)\( \beta_{4} - \)\(45\!\cdots\!24\)\( \beta_{5} - \)\(58\!\cdots\!86\)\( \beta_{6} - \)\(37\!\cdots\!12\)\( \beta_{7}) q^{79}\) \(+(\)\(23\!\cdots\!20\)\( + \)\(73\!\cdots\!16\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(29\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!36\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(69\!\cdots\!00\)\( \beta_{7}) q^{80}\) \(+(\)\(15\!\cdots\!21\)\( + \)\(63\!\cdots\!88\)\( \beta_{1} - \)\(57\!\cdots\!22\)\( \beta_{2} - \)\(51\!\cdots\!56\)\( \beta_{3} + \)\(55\!\cdots\!30\)\( \beta_{4} + \)\(28\!\cdots\!68\)\( \beta_{5} + \)\(17\!\cdots\!16\)\( \beta_{6} + \)\(18\!\cdots\!72\)\( \beta_{7}) q^{81}\) \(+(\)\(40\!\cdots\!30\)\( - \)\(26\!\cdots\!06\)\( \beta_{1} + \)\(91\!\cdots\!44\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(22\!\cdots\!28\)\( \beta_{4} + \)\(14\!\cdots\!56\)\( \beta_{5} - \)\(43\!\cdots\!40\)\( \beta_{6} - \)\(30\!\cdots\!72\)\( \beta_{7}) q^{82}\) \(+(\)\(42\!\cdots\!60\)\( - \)\(26\!\cdots\!51\)\( \beta_{1} - \)\(88\!\cdots\!73\)\( \beta_{2} - \)\(99\!\cdots\!00\)\( \beta_{3} - \)\(21\!\cdots\!00\)\( \beta_{4} - \)\(54\!\cdots\!20\)\( \beta_{5} + \)\(57\!\cdots\!60\)\( \beta_{6} + \)\(47\!\cdots\!20\)\( \beta_{7}) q^{83}\) \(+(\)\(21\!\cdots\!16\)\( - \)\(62\!\cdots\!64\)\( \beta_{1} + \)\(39\!\cdots\!32\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3} - \)\(78\!\cdots\!72\)\( \beta_{4} + \)\(75\!\cdots\!08\)\( \beta_{5} + \)\(19\!\cdots\!40\)\( \beta_{6} + \)\(10\!\cdots\!80\)\( \beta_{7}) q^{84}\) \(+(\)\(73\!\cdots\!20\)\( - \)\(69\!\cdots\!04\)\( \beta_{1} - \)\(36\!\cdots\!74\)\( \beta_{2} + \)\(71\!\cdots\!80\)\( \beta_{3} + \)\(24\!\cdots\!74\)\( \beta_{4} - \)\(13\!\cdots\!24\)\( \beta_{5} - \)\(10\!\cdots\!08\)\( \beta_{6} - \)\(12\!\cdots\!76\)\( \beta_{7}) q^{85}\) \(+(-\)\(19\!\cdots\!68\)\( + \)\(15\!\cdots\!58\)\( \beta_{1} + \)\(36\!\cdots\!55\)\( \beta_{2} - \)\(39\!\cdots\!88\)\( \beta_{3} + \)\(73\!\cdots\!77\)\( \beta_{4} + \)\(21\!\cdots\!63\)\( \beta_{5} + \)\(73\!\cdots\!72\)\( \beta_{6} + \)\(10\!\cdots\!24\)\( \beta_{7}) q^{86}\) \(+(-\)\(68\!\cdots\!40\)\( + \)\(36\!\cdots\!86\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} - \)\(50\!\cdots\!71\)\( \beta_{3} + \)\(28\!\cdots\!53\)\( \beta_{4} + \)\(31\!\cdots\!08\)\( \beta_{5} - \)\(90\!\cdots\!81\)\( \beta_{6} + \)\(88\!\cdots\!96\)\( \beta_{7}) q^{87}\) \(+(-\)\(15\!\cdots\!40\)\( + \)\(51\!\cdots\!12\)\( \beta_{1} - \)\(51\!\cdots\!32\)\( \beta_{2} - \)\(19\!\cdots\!84\)\( \beta_{3} - \)\(48\!\cdots\!68\)\( \beta_{4} - \)\(14\!\cdots\!44\)\( \beta_{5} + \)\(52\!\cdots\!04\)\( \beta_{6} + \)\(21\!\cdots\!60\)\( \beta_{7}) q^{88}\) \(+(-\)\(42\!\cdots\!10\)\( - \)\(43\!\cdots\!92\)\( \beta_{1} - \)\(85\!\cdots\!98\)\( \beta_{2} + \)\(60\!\cdots\!20\)\( \beta_{3} - \)\(94\!\cdots\!26\)\( \beta_{4} + \)\(19\!\cdots\!48\)\( \beta_{5} - \)\(68\!\cdots\!92\)\( \beta_{6} - \)\(72\!\cdots\!64\)\( \beta_{7}) q^{89}\) \(+(-\)\(43\!\cdots\!10\)\( - \)\(83\!\cdots\!78\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(95\!\cdots\!60\)\( \beta_{3} + \)\(25\!\cdots\!68\)\( \beta_{4} + \)\(44\!\cdots\!32\)\( \beta_{5} - \)\(12\!\cdots\!56\)\( \beta_{6} + \)\(31\!\cdots\!68\)\( \beta_{7}) q^{90}\) \(+(-\)\(43\!\cdots\!68\)\( - \)\(71\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(19\!\cdots\!52\)\( \beta_{4} - \)\(24\!\cdots\!28\)\( \beta_{5} + \)\(48\!\cdots\!48\)\( \beta_{6} + \)\(36\!\cdots\!16\)\( \beta_{7}) q^{91}\) \(+(-\)\(15\!\cdots\!80\)\( - \)\(52\!\cdots\!80\)\( \beta_{1} - \)\(18\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!56\)\( \beta_{3} + \)\(21\!\cdots\!48\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(51\!\cdots\!40\)\( \beta_{6} - \)\(69\!\cdots\!52\)\( \beta_{7}) q^{92}\) \(+(\)\(24\!\cdots\!80\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(89\!\cdots\!88\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} + \)\(61\!\cdots\!96\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5} - \)\(78\!\cdots\!24\)\( \beta_{6} - \)\(22\!\cdots\!92\)\( \beta_{7}) q^{93}\) \(+(\)\(19\!\cdots\!56\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(57\!\cdots\!32\)\( \beta_{2} + \)\(59\!\cdots\!36\)\( \beta_{3} - \)\(18\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(98\!\cdots\!24\)\( \beta_{6} + \)\(21\!\cdots\!08\)\( \beta_{7}) q^{94}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(63\!\cdots\!30\)\( \beta_{1} + \)\(82\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!25\)\( \beta_{3} + \)\(23\!\cdots\!45\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} - \)\(23\!\cdots\!25\)\( \beta_{6} - \)\(16\!\cdots\!00\)\( \beta_{7}) q^{95}\) \(+(\)\(12\!\cdots\!72\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} + \)\(33\!\cdots\!64\)\( \beta_{2} + \)\(64\!\cdots\!08\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4} + \)\(65\!\cdots\!48\)\( \beta_{5} + \)\(44\!\cdots\!40\)\( \beta_{6} - \)\(28\!\cdots\!20\)\( \beta_{7}) q^{96}\) \(+(\)\(12\!\cdots\!30\)\( + \)\(19\!\cdots\!28\)\( \beta_{1} + \)\(26\!\cdots\!18\)\( \beta_{2} - \)\(35\!\cdots\!68\)\( \beta_{3} - \)\(26\!\cdots\!06\)\( \beta_{4} - \)\(80\!\cdots\!12\)\( \beta_{5} - \)\(28\!\cdots\!20\)\( \beta_{6} + \)\(29\!\cdots\!44\)\( \beta_{7}) q^{97}\) \(+(\)\(18\!\cdots\!95\)\( - \)\(14\!\cdots\!95\)\( \beta_{1} - \)\(38\!\cdots\!96\)\( \beta_{2} + \)\(15\!\cdots\!84\)\( \beta_{3} - \)\(68\!\cdots\!72\)\( \beta_{4} + \)\(44\!\cdots\!16\)\( \beta_{5} - \)\(85\!\cdots\!20\)\( \beta_{6} + \)\(85\!\cdots\!68\)\( \beta_{7}) q^{98}\) \(+(\)\(38\!\cdots\!84\)\( - \)\(10\!\cdots\!21\)\( \beta_{1} + \)\(97\!\cdots\!57\)\( \beta_{2} - \)\(83\!\cdots\!20\)\( \beta_{3} + \)\(72\!\cdots\!52\)\( \beta_{4} + \)\(31\!\cdots\!48\)\( \beta_{5} + \)\(20\!\cdots\!92\)\( \beta_{6} - \)\(79\!\cdots\!36\)\( \beta_{7}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 5835659138280q^{2} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!80\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!84\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!36\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 5835659138280q^{2} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!80\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!84\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!36\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!40\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!16\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!08\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!20\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!28\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!40\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!80\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!40\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!36\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!40\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!80\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!04\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!40\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!40\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!80\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!16\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!20\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!32\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!60\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!28\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!32\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!84\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!68\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!80\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!84\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!40\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!60\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!44\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!56\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!80\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!20\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!40\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!80\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!16\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!24\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!20\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!52\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!40\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!20\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!28\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!84\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!20\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!40\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!12\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!20\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!68\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!40\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!80\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!28\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!60\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!44\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!20\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!80\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!44\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!40\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!48\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!76\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!60\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!72\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(45\!\cdots\!12\) \(x^{6}\mathstrut +\mathstrut \) \(33\!\cdots\!60\) \(x^{5}\mathstrut +\mathstrut \) \(59\!\cdots\!16\) \(x^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!20\) \(x^{3}\mathstrut -\mathstrut \) \(20\!\cdots\!88\) \(x^{2}\mathstrut +\mathstrut \) \(46\!\cdots\!68\) \(x\mathstrut +\mathstrut \) \(12\!\cdots\!76\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 3 \)
\(\beta_{2}\)\(=\)\((\)\(76\!\cdots\!51\) \(\nu^{7}\mathstrut +\mathstrut \) \(14\!\cdots\!81\) \(\nu^{6}\mathstrut -\mathstrut \) \(52\!\cdots\!16\) \(\nu^{5}\mathstrut -\mathstrut \) \(40\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(90\!\cdots\!96\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!32\) \(\nu^{2}\mathstrut -\mathstrut \) \(29\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(69\!\cdots\!36\)\()/\)\(58\!\cdots\!72\)
\(\beta_{3}\)\(=\)\((\)\(52\!\cdots\!49\) \(\nu^{7}\mathstrut +\mathstrut \) \(10\!\cdots\!19\) \(\nu^{6}\mathstrut -\mathstrut \) \(36\!\cdots\!84\) \(\nu^{5}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(62\!\cdots\!04\) \(\nu^{3}\mathstrut +\mathstrut \) \(47\!\cdots\!76\) \(\nu^{2}\mathstrut -\mathstrut \) \(16\!\cdots\!16\) \(\nu\mathstrut -\mathstrut \) \(43\!\cdots\!12\)\()/\)\(64\!\cdots\!08\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(13\!\cdots\!97\) \(\nu^{7}\mathstrut -\mathstrut \) \(63\!\cdots\!15\) \(\nu^{6}\mathstrut +\mathstrut \) \(51\!\cdots\!24\) \(\nu^{5}\mathstrut +\mathstrut \) \(41\!\cdots\!84\) \(\nu^{4}\mathstrut -\mathstrut \) \(50\!\cdots\!88\) \(\nu^{3}\mathstrut -\mathstrut \) \(67\!\cdots\!08\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!68\) \(\nu\mathstrut -\mathstrut \) \(31\!\cdots\!68\)\()/\)\(73\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(60\!\cdots\!17\) \(\nu^{7}\mathstrut +\mathstrut \) \(10\!\cdots\!85\) \(\nu^{6}\mathstrut +\mathstrut \) \(17\!\cdots\!64\) \(\nu^{5}\mathstrut -\mathstrut \) \(26\!\cdots\!76\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!68\) \(\nu^{3}\mathstrut +\mathstrut \) \(76\!\cdots\!12\) \(\nu^{2}\mathstrut +\mathstrut \) \(33\!\cdots\!48\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!52\)\()/\)\(18\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(14\!\cdots\!57\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!15\) \(\nu^{6}\mathstrut +\mathstrut \) \(58\!\cdots\!44\) \(\nu^{5}\mathstrut +\mathstrut \) \(43\!\cdots\!04\) \(\nu^{4}\mathstrut -\mathstrut \) \(62\!\cdots\!28\) \(\nu^{3}\mathstrut -\mathstrut \) \(32\!\cdots\!48\) \(\nu^{2}\mathstrut +\mathstrut \) \(15\!\cdots\!08\) \(\nu\mathstrut +\mathstrut \) \(49\!\cdots\!92\)\()/\)\(12\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(37\!\cdots\!87\) \(\nu^{7}\mathstrut +\mathstrut \) \(30\!\cdots\!65\) \(\nu^{6}\mathstrut -\mathstrut \) \(21\!\cdots\!04\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!64\) \(\nu^{4}\mathstrut +\mathstrut \) \(51\!\cdots\!48\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(54\!\cdots\!28\) \(\nu\mathstrut -\mathstrut \) \(15\!\cdots\!72\)\()/\)\(18\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(62091\) \(\beta_{2}\mathstrut -\mathstrut \) \(26761022262152\) \(\beta_{1}\mathstrut +\mathstrut \) \(65733123937758428360847400200\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(295\) \(\beta_{6}\mathstrut +\mathstrut \) \(160002\) \(\beta_{5}\mathstrut -\mathstrut \) \(2004356999\) \(\beta_{4}\mathstrut +\mathstrut \) \(51851570017067\) \(\beta_{3}\mathstrut +\mathstrut \) \(3276903991940428836\) \(\beta_{2}\mathstrut +\mathstrut \) \(112986517961279136356684832712\) \(\beta_{1}\mathstrut -\mathstrut \) \(1759085591639276897686588409592076569213504\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(3926523411123\) \(\beta_{7}\mathstrut +\mathstrut \) \(6959296706126917\) \(\beta_{6}\mathstrut +\mathstrut \) \(10823831194312423526\) \(\beta_{5}\mathstrut -\mathstrut \) \(64844235599751336866149\) \(\beta_{4}\mathstrut +\mathstrut \) \(19321453573813038341957137905\) \(\beta_{3}\mathstrut -\mathstrut \) \(1394410694473382624668011668515956\) \(\beta_{2}\mathstrut -\mathstrut \) \(298710901026801058381966898320637168298536\) \(\beta_{1}\mathstrut +\mathstrut \) \(928369598546200014171379235704963029862290826589385463872\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(2621067272289636342635857733\) \(\beta_{7}\mathstrut +\mathstrut \) \(781805958213198104016871956611\) \(\beta_{6}\mathstrut +\mathstrut \) \(626720668829570339117854559945610\) \(\beta_{5}\mathstrut -\mathstrut \) \(6015663869305187979638343702232227939\) \(\beta_{4}\mathstrut +\mathstrut \) \(205850449531108259889513399496641754951863\) \(\beta_{3}\mathstrut -\mathstrut \) \(41887807299957782555913661895122244965673155244\) \(\beta_{2}\mathstrut +\mathstrut \) \(229515919805726057201751531896339304137050604018359815144\) \(\beta_{1}\mathstrut -\mathstrut \) \(2454400080858624658714224866616934605524010680291289642149797035267648\)\()/124416\)
\(\nu^{6}\)\(=\)\((\)\(2456698967573154102027892526992622040587\) \(\beta_{7}\mathstrut +\mathstrut \) \(2287336922187280642817740919038569124333229\) \(\beta_{6}\mathstrut +\mathstrut \) \(4173765778599191392471236993532318130284922646\) \(\beta_{5}\mathstrut -\mathstrut \) \(22719176458145430265227683260813234396944017761485\) \(\beta_{4}\mathstrut +\mathstrut \) \(4820608307462255271214311913264579460921736185013012921\) \(\beta_{3}\mathstrut -\mathstrut \) \(421824107166754602266817224392645212904577134947783234195476\) \(\beta_{2}\mathstrut +\mathstrut \) \(14896020318557401322538306359430870429741697951273163415269325560472\) \(\beta_{1}\mathstrut +\mathstrut \) \(209538866711613197149097879153829515723664322115869508558284660196534474158205620800\)\()/41472\)
\(\nu^{7}\)\(=\)\((\)\(22\!\cdots\!91\) \(\beta_{7}\mathstrut +\mathstrut \) \(67\!\cdots\!33\) \(\beta_{6}\mathstrut +\mathstrut \) \(63\!\cdots\!18\) \(\beta_{5}\mathstrut -\mathstrut \) \(57\!\cdots\!61\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\!\cdots\!85\) \(\beta_{3}\mathstrut -\mathstrut \) \(59\!\cdots\!32\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\!\cdots\!88\) \(\beta_{1}\mathstrut +\mathstrut \) \(40\!\cdots\!88\)\()/41472\)

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
−1.48943e13
−1.46162e13
−6.56924e12
−1.67307e12
4.79736e12
6.86315e12
9.80521e12
1.62871e13
−3.58194e14 −7.12911e22 8.86885e28 −1.64420e33 2.55360e37 −1.35904e40 −1.75781e43 2.96152e45 5.88943e47
1.2 −3.51519e14 6.56090e22 8.39514e28 2.79946e33 −2.30628e37 5.53054e39 −1.55854e43 2.18364e45 −9.84062e47
1.3 −1.58391e14 9.80327e21 −1.45263e28 −7.31691e32 −1.55275e36 −1.37055e39 8.57536e42 −2.02479e45 1.15893e47
1.4 −4.08831e13 −7.46060e22 −3.79427e28 2.52173e33 3.05013e36 2.11876e40 3.17076e42 3.44515e45 −1.03096e47
1.5 1.14407e14 8.75958e22 −2.65251e28 −1.07741e33 1.00216e37 1.86215e40 −7.56680e42 5.55214e45 −1.23264e47
1.6 1.63986e14 6.46739e21 −1.27226e28 1.20020e33 1.06056e36 −2.14893e40 −8.58249e42 −2.07907e45 1.96817e47
1.7 2.34596e14 −5.48368e22 1.54210e28 −2.52565e33 −1.28645e37 4.80687e39 −5.67559e42 8.86182e44 −5.92506e47
1.8 3.90162e14 2.16924e22 1.12612e29 1.39914e33 8.46353e36 1.75361e40 2.84812e43 −1.65034e45 5.45891e47
\(n\): 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 \(S_{96}^{\mathrm{new}}(\Gamma_0(1))\).