Properties

Label 1.108.a.a
Level 1
Weight 108
Character orbit 1.a
Self dual Yes
Analytic conductor 72.504
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 108 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(72.5037502298\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{143}\cdot 3^{48}\cdot 5^{18}\cdot 7^{8}\cdot 11^{2}\cdot 13^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(+(607308845937277 - \beta_{1}) q^{2}\) \(+(\)\(17\!\cdots\!33\)\( + 62113396 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(71\!\cdots\!70\)\( + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(27\!\cdots\!09\)\( + \)\(50\!\cdots\!29\)\( \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(13\!\cdots\!31\)\( + \)\(25\!\cdots\!77\)\( \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(10\!\cdots\!24\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(11\!\cdots\!77\)\( - \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(51\!\cdots\!47\)\( \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(41\!\cdots\!23\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(17\!\cdots\!19\)\( \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(607308845937277 - \beta_{1}) q^{2}\) \(+(\)\(17\!\cdots\!33\)\( + 62113396 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(71\!\cdots\!70\)\( + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(27\!\cdots\!09\)\( + \)\(50\!\cdots\!29\)\( \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(13\!\cdots\!31\)\( + \)\(25\!\cdots\!77\)\( \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(10\!\cdots\!24\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(11\!\cdots\!77\)\( - \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(51\!\cdots\!47\)\( \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(41\!\cdots\!23\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(17\!\cdots\!19\)\( \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9}\) \(+(-\)\(11\!\cdots\!14\)\( - \)\(85\!\cdots\!98\)\( \beta_{1} + \)\(34\!\cdots\!28\)\( \beta_{2} - \)\(42\!\cdots\!40\)\( \beta_{3} + 3035644428618332 \beta_{4} + 137787538404 \beta_{5} + 147169648 \beta_{6} - 120312 \beta_{7} - 216 \beta_{8}) q^{10}\) \(+(\)\(83\!\cdots\!59\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(26\!\cdots\!77\)\( \beta_{2} - \)\(30\!\cdots\!24\)\( \beta_{3} + 39655237059879084 \beta_{4} + 18943495258882 \beta_{5} + 20412816178 \beta_{6} + 12038260 \beta_{7} + 23004 \beta_{8}) q^{11}\) \(+(-\)\(88\!\cdots\!96\)\( + \)\(84\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!32\)\( \beta_{3} + 17417687351454383040 \beta_{4} + 8277620636103552 \beta_{5} + 1126938004992 \beta_{6} - 91849536 \beta_{7} - 1609920 \beta_{8}) q^{12}\) \(+(\)\(42\!\cdots\!13\)\( - \)\(29\!\cdots\!41\)\( \beta_{1} + \)\(84\!\cdots\!80\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3} - 1292007267852162005 \beta_{4} + 688583074305208092 \beta_{5} - 165811929566128 \beta_{6} - 46020832614 \beta_{7} + 83255310 \beta_{8}) q^{13}\) \(+(-\)\(82\!\cdots\!78\)\( + \)\(43\!\cdots\!78\)\( \beta_{1} + \)\(78\!\cdots\!18\)\( \beta_{2} + \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(34\!\cdots\!22\)\( \beta_{4} + 17161982422390714858 \beta_{5} + 4494593484494656 \beta_{6} + 3819461803360 \beta_{7} - 3391893792 \beta_{8}) q^{14}\) \(+(\)\(12\!\cdots\!72\)\( - \)\(49\!\cdots\!31\)\( \beta_{1} - \)\(43\!\cdots\!19\)\( \beta_{2} + \)\(11\!\cdots\!20\)\( \beta_{3} + \)\(28\!\cdots\!84\)\( \beta_{4} - \)\(14\!\cdots\!57\)\( \beta_{5} + 71321094534085191 \beta_{6} - 177279761520504 \beta_{7} + 113342680728 \beta_{8}) q^{15}\) \(+(\)\(32\!\cdots\!36\)\( + \)\(77\!\cdots\!76\)\( \beta_{1} - \)\(67\!\cdots\!24\)\( \beta_{2} + \)\(88\!\cdots\!92\)\( \beta_{3} + \)\(73\!\cdots\!32\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5} - 7893052997101223272 \beta_{6} + 5829377304694680 \beta_{7} - 3193371357696 \beta_{8}) q^{16}\) \(+(\)\(61\!\cdots\!04\)\( - \)\(51\!\cdots\!51\)\( \beta_{1} + \)\(30\!\cdots\!87\)\( \beta_{2} - \)\(44\!\cdots\!23\)\( \beta_{3} + \)\(45\!\cdots\!85\)\( \beta_{4} + \)\(86\!\cdots\!14\)\( \beta_{5} + \)\(25\!\cdots\!84\)\( \beta_{6} - 147577032261572145 \beta_{7} + 77391112194765 \beta_{8}) q^{17}\) \(+(\)\(26\!\cdots\!33\)\( - \)\(54\!\cdots\!45\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(56\!\cdots\!08\)\( \beta_{3} - \)\(44\!\cdots\!20\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(45\!\cdots\!64\)\( \beta_{6} + 2995387893492815280 \beta_{7} - 1637786543560080 \beta_{8}) q^{18}\) \(+(\)\(95\!\cdots\!49\)\( - \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(75\!\cdots\!37\)\( \beta_{2} + \)\(95\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!68\)\( \beta_{4} - \)\(67\!\cdots\!94\)\( \beta_{5} + \)\(45\!\cdots\!14\)\( \beta_{6} - 49760167465115702820 \beta_{7} + 30620433423116052 \beta_{8}) q^{19}\) \(+(\)\(14\!\cdots\!52\)\( + \)\(10\!\cdots\!32\)\( \beta_{1} + \)\(59\!\cdots\!86\)\( \beta_{2} + \)\(26\!\cdots\!70\)\( \beta_{3} + \)\(90\!\cdots\!88\)\( \beta_{4} + \)\(51\!\cdots\!80\)\( \beta_{5} + \)\(96\!\cdots\!60\)\( \beta_{6} + \)\(68\!\cdots\!60\)\( \beta_{7} - 510445792877080320 \beta_{8}) q^{20}\) \(+(\)\(10\!\cdots\!08\)\( - \)\(50\!\cdots\!94\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} + \)\(67\!\cdots\!70\)\( \beta_{3} + \)\(16\!\cdots\!94\)\( \beta_{4} - \)\(69\!\cdots\!64\)\( \beta_{5} - \)\(89\!\cdots\!68\)\( \beta_{6} - \)\(76\!\cdots\!30\)\( \beta_{7} + 7643011086349644726 \beta_{8}) q^{21}\) \(+(-\)\(23\!\cdots\!17\)\( - \)\(35\!\cdots\!21\)\( \beta_{1} + \)\(45\!\cdots\!75\)\( \beta_{2} - \)\(25\!\cdots\!70\)\( \beta_{3} - \)\(13\!\cdots\!85\)\( \beta_{4} + \)\(23\!\cdots\!47\)\( \beta_{5} + \)\(17\!\cdots\!32\)\( \beta_{6} + \)\(66\!\cdots\!56\)\( \beta_{7} - \)\(10\!\cdots\!60\)\( \beta_{8}) q^{22}\) \(+(-\)\(11\!\cdots\!80\)\( + \)\(57\!\cdots\!65\)\( \beta_{1} + \)\(88\!\cdots\!29\)\( \beta_{2} - \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(44\!\cdots\!95\)\( \beta_{5} - \)\(20\!\cdots\!45\)\( \beta_{6} - \)\(39\!\cdots\!00\)\( \beta_{7} + \)\(12\!\cdots\!60\)\( \beta_{8}) q^{23}\) \(+(-\)\(32\!\cdots\!60\)\( + \)\(54\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} + \)\(46\!\cdots\!44\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4} - \)\(63\!\cdots\!52\)\( \beta_{5} + \)\(16\!\cdots\!08\)\( \beta_{6} + \)\(18\!\cdots\!20\)\( \beta_{7} - \)\(14\!\cdots\!56\)\( \beta_{8}) q^{24}\) \(+(\)\(23\!\cdots\!75\)\( + \)\(76\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3} - \)\(96\!\cdots\!50\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!50\)\( \beta_{7} + \)\(14\!\cdots\!50\)\( \beta_{8}) q^{25}\) \(+(\)\(33\!\cdots\!98\)\( - \)\(58\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(65\!\cdots\!56\)\( \beta_{3} - \)\(55\!\cdots\!60\)\( \beta_{4} + \)\(29\!\cdots\!96\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} - \)\(53\!\cdots\!60\)\( \beta_{7} - \)\(13\!\cdots\!72\)\( \beta_{8}) q^{26}\) \(+(\)\(13\!\cdots\!82\)\( + \)\(74\!\cdots\!46\)\( \beta_{1} + \)\(91\!\cdots\!08\)\( \beta_{2} - \)\(84\!\cdots\!72\)\( \beta_{3} + \)\(70\!\cdots\!40\)\( \beta_{4} - \)\(42\!\cdots\!42\)\( \beta_{5} + \)\(48\!\cdots\!38\)\( \beta_{6} + \)\(58\!\cdots\!36\)\( \beta_{7} + \)\(11\!\cdots\!40\)\( \beta_{8}) q^{27}\) \(+(-\)\(85\!\cdots\!36\)\( - \)\(32\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!92\)\( \beta_{2} - \)\(63\!\cdots\!32\)\( \beta_{3} + \)\(65\!\cdots\!00\)\( \beta_{4} + \)\(21\!\cdots\!48\)\( \beta_{5} - \)\(38\!\cdots\!32\)\( \beta_{6} - \)\(48\!\cdots\!24\)\( \beta_{7} - \)\(91\!\cdots\!20\)\( \beta_{8}) q^{28}\) \(+(\)\(39\!\cdots\!69\)\( + \)\(80\!\cdots\!49\)\( \beta_{1} - \)\(17\!\cdots\!98\)\( \beta_{2} + \)\(19\!\cdots\!10\)\( \beta_{3} - \)\(36\!\cdots\!15\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{6} + \)\(33\!\cdots\!20\)\( \beta_{7} + \)\(66\!\cdots\!48\)\( \beta_{8}) q^{29}\) \(+(\)\(19\!\cdots\!58\)\( - \)\(14\!\cdots\!22\)\( \beta_{1} - \)\(25\!\cdots\!06\)\( \beta_{2} + \)\(33\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!02\)\( \beta_{4} - \)\(79\!\cdots\!30\)\( \beta_{5} + \)\(90\!\cdots\!40\)\( \beta_{6} - \)\(18\!\cdots\!60\)\( \beta_{7} - \)\(43\!\cdots\!80\)\( \beta_{8}) q^{30}\) \(+(\)\(88\!\cdots\!68\)\( + \)\(88\!\cdots\!24\)\( \beta_{1} - \)\(76\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(56\!\cdots\!48\)\( \beta_{4} + \)\(36\!\cdots\!20\)\( \beta_{5} - \)\(14\!\cdots\!28\)\( \beta_{6} + \)\(75\!\cdots\!40\)\( \beta_{7} + \)\(26\!\cdots\!96\)\( \beta_{8}) q^{31}\) \(+(-\)\(16\!\cdots\!28\)\( - \)\(69\!\cdots\!40\)\( \beta_{1} + \)\(84\!\cdots\!44\)\( \beta_{2} - \)\(13\!\cdots\!08\)\( \beta_{3} - \)\(36\!\cdots\!20\)\( \beta_{4} + \)\(96\!\cdots\!04\)\( \beta_{5} + \)\(10\!\cdots\!04\)\( \beta_{6} - \)\(19\!\cdots\!20\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{32}\) \(+(\)\(43\!\cdots\!10\)\( - \)\(26\!\cdots\!29\)\( \beta_{1} + \)\(21\!\cdots\!69\)\( \beta_{2} + \)\(86\!\cdots\!91\)\( \beta_{3} - \)\(15\!\cdots\!25\)\( \beta_{4} - \)\(19\!\cdots\!86\)\( \beta_{5} - \)\(49\!\cdots\!96\)\( \beta_{6} - \)\(20\!\cdots\!79\)\( \beta_{7} + \)\(69\!\cdots\!55\)\( \beta_{8}) q^{33}\) \(+(\)\(12\!\cdots\!46\)\( - \)\(37\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + \)\(36\!\cdots\!80\)\( \beta_{3} + \)\(16\!\cdots\!92\)\( \beta_{4} + \)\(10\!\cdots\!96\)\( \beta_{5} + \)\(11\!\cdots\!44\)\( \beta_{6} + \)\(52\!\cdots\!60\)\( \beta_{7} - \)\(29\!\cdots\!08\)\( \beta_{8}) q^{34}\) \(+(\)\(54\!\cdots\!76\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} + \)\(11\!\cdots\!48\)\( \beta_{2} - \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(11\!\cdots\!68\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(30\!\cdots\!88\)\( \beta_{6} - \)\(27\!\cdots\!72\)\( \beta_{7} + \)\(10\!\cdots\!04\)\( \beta_{8}) q^{35}\) \(+(-\)\(52\!\cdots\!14\)\( - \)\(10\!\cdots\!58\)\( \beta_{1} - \)\(29\!\cdots\!21\)\( \beta_{2} - \)\(51\!\cdots\!11\)\( \beta_{3} - \)\(32\!\cdots\!04\)\( \beta_{4} - \)\(13\!\cdots\!44\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} + \)\(41\!\cdots\!80\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{36}\) \(+(-\)\(23\!\cdots\!87\)\( - \)\(34\!\cdots\!45\)\( \beta_{1} - \)\(84\!\cdots\!56\)\( \beta_{2} - \)\(51\!\cdots\!04\)\( \beta_{3} + \)\(94\!\cdots\!35\)\( \beta_{4} + \)\(75\!\cdots\!56\)\( \beta_{5} + \)\(17\!\cdots\!36\)\( \beta_{6} + \)\(57\!\cdots\!82\)\( \beta_{7} + \)\(39\!\cdots\!50\)\( \beta_{8}) q^{37}\) \(+(\)\(91\!\cdots\!89\)\( - \)\(23\!\cdots\!71\)\( \beta_{1} - \)\(13\!\cdots\!27\)\( \beta_{2} + \)\(81\!\cdots\!98\)\( \beta_{3} + \)\(38\!\cdots\!25\)\( \beta_{4} - \)\(78\!\cdots\!31\)\( \beta_{5} - \)\(26\!\cdots\!36\)\( \beta_{6} - \)\(63\!\cdots\!76\)\( \beta_{7} + \)\(18\!\cdots\!20\)\( \beta_{8}) q^{38}\) \(+(\)\(13\!\cdots\!32\)\( - \)\(79\!\cdots\!55\)\( \beta_{1} + \)\(11\!\cdots\!41\)\( \beta_{2} + \)\(29\!\cdots\!84\)\( \beta_{3} - \)\(24\!\cdots\!36\)\( \beta_{4} - \)\(81\!\cdots\!53\)\( \beta_{5} - \)\(12\!\cdots\!37\)\( \beta_{6} + \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(18\!\cdots\!16\)\( \beta_{8}) q^{39}\) \(+(-\)\(51\!\cdots\!50\)\( - \)\(42\!\cdots\!30\)\( \beta_{1} + \)\(96\!\cdots\!50\)\( \beta_{2} - \)\(22\!\cdots\!50\)\( \beta_{3} - \)\(20\!\cdots\!40\)\( \beta_{4} + \)\(33\!\cdots\!80\)\( \beta_{5} + \)\(86\!\cdots\!10\)\( \beta_{6} - \)\(16\!\cdots\!90\)\( \beta_{7} + \)\(88\!\cdots\!80\)\( \beta_{8}) q^{40}\) \(+(-\)\(16\!\cdots\!62\)\( - \)\(41\!\cdots\!22\)\( \beta_{1} + \)\(51\!\cdots\!10\)\( \beta_{2} + \)\(35\!\cdots\!34\)\( \beta_{3} + \)\(27\!\cdots\!74\)\( \beta_{4} + \)\(35\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(51\!\cdots\!70\)\( \beta_{7} - \)\(29\!\cdots\!50\)\( \beta_{8}) q^{41}\) \(+(\)\(12\!\cdots\!80\)\( - \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{3} - \)\(22\!\cdots\!60\)\( \beta_{4} - \)\(65\!\cdots\!84\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} - \)\(81\!\cdots\!60\)\( \beta_{7} + \)\(58\!\cdots\!60\)\( \beta_{8}) q^{42}\) \(+(\)\(65\!\cdots\!03\)\( + \)\(28\!\cdots\!76\)\( \beta_{1} - \)\(27\!\cdots\!41\)\( \beta_{2} + \)\(89\!\cdots\!76\)\( \beta_{3} - \)\(26\!\cdots\!80\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5} + \)\(11\!\cdots\!60\)\( \beta_{6} - \)\(18\!\cdots\!76\)\( \beta_{7} + \)\(34\!\cdots\!00\)\( \beta_{8}) q^{43}\) \(+(-\)\(68\!\cdots\!44\)\( + \)\(47\!\cdots\!44\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} + \)\(78\!\cdots\!08\)\( \beta_{3} + \)\(29\!\cdots\!68\)\( \beta_{4} + \)\(37\!\cdots\!60\)\( \beta_{5} - \)\(44\!\cdots\!08\)\( \beta_{6} + \)\(19\!\cdots\!00\)\( \beta_{7} - \)\(90\!\cdots\!44\)\( \beta_{8}) q^{44}\) \(+(-\)\(35\!\cdots\!67\)\( + \)\(11\!\cdots\!63\)\( \beta_{1} + \)\(46\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} + \)\(54\!\cdots\!07\)\( \beta_{4} - \)\(33\!\cdots\!40\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} - \)\(81\!\cdots\!30\)\( \beta_{7} + \)\(44\!\cdots\!10\)\( \beta_{8}) q^{45}\) \(+(-\)\(20\!\cdots\!62\)\( + \)\(53\!\cdots\!18\)\( \beta_{1} + \)\(92\!\cdots\!90\)\( \beta_{2} + \)\(14\!\cdots\!44\)\( \beta_{3} - \)\(13\!\cdots\!66\)\( \beta_{4} + \)\(78\!\cdots\!34\)\( \beta_{5} - \)\(11\!\cdots\!40\)\( \beta_{6} + \)\(19\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!20\)\( \beta_{8}) q^{46}\) \(+(\)\(45\!\cdots\!20\)\( + \)\(64\!\cdots\!90\)\( \beta_{1} - \)\(18\!\cdots\!22\)\( \beta_{2} + \)\(60\!\cdots\!72\)\( \beta_{3} - \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!50\)\( \beta_{5} - \)\(12\!\cdots\!90\)\( \beta_{6} - \)\(12\!\cdots\!32\)\( \beta_{7} + \)\(15\!\cdots\!20\)\( \beta_{8}) q^{47}\) \(+(\)\(15\!\cdots\!00\)\( - \)\(36\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} - \)\(25\!\cdots\!28\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!76\)\( \beta_{5} + \)\(34\!\cdots\!04\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(56\!\cdots\!00\)\( \beta_{8}) q^{48}\) \(+(-\)\(55\!\cdots\!51\)\( - \)\(45\!\cdots\!52\)\( \beta_{1} - \)\(21\!\cdots\!20\)\( \beta_{2} - \)\(37\!\cdots\!16\)\( \beta_{3} + \)\(16\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!56\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6} + \)\(59\!\cdots\!80\)\( \beta_{7} - \)\(42\!\cdots\!80\)\( \beta_{8}) q^{49}\) \(+(-\)\(36\!\cdots\!25\)\( - \)\(48\!\cdots\!75\)\( \beta_{1} + \)\(70\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} - \)\(51\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{50}\) \(+(\)\(47\!\cdots\!38\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(63\!\cdots\!88\)\( \beta_{3} - \)\(39\!\cdots\!72\)\( \beta_{4} - \)\(32\!\cdots\!86\)\( \beta_{5} - \)\(22\!\cdots\!54\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7} - \)\(22\!\cdots\!72\)\( \beta_{8}) q^{51}\) \(+(\)\(65\!\cdots\!08\)\( - \)\(43\!\cdots\!24\)\( \beta_{1} + \)\(62\!\cdots\!50\)\( \beta_{2} + \)\(10\!\cdots\!42\)\( \beta_{3} + \)\(98\!\cdots\!40\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(52\!\cdots\!08\)\( \beta_{7} - \)\(32\!\cdots\!00\)\( \beta_{8}) q^{52}\) \(+(-\)\(59\!\cdots\!91\)\( + \)\(22\!\cdots\!71\)\( \beta_{1} - \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(31\!\cdots\!68\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} - \)\(90\!\cdots\!88\)\( \beta_{5} - \)\(66\!\cdots\!08\)\( \beta_{6} - \)\(22\!\cdots\!46\)\( \beta_{7} + \)\(33\!\cdots\!70\)\( \beta_{8}) q^{53}\) \(+(-\)\(17\!\cdots\!66\)\( + \)\(14\!\cdots\!14\)\( \beta_{1} - \)\(52\!\cdots\!30\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(89\!\cdots\!22\)\( \beta_{4} + \)\(55\!\cdots\!10\)\( \beta_{5} + \)\(81\!\cdots\!32\)\( \beta_{6} + \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(10\!\cdots\!24\)\( \beta_{8}) q^{54}\) \(+(\)\(16\!\cdots\!48\)\( + \)\(59\!\cdots\!63\)\( \beta_{1} - \)\(77\!\cdots\!61\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!72\)\( \beta_{4} - \)\(18\!\cdots\!75\)\( \beta_{5} + \)\(18\!\cdots\!25\)\( \beta_{6} + \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(15\!\cdots\!00\)\( \beta_{8}) q^{55}\) \(+(\)\(82\!\cdots\!64\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(25\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!48\)\( \beta_{3} + \)\(12\!\cdots\!36\)\( \beta_{4} + \)\(14\!\cdots\!60\)\( \beta_{5} - \)\(10\!\cdots\!76\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(26\!\cdots\!32\)\( \beta_{8}) q^{56}\) \(+(-\)\(10\!\cdots\!98\)\( + \)\(96\!\cdots\!49\)\( \beta_{1} + \)\(13\!\cdots\!91\)\( \beta_{2} - \)\(11\!\cdots\!99\)\( \beta_{3} + \)\(83\!\cdots\!85\)\( \beta_{4} + \)\(80\!\cdots\!50\)\( \beta_{5} + \)\(20\!\cdots\!80\)\( \beta_{6} - \)\(13\!\cdots\!81\)\( \beta_{7} - \)\(22\!\cdots\!15\)\( \beta_{8}) q^{57}\) \(+(-\)\(16\!\cdots\!70\)\( - \)\(78\!\cdots\!94\)\( \beta_{1} + \)\(96\!\cdots\!08\)\( \beta_{2} - \)\(77\!\cdots\!60\)\( \beta_{3} - \)\(60\!\cdots\!80\)\( \beta_{4} - \)\(25\!\cdots\!16\)\( \beta_{5} - \)\(14\!\cdots\!36\)\( \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{7} + \)\(61\!\cdots\!40\)\( \beta_{8}) q^{58}\) \(+(-\)\(19\!\cdots\!21\)\( - \)\(18\!\cdots\!36\)\( \beta_{1} - \)\(46\!\cdots\!17\)\( \beta_{2} + \)\(54\!\cdots\!20\)\( \beta_{3} + \)\(98\!\cdots\!56\)\( \beta_{4} + \)\(18\!\cdots\!64\)\( \beta_{5} - \)\(52\!\cdots\!32\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7} - \)\(58\!\cdots\!76\)\( \beta_{8}) q^{59}\) \(+(\)\(13\!\cdots\!16\)\( - \)\(61\!\cdots\!28\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(29\!\cdots\!60\)\( \beta_{3} - \)\(12\!\cdots\!28\)\( \beta_{4} + \)\(63\!\cdots\!64\)\( \beta_{5} - \)\(41\!\cdots\!32\)\( \beta_{6} - \)\(10\!\cdots\!92\)\( \beta_{7} - \)\(23\!\cdots\!56\)\( \beta_{8}) q^{60}\) \(+(\)\(11\!\cdots\!97\)\( + \)\(15\!\cdots\!03\)\( \beta_{1} + \)\(77\!\cdots\!84\)\( \beta_{2} + \)\(66\!\cdots\!56\)\( \beta_{3} + \)\(35\!\cdots\!07\)\( \beta_{4} + \)\(22\!\cdots\!28\)\( \beta_{5} + \)\(38\!\cdots\!16\)\( \beta_{6} + \)\(50\!\cdots\!50\)\( \beta_{7} + \)\(11\!\cdots\!38\)\( \beta_{8}) q^{61}\) \(+(-\)\(20\!\cdots\!56\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(52\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!92\)\( \beta_{3} - \)\(21\!\cdots\!20\)\( \beta_{4} - \)\(95\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!96\)\( \beta_{6} - \)\(83\!\cdots\!40\)\( \beta_{7} - \)\(24\!\cdots\!80\)\( \beta_{8}) q^{62}\) \(+(-\)\(58\!\cdots\!92\)\( + \)\(79\!\cdots\!67\)\( \beta_{1} + \)\(24\!\cdots\!71\)\( \beta_{2} - \)\(22\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!60\)\( \beta_{4} + \)\(80\!\cdots\!09\)\( \beta_{5} - \)\(54\!\cdots\!91\)\( \beta_{6} - \)\(11\!\cdots\!12\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{63}\) \(+(\)\(98\!\cdots\!04\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!76\)\( \beta_{2} + \)\(20\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!32\)\( \beta_{4} + \)\(29\!\cdots\!68\)\( \beta_{5} + \)\(79\!\cdots\!20\)\( \beta_{6} + \)\(94\!\cdots\!60\)\( \beta_{7} + \)\(14\!\cdots\!60\)\( \beta_{8}) q^{64}\) \(+(\)\(37\!\cdots\!12\)\( + \)\(10\!\cdots\!34\)\( \beta_{1} - \)\(14\!\cdots\!74\)\( \beta_{2} + \)\(19\!\cdots\!70\)\( \beta_{3} + \)\(15\!\cdots\!94\)\( \beta_{4} - \)\(61\!\cdots\!32\)\( \beta_{5} + \)\(12\!\cdots\!16\)\( \beta_{6} - \)\(20\!\cdots\!54\)\( \beta_{7} - \)\(45\!\cdots\!22\)\( \beta_{8}) q^{65}\) \(+(\)\(64\!\cdots\!12\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(31\!\cdots\!44\)\( \beta_{2} + \)\(15\!\cdots\!04\)\( \beta_{3} - \)\(62\!\cdots\!36\)\( \beta_{4} - \)\(37\!\cdots\!28\)\( \beta_{5} - \)\(66\!\cdots\!12\)\( \beta_{6} + \)\(38\!\cdots\!20\)\( \beta_{7} + \)\(51\!\cdots\!84\)\( \beta_{8}) q^{66}\) \(+(\)\(74\!\cdots\!65\)\( - \)\(59\!\cdots\!54\)\( \beta_{1} + \)\(26\!\cdots\!79\)\( \beta_{2} - \)\(10\!\cdots\!92\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4} - \)\(33\!\cdots\!66\)\( \beta_{5} + \)\(71\!\cdots\!14\)\( \beta_{6} + \)\(97\!\cdots\!24\)\( \beta_{7} + \)\(10\!\cdots\!80\)\( \beta_{8}) q^{67}\) \(+(-\)\(58\!\cdots\!72\)\( - \)\(52\!\cdots\!72\)\( \beta_{1} + \)\(15\!\cdots\!18\)\( \beta_{2} - \)\(94\!\cdots\!02\)\( \beta_{3} + \)\(18\!\cdots\!20\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(14\!\cdots\!68\)\( \beta_{6} - \)\(26\!\cdots\!24\)\( \beta_{7} - \)\(57\!\cdots\!40\)\( \beta_{8}) q^{68}\) \(+(\)\(13\!\cdots\!76\)\( - \)\(60\!\cdots\!94\)\( \beta_{1} + \)\(17\!\cdots\!34\)\( \beta_{2} - \)\(43\!\cdots\!02\)\( \beta_{3} - \)\(15\!\cdots\!98\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(35\!\cdots\!52\)\( \beta_{6} + \)\(21\!\cdots\!30\)\( \beta_{7} + \)\(10\!\cdots\!14\)\( \beta_{8}) q^{69}\) \(+(\)\(33\!\cdots\!64\)\( + \)\(21\!\cdots\!44\)\( \beta_{1} - \)\(27\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!40\)\( \beta_{3} - \)\(39\!\cdots\!24\)\( \beta_{4} - \)\(67\!\cdots\!60\)\( \beta_{5} - \)\(57\!\cdots\!20\)\( \beta_{6} + \)\(53\!\cdots\!80\)\( \beta_{7} + \)\(89\!\cdots\!40\)\( \beta_{8}) q^{70}\) \(+(\)\(61\!\cdots\!12\)\( + \)\(59\!\cdots\!79\)\( \beta_{1} - \)\(10\!\cdots\!93\)\( \beta_{2} - \)\(22\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!84\)\( \beta_{4} + \)\(14\!\cdots\!89\)\( \beta_{5} + \)\(28\!\cdots\!33\)\( \beta_{6} - \)\(17\!\cdots\!20\)\( \beta_{7} - \)\(52\!\cdots\!56\)\( \beta_{8}) q^{71}\) \(+(\)\(18\!\cdots\!19\)\( + \)\(13\!\cdots\!31\)\( \beta_{1} - \)\(16\!\cdots\!17\)\( \beta_{2} + \)\(35\!\cdots\!41\)\( \beta_{3} + \)\(57\!\cdots\!20\)\( \beta_{4} - \)\(17\!\cdots\!38\)\( \beta_{5} - \)\(81\!\cdots\!93\)\( \beta_{6} + \)\(18\!\cdots\!55\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{72}\) \(+(\)\(14\!\cdots\!56\)\( + \)\(12\!\cdots\!33\)\( \beta_{1} + \)\(26\!\cdots\!15\)\( \beta_{2} - \)\(14\!\cdots\!07\)\( \beta_{3} - \)\(55\!\cdots\!75\)\( \beta_{4} - \)\(52\!\cdots\!06\)\( \beta_{5} - \)\(14\!\cdots\!36\)\( \beta_{6} + \)\(42\!\cdots\!79\)\( \beta_{7} - \)\(86\!\cdots\!55\)\( \beta_{8}) q^{73}\) \(+(\)\(80\!\cdots\!30\)\( + \)\(93\!\cdots\!34\)\( \beta_{1} + \)\(88\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!44\)\( \beta_{4} - \)\(27\!\cdots\!52\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(15\!\cdots\!40\)\( \beta_{7} - \)\(34\!\cdots\!68\)\( \beta_{8}) q^{74}\) \(+(\)\(20\!\cdots\!75\)\( - \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(25\!\cdots\!75\)\( \beta_{2} - \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!00\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5} - \)\(35\!\cdots\!00\)\( \beta_{6} - \)\(24\!\cdots\!00\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8}) q^{75}\) \(+(\)\(41\!\cdots\!68\)\( - \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3} + \)\(59\!\cdots\!52\)\( \beta_{4} - \)\(57\!\cdots\!40\)\( \beta_{5} - \)\(72\!\cdots\!92\)\( \beta_{6} - \)\(27\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!56\)\( \beta_{8}) q^{76}\) \(+(\)\(43\!\cdots\!92\)\( - \)\(19\!\cdots\!50\)\( \beta_{1} - \)\(70\!\cdots\!54\)\( \beta_{2} - \)\(31\!\cdots\!18\)\( \beta_{3} + \)\(96\!\cdots\!50\)\( \beta_{4} - \)\(65\!\cdots\!36\)\( \beta_{5} + \)\(78\!\cdots\!44\)\( \beta_{6} + \)\(13\!\cdots\!90\)\( \beta_{7} - \)\(14\!\cdots\!50\)\( \beta_{8}) q^{77}\) \(+(\)\(18\!\cdots\!66\)\( - \)\(46\!\cdots\!86\)\( \beta_{1} - \)\(16\!\cdots\!74\)\( \beta_{2} - \)\(37\!\cdots\!64\)\( \beta_{3} - \)\(25\!\cdots\!10\)\( \beta_{4} + \)\(65\!\cdots\!58\)\( \beta_{5} - \)\(29\!\cdots\!92\)\( \beta_{6} - \)\(18\!\cdots\!92\)\( \beta_{7} + \)\(75\!\cdots\!00\)\( \beta_{8}) q^{78}\) \(+(\)\(19\!\cdots\!28\)\( + \)\(83\!\cdots\!14\)\( \beta_{1} + \)\(61\!\cdots\!46\)\( \beta_{2} - \)\(58\!\cdots\!68\)\( \beta_{3} + \)\(46\!\cdots\!68\)\( \beta_{4} + \)\(89\!\cdots\!30\)\( \beta_{5} + \)\(50\!\cdots\!62\)\( \beta_{6} - \)\(34\!\cdots\!00\)\( \beta_{7} - \)\(10\!\cdots\!84\)\( \beta_{8}) q^{79}\) \(+(\)\(74\!\cdots\!44\)\( + \)\(29\!\cdots\!24\)\( \beta_{1} + \)\(37\!\cdots\!92\)\( \beta_{2} + \)\(37\!\cdots\!40\)\( \beta_{3} - \)\(77\!\cdots\!04\)\( \beta_{4} + \)\(25\!\cdots\!40\)\( \beta_{5} - \)\(15\!\cdots\!20\)\( \beta_{6} + \)\(16\!\cdots\!80\)\( \beta_{7} - \)\(17\!\cdots\!60\)\( \beta_{8}) q^{80}\) \(+(\)\(95\!\cdots\!79\)\( + \)\(50\!\cdots\!19\)\( \beta_{1} + \)\(15\!\cdots\!29\)\( \beta_{2} - \)\(23\!\cdots\!13\)\( \beta_{3} + \)\(32\!\cdots\!71\)\( \beta_{4} - \)\(15\!\cdots\!06\)\( \beta_{5} - \)\(19\!\cdots\!92\)\( \beta_{6} - \)\(15\!\cdots\!15\)\( \beta_{7} + \)\(31\!\cdots\!19\)\( \beta_{8}) q^{81}\) \(+(\)\(95\!\cdots\!02\)\( + \)\(17\!\cdots\!38\)\( \beta_{1} + \)\(40\!\cdots\!32\)\( \beta_{2} - \)\(38\!\cdots\!04\)\( \beta_{3} - \)\(13\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!36\)\( \beta_{5} + \)\(86\!\cdots\!76\)\( \beta_{6} - \)\(39\!\cdots\!88\)\( \beta_{7} - \)\(50\!\cdots\!40\)\( \beta_{8}) q^{82}\) \(+(\)\(11\!\cdots\!81\)\( + \)\(49\!\cdots\!52\)\( \beta_{1} - \)\(32\!\cdots\!47\)\( \beta_{2} - \)\(79\!\cdots\!20\)\( \beta_{3} - \)\(58\!\cdots\!80\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} - \)\(14\!\cdots\!40\)\( \beta_{6} + \)\(11\!\cdots\!80\)\( \beta_{7} + \)\(25\!\cdots\!80\)\( \beta_{8}) q^{83}\) \(+(\)\(32\!\cdots\!20\)\( - \)\(26\!\cdots\!24\)\( \beta_{1} - \)\(13\!\cdots\!12\)\( \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(50\!\cdots\!72\)\( \beta_{4} - \)\(67\!\cdots\!72\)\( \beta_{5} - \)\(48\!\cdots\!80\)\( \beta_{6} + \)\(70\!\cdots\!80\)\( \beta_{7} + \)\(19\!\cdots\!60\)\( \beta_{8}) q^{84}\) \(+(\)\(31\!\cdots\!06\)\( - \)\(78\!\cdots\!88\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} + \)\(16\!\cdots\!10\)\( \beta_{3} + \)\(21\!\cdots\!32\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5} + \)\(32\!\cdots\!68\)\( \beta_{6} - \)\(42\!\cdots\!42\)\( \beta_{7} - \)\(31\!\cdots\!06\)\( \beta_{8}) q^{85}\) \(+(-\)\(62\!\cdots\!33\)\( - \)\(32\!\cdots\!33\)\( \beta_{1} + \)\(23\!\cdots\!07\)\( \beta_{2} + \)\(36\!\cdots\!54\)\( \beta_{3} + \)\(45\!\cdots\!39\)\( \beta_{4} - \)\(97\!\cdots\!45\)\( \beta_{5} + \)\(50\!\cdots\!16\)\( \beta_{6} + \)\(44\!\cdots\!60\)\( \beta_{7} + \)\(19\!\cdots\!88\)\( \beta_{8}) q^{86}\) \(+(-\)\(26\!\cdots\!60\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + \)\(76\!\cdots\!49\)\( \beta_{2} - \)\(79\!\cdots\!16\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4} + \)\(64\!\cdots\!31\)\( \beta_{5} - \)\(64\!\cdots\!89\)\( \beta_{6} + \)\(12\!\cdots\!44\)\( \beta_{7} - \)\(93\!\cdots\!60\)\( \beta_{8}) q^{87}\) \(+(-\)\(72\!\cdots\!36\)\( - \)\(31\!\cdots\!56\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} - \)\(47\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(37\!\cdots\!76\)\( \beta_{5} - \)\(18\!\cdots\!76\)\( \beta_{6} - \)\(34\!\cdots\!60\)\( \beta_{7} + \)\(17\!\cdots\!20\)\( \beta_{8}) q^{88}\) \(+(-\)\(98\!\cdots\!68\)\( + \)\(59\!\cdots\!65\)\( \beta_{1} - \)\(10\!\cdots\!05\)\( \beta_{2} + \)\(23\!\cdots\!41\)\( \beta_{3} - \)\(41\!\cdots\!43\)\( \beta_{4} - \)\(18\!\cdots\!34\)\( \beta_{5} + \)\(74\!\cdots\!24\)\( \beta_{6} + \)\(42\!\cdots\!35\)\( \beta_{7} + \)\(63\!\cdots\!57\)\( \beta_{8}) q^{89}\) \(+(-\)\(26\!\cdots\!18\)\( + \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(45\!\cdots\!64\)\( \beta_{2} + \)\(89\!\cdots\!20\)\( \beta_{3} + \)\(58\!\cdots\!04\)\( \beta_{4} + \)\(30\!\cdots\!08\)\( \beta_{5} - \)\(38\!\cdots\!04\)\( \beta_{6} + \)\(11\!\cdots\!76\)\( \beta_{7} - \)\(86\!\cdots\!32\)\( \beta_{8}) q^{90}\) \(+(-\)\(22\!\cdots\!24\)\( - \)\(26\!\cdots\!72\)\( \beta_{1} - \)\(38\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!48\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} - \)\(20\!\cdots\!36\)\( \beta_{6} - \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!52\)\( \beta_{8}) q^{91}\) \(+(-\)\(10\!\cdots\!88\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} + \)\(53\!\cdots\!60\)\( \beta_{2} - \)\(52\!\cdots\!04\)\( \beta_{3} - \)\(12\!\cdots\!40\)\( \beta_{4} + \)\(67\!\cdots\!96\)\( \beta_{5} + \)\(43\!\cdots\!76\)\( \beta_{6} - \)\(53\!\cdots\!48\)\( \beta_{7} + \)\(12\!\cdots\!00\)\( \beta_{8}) q^{92}\) \(+(-\)\(11\!\cdots\!56\)\( - \)\(35\!\cdots\!64\)\( \beta_{1} + \)\(29\!\cdots\!80\)\( \beta_{2} + \)\(17\!\cdots\!84\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4} - \)\(12\!\cdots\!48\)\( \beta_{5} - \)\(20\!\cdots\!28\)\( \beta_{6} + \)\(46\!\cdots\!32\)\( \beta_{7} - \)\(40\!\cdots\!80\)\( \beta_{8}) q^{93}\) \(+(-\)\(14\!\cdots\!80\)\( - \)\(15\!\cdots\!96\)\( \beta_{1} + \)\(38\!\cdots\!60\)\( \beta_{2} - \)\(77\!\cdots\!40\)\( \beta_{3} - \)\(40\!\cdots\!72\)\( \beta_{4} - \)\(19\!\cdots\!60\)\( \beta_{5} - \)\(34\!\cdots\!88\)\( \beta_{6} - \)\(39\!\cdots\!20\)\( \beta_{7} + \)\(52\!\cdots\!16\)\( \beta_{8}) q^{94}\) \(+(\)\(21\!\cdots\!00\)\( - \)\(46\!\cdots\!95\)\( \beta_{1} + \)\(54\!\cdots\!25\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!40\)\( \beta_{4} + \)\(68\!\cdots\!95\)\( \beta_{5} + \)\(89\!\cdots\!15\)\( \beta_{6} - \)\(13\!\cdots\!60\)\( \beta_{7} + \)\(12\!\cdots\!20\)\( \beta_{8}) q^{95}\) \(+(\)\(90\!\cdots\!12\)\( - \)\(11\!\cdots\!84\)\( \beta_{1} - \)\(25\!\cdots\!28\)\( \beta_{2} + \)\(18\!\cdots\!72\)\( \beta_{3} - \)\(16\!\cdots\!92\)\( \beta_{4} - \)\(21\!\cdots\!92\)\( \beta_{5} - \)\(20\!\cdots\!80\)\( \beta_{6} - \)\(52\!\cdots\!20\)\( \beta_{7} - \)\(49\!\cdots\!40\)\( \beta_{8}) q^{96}\) \(+(\)\(20\!\cdots\!52\)\( + \)\(52\!\cdots\!41\)\( \beta_{1} + \)\(93\!\cdots\!15\)\( \beta_{2} - \)\(15\!\cdots\!47\)\( \beta_{3} - \)\(10\!\cdots\!55\)\( \beta_{4} - \)\(49\!\cdots\!82\)\( \beta_{5} + \)\(22\!\cdots\!68\)\( \beta_{6} + \)\(21\!\cdots\!71\)\( \beta_{7} - \)\(10\!\cdots\!15\)\( \beta_{8}) q^{97}\) \(+(\)\(10\!\cdots\!89\)\( + \)\(10\!\cdots\!67\)\( \beta_{1} - \)\(38\!\cdots\!28\)\( \beta_{2} - \)\(36\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(27\!\cdots\!84\)\( \beta_{5} + \)\(50\!\cdots\!76\)\( \beta_{6} + \)\(18\!\cdots\!92\)\( \beta_{7} + \)\(11\!\cdots\!80\)\( \beta_{8}) q^{98}\) \(+(\)\(23\!\cdots\!75\)\( + \)\(97\!\cdots\!92\)\( \beta_{1} + \)\(12\!\cdots\!91\)\( \beta_{2} - \)\(20\!\cdots\!88\)\( \beta_{3} + \)\(46\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!60\)\( \beta_{5} - \)\(13\!\cdots\!84\)\( \beta_{6} - \)\(15\!\cdots\!20\)\( \beta_{7} + \)\(85\!\cdots\!88\)\( \beta_{8}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 5465779613435496q^{2} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!12\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!92\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!32\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!44\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!13\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 5465779613435496q^{2} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!12\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!92\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!32\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!44\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!13\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!48\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!44\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!82\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!16\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!64\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!26\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!72\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!60\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!48\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!88\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!48\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!48\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!72\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!90\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!08\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!64\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!64\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!64\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!56\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!34\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!56\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!62\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!12\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!92\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!76\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!72\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!36\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!52\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!63\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!08\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!16\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!38\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!40\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!20\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!98\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!48\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!08\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!12\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!96\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!76\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!12\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!16\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!28\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!02\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!24\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!80\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!32\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!64\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!89\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!72\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!24\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!12\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!80\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!30\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!72\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!24\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!56\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!96\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!88\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!86\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!28\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!36\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(18\!\cdots\!64\) \(x^{7}\mathstrut -\mathstrut \) \(73\!\cdots\!04\) \(x^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!46\) \(x^{5}\mathstrut +\mathstrut \) \(37\!\cdots\!92\) \(x^{4}\mathstrut -\mathstrut \) \(21\!\cdots\!48\) \(x^{3}\mathstrut -\mathstrut \) \(45\!\cdots\!88\) \(x^{2}\mathstrut +\mathstrut \) \(13\!\cdots\!17\) \(x\mathstrut +\mathstrut \) \(27\!\cdots\!52\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 11 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(12\!\cdots\!79\) \(\nu^{8}\mathstrut -\mathstrut \) \(17\!\cdots\!29\) \(\nu^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!45\) \(\nu^{6}\mathstrut +\mathstrut \) \(27\!\cdots\!23\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!49\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!03\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!11\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!21\) \(\nu\mathstrut +\mathstrut \) \(21\!\cdots\!68\)\()/\)\(18\!\cdots\!24\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(47\!\cdots\!19\) \(\nu^{8}\mathstrut -\mathstrut \) \(66\!\cdots\!69\) \(\nu^{7}\mathstrut +\mathstrut \) \(86\!\cdots\!45\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!03\) \(\nu^{5}\mathstrut -\mathstrut \) \(40\!\cdots\!89\) \(\nu^{4}\mathstrut -\mathstrut \) \(44\!\cdots\!83\) \(\nu^{3}\mathstrut +\mathstrut \) \(10\!\cdots\!43\) \(\nu^{2}\mathstrut +\mathstrut \) \(42\!\cdots\!73\) \(\nu\mathstrut -\mathstrut \) \(24\!\cdots\!20\)\()/\)\(10\!\cdots\!72\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(32\!\cdots\!89\) \(\nu^{8}\mathstrut +\mathstrut \) \(90\!\cdots\!09\) \(\nu^{7}\mathstrut +\mathstrut \) \(58\!\cdots\!03\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!75\) \(\nu^{5}\mathstrut -\mathstrut \) \(29\!\cdots\!19\) \(\nu^{4}\mathstrut +\mathstrut \) \(41\!\cdots\!75\) \(\nu^{3}\mathstrut +\mathstrut \) \(28\!\cdots\!97\) \(\nu^{2}\mathstrut -\mathstrut \) \(53\!\cdots\!37\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!36\)\()/\)\(96\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(59\!\cdots\!89\) \(\nu^{8}\mathstrut +\mathstrut \) \(35\!\cdots\!91\) \(\nu^{7}\mathstrut -\mathstrut \) \(91\!\cdots\!03\) \(\nu^{6}\mathstrut -\mathstrut \) \(13\!\cdots\!25\) \(\nu^{5}\mathstrut +\mathstrut \) \(38\!\cdots\!19\) \(\nu^{4}\mathstrut +\mathstrut \) \(73\!\cdots\!25\) \(\nu^{3}\mathstrut -\mathstrut \) \(40\!\cdots\!97\) \(\nu^{2}\mathstrut -\mathstrut \) \(91\!\cdots\!63\) \(\nu\mathstrut -\mathstrut \) \(93\!\cdots\!36\)\()/\)\(25\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(50\!\cdots\!53\) \(\nu^{8}\mathstrut +\mathstrut \) \(54\!\cdots\!93\) \(\nu^{7}\mathstrut +\mathstrut \) \(89\!\cdots\!31\) \(\nu^{6}\mathstrut -\mathstrut \) \(76\!\cdots\!75\) \(\nu^{5}\mathstrut -\mathstrut \) \(47\!\cdots\!63\) \(\nu^{4}\mathstrut +\mathstrut \) \(50\!\cdots\!75\) \(\nu^{3}\mathstrut +\mathstrut \) \(84\!\cdots\!69\) \(\nu^{2}\mathstrut -\mathstrut \) \(88\!\cdots\!49\) \(\nu\mathstrut -\mathstrut \) \(31\!\cdots\!28\)\()/\)\(11\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(26\!\cdots\!07\) \(\nu^{8}\mathstrut +\mathstrut \) \(47\!\cdots\!67\) \(\nu^{7}\mathstrut +\mathstrut \) \(44\!\cdots\!89\) \(\nu^{6}\mathstrut -\mathstrut \) \(57\!\cdots\!25\) \(\nu^{5}\mathstrut -\mathstrut \) \(21\!\cdots\!97\) \(\nu^{4}\mathstrut +\mathstrut \) \(41\!\cdots\!25\) \(\nu^{3}\mathstrut +\mathstrut \) \(28\!\cdots\!11\) \(\nu^{2}\mathstrut -\mathstrut \) \(13\!\cdots\!31\) \(\nu\mathstrut +\mathstrut \) \(53\!\cdots\!68\)\()/\)\(11\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(45\!\cdots\!37\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!97\) \(\nu^{7}\mathstrut +\mathstrut \) \(10\!\cdots\!99\) \(\nu^{6}\mathstrut +\mathstrut \) \(31\!\cdots\!25\) \(\nu^{5}\mathstrut -\mathstrut \) \(90\!\cdots\!27\) \(\nu^{4}\mathstrut -\mathstrut \) \(18\!\cdots\!25\) \(\nu^{3}\mathstrut +\mathstrut \) \(26\!\cdots\!01\) \(\nu^{2}\mathstrut +\mathstrut \) \(43\!\cdots\!79\) \(\nu\mathstrut -\mathstrut \) \(14\!\cdots\!12\)\()/\)\(76\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(11\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(664037\) \(\beta_{2}\mathstrut +\mathstrut \) \(1454496237795986\) \(\beta_{1}\mathstrut +\mathstrut \) \(233205805914080463879155578823890\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(287\) \(\beta_{6}\mathstrut +\mathstrut \) \(326098\) \(\beta_{5}\mathstrut -\mathstrut \) \(11536093720\) \(\beta_{4}\mathstrut +\mathstrut \) \(6080152289344487\) \(\beta_{3}\mathstrut -\mathstrut \) \(1729190174695774540615\) \(\beta_{2}\mathstrut +\mathstrut \) \(389982933498588803529772202388069\) \(\beta_{1}\mathstrut +\mathstrut \) \(339196967390022887798427489622681198989944874045\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(399171419712\) \(\beta_{8}\mathstrut +\mathstrut \) \(1032326586055479\) \(\beta_{7}\mathstrut -\mathstrut \) \(1073780444029653737\) \(\beta_{6}\mathstrut +\mathstrut \) \(398942241277981412382\) \(\beta_{5}\mathstrut +\mathstrut \) \(88256816768613430614080024\) \(\beta_{4}\mathstrut +\mathstrut \) \(73419319872788947466478796229417\) \(\beta_{3}\mathstrut -\mathstrut \) \(40830610558608177706493963427534012393\) \(\beta_{2}\mathstrut +\mathstrut \) \(229589003907921368619341708370163101878428065411\) \(\beta_{1}\mathstrut +\mathstrut \) \(11368285537427622943708687074161312670632101143477642887442636587\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(1040276766472715976193601280\) \(\beta_{8}\mathstrut +\mathstrut \) \(5386861631393577261729528541249\) \(\beta_{7}\mathstrut -\mathstrut \) \(2503564908944438757999586187538911\) \(\beta_{6}\mathstrut +\mathstrut \) \(965686401281995211115857145364280914\) \(\beta_{5}\mathstrut -\mathstrut \) \(13058856798979542643874920464718010944280\) \(\beta_{4}\mathstrut +\mathstrut \) \(45718502092925457823634876203711469669762023863\) \(\beta_{3}\mathstrut -\mathstrut \) \(16942881902956622105520393094251233220884436358727511\) \(\beta_{2}\mathstrut +\mathstrut \) \(1438922850832020154332150699622892850112530279258047393493043109\) \(\beta_{1}\mathstrut +\mathstrut \) \(3346343043049345609121544223011877028625201201516347548008693867606683570380605\)\()/62208\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(69\!\cdots\!00\) \(\beta_{8}\mathstrut +\mathstrut \) \(88\!\cdots\!81\) \(\beta_{7}\mathstrut -\mathstrut \) \(73\!\cdots\!19\) \(\beta_{6}\mathstrut +\mathstrut \) \(58\!\cdots\!94\) \(\beta_{5}\mathstrut +\mathstrut \) \(61\!\cdots\!88\) \(\beta_{4}\mathstrut +\mathstrut \) \(35\!\cdots\!43\) \(\beta_{3}\mathstrut -\mathstrut \) \(16\!\cdots\!35\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\!\cdots\!85\) \(\beta_{1}\mathstrut +\mathstrut \) \(46\!\cdots\!93\)\()/20736\)
\(\nu^{7}\)\(=\)\((\)\(10\!\cdots\!80\) \(\beta_{8}\mathstrut +\mathstrut \) \(37\!\cdots\!81\) \(\beta_{7}\mathstrut -\mathstrut \) \(21\!\cdots\!15\) \(\beta_{6}\mathstrut +\mathstrut \) \(41\!\cdots\!14\) \(\beta_{5}\mathstrut +\mathstrut \) \(20\!\cdots\!24\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\!\cdots\!51\) \(\beta_{3}\mathstrut -\mathstrut \) \(20\!\cdots\!27\) \(\beta_{2}\mathstrut +\mathstrut \) \(86\!\cdots\!33\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\!\cdots\!45\)\()/41472\)
\(\nu^{8}\)\(=\)\((\)\(42\!\cdots\!96\) \(\beta_{8}\mathstrut +\mathstrut \) \(69\!\cdots\!03\) \(\beta_{7}\mathstrut -\mathstrut \) \(51\!\cdots\!17\) \(\beta_{6}\mathstrut +\mathstrut \) \(51\!\cdots\!14\) \(\beta_{5}\mathstrut +\mathstrut \) \(40\!\cdots\!76\) \(\beta_{4}\mathstrut +\mathstrut \) \(21\!\cdots\!81\) \(\beta_{3}\mathstrut -\mathstrut \) \(85\!\cdots\!13\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\!\cdots\!47\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\!\cdots\!07\)\()/124416\)

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
1.03210e15
6.73733e14
4.82720e14
3.47594e14
−2.01415e13
−3.22655e14
−5.32911e14
−7.86617e14
−8.73826e14
−2.41632e16 −3.68177e24 4.21601e32 3.92322e37 8.89632e40 −1.24740e45 −6.26652e48 −1.11358e51 −9.47976e53
1.2 −1.55623e16 1.35366e25 7.99254e31 −3.05298e37 −2.10660e41 6.30256e44 1.28130e48 −9.43892e50 4.75113e53
1.3 −1.09780e16 −5.69800e25 −4.17435e31 2.81503e36 6.25524e41 3.42403e44 2.23954e48 2.11959e51 −3.09033e52
1.4 −7.73495e15 6.57221e25 −1.02430e32 3.43777e37 −5.08357e41 9.22106e44 2.04736e48 3.19227e51 −2.65909e53
1.5 1.09070e15 6.73736e23 −1.61070e32 1.05761e37 7.34847e38 −2.03867e45 −3.52656e47 −1.12668e51 1.15354e52
1.6 8.35103e15 −1.73817e25 −9.25196e31 3.17576e36 −1.45155e41 2.61780e45 −2.12767e48 −8.25006e50 2.65209e52
1.7 1.33972e16 4.78035e25 1.72251e31 −4.75138e37 6.40432e41 −7.19244e44 −1.94305e48 1.15805e51 −6.36552e53
1.8 1.94861e16 −5.50693e25 2.17450e32 −2.12470e37 −1.07309e42 −2.04209e45 1.07545e48 1.90550e51 −4.14022e53
1.9 2.15791e16 2.13607e25 3.03400e32 3.38441e37 4.60946e41 5.98905e44 3.04570e48 −6.70851e50 7.30327e53
\(n\): 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 \(S_{108}^{\mathrm{new}}(\Gamma_0(1))\).