Properties

Label 1.118.a.a
Level 1
Weight 118
Character orbit 1.a
Self dual Yes
Analytic conductor 86.689
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 118 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(86.6887159558\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{151}\cdot 3^{56}\cdot 5^{18}\cdot 7^{7}\cdot 11^{4}\cdot 13^{4}\cdot 17^{2}\cdot 19\cdot 23\cdot 29^{2} \)
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.

\(f(q)\) \(=\) \( q +(4490781853906528 - \beta_{1}) q^{2} +(\)\(11\!\cdots\!04\)\( + 2251433924 \beta_{1} + \beta_{2}) q^{3} +(\)\(75\!\cdots\!32\)\( - 40679477561379210 \beta_{1} + 3291802 \beta_{2} + \beta_{3}) q^{4} +(\)\(42\!\cdots\!50\)\( + \)\(53\!\cdots\!23\)\( \beta_{1} - 1520180837375 \beta_{2} + 10132 \beta_{3} + \beta_{4}) q^{5} +(-\)\(53\!\cdots\!28\)\( - \)\(22\!\cdots\!04\)\( \beta_{1} - 27746238674528446 \beta_{2} - 10032502536 \beta_{3} - 12522 \beta_{4} + \beta_{5}) q^{6} +(-\)\(37\!\cdots\!12\)\( - \)\(16\!\cdots\!40\)\( \beta_{1} - \)\(25\!\cdots\!64\)\( \beta_{2} - 12981069049497 \beta_{3} - 57565343 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(\)\(94\!\cdots\!80\)\( - \)\(80\!\cdots\!44\)\( \beta_{1} - \)\(34\!\cdots\!76\)\( \beta_{2} + 43805503341968276 \beta_{3} + 13791262570 \beta_{4} + 348895 \beta_{5} + 323 \beta_{6} - \beta_{7}) q^{8} +(\)\(25\!\cdots\!33\)\( + \)\(19\!\cdots\!04\)\( \beta_{1} + \)\(26\!\cdots\!24\)\( \beta_{2} + 62596919628984986927 \beta_{3} + 102199893767245 \beta_{4} + 1889573813 \beta_{5} - 601468 \beta_{6} + 120 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\) \( q +(4490781853906528 - \beta_{1}) q^{2} +(\)\(11\!\cdots\!04\)\( + 2251433924 \beta_{1} + \beta_{2}) q^{3} +(\)\(75\!\cdots\!32\)\( - 40679477561379210 \beta_{1} + 3291802 \beta_{2} + \beta_{3}) q^{4} +(\)\(42\!\cdots\!50\)\( + \)\(53\!\cdots\!23\)\( \beta_{1} - 1520180837375 \beta_{2} + 10132 \beta_{3} + \beta_{4}) q^{5} +(-\)\(53\!\cdots\!28\)\( - \)\(22\!\cdots\!04\)\( \beta_{1} - 27746238674528446 \beta_{2} - 10032502536 \beta_{3} - 12522 \beta_{4} + \beta_{5}) q^{6} +(-\)\(37\!\cdots\!12\)\( - \)\(16\!\cdots\!40\)\( \beta_{1} - \)\(25\!\cdots\!64\)\( \beta_{2} - 12981069049497 \beta_{3} - 57565343 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(\)\(94\!\cdots\!80\)\( - \)\(80\!\cdots\!44\)\( \beta_{1} - \)\(34\!\cdots\!76\)\( \beta_{2} + 43805503341968276 \beta_{3} + 13791262570 \beta_{4} + 348895 \beta_{5} + 323 \beta_{6} - \beta_{7}) q^{8} +(\)\(25\!\cdots\!33\)\( + \)\(19\!\cdots\!04\)\( \beta_{1} + \)\(26\!\cdots\!24\)\( \beta_{2} + 62596919628984986927 \beta_{3} + 102199893767245 \beta_{4} + 1889573813 \beta_{5} - 601468 \beta_{6} + 120 \beta_{7} + \beta_{8}) q^{9} +(-\)\(12\!\cdots\!00\)\( - \)\(21\!\cdots\!74\)\( \beta_{1} - \)\(98\!\cdots\!80\)\( \beta_{2} - \)\(33\!\cdots\!56\)\( \beta_{3} - 49701743452609928 \beta_{4} - 942583637820 \beta_{5} + 250069240 \beta_{6} + 31320 \beta_{7} - 480 \beta_{8}) q^{10} +(-\)\(20\!\cdots\!28\)\( + \)\(59\!\cdots\!20\)\( \beta_{1} + \)\(14\!\cdots\!07\)\( \beta_{2} - \)\(67\!\cdots\!62\)\( \beta_{3} - 5338057975494288042 \beta_{4} - 147114746596186 \beta_{5} - 18276323078 \beta_{6} + 76582240 \beta_{7} - 55404 \beta_{8}) q^{11} +(\)\(35\!\cdots\!28\)\( + \)\(17\!\cdots\!96\)\( \beta_{1} + \)\(96\!\cdots\!80\)\( \beta_{2} + \)\(72\!\cdots\!04\)\( \beta_{3} + \)\(26\!\cdots\!20\)\( \beta_{4} + 11832517347912840 \beta_{5} - 33453891091608 \beta_{6} + 16020396936 \beta_{7} + 16373760 \beta_{8}) q^{12} +(-\)\(10\!\cdots\!26\)\( - \)\(26\!\cdots\!25\)\( \beta_{1} - \)\(62\!\cdots\!99\)\( \beta_{2} + \)\(50\!\cdots\!82\)\( \beta_{3} + \)\(16\!\cdots\!87\)\( \beta_{4} - 8951074590109799394 \beta_{5} - 3186407966253352 \beta_{6} - 1166385993264 \beta_{7} - 1554084570 \beta_{8}) q^{13} +(\)\(39\!\cdots\!84\)\( + \)\(56\!\cdots\!36\)\( \beta_{1} + \)\(22\!\cdots\!64\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!32\)\( \beta_{4} - \)\(11\!\cdots\!82\)\( \beta_{5} - 241032807680032544 \beta_{6} - 16831870916000 \beta_{7} + 87583483008 \beta_{8}) q^{14} +(-\)\(10\!\cdots\!00\)\( + \)\(42\!\cdots\!28\)\( \beta_{1} + \)\(86\!\cdots\!60\)\( \beta_{2} + \)\(90\!\cdots\!57\)\( \beta_{3} + \)\(33\!\cdots\!91\)\( \beta_{4} - \)\(15\!\cdots\!85\)\( \beta_{5} + 13655797526221366095 \beta_{6} + 4410243118560960 \beta_{7} - 3386980324440 \beta_{8}) q^{15} +(\)\(68\!\cdots\!36\)\( - \)\(86\!\cdots\!92\)\( \beta_{1} - \)\(30\!\cdots\!48\)\( \beta_{2} + \)\(72\!\cdots\!64\)\( \beta_{3} + \)\(16\!\cdots\!52\)\( \beta_{4} - \)\(56\!\cdots\!28\)\( \beta_{5} + \)\(17\!\cdots\!72\)\( \beta_{6} - 215521236052380000 \beta_{7} + 93950484381696 \beta_{8}) q^{16} +(-\)\(75\!\cdots\!42\)\( + \)\(33\!\cdots\!60\)\( \beta_{1} + \)\(26\!\cdots\!20\)\( \beta_{2} - \)\(67\!\cdots\!33\)\( \beta_{3} - \)\(53\!\cdots\!07\)\( \beta_{4} + \)\(35\!\cdots\!89\)\( \beta_{5} - \)\(20\!\cdots\!36\)\( \beta_{6} + 5500046597270479320 \beta_{7} - 1805538678018795 \beta_{8}) q^{17} +(-\)\(46\!\cdots\!76\)\( - \)\(38\!\cdots\!49\)\( \beta_{1} - \)\(21\!\cdots\!36\)\( \beta_{2} - \)\(59\!\cdots\!08\)\( \beta_{3} - \)\(36\!\cdots\!52\)\( \beta_{4} + \)\(54\!\cdots\!24\)\( \beta_{5} + \)\(41\!\cdots\!24\)\( \beta_{6} - 65484969797440419120 \beta_{7} + 17746352259341760 \beta_{8}) q^{18} +(-\)\(17\!\cdots\!60\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} + \)\(81\!\cdots\!77\)\( \beta_{2} + \)\(10\!\cdots\!10\)\( \beta_{3} - \)\(61\!\cdots\!74\)\( \beta_{4} - \)\(36\!\cdots\!22\)\( \beta_{5} + \)\(15\!\cdots\!14\)\( \beta_{6} - \)\(67\!\cdots\!20\)\( \beta_{7} + 260371001965615452 \beta_{8}) q^{19} +(-\)\(24\!\cdots\!00\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} + \)\(53\!\cdots\!34\)\( \beta_{3} + \)\(56\!\cdots\!12\)\( \beta_{4} - \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(21\!\cdots\!00\)\( \beta_{6} + \)\(50\!\cdots\!00\)\( \beta_{7} - 17753514941131776000 \beta_{8}) q^{20} +(-\)\(36\!\cdots\!88\)\( - \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(92\!\cdots\!12\)\( \beta_{2} - \)\(73\!\cdots\!82\)\( \beta_{3} + \)\(11\!\cdots\!26\)\( \beta_{4} + \)\(30\!\cdots\!14\)\( \beta_{5} + \)\(41\!\cdots\!68\)\( \beta_{6} - \)\(13\!\cdots\!00\)\( \beta_{7} + \)\(51\!\cdots\!74\)\( \beta_{8}) q^{21} +(-\)\(14\!\cdots\!84\)\( + \)\(32\!\cdots\!20\)\( \beta_{1} - \)\(25\!\cdots\!86\)\( \beta_{2} - \)\(17\!\cdots\!52\)\( \beta_{3} + \)\(17\!\cdots\!58\)\( \beta_{4} + \)\(19\!\cdots\!19\)\( \beta_{5} - \)\(31\!\cdots\!28\)\( \beta_{6} + \)\(21\!\cdots\!44\)\( \beta_{7} - \)\(11\!\cdots\!20\)\( \beta_{8}) q^{22} +(-\)\(64\!\cdots\!56\)\( - \)\(79\!\cdots\!80\)\( \beta_{1} - \)\(70\!\cdots\!76\)\( \beta_{2} - \)\(13\!\cdots\!35\)\( \beta_{3} - \)\(80\!\cdots\!25\)\( \beta_{4} - \)\(68\!\cdots\!05\)\( \beta_{5} - \)\(23\!\cdots\!25\)\( \beta_{6} - \)\(26\!\cdots\!00\)\( \beta_{7} + \)\(19\!\cdots\!80\)\( \beta_{8}) q^{23} +(-\)\(33\!\cdots\!80\)\( - \)\(15\!\cdots\!36\)\( \beta_{1} - \)\(53\!\cdots\!60\)\( \beta_{2} - \)\(30\!\cdots\!64\)\( \beta_{3} - \)\(47\!\cdots\!56\)\( \beta_{4} + \)\(24\!\cdots\!60\)\( \beta_{5} + \)\(91\!\cdots\!08\)\( \beta_{6} + \)\(21\!\cdots\!80\)\( \beta_{7} - \)\(28\!\cdots\!56\)\( \beta_{8}) q^{24} +(\)\(61\!\cdots\!75\)\( + \)\(11\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!00\)\( \beta_{2} - \)\(85\!\cdots\!50\)\( \beta_{3} + \)\(21\!\cdots\!50\)\( \beta_{4} + \)\(69\!\cdots\!50\)\( \beta_{5} - \)\(11\!\cdots\!00\)\( \beta_{6} - \)\(73\!\cdots\!00\)\( \beta_{7} + \)\(35\!\cdots\!50\)\( \beta_{8}) q^{25} +(\)\(62\!\cdots\!32\)\( + \)\(93\!\cdots\!82\)\( \beta_{1} - \)\(19\!\cdots\!56\)\( \beta_{2} + \)\(22\!\cdots\!20\)\( \beta_{3} + \)\(40\!\cdots\!32\)\( \beta_{4} - \)\(87\!\cdots\!00\)\( \beta_{5} + \)\(75\!\cdots\!04\)\( \beta_{6} - \)\(10\!\cdots\!60\)\( \beta_{7} - \)\(39\!\cdots\!28\)\( \beta_{8}) q^{26} +(\)\(20\!\cdots\!80\)\( - \)\(28\!\cdots\!84\)\( \beta_{1} + \)\(12\!\cdots\!34\)\( \beta_{2} + \)\(21\!\cdots\!26\)\( \beta_{3} - \)\(17\!\cdots\!30\)\( \beta_{4} + \)\(12\!\cdots\!90\)\( \beta_{5} - \)\(76\!\cdots\!22\)\( \beta_{6} + \)\(24\!\cdots\!64\)\( \beta_{7} + \)\(39\!\cdots\!80\)\( \beta_{8}) q^{27} +(-\)\(71\!\cdots\!84\)\( - \)\(49\!\cdots\!00\)\( \beta_{1} - \)\(33\!\cdots\!20\)\( \beta_{2} - \)\(17\!\cdots\!76\)\( \beta_{3} - \)\(32\!\cdots\!00\)\( \beta_{4} + \)\(48\!\cdots\!00\)\( \beta_{5} - \)\(34\!\cdots\!88\)\( \beta_{6} - \)\(29\!\cdots\!24\)\( \beta_{7} - \)\(33\!\cdots\!60\)\( \beta_{8}) q^{28} +(-\)\(53\!\cdots\!90\)\( + \)\(28\!\cdots\!43\)\( \beta_{1} - \)\(17\!\cdots\!71\)\( \beta_{2} - \)\(59\!\cdots\!76\)\( \beta_{3} + \)\(18\!\cdots\!13\)\( \beta_{4} - \)\(34\!\cdots\!40\)\( \beta_{5} + \)\(31\!\cdots\!36\)\( \beta_{6} + \)\(23\!\cdots\!20\)\( \beta_{7} + \)\(25\!\cdots\!48\)\( \beta_{8}) q^{29} +(-\)\(10\!\cdots\!00\)\( - \)\(43\!\cdots\!04\)\( \beta_{1} - \)\(37\!\cdots\!00\)\( \beta_{2} - \)\(75\!\cdots\!36\)\( \beta_{3} + \)\(12\!\cdots\!52\)\( \beta_{4} - \)\(85\!\cdots\!50\)\( \beta_{5} - \)\(48\!\cdots\!00\)\( \beta_{6} - \)\(13\!\cdots\!00\)\( \beta_{7} - \)\(16\!\cdots\!00\)\( \beta_{8}) q^{30} +(-\)\(69\!\cdots\!88\)\( - \)\(84\!\cdots\!08\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(76\!\cdots\!72\)\( \beta_{3} - \)\(98\!\cdots\!52\)\( \beta_{4} + \)\(23\!\cdots\!88\)\( \beta_{5} - \)\(15\!\cdots\!72\)\( \beta_{6} + \)\(44\!\cdots\!60\)\( \beta_{7} + \)\(85\!\cdots\!04\)\( \beta_{8}) q^{31} +(\)\(55\!\cdots\!48\)\( - \)\(54\!\cdots\!64\)\( \beta_{1} - \)\(84\!\cdots\!32\)\( \beta_{2} + \)\(42\!\cdots\!12\)\( \beta_{3} - \)\(39\!\cdots\!72\)\( \beta_{4} - \)\(10\!\cdots\!16\)\( \beta_{5} + \)\(16\!\cdots\!44\)\( \beta_{6} + \)\(11\!\cdots\!20\)\( \beta_{7} - \)\(30\!\cdots\!60\)\( \beta_{8}) q^{32} +(\)\(13\!\cdots\!88\)\( + \)\(93\!\cdots\!60\)\( \beta_{1} - \)\(36\!\cdots\!56\)\( \beta_{2} + \)\(15\!\cdots\!01\)\( \beta_{3} + \)\(46\!\cdots\!43\)\( \beta_{4} - \)\(35\!\cdots\!81\)\( \beta_{5} - \)\(82\!\cdots\!24\)\( \beta_{6} - \)\(28\!\cdots\!04\)\( \beta_{7} + \)\(39\!\cdots\!15\)\( \beta_{8}) q^{33} +(-\)\(82\!\cdots\!56\)\( + \)\(16\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!48\)\( \beta_{2} - \)\(14\!\cdots\!72\)\( \beta_{3} + \)\(24\!\cdots\!96\)\( \beta_{4} + \)\(52\!\cdots\!48\)\( \beta_{5} + \)\(73\!\cdots\!44\)\( \beta_{6} + \)\(23\!\cdots\!40\)\( \beta_{7} + \)\(11\!\cdots\!92\)\( \beta_{8}) q^{34} +(-\)\(52\!\cdots\!00\)\( - \)\(11\!\cdots\!64\)\( \beta_{1} + \)\(78\!\cdots\!20\)\( \beta_{2} - \)\(50\!\cdots\!16\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(13\!\cdots\!20\)\( \beta_{5} + \)\(28\!\cdots\!40\)\( \beta_{6} - \)\(11\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!80\)\( \beta_{8}) q^{35} +(\)\(50\!\cdots\!56\)\( + \)\(98\!\cdots\!26\)\( \beta_{1} + \)\(13\!\cdots\!46\)\( \beta_{2} + \)\(26\!\cdots\!65\)\( \beta_{3} + \)\(19\!\cdots\!08\)\( \beta_{4} - \)\(75\!\cdots\!24\)\( \beta_{5} - \)\(22\!\cdots\!00\)\( \beta_{6} + \)\(35\!\cdots\!40\)\( \beta_{7} + \)\(93\!\cdots\!00\)\( \beta_{8}) q^{36} +(-\)\(38\!\cdots\!02\)\( - \)\(88\!\cdots\!45\)\( \beta_{1} + \)\(69\!\cdots\!05\)\( \beta_{2} + \)\(58\!\cdots\!42\)\( \beta_{3} + \)\(14\!\cdots\!55\)\( \beta_{4} + \)\(57\!\cdots\!10\)\( \beta_{5} + \)\(82\!\cdots\!56\)\( \beta_{6} + \)\(77\!\cdots\!68\)\( \beta_{7} - \)\(55\!\cdots\!50\)\( \beta_{8}) q^{37} +(\)\(80\!\cdots\!20\)\( - \)\(20\!\cdots\!52\)\( \beta_{1} - \)\(21\!\cdots\!26\)\( \beta_{2} + \)\(36\!\cdots\!80\)\( \beta_{3} + \)\(19\!\cdots\!26\)\( \beta_{4} + \)\(13\!\cdots\!33\)\( \beta_{5} - \)\(58\!\cdots\!44\)\( \beta_{6} - \)\(76\!\cdots\!76\)\( \beta_{7} + \)\(27\!\cdots\!60\)\( \beta_{8}) q^{38} +(-\)\(58\!\cdots\!24\)\( + \)\(66\!\cdots\!40\)\( \beta_{1} - \)\(86\!\cdots\!36\)\( \beta_{2} - \)\(28\!\cdots\!75\)\( \beta_{3} - \)\(54\!\cdots\!33\)\( \beta_{4} - \)\(15\!\cdots\!89\)\( \beta_{5} - \)\(10\!\cdots\!37\)\( \beta_{6} + \)\(43\!\cdots\!20\)\( \beta_{7} - \)\(11\!\cdots\!16\)\( \beta_{8}) q^{39} +(-\)\(44\!\cdots\!00\)\( - \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(34\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!40\)\( \beta_{3} + \)\(24\!\cdots\!80\)\( \beta_{4} + \)\(60\!\cdots\!50\)\( \beta_{5} + \)\(45\!\cdots\!50\)\( \beta_{6} - \)\(82\!\cdots\!50\)\( \beta_{7} + \)\(37\!\cdots\!00\)\( \beta_{8}) q^{40} +(\)\(11\!\cdots\!82\)\( + \)\(11\!\cdots\!24\)\( \beta_{1} - \)\(65\!\cdots\!12\)\( \beta_{2} + \)\(32\!\cdots\!42\)\( \beta_{3} + \)\(97\!\cdots\!02\)\( \beta_{4} + \)\(58\!\cdots\!94\)\( \beta_{5} + \)\(35\!\cdots\!00\)\( \beta_{6} - \)\(55\!\cdots\!40\)\( \beta_{7} - \)\(97\!\cdots\!50\)\( \beta_{8}) q^{41} +(\)\(28\!\cdots\!36\)\( + \)\(16\!\cdots\!08\)\( \beta_{1} + \)\(85\!\cdots\!08\)\( \beta_{2} + \)\(32\!\cdots\!88\)\( \beta_{3} - \)\(37\!\cdots\!28\)\( \beta_{4} - \)\(11\!\cdots\!44\)\( \beta_{5} - \)\(11\!\cdots\!84\)\( \beta_{6} + \)\(61\!\cdots\!60\)\( \beta_{7} + \)\(14\!\cdots\!20\)\( \beta_{8}) q^{42} +(\)\(56\!\cdots\!84\)\( + \)\(73\!\cdots\!12\)\( \beta_{1} + \)\(13\!\cdots\!07\)\( \beta_{2} - \)\(38\!\cdots\!68\)\( \beta_{3} - \)\(20\!\cdots\!16\)\( \beta_{4} + \)\(20\!\cdots\!32\)\( \beta_{5} + \)\(52\!\cdots\!00\)\( \beta_{6} - \)\(32\!\cdots\!76\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{43} +(-\)\(44\!\cdots\!96\)\( + \)\(34\!\cdots\!08\)\( \beta_{1} + \)\(52\!\cdots\!92\)\( \beta_{2} - \)\(55\!\cdots\!96\)\( \beta_{3} + \)\(28\!\cdots\!72\)\( \beta_{4} + \)\(10\!\cdots\!52\)\( \beta_{5} - \)\(54\!\cdots\!08\)\( \beta_{6} + \)\(11\!\cdots\!60\)\( \beta_{7} - \)\(17\!\cdots\!44\)\( \beta_{8}) q^{44} +(\)\(60\!\cdots\!50\)\( + \)\(10\!\cdots\!39\)\( \beta_{1} - \)\(26\!\cdots\!75\)\( \beta_{2} + \)\(57\!\cdots\!26\)\( \beta_{3} + \)\(31\!\cdots\!43\)\( \beta_{4} - \)\(52\!\cdots\!50\)\( \beta_{5} - \)\(53\!\cdots\!00\)\( \beta_{6} - \)\(27\!\cdots\!00\)\( \beta_{7} + \)\(56\!\cdots\!50\)\( \beta_{8}) q^{45} +(\)\(16\!\cdots\!92\)\( + \)\(30\!\cdots\!24\)\( \beta_{1} - \)\(88\!\cdots\!52\)\( \beta_{2} + \)\(44\!\cdots\!12\)\( \beta_{3} - \)\(84\!\cdots\!68\)\( \beta_{4} - \)\(31\!\cdots\!06\)\( \beta_{5} + \)\(29\!\cdots\!40\)\( \beta_{6} + \)\(42\!\cdots\!40\)\( \beta_{7} - \)\(79\!\cdots\!80\)\( \beta_{8}) q^{46} +(\)\(18\!\cdots\!68\)\( + \)\(99\!\cdots\!84\)\( \beta_{1} - \)\(22\!\cdots\!64\)\( \beta_{2} + \)\(32\!\cdots\!86\)\( \beta_{3} - \)\(28\!\cdots\!78\)\( \beta_{4} + \)\(73\!\cdots\!66\)\( \beta_{5} - \)\(55\!\cdots\!70\)\( \beta_{6} - \)\(80\!\cdots\!68\)\( \beta_{7} + \)\(80\!\cdots\!40\)\( \beta_{8}) q^{47} +(-\)\(24\!\cdots\!56\)\( + \)\(55\!\cdots\!68\)\( \beta_{1} + \)\(42\!\cdots\!16\)\( \beta_{2} - \)\(31\!\cdots\!72\)\( \beta_{3} + \)\(10\!\cdots\!52\)\( \beta_{4} - \)\(11\!\cdots\!84\)\( \beta_{5} - \)\(63\!\cdots\!04\)\( \beta_{6} + \)\(63\!\cdots\!80\)\( \beta_{7} - \)\(86\!\cdots\!00\)\( \beta_{8}) q^{48} +(\)\(18\!\cdots\!57\)\( + \)\(97\!\cdots\!36\)\( \beta_{1} + \)\(26\!\cdots\!32\)\( \beta_{2} - \)\(85\!\cdots\!72\)\( \beta_{3} + \)\(30\!\cdots\!68\)\( \beta_{4} - \)\(57\!\cdots\!24\)\( \beta_{5} + \)\(30\!\cdots\!40\)\( \beta_{6} - \)\(40\!\cdots\!40\)\( \beta_{7} + \)\(10\!\cdots\!20\)\( \beta_{8}) q^{49} +(-\)\(27\!\cdots\!00\)\( + \)\(87\!\cdots\!25\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!00\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5} + \)\(21\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!00\)\( \beta_{7} - \)\(59\!\cdots\!00\)\( \beta_{8}) q^{50} +(\)\(25\!\cdots\!92\)\( - \)\(45\!\cdots\!20\)\( \beta_{1} + \)\(24\!\cdots\!82\)\( \beta_{2} + \)\(20\!\cdots\!30\)\( \beta_{3} - \)\(12\!\cdots\!14\)\( \beta_{4} + \)\(34\!\cdots\!18\)\( \beta_{5} - \)\(19\!\cdots\!46\)\( \beta_{6} - \)\(18\!\cdots\!60\)\( \beta_{7} + \)\(18\!\cdots\!72\)\( \beta_{8}) q^{51} +(-\)\(40\!\cdots\!32\)\( - \)\(36\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(37\!\cdots\!06\)\( \beta_{3} + \)\(10\!\cdots\!72\)\( \beta_{4} - \)\(27\!\cdots\!44\)\( \beta_{5} + \)\(67\!\cdots\!00\)\( \beta_{6} - \)\(87\!\cdots\!08\)\( \beta_{7} - \)\(12\!\cdots\!00\)\( \beta_{8}) q^{52} +(-\)\(23\!\cdots\!46\)\( - \)\(47\!\cdots\!81\)\( \beta_{1} - \)\(24\!\cdots\!43\)\( \beta_{2} - \)\(22\!\cdots\!14\)\( \beta_{3} + \)\(18\!\cdots\!35\)\( \beta_{4} + \)\(18\!\cdots\!30\)\( \beta_{5} - \)\(12\!\cdots\!72\)\( \beta_{6} + \)\(63\!\cdots\!04\)\( \beta_{7} - \)\(17\!\cdots\!90\)\( \beta_{8}) q^{53} +(\)\(78\!\cdots\!40\)\( - \)\(57\!\cdots\!32\)\( \beta_{1} - \)\(36\!\cdots\!64\)\( \beta_{2} - \)\(36\!\cdots\!24\)\( \beta_{3} - \)\(10\!\cdots\!08\)\( \beta_{4} + \)\(18\!\cdots\!82\)\( \beta_{5} + \)\(75\!\cdots\!32\)\( \beta_{6} - \)\(19\!\cdots\!60\)\( \beta_{7} + \)\(10\!\cdots\!76\)\( \beta_{8}) q^{54} +(\)\(16\!\cdots\!00\)\( - \)\(55\!\cdots\!44\)\( \beta_{1} + \)\(41\!\cdots\!00\)\( \beta_{2} + \)\(77\!\cdots\!79\)\( \beta_{3} - \)\(16\!\cdots\!03\)\( \beta_{4} - \)\(17\!\cdots\!75\)\( \beta_{5} - \)\(15\!\cdots\!75\)\( \beta_{6} + \)\(25\!\cdots\!00\)\( \beta_{7} - \)\(31\!\cdots\!00\)\( \beta_{8}) q^{55} +(\)\(54\!\cdots\!40\)\( + \)\(31\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(65\!\cdots\!40\)\( \beta_{3} + \)\(99\!\cdots\!76\)\( \beta_{4} - \)\(23\!\cdots\!84\)\( \beta_{5} + \)\(21\!\cdots\!76\)\( \beta_{6} + \)\(45\!\cdots\!20\)\( \beta_{7} + \)\(31\!\cdots\!68\)\( \beta_{8}) q^{56} +(\)\(52\!\cdots\!60\)\( + \)\(47\!\cdots\!36\)\( \beta_{1} - \)\(75\!\cdots\!04\)\( \beta_{2} - \)\(65\!\cdots\!37\)\( \beta_{3} + \)\(86\!\cdots\!01\)\( \beta_{4} + \)\(85\!\cdots\!53\)\( \beta_{5} - \)\(84\!\cdots\!60\)\( \beta_{6} - \)\(29\!\cdots\!44\)\( \beta_{7} + \)\(17\!\cdots\!45\)\( \beta_{8}) q^{57} +(-\)\(68\!\cdots\!20\)\( + \)\(18\!\cdots\!82\)\( \beta_{1} - \)\(72\!\cdots\!00\)\( \beta_{2} - \)\(38\!\cdots\!76\)\( \beta_{3} - \)\(63\!\cdots\!76\)\( \beta_{4} - \)\(90\!\cdots\!48\)\( \beta_{5} + \)\(79\!\cdots\!36\)\( \beta_{6} + \)\(60\!\cdots\!92\)\( \beta_{7} - \)\(10\!\cdots\!80\)\( \beta_{8}) q^{58} +(-\)\(86\!\cdots\!80\)\( + \)\(39\!\cdots\!08\)\( \beta_{1} - \)\(15\!\cdots\!77\)\( \beta_{2} - \)\(27\!\cdots\!12\)\( \beta_{3} - \)\(11\!\cdots\!44\)\( \beta_{4} + \)\(87\!\cdots\!04\)\( \beta_{5} + \)\(47\!\cdots\!68\)\( \beta_{6} - \)\(18\!\cdots\!20\)\( \beta_{7} + \)\(27\!\cdots\!24\)\( \beta_{8}) q^{59} +(\)\(26\!\cdots\!00\)\( + \)\(20\!\cdots\!36\)\( \beta_{1} - \)\(71\!\cdots\!80\)\( \beta_{2} + \)\(35\!\cdots\!84\)\( \beta_{3} + \)\(71\!\cdots\!92\)\( \beta_{4} - \)\(14\!\cdots\!20\)\( \beta_{5} - \)\(20\!\cdots\!60\)\( \beta_{6} - \)\(10\!\cdots\!80\)\( \beta_{7} - \)\(18\!\cdots\!80\)\( \beta_{8}) q^{60} +(-\)\(24\!\cdots\!78\)\( - \)\(23\!\cdots\!81\)\( \beta_{1} + \)\(54\!\cdots\!37\)\( \beta_{2} - \)\(21\!\cdots\!02\)\( \beta_{3} - \)\(42\!\cdots\!37\)\( \beta_{4} + \)\(62\!\cdots\!02\)\( \beta_{5} + \)\(37\!\cdots\!84\)\( \beta_{6} - \)\(31\!\cdots\!80\)\( \beta_{7} - \)\(14\!\cdots\!38\)\( \beta_{8}) q^{61} +(\)\(20\!\cdots\!36\)\( + \)\(33\!\cdots\!96\)\( \beta_{1} + \)\(66\!\cdots\!52\)\( \beta_{2} + \)\(83\!\cdots\!48\)\( \beta_{3} - \)\(11\!\cdots\!08\)\( \beta_{4} - \)\(97\!\cdots\!04\)\( \beta_{5} - \)\(42\!\cdots\!64\)\( \beta_{6} + \)\(25\!\cdots\!40\)\( \beta_{7} + \)\(67\!\cdots\!40\)\( \beta_{8}) q^{62} +(-\)\(68\!\cdots\!96\)\( - \)\(26\!\cdots\!32\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} - \)\(56\!\cdots\!93\)\( \beta_{3} - \)\(96\!\cdots\!35\)\( \beta_{4} + \)\(27\!\cdots\!25\)\( \beta_{5} + \)\(13\!\cdots\!01\)\( \beta_{6} - \)\(38\!\cdots\!12\)\( \beta_{7} - \)\(13\!\cdots\!60\)\( \beta_{8}) q^{63} +(\)\(19\!\cdots\!72\)\( - \)\(53\!\cdots\!68\)\( \beta_{1} - \)\(24\!\cdots\!12\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3} + \)\(25\!\cdots\!36\)\( \beta_{4} - \)\(28\!\cdots\!88\)\( \beta_{5} - \)\(45\!\cdots\!80\)\( \beta_{6} - \)\(16\!\cdots\!80\)\( \beta_{7} - \)\(13\!\cdots\!40\)\( \beta_{8}) q^{64} +(\)\(14\!\cdots\!00\)\( - \)\(11\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(54\!\cdots\!02\)\( \beta_{3} + \)\(45\!\cdots\!26\)\( \beta_{4} + \)\(20\!\cdots\!90\)\( \beta_{5} + \)\(29\!\cdots\!20\)\( \beta_{6} + \)\(94\!\cdots\!60\)\( \beta_{7} + \)\(92\!\cdots\!10\)\( \beta_{8}) q^{65} +(-\)\(22\!\cdots\!16\)\( - \)\(34\!\cdots\!80\)\( \beta_{1} - \)\(16\!\cdots\!64\)\( \beta_{2} + \)\(65\!\cdots\!20\)\( \beta_{3} - \)\(20\!\cdots\!52\)\( \beta_{4} - \)\(36\!\cdots\!16\)\( \beta_{5} + \)\(50\!\cdots\!12\)\( \beta_{6} - \)\(18\!\cdots\!80\)\( \beta_{7} - \)\(28\!\cdots\!84\)\( \beta_{8}) q^{66} +(\)\(11\!\cdots\!08\)\( + \)\(32\!\cdots\!24\)\( \beta_{1} + \)\(34\!\cdots\!85\)\( \beta_{2} + \)\(10\!\cdots\!90\)\( \beta_{3} - \)\(26\!\cdots\!86\)\( \beta_{4} - \)\(36\!\cdots\!58\)\( \beta_{5} - \)\(16\!\cdots\!86\)\( \beta_{6} - \)\(67\!\cdots\!24\)\( \beta_{7} + \)\(31\!\cdots\!60\)\( \beta_{8}) q^{67} +(-\)\(27\!\cdots\!44\)\( + \)\(26\!\cdots\!12\)\( \beta_{1} + \)\(10\!\cdots\!36\)\( \beta_{2} - \)\(56\!\cdots\!86\)\( \beta_{3} + \)\(45\!\cdots\!80\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(14\!\cdots\!52\)\( \beta_{6} + \)\(12\!\cdots\!76\)\( \beta_{7} + \)\(64\!\cdots\!80\)\( \beta_{8}) q^{68} +(-\)\(70\!\cdots\!44\)\( + \)\(25\!\cdots\!72\)\( \beta_{1} - \)\(91\!\cdots\!40\)\( \beta_{2} - \)\(24\!\cdots\!58\)\( \beta_{3} - \)\(21\!\cdots\!02\)\( \beta_{4} - \)\(76\!\cdots\!42\)\( \beta_{5} + \)\(36\!\cdots\!48\)\( \beta_{6} - \)\(28\!\cdots\!00\)\( \beta_{7} - \)\(34\!\cdots\!86\)\( \beta_{8}) q^{69} +(\)\(27\!\cdots\!00\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} + \)\(48\!\cdots\!00\)\( \beta_{2} + \)\(21\!\cdots\!68\)\( \beta_{3} + \)\(37\!\cdots\!24\)\( \beta_{4} + \)\(83\!\cdots\!00\)\( \beta_{5} - \)\(83\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!00\)\( \beta_{7} + \)\(61\!\cdots\!00\)\( \beta_{8}) q^{70} +(\)\(16\!\cdots\!92\)\( + \)\(73\!\cdots\!32\)\( \beta_{1} - \)\(31\!\cdots\!44\)\( \beta_{2} + \)\(58\!\cdots\!19\)\( \beta_{3} - \)\(14\!\cdots\!91\)\( \beta_{4} - \)\(28\!\cdots\!79\)\( \beta_{5} - \)\(93\!\cdots\!83\)\( \beta_{6} + \)\(63\!\cdots\!00\)\( \beta_{7} + \)\(57\!\cdots\!56\)\( \beta_{8}) q^{71} +(-\)\(15\!\cdots\!60\)\( - \)\(43\!\cdots\!36\)\( \beta_{1} - \)\(32\!\cdots\!12\)\( \beta_{2} - \)\(14\!\cdots\!84\)\( \beta_{3} + \)\(75\!\cdots\!14\)\( \beta_{4} + \)\(80\!\cdots\!27\)\( \beta_{5} + \)\(34\!\cdots\!67\)\( \beta_{6} - \)\(12\!\cdots\!05\)\( \beta_{7} - \)\(27\!\cdots\!40\)\( \beta_{8}) q^{72} +(-\)\(40\!\cdots\!06\)\( - \)\(10\!\cdots\!56\)\( \beta_{1} + \)\(21\!\cdots\!76\)\( \beta_{2} - \)\(15\!\cdots\!05\)\( \beta_{3} - \)\(53\!\cdots\!19\)\( \beta_{4} - \)\(31\!\cdots\!27\)\( \beta_{5} + \)\(85\!\cdots\!56\)\( \beta_{6} + \)\(14\!\cdots\!04\)\( \beta_{7} + \)\(56\!\cdots\!85\)\( \beta_{8}) q^{73} +(\)\(21\!\cdots\!64\)\( - \)\(69\!\cdots\!82\)\( \beta_{1} + \)\(90\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(19\!\cdots\!80\)\( \beta_{4} - \)\(12\!\cdots\!44\)\( \beta_{5} - \)\(45\!\cdots\!76\)\( \beta_{6} - \)\(69\!\cdots\!00\)\( \beta_{7} - \)\(15\!\cdots\!68\)\( \beta_{8}) q^{74} +(-\)\(66\!\cdots\!00\)\( - \)\(58\!\cdots\!00\)\( \beta_{1} + \)\(31\!\cdots\!75\)\( \beta_{2} + \)\(86\!\cdots\!00\)\( \beta_{3} - \)\(48\!\cdots\!00\)\( \beta_{4} - \)\(19\!\cdots\!00\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6} + \)\(18\!\cdots\!00\)\( \beta_{7} - \)\(13\!\cdots\!00\)\( \beta_{8}) q^{75} +(\)\(83\!\cdots\!80\)\( - \)\(84\!\cdots\!92\)\( \beta_{1} - \)\(23\!\cdots\!56\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!92\)\( \beta_{4} + \)\(77\!\cdots\!52\)\( \beta_{5} + \)\(10\!\cdots\!92\)\( \beta_{6} - \)\(50\!\cdots\!80\)\( \beta_{7} + \)\(15\!\cdots\!56\)\( \beta_{8}) q^{76} +(-\)\(26\!\cdots\!64\)\( + \)\(94\!\cdots\!76\)\( \beta_{1} - \)\(92\!\cdots\!64\)\( \beta_{2} - \)\(20\!\cdots\!38\)\( \beta_{3} - \)\(37\!\cdots\!82\)\( \beta_{4} + \)\(19\!\cdots\!94\)\( \beta_{5} - \)\(26\!\cdots\!56\)\( \beta_{6} - \)\(70\!\cdots\!40\)\( \beta_{7} + \)\(44\!\cdots\!50\)\( \beta_{8}) q^{77} +(-\)\(16\!\cdots\!72\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} - \)\(18\!\cdots\!32\)\( \beta_{2} - \)\(52\!\cdots\!20\)\( \beta_{3} + \)\(44\!\cdots\!12\)\( \beta_{4} - \)\(89\!\cdots\!34\)\( \beta_{5} - \)\(47\!\cdots\!88\)\( \beta_{6} + \)\(32\!\cdots\!08\)\( \beta_{7} - \)\(93\!\cdots\!00\)\( \beta_{8}) q^{78} +(-\)\(25\!\cdots\!40\)\( + \)\(20\!\cdots\!28\)\( \beta_{1} - \)\(31\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!42\)\( \beta_{3} - \)\(54\!\cdots\!18\)\( \beta_{4} + \)\(66\!\cdots\!02\)\( \beta_{5} + \)\(30\!\cdots\!62\)\( \beta_{6} - \)\(72\!\cdots\!40\)\( \beta_{7} - \)\(28\!\cdots\!84\)\( \beta_{8}) q^{79} +(\)\(81\!\cdots\!00\)\( + \)\(97\!\cdots\!08\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(21\!\cdots\!72\)\( \beta_{3} - \)\(71\!\cdots\!04\)\( \beta_{4} + \)\(73\!\cdots\!00\)\( \beta_{5} + \)\(19\!\cdots\!00\)\( \beta_{6} + \)\(22\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{80} +(-\)\(39\!\cdots\!59\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} + \)\(26\!\cdots\!88\)\( \beta_{2} + \)\(74\!\cdots\!65\)\( \beta_{3} + \)\(34\!\cdots\!39\)\( \beta_{4} + \)\(25\!\cdots\!71\)\( \beta_{5} - \)\(57\!\cdots\!08\)\( \beta_{6} + \)\(17\!\cdots\!40\)\( \beta_{7} - \)\(17\!\cdots\!69\)\( \beta_{8}) q^{81} +(-\)\(28\!\cdots\!04\)\( - \)\(53\!\cdots\!22\)\( \beta_{1} - \)\(93\!\cdots\!08\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3} + \)\(34\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!40\)\( \beta_{5} - \)\(68\!\cdots\!84\)\( \beta_{6} - \)\(29\!\cdots\!12\)\( \beta_{7} - \)\(42\!\cdots\!80\)\( \beta_{8}) q^{82} +(-\)\(29\!\cdots\!36\)\( - \)\(17\!\cdots\!72\)\( \beta_{1} - \)\(28\!\cdots\!87\)\( \beta_{2} + \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(90\!\cdots\!40\)\( \beta_{4} - \)\(53\!\cdots\!20\)\( \beta_{5} + \)\(19\!\cdots\!80\)\( \beta_{6} + \)\(95\!\cdots\!80\)\( \beta_{7} + \)\(20\!\cdots\!40\)\( \beta_{8}) q^{83} +(-\)\(33\!\cdots\!16\)\( - \)\(66\!\cdots\!08\)\( \beta_{1} - \)\(26\!\cdots\!24\)\( \beta_{2} - \)\(17\!\cdots\!88\)\( \beta_{3} - \)\(35\!\cdots\!04\)\( \beta_{4} + \)\(49\!\cdots\!72\)\( \beta_{5} - \)\(13\!\cdots\!80\)\( \beta_{6} - \)\(54\!\cdots\!40\)\( \beta_{7} - \)\(28\!\cdots\!40\)\( \beta_{8}) q^{84} +(-\)\(30\!\cdots\!00\)\( - \)\(43\!\cdots\!94\)\( \beta_{1} - \)\(86\!\cdots\!30\)\( \beta_{2} + \)\(16\!\cdots\!14\)\( \beta_{3} + \)\(12\!\cdots\!32\)\( \beta_{4} - \)\(39\!\cdots\!70\)\( \beta_{5} - \)\(11\!\cdots\!60\)\( \beta_{6} + \)\(30\!\cdots\!20\)\( \beta_{7} - \)\(25\!\cdots\!30\)\( \beta_{8}) q^{85} +(-\)\(17\!\cdots\!88\)\( + \)\(44\!\cdots\!36\)\( \beta_{1} + \)\(66\!\cdots\!74\)\( \beta_{2} + \)\(22\!\cdots\!68\)\( \beta_{3} - \)\(24\!\cdots\!26\)\( \beta_{4} + \)\(29\!\cdots\!19\)\( \beta_{5} - \)\(14\!\cdots\!16\)\( \beta_{6} - \)\(27\!\cdots\!00\)\( \beta_{7} + \)\(17\!\cdots\!12\)\( \beta_{8}) q^{86} +(-\)\(15\!\cdots\!60\)\( + \)\(25\!\cdots\!00\)\( \beta_{1} + \)\(66\!\cdots\!96\)\( \beta_{2} + \)\(93\!\cdots\!41\)\( \beta_{3} + \)\(21\!\cdots\!63\)\( \beta_{4} - \)\(36\!\cdots\!61\)\( \beta_{5} + \)\(53\!\cdots\!31\)\( \beta_{6} - \)\(15\!\cdots\!44\)\( \beta_{7} - \)\(29\!\cdots\!20\)\( \beta_{8}) q^{87} +(-\)\(59\!\cdots\!40\)\( + \)\(88\!\cdots\!40\)\( \beta_{1} + \)\(31\!\cdots\!88\)\( \beta_{2} - \)\(28\!\cdots\!92\)\( \beta_{3} - \)\(30\!\cdots\!08\)\( \beta_{4} + \)\(62\!\cdots\!76\)\( \beta_{5} - \)\(10\!\cdots\!64\)\( \beta_{6} + \)\(49\!\cdots\!40\)\( \beta_{7} + \)\(22\!\cdots\!60\)\( \beta_{8}) q^{88} +(-\)\(48\!\cdots\!70\)\( + \)\(74\!\cdots\!40\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(60\!\cdots\!99\)\( \beta_{3} - \)\(12\!\cdots\!59\)\( \beta_{4} + \)\(76\!\cdots\!33\)\( \beta_{5} - \)\(24\!\cdots\!76\)\( \beta_{6} - \)\(32\!\cdots\!60\)\( \beta_{7} + \)\(93\!\cdots\!57\)\( \beta_{8}) q^{89} +(-\)\(26\!\cdots\!00\)\( - \)\(84\!\cdots\!82\)\( \beta_{1} - \)\(18\!\cdots\!40\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3} - \)\(40\!\cdots\!04\)\( \beta_{4} - \)\(25\!\cdots\!60\)\( \beta_{5} + \)\(43\!\cdots\!20\)\( \beta_{6} - \)\(11\!\cdots\!40\)\( \beta_{7} - \)\(20\!\cdots\!40\)\( \beta_{8}) q^{90} +(-\)\(19\!\cdots\!28\)\( - \)\(74\!\cdots\!36\)\( \beta_{1} - \)\(44\!\cdots\!68\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(11\!\cdots\!16\)\( \beta_{4} + \)\(34\!\cdots\!96\)\( \beta_{5} - \)\(71\!\cdots\!64\)\( \beta_{6} + \)\(25\!\cdots\!80\)\( \beta_{7} + \)\(18\!\cdots\!48\)\( \beta_{8}) q^{91} +(-\)\(61\!\cdots\!92\)\( - \)\(51\!\cdots\!72\)\( \beta_{1} - \)\(11\!\cdots\!56\)\( \beta_{2} - \)\(31\!\cdots\!48\)\( \beta_{3} + \)\(44\!\cdots\!40\)\( \beta_{4} - \)\(15\!\cdots\!60\)\( \beta_{5} - \)\(33\!\cdots\!04\)\( \beta_{6} - \)\(72\!\cdots\!52\)\( \beta_{7} + \)\(17\!\cdots\!00\)\( \beta_{8}) q^{92} +(-\)\(12\!\cdots\!52\)\( - \)\(26\!\cdots\!60\)\( \beta_{1} + \)\(30\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!12\)\( \beta_{4} + \)\(19\!\cdots\!24\)\( \beta_{5} - \)\(20\!\cdots\!32\)\( \beta_{6} - \)\(31\!\cdots\!68\)\( \beta_{7} - \)\(96\!\cdots\!40\)\( \beta_{8}) q^{93} +(-\)\(23\!\cdots\!76\)\( - \)\(85\!\cdots\!72\)\( \beta_{1} + \)\(13\!\cdots\!96\)\( \beta_{2} - \)\(85\!\cdots\!44\)\( \beta_{3} + \)\(45\!\cdots\!12\)\( \beta_{4} - \)\(19\!\cdots\!28\)\( \beta_{5} - \)\(12\!\cdots\!88\)\( \beta_{6} + \)\(69\!\cdots\!20\)\( \beta_{7} + \)\(18\!\cdots\!16\)\( \beta_{8}) q^{94} +(-\)\(40\!\cdots\!00\)\( - \)\(74\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(69\!\cdots\!45\)\( \beta_{3} - \)\(20\!\cdots\!65\)\( \beta_{4} - \)\(36\!\cdots\!25\)\( \beta_{5} + \)\(88\!\cdots\!75\)\( \beta_{6} + \)\(57\!\cdots\!00\)\( \beta_{7} - \)\(15\!\cdots\!00\)\( \beta_{8}) q^{95} +(-\)\(77\!\cdots\!48\)\( + \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(31\!\cdots\!52\)\( \beta_{2} - \)\(14\!\cdots\!40\)\( \beta_{3} + \)\(65\!\cdots\!44\)\( \beta_{4} + \)\(31\!\cdots\!08\)\( \beta_{5} - \)\(10\!\cdots\!20\)\( \beta_{6} + \)\(57\!\cdots\!40\)\( \beta_{7} - \)\(25\!\cdots\!60\)\( \beta_{8}) q^{96} +(-\)\(33\!\cdots\!82\)\( + \)\(12\!\cdots\!96\)\( \beta_{1} - \)\(37\!\cdots\!48\)\( \beta_{2} + \)\(68\!\cdots\!51\)\( \beta_{3} + \)\(29\!\cdots\!25\)\( \beta_{4} + \)\(75\!\cdots\!65\)\( \beta_{5} - \)\(11\!\cdots\!52\)\( \beta_{6} - \)\(21\!\cdots\!96\)\( \beta_{7} + \)\(10\!\cdots\!45\)\( \beta_{8}) q^{97} +(-\)\(23\!\cdots\!04\)\( + \)\(12\!\cdots\!03\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2} - \)\(56\!\cdots\!92\)\( \beta_{3} - \)\(11\!\cdots\!20\)\( \beta_{4} + \)\(21\!\cdots\!80\)\( \beta_{5} + \)\(26\!\cdots\!44\)\( \beta_{6} - \)\(20\!\cdots\!08\)\( \beta_{7} - \)\(14\!\cdots\!60\)\( \beta_{8}) q^{98} +(-\)\(16\!\cdots\!24\)\( + \)\(28\!\cdots\!64\)\( \beta_{1} + \)\(27\!\cdots\!87\)\( \beta_{2} + \)\(67\!\cdots\!60\)\( \beta_{3} - \)\(77\!\cdots\!24\)\( \beta_{4} - \)\(89\!\cdots\!04\)\( \beta_{5} + \)\(22\!\cdots\!16\)\( \beta_{6} - \)\(26\!\cdots\!60\)\( \beta_{7} - \)\(12\!\cdots\!12\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 40417036685158752q^{2} + \)\(10\!\cdots\!36\)\(q^{3} + \)\(67\!\cdots\!88\)\(q^{4} + \)\(38\!\cdots\!50\)\(q^{5} - \)\(48\!\cdots\!52\)\(q^{6} - \)\(33\!\cdots\!08\)\(q^{7} + \)\(84\!\cdots\!20\)\(q^{8} + \)\(22\!\cdots\!97\)\(q^{9} + O(q^{10}) \) \( 9q + 40417036685158752q^{2} + \)\(10\!\cdots\!36\)\(q^{3} + \)\(67\!\cdots\!88\)\(q^{4} + \)\(38\!\cdots\!50\)\(q^{5} - \)\(48\!\cdots\!52\)\(q^{6} - \)\(33\!\cdots\!08\)\(q^{7} + \)\(84\!\cdots\!20\)\(q^{8} + \)\(22\!\cdots\!97\)\(q^{9} - \)\(11\!\cdots\!00\)\(q^{10} - \)\(18\!\cdots\!52\)\(q^{11} + \)\(32\!\cdots\!52\)\(q^{12} - \)\(98\!\cdots\!34\)\(q^{13} + \)\(35\!\cdots\!56\)\(q^{14} - \)\(91\!\cdots\!00\)\(q^{15} + \)\(61\!\cdots\!24\)\(q^{16} - \)\(68\!\cdots\!78\)\(q^{17} - \)\(41\!\cdots\!84\)\(q^{18} - \)\(16\!\cdots\!40\)\(q^{19} - \)\(22\!\cdots\!00\)\(q^{20} - \)\(32\!\cdots\!92\)\(q^{21} - \)\(13\!\cdots\!56\)\(q^{22} - \)\(58\!\cdots\!04\)\(q^{23} - \)\(30\!\cdots\!20\)\(q^{24} + \)\(55\!\cdots\!75\)\(q^{25} + \)\(56\!\cdots\!88\)\(q^{26} + \)\(18\!\cdots\!20\)\(q^{27} - \)\(64\!\cdots\!56\)\(q^{28} - \)\(48\!\cdots\!10\)\(q^{29} - \)\(93\!\cdots\!00\)\(q^{30} - \)\(62\!\cdots\!92\)\(q^{31} + \)\(50\!\cdots\!32\)\(q^{32} + \)\(12\!\cdots\!92\)\(q^{33} - \)\(73\!\cdots\!04\)\(q^{34} - \)\(47\!\cdots\!00\)\(q^{35} + \)\(45\!\cdots\!04\)\(q^{36} - \)\(34\!\cdots\!18\)\(q^{37} + \)\(72\!\cdots\!80\)\(q^{38} - \)\(52\!\cdots\!16\)\(q^{39} - \)\(39\!\cdots\!00\)\(q^{40} + \)\(10\!\cdots\!38\)\(q^{41} + \)\(25\!\cdots\!24\)\(q^{42} + \)\(50\!\cdots\!56\)\(q^{43} - \)\(40\!\cdots\!64\)\(q^{44} + \)\(54\!\cdots\!50\)\(q^{45} + \)\(14\!\cdots\!28\)\(q^{46} + \)\(16\!\cdots\!12\)\(q^{47} - \)\(21\!\cdots\!04\)\(q^{48} + \)\(16\!\cdots\!13\)\(q^{49} - \)\(24\!\cdots\!00\)\(q^{50} + \)\(22\!\cdots\!28\)\(q^{51} - \)\(36\!\cdots\!88\)\(q^{52} - \)\(21\!\cdots\!14\)\(q^{53} + \)\(70\!\cdots\!60\)\(q^{54} + \)\(14\!\cdots\!00\)\(q^{55} + \)\(49\!\cdots\!60\)\(q^{56} + \)\(47\!\cdots\!40\)\(q^{57} - \)\(61\!\cdots\!80\)\(q^{58} - \)\(77\!\cdots\!20\)\(q^{59} + \)\(24\!\cdots\!00\)\(q^{60} - \)\(22\!\cdots\!02\)\(q^{61} + \)\(18\!\cdots\!24\)\(q^{62} - \)\(61\!\cdots\!64\)\(q^{63} + \)\(17\!\cdots\!48\)\(q^{64} + \)\(12\!\cdots\!00\)\(q^{65} - \)\(19\!\cdots\!44\)\(q^{66} + \)\(10\!\cdots\!72\)\(q^{67} - \)\(25\!\cdots\!96\)\(q^{68} - \)\(63\!\cdots\!96\)\(q^{69} + \)\(24\!\cdots\!00\)\(q^{70} + \)\(15\!\cdots\!28\)\(q^{71} - \)\(14\!\cdots\!40\)\(q^{72} - \)\(36\!\cdots\!54\)\(q^{73} + \)\(18\!\cdots\!76\)\(q^{74} - \)\(59\!\cdots\!00\)\(q^{75} + \)\(75\!\cdots\!20\)\(q^{76} - \)\(24\!\cdots\!76\)\(q^{77} - \)\(14\!\cdots\!48\)\(q^{78} - \)\(22\!\cdots\!60\)\(q^{79} + \)\(73\!\cdots\!00\)\(q^{80} - \)\(35\!\cdots\!31\)\(q^{81} - \)\(25\!\cdots\!36\)\(q^{82} - \)\(26\!\cdots\!24\)\(q^{83} - \)\(30\!\cdots\!44\)\(q^{84} - \)\(27\!\cdots\!00\)\(q^{85} - \)\(15\!\cdots\!92\)\(q^{86} - \)\(13\!\cdots\!40\)\(q^{87} - \)\(53\!\cdots\!60\)\(q^{88} - \)\(43\!\cdots\!30\)\(q^{89} - \)\(23\!\cdots\!00\)\(q^{90} - \)\(17\!\cdots\!52\)\(q^{91} - \)\(55\!\cdots\!28\)\(q^{92} - \)\(11\!\cdots\!68\)\(q^{93} - \)\(21\!\cdots\!84\)\(q^{94} - \)\(36\!\cdots\!00\)\(q^{95} - \)\(70\!\cdots\!32\)\(q^{96} - \)\(29\!\cdots\!38\)\(q^{97} - \)\(21\!\cdots\!36\)\(q^{98} - \)\(15\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - \)\(13\!\cdots\!64\)\( x^{7} + \)\(96\!\cdots\!96\)\( x^{6} + \)\(53\!\cdots\!46\)\( x^{5} - \)\(78\!\cdots\!08\)\( x^{4} - \)\(72\!\cdots\!48\)\( x^{3} + \)\(22\!\cdots\!12\)\( x^{2} + \)\(24\!\cdots\!17\)\( x - \)\(93\!\cdots\!48\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 288 \nu - 128 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(15\!\cdots\!87\)\( \nu^{8} - \)\(78\!\cdots\!23\)\( \nu^{7} - \)\(35\!\cdots\!45\)\( \nu^{6} + \)\(11\!\cdots\!21\)\( \nu^{5} + \)\(25\!\cdots\!97\)\( \nu^{4} - \)\(47\!\cdots\!61\)\( \nu^{3} - \)\(57\!\cdots\!43\)\( \nu^{2} + \)\(39\!\cdots\!27\)\( \nu + \)\(12\!\cdots\!40\)\(\)\()/ \)\(21\!\cdots\!16\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(25\!\cdots\!87\)\( \nu^{8} + \)\(12\!\cdots\!23\)\( \nu^{7} + \)\(58\!\cdots\!45\)\( \nu^{6} - \)\(19\!\cdots\!21\)\( \nu^{5} - \)\(41\!\cdots\!97\)\( \nu^{4} + \)\(78\!\cdots\!61\)\( \nu^{3} + \)\(18\!\cdots\!95\)\( \nu^{2} - \)\(54\!\cdots\!35\)\( \nu - \)\(27\!\cdots\!24\)\(\)\()/ \)\(10\!\cdots\!08\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(21\!\cdots\!11\)\( \nu^{8} + \)\(99\!\cdots\!29\)\( \nu^{7} - \)\(26\!\cdots\!57\)\( \nu^{6} - \)\(10\!\cdots\!95\)\( \nu^{5} + \)\(98\!\cdots\!21\)\( \nu^{4} + \)\(32\!\cdots\!15\)\( \nu^{3} - \)\(11\!\cdots\!83\)\( \nu^{2} - \)\(16\!\cdots\!37\)\( \nu + \)\(27\!\cdots\!96\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(82\!\cdots\!23\)\( \nu^{8} + \)\(63\!\cdots\!97\)\( \nu^{7} - \)\(10\!\cdots\!01\)\( \nu^{6} - \)\(69\!\cdots\!35\)\( \nu^{5} + \)\(40\!\cdots\!53\)\( \nu^{4} + \)\(20\!\cdots\!95\)\( \nu^{3} - \)\(41\!\cdots\!19\)\( \nu^{2} - \)\(63\!\cdots\!41\)\( \nu + \)\(54\!\cdots\!28\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(83\!\cdots\!21\)\( \nu^{8} + \)\(41\!\cdots\!19\)\( \nu^{7} - \)\(10\!\cdots\!27\)\( \nu^{6} - \)\(46\!\cdots\!45\)\( \nu^{5} + \)\(43\!\cdots\!31\)\( \nu^{4} + \)\(15\!\cdots\!65\)\( \nu^{3} - \)\(55\!\cdots\!13\)\( \nu^{2} - \)\(95\!\cdots\!07\)\( \nu + \)\(16\!\cdots\!56\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(29\!\cdots\!93\)\( \nu^{8} + \)\(23\!\cdots\!27\)\( \nu^{7} - \)\(36\!\cdots\!91\)\( \nu^{6} - \)\(29\!\cdots\!85\)\( \nu^{5} + \)\(13\!\cdots\!23\)\( \nu^{4} + \)\(12\!\cdots\!45\)\( \nu^{3} - \)\(13\!\cdots\!29\)\( \nu^{2} - \)\(14\!\cdots\!31\)\( \nu + \)\(43\!\cdots\!48\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(75\!\cdots\!27\)\( \nu^{8} + \)\(98\!\cdots\!53\)\( \nu^{7} - \)\(97\!\cdots\!49\)\( \nu^{6} - \)\(10\!\cdots\!15\)\( \nu^{5} + \)\(38\!\cdots\!97\)\( \nu^{4} + \)\(33\!\cdots\!55\)\( \nu^{3} - \)\(49\!\cdots\!31\)\( \nu^{2} - \)\(19\!\cdots\!09\)\( \nu + \)\(15\!\cdots\!72\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 128\)\()/288\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3291802 \beta_{2} - 31697913853565898 \beta_{1} + 241647385072183322446125411887407104\)\()/82944\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 323 \beta_{6} - 348895 \beta_{5} - 13791262570 \beta_{4} - 30333157780248308 \beta_{3} + 3453919534744843934634612 \beta_{2} + 411937983751109082066853391940780016 \beta_{1} - 7659717995331047826588782256951221276922315975426048\)\()/23887872\)
\(\nu^{4}\)\(=\)\((\)\(2935952636928 \beta_{8} - 6173690894898543 \beta_{7} + 5271382138869910285 \beta_{6} - 175925030717144398461519 \beta_{5} + 505510440680618600979942646 \beta_{4} + 17825879827662828203660365354479596 \beta_{3} + 43705305852150727761155612929278408101620 \beta_{2} - 672651647482282139055958709904687902256114887853648 \beta_{1} + 3110741768280347164138266563663772046755894763808999805634506899324928\)\()/ 214990848 \)
\(\nu^{5}\)\(=\)\((\)\(16117974269151869520488263680 \beta_{8} + 316648969337027161375748907209761 \beta_{7} - 185910836485464434316604712146855331 \beta_{6} + 324478170954017801218863684552889374017 \beta_{5} + 15067640679744040387565609726048703977923414 \beta_{4} - 28750832803688675086637561894187138017280563877300 \beta_{3} + 1534486564822527537410674011578631123063053364871115303540 \beta_{2} + 95804667762395776902642697290127011197893083981071037632012294274672 \beta_{1} - 2539757995055448734842476481936024965671589449166256767371755840451543396379725266944\)\()/ 967458816 \)
\(\nu^{6}\)\(=\)\((\)\(\)\(13\!\cdots\!80\)\( \beta_{8} - \)\(30\!\cdots\!13\)\( \beta_{7} + \)\(15\!\cdots\!63\)\( \beta_{6} - \)\(83\!\cdots\!53\)\( \beta_{5} + \)\(68\!\cdots\!82\)\( \beta_{4} + \)\(52\!\cdots\!16\)\( \beta_{3} + \)\(68\!\cdots\!00\)\( \beta_{2} - \)\(29\!\cdots\!04\)\( \beta_{1} + \)\(80\!\cdots\!16\)\(\)\()/ 967458816 \)
\(\nu^{7}\)\(=\)\((\)\(\)\(52\!\cdots\!80\)\( \beta_{8} + \)\(44\!\cdots\!87\)\( \beta_{7} - \)\(34\!\cdots\!25\)\( \beta_{6} + \)\(92\!\cdots\!35\)\( \beta_{5} + \)\(34\!\cdots\!14\)\( \beta_{4} - \)\(61\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!64\)\( \beta_{2} + \)\(11\!\cdots\!48\)\( \beta_{1} - \)\(49\!\cdots\!68\)\(\)\()/ 1934917632 \)
\(\nu^{8}\)\(=\)\((\)\(\)\(65\!\cdots\!76\)\( \beta_{8} - \)\(17\!\cdots\!91\)\( \beta_{7} + \)\(56\!\cdots\!09\)\( \beta_{6} - \)\(45\!\cdots\!67\)\( \beta_{5} + \)\(51\!\cdots\!26\)\( \beta_{4} + \)\(22\!\cdots\!28\)\( \beta_{3} + \)\(75\!\cdots\!04\)\( \beta_{2} - \)\(18\!\cdots\!40\)\( \beta_{1} + \)\(32\!\cdots\!24\)\(\)\()/ 644972544 \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.49518e15
2.13220e15
1.10803e15
6.27063e14
4.58648e14
−7.00565e14
−1.52687e15
−1.94766e15
−2.64602e15
−7.14121e17 1.43031e28 3.43815e35 7.63780e40 −1.02141e46 −4.21519e49 −1.26872e53 1.38022e56 −5.45431e58
1.2 −6.09583e17 −4.19687e27 2.05438e35 −4.17042e40 2.55834e45 1.99165e49 −2.39470e52 −4.89422e55 2.54222e58
1.3 −3.14621e17 −1.19854e28 −6.71671e34 1.37379e41 3.77084e45 −3.15157e49 7.34076e52 7.70927e55 −4.32224e58
1.4 −1.76103e17 5.15707e27 −1.35141e35 −8.41896e40 −9.08178e44 −3.65434e49 5.30590e52 −3.99606e55 1.48261e58
1.5 −1.27600e17 8.60094e27 −1.49872e35 5.94769e40 −1.09748e45 4.46500e49 4.03248e52 7.42032e54 −7.58924e57
1.6 2.06254e17 −1.02253e28 −1.23613e35 −6.31988e40 −2.10900e45 1.11715e49 −5.97654e52 3.80005e55 −1.30350e58
1.7 4.44230e17 2.03993e27 3.11870e34 7.97097e40 9.06197e44 −1.30568e49 −5.99562e52 −6.23946e55 3.54095e58
1.8 5.65415e17 1.40368e28 1.53541e35 −1.09678e41 7.93660e45 1.38759e48 −7.13120e51 1.30475e56 −6.20137e58
1.9 7.66546e17 −7.42736e27 4.21439e35 −1.57195e40 −5.69341e45 1.22056e49 1.95688e53 −1.13902e55 −1.20497e58
\(n\): 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 \(S_{118}^{\mathrm{new}}(\Gamma_0(1))\).