Properties

Label 1.114.a.a
Level 1
Weight 114
Character orbit 1.a
Self dual Yes
Analytic conductor 80.863
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 114 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(80.8627478904\)
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 \( 2^{144}\cdot 3^{48}\cdot 5^{19}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 19^{3}\cdot 23 \)
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.

\(f(q)\) \(=\) \( q\) \(+(552636039763781 + \beta_{1}) q^{2}\) \(+(-\)\(11\!\cdots\!03\)\( - 53786200 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(63\!\cdots\!76\)\( - 7521194705788522 \beta_{1} + 233291 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(34\!\cdots\!43\)\( - \)\(10\!\cdots\!25\)\( \beta_{1} - 149841320469 \beta_{2} - 84572 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(96\!\cdots\!48\)\( - \)\(10\!\cdots\!51\)\( \beta_{1} + 3078932092137945 \beta_{2} + 522089023 \beta_{3} - 13268 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(19\!\cdots\!47\)\( + \)\(15\!\cdots\!31\)\( \beta_{1} - \)\(26\!\cdots\!43\)\( \beta_{2} - 8203997863513 \beta_{3} - 19535798 \beta_{4} - 1095 \beta_{5} - \beta_{6}) q^{7}\) \(+(-\)\(12\!\cdots\!17\)\( + \)\(46\!\cdots\!51\)\( \beta_{1} + \)\(94\!\cdots\!50\)\( \beta_{2} - 15716006786227561 \beta_{3} - 49231023484 \beta_{4} + 1827020 \beta_{5} + 54 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(61\!\cdots\!67\)\( + \)\(14\!\cdots\!41\)\( \beta_{1} - \)\(26\!\cdots\!17\)\( \beta_{2} - 111141217798568264 \beta_{3} - 1279755571357 \beta_{4} + 352669740 \beta_{5} - 209767 \beta_{6} + 93 \beta_{7} - \beta_{8}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(552636039763781 + \beta_{1}) q^{2}\) \(+(-\)\(11\!\cdots\!03\)\( - 53786200 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(63\!\cdots\!76\)\( - 7521194705788522 \beta_{1} + 233291 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(34\!\cdots\!43\)\( - \)\(10\!\cdots\!25\)\( \beta_{1} - 149841320469 \beta_{2} - 84572 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(96\!\cdots\!48\)\( - \)\(10\!\cdots\!51\)\( \beta_{1} + 3078932092137945 \beta_{2} + 522089023 \beta_{3} - 13268 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(19\!\cdots\!47\)\( + \)\(15\!\cdots\!31\)\( \beta_{1} - \)\(26\!\cdots\!43\)\( \beta_{2} - 8203997863513 \beta_{3} - 19535798 \beta_{4} - 1095 \beta_{5} - \beta_{6}) q^{7}\) \(+(-\)\(12\!\cdots\!17\)\( + \)\(46\!\cdots\!51\)\( \beta_{1} + \)\(94\!\cdots\!50\)\( \beta_{2} - 15716006786227561 \beta_{3} - 49231023484 \beta_{4} + 1827020 \beta_{5} + 54 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(61\!\cdots\!67\)\( + \)\(14\!\cdots\!41\)\( \beta_{1} - \)\(26\!\cdots\!17\)\( \beta_{2} - 111141217798568264 \beta_{3} - 1279755571357 \beta_{4} + 352669740 \beta_{5} - 209767 \beta_{6} + 93 \beta_{7} - \beta_{8}) q^{9}\) \(+(-\)\(17\!\cdots\!46\)\( - \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(35\!\cdots\!56\)\( \beta_{3} + 11752642414485088 \beta_{4} - 598794841860 \beta_{5} + 172191680 \beta_{6} - 273240 \beta_{7} + 720 \beta_{8}) q^{10}\) \(+(\)\(13\!\cdots\!29\)\( - \)\(20\!\cdots\!34\)\( \beta_{1} - \)\(25\!\cdots\!95\)\( \beta_{2} + \)\(10\!\cdots\!66\)\( \beta_{3} + 3016148434268673000 \beta_{4} - 13973036603174 \beta_{5} + 10347763338 \beta_{6} - 36310252 \beta_{7} - 115236 \beta_{8}) q^{11}\) \(+(-\)\(52\!\cdots\!16\)\( + \)\(49\!\cdots\!08\)\( \beta_{1} + \)\(62\!\cdots\!04\)\( \beta_{2} + \)\(81\!\cdots\!56\)\( \beta_{3} + \)\(32\!\cdots\!44\)\( \beta_{4} + 4905906456797280 \beta_{5} + 15568162228656 \beta_{6} + 700932744 \beta_{7} + 9734400 \beta_{8}) q^{12}\) \(+(\)\(13\!\cdots\!93\)\( + \)\(18\!\cdots\!65\)\( \beta_{1} + \)\(11\!\cdots\!41\)\( \beta_{2} + \)\(61\!\cdots\!48\)\( \beta_{3} + \)\(49\!\cdots\!41\)\( \beta_{4} - 1705651472282584440 \beta_{5} + 804733821555910 \beta_{6} + 374426482014 \beta_{7} - 538082550 \beta_{8}) q^{13}\) \(+(\)\(25\!\cdots\!48\)\( - \)\(94\!\cdots\!42\)\( \beta_{1} - \)\(13\!\cdots\!58\)\( \beta_{2} - \)\(11\!\cdots\!82\)\( \beta_{3} + \)\(80\!\cdots\!52\)\( \beta_{4} - \)\(61\!\cdots\!46\)\( \beta_{5} - 52162133780342016 \beta_{6} - 34620392506336 \beta_{7} + 21291826752 \beta_{8}) q^{14}\) \(+(-\)\(89\!\cdots\!53\)\( + \)\(95\!\cdots\!85\)\( \beta_{1} - \)\(29\!\cdots\!69\)\( \beta_{2} - \)\(26\!\cdots\!67\)\( \beta_{3} + \)\(37\!\cdots\!34\)\( \beta_{4} - \)\(19\!\cdots\!05\)\( \beta_{5} - 928742678667824535 \beta_{6} + 1551521939483880 \beta_{7} - 624217733640 \beta_{8}) q^{15}\) \(+(\)\(12\!\cdots\!24\)\( - \)\(22\!\cdots\!60\)\( \beta_{1} + \)\(32\!\cdots\!64\)\( \beta_{2} + \)\(31\!\cdots\!64\)\( \beta_{3} + \)\(18\!\cdots\!36\)\( \beta_{4} + \)\(34\!\cdots\!04\)\( \beta_{5} + \)\(11\!\cdots\!28\)\( \beta_{6} - 42031723946208912 \beta_{7} + 13320235229184 \beta_{8}) q^{16}\) \(+(\)\(11\!\cdots\!28\)\( + \)\(21\!\cdots\!13\)\( \beta_{1} - \)\(51\!\cdots\!21\)\( \beta_{2} + \)\(46\!\cdots\!48\)\( \beta_{3} + \)\(52\!\cdots\!03\)\( \beta_{4} + \)\(19\!\cdots\!20\)\( \beta_{5} - \)\(27\!\cdots\!19\)\( \beta_{6} + 644440847452975185 \beta_{7} - 176948799905925 \beta_{8}) q^{17}\) \(+(\)\(24\!\cdots\!41\)\( + \)\(55\!\cdots\!93\)\( \beta_{1} + \)\(44\!\cdots\!28\)\( \beta_{2} + \)\(78\!\cdots\!88\)\( \beta_{3} + \)\(23\!\cdots\!08\)\( \beta_{4} - \)\(31\!\cdots\!80\)\( \beta_{5} + \)\(10\!\cdots\!56\)\( \beta_{6} - 184252463709552720 \beta_{7} - 212945718813600 \beta_{8}) q^{18}\) \(+(-\)\(22\!\cdots\!65\)\( - \)\(53\!\cdots\!46\)\( \beta_{1} + \)\(22\!\cdots\!79\)\( \beta_{2} - \)\(37\!\cdots\!74\)\( \beta_{3} + \)\(77\!\cdots\!20\)\( \beta_{4} - \)\(63\!\cdots\!58\)\( \beta_{5} + \)\(10\!\cdots\!66\)\( \beta_{6} - \)\(29\!\cdots\!64\)\( \beta_{7} + 93757583067879348 \beta_{8}) q^{19}\) \(+(-\)\(14\!\cdots\!76\)\( + \)\(35\!\cdots\!00\)\( \beta_{1} - \)\(84\!\cdots\!58\)\( \beta_{2} - \)\(24\!\cdots\!54\)\( \beta_{3} - \)\(10\!\cdots\!32\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5} - \)\(29\!\cdots\!00\)\( \beta_{6} + \)\(99\!\cdots\!00\)\( \beta_{7} - 3314985135660057600 \beta_{8}) q^{20}\) \(+(-\)\(22\!\cdots\!04\)\( - \)\(69\!\cdots\!82\)\( \beta_{1} + \)\(88\!\cdots\!54\)\( \beta_{2} - \)\(35\!\cdots\!68\)\( \beta_{3} - \)\(57\!\cdots\!94\)\( \beta_{4} - \)\(35\!\cdots\!68\)\( \beta_{5} + \)\(39\!\cdots\!62\)\( \beta_{6} - \)\(20\!\cdots\!98\)\( \beta_{7} + 79592770128386681586 \beta_{8}) q^{21}\) \(+(-\)\(34\!\cdots\!48\)\( + \)\(24\!\cdots\!95\)\( \beta_{1} - \)\(79\!\cdots\!61\)\( \beta_{2} - \)\(54\!\cdots\!03\)\( \beta_{3} + \)\(15\!\cdots\!44\)\( \beta_{4} - \)\(29\!\cdots\!85\)\( \beta_{5} - \)\(22\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!56\)\( \beta_{7} - \)\(15\!\cdots\!00\)\( \beta_{8}) q^{22}\) \(+(-\)\(18\!\cdots\!01\)\( - \)\(18\!\cdots\!79\)\( \beta_{1} - \)\(57\!\cdots\!21\)\( \beta_{2} - \)\(26\!\cdots\!55\)\( \beta_{3} + \)\(11\!\cdots\!30\)\( \beta_{4} + \)\(31\!\cdots\!75\)\( \beta_{5} - \)\(21\!\cdots\!55\)\( \beta_{6} - \)\(39\!\cdots\!20\)\( \beta_{7} + \)\(24\!\cdots\!00\)\( \beta_{8}) q^{23}\) \(+(\)\(93\!\cdots\!00\)\( + \)\(12\!\cdots\!24\)\( \beta_{1} + \)\(44\!\cdots\!68\)\( \beta_{2} + \)\(94\!\cdots\!24\)\( \beta_{3} - \)\(17\!\cdots\!72\)\( \beta_{4} + \)\(15\!\cdots\!28\)\( \beta_{5} + \)\(63\!\cdots\!12\)\( \beta_{6} + \)\(39\!\cdots\!52\)\( \beta_{7} - \)\(35\!\cdots\!64\)\( \beta_{8}) q^{24}\) \(+(\)\(40\!\cdots\!75\)\( - \)\(11\!\cdots\!50\)\( \beta_{1} - \)\(32\!\cdots\!50\)\( \beta_{2} + \)\(54\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!50\)\( \beta_{4} - \)\(59\!\cdots\!00\)\( \beta_{5} - \)\(68\!\cdots\!50\)\( \beta_{6} - \)\(30\!\cdots\!50\)\( \beta_{7} + \)\(44\!\cdots\!50\)\( \beta_{8}) q^{25}\) \(+(\)\(30\!\cdots\!70\)\( + \)\(51\!\cdots\!98\)\( \beta_{1} - \)\(32\!\cdots\!84\)\( \beta_{2} + \)\(46\!\cdots\!24\)\( \beta_{3} + \)\(10\!\cdots\!04\)\( \beta_{4} + \)\(48\!\cdots\!84\)\( \beta_{5} + \)\(34\!\cdots\!56\)\( \beta_{6} + \)\(15\!\cdots\!76\)\( \beta_{7} - \)\(50\!\cdots\!32\)\( \beta_{8}) q^{26}\) \(+(-\)\(14\!\cdots\!72\)\( + \)\(26\!\cdots\!46\)\( \beta_{1} - \)\(34\!\cdots\!80\)\( \beta_{2} - \)\(26\!\cdots\!66\)\( \beta_{3} - \)\(57\!\cdots\!04\)\( \beta_{4} - \)\(99\!\cdots\!30\)\( \beta_{5} + \)\(12\!\cdots\!74\)\( \beta_{6} - \)\(21\!\cdots\!44\)\( \beta_{7} + \)\(51\!\cdots\!00\)\( \beta_{8}) q^{27}\) \(+(-\)\(13\!\cdots\!20\)\( + \)\(59\!\cdots\!84\)\( \beta_{1} - \)\(75\!\cdots\!36\)\( \beta_{2} - \)\(42\!\cdots\!64\)\( \beta_{3} - \)\(61\!\cdots\!76\)\( \beta_{4} - \)\(28\!\cdots\!20\)\( \beta_{5} - \)\(41\!\cdots\!84\)\( \beta_{6} - \)\(61\!\cdots\!56\)\( \beta_{7} - \)\(47\!\cdots\!00\)\( \beta_{8}) q^{28}\) \(+(\)\(18\!\cdots\!13\)\( + \)\(30\!\cdots\!99\)\( \beta_{1} - \)\(19\!\cdots\!17\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} + \)\(56\!\cdots\!91\)\( \beta_{4} + \)\(22\!\cdots\!16\)\( \beta_{5} + \)\(41\!\cdots\!64\)\( \beta_{6} + \)\(81\!\cdots\!44\)\( \beta_{7} + \)\(39\!\cdots\!92\)\( \beta_{8}) q^{29}\) \(+(\)\(16\!\cdots\!24\)\( - \)\(40\!\cdots\!50\)\( \beta_{1} - \)\(34\!\cdots\!58\)\( \beta_{2} + \)\(18\!\cdots\!46\)\( \beta_{3} + \)\(39\!\cdots\!68\)\( \beta_{4} - \)\(22\!\cdots\!50\)\( \beta_{5} - \)\(23\!\cdots\!00\)\( \beta_{6} - \)\(59\!\cdots\!00\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{30}\) \(+(\)\(14\!\cdots\!72\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(41\!\cdots\!08\)\( \beta_{2} + \)\(25\!\cdots\!48\)\( \beta_{3} - \)\(95\!\cdots\!36\)\( \beta_{4} - \)\(73\!\cdots\!84\)\( \beta_{5} + \)\(62\!\cdots\!32\)\( \beta_{6} + \)\(24\!\cdots\!72\)\( \beta_{7} + \)\(20\!\cdots\!96\)\( \beta_{8}) q^{31}\) \(+(-\)\(23\!\cdots\!60\)\( - \)\(68\!\cdots\!96\)\( \beta_{1} + \)\(75\!\cdots\!92\)\( \beta_{2} - \)\(42\!\cdots\!72\)\( \beta_{3} - \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(51\!\cdots\!20\)\( \beta_{5} + \)\(21\!\cdots\!76\)\( \beta_{6} + \)\(21\!\cdots\!20\)\( \beta_{7} - \)\(12\!\cdots\!00\)\( \beta_{8}) q^{32}\) \(+(-\)\(23\!\cdots\!38\)\( - \)\(17\!\cdots\!57\)\( \beta_{1} + \)\(50\!\cdots\!77\)\( \beta_{2} - \)\(22\!\cdots\!56\)\( \beta_{3} + \)\(74\!\cdots\!17\)\( \beta_{4} - \)\(66\!\cdots\!00\)\( \beta_{5} - \)\(29\!\cdots\!93\)\( \beta_{6} - \)\(13\!\cdots\!81\)\( \beta_{7} + \)\(72\!\cdots\!25\)\( \beta_{8}) q^{33}\) \(+(\)\(35\!\cdots\!06\)\( + \)\(49\!\cdots\!38\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(69\!\cdots\!64\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4} - \)\(91\!\cdots\!08\)\( \beta_{5} + \)\(10\!\cdots\!36\)\( \beta_{6} + \)\(13\!\cdots\!56\)\( \beta_{7} - \)\(37\!\cdots\!92\)\( \beta_{8}) q^{34}\) \(+(\)\(38\!\cdots\!24\)\( - \)\(57\!\cdots\!80\)\( \beta_{1} + \)\(71\!\cdots\!52\)\( \beta_{2} + \)\(76\!\cdots\!36\)\( \beta_{3} - \)\(29\!\cdots\!72\)\( \beta_{4} + \)\(42\!\cdots\!40\)\( \beta_{5} + \)\(29\!\cdots\!80\)\( \beta_{6} - \)\(82\!\cdots\!40\)\( \beta_{7} + \)\(16\!\cdots\!20\)\( \beta_{8}) q^{35}\) \(+(\)\(30\!\cdots\!20\)\( + \)\(72\!\cdots\!02\)\( \beta_{1} - \)\(60\!\cdots\!29\)\( \beta_{2} - \)\(82\!\cdots\!11\)\( \beta_{3} - \)\(36\!\cdots\!88\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} - \)\(56\!\cdots\!20\)\( \beta_{6} + \)\(36\!\cdots\!80\)\( \beta_{7} - \)\(66\!\cdots\!60\)\( \beta_{8}) q^{36}\) \(+(-\)\(66\!\cdots\!95\)\( - \)\(62\!\cdots\!43\)\( \beta_{1} - \)\(14\!\cdots\!23\)\( \beta_{2} - \)\(11\!\cdots\!72\)\( \beta_{3} + \)\(88\!\cdots\!37\)\( \beta_{4} - \)\(16\!\cdots\!60\)\( \beta_{5} + \)\(32\!\cdots\!98\)\( \beta_{6} - \)\(90\!\cdots\!58\)\( \beta_{7} + \)\(22\!\cdots\!50\)\( \beta_{8}) q^{37}\) \(+(-\)\(91\!\cdots\!48\)\( - \)\(54\!\cdots\!15\)\( \beta_{1} - \)\(10\!\cdots\!67\)\( \beta_{2} - \)\(36\!\cdots\!85\)\( \beta_{3} - \)\(15\!\cdots\!08\)\( \beta_{4} + \)\(40\!\cdots\!85\)\( \beta_{5} - \)\(82\!\cdots\!04\)\( \beta_{6} - \)\(13\!\cdots\!84\)\( \beta_{7} - \)\(58\!\cdots\!00\)\( \beta_{8}) q^{38}\) \(+(\)\(96\!\cdots\!51\)\( - \)\(19\!\cdots\!07\)\( \beta_{1} + \)\(16\!\cdots\!63\)\( \beta_{2} + \)\(17\!\cdots\!17\)\( \beta_{3} - \)\(13\!\cdots\!90\)\( \beta_{4} + \)\(18\!\cdots\!99\)\( \beta_{5} - \)\(11\!\cdots\!03\)\( \beta_{6} + \)\(29\!\cdots\!12\)\( \beta_{7} + \)\(91\!\cdots\!16\)\( \beta_{8}) q^{39}\) \(+(\)\(75\!\cdots\!10\)\( - \)\(24\!\cdots\!50\)\( \beta_{1} + \)\(23\!\cdots\!80\)\( \beta_{2} + \)\(15\!\cdots\!90\)\( \beta_{3} + \)\(58\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(19\!\cdots\!50\)\( \beta_{7} + \)\(10\!\cdots\!00\)\( \beta_{8}) q^{40}\) \(+(-\)\(38\!\cdots\!94\)\( - \)\(45\!\cdots\!86\)\( \beta_{1} + \)\(51\!\cdots\!66\)\( \beta_{2} - \)\(44\!\cdots\!92\)\( \beta_{3} + \)\(52\!\cdots\!98\)\( \beta_{4} + \)\(28\!\cdots\!04\)\( \beta_{5} - \)\(61\!\cdots\!30\)\( \beta_{6} + \)\(82\!\cdots\!70\)\( \beta_{7} - \)\(13\!\cdots\!90\)\( \beta_{8}) q^{41}\) \(+(-\)\(11\!\cdots\!28\)\( - \)\(52\!\cdots\!84\)\( \beta_{1} - \)\(72\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!28\)\( \beta_{3} - \)\(64\!\cdots\!68\)\( \beta_{4} + \)\(10\!\cdots\!80\)\( \beta_{5} - \)\(60\!\cdots\!96\)\( \beta_{6} - \)\(25\!\cdots\!40\)\( \beta_{7} + \)\(57\!\cdots\!00\)\( \beta_{8}) q^{42}\) \(+(\)\(23\!\cdots\!19\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} - \)\(70\!\cdots\!85\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3} - \)\(38\!\cdots\!20\)\( \beta_{4} - \)\(48\!\cdots\!20\)\( \beta_{5} + \)\(14\!\cdots\!92\)\( \beta_{6} + \)\(51\!\cdots\!36\)\( \beta_{7} - \)\(14\!\cdots\!00\)\( \beta_{8}) q^{43}\) \(+(\)\(25\!\cdots\!32\)\( - \)\(62\!\cdots\!20\)\( \beta_{1} - \)\(49\!\cdots\!96\)\( \beta_{2} + \)\(33\!\cdots\!24\)\( \beta_{3} + \)\(80\!\cdots\!36\)\( \beta_{4} - \)\(10\!\cdots\!76\)\( \beta_{5} - \)\(63\!\cdots\!12\)\( \beta_{6} - \)\(63\!\cdots\!52\)\( \beta_{7} + \)\(19\!\cdots\!64\)\( \beta_{8}) q^{44}\) \(+(\)\(28\!\cdots\!21\)\( - \)\(18\!\cdots\!75\)\( \beta_{1} + \)\(46\!\cdots\!93\)\( \beta_{2} + \)\(24\!\cdots\!84\)\( \beta_{3} - \)\(10\!\cdots\!03\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!50\)\( \beta_{6} + \)\(11\!\cdots\!50\)\( \beta_{7} - \)\(26\!\cdots\!50\)\( \beta_{8}) q^{45}\) \(+(-\)\(31\!\cdots\!24\)\( - \)\(39\!\cdots\!26\)\( \beta_{1} + \)\(39\!\cdots\!66\)\( \beta_{2} - \)\(46\!\cdots\!62\)\( \beta_{3} - \)\(76\!\cdots\!92\)\( \beta_{4} - \)\(25\!\cdots\!26\)\( \beta_{5} - \)\(29\!\cdots\!60\)\( \beta_{6} - \)\(11\!\cdots\!60\)\( \beta_{7} + \)\(29\!\cdots\!20\)\( \beta_{8}) q^{46}\) \(+(-\)\(54\!\cdots\!66\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + \)\(13\!\cdots\!10\)\( \beta_{2} - \)\(64\!\cdots\!86\)\( \beta_{3} - \)\(53\!\cdots\!60\)\( \beta_{4} - \)\(47\!\cdots\!10\)\( \beta_{5} - \)\(20\!\cdots\!54\)\( \beta_{6} + \)\(71\!\cdots\!68\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{47}\) \(+(\)\(27\!\cdots\!20\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(45\!\cdots\!72\)\( \beta_{3} + \)\(81\!\cdots\!52\)\( \beta_{4} + \)\(23\!\cdots\!80\)\( \beta_{5} - \)\(52\!\cdots\!36\)\( \beta_{6} - \)\(20\!\cdots\!80\)\( \beta_{7} + \)\(94\!\cdots\!00\)\( \beta_{8}) q^{48}\) \(+(\)\(83\!\cdots\!33\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} - \)\(49\!\cdots\!76\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3} + \)\(37\!\cdots\!92\)\( \beta_{4} + \)\(63\!\cdots\!16\)\( \beta_{5} + \)\(37\!\cdots\!80\)\( \beta_{6} - \)\(36\!\cdots\!20\)\( \beta_{7} - \)\(16\!\cdots\!60\)\( \beta_{8}) q^{49}\) \(+(-\)\(19\!\cdots\!25\)\( + \)\(87\!\cdots\!75\)\( \beta_{1} - \)\(83\!\cdots\!00\)\( \beta_{2} - \)\(52\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} - \)\(56\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(33\!\cdots\!00\)\( \beta_{7} - \)\(55\!\cdots\!00\)\( \beta_{8}) q^{50}\) \(+(-\)\(46\!\cdots\!48\)\( + \)\(91\!\cdots\!34\)\( \beta_{1} + \)\(64\!\cdots\!24\)\( \beta_{2} - \)\(55\!\cdots\!94\)\( \beta_{3} + \)\(43\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!82\)\( \beta_{5} + \)\(24\!\cdots\!06\)\( \beta_{6} - \)\(17\!\cdots\!24\)\( \beta_{7} + \)\(56\!\cdots\!68\)\( \beta_{8}) q^{51}\) \(+(-\)\(33\!\cdots\!84\)\( + \)\(51\!\cdots\!56\)\( \beta_{1} + \)\(40\!\cdots\!58\)\( \beta_{2} + \)\(99\!\cdots\!94\)\( \beta_{3} + \)\(79\!\cdots\!40\)\( \beta_{4} + \)\(13\!\cdots\!40\)\( \beta_{5} - \)\(56\!\cdots\!84\)\( \beta_{6} + \)\(45\!\cdots\!28\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{52}\) \(+(\)\(50\!\cdots\!53\)\( - \)\(22\!\cdots\!15\)\( \beta_{1} + \)\(11\!\cdots\!73\)\( \beta_{2} + \)\(97\!\cdots\!04\)\( \beta_{3} - \)\(41\!\cdots\!19\)\( \beta_{4} - \)\(38\!\cdots\!80\)\( \beta_{5} - \)\(61\!\cdots\!66\)\( \beta_{6} - \)\(31\!\cdots\!74\)\( \beta_{7} + \)\(45\!\cdots\!50\)\( \beta_{8}) q^{53}\) \(+(\)\(42\!\cdots\!56\)\( - \)\(37\!\cdots\!54\)\( \beta_{1} - \)\(38\!\cdots\!94\)\( \beta_{2} + \)\(13\!\cdots\!86\)\( \beta_{3} - \)\(39\!\cdots\!44\)\( \beta_{4} - \)\(35\!\cdots\!06\)\( \beta_{5} - \)\(17\!\cdots\!32\)\( \beta_{6} - \)\(27\!\cdots\!72\)\( \beta_{7} + \)\(40\!\cdots\!04\)\( \beta_{8}) q^{54}\) \(+(-\)\(48\!\cdots\!51\)\( - \)\(70\!\cdots\!25\)\( \beta_{1} - \)\(80\!\cdots\!83\)\( \beta_{2} - \)\(76\!\cdots\!29\)\( \beta_{3} - \)\(17\!\cdots\!82\)\( \beta_{4} + \)\(19\!\cdots\!25\)\( \beta_{5} + \)\(53\!\cdots\!75\)\( \beta_{6} + \)\(13\!\cdots\!00\)\( \beta_{7} - \)\(67\!\cdots\!00\)\( \beta_{8}) q^{55}\) \(+(-\)\(17\!\cdots\!04\)\( - \)\(41\!\cdots\!56\)\( \beta_{1} - \)\(21\!\cdots\!00\)\( \beta_{2} - \)\(60\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(37\!\cdots\!08\)\( \beta_{5} + \)\(63\!\cdots\!44\)\( \beta_{6} - \)\(29\!\cdots\!76\)\( \beta_{7} + \)\(25\!\cdots\!32\)\( \beta_{8}) q^{56}\) \(+(\)\(22\!\cdots\!46\)\( - \)\(55\!\cdots\!67\)\( \beta_{1} - \)\(44\!\cdots\!17\)\( \beta_{2} + \)\(24\!\cdots\!92\)\( \beta_{3} - \)\(10\!\cdots\!05\)\( \beta_{4} + \)\(97\!\cdots\!20\)\( \beta_{5} - \)\(13\!\cdots\!87\)\( \beta_{6} + \)\(13\!\cdots\!29\)\( \beta_{7} - \)\(46\!\cdots\!25\)\( \beta_{8}) q^{57}\) \(+(\)\(51\!\cdots\!62\)\( + \)\(42\!\cdots\!10\)\( \beta_{1} + \)\(31\!\cdots\!40\)\( \beta_{2} + \)\(11\!\cdots\!76\)\( \beta_{3} - \)\(39\!\cdots\!88\)\( \beta_{4} - \)\(18\!\cdots\!80\)\( \beta_{5} + \)\(49\!\cdots\!80\)\( \beta_{6} + \)\(13\!\cdots\!28\)\( \beta_{7} - \)\(45\!\cdots\!00\)\( \beta_{8}) q^{58}\) \(+(-\)\(88\!\cdots\!01\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(44\!\cdots\!59\)\( \beta_{2} + \)\(21\!\cdots\!32\)\( \beta_{3} - \)\(25\!\cdots\!04\)\( \beta_{4} + \)\(59\!\cdots\!12\)\( \beta_{5} - \)\(63\!\cdots\!08\)\( \beta_{6} - \)\(48\!\cdots\!68\)\( \beta_{7} + \)\(59\!\cdots\!76\)\( \beta_{8}) q^{59}\) \(+(-\)\(57\!\cdots\!96\)\( + \)\(21\!\cdots\!20\)\( \beta_{1} - \)\(70\!\cdots\!08\)\( \beta_{2} - \)\(11\!\cdots\!44\)\( \beta_{3} + \)\(46\!\cdots\!88\)\( \beta_{4} - \)\(19\!\cdots\!60\)\( \beta_{5} - \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(73\!\cdots\!60\)\( \beta_{7} - \)\(19\!\cdots\!80\)\( \beta_{8}) q^{60}\) \(+(-\)\(22\!\cdots\!95\)\( + \)\(37\!\cdots\!01\)\( \beta_{1} - \)\(11\!\cdots\!39\)\( \beta_{2} - \)\(33\!\cdots\!40\)\( \beta_{3} + \)\(52\!\cdots\!53\)\( \beta_{4} + \)\(16\!\cdots\!36\)\( \beta_{5} + \)\(55\!\cdots\!66\)\( \beta_{6} - \)\(41\!\cdots\!14\)\( \beta_{7} + \)\(24\!\cdots\!98\)\( \beta_{8}) q^{61}\) \(+(\)\(32\!\cdots\!28\)\( + \)\(34\!\cdots\!84\)\( \beta_{1} - \)\(83\!\cdots\!12\)\( \beta_{2} + \)\(32\!\cdots\!12\)\( \beta_{3} - \)\(37\!\cdots\!48\)\( \beta_{4} + \)\(29\!\cdots\!80\)\( \beta_{5} - \)\(49\!\cdots\!76\)\( \beta_{6} - \)\(17\!\cdots\!00\)\( \beta_{7} + \)\(58\!\cdots\!00\)\( \beta_{8}) q^{62}\) \(+(\)\(12\!\cdots\!17\)\( - \)\(39\!\cdots\!41\)\( \beta_{1} - \)\(10\!\cdots\!63\)\( \beta_{2} + \)\(16\!\cdots\!43\)\( \beta_{3} + \)\(41\!\cdots\!42\)\( \beta_{4} + \)\(13\!\cdots\!65\)\( \beta_{5} - \)\(10\!\cdots\!77\)\( \beta_{6} + \)\(20\!\cdots\!12\)\( \beta_{7} - \)\(39\!\cdots\!00\)\( \beta_{8}) q^{63}\) \(+(-\)\(24\!\cdots\!88\)\( - \)\(36\!\cdots\!12\)\( \beta_{1} + \)\(64\!\cdots\!40\)\( \beta_{2} - \)\(20\!\cdots\!64\)\( \beta_{3} + \)\(62\!\cdots\!64\)\( \beta_{4} - \)\(29\!\cdots\!08\)\( \beta_{5} - \)\(27\!\cdots\!80\)\( \beta_{6} - \)\(32\!\cdots\!80\)\( \beta_{7} + \)\(94\!\cdots\!60\)\( \beta_{8}) q^{64}\) \(+(-\)\(67\!\cdots\!88\)\( - \)\(11\!\cdots\!90\)\( \beta_{1} + \)\(15\!\cdots\!26\)\( \beta_{2} - \)\(69\!\cdots\!32\)\( \beta_{3} + \)\(31\!\cdots\!14\)\( \beta_{4} - \)\(14\!\cdots\!80\)\( \beta_{5} + \)\(18\!\cdots\!90\)\( \beta_{6} + \)\(39\!\cdots\!30\)\( \beta_{7} - \)\(46\!\cdots\!90\)\( \beta_{8}) q^{65}\) \(+(-\)\(31\!\cdots\!16\)\( - \)\(41\!\cdots\!08\)\( \beta_{1} - \)\(23\!\cdots\!28\)\( \beta_{2} - \)\(12\!\cdots\!72\)\( \beta_{3} - \)\(20\!\cdots\!20\)\( \beta_{4} + \)\(36\!\cdots\!96\)\( \beta_{5} - \)\(50\!\cdots\!72\)\( \beta_{6} - \)\(16\!\cdots\!12\)\( \beta_{7} - \)\(47\!\cdots\!16\)\( \beta_{8}) q^{66}\) \(+(\)\(27\!\cdots\!15\)\( + \)\(74\!\cdots\!74\)\( \beta_{1} - \)\(14\!\cdots\!69\)\( \beta_{2} + \)\(54\!\cdots\!30\)\( \beta_{3} + \)\(26\!\cdots\!28\)\( \beta_{4} - \)\(83\!\cdots\!10\)\( \beta_{5} + \)\(79\!\cdots\!94\)\( \beta_{6} + \)\(23\!\cdots\!84\)\( \beta_{7} + \)\(18\!\cdots\!00\)\( \beta_{8}) q^{67}\) \(+(\)\(70\!\cdots\!04\)\( + \)\(72\!\cdots\!16\)\( \beta_{1} - \)\(12\!\cdots\!90\)\( \beta_{2} + \)\(88\!\cdots\!66\)\( \beta_{3} - \)\(44\!\cdots\!96\)\( \beta_{4} + \)\(61\!\cdots\!80\)\( \beta_{5} + \)\(27\!\cdots\!76\)\( \beta_{6} + \)\(55\!\cdots\!44\)\( \beta_{7} - \)\(27\!\cdots\!00\)\( \beta_{8}) q^{68}\) \(+(-\)\(26\!\cdots\!08\)\( + \)\(23\!\cdots\!18\)\( \beta_{1} - \)\(28\!\cdots\!58\)\( \beta_{2} - \)\(40\!\cdots\!68\)\( \beta_{3} + \)\(13\!\cdots\!74\)\( \beta_{4} - \)\(11\!\cdots\!84\)\( \beta_{5} - \)\(63\!\cdots\!58\)\( \beta_{6} - \)\(30\!\cdots\!18\)\( \beta_{7} - \)\(26\!\cdots\!74\)\( \beta_{8}) q^{69}\) \(+(-\)\(95\!\cdots\!92\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!64\)\( \beta_{2} - \)\(10\!\cdots\!68\)\( \beta_{3} - \)\(54\!\cdots\!44\)\( \beta_{4} + \)\(65\!\cdots\!00\)\( \beta_{5} + \)\(49\!\cdots\!00\)\( \beta_{6} + \)\(47\!\cdots\!00\)\( \beta_{7} + \)\(23\!\cdots\!00\)\( \beta_{8}) q^{70}\) \(+(\)\(10\!\cdots\!21\)\( + \)\(13\!\cdots\!83\)\( \beta_{1} + \)\(57\!\cdots\!13\)\( \beta_{2} + \)\(46\!\cdots\!55\)\( \beta_{3} + \)\(46\!\cdots\!74\)\( \beta_{4} - \)\(57\!\cdots\!87\)\( \beta_{5} - \)\(26\!\cdots\!97\)\( \beta_{6} + \)\(42\!\cdots\!88\)\( \beta_{7} - \)\(51\!\cdots\!16\)\( \beta_{8}) q^{71}\) \(+(\)\(97\!\cdots\!83\)\( - \)\(11\!\cdots\!93\)\( \beta_{1} + \)\(15\!\cdots\!50\)\( \beta_{2} + \)\(28\!\cdots\!99\)\( \beta_{3} + \)\(49\!\cdots\!84\)\( \beta_{4} - \)\(38\!\cdots\!40\)\( \beta_{5} + \)\(19\!\cdots\!38\)\( \beta_{6} - \)\(27\!\cdots\!35\)\( \beta_{7} + \)\(22\!\cdots\!00\)\( \beta_{8}) q^{72}\) \(+(\)\(60\!\cdots\!88\)\( - \)\(53\!\cdots\!63\)\( \beta_{1} - \)\(53\!\cdots\!85\)\( \beta_{2} + \)\(70\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!87\)\( \beta_{4} + \)\(10\!\cdots\!60\)\( \beta_{5} + \)\(39\!\cdots\!21\)\( \beta_{6} + \)\(24\!\cdots\!21\)\( \beta_{7} + \)\(20\!\cdots\!75\)\( \beta_{8}) q^{73}\) \(+(-\)\(10\!\cdots\!34\)\( - \)\(16\!\cdots\!70\)\( \beta_{1} - \)\(41\!\cdots\!88\)\( \beta_{2} - \)\(18\!\cdots\!92\)\( \beta_{3} - \)\(54\!\cdots\!64\)\( \beta_{4} - \)\(56\!\cdots\!40\)\( \beta_{5} - \)\(33\!\cdots\!44\)\( \beta_{6} + \)\(80\!\cdots\!76\)\( \beta_{7} - \)\(64\!\cdots\!32\)\( \beta_{8}) q^{74}\) \(+(-\)\(32\!\cdots\!25\)\( - \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(27\!\cdots\!75\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!00\)\( \beta_{4} + \)\(34\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!00\)\( \beta_{7} + \)\(71\!\cdots\!00\)\( \beta_{8}) q^{75}\) \(+(-\)\(67\!\cdots\!88\)\( - \)\(31\!\cdots\!92\)\( \beta_{1} + \)\(54\!\cdots\!80\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(21\!\cdots\!96\)\( \beta_{4} - \)\(80\!\cdots\!96\)\( \beta_{5} - \)\(65\!\cdots\!12\)\( \beta_{6} - \)\(32\!\cdots\!52\)\( \beta_{7} + \)\(10\!\cdots\!64\)\( \beta_{8}) q^{76}\) \(+(\)\(11\!\cdots\!48\)\( + \)\(98\!\cdots\!46\)\( \beta_{1} - \)\(55\!\cdots\!90\)\( \beta_{2} + \)\(17\!\cdots\!88\)\( \beta_{3} - \)\(88\!\cdots\!22\)\( \beta_{4} + \)\(16\!\cdots\!20\)\( \beta_{5} - \)\(28\!\cdots\!34\)\( \beta_{6} + \)\(16\!\cdots\!90\)\( \beta_{7} - \)\(57\!\cdots\!50\)\( \beta_{8}) q^{77}\) \(+(-\)\(31\!\cdots\!40\)\( + \)\(26\!\cdots\!26\)\( \beta_{1} + \)\(54\!\cdots\!46\)\( \beta_{2} + \)\(12\!\cdots\!70\)\( \beta_{3} + \)\(27\!\cdots\!24\)\( \beta_{4} + \)\(19\!\cdots\!70\)\( \beta_{5} + \)\(67\!\cdots\!72\)\( \beta_{6} - \)\(12\!\cdots\!68\)\( \beta_{7} + \)\(92\!\cdots\!00\)\( \beta_{8}) q^{78}\) \(+(-\)\(66\!\cdots\!26\)\( + \)\(49\!\cdots\!82\)\( \beta_{1} - \)\(52\!\cdots\!22\)\( \beta_{2} - \)\(16\!\cdots\!22\)\( \beta_{3} - \)\(18\!\cdots\!24\)\( \beta_{4} - \)\(10\!\cdots\!26\)\( \beta_{5} - \)\(14\!\cdots\!22\)\( \beta_{6} - \)\(68\!\cdots\!12\)\( \beta_{7} + \)\(66\!\cdots\!84\)\( \beta_{8}) q^{79}\) \(+(-\)\(26\!\cdots\!88\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!04\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} + \)\(17\!\cdots\!84\)\( \beta_{4} + \)\(97\!\cdots\!00\)\( \beta_{5} - \)\(72\!\cdots\!00\)\( \beta_{6} + \)\(20\!\cdots\!00\)\( \beta_{7} - \)\(35\!\cdots\!00\)\( \beta_{8}) q^{80}\) \(+(-\)\(33\!\cdots\!61\)\( - \)\(31\!\cdots\!69\)\( \beta_{1} - \)\(58\!\cdots\!91\)\( \beta_{2} + \)\(12\!\cdots\!56\)\( \beta_{3} - \)\(58\!\cdots\!31\)\( \beta_{4} - \)\(59\!\cdots\!12\)\( \beta_{5} - \)\(19\!\cdots\!77\)\( \beta_{6} - \)\(10\!\cdots\!17\)\( \beta_{7} + \)\(77\!\cdots\!69\)\( \beta_{8}) q^{81}\) \(+(-\)\(77\!\cdots\!66\)\( - \)\(72\!\cdots\!38\)\( \beta_{1} + \)\(26\!\cdots\!04\)\( \beta_{2} + \)\(29\!\cdots\!08\)\( \beta_{3} + \)\(15\!\cdots\!12\)\( \beta_{4} + \)\(48\!\cdots\!40\)\( \beta_{5} + \)\(65\!\cdots\!68\)\( \beta_{6} - \)\(56\!\cdots\!48\)\( \beta_{7} - \)\(51\!\cdots\!00\)\( \beta_{8}) q^{82}\) \(+(-\)\(12\!\cdots\!75\)\( - \)\(70\!\cdots\!96\)\( \beta_{1} + \)\(10\!\cdots\!93\)\( \beta_{2} + \)\(11\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!00\)\( \beta_{4} - \)\(77\!\cdots\!00\)\( \beta_{5} + \)\(49\!\cdots\!20\)\( \beta_{6} + \)\(12\!\cdots\!60\)\( \beta_{7} - \)\(13\!\cdots\!00\)\( \beta_{8}) q^{83}\) \(+(-\)\(64\!\cdots\!20\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} - \)\(31\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!56\)\( \beta_{4} - \)\(16\!\cdots\!08\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} - \)\(39\!\cdots\!00\)\( \beta_{7} + \)\(44\!\cdots\!00\)\( \beta_{8}) q^{84}\) \(+(-\)\(71\!\cdots\!86\)\( - \)\(33\!\cdots\!80\)\( \beta_{1} - \)\(92\!\cdots\!28\)\( \beta_{2} + \)\(23\!\cdots\!96\)\( \beta_{3} + \)\(28\!\cdots\!08\)\( \beta_{4} + \)\(50\!\cdots\!40\)\( \beta_{5} + \)\(35\!\cdots\!30\)\( \beta_{6} - \)\(21\!\cdots\!90\)\( \beta_{7} - \)\(57\!\cdots\!30\)\( \beta_{8}) q^{85}\) \(+(-\)\(23\!\cdots\!92\)\( + \)\(48\!\cdots\!55\)\( \beta_{1} - \)\(64\!\cdots\!17\)\( \beta_{2} - \)\(10\!\cdots\!87\)\( \beta_{3} + \)\(12\!\cdots\!32\)\( \beta_{4} - \)\(11\!\cdots\!97\)\( \beta_{5} + \)\(42\!\cdots\!76\)\( \beta_{6} + \)\(14\!\cdots\!96\)\( \beta_{7} + \)\(45\!\cdots\!28\)\( \beta_{8}) q^{86}\) \(+(-\)\(17\!\cdots\!81\)\( + \)\(15\!\cdots\!41\)\( \beta_{1} - \)\(93\!\cdots\!41\)\( \beta_{2} + \)\(24\!\cdots\!29\)\( \beta_{3} + \)\(17\!\cdots\!02\)\( \beta_{4} + \)\(74\!\cdots\!75\)\( \beta_{5} + \)\(16\!\cdots\!77\)\( \beta_{6} + \)\(46\!\cdots\!44\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{87}\) \(+(-\)\(67\!\cdots\!40\)\( + \)\(34\!\cdots\!64\)\( \beta_{1} - \)\(21\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!08\)\( \beta_{3} - \)\(10\!\cdots\!28\)\( \beta_{4} - \)\(33\!\cdots\!20\)\( \beta_{5} - \)\(34\!\cdots\!96\)\( \beta_{6} + \)\(12\!\cdots\!20\)\( \beta_{7} - \)\(36\!\cdots\!00\)\( \beta_{8}) q^{88}\) \(+(-\)\(72\!\cdots\!84\)\( + \)\(12\!\cdots\!21\)\( \beta_{1} + \)\(54\!\cdots\!23\)\( \beta_{2} + \)\(73\!\cdots\!40\)\( \beta_{3} + \)\(44\!\cdots\!15\)\( \beta_{4} + \)\(42\!\cdots\!52\)\( \beta_{5} - \)\(29\!\cdots\!59\)\( \beta_{6} - \)\(42\!\cdots\!39\)\( \beta_{7} + \)\(46\!\cdots\!23\)\( \beta_{8}) q^{89}\) \(+(-\)\(30\!\cdots\!38\)\( + \)\(39\!\cdots\!10\)\( \beta_{1} + \)\(17\!\cdots\!76\)\( \beta_{2} - \)\(15\!\cdots\!32\)\( \beta_{3} - \)\(65\!\cdots\!36\)\( \beta_{4} + \)\(10\!\cdots\!20\)\( \beta_{5} + \)\(22\!\cdots\!40\)\( \beta_{6} + \)\(78\!\cdots\!80\)\( \beta_{7} - \)\(26\!\cdots\!40\)\( \beta_{8}) q^{90}\) \(+(-\)\(25\!\cdots\!92\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} - \)\(60\!\cdots\!20\)\( \beta_{2} + \)\(27\!\cdots\!60\)\( \beta_{3} + \)\(80\!\cdots\!88\)\( \beta_{4} - \)\(20\!\cdots\!68\)\( \beta_{5} - \)\(35\!\cdots\!76\)\( \beta_{6} + \)\(61\!\cdots\!04\)\( \beta_{7} - \)\(13\!\cdots\!28\)\( \beta_{8}) q^{91}\) \(+(-\)\(47\!\cdots\!04\)\( - \)\(48\!\cdots\!80\)\( \beta_{1} - \)\(19\!\cdots\!24\)\( \beta_{2} - \)\(16\!\cdots\!72\)\( \beta_{3} - \)\(71\!\cdots\!68\)\( \beta_{4} - \)\(85\!\cdots\!60\)\( \beta_{5} - \)\(10\!\cdots\!92\)\( \beta_{6} - \)\(23\!\cdots\!48\)\( \beta_{7} + \)\(31\!\cdots\!00\)\( \beta_{8}) q^{92}\) \(+(\)\(17\!\cdots\!76\)\( - \)\(45\!\cdots\!92\)\( \beta_{1} - \)\(61\!\cdots\!20\)\( \beta_{2} + \)\(57\!\cdots\!60\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!20\)\( \beta_{5} + \)\(57\!\cdots\!28\)\( \beta_{6} + \)\(51\!\cdots\!08\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{93}\) \(+(\)\(20\!\cdots\!80\)\( - \)\(10\!\cdots\!84\)\( \beta_{1} - \)\(66\!\cdots\!44\)\( \beta_{2} + \)\(78\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!04\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(89\!\cdots\!12\)\( \beta_{6} - \)\(93\!\cdots\!52\)\( \beta_{7} - \)\(12\!\cdots\!36\)\( \beta_{8}) q^{94}\) \(+(-\)\(45\!\cdots\!05\)\( + \)\(15\!\cdots\!25\)\( \beta_{1} + \)\(11\!\cdots\!35\)\( \beta_{2} - \)\(11\!\cdots\!95\)\( \beta_{3} + \)\(30\!\cdots\!90\)\( \beta_{4} - \)\(96\!\cdots\!25\)\( \beta_{5} + \)\(15\!\cdots\!25\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7} + \)\(71\!\cdots\!00\)\( \beta_{8}) q^{95}\) \(+(\)\(93\!\cdots\!64\)\( + \)\(52\!\cdots\!16\)\( \beta_{1} + \)\(21\!\cdots\!88\)\( \beta_{2} - \)\(20\!\cdots\!08\)\( \beta_{3} + \)\(14\!\cdots\!56\)\( \beta_{4} - \)\(37\!\cdots\!92\)\( \beta_{5} - \)\(84\!\cdots\!00\)\( \beta_{6} + \)\(20\!\cdots\!00\)\( \beta_{7} - \)\(12\!\cdots\!00\)\( \beta_{8}) q^{96}\) \(+(\)\(40\!\cdots\!16\)\( + \)\(98\!\cdots\!73\)\( \beta_{1} + \)\(56\!\cdots\!63\)\( \beta_{2} - \)\(50\!\cdots\!16\)\( \beta_{3} - \)\(26\!\cdots\!69\)\( \beta_{4} + \)\(13\!\cdots\!20\)\( \beta_{5} + \)\(71\!\cdots\!29\)\( \beta_{6} - \)\(13\!\cdots\!39\)\( \beta_{7} + \)\(28\!\cdots\!75\)\( \beta_{8}) q^{97}\) \(+(\)\(25\!\cdots\!05\)\( + \)\(17\!\cdots\!77\)\( \beta_{1} + \)\(95\!\cdots\!16\)\( \beta_{2} + \)\(14\!\cdots\!72\)\( \beta_{3} - \)\(11\!\cdots\!92\)\( \beta_{4} + \)\(97\!\cdots\!60\)\( \beta_{5} + \)\(23\!\cdots\!12\)\( \beta_{6} + \)\(22\!\cdots\!68\)\( \beta_{7} - \)\(53\!\cdots\!00\)\( \beta_{8}) q^{98}\) \(+(\)\(35\!\cdots\!87\)\( + \)\(49\!\cdots\!24\)\( \beta_{1} - \)\(35\!\cdots\!45\)\( \beta_{2} - \)\(52\!\cdots\!40\)\( \beta_{3} - \)\(45\!\cdots\!92\)\( \beta_{4} + \)\(13\!\cdots\!12\)\( \beta_{5} - \)\(46\!\cdots\!16\)\( \beta_{6} + \)\(91\!\cdots\!64\)\( \beta_{7} - \)\(74\!\cdots\!48\)\( \beta_{8}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 4973724357874032q^{2} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!24\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!88\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!12\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!08\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!77\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 4973724357874032q^{2} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!24\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!88\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!12\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!08\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!77\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!88\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!68\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!46\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!96\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!62\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!96\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!76\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!84\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!48\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!56\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!10\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!28\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!68\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!56\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!64\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!98\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!80\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!64\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!02\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!56\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!56\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!16\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!92\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!28\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!16\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!13\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!92\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!28\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!26\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!40\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!80\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!62\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!44\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!72\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!84\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!12\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!84\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!56\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!08\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!40\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!66\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!24\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!20\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!28\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!40\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!51\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!96\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!64\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!04\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!72\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!40\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!70\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!12\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!88\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!92\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!36\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!68\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!22\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!24\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!64\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut -\mathstrut \) \(32\!\cdots\!04\) \(x^{7}\mathstrut +\mathstrut \) \(39\!\cdots\!64\) \(x^{6}\mathstrut +\mathstrut \) \(35\!\cdots\!16\) \(x^{5}\mathstrut -\mathstrut \) \(67\!\cdots\!92\) \(x^{4}\mathstrut -\mathstrut \) \(14\!\cdots\!88\) \(x^{3}\mathstrut +\mathstrut \) \(31\!\cdots\!68\) \(x^{2}\mathstrut +\mathstrut \) \(14\!\cdots\!72\) \(x\mathstrut -\mathstrut \) \(66\!\cdots\!92\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(83\!\cdots\!59\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!81\) \(\nu^{7}\mathstrut +\mathstrut \) \(36\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(56\!\cdots\!12\) \(\nu^{5}\mathstrut -\mathstrut \) \(51\!\cdots\!24\) \(\nu^{4}\mathstrut -\mathstrut \) \(58\!\cdots\!72\) \(\nu^{3}\mathstrut +\mathstrut \) \(23\!\cdots\!16\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!08\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!40\)\()/\)\(77\!\cdots\!28\)
\(\beta_{3}\)\(=\)\((\)\(11\!\cdots\!57\) \(\nu^{8}\mathstrut +\mathstrut \) \(22\!\cdots\!63\) \(\nu^{7}\mathstrut -\mathstrut \) \(50\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(77\!\cdots\!76\) \(\nu^{5}\mathstrut +\mathstrut \) \(70\!\cdots\!52\) \(\nu^{4}\mathstrut +\mathstrut \) \(80\!\cdots\!56\) \(\nu^{3}\mathstrut +\mathstrut \) \(72\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(58\!\cdots\!16\) \(\nu\mathstrut -\mathstrut \) \(73\!\cdots\!48\)\()/\)\(45\!\cdots\!84\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(22\!\cdots\!33\) \(\nu^{8}\mathstrut -\mathstrut \) \(36\!\cdots\!43\) \(\nu^{7}\mathstrut +\mathstrut \) \(64\!\cdots\!36\) \(\nu^{6}\mathstrut +\mathstrut \) \(76\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(54\!\cdots\!68\) \(\nu^{4}\mathstrut -\mathstrut \) \(40\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!84\) \(\nu^{2}\mathstrut +\mathstrut \) \(40\!\cdots\!04\) \(\nu\mathstrut -\mathstrut \) \(49\!\cdots\!88\)\()/\)\(19\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(61\!\cdots\!93\) \(\nu^{8}\mathstrut +\mathstrut \) \(88\!\cdots\!97\) \(\nu^{7}\mathstrut +\mathstrut \) \(19\!\cdots\!56\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(51\!\cdots\!64\) \(\nu^{2}\mathstrut -\mathstrut \) \(88\!\cdots\!16\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!48\)\()/\)\(24\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(13\!\cdots\!91\) \(\nu^{8}\mathstrut +\mathstrut \) \(74\!\cdots\!61\) \(\nu^{7}\mathstrut -\mathstrut \) \(43\!\cdots\!72\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!60\) \(\nu^{5}\mathstrut +\mathstrut \) \(44\!\cdots\!36\) \(\nu^{4}\mathstrut +\mathstrut \) \(14\!\cdots\!20\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(52\!\cdots\!08\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!76\)\()/\)\(48\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(32\!\cdots\!73\) \(\nu^{8}\mathstrut -\mathstrut \) \(28\!\cdots\!83\) \(\nu^{7}\mathstrut +\mathstrut \) \(36\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(57\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(29\!\cdots\!92\) \(\nu^{4}\mathstrut -\mathstrut \) \(50\!\cdots\!60\) \(\nu^{3}\mathstrut -\mathstrut \) \(24\!\cdots\!96\) \(\nu^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!76\) \(\nu\mathstrut -\mathstrut \) \(85\!\cdots\!28\)\()/\)\(19\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(50\!\cdots\!73\) \(\nu^{8}\mathstrut +\mathstrut \) \(24\!\cdots\!83\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!16\) \(\nu^{6}\mathstrut -\mathstrut \) \(65\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(89\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(52\!\cdots\!60\) \(\nu^{3}\mathstrut -\mathstrut \) \(25\!\cdots\!04\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!24\) \(\nu\mathstrut +\mathstrut \) \(44\!\cdots\!28\)\()/\)\(64\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(233291\) \(\beta_{2}\mathstrut -\mathstrut \) \(8626466785316074\) \(\beta_{1}\mathstrut +\mathstrut \) \(16733190786889893764973572264708832\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(54\) \(\beta_{6}\mathstrut +\mathstrut \) \(1827020\) \(\beta_{5}\mathstrut -\mathstrut \) \(49231023484\) \(\beta_{4}\mathstrut -\mathstrut \) \(17373914905518889\) \(\beta_{3}\mathstrut +\mathstrut \) \(94357237491658779056302\) \(\beta_{2}\mathstrut +\mathstrut \) \(25450765450947347315780708407499879\) \(\beta_{1}\mathstrut -\mathstrut \) \(144348314535107460480424942609861832895932180528325\)\()/110592\)
\(\nu^{4}\)\(=\)\((\)\(832514701824\) \(\beta_{8}\mathstrut -\mathstrut \) \(2765141756579001\) \(\beta_{7}\mathstrut +\mathstrut \) \(6946973400136530682\) \(\beta_{6}\mathstrut +\mathstrut \) \(21141823508226342818964\) \(\beta_{5}\mathstrut +\mathstrut \) \(1139173923285625331184168092\) \(\beta_{4}\mathstrut +\mathstrut \) \(2147164449978826711610930479658465\) \(\beta_{3}\mathstrut +\mathstrut \) \(2477893574206486168252478874096488413186\) \(\beta_{2}\mathstrut -\mathstrut \) \(32054464969139057704892063162013178160964878990063\) \(\beta_{1}\mathstrut +\mathstrut \) \(26617032122701961034418401648430041724999585885114115987802567375965\)\()/331776\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(50427023822868726326442078720\) \(\beta_{8}\mathstrut +\mathstrut \) \(163551734811077361753504119093093\) \(\beta_{7}\mathstrut +\mathstrut \) \(15838759907024084470952284692122958\) \(\beta_{6}\mathstrut +\mathstrut \) \(495340693749304401081259955541927054460\) \(\beta_{5}\mathstrut -\mathstrut \) \(14190224035519964910538321026303280867605484\) \(\beta_{4}\mathstrut -\mathstrut \) \(4585280352109805668271447470351863471821186041965\) \(\beta_{3}\mathstrut +\mathstrut \) \(17892103407323365203630525268140012620340148590969230822\) \(\beta_{2}\mathstrut +\mathstrut \) \(2842237229076009581805829080353737246520310347603912133999408914467\) \(\beta_{1}\mathstrut -\mathstrut \) \(33523342368531939213043617462018388045570540823503359218626770883062512783505091257\)\()/995328\)
\(\nu^{6}\)\(=\)\((\)\(45\!\cdots\!40\) \(\beta_{8}\mathstrut -\mathstrut \) \(63\!\cdots\!77\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\!\cdots\!18\) \(\beta_{6}\mathstrut +\mathstrut \) \(38\!\cdots\!08\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\!\cdots\!32\) \(\beta_{4}\mathstrut +\mathstrut \) \(30\!\cdots\!13\) \(\beta_{3}\mathstrut +\mathstrut \) \(69\!\cdots\!06\) \(\beta_{2}\mathstrut -\mathstrut \) \(67\!\cdots\!43\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\!\cdots\!81\)\()/331776\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(40\!\cdots\!20\) \(\beta_{8}\mathstrut +\mathstrut \) \(85\!\cdots\!91\) \(\beta_{7}\mathstrut +\mathstrut \) \(60\!\cdots\!42\) \(\beta_{6}\mathstrut +\mathstrut \) \(33\!\cdots\!68\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\!\cdots\!44\) \(\beta_{4}\mathstrut -\mathstrut \) \(30\!\cdots\!67\) \(\beta_{3}\mathstrut +\mathstrut \) \(66\!\cdots\!18\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\!\cdots\!53\) \(\beta_{1}\mathstrut -\mathstrut \) \(23\!\cdots\!35\)\()/331776\)
\(\nu^{8}\)\(=\)\((\)\(35\!\cdots\!88\) \(\beta_{8}\mathstrut -\mathstrut \) \(36\!\cdots\!47\) \(\beta_{7}\mathstrut +\mathstrut \) \(82\!\cdots\!66\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\!\cdots\!92\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\!\cdots\!88\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\!\cdots\!95\) \(\beta_{3}\mathstrut +\mathstrut \) \(42\!\cdots\!90\) \(\beta_{2}\mathstrut -\mathstrut \) \(37\!\cdots\!09\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\!\cdots\!55\)\()/995328\)

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
−3.94089e15
−3.29839e15
−2.30101e15
−1.42245e15
5.24199e14
9.66339e14
2.64772e15
3.15688e15
3.66758e15
−1.88610e17 4.27041e26 2.51891e34 −5.60099e39 −8.05442e43 −6.14078e47 −2.79228e51 −6.39314e53 1.05640e57
1.2 −1.57770e17 −1.34800e27 1.45067e34 3.80102e39 2.12674e44 5.08173e47 −6.50350e50 9.95431e53 −5.99686e56
1.3 −1.09896e17 1.02220e27 1.69249e33 3.22293e39 −1.12336e44 −9.90814e46 9.55225e50 2.23221e53 −3.54186e56
1.4 −6.77248e16 −4.98640e26 −5.79795e33 −2.60538e39 3.37703e43 3.08065e47 1.09596e51 −5.73036e53 1.76449e56
1.5 2.57142e16 −9.84309e26 −9.72337e33 1.11425e39 −2.53107e43 −8.63240e47 −5.17060e50 1.47185e53 2.86521e55
1.6 4.69369e16 1.07652e27 −8.18152e33 −1.53406e39 5.05287e43 2.72254e47 −8.71436e50 3.37224e53 −7.20041e55
1.7 1.27643e17 −2.19803e26 5.90823e33 5.13288e39 −2.80563e43 6.06875e47 −5.71378e50 −7.73365e53 6.55179e56
1.8 1.52083e17 −1.34317e27 1.27446e34 −5.29218e39 −2.04274e44 7.47667e47 3.58920e50 9.82432e53 −8.04850e56
1.9 1.76597e17 8.20335e26 2.08018e34 −1.35461e39 1.44868e44 −1.04223e48 1.83964e51 −1.48729e53 −2.39220e56
\(n\): 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 \(S_{114}^{\mathrm{new}}(\Gamma_0(1))\).