Properties

Label 1.124.a.a
Level 1
Weight 124
Character orbit 1.a
Self dual Yes
Analytic conductor 95.808
Analytic rank 0
Dimension 10
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 124 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(95.8076224914\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{178}\cdot 3^{70}\cdot 5^{22}\cdot 7^{9}\cdot 11^{6}\cdot 13^{2}\cdot 17\cdot 31^{2}\cdot 41^{3} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(220236429173363460 - \beta_{1}) q^{2}\) \(+(\)\(13\!\cdots\!60\)\( - 10279317013 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(57\!\cdots\!28\)\( - 259966369609758270 \beta_{1} - 4944179 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(11\!\cdots\!02\)\( - \)\(35\!\cdots\!43\)\( \beta_{1} + 694348080269 \beta_{2} - 165292 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(17\!\cdots\!52\)\( - \)\(31\!\cdots\!96\)\( \beta_{1} - 49080586918974807 \beta_{2} + 17153980966 \beta_{3} - 12917 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(58\!\cdots\!00\)\( - \)\(17\!\cdots\!77\)\( \beta_{1} - \)\(77\!\cdots\!56\)\( \beta_{2} + 31074175206452 \beta_{3} - 84285805 \beta_{4} - 2393 \beta_{5} + \beta_{6}) q^{7}\) \(+(\)\(31\!\cdots\!20\)\( - \)\(47\!\cdots\!75\)\( \beta_{1} - \)\(19\!\cdots\!07\)\( \beta_{2} + 1219253674561692790 \beta_{3} - 883856762568 \beta_{4} - 2345239 \beta_{5} - 597 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(11\!\cdots\!57\)\( + \)\(16\!\cdots\!58\)\( \beta_{1} - \)\(22\!\cdots\!09\)\( \beta_{2} - \)\(10\!\cdots\!79\)\( \beta_{3} - 391080143972015 \beta_{4} - 3531087761 \beta_{5} + 718715 \beta_{6} - 384 \beta_{7} + \beta_{8}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(220236429173363460 - \beta_{1}) q^{2}\) \(+(\)\(13\!\cdots\!60\)\( - 10279317013 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(57\!\cdots\!28\)\( - 259966369609758270 \beta_{1} - 4944179 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(11\!\cdots\!02\)\( - \)\(35\!\cdots\!43\)\( \beta_{1} + 694348080269 \beta_{2} - 165292 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(17\!\cdots\!52\)\( - \)\(31\!\cdots\!96\)\( \beta_{1} - 49080586918974807 \beta_{2} + 17153980966 \beta_{3} - 12917 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(58\!\cdots\!00\)\( - \)\(17\!\cdots\!77\)\( \beta_{1} - \)\(77\!\cdots\!56\)\( \beta_{2} + 31074175206452 \beta_{3} - 84285805 \beta_{4} - 2393 \beta_{5} + \beta_{6}) q^{7}\) \(+(\)\(31\!\cdots\!20\)\( - \)\(47\!\cdots\!75\)\( \beta_{1} - \)\(19\!\cdots\!07\)\( \beta_{2} + 1219253674561692790 \beta_{3} - 883856762568 \beta_{4} - 2345239 \beta_{5} - 597 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(11\!\cdots\!57\)\( + \)\(16\!\cdots\!58\)\( \beta_{1} - \)\(22\!\cdots\!09\)\( \beta_{2} - \)\(10\!\cdots\!79\)\( \beta_{3} - 391080143972015 \beta_{4} - 3531087761 \beta_{5} + 718715 \beta_{6} - 384 \beta_{7} + \beta_{8}) q^{9}\) \(+(\)\(60\!\cdots\!92\)\( + \)\(15\!\cdots\!17\)\( \beta_{1} + \)\(34\!\cdots\!53\)\( \beta_{2} - \)\(88\!\cdots\!06\)\( \beta_{3} - 262460983776109564 \beta_{4} + 6088979608589 \beta_{5} + 158731213 \beta_{6} + 272350 \beta_{7} + 251 \beta_{8} - \beta_{9}) q^{10}\) \(+(\)\(20\!\cdots\!72\)\( - \)\(19\!\cdots\!49\)\( \beta_{1} - \)\(48\!\cdots\!75\)\( \beta_{2} - \)\(12\!\cdots\!92\)\( \beta_{3} - \)\(10\!\cdots\!62\)\( \beta_{4} - 727331012694110 \beta_{5} + 100824633030 \beta_{6} - 112318976 \beta_{7} + 169884 \beta_{8}) q^{11}\) \(+(\)\(40\!\cdots\!20\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(23\!\cdots\!88\)\( \beta_{2} + \)\(45\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!68\)\( \beta_{4} - 382254920140204344 \beta_{5} + 4151057026728 \beta_{6} - 56598374736 \beta_{7} + 18812760 \beta_{8} + 27000 \beta_{9}) q^{12}\) \(+(-\)\(29\!\cdots\!30\)\( - \)\(31\!\cdots\!95\)\( \beta_{1} + \)\(27\!\cdots\!79\)\( \beta_{2} + \)\(10\!\cdots\!58\)\( \beta_{3} - \)\(66\!\cdots\!17\)\( \beta_{4} - 1655180579307371118 \beta_{5} + 2053178926231226 \beta_{6} - 4517046842624 \beta_{7} - 3767924530 \beta_{8} - 4096000 \beta_{9}) q^{13}\) \(+(\)\(27\!\cdots\!36\)\( - \)\(37\!\cdots\!72\)\( \beta_{1} - \)\(33\!\cdots\!46\)\( \beta_{2} - \)\(19\!\cdots\!88\)\( \beta_{3} - \)\(12\!\cdots\!62\)\( \beta_{4} - \)\(10\!\cdots\!54\)\( \beta_{5} + 2509046384125715580 \beta_{6} - 71759244972632 \beta_{7} + 180182388708 \beta_{8} + 344452500 \beta_{9}) q^{14}\) \(+(-\)\(39\!\cdots\!96\)\( + \)\(13\!\cdots\!29\)\( \beta_{1} + \)\(64\!\cdots\!96\)\( \beta_{2} - \)\(37\!\cdots\!32\)\( \beta_{3} + \)\(25\!\cdots\!57\)\( \beta_{4} - \)\(86\!\cdots\!47\)\( \beta_{5} + \)\(19\!\cdots\!51\)\( \beta_{6} + 39791982366489600 \beta_{7} + 1391792694552 \beta_{8} - 20288765952 \beta_{9}) q^{15}\) \(+(\)\(17\!\cdots\!76\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} - \)\(49\!\cdots\!00\)\( \beta_{2} + \)\(48\!\cdots\!92\)\( \beta_{3} - \)\(20\!\cdots\!08\)\( \beta_{4} - \)\(95\!\cdots\!68\)\( \beta_{5} - \)\(32\!\cdots\!60\)\( \beta_{6} - 1904789829199852456 \beta_{7} - 608240109734336 \beta_{8} + 919045960000 \beta_{9}) q^{16}\) \(+(\)\(12\!\cdots\!30\)\( + \)\(54\!\cdots\!06\)\( \beta_{1} + \)\(30\!\cdots\!87\)\( \beta_{2} - \)\(12\!\cdots\!47\)\( \beta_{3} - \)\(10\!\cdots\!95\)\( \beta_{4} - \)\(46\!\cdots\!97\)\( \beta_{5} - \)\(26\!\cdots\!01\)\( \beta_{6} + 22426747470803233920 \beta_{7} + 38826615922472205 \beta_{8} - 33699815424000 \beta_{9}) q^{17}\) \(+(-\)\(32\!\cdots\!40\)\( - \)\(13\!\cdots\!63\)\( \beta_{1} - \)\(34\!\cdots\!66\)\( \beta_{2} + \)\(23\!\cdots\!52\)\( \beta_{3} + \)\(35\!\cdots\!40\)\( \beta_{4} - \)\(11\!\cdots\!18\)\( \beta_{5} + \)\(54\!\cdots\!86\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7} - 1538837366258255310 \beta_{8} + 1033386534794250 \beta_{9}) q^{18}\) \(+(\)\(16\!\cdots\!00\)\( - \)\(17\!\cdots\!27\)\( \beta_{1} - \)\(18\!\cdots\!41\)\( \beta_{2} - \)\(70\!\cdots\!32\)\( \beta_{3} + \)\(18\!\cdots\!54\)\( \beta_{4} + \)\(21\!\cdots\!30\)\( \beta_{5} + \)\(13\!\cdots\!90\)\( \beta_{6} - \)\(65\!\cdots\!68\)\( \beta_{7} + 44936514964329616212 \beta_{8} - 27102607933440000 \beta_{9}) q^{19}\) \(+(-\)\(25\!\cdots\!84\)\( + \)\(90\!\cdots\!56\)\( \beta_{1} + \)\(82\!\cdots\!22\)\( \beta_{2} - \)\(30\!\cdots\!06\)\( \beta_{3} + \)\(11\!\cdots\!08\)\( \beta_{4} + \)\(92\!\cdots\!20\)\( \beta_{5} - \)\(58\!\cdots\!60\)\( \beta_{6} + \)\(16\!\cdots\!00\)\( \beta_{7} - \)\(10\!\cdots\!20\)\( \beta_{8} + 617995773693997920 \beta_{9}) q^{20}\) \(+(\)\(47\!\cdots\!72\)\( - \)\(11\!\cdots\!04\)\( \beta_{1} - \)\(74\!\cdots\!30\)\( \beta_{2} - \)\(56\!\cdots\!34\)\( \beta_{3} + \)\(11\!\cdots\!46\)\( \beta_{4} - \)\(11\!\cdots\!38\)\( \beta_{5} + \)\(55\!\cdots\!10\)\( \beta_{6} - \)\(22\!\cdots\!24\)\( \beta_{7} + \)\(19\!\cdots\!06\)\( \beta_{8} - 12403404875796480000 \beta_{9}) q^{21}\) \(+(\)\(36\!\cdots\!20\)\( - \)\(95\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!01\)\( \beta_{2} + \)\(11\!\cdots\!18\)\( \beta_{3} + \)\(50\!\cdots\!33\)\( \beta_{4} - \)\(24\!\cdots\!23\)\( \beta_{5} + \)\(78\!\cdots\!76\)\( \beta_{6} + \)\(11\!\cdots\!16\)\( \beta_{7} - \)\(28\!\cdots\!20\)\( \beta_{8} + \)\(22\!\cdots\!00\)\( \beta_{9}) q^{22}\) \(+(\)\(60\!\cdots\!80\)\( + \)\(31\!\cdots\!49\)\( \beta_{1} + \)\(25\!\cdots\!56\)\( \beta_{2} + \)\(59\!\cdots\!40\)\( \beta_{3} - \)\(23\!\cdots\!35\)\( \beta_{4} + \)\(43\!\cdots\!05\)\( \beta_{5} - \)\(25\!\cdots\!05\)\( \beta_{6} + \)\(28\!\cdots\!60\)\( \beta_{7} + \)\(35\!\cdots\!20\)\( \beta_{8} - \)\(35\!\cdots\!00\)\( \beta_{9}) q^{23}\) \(+(\)\(19\!\cdots\!60\)\( - \)\(55\!\cdots\!80\)\( \beta_{1} - \)\(72\!\cdots\!24\)\( \beta_{2} + \)\(61\!\cdots\!12\)\( \beta_{3} - \)\(46\!\cdots\!04\)\( \beta_{4} + \)\(12\!\cdots\!64\)\( \beta_{5} + \)\(20\!\cdots\!60\)\( \beta_{6} - \)\(88\!\cdots\!76\)\( \beta_{7} - \)\(33\!\cdots\!36\)\( \beta_{8} + \)\(50\!\cdots\!00\)\( \beta_{9}) q^{24}\) \(+(\)\(48\!\cdots\!15\)\( + \)\(26\!\cdots\!40\)\( \beta_{1} + \)\(26\!\cdots\!70\)\( \beta_{2} + \)\(58\!\cdots\!70\)\( \beta_{3} + \)\(19\!\cdots\!70\)\( \beta_{4} - \)\(62\!\cdots\!10\)\( \beta_{5} + \)\(14\!\cdots\!30\)\( \beta_{6} + \)\(13\!\cdots\!00\)\( \beta_{7} + \)\(19\!\cdots\!10\)\( \beta_{8} - \)\(65\!\cdots\!60\)\( \beta_{9}) q^{25}\) \(+(\)\(50\!\cdots\!32\)\( - \)\(76\!\cdots\!15\)\( \beta_{1} - \)\(58\!\cdots\!95\)\( \beta_{2} + \)\(84\!\cdots\!62\)\( \beta_{3} + \)\(58\!\cdots\!12\)\( \beta_{4} + \)\(12\!\cdots\!73\)\( \beta_{5} - \)\(50\!\cdots\!55\)\( \beta_{6} - \)\(14\!\cdots\!62\)\( \beta_{7} + \)\(36\!\cdots\!43\)\( \beta_{8} + \)\(77\!\cdots\!75\)\( \beta_{9}) q^{26}\) \(+(\)\(77\!\cdots\!80\)\( + \)\(38\!\cdots\!60\)\( \beta_{1} - \)\(17\!\cdots\!58\)\( \beta_{2} + \)\(27\!\cdots\!20\)\( \beta_{3} + \)\(72\!\cdots\!78\)\( \beta_{4} + \)\(63\!\cdots\!34\)\( \beta_{5} + \)\(49\!\cdots\!02\)\( \beta_{6} + \)\(97\!\cdots\!16\)\( \beta_{7} - \)\(28\!\cdots\!20\)\( \beta_{8} - \)\(82\!\cdots\!00\)\( \beta_{9}) q^{27}\) \(+(\)\(74\!\cdots\!20\)\( - \)\(13\!\cdots\!92\)\( \beta_{1} - \)\(51\!\cdots\!32\)\( \beta_{2} + \)\(83\!\cdots\!60\)\( \beta_{3} - \)\(60\!\cdots\!52\)\( \beta_{4} - \)\(50\!\cdots\!36\)\( \beta_{5} - \)\(99\!\cdots\!88\)\( \beta_{6} - \)\(29\!\cdots\!24\)\( \beta_{7} + \)\(47\!\cdots\!60\)\( \beta_{8} + \)\(80\!\cdots\!00\)\( \beta_{9}) q^{28}\) \(+(-\)\(28\!\cdots\!90\)\( - \)\(19\!\cdots\!15\)\( \beta_{1} + \)\(20\!\cdots\!89\)\( \beta_{2} - \)\(34\!\cdots\!44\)\( \beta_{3} - \)\(12\!\cdots\!43\)\( \beta_{4} - \)\(14\!\cdots\!32\)\( \beta_{5} - \)\(30\!\cdots\!40\)\( \beta_{6} - \)\(24\!\cdots\!72\)\( \beta_{7} - \)\(54\!\cdots\!52\)\( \beta_{8} - \)\(70\!\cdots\!00\)\( \beta_{9}) q^{29}\) \(+(-\)\(22\!\cdots\!16\)\( + \)\(40\!\cdots\!44\)\( \beta_{1} - \)\(90\!\cdots\!22\)\( \beta_{2} - \)\(58\!\cdots\!44\)\( \beta_{3} + \)\(46\!\cdots\!42\)\( \beta_{4} + \)\(33\!\cdots\!30\)\( \beta_{5} + \)\(40\!\cdots\!60\)\( \beta_{6} + \)\(40\!\cdots\!00\)\( \beta_{7} + \)\(48\!\cdots\!20\)\( \beta_{8} + \)\(55\!\cdots\!80\)\( \beta_{9}) q^{30}\) \(+(\)\(87\!\cdots\!12\)\( - \)\(29\!\cdots\!68\)\( \beta_{1} - \)\(67\!\cdots\!52\)\( \beta_{2} + \)\(10\!\cdots\!16\)\( \beta_{3} + \)\(12\!\cdots\!48\)\( \beta_{4} - \)\(15\!\cdots\!92\)\( \beta_{5} - \)\(21\!\cdots\!80\)\( \beta_{6} - \)\(26\!\cdots\!64\)\( \beta_{7} - \)\(34\!\cdots\!24\)\( \beta_{8} - \)\(38\!\cdots\!00\)\( \beta_{9}) q^{31}\) \(+(\)\(17\!\cdots\!60\)\( - \)\(14\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(22\!\cdots\!88\)\( \beta_{3} - \)\(19\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!92\)\( \beta_{5} - \)\(15\!\cdots\!96\)\( \beta_{6} + \)\(29\!\cdots\!00\)\( \beta_{7} + \)\(18\!\cdots\!40\)\( \beta_{8} + \)\(23\!\cdots\!00\)\( \beta_{9}) q^{32}\) \(+(\)\(33\!\cdots\!20\)\( - \)\(46\!\cdots\!94\)\( \beta_{1} - \)\(88\!\cdots\!03\)\( \beta_{2} + \)\(76\!\cdots\!67\)\( \beta_{3} - \)\(86\!\cdots\!57\)\( \beta_{4} + \)\(11\!\cdots\!01\)\( \beta_{5} + \)\(93\!\cdots\!53\)\( \beta_{6} + \)\(12\!\cdots\!76\)\( \beta_{7} - \)\(63\!\cdots\!65\)\( \beta_{8} - \)\(11\!\cdots\!00\)\( \beta_{9}) q^{33}\) \(+(-\)\(62\!\cdots\!44\)\( - \)\(10\!\cdots\!32\)\( \beta_{1} - \)\(56\!\cdots\!98\)\( \beta_{2} - \)\(24\!\cdots\!84\)\( \beta_{3} + \)\(72\!\cdots\!24\)\( \beta_{4} + \)\(33\!\cdots\!50\)\( \beta_{5} - \)\(68\!\cdots\!70\)\( \beta_{6} - \)\(14\!\cdots\!28\)\( \beta_{7} - \)\(80\!\cdots\!58\)\( \beta_{8} + \)\(39\!\cdots\!50\)\( \beta_{9}) q^{34}\) \(+(-\)\(12\!\cdots\!88\)\( + \)\(94\!\cdots\!12\)\( \beta_{1} - \)\(28\!\cdots\!32\)\( \beta_{2} - \)\(54\!\cdots\!76\)\( \beta_{3} + \)\(60\!\cdots\!96\)\( \beta_{4} - \)\(55\!\cdots\!36\)\( \beta_{5} + \)\(15\!\cdots\!88\)\( \beta_{6} + \)\(87\!\cdots\!00\)\( \beta_{7} + \)\(33\!\cdots\!76\)\( \beta_{8} + \)\(18\!\cdots\!24\)\( \beta_{9}) q^{35}\) \(+(-\)\(96\!\cdots\!44\)\( - \)\(24\!\cdots\!46\)\( \beta_{1} - \)\(10\!\cdots\!23\)\( \beta_{2} - \)\(10\!\cdots\!15\)\( \beta_{3} - \)\(27\!\cdots\!72\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5} + \)\(10\!\cdots\!40\)\( \beta_{6} - \)\(24\!\cdots\!60\)\( \beta_{7} - \)\(30\!\cdots\!20\)\( \beta_{8} - \)\(17\!\cdots\!00\)\( \beta_{9}) q^{36}\) \(+(\)\(62\!\cdots\!90\)\( + \)\(19\!\cdots\!21\)\( \beta_{1} - \)\(81\!\cdots\!89\)\( \beta_{2} + \)\(62\!\cdots\!10\)\( \beta_{3} + \)\(13\!\cdots\!31\)\( \beta_{4} + \)\(12\!\cdots\!78\)\( \beta_{5} - \)\(98\!\cdots\!06\)\( \beta_{6} - \)\(10\!\cdots\!08\)\( \beta_{7} + \)\(19\!\cdots\!50\)\( \beta_{8} + \)\(19\!\cdots\!00\)\( \beta_{9}) q^{37}\) \(+(\)\(28\!\cdots\!80\)\( + \)\(46\!\cdots\!76\)\( \beta_{1} + \)\(32\!\cdots\!29\)\( \beta_{2} + \)\(54\!\cdots\!34\)\( \beta_{3} + \)\(64\!\cdots\!87\)\( \beta_{4} - \)\(64\!\cdots\!25\)\( \beta_{5} + \)\(23\!\cdots\!40\)\( \beta_{6} + \)\(17\!\cdots\!64\)\( \beta_{7} - \)\(87\!\cdots\!60\)\( \beta_{8} - \)\(14\!\cdots\!00\)\( \beta_{9}) q^{38}\) \(+(\)\(32\!\cdots\!24\)\( - \)\(32\!\cdots\!39\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2} + \)\(88\!\cdots\!76\)\( \beta_{3} - \)\(79\!\cdots\!67\)\( \beta_{4} - \)\(18\!\cdots\!95\)\( \beta_{5} + \)\(13\!\cdots\!95\)\( \beta_{6} - \)\(11\!\cdots\!56\)\( \beta_{7} + \)\(26\!\cdots\!44\)\( \beta_{8} + \)\(89\!\cdots\!00\)\( \beta_{9}) q^{39}\) \(+(-\)\(15\!\cdots\!00\)\( + \)\(39\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!30\)\( \beta_{2} - \)\(22\!\cdots\!80\)\( \beta_{3} - \)\(98\!\cdots\!00\)\( \beta_{4} + \)\(26\!\cdots\!30\)\( \beta_{5} - \)\(12\!\cdots\!90\)\( \beta_{6} + \)\(46\!\cdots\!50\)\( \beta_{7} - \)\(16\!\cdots\!80\)\( \beta_{8} - \)\(46\!\cdots\!20\)\( \beta_{9}) q^{40}\) \(+(-\)\(10\!\cdots\!58\)\( - \)\(59\!\cdots\!44\)\( \beta_{1} - \)\(68\!\cdots\!34\)\( \beta_{2} - \)\(72\!\cdots\!62\)\( \beta_{3} - \)\(15\!\cdots\!38\)\( \beta_{4} - \)\(62\!\cdots\!74\)\( \beta_{5} + \)\(29\!\cdots\!10\)\( \beta_{6} - \)\(86\!\cdots\!40\)\( \beta_{7} - \)\(33\!\cdots\!30\)\( \beta_{8} + \)\(20\!\cdots\!00\)\( \beta_{9}) q^{41}\) \(+(\)\(20\!\cdots\!80\)\( - \)\(48\!\cdots\!76\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2} + \)\(20\!\cdots\!12\)\( \beta_{3} + \)\(19\!\cdots\!80\)\( \beta_{4} - \)\(25\!\cdots\!08\)\( \beta_{5} + \)\(10\!\cdots\!36\)\( \beta_{6} - \)\(26\!\cdots\!00\)\( \beta_{7} + \)\(19\!\cdots\!20\)\( \beta_{8} - \)\(73\!\cdots\!00\)\( \beta_{9}) q^{42}\) \(+(\)\(46\!\cdots\!00\)\( - \)\(41\!\cdots\!71\)\( \beta_{1} - \)\(73\!\cdots\!91\)\( \beta_{2} + \)\(64\!\cdots\!36\)\( \beta_{3} + \)\(46\!\cdots\!12\)\( \beta_{4} + \)\(16\!\cdots\!32\)\( \beta_{5} - \)\(95\!\cdots\!44\)\( \beta_{6} + \)\(26\!\cdots\!64\)\( \beta_{7} - \)\(27\!\cdots\!00\)\( \beta_{8} + \)\(19\!\cdots\!00\)\( \beta_{9}) q^{43}\) \(+(-\)\(52\!\cdots\!84\)\( - \)\(12\!\cdots\!28\)\( \beta_{1} - \)\(25\!\cdots\!40\)\( \beta_{2} + \)\(14\!\cdots\!48\)\( \beta_{3} - \)\(60\!\cdots\!32\)\( \beta_{4} + \)\(67\!\cdots\!28\)\( \beta_{5} + \)\(14\!\cdots\!00\)\( \beta_{6} - \)\(89\!\cdots\!24\)\( \beta_{7} - \)\(43\!\cdots\!04\)\( \beta_{8} - \)\(18\!\cdots\!00\)\( \beta_{9}) q^{44}\) \(+(-\)\(27\!\cdots\!46\)\( - \)\(26\!\cdots\!11\)\( \beta_{1} + \)\(55\!\cdots\!03\)\( \beta_{2} - \)\(23\!\cdots\!74\)\( \beta_{3} - \)\(17\!\cdots\!73\)\( \beta_{4} - \)\(27\!\cdots\!10\)\( \beta_{5} + \)\(13\!\cdots\!30\)\( \beta_{6} + \)\(28\!\cdots\!00\)\( \beta_{7} + \)\(44\!\cdots\!10\)\( \beta_{8} - \)\(16\!\cdots\!60\)\( \beta_{9}) q^{45}\) \(+(-\)\(49\!\cdots\!68\)\( - \)\(11\!\cdots\!64\)\( \beta_{1} + \)\(45\!\cdots\!46\)\( \beta_{2} - \)\(22\!\cdots\!92\)\( \beta_{3} + \)\(10\!\cdots\!62\)\( \beta_{4} + \)\(51\!\cdots\!66\)\( \beta_{5} - \)\(86\!\cdots\!60\)\( \beta_{6} + \)\(10\!\cdots\!20\)\( \beta_{7} - \)\(25\!\cdots\!40\)\( \beta_{8} + \)\(13\!\cdots\!00\)\( \beta_{9}) q^{46}\) \(+(\)\(17\!\cdots\!20\)\( - \)\(74\!\cdots\!34\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2} + \)\(69\!\cdots\!12\)\( \beta_{3} - \)\(91\!\cdots\!46\)\( \beta_{4} + \)\(26\!\cdots\!34\)\( \beta_{5} + \)\(18\!\cdots\!82\)\( \beta_{6} - \)\(37\!\cdots\!52\)\( \beta_{7} + \)\(10\!\cdots\!40\)\( \beta_{8} - \)\(60\!\cdots\!00\)\( \beta_{9}) q^{47}\) \(+(\)\(51\!\cdots\!80\)\( - \)\(56\!\cdots\!16\)\( \beta_{1} + \)\(12\!\cdots\!16\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4} - \)\(11\!\cdots\!92\)\( \beta_{5} + \)\(24\!\cdots\!44\)\( \beta_{6} - \)\(76\!\cdots\!20\)\( \beta_{7} - \)\(26\!\cdots\!00\)\( \beta_{8} + \)\(18\!\cdots\!00\)\( \beta_{9}) q^{48}\) \(+(\)\(48\!\cdots\!13\)\( - \)\(13\!\cdots\!04\)\( \beta_{1} - \)\(39\!\cdots\!64\)\( \beta_{2} - \)\(30\!\cdots\!72\)\( \beta_{3} + \)\(16\!\cdots\!92\)\( \beta_{4} - \)\(99\!\cdots\!84\)\( \beta_{5} - \)\(19\!\cdots\!60\)\( \beta_{6} + \)\(13\!\cdots\!60\)\( \beta_{7} + \)\(24\!\cdots\!00\)\( \beta_{8} - \)\(28\!\cdots\!00\)\( \beta_{9}) q^{49}\) \(+(-\)\(33\!\cdots\!60\)\( - \)\(10\!\cdots\!35\)\( \beta_{1} - \)\(78\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3} + \)\(78\!\cdots\!20\)\( \beta_{4} + \)\(13\!\cdots\!40\)\( \beta_{5} - \)\(17\!\cdots\!20\)\( \beta_{6} - \)\(70\!\cdots\!00\)\( \beta_{7} + \)\(19\!\cdots\!60\)\( \beta_{8} - \)\(64\!\cdots\!60\)\( \beta_{9}) q^{50}\) \(+(-\)\(16\!\cdots\!48\)\( - \)\(21\!\cdots\!88\)\( \beta_{1} - \)\(16\!\cdots\!94\)\( \beta_{2} - \)\(14\!\cdots\!48\)\( \beta_{3} + \)\(13\!\cdots\!46\)\( \beta_{4} - \)\(38\!\cdots\!50\)\( \beta_{5} + \)\(43\!\cdots\!70\)\( \beta_{6} + \)\(21\!\cdots\!88\)\( \beta_{7} - \)\(12\!\cdots\!32\)\( \beta_{8} + \)\(68\!\cdots\!00\)\( \beta_{9}) q^{51}\) \(+(\)\(16\!\cdots\!00\)\( - \)\(99\!\cdots\!76\)\( \beta_{1} + \)\(10\!\cdots\!06\)\( \beta_{2} + \)\(14\!\cdots\!22\)\( \beta_{3} - \)\(19\!\cdots\!76\)\( \beta_{4} - \)\(13\!\cdots\!36\)\( \beta_{5} - \)\(15\!\cdots\!88\)\( \beta_{6} - \)\(37\!\cdots\!72\)\( \beta_{7} + \)\(43\!\cdots\!00\)\( \beta_{8} - \)\(28\!\cdots\!00\)\( \beta_{9}) q^{52}\) \(+(\)\(98\!\cdots\!10\)\( - \)\(22\!\cdots\!55\)\( \beta_{1} + \)\(18\!\cdots\!95\)\( \beta_{2} + \)\(41\!\cdots\!30\)\( \beta_{3} + \)\(11\!\cdots\!67\)\( \beta_{4} + \)\(16\!\cdots\!46\)\( \beta_{5} + \)\(13\!\cdots\!18\)\( \beta_{6} + \)\(13\!\cdots\!04\)\( \beta_{7} - \)\(82\!\cdots\!10\)\( \beta_{8} + \)\(70\!\cdots\!00\)\( \beta_{9}) q^{53}\) \(+(-\)\(60\!\cdots\!00\)\( - \)\(34\!\cdots\!88\)\( \beta_{1} + \)\(75\!\cdots\!02\)\( \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3} + \)\(15\!\cdots\!18\)\( \beta_{4} + \)\(13\!\cdots\!18\)\( \beta_{5} + \)\(33\!\cdots\!20\)\( \beta_{6} + \)\(54\!\cdots\!36\)\( \beta_{7} - \)\(18\!\cdots\!24\)\( \beta_{8} - \)\(44\!\cdots\!00\)\( \beta_{9}) q^{54}\) \(+(-\)\(12\!\cdots\!36\)\( - \)\(26\!\cdots\!01\)\( \beta_{1} + \)\(65\!\cdots\!08\)\( \beta_{2} - \)\(51\!\cdots\!44\)\( \beta_{3} - \)\(75\!\cdots\!93\)\( \beta_{4} - \)\(35\!\cdots\!25\)\( \beta_{5} + \)\(16\!\cdots\!25\)\( \beta_{6} + \)\(27\!\cdots\!00\)\( \beta_{7} + \)\(74\!\cdots\!00\)\( \beta_{8} - \)\(50\!\cdots\!00\)\( \beta_{9}) q^{55}\) \(+(-\)\(57\!\cdots\!40\)\( - \)\(51\!\cdots\!96\)\( \beta_{1} - \)\(45\!\cdots\!08\)\( \beta_{2} + \)\(86\!\cdots\!88\)\( \beta_{3} - \)\(13\!\cdots\!24\)\( \beta_{4} - \)\(80\!\cdots\!84\)\( \beta_{5} - \)\(21\!\cdots\!60\)\( \beta_{6} - \)\(20\!\cdots\!88\)\( \beta_{7} - \)\(28\!\cdots\!08\)\( \beta_{8} + \)\(27\!\cdots\!00\)\( \beta_{9}) q^{56}\) \(+(\)\(13\!\cdots\!60\)\( + \)\(11\!\cdots\!74\)\( \beta_{1} - \)\(25\!\cdots\!33\)\( \beta_{2} + \)\(15\!\cdots\!61\)\( \beta_{3} - \)\(39\!\cdots\!63\)\( \beta_{4} + \)\(34\!\cdots\!27\)\( \beta_{5} + \)\(79\!\cdots\!71\)\( \beta_{6} + \)\(24\!\cdots\!44\)\( \beta_{7} + \)\(53\!\cdots\!45\)\( \beta_{8} - \)\(77\!\cdots\!00\)\( \beta_{9}) q^{57}\) \(+(\)\(25\!\cdots\!20\)\( + \)\(62\!\cdots\!61\)\( \beta_{1} - \)\(16\!\cdots\!15\)\( \beta_{2} + \)\(23\!\cdots\!66\)\( \beta_{3} + \)\(15\!\cdots\!76\)\( \beta_{4} + \)\(56\!\cdots\!09\)\( \beta_{5} - \)\(12\!\cdots\!03\)\( \beta_{6} + \)\(24\!\cdots\!82\)\( \beta_{7} - \)\(12\!\cdots\!45\)\( \beta_{8} + \)\(85\!\cdots\!75\)\( \beta_{9}) q^{58}\) \(+(-\)\(51\!\cdots\!60\)\( + \)\(87\!\cdots\!69\)\( \beta_{1} + \)\(44\!\cdots\!85\)\( \beta_{2} + \)\(11\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!76\)\( \beta_{4} - \)\(37\!\cdots\!12\)\( \beta_{5} + \)\(63\!\cdots\!60\)\( \beta_{6} - \)\(14\!\cdots\!16\)\( \beta_{7} - \)\(22\!\cdots\!76\)\( \beta_{8} + \)\(32\!\cdots\!00\)\( \beta_{9}) q^{59}\) \(+(-\)\(66\!\cdots\!68\)\( + \)\(62\!\cdots\!32\)\( \beta_{1} + \)\(67\!\cdots\!28\)\( \beta_{2} - \)\(12\!\cdots\!16\)\( \beta_{3} - \)\(63\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!16\)\( \beta_{5} - \)\(77\!\cdots\!72\)\( \beta_{6} + \)\(39\!\cdots\!00\)\( \beta_{7} + \)\(60\!\cdots\!56\)\( \beta_{8} - \)\(20\!\cdots\!56\)\( \beta_{9}) q^{60}\) \(+(-\)\(10\!\cdots\!18\)\( + \)\(55\!\cdots\!53\)\( \beta_{1} + \)\(42\!\cdots\!91\)\( \beta_{2} - \)\(28\!\cdots\!26\)\( \beta_{3} - \)\(45\!\cdots\!57\)\( \beta_{4} + \)\(32\!\cdots\!86\)\( \beta_{5} + \)\(28\!\cdots\!50\)\( \beta_{6} - \)\(34\!\cdots\!32\)\( \beta_{7} - \)\(35\!\cdots\!22\)\( \beta_{8} + \)\(56\!\cdots\!00\)\( \beta_{9}) q^{61}\) \(+(\)\(49\!\cdots\!20\)\( - \)\(11\!\cdots\!32\)\( \beta_{1} - \)\(49\!\cdots\!60\)\( \beta_{2} + \)\(72\!\cdots\!32\)\( \beta_{3} + \)\(44\!\cdots\!80\)\( \beta_{4} - \)\(39\!\cdots\!68\)\( \beta_{5} - \)\(10\!\cdots\!64\)\( \beta_{6} - \)\(13\!\cdots\!20\)\( \beta_{7} - \)\(60\!\cdots\!60\)\( \beta_{8} - \)\(55\!\cdots\!00\)\( \beta_{9}) q^{62}\) \(+(\)\(71\!\cdots\!40\)\( - \)\(36\!\cdots\!29\)\( \beta_{1} - \)\(77\!\cdots\!36\)\( \beta_{2} + \)\(48\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!99\)\( \beta_{4} - \)\(40\!\cdots\!93\)\( \beta_{5} + \)\(82\!\cdots\!01\)\( \beta_{6} + \)\(61\!\cdots\!08\)\( \beta_{7} - \)\(61\!\cdots\!40\)\( \beta_{8} - \)\(22\!\cdots\!00\)\( \beta_{9}) q^{63}\) \(+(\)\(97\!\cdots\!68\)\( - \)\(27\!\cdots\!28\)\( \beta_{1} - \)\(30\!\cdots\!96\)\( \beta_{2} + \)\(10\!\cdots\!68\)\( \beta_{3} - \)\(50\!\cdots\!96\)\( \beta_{4} - \)\(48\!\cdots\!28\)\( \beta_{5} + \)\(47\!\cdots\!80\)\( \beta_{6} - \)\(11\!\cdots\!60\)\( \beta_{7} + \)\(46\!\cdots\!20\)\( \beta_{8} + \)\(12\!\cdots\!00\)\( \beta_{9}) q^{64}\) \(+(-\)\(11\!\cdots\!76\)\( - \)\(37\!\cdots\!76\)\( \beta_{1} + \)\(30\!\cdots\!66\)\( \beta_{2} - \)\(15\!\cdots\!82\)\( \beta_{3} - \)\(26\!\cdots\!58\)\( \beta_{4} + \)\(33\!\cdots\!58\)\( \beta_{5} - \)\(14\!\cdots\!14\)\( \beta_{6} + \)\(13\!\cdots\!00\)\( \beta_{7} - \)\(11\!\cdots\!78\)\( \beta_{8} - \)\(28\!\cdots\!72\)\( \beta_{9}) q^{65}\) \(+(\)\(83\!\cdots\!44\)\( - \)\(12\!\cdots\!74\)\( \beta_{1} + \)\(11\!\cdots\!58\)\( \beta_{2} + \)\(61\!\cdots\!96\)\( \beta_{3} + \)\(88\!\cdots\!88\)\( \beta_{4} - \)\(66\!\cdots\!30\)\( \beta_{5} + \)\(70\!\cdots\!10\)\( \beta_{6} - \)\(22\!\cdots\!56\)\( \beta_{7} - \)\(60\!\cdots\!66\)\( \beta_{8} + \)\(14\!\cdots\!50\)\( \beta_{9}) q^{66}\) \(+(\)\(39\!\cdots\!80\)\( + \)\(26\!\cdots\!89\)\( \beta_{1} + \)\(30\!\cdots\!83\)\( \beta_{2} + \)\(21\!\cdots\!44\)\( \beta_{3} + \)\(82\!\cdots\!42\)\( \beta_{4} + \)\(30\!\cdots\!70\)\( \beta_{5} + \)\(14\!\cdots\!30\)\( \beta_{6} + \)\(57\!\cdots\!04\)\( \beta_{7} + \)\(13\!\cdots\!60\)\( \beta_{8} + \)\(14\!\cdots\!00\)\( \beta_{9}) q^{67}\) \(+(\)\(39\!\cdots\!60\)\( + \)\(13\!\cdots\!52\)\( \beta_{1} + \)\(29\!\cdots\!02\)\( \beta_{2} + \)\(23\!\cdots\!50\)\( \beta_{3} - \)\(16\!\cdots\!12\)\( \beta_{4} - \)\(13\!\cdots\!16\)\( \beta_{5} + \)\(13\!\cdots\!12\)\( \beta_{6} + \)\(92\!\cdots\!96\)\( \beta_{7} - \)\(42\!\cdots\!80\)\( \beta_{8} - \)\(56\!\cdots\!00\)\( \beta_{9}) q^{68}\) \(+(-\)\(19\!\cdots\!96\)\( + \)\(15\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} - \)\(20\!\cdots\!42\)\( \beta_{3} - \)\(22\!\cdots\!18\)\( \beta_{4} + \)\(11\!\cdots\!62\)\( \beta_{5} - \)\(71\!\cdots\!10\)\( \beta_{6} - \)\(12\!\cdots\!16\)\( \beta_{7} + \)\(43\!\cdots\!54\)\( \beta_{8} + \)\(92\!\cdots\!00\)\( \beta_{9}) q^{69}\) \(+(-\)\(18\!\cdots\!48\)\( + \)\(59\!\cdots\!32\)\( \beta_{1} - \)\(54\!\cdots\!96\)\( \beta_{2} - \)\(13\!\cdots\!52\)\( \beta_{3} + \)\(54\!\cdots\!76\)\( \beta_{4} - \)\(27\!\cdots\!40\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} + \)\(40\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!40\)\( \beta_{8} + \)\(51\!\cdots\!60\)\( \beta_{9}) q^{70}\) \(+(-\)\(14\!\cdots\!88\)\( + \)\(62\!\cdots\!59\)\( \beta_{1} - \)\(56\!\cdots\!52\)\( \beta_{2} - \)\(12\!\cdots\!48\)\( \beta_{3} + \)\(25\!\cdots\!39\)\( \beta_{4} + \)\(97\!\cdots\!23\)\( \beta_{5} + \)\(13\!\cdots\!65\)\( \beta_{6} - \)\(37\!\cdots\!56\)\( \beta_{7} - \)\(69\!\cdots\!36\)\( \beta_{8} - \)\(68\!\cdots\!00\)\( \beta_{9}) q^{71}\) \(+(\)\(38\!\cdots\!40\)\( + \)\(10\!\cdots\!49\)\( \beta_{1} + \)\(70\!\cdots\!29\)\( \beta_{2} + \)\(44\!\cdots\!14\)\( \beta_{3} - \)\(13\!\cdots\!00\)\( \beta_{4} + \)\(73\!\cdots\!89\)\( \beta_{5} - \)\(60\!\cdots\!33\)\( \beta_{6} - \)\(12\!\cdots\!65\)\( \beta_{7} + \)\(11\!\cdots\!60\)\( \beta_{8} + \)\(16\!\cdots\!00\)\( \beta_{9}) q^{72}\) \(+(\)\(21\!\cdots\!30\)\( + \)\(18\!\cdots\!22\)\( \beta_{1} + \)\(77\!\cdots\!47\)\( \beta_{2} + \)\(47\!\cdots\!69\)\( \beta_{3} - \)\(29\!\cdots\!03\)\( \beta_{4} + \)\(18\!\cdots\!15\)\( \beta_{5} + \)\(56\!\cdots\!35\)\( \beta_{6} + \)\(42\!\cdots\!84\)\( \beta_{7} + \)\(39\!\cdots\!65\)\( \beta_{8} - \)\(11\!\cdots\!00\)\( \beta_{9}) q^{73}\) \(+(-\)\(17\!\cdots\!84\)\( - \)\(12\!\cdots\!99\)\( \beta_{1} + \)\(14\!\cdots\!53\)\( \beta_{2} + \)\(17\!\cdots\!18\)\( \beta_{3} - \)\(36\!\cdots\!00\)\( \beta_{4} - \)\(29\!\cdots\!87\)\( \beta_{5} - \)\(99\!\cdots\!55\)\( \beta_{6} - \)\(20\!\cdots\!38\)\( \beta_{7} - \)\(60\!\cdots\!53\)\( \beta_{8} - \)\(51\!\cdots\!25\)\( \beta_{9}) q^{74}\) \(+(\)\(50\!\cdots\!80\)\( - \)\(38\!\cdots\!95\)\( \beta_{1} - \)\(18\!\cdots\!35\)\( \beta_{2} - \)\(72\!\cdots\!60\)\( \beta_{3} + \)\(90\!\cdots\!40\)\( \beta_{4} - \)\(12\!\cdots\!20\)\( \beta_{5} + \)\(85\!\cdots\!60\)\( \beta_{6} - \)\(11\!\cdots\!00\)\( \beta_{7} + \)\(11\!\cdots\!20\)\( \beta_{8} + \)\(18\!\cdots\!80\)\( \beta_{9}) q^{75}\) \(+(-\)\(87\!\cdots\!80\)\( - \)\(56\!\cdots\!92\)\( \beta_{1} - \)\(46\!\cdots\!96\)\( \beta_{2} - \)\(16\!\cdots\!64\)\( \beta_{3} + \)\(38\!\cdots\!52\)\( \beta_{4} + \)\(38\!\cdots\!32\)\( \beta_{5} - \)\(22\!\cdots\!40\)\( \beta_{6} + \)\(73\!\cdots\!24\)\( \beta_{7} - \)\(28\!\cdots\!36\)\( \beta_{8} - \)\(21\!\cdots\!00\)\( \beta_{9}) q^{76}\) \(+(\)\(12\!\cdots\!00\)\( - \)\(65\!\cdots\!04\)\( \beta_{1} - \)\(15\!\cdots\!82\)\( \beta_{2} + \)\(12\!\cdots\!38\)\( \beta_{3} + \)\(13\!\cdots\!50\)\( \beta_{4} - \)\(10\!\cdots\!82\)\( \beta_{5} + \)\(38\!\cdots\!74\)\( \beta_{6} + \)\(95\!\cdots\!40\)\( \beta_{7} - \)\(23\!\cdots\!50\)\( \beta_{8} - \)\(25\!\cdots\!00\)\( \beta_{9}) q^{77}\) \(+(\)\(53\!\cdots\!00\)\( - \)\(93\!\cdots\!32\)\( \beta_{1} - \)\(27\!\cdots\!42\)\( \beta_{2} + \)\(13\!\cdots\!68\)\( \beta_{3} - \)\(17\!\cdots\!46\)\( \beta_{4} - \)\(18\!\cdots\!90\)\( \beta_{5} + \)\(96\!\cdots\!40\)\( \beta_{6} - \)\(22\!\cdots\!32\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8} + \)\(13\!\cdots\!00\)\( \beta_{9}) q^{78}\) \(+(\)\(79\!\cdots\!40\)\( + \)\(43\!\cdots\!82\)\( \beta_{1} + \)\(78\!\cdots\!44\)\( \beta_{2} + \)\(54\!\cdots\!52\)\( \beta_{3} + \)\(19\!\cdots\!18\)\( \beta_{4} - \)\(31\!\cdots\!02\)\( \beta_{5} - \)\(17\!\cdots\!50\)\( \beta_{6} - \)\(19\!\cdots\!44\)\( \beta_{7} + \)\(93\!\cdots\!76\)\( \beta_{8} - \)\(13\!\cdots\!00\)\( \beta_{9}) q^{79}\) \(+(-\)\(40\!\cdots\!28\)\( + \)\(29\!\cdots\!52\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2} - \)\(64\!\cdots\!72\)\( \beta_{3} + \)\(16\!\cdots\!36\)\( \beta_{4} + \)\(84\!\cdots\!60\)\( \beta_{5} + \)\(90\!\cdots\!20\)\( \beta_{6} + \)\(17\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!40\)\( \beta_{8} - \)\(22\!\cdots\!40\)\( \beta_{9}) q^{80}\) \(+(-\)\(49\!\cdots\!99\)\( + \)\(22\!\cdots\!94\)\( \beta_{1} + \)\(19\!\cdots\!97\)\( \beta_{2} - \)\(41\!\cdots\!89\)\( \beta_{3} + \)\(94\!\cdots\!59\)\( \beta_{4} + \)\(10\!\cdots\!73\)\( \beta_{5} - \)\(47\!\cdots\!95\)\( \beta_{6} - \)\(24\!\cdots\!96\)\( \beta_{7} - \)\(11\!\cdots\!61\)\( \beta_{8} + \)\(58\!\cdots\!00\)\( \beta_{9}) q^{81}\) \(+(\)\(94\!\cdots\!20\)\( + \)\(17\!\cdots\!22\)\( \beta_{1} - \)\(77\!\cdots\!32\)\( \beta_{2} + \)\(11\!\cdots\!40\)\( \beta_{3} - \)\(43\!\cdots\!84\)\( \beta_{4} - \)\(19\!\cdots\!92\)\( \beta_{5} + \)\(22\!\cdots\!64\)\( \beta_{6} - \)\(26\!\cdots\!08\)\( \beta_{7} + \)\(31\!\cdots\!20\)\( \beta_{8} + \)\(97\!\cdots\!00\)\( \beta_{9}) q^{82}\) \(+(\)\(25\!\cdots\!40\)\( - \)\(22\!\cdots\!45\)\( \beta_{1} - \)\(14\!\cdots\!93\)\( \beta_{2} + \)\(17\!\cdots\!80\)\( \beta_{3} + \)\(32\!\cdots\!60\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6} + \)\(11\!\cdots\!60\)\( \beta_{7} - \)\(34\!\cdots\!40\)\( \beta_{8} - \)\(55\!\cdots\!00\)\( \beta_{9}) q^{83}\) \(+(\)\(33\!\cdots\!56\)\( - \)\(29\!\cdots\!88\)\( \beta_{1} - \)\(23\!\cdots\!72\)\( \beta_{2} + \)\(78\!\cdots\!72\)\( \beta_{3} - \)\(21\!\cdots\!16\)\( \beta_{4} + \)\(29\!\cdots\!52\)\( \beta_{5} - \)\(92\!\cdots\!60\)\( \beta_{6} - \)\(15\!\cdots\!00\)\( \beta_{7} - \)\(16\!\cdots\!60\)\( \beta_{8} + \)\(34\!\cdots\!00\)\( \beta_{9}) q^{84}\) \(+(-\)\(96\!\cdots\!48\)\( - \)\(64\!\cdots\!98\)\( \beta_{1} - \)\(18\!\cdots\!52\)\( \beta_{2} - \)\(19\!\cdots\!66\)\( \beta_{3} + \)\(28\!\cdots\!16\)\( \beta_{4} + \)\(23\!\cdots\!14\)\( \beta_{5} + \)\(24\!\cdots\!38\)\( \beta_{6} - \)\(33\!\cdots\!00\)\( \beta_{7} + \)\(37\!\cdots\!26\)\( \beta_{8} + \)\(32\!\cdots\!24\)\( \beta_{9}) q^{85}\) \(+(\)\(10\!\cdots\!12\)\( - \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(20\!\cdots\!35\)\( \beta_{2} - \)\(10\!\cdots\!46\)\( \beta_{3} - \)\(48\!\cdots\!31\)\( \beta_{4} - \)\(51\!\cdots\!91\)\( \beta_{5} - \)\(12\!\cdots\!20\)\( \beta_{6} + \)\(27\!\cdots\!48\)\( \beta_{7} - \)\(13\!\cdots\!12\)\( \beta_{8} - \)\(10\!\cdots\!00\)\( \beta_{9}) q^{86}\) \(+(-\)\(12\!\cdots\!60\)\( + \)\(26\!\cdots\!97\)\( \beta_{1} + \)\(36\!\cdots\!60\)\( \beta_{2} - \)\(36\!\cdots\!52\)\( \beta_{3} - \)\(54\!\cdots\!83\)\( \beta_{4} + \)\(67\!\cdots\!09\)\( \beta_{5} - \)\(18\!\cdots\!13\)\( \beta_{6} + \)\(21\!\cdots\!04\)\( \beta_{7} - \)\(97\!\cdots\!20\)\( \beta_{8} + \)\(66\!\cdots\!00\)\( \beta_{9}) q^{87}\) \(+(\)\(16\!\cdots\!40\)\( - \)\(36\!\cdots\!16\)\( \beta_{1} - \)\(78\!\cdots\!96\)\( \beta_{2} + \)\(15\!\cdots\!28\)\( \beta_{3} - \)\(57\!\cdots\!20\)\( \beta_{4} - \)\(58\!\cdots\!12\)\( \beta_{5} - \)\(98\!\cdots\!56\)\( \beta_{6} - \)\(35\!\cdots\!40\)\( \beta_{7} + \)\(18\!\cdots\!40\)\( \beta_{8} + \)\(39\!\cdots\!00\)\( \beta_{9}) q^{88}\) \(+(\)\(14\!\cdots\!30\)\( + \)\(58\!\cdots\!38\)\( \beta_{1} - \)\(76\!\cdots\!37\)\( \beta_{2} + \)\(88\!\cdots\!77\)\( \beta_{3} + \)\(18\!\cdots\!69\)\( \beta_{4} + \)\(45\!\cdots\!75\)\( \beta_{5} + \)\(15\!\cdots\!55\)\( \beta_{6} - \)\(31\!\cdots\!68\)\( \beta_{7} - \)\(48\!\cdots\!23\)\( \beta_{8} - \)\(12\!\cdots\!00\)\( \beta_{9}) q^{89}\) \(+(\)\(42\!\cdots\!84\)\( + \)\(25\!\cdots\!09\)\( \beta_{1} - \)\(30\!\cdots\!59\)\( \beta_{2} - \)\(45\!\cdots\!22\)\( \beta_{3} + \)\(61\!\cdots\!72\)\( \beta_{4} - \)\(25\!\cdots\!87\)\( \beta_{5} + \)\(52\!\cdots\!21\)\( \beta_{6} + \)\(16\!\cdots\!50\)\( \beta_{7} + \)\(10\!\cdots\!67\)\( \beta_{8} + \)\(10\!\cdots\!83\)\( \beta_{9}) q^{90}\) \(+(\)\(37\!\cdots\!32\)\( + \)\(25\!\cdots\!24\)\( \beta_{1} - \)\(23\!\cdots\!68\)\( \beta_{2} - \)\(39\!\cdots\!92\)\( \beta_{3} - \)\(13\!\cdots\!24\)\( \beta_{4} - \)\(16\!\cdots\!24\)\( \beta_{5} - \)\(17\!\cdots\!00\)\( \beta_{6} - \)\(16\!\cdots\!48\)\( \beta_{7} - \)\(99\!\cdots\!08\)\( \beta_{8} + \)\(30\!\cdots\!00\)\( \beta_{9}) q^{91}\) \(+(\)\(10\!\cdots\!80\)\( + \)\(45\!\cdots\!20\)\( \beta_{1} + \)\(66\!\cdots\!40\)\( \beta_{2} - \)\(13\!\cdots\!80\)\( \beta_{3} - \)\(48\!\cdots\!64\)\( \beta_{4} + \)\(22\!\cdots\!28\)\( \beta_{5} + \)\(93\!\cdots\!44\)\( \beta_{6} + \)\(74\!\cdots\!52\)\( \beta_{7} + \)\(13\!\cdots\!00\)\( \beta_{8} - \)\(10\!\cdots\!00\)\( \beta_{9}) q^{92}\) \(+(\)\(44\!\cdots\!20\)\( - \)\(27\!\cdots\!72\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} - \)\(30\!\cdots\!64\)\( \beta_{4} - \)\(68\!\cdots\!40\)\( \beta_{5} + \)\(16\!\cdots\!80\)\( \beta_{6} - \)\(57\!\cdots\!28\)\( \beta_{7} + \)\(66\!\cdots\!40\)\( \beta_{8} + \)\(11\!\cdots\!00\)\( \beta_{9}) q^{93}\) \(+(\)\(12\!\cdots\!56\)\( - \)\(22\!\cdots\!28\)\( \beta_{1} + \)\(24\!\cdots\!52\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!48\)\( \beta_{4} + \)\(91\!\cdots\!68\)\( \beta_{5} + \)\(20\!\cdots\!40\)\( \beta_{6} + \)\(80\!\cdots\!76\)\( \beta_{7} - \)\(25\!\cdots\!64\)\( \beta_{8} + \)\(16\!\cdots\!00\)\( \beta_{9}) q^{94}\) \(+(\)\(26\!\cdots\!00\)\( - \)\(43\!\cdots\!75\)\( \beta_{1} + \)\(37\!\cdots\!80\)\( \beta_{2} - \)\(42\!\cdots\!80\)\( \beta_{3} - \)\(10\!\cdots\!75\)\( \beta_{4} - \)\(22\!\cdots\!95\)\( \beta_{5} - \)\(76\!\cdots\!65\)\( \beta_{6} + \)\(43\!\cdots\!00\)\( \beta_{7} + \)\(17\!\cdots\!20\)\( \beta_{8} - \)\(64\!\cdots\!20\)\( \beta_{9}) q^{95}\) \(+(\)\(71\!\cdots\!32\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} + \)\(18\!\cdots\!80\)\( \beta_{3} - \)\(84\!\cdots\!16\)\( \beta_{4} - \)\(80\!\cdots\!48\)\( \beta_{5} - \)\(20\!\cdots\!60\)\( \beta_{6} - \)\(15\!\cdots\!00\)\( \beta_{7} + \)\(82\!\cdots\!40\)\( \beta_{8} + \)\(69\!\cdots\!00\)\( \beta_{9}) q^{96}\) \(+(\)\(15\!\cdots\!70\)\( - \)\(53\!\cdots\!38\)\( \beta_{1} - \)\(23\!\cdots\!85\)\( \beta_{2} - \)\(71\!\cdots\!95\)\( \beta_{3} - \)\(58\!\cdots\!47\)\( \beta_{4} + \)\(40\!\cdots\!59\)\( \beta_{5} + \)\(71\!\cdots\!87\)\( \beta_{6} + \)\(11\!\cdots\!76\)\( \beta_{7} - \)\(21\!\cdots\!55\)\( \beta_{8} + \)\(43\!\cdots\!00\)\( \beta_{9}) q^{97}\) \(+(\)\(21\!\cdots\!80\)\( + \)\(27\!\cdots\!51\)\( \beta_{1} - \)\(78\!\cdots\!52\)\( \beta_{2} - \)\(17\!\cdots\!80\)\( \beta_{3} + \)\(88\!\cdots\!16\)\( \beta_{4} + \)\(64\!\cdots\!28\)\( \beta_{5} - \)\(28\!\cdots\!16\)\( \beta_{6} + \)\(35\!\cdots\!52\)\( \beta_{7} + \)\(29\!\cdots\!60\)\( \beta_{8} - \)\(19\!\cdots\!00\)\( \beta_{9}) q^{98}\) \(+(\)\(42\!\cdots\!04\)\( + \)\(44\!\cdots\!01\)\( \beta_{1} + \)\(17\!\cdots\!33\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} + \)\(12\!\cdots\!04\)\( \beta_{4} - \)\(87\!\cdots\!56\)\( \beta_{5} - \)\(50\!\cdots\!60\)\( \beta_{6} - \)\(78\!\cdots\!32\)\( \beta_{7} + \)\(67\!\cdots\!68\)\( \beta_{8} + \)\(17\!\cdots\!00\)\( \beta_{9}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 2202364291733634600q^{2} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!80\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!70\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 2202364291733634600q^{2} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!80\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!70\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!20\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!60\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!60\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!40\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!20\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!00\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!20\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!00\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!60\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!20\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!40\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!40\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!00\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!40\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!40\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!60\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!80\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!00\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!30\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!80\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!00\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!60\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!80\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!80\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!40\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!60\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!80\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!80\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!00\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!40\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!80\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!90\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!00\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!60\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!80\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!40\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!20\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!20\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!40\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(5\) \(x^{9}\mathstrut -\mathstrut \) \(15\!\cdots\!40\) \(x^{8}\mathstrut -\mathstrut \) \(26\!\cdots\!60\) \(x^{7}\mathstrut +\mathstrut \) \(84\!\cdots\!30\) \(x^{6}\mathstrut +\mathstrut \) \(37\!\cdots\!74\) \(x^{5}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(x^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!00\) \(x^{3}\mathstrut +\mathstrut \) \(12\!\cdots\!25\) \(x^{2}\mathstrut +\mathstrut \) \(52\!\cdots\!75\) \(x\mathstrut -\mathstrut \) \(30\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 36 \)
\(\beta_{2}\)\(=\)\((\)\(72\!\cdots\!23\) \(\nu^{9}\mathstrut +\mathstrut \) \(45\!\cdots\!04\) \(\nu^{8}\mathstrut -\mathstrut \) \(12\!\cdots\!92\) \(\nu^{7}\mathstrut -\mathstrut \) \(72\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(71\!\cdots\!10\) \(\nu^{5}\mathstrut +\mathstrut \) \(38\!\cdots\!76\) \(\nu^{4}\mathstrut -\mathstrut \) \(15\!\cdots\!56\) \(\nu^{3}\mathstrut -\mathstrut \) \(71\!\cdots\!08\) \(\nu^{2}\mathstrut +\mathstrut \) \(60\!\cdots\!91\) \(\nu\mathstrut +\mathstrut \) \(25\!\cdots\!76\)\()/\)\(13\!\cdots\!76\)
\(\beta_{3}\)\(=\)\((\)\(35\!\cdots\!17\) \(\nu^{9}\mathstrut +\mathstrut \) \(22\!\cdots\!16\) \(\nu^{8}\mathstrut -\mathstrut \) \(61\!\cdots\!68\) \(\nu^{7}\mathstrut -\mathstrut \) \(35\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(35\!\cdots\!90\) \(\nu^{5}\mathstrut +\mathstrut \) \(18\!\cdots\!04\) \(\nu^{4}\mathstrut -\mathstrut \) \(74\!\cdots\!24\) \(\nu^{3}\mathstrut -\mathstrut \) \(28\!\cdots\!48\) \(\nu^{2}\mathstrut +\mathstrut \) \(28\!\cdots\!05\) \(\nu\mathstrut -\mathstrut \) \(88\!\cdots\!36\)\()/\)\(13\!\cdots\!76\)
\(\beta_{4}\)\(=\)\((\)\(30\!\cdots\!95\) \(\nu^{9}\mathstrut -\mathstrut \) \(38\!\cdots\!92\) \(\nu^{8}\mathstrut -\mathstrut \) \(42\!\cdots\!72\) \(\nu^{7}\mathstrut +\mathstrut \) \(53\!\cdots\!68\) \(\nu^{6}\mathstrut +\mathstrut \) \(19\!\cdots\!98\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!72\) \(\nu^{4}\mathstrut -\mathstrut \) \(32\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(69\!\cdots\!00\)\()/\)\(54\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(18\!\cdots\!65\) \(\nu^{9}\mathstrut +\mathstrut \) \(42\!\cdots\!64\) \(\nu^{8}\mathstrut +\mathstrut \) \(27\!\cdots\!24\) \(\nu^{7}\mathstrut -\mathstrut \) \(50\!\cdots\!56\) \(\nu^{6}\mathstrut -\mathstrut \) \(13\!\cdots\!66\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!24\) \(\nu^{4}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(89\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(81\!\cdots\!25\) \(\nu\mathstrut -\mathstrut \) \(38\!\cdots\!00\)\()/\)\(58\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(44\!\cdots\!95\) \(\nu^{9}\mathstrut -\mathstrut \) \(52\!\cdots\!08\) \(\nu^{8}\mathstrut +\mathstrut \) \(62\!\cdots\!72\) \(\nu^{7}\mathstrut +\mathstrut \) \(83\!\cdots\!32\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!98\) \(\nu^{5}\mathstrut -\mathstrut \) \(44\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(33\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(86\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(20\!\cdots\!25\) \(\nu\mathstrut -\mathstrut \) \(35\!\cdots\!00\)\()/\)\(54\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(36\!\cdots\!65\) \(\nu^{9}\mathstrut +\mathstrut \) \(44\!\cdots\!04\) \(\nu^{8}\mathstrut +\mathstrut \) \(52\!\cdots\!64\) \(\nu^{7}\mathstrut -\mathstrut \) \(65\!\cdots\!16\) \(\nu^{6}\mathstrut -\mathstrut \) \(25\!\cdots\!26\) \(\nu^{5}\mathstrut +\mathstrut \) \(26\!\cdots\!64\) \(\nu^{4}\mathstrut +\mathstrut \) \(48\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(32\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(36\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!00\)\()/\)\(82\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(20\!\cdots\!65\) \(\nu^{9}\mathstrut +\mathstrut \) \(31\!\cdots\!76\) \(\nu^{8}\mathstrut -\mathstrut \) \(31\!\cdots\!84\) \(\nu^{7}\mathstrut -\mathstrut \) \(54\!\cdots\!04\) \(\nu^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!06\) \(\nu^{5}\mathstrut +\mathstrut \) \(32\!\cdots\!16\) \(\nu^{4}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(70\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(91\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(24\!\cdots\!00\)\()/\)\(41\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(14\!\cdots\!07\) \(\nu^{9}\mathstrut +\mathstrut \) \(39\!\cdots\!72\) \(\nu^{8}\mathstrut -\mathstrut \) \(23\!\cdots\!68\) \(\nu^{7}\mathstrut -\mathstrut \) \(54\!\cdots\!48\) \(\nu^{6}\mathstrut +\mathstrut \) \(13\!\cdots\!02\) \(\nu^{5}\mathstrut +\mathstrut \) \(24\!\cdots\!60\) \(\nu^{4}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(38\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(46\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!00\)\()/\)\(41\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(36\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(4944179\) \(\beta_{2}\mathstrut +\mathstrut \) \(180506488736968722\) \(\beta_{1}\mathstrut +\mathstrut \) \(16293782570364898259424186091603158432\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(597\) \(\beta_{6}\mathstrut +\mathstrut \) \(2345239\) \(\beta_{5}\mathstrut +\mathstrut \) \(883856762568\) \(\beta_{4}\mathstrut -\mathstrut \) \(558544387041602302\) \(\beta_{3}\mathstrut +\mathstrut \) \(16571801909522981147985755\) \(\beta_{2}\mathstrut +\mathstrut \) \(25970965870877190779900608416506365579\) \(\beta_{1}\mathstrut +\mathstrut \) \(2941133480019533289781380345542542927474747837455884544\)\()/373248\)
\(\nu^{4}\)\(=\)\((\)\(114880745000\) \(\beta_{9}\mathstrut -\mathstrut \) \(76030013716792\) \(\beta_{8}\mathstrut -\mathstrut \) \(127980514063299809\) \(\beta_{7}\mathstrut -\mathstrut \) \(340103716747196954189\) \(\beta_{6}\mathstrut -\mathstrut \) \(11651555598596332149933199\) \(\beta_{5}\mathstrut -\mathstrut \) \(157450884945762875580599909512\) \(\beta_{4}\mathstrut +\mathstrut \) \(4499771064831813638120901944135941494\) \(\beta_{3}\mathstrut -\mathstrut \) \(23868372656995497542451497739269630385861899\) \(\beta_{2}\mathstrut +\mathstrut \) \(250050996580689934310629507085458705927428041253244157\) \(\beta_{1}\mathstrut +\mathstrut \) \(52895658880305325138338583060549595275650196955334736341867049065107318016\)\()/3359232\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(10\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(19\!\cdots\!40\) \(\beta_{8}\mathstrut +\mathstrut \) \(31\!\cdots\!79\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\!\cdots\!15\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\!\cdots\!05\) \(\beta_{5}\mathstrut +\mathstrut \) \(43\!\cdots\!72\) \(\beta_{4}\mathstrut -\mathstrut \) \(26\!\cdots\!26\) \(\beta_{3}\mathstrut +\mathstrut \) \(13\!\cdots\!93\) \(\beta_{2}\mathstrut +\mathstrut \) \(60\!\cdots\!97\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\!\cdots\!72\)\()/15116544\)
\(\nu^{6}\)\(=\)\((\)\(19\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(99\!\cdots\!00\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\!\cdots\!29\) \(\beta_{7}\mathstrut -\mathstrut \) \(30\!\cdots\!65\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\!\cdots\!59\) \(\beta_{5}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\!\cdots\!38\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\!\cdots\!23\) \(\beta_{2}\mathstrut -\mathstrut \) \(60\!\cdots\!43\) \(\beta_{1}\mathstrut +\mathstrut \) \(40\!\cdots\!96\)\()/45349632\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(30\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(43\!\cdots\!60\) \(\beta_{8}\mathstrut +\mathstrut \) \(44\!\cdots\!11\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\!\cdots\!43\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\!\cdots\!85\) \(\beta_{5}\mathstrut +\mathstrut \) \(76\!\cdots\!36\) \(\beta_{4}\mathstrut -\mathstrut \) \(49\!\cdots\!34\) \(\beta_{3}\mathstrut +\mathstrut \) \(26\!\cdots\!45\) \(\beta_{2}\mathstrut +\mathstrut \) \(72\!\cdots\!85\) \(\beta_{1}\mathstrut -\mathstrut \) \(91\!\cdots\!96\)\()/30233088\)
\(\nu^{8}\)\(=\)\((\)\(34\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\!\cdots\!76\) \(\beta_{8}\mathstrut -\mathstrut \) \(33\!\cdots\!21\) \(\beta_{7}\mathstrut -\mathstrut \) \(30\!\cdots\!81\) \(\beta_{6}\mathstrut -\mathstrut \) \(27\!\cdots\!19\) \(\beta_{5}\mathstrut -\mathstrut \) \(53\!\cdots\!76\) \(\beta_{4}\mathstrut +\mathstrut \) \(50\!\cdots\!94\) \(\beta_{3}\mathstrut -\mathstrut \) \(43\!\cdots\!79\) \(\beta_{2}\mathstrut -\mathstrut \) \(19\!\cdots\!75\) \(\beta_{1}\mathstrut +\mathstrut \) \(49\!\cdots\!92\)\()/90699264\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(25\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(29\!\cdots\!24\) \(\beta_{8}\mathstrut +\mathstrut \) \(24\!\cdots\!77\) \(\beta_{7}\mathstrut +\mathstrut \) \(57\!\cdots\!57\) \(\beta_{6}\mathstrut +\mathstrut \) \(27\!\cdots\!07\) \(\beta_{5}\mathstrut +\mathstrut \) \(50\!\cdots\!96\) \(\beta_{4}\mathstrut -\mathstrut \) \(34\!\cdots\!62\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\!\cdots\!87\) \(\beta_{2}\mathstrut +\mathstrut \) \(38\!\cdots\!79\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\!\cdots\!88\)\()/2519424\)

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
7.85561e16
7.40535e16
5.97127e16
2.18088e16
1.97587e16
−2.24732e16
−3.52603e16
−4.50246e16
−6.69543e16
−8.41774e16
−5.43581e18 −3.35417e28 1.89142e37 1.63130e43 1.82326e47 −3.09666e51 −4.50103e55 −4.73942e58 −8.86742e61
1.2 −5.11162e18 −2.22156e29 1.54948e37 −1.83710e43 1.13558e48 3.56438e51 −2.48476e55 8.34100e56 9.39057e61
1.3 −4.07908e18 3.50571e29 6.00505e36 −4.74746e42 −1.43001e48 −1.17353e51 1.88811e55 7.43807e58 1.93653e61
1.4 −1.35000e18 1.15541e28 −8.81132e36 1.19036e42 −1.55981e46 4.29022e51 2.62510e55 −4.83858e58 −1.60698e60
1.5 −1.20239e18 −3.80552e29 −9.18808e36 5.95717e42 4.57573e47 −7.49927e51 2.38337e55 9.63009e58 −7.16285e60
1.6 1.83831e18 1.91755e29 −7.25444e36 −8.31906e42 3.52506e47 −1.53178e52 −3.28842e55 −1.17492e58 −1.52930e61
1.7 2.75898e18 3.25623e29 −3.02185e36 1.58514e43 8.98387e47 1.53160e52 −3.76757e55 5.75108e58 4.37336e61
1.8 3.46200e18 −2.27154e29 1.35165e36 −7.89778e42 −7.86409e47 1.06526e52 −3.21349e55 3.07977e57 −2.73421e61
1.9 5.04095e18 −1.36074e29 1.47773e37 1.38517e43 −6.85941e47 −1.32080e52 2.08873e55 −3.00032e58 6.98256e61
1.10 6.28101e18 2.54101e29 2.88173e37 −1.37149e43 1.59601e48 5.88359e51 1.14211e56 1.60483e58 −8.61436e61
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{124}^{\mathrm{new}}(\Gamma_0(1))\).