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 \( 2^{178}\cdot 3^{70}\cdot 5^{22}\cdot 7^{9}\cdot 11^{6}\cdot 13^{2}\cdot 17\cdot 31^{2}\cdot 41^{3} \)
Twist minimal: yes
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.

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.124.a.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.124.a.a 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{124}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2202364291733634600 T + \)\(27\!\cdots\!00\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(94\!\cdots\!60\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} - \)\(30\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} - \)\(27\!\cdots\!00\)\( T^{13} + \)\(91\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!80\)\( T^{16} - \)\(90\!\cdots\!00\)\( T^{17} + \)\(44\!\cdots\!00\)\( T^{18} - \)\(38\!\cdots\!00\)\( T^{19} + \)\(18\!\cdots\!24\)\( T^{20} \)
$3$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(19\!\cdots\!50\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!45\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(75\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!00\)\( T^{10} - \)\(36\!\cdots\!00\)\( T^{11} + \)\(24\!\cdots\!90\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} + \)\(94\!\cdots\!00\)\( T^{14} - \)\(64\!\cdots\!00\)\( T^{15} + \)\(28\!\cdots\!05\)\( T^{16} - \)\(16\!\cdots\!00\)\( T^{17} + \)\(60\!\cdots\!50\)\( T^{18} - \)\(19\!\cdots\!00\)\( T^{19} + \)\(72\!\cdots\!49\)\( T^{20} \)
$5$ \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(22\!\cdots\!50\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!25\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(35\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(35\!\cdots\!50\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!25\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(65\!\cdots\!00\)\( T^{19} + \)\(54\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 + \)\(58\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(61\!\cdots\!00\)\( T^{3} + \)\(97\!\cdots\!45\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} + \)\(25\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!90\)\( T^{12} + \)\(16\!\cdots\!00\)\( T^{13} + \)\(90\!\cdots\!00\)\( T^{14} + \)\(90\!\cdots\!00\)\( T^{15} + \)\(46\!\cdots\!05\)\( T^{16} + \)\(26\!\cdots\!00\)\( T^{17} + \)\(16\!\cdots\!50\)\( T^{18} + \)\(19\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!45\)\( T^{4} - \)\(76\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!60\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!70\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{9} + \)\(45\!\cdots\!56\)\( T^{10} - \)\(43\!\cdots\!60\)\( T^{11} + \)\(48\!\cdots\!70\)\( T^{12} - \)\(37\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!60\)\( T^{14} - \)\(21\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!45\)\( T^{16} - \)\(79\!\cdots\!40\)\( T^{17} + \)\(55\!\cdots\!90\)\( T^{18} - \)\(13\!\cdots\!20\)\( T^{19} + \)\(81\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + \)\(29\!\cdots\!00\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(54\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!45\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!10\)\( T^{8} + \)\(23\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!90\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(14\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!05\)\( T^{16} + \)\(69\!\cdots\!00\)\( T^{17} + \)\(48\!\cdots\!50\)\( T^{18} + \)\(39\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(14\!\cdots\!50\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(90\!\cdots\!45\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(97\!\cdots\!10\)\( T^{8} - \)\(46\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(47\!\cdots\!90\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{13} + \)\(82\!\cdots\!00\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(30\!\cdots\!00\)\( T^{17} + \)\(86\!\cdots\!50\)\( T^{18} - \)\(15\!\cdots\!00\)\( T^{19} + \)\(28\!\cdots\!49\)\( T^{20} \)
$19$ \( 1 - \)\(16\!\cdots\!00\)\( T + \)\(83\!\cdots\!90\)\( T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!45\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!10\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!48\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!10\)\( T^{12} + \)\(14\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!80\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(19\!\cdots\!45\)\( T^{16} - \)\(82\!\cdots\!00\)\( T^{17} + \)\(16\!\cdots\!90\)\( T^{18} - \)\(63\!\cdots\!00\)\( T^{19} + \)\(73\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 - \)\(60\!\cdots\!00\)\( T + \)\(15\!\cdots\!50\)\( T^{2} - \)\(89\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!45\)\( T^{4} - \)\(68\!\cdots\!00\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} - \)\(35\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!10\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!90\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(66\!\cdots\!00\)\( T^{14} - \)\(19\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!05\)\( T^{16} - \)\(25\!\cdots\!00\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(16\!\cdots\!00\)\( T^{19} + \)\(84\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(87\!\cdots\!90\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!45\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{5} + \)\(59\!\cdots\!80\)\( T^{6} + \)\(69\!\cdots\!00\)\( T^{7} + \)\(76\!\cdots\!10\)\( T^{8} + \)\(74\!\cdots\!00\)\( T^{9} + \)\(68\!\cdots\!48\)\( T^{10} + \)\(55\!\cdots\!00\)\( T^{11} + \)\(42\!\cdots\!10\)\( T^{12} + \)\(29\!\cdots\!00\)\( T^{13} + \)\(18\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!00\)\( T^{15} + \)\(53\!\cdots\!45\)\( T^{16} + \)\(22\!\cdots\!00\)\( T^{17} + \)\(87\!\cdots\!90\)\( T^{18} + \)\(21\!\cdots\!00\)\( T^{19} + \)\(56\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - \)\(87\!\cdots\!20\)\( T + \)\(18\!\cdots\!90\)\( T^{2} - \)\(81\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{5} + \)\(98\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!80\)\( T^{7} + \)\(40\!\cdots\!70\)\( T^{8} + \)\(96\!\cdots\!40\)\( T^{9} + \)\(12\!\cdots\!56\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!70\)\( T^{12} + \)\(29\!\cdots\!80\)\( T^{13} + \)\(55\!\cdots\!60\)\( T^{14} - \)\(21\!\cdots\!04\)\( T^{15} + \)\(72\!\cdots\!45\)\( T^{16} - \)\(94\!\cdots\!40\)\( T^{17} + \)\(60\!\cdots\!90\)\( T^{18} - \)\(76\!\cdots\!20\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 - \)\(62\!\cdots\!00\)\( T + \)\(72\!\cdots\!50\)\( T^{2} - \)\(34\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!45\)\( T^{4} - \)\(88\!\cdots\!00\)\( T^{5} + \)\(43\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(51\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!00\)\( T^{11} + \)\(33\!\cdots\!90\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(24\!\cdots\!00\)\( T^{15} + \)\(49\!\cdots\!05\)\( T^{16} - \)\(57\!\cdots\!00\)\( T^{17} + \)\(93\!\cdots\!50\)\( T^{18} - \)\(62\!\cdots\!00\)\( T^{19} + \)\(77\!\cdots\!49\)\( T^{20} \)
$41$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(15\!\cdots\!90\)\( T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!45\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!60\)\( T^{6} + \)\(73\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!70\)\( T^{8} + \)\(24\!\cdots\!40\)\( T^{9} + \)\(62\!\cdots\!56\)\( T^{10} + \)\(57\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!70\)\( T^{12} + \)\(95\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!96\)\( T^{15} + \)\(21\!\cdots\!45\)\( T^{16} + \)\(77\!\cdots\!60\)\( T^{17} + \)\(14\!\cdots\!90\)\( T^{18} + \)\(24\!\cdots\!80\)\( T^{19} + \)\(52\!\cdots\!01\)\( T^{20} \)
$43$ \( 1 - \)\(46\!\cdots\!00\)\( T + \)\(48\!\cdots\!50\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!45\)\( T^{4} - \)\(37\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(52\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!10\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(45\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!90\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{13} + \)\(85\!\cdots\!00\)\( T^{14} - \)\(14\!\cdots\!00\)\( T^{15} + \)\(36\!\cdots\!05\)\( T^{16} - \)\(48\!\cdots\!00\)\( T^{17} + \)\(10\!\cdots\!50\)\( T^{18} - \)\(82\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 - \)\(17\!\cdots\!00\)\( T + \)\(30\!\cdots\!50\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!45\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!10\)\( T^{8} - \)\(50\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!00\)\( T^{10} - \)\(23\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!90\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{13} + \)\(87\!\cdots\!00\)\( T^{14} - \)\(48\!\cdots\!00\)\( T^{15} + \)\(30\!\cdots\!05\)\( T^{16} - \)\(14\!\cdots\!00\)\( T^{17} + \)\(67\!\cdots\!50\)\( T^{18} - \)\(18\!\cdots\!00\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 - \)\(98\!\cdots\!00\)\( T + \)\(47\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!10\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{9} + \)\(52\!\cdots\!00\)\( T^{10} - \)\(49\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!90\)\( T^{12} - \)\(50\!\cdots\!00\)\( T^{13} + \)\(60\!\cdots\!00\)\( T^{14} - \)\(44\!\cdots\!00\)\( T^{15} + \)\(43\!\cdots\!05\)\( T^{16} - \)\(25\!\cdots\!00\)\( T^{17} + \)\(23\!\cdots\!50\)\( T^{18} - \)\(58\!\cdots\!00\)\( T^{19} + \)\(72\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 + \)\(51\!\cdots\!00\)\( T + \)\(36\!\cdots\!90\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!45\)\( T^{4} + \)\(36\!\cdots\!00\)\( T^{5} + \)\(96\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!00\)\( T^{7} + \)\(94\!\cdots\!10\)\( T^{8} + \)\(39\!\cdots\!00\)\( T^{9} + \)\(70\!\cdots\!48\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{11} + \)\(40\!\cdots\!10\)\( T^{12} + \)\(12\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!80\)\( T^{14} + \)\(43\!\cdots\!00\)\( T^{15} + \)\(55\!\cdots\!45\)\( T^{16} + \)\(98\!\cdots\!00\)\( T^{17} + \)\(11\!\cdots\!90\)\( T^{18} + \)\(11\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(28\!\cdots\!90\)\( T^{2} + \)\(23\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!45\)\( T^{4} + \)\(27\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!60\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} + \)\(10\!\cdots\!40\)\( T^{9} + \)\(92\!\cdots\!56\)\( T^{10} + \)\(42\!\cdots\!40\)\( T^{11} + \)\(31\!\cdots\!70\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{13} + \)\(81\!\cdots\!60\)\( T^{14} + \)\(26\!\cdots\!96\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(35\!\cdots\!60\)\( T^{17} + \)\(16\!\cdots\!90\)\( T^{18} + \)\(23\!\cdots\!80\)\( T^{19} + \)\(90\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 - \)\(39\!\cdots\!00\)\( T + \)\(34\!\cdots\!50\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!45\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!10\)\( T^{8} - \)\(77\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(31\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!90\)\( T^{12} - \)\(88\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!05\)\( T^{16} - \)\(20\!\cdots\!00\)\( T^{17} + \)\(25\!\cdots\!50\)\( T^{18} - \)\(11\!\cdots\!00\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \)
$71$ \( 1 + \)\(14\!\cdots\!80\)\( T + \)\(39\!\cdots\!90\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{3} + \)\(71\!\cdots\!45\)\( T^{4} + \)\(67\!\cdots\!96\)\( T^{5} + \)\(79\!\cdots\!60\)\( T^{6} + \)\(63\!\cdots\!80\)\( T^{7} + \)\(61\!\cdots\!70\)\( T^{8} + \)\(42\!\cdots\!40\)\( T^{9} + \)\(35\!\cdots\!56\)\( T^{10} + \)\(21\!\cdots\!40\)\( T^{11} + \)\(15\!\cdots\!70\)\( T^{12} + \)\(82\!\cdots\!80\)\( T^{13} + \)\(52\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!96\)\( T^{15} + \)\(12\!\cdots\!45\)\( T^{16} + \)\(39\!\cdots\!60\)\( T^{17} + \)\(17\!\cdots\!90\)\( T^{18} + \)\(32\!\cdots\!80\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$73$ \( 1 - \)\(21\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(41\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!10\)\( T^{8} - \)\(89\!\cdots\!00\)\( T^{9} + \)\(80\!\cdots\!00\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!90\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(75\!\cdots\!05\)\( T^{16} - \)\(52\!\cdots\!00\)\( T^{17} + \)\(34\!\cdots\!50\)\( T^{18} - \)\(10\!\cdots\!00\)\( T^{19} + \)\(77\!\cdots\!49\)\( T^{20} \)
$79$ \( 1 - \)\(79\!\cdots\!00\)\( T + \)\(16\!\cdots\!90\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} - \)\(73\!\cdots\!00\)\( T^{5} + \)\(66\!\cdots\!80\)\( T^{6} - \)\(32\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(68\!\cdots\!48\)\( T^{10} - \)\(26\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!10\)\( T^{12} - \)\(53\!\cdots\!00\)\( T^{13} + \)\(28\!\cdots\!80\)\( T^{14} - \)\(80\!\cdots\!00\)\( T^{15} + \)\(37\!\cdots\!45\)\( T^{16} - \)\(79\!\cdots\!00\)\( T^{17} + \)\(31\!\cdots\!90\)\( T^{18} - \)\(37\!\cdots\!00\)\( T^{19} + \)\(12\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 - \)\(25\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!45\)\( T^{4} - \)\(80\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!10\)\( T^{8} - \)\(30\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!00\)\( T^{10} - \)\(33\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!90\)\( T^{12} - \)\(26\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(13\!\cdots\!00\)\( T^{15} + \)\(99\!\cdots\!05\)\( T^{16} - \)\(44\!\cdots\!00\)\( T^{17} + \)\(25\!\cdots\!50\)\( T^{18} - \)\(67\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!49\)\( T^{20} \)
$89$ \( 1 - \)\(14\!\cdots\!00\)\( T + \)\(30\!\cdots\!90\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!45\)\( T^{4} - \)\(62\!\cdots\!00\)\( T^{5} + \)\(65\!\cdots\!80\)\( T^{6} - \)\(64\!\cdots\!00\)\( T^{7} + \)\(58\!\cdots\!10\)\( T^{8} - \)\(49\!\cdots\!00\)\( T^{9} + \)\(40\!\cdots\!48\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{13} + \)\(82\!\cdots\!80\)\( T^{14} - \)\(46\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!45\)\( T^{16} - \)\(10\!\cdots\!00\)\( T^{17} + \)\(48\!\cdots\!90\)\( T^{18} - \)\(13\!\cdots\!00\)\( T^{19} + \)\(56\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(85\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!45\)\( T^{4} - \)\(79\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!00\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!10\)\( T^{8} - \)\(77\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!00\)\( T^{10} - \)\(18\!\cdots\!00\)\( T^{11} + \)\(28\!\cdots\!90\)\( T^{12} - \)\(35\!\cdots\!00\)\( T^{13} + \)\(51\!\cdots\!00\)\( T^{14} - \)\(57\!\cdots\!00\)\( T^{15} + \)\(74\!\cdots\!05\)\( T^{16} - \)\(65\!\cdots\!00\)\( T^{17} + \)\(82\!\cdots\!50\)\( T^{18} - \)\(34\!\cdots\!00\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \)
show more
show less