Properties

Label 3.66.a.b
Level 3
Weight 66
Character orbit 3.a
Self dual Yes
Analytic conductor 80.272
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 66 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(80.2717069417\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{43}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11\cdot 13 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(1035163827 + \beta_{1}) q^{2}\) \(+1853020188851841 q^{3}\) \(+(15223484176714109398 + 2766449869 \beta_{1} + \beta_{2}) q^{4}\) \(+(\)\(58\!\cdots\!50\)\( + 4058291886090 \beta_{1} - 115 \beta_{2} + \beta_{3}) q^{5}\) \(+(\)\(19\!\cdots\!07\)\( + 1853020188851841 \beta_{1}) q^{6}\) \(+(\)\(21\!\cdots\!84\)\( - 79972783738454934 \beta_{1} + 20360674 \beta_{2} - 13013 \beta_{3} + 2 \beta_{4} - 11 \beta_{5}) q^{7}\) \(+(\)\(11\!\cdots\!36\)\( + 21078861912981178230 \beta_{1} + 5208391705 \beta_{2} + 1535423 \beta_{3} + 957 \beta_{4} + 902 \beta_{5}) q^{8}\) \(+\)\(34\!\cdots\!81\)\( q^{9}\) \(+O(q^{10})\) \( q\) \(+(1035163827 + \beta_{1}) q^{2}\) \(+1853020188851841 q^{3}\) \(+(15223484176714109398 + 2766449869 \beta_{1} + \beta_{2}) q^{4}\) \(+(\)\(58\!\cdots\!50\)\( + 4058291886090 \beta_{1} - 115 \beta_{2} + \beta_{3}) q^{5}\) \(+(\)\(19\!\cdots\!07\)\( + 1853020188851841 \beta_{1}) q^{6}\) \(+(\)\(21\!\cdots\!84\)\( - 79972783738454934 \beta_{1} + 20360674 \beta_{2} - 13013 \beta_{3} + 2 \beta_{4} - 11 \beta_{5}) q^{7}\) \(+(\)\(11\!\cdots\!36\)\( + 21078861912981178230 \beta_{1} + 5208391705 \beta_{2} + 1535423 \beta_{3} + 957 \beta_{4} + 902 \beta_{5}) q^{8}\) \(+\)\(34\!\cdots\!81\)\( q^{9}\) \(+(\)\(21\!\cdots\!50\)\( + \)\(85\!\cdots\!74\)\( \beta_{1} + 5991131641124 \beta_{2} + 914975956 \beta_{3} - 2231044 \beta_{4} + 538648 \beta_{5}) q^{10}\) \(+(-\)\(38\!\cdots\!96\)\( + \)\(36\!\cdots\!40\)\( \beta_{1} + 27075971338498 \beta_{2} + 12902997746 \beta_{3} + 83434704 \beta_{4} + 3069064 \beta_{5}) q^{11}\) \(+(\)\(28\!\cdots\!18\)\( + \)\(51\!\cdots\!29\)\( \beta_{1} + 1853020188851841 \beta_{2}) q^{12}\) \(+(\)\(16\!\cdots\!58\)\( - \)\(44\!\cdots\!48\)\( \beta_{1} + 4434013985876935 \beta_{2} - 2482793608804 \beta_{3} - 8774881934 \beta_{4} - 5647118323 \beta_{5}) q^{13}\) \(+(-\)\(38\!\cdots\!68\)\( + \)\(85\!\cdots\!04\)\( \beta_{1} - 242504447544261548 \beta_{2} - 97016561333692 \beta_{3} + 62640614220 \beta_{4} - 29679396296 \beta_{5}) q^{14}\) \(+(\)\(10\!\cdots\!50\)\( + \)\(75\!\cdots\!90\)\( \beta_{1} - 213097321717961715 \beta_{2} + 1853020188851841 \beta_{3}) q^{15}\) \(+(\)\(63\!\cdots\!60\)\( + \)\(25\!\cdots\!92\)\( \beta_{1} + 21369434725528966530 \beta_{2} + 9632904364827470 \beta_{3} - 3565193201942 \beta_{4} + 8242560826412 \beta_{5}) q^{16}\) \(+(-\)\(22\!\cdots\!22\)\( - \)\(59\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} + 150414742067361454 \beta_{3} + 29629607763420 \beta_{4} - 40885437728058 \beta_{5}) q^{17}\) \(+(\)\(35\!\cdots\!87\)\( + \)\(34\!\cdots\!81\)\( \beta_{1}) q^{18}\) \(+(\)\(93\!\cdots\!32\)\( - \)\(17\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!12\)\( \beta_{2} + 5776526584106469454 \beta_{3} - 1710096887588356 \beta_{4} - 778455099992042 \beta_{5}) q^{19}\) \(+(\)\(43\!\cdots\!00\)\( + \)\(30\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!26\)\( \beta_{2} - 3233815714682863936 \beta_{3} + 10675902499527744 \beta_{4} + 9259352738477952 \beta_{5}) q^{20}\) \(+(\)\(40\!\cdots\!44\)\( - \)\(14\!\cdots\!94\)\( \beta_{1} + \)\(37\!\cdots\!34\)\( \beta_{2} - 24113351717529006933 \beta_{3} + 3706040377703682 \beta_{4} - 20383222077370251 \beta_{5}) q^{21}\) \(+(\)\(18\!\cdots\!36\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + \)\(96\!\cdots\!12\)\( \beta_{2} - \)\(61\!\cdots\!76\)\( \beta_{3} - 306812746499247368 \beta_{4} - 8360672652871120 \beta_{5}) q^{22}\) \(+(\)\(79\!\cdots\!32\)\( - \)\(29\!\cdots\!40\)\( \beta_{1} + \)\(26\!\cdots\!28\)\( \beta_{2} - \)\(47\!\cdots\!50\)\( \beta_{3} + 1084590165223127604 \beta_{4} - 283796188057375774 \beta_{5}) q^{23}\) \(+(\)\(22\!\cdots\!76\)\( + \)\(39\!\cdots\!30\)\( \beta_{1} + \)\(96\!\cdots\!05\)\( \beta_{2} + \)\(28\!\cdots\!43\)\( \beta_{3} + 1773340320731211837 \beta_{4} + 1671424210344360582 \beta_{5}) q^{24}\) \(+(\)\(98\!\cdots\!75\)\( + \)\(81\!\cdots\!00\)\( \beta_{1} - \)\(33\!\cdots\!50\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3} - 21856701789537085300 \beta_{4} + 424227633178820350 \beta_{5}) q^{25}\) \(+(-\)\(21\!\cdots\!38\)\( + \)\(25\!\cdots\!94\)\( \beta_{1} - \)\(18\!\cdots\!32\)\( \beta_{2} + \)\(27\!\cdots\!68\)\( \beta_{3} + 52622463852183003336 \beta_{4} - 11883871272460988592 \beta_{5}) q^{26}\) \(+\)\(63\!\cdots\!21\)\( q^{27}\) \(+(\)\(31\!\cdots\!88\)\( - \)\(86\!\cdots\!00\)\( \beta_{1} - \)\(89\!\cdots\!32\)\( \beta_{2} - \)\(78\!\cdots\!76\)\( \beta_{3} - \)\(26\!\cdots\!12\)\( \beta_{4} - 17846740719351029376 \beta_{5}) q^{28}\) \(+(\)\(30\!\cdots\!34\)\( - \)\(20\!\cdots\!50\)\( \beta_{1} - \)\(41\!\cdots\!75\)\( \beta_{2} - \)\(17\!\cdots\!01\)\( \beta_{3} + \)\(10\!\cdots\!84\)\( \beta_{4} + \)\(30\!\cdots\!98\)\( \beta_{5}) q^{29}\) \(+(\)\(39\!\cdots\!50\)\( + \)\(15\!\cdots\!34\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(16\!\cdots\!96\)\( \beta_{3} - \)\(41\!\cdots\!04\)\( \beta_{4} + \)\(99\!\cdots\!68\)\( \beta_{5}) q^{30}\) \(+(\)\(43\!\cdots\!16\)\( + \)\(10\!\cdots\!54\)\( \beta_{1} - \)\(17\!\cdots\!14\)\( \beta_{2} + \)\(24\!\cdots\!11\)\( \beta_{3} + \)\(12\!\cdots\!22\)\( \beta_{4} - \)\(10\!\cdots\!59\)\( \beta_{5}) q^{31}\) \(+(\)\(90\!\cdots\!24\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} + \)\(89\!\cdots\!56\)\( \beta_{3} - \)\(39\!\cdots\!64\)\( \beta_{4} + \)\(23\!\cdots\!12\)\( \beta_{5}) q^{32}\) \(+(-\)\(71\!\cdots\!36\)\( + \)\(67\!\cdots\!40\)\( \beta_{1} + \)\(50\!\cdots\!18\)\( \beta_{2} + \)\(23\!\cdots\!86\)\( \beta_{3} + \)\(15\!\cdots\!64\)\( \beta_{4} + \)\(56\!\cdots\!24\)\( \beta_{5}) q^{33}\) \(+(-\)\(32\!\cdots\!14\)\( - \)\(76\!\cdots\!18\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} - \)\(62\!\cdots\!08\)\( \beta_{3} - \)\(45\!\cdots\!92\)\( \beta_{4} - \)\(16\!\cdots\!92\)\( \beta_{5}) q^{34}\) \(+(-\)\(56\!\cdots\!00\)\( - \)\(91\!\cdots\!44\)\( \beta_{1} - \)\(14\!\cdots\!74\)\( \beta_{2} - \)\(31\!\cdots\!10\)\( \beta_{3} + \)\(59\!\cdots\!04\)\( \beta_{4} + \)\(44\!\cdots\!32\)\( \beta_{5}) q^{35}\) \(+(\)\(52\!\cdots\!38\)\( + \)\(94\!\cdots\!89\)\( \beta_{1} + \)\(34\!\cdots\!81\)\( \beta_{2}) q^{36}\) \(+(-\)\(32\!\cdots\!46\)\( + \)\(37\!\cdots\!16\)\( \beta_{1} - \)\(71\!\cdots\!61\)\( \beta_{2} + \)\(28\!\cdots\!94\)\( \beta_{3} + \)\(17\!\cdots\!58\)\( \beta_{4} - \)\(27\!\cdots\!21\)\( \beta_{5}) q^{37}\) \(+(-\)\(78\!\cdots\!48\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2} + \)\(21\!\cdots\!48\)\( \beta_{3} + \)\(51\!\cdots\!68\)\( \beta_{4} + \)\(63\!\cdots\!76\)\( \beta_{5}) q^{38}\) \(+(\)\(29\!\cdots\!78\)\( - \)\(83\!\cdots\!68\)\( \beta_{1} + \)\(82\!\cdots\!35\)\( \beta_{2} - \)\(46\!\cdots\!64\)\( \beta_{3} - \)\(16\!\cdots\!94\)\( \beta_{4} - \)\(10\!\cdots\!43\)\( \beta_{5}) q^{39}\) \(+(\)\(81\!\cdots\!00\)\( + \)\(16\!\cdots\!56\)\( \beta_{1} + \)\(36\!\cdots\!66\)\( \beta_{2} - \)\(34\!\cdots\!18\)\( \beta_{3} + \)\(57\!\cdots\!34\)\( \beta_{4} + \)\(28\!\cdots\!72\)\( \beta_{5}) q^{40}\) \(+(\)\(74\!\cdots\!86\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!98\)\( \beta_{3} - \)\(83\!\cdots\!16\)\( \beta_{4} - \)\(28\!\cdots\!78\)\( \beta_{5}) q^{41}\) \(+(-\)\(71\!\cdots\!88\)\( + \)\(15\!\cdots\!64\)\( \beta_{1} - \)\(44\!\cdots\!68\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4} - \)\(54\!\cdots\!36\)\( \beta_{5}) q^{42}\) \(+(\)\(25\!\cdots\!40\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(13\!\cdots\!66\)\( \beta_{3} - \)\(26\!\cdots\!64\)\( \beta_{4} - \)\(58\!\cdots\!54\)\( \beta_{5}) q^{43}\) \(+(\)\(97\!\cdots\!04\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} - \)\(54\!\cdots\!92\)\( \beta_{2} + \)\(28\!\cdots\!88\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4} + \)\(51\!\cdots\!44\)\( \beta_{5}) q^{44}\) \(+(\)\(20\!\cdots\!50\)\( + \)\(13\!\cdots\!90\)\( \beta_{1} - \)\(39\!\cdots\!15\)\( \beta_{2} + \)\(34\!\cdots\!81\)\( \beta_{3}) q^{45}\) \(+(-\)\(14\!\cdots\!24\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(33\!\cdots\!32\)\( \beta_{2} - \)\(80\!\cdots\!20\)\( \beta_{3} + \)\(28\!\cdots\!36\)\( \beta_{4} + \)\(44\!\cdots\!04\)\( \beta_{5}) q^{46}\) \(+(-\)\(39\!\cdots\!80\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(20\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!18\)\( \beta_{3} + \)\(16\!\cdots\!40\)\( \beta_{4} - \)\(92\!\cdots\!94\)\( \beta_{5}) q^{47}\) \(+(\)\(11\!\cdots\!60\)\( + \)\(46\!\cdots\!72\)\( \beta_{1} + \)\(39\!\cdots\!30\)\( \beta_{2} + \)\(17\!\cdots\!70\)\( \beta_{3} - \)\(66\!\cdots\!22\)\( \beta_{4} + \)\(15\!\cdots\!92\)\( \beta_{5}) q^{48}\) \(+(-\)\(22\!\cdots\!55\)\( - \)\(58\!\cdots\!08\)\( \beta_{1} - \)\(26\!\cdots\!02\)\( \beta_{2} - \)\(92\!\cdots\!52\)\( \beta_{3} - \)\(64\!\cdots\!56\)\( \beta_{4} - \)\(33\!\cdots\!10\)\( \beta_{5}) q^{49}\) \(+(\)\(14\!\cdots\!25\)\( + \)\(87\!\cdots\!75\)\( \beta_{1} - \)\(56\!\cdots\!00\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4} + \)\(27\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(-\)\(41\!\cdots\!02\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} - \)\(21\!\cdots\!68\)\( \beta_{2} + \)\(27\!\cdots\!14\)\( \beta_{3} + \)\(54\!\cdots\!20\)\( \beta_{4} - \)\(75\!\cdots\!78\)\( \beta_{5}) q^{51}\) \(+(\)\(46\!\cdots\!68\)\( - \)\(71\!\cdots\!54\)\( \beta_{1} - \)\(23\!\cdots\!22\)\( \beta_{2} - \)\(71\!\cdots\!92\)\( \beta_{3} - \)\(77\!\cdots\!28\)\( \beta_{4} - \)\(16\!\cdots\!88\)\( \beta_{5}) q^{52}\) \(+(\)\(16\!\cdots\!22\)\( - \)\(96\!\cdots\!02\)\( \beta_{1} - \)\(95\!\cdots\!35\)\( \beta_{2} + \)\(45\!\cdots\!07\)\( \beta_{3} - \)\(25\!\cdots\!56\)\( \beta_{4} + \)\(55\!\cdots\!02\)\( \beta_{5}) q^{53}\) \(+(\)\(65\!\cdots\!67\)\( + \)\(63\!\cdots\!21\)\( \beta_{1}) q^{54}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(32\!\cdots\!48\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(43\!\cdots\!84\)\( \beta_{3} + \)\(15\!\cdots\!08\)\( \beta_{4} + \)\(58\!\cdots\!64\)\( \beta_{5}) q^{55}\) \(+(-\)\(26\!\cdots\!40\)\( - \)\(48\!\cdots\!16\)\( \beta_{1} - \)\(53\!\cdots\!28\)\( \beta_{2} + \)\(34\!\cdots\!76\)\( \beta_{3} - \)\(51\!\cdots\!20\)\( \beta_{4} - \)\(11\!\cdots\!72\)\( \beta_{5}) q^{56}\) \(+(\)\(17\!\cdots\!12\)\( - \)\(31\!\cdots\!28\)\( \beta_{1} + \)\(88\!\cdots\!92\)\( \beta_{2} + \)\(10\!\cdots\!14\)\( \beta_{3} - \)\(31\!\cdots\!96\)\( \beta_{4} - \)\(14\!\cdots\!22\)\( \beta_{5}) q^{57}\) \(+(-\)\(10\!\cdots\!26\)\( - \)\(16\!\cdots\!78\)\( \beta_{1} - \)\(23\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!68\)\( \beta_{3} - \)\(47\!\cdots\!96\)\( \beta_{4} - \)\(43\!\cdots\!28\)\( \beta_{5}) q^{58}\) \(+(-\)\(11\!\cdots\!92\)\( - \)\(30\!\cdots\!56\)\( \beta_{1} - \)\(47\!\cdots\!00\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!76\)\( \beta_{4} + \)\(54\!\cdots\!44\)\( \beta_{5}) q^{59}\) \(+(\)\(81\!\cdots\!00\)\( + \)\(56\!\cdots\!66\)\( \beta_{1} + \)\(49\!\cdots\!66\)\( \beta_{2} - \)\(59\!\cdots\!76\)\( \beta_{3} + \)\(19\!\cdots\!04\)\( \beta_{4} + \)\(17\!\cdots\!32\)\( \beta_{5}) q^{60}\) \(+(\)\(67\!\cdots\!50\)\( + \)\(36\!\cdots\!44\)\( \beta_{1} - \)\(36\!\cdots\!45\)\( \beta_{2} - \)\(11\!\cdots\!98\)\( \beta_{3} - \)\(39\!\cdots\!78\)\( \beta_{4} + \)\(23\!\cdots\!59\)\( \beta_{5}) q^{61}\) \(+(\)\(58\!\cdots\!48\)\( - \)\(45\!\cdots\!16\)\( \beta_{1} - \)\(43\!\cdots\!88\)\( \beta_{2} - \)\(22\!\cdots\!40\)\( \beta_{3} - \)\(47\!\cdots\!32\)\( \beta_{4} - \)\(58\!\cdots\!48\)\( \beta_{5}) q^{62}\) \(+(\)\(75\!\cdots\!04\)\( - \)\(27\!\cdots\!54\)\( \beta_{1} + \)\(69\!\cdots\!94\)\( \beta_{2} - \)\(44\!\cdots\!53\)\( \beta_{3} + \)\(68\!\cdots\!62\)\( \beta_{4} - \)\(37\!\cdots\!91\)\( \beta_{5}) q^{63}\) \(+(\)\(42\!\cdots\!32\)\( + \)\(11\!\cdots\!32\)\( \beta_{1} + \)\(13\!\cdots\!96\)\( \beta_{2} + \)\(67\!\cdots\!88\)\( \beta_{3} + \)\(28\!\cdots\!00\)\( \beta_{4} + \)\(92\!\cdots\!84\)\( \beta_{5}) q^{64}\) \(+(-\)\(16\!\cdots\!00\)\( + \)\(34\!\cdots\!24\)\( \beta_{1} - \)\(18\!\cdots\!56\)\( \beta_{2} + \)\(48\!\cdots\!42\)\( \beta_{3} - \)\(76\!\cdots\!04\)\( \beta_{4} + \)\(14\!\cdots\!18\)\( \beta_{5}) q^{65}\) \(+(\)\(33\!\cdots\!76\)\( + \)\(23\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!92\)\( \beta_{2} - \)\(11\!\cdots\!16\)\( \beta_{3} - \)\(56\!\cdots\!88\)\( \beta_{4} - \)\(15\!\cdots\!20\)\( \beta_{5}) q^{66}\) \(+(\)\(24\!\cdots\!68\)\( + \)\(13\!\cdots\!52\)\( \beta_{1} + \)\(62\!\cdots\!20\)\( \beta_{2} + \)\(22\!\cdots\!96\)\( \beta_{3} - \)\(21\!\cdots\!76\)\( \beta_{4} - \)\(20\!\cdots\!16\)\( \beta_{5}) q^{67}\) \(+(-\)\(34\!\cdots\!36\)\( - \)\(66\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!06\)\( \beta_{2} - \)\(56\!\cdots\!04\)\( \beta_{3} + \)\(11\!\cdots\!88\)\( \beta_{4} - \)\(19\!\cdots\!00\)\( \beta_{5}) q^{68}\) \(+(\)\(14\!\cdots\!12\)\( - \)\(54\!\cdots\!40\)\( \beta_{1} + \)\(49\!\cdots\!48\)\( \beta_{2} - \)\(88\!\cdots\!50\)\( \beta_{3} + \)\(20\!\cdots\!64\)\( \beta_{4} - \)\(52\!\cdots\!34\)\( \beta_{5}) q^{69}\) \(+(-\)\(62\!\cdots\!00\)\( - \)\(10\!\cdots\!32\)\( \beta_{1} + \)\(67\!\cdots\!28\)\( \beta_{2} - \)\(35\!\cdots\!20\)\( \beta_{3} - \)\(37\!\cdots\!88\)\( \beta_{4} - \)\(28\!\cdots\!04\)\( \beta_{5}) q^{70}\) \(+(\)\(76\!\cdots\!52\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(21\!\cdots\!86\)\( \beta_{3} - \)\(37\!\cdots\!52\)\( \beta_{4} + \)\(38\!\cdots\!70\)\( \beta_{5}) q^{71}\) \(+(\)\(40\!\cdots\!16\)\( + \)\(72\!\cdots\!30\)\( \beta_{1} + \)\(17\!\cdots\!05\)\( \beta_{2} + \)\(52\!\cdots\!63\)\( \beta_{3} + \)\(32\!\cdots\!17\)\( \beta_{4} + \)\(30\!\cdots\!62\)\( \beta_{5}) q^{72}\) \(+(\)\(88\!\cdots\!42\)\( - \)\(39\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!26\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3} + \)\(14\!\cdots\!56\)\( \beta_{4} - \)\(66\!\cdots\!02\)\( \beta_{5}) q^{73}\) \(+(\)\(15\!\cdots\!02\)\( - \)\(53\!\cdots\!94\)\( \beta_{1} - \)\(27\!\cdots\!92\)\( \beta_{2} - \)\(35\!\cdots\!60\)\( \beta_{3} - \)\(17\!\cdots\!88\)\( \beta_{4} - \)\(15\!\cdots\!52\)\( \beta_{5}) q^{74}\) \(+(\)\(18\!\cdots\!75\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(62\!\cdots\!50\)\( \beta_{2} + \)\(45\!\cdots\!00\)\( \beta_{3} - \)\(40\!\cdots\!00\)\( \beta_{4} + \)\(78\!\cdots\!50\)\( \beta_{5}) q^{75}\) \(+(\)\(82\!\cdots\!88\)\( - \)\(31\!\cdots\!16\)\( \beta_{1} + \)\(18\!\cdots\!28\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(99\!\cdots\!24\)\( \beta_{4} + \)\(50\!\cdots\!04\)\( \beta_{5}) q^{76}\) \(+(-\)\(30\!\cdots\!72\)\( + \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(33\!\cdots\!88\)\( \beta_{2} + \)\(81\!\cdots\!52\)\( \beta_{3} + \)\(72\!\cdots\!08\)\( \beta_{4} - \)\(10\!\cdots\!12\)\( \beta_{5}) q^{77}\) \(+(-\)\(39\!\cdots\!58\)\( + \)\(46\!\cdots\!54\)\( \beta_{1} - \)\(34\!\cdots\!12\)\( \beta_{2} + \)\(51\!\cdots\!88\)\( \beta_{3} + \)\(97\!\cdots\!76\)\( \beta_{4} - \)\(22\!\cdots\!72\)\( \beta_{5}) q^{78}\) \(+(\)\(91\!\cdots\!40\)\( + \)\(26\!\cdots\!98\)\( \beta_{1} - \)\(54\!\cdots\!90\)\( \beta_{2} + \)\(15\!\cdots\!23\)\( \beta_{3} - \)\(11\!\cdots\!22\)\( \beta_{4} - \)\(50\!\cdots\!19\)\( \beta_{5}) q^{79}\) \(+(\)\(77\!\cdots\!00\)\( + \)\(13\!\cdots\!04\)\( \beta_{1} + \)\(23\!\cdots\!64\)\( \beta_{2} + \)\(44\!\cdots\!84\)\( \beta_{3} - \)\(30\!\cdots\!04\)\( \beta_{4} + \)\(47\!\cdots\!68\)\( \beta_{5}) q^{80}\) \(+\)\(11\!\cdots\!61\)\( q^{81}\) \(+(\)\(84\!\cdots\!38\)\( + \)\(57\!\cdots\!26\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2} + \)\(56\!\cdots\!28\)\( \beta_{3} + \)\(65\!\cdots\!08\)\( \beta_{4} - \)\(27\!\cdots\!84\)\( \beta_{5}) q^{82}\) \(+(\)\(20\!\cdots\!76\)\( - \)\(90\!\cdots\!20\)\( \beta_{1} + \)\(17\!\cdots\!82\)\( \beta_{2} - \)\(14\!\cdots\!06\)\( \beta_{3} + \)\(46\!\cdots\!48\)\( \beta_{4} + \)\(66\!\cdots\!44\)\( \beta_{5}) q^{83}\) \(+(\)\(58\!\cdots\!08\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3} - \)\(48\!\cdots\!92\)\( \beta_{4} - \)\(33\!\cdots\!16\)\( \beta_{5}) q^{84}\) \(+(\)\(30\!\cdots\!00\)\( - \)\(70\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!90\)\( \beta_{2} + \)\(16\!\cdots\!98\)\( \beta_{3} - \)\(19\!\cdots\!80\)\( \beta_{4} - \)\(85\!\cdots\!40\)\( \beta_{5}) q^{85}\) \(+(\)\(63\!\cdots\!44\)\( - \)\(18\!\cdots\!60\)\( \beta_{1} + \)\(50\!\cdots\!32\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} - \)\(27\!\cdots\!68\)\( \beta_{4} - \)\(16\!\cdots\!36\)\( \beta_{5}) q^{86}\) \(+(\)\(57\!\cdots\!94\)\( - \)\(38\!\cdots\!50\)\( \beta_{1} - \)\(76\!\cdots\!75\)\( \beta_{2} - \)\(31\!\cdots\!41\)\( \beta_{3} + \)\(19\!\cdots\!44\)\( \beta_{4} + \)\(57\!\cdots\!18\)\( \beta_{5}) q^{87}\) \(+(\)\(16\!\cdots\!16\)\( + \)\(10\!\cdots\!36\)\( \beta_{1} + \)\(23\!\cdots\!36\)\( \beta_{2} + \)\(27\!\cdots\!48\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} + \)\(32\!\cdots\!64\)\( \beta_{5}) q^{88}\) \(+(\)\(10\!\cdots\!02\)\( - \)\(23\!\cdots\!32\)\( \beta_{1} - \)\(70\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!76\)\( \beta_{3} + \)\(44\!\cdots\!12\)\( \beta_{4} - \)\(43\!\cdots\!20\)\( \beta_{5}) q^{89}\) \(+(\)\(73\!\cdots\!50\)\( + \)\(29\!\cdots\!94\)\( \beta_{1} + \)\(20\!\cdots\!44\)\( \beta_{2} + \)\(31\!\cdots\!36\)\( \beta_{3} - \)\(76\!\cdots\!64\)\( \beta_{4} + \)\(18\!\cdots\!88\)\( \beta_{5}) q^{90}\) \(+(\)\(26\!\cdots\!16\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!84\)\( \beta_{2} - \)\(56\!\cdots\!02\)\( \beta_{3} - \)\(68\!\cdots\!28\)\( \beta_{4} + \)\(16\!\cdots\!22\)\( \beta_{5}) q^{91}\) \(+(\)\(49\!\cdots\!84\)\( + \)\(27\!\cdots\!76\)\( \beta_{1} + \)\(57\!\cdots\!40\)\( \beta_{2} + \)\(15\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!64\)\( \beta_{4} + \)\(97\!\cdots\!24\)\( \beta_{5}) q^{92}\) \(+(\)\(81\!\cdots\!56\)\( + \)\(19\!\cdots\!14\)\( \beta_{1} - \)\(32\!\cdots\!74\)\( \beta_{2} + \)\(45\!\cdots\!51\)\( \beta_{3} + \)\(23\!\cdots\!02\)\( \beta_{4} - \)\(18\!\cdots\!19\)\( \beta_{5}) q^{93}\) \(+(\)\(93\!\cdots\!88\)\( + \)\(69\!\cdots\!08\)\( \beta_{1} + \)\(50\!\cdots\!72\)\( \beta_{2} - \)\(81\!\cdots\!24\)\( \beta_{3} + \)\(66\!\cdots\!44\)\( \beta_{4} - \)\(25\!\cdots\!96\)\( \beta_{5}) q^{94}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(18\!\cdots\!32\)\( \beta_{1} - \)\(49\!\cdots\!48\)\( \beta_{2} + \)\(18\!\cdots\!84\)\( \beta_{3} + \)\(52\!\cdots\!48\)\( \beta_{4} + \)\(39\!\cdots\!84\)\( \beta_{5}) q^{95}\) \(+(\)\(16\!\cdots\!84\)\( + \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(47\!\cdots\!40\)\( \beta_{2} + \)\(16\!\cdots\!96\)\( \beta_{3} - \)\(73\!\cdots\!24\)\( \beta_{4} + \)\(43\!\cdots\!92\)\( \beta_{5}) q^{96}\) \(+(\)\(21\!\cdots\!38\)\( - \)\(68\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(86\!\cdots\!04\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4} - \)\(24\!\cdots\!32\)\( \beta_{5}) q^{97}\) \(+(-\)\(53\!\cdots\!33\)\( - \)\(33\!\cdots\!35\)\( \beta_{1} - \)\(42\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!48\)\( \beta_{3} + \)\(20\!\cdots\!04\)\( \beta_{4} - \)\(90\!\cdots\!48\)\( \beta_{5}) q^{98}\) \(+(-\)\(13\!\cdots\!76\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(92\!\cdots\!38\)\( \beta_{2} + \)\(44\!\cdots\!26\)\( \beta_{3} + \)\(28\!\cdots\!24\)\( \beta_{4} + \)\(10\!\cdots\!84\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 6210982962q^{2} \) \(\mathstrut +\mathstrut 11118121133111046q^{3} \) \(\mathstrut +\mathstrut 91340905060284656388q^{4} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!42\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!04\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!16\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!86\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 6210982962q^{2} \) \(\mathstrut +\mathstrut 11118121133111046q^{3} \) \(\mathstrut +\mathstrut 91340905060284656388q^{4} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!42\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!04\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!16\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!86\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!76\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!08\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!48\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!08\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!60\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!32\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!22\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!92\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!64\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!16\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!92\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!56\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!28\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!26\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!28\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!04\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!96\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!44\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!16\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!84\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!28\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!76\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!88\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!68\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!16\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!28\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!40\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!24\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!44\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!60\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!30\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!50\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!12\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!08\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!32\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!02\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!56\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!52\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!88\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!24\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!92\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!56\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!08\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!16\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!72\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!12\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!96\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!52\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!12\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!50\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!28\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!32\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!48\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!66\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!28\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!56\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!48\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!64\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!64\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!96\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!12\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!96\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!04\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!36\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!28\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!04\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!28\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!98\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!56\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(4253784014616993888\) \(x^{4}\mathstrut -\mathstrut \) \(329017052684987797861154944\) \(x^{3}\mathstrut +\mathstrut \) \(3760278921468113125234877358788007936\) \(x^{2}\mathstrut +\mathstrut \) \(22551593102849743113985540842573617614848\) \(x\mathstrut -\mathstrut \) \(272928737776547549611547193931962354302098378900635648\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 3 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} - 4176733326 \nu - 51045408173315560047 \)
\(\beta_{3}\)\(=\)\((\)\(34220984678163\) \(\nu^{5}\mathstrut -\mathstrut \) \(204624719522368468497\) \(\nu^{4}\mathstrut -\mathstrut \) \(129792467407002101837559992998872\) \(\nu^{3}\mathstrut -\mathstrut \) \(15397563421483621154453628474844242720960\) \(\nu^{2}\mathstrut +\mathstrut \) \(83525140826605298574632007994422423386411159674368\) \(\nu\mathstrut +\mathstrut \) \(3994341035062746726204001760515400120170047068531324092416\)\()/\)\(32\!\cdots\!40\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(85579356314943207\) \(\nu^{5}\mathstrut -\mathstrut \) \(235898923333219397441125587\) \(\nu^{4}\mathstrut +\mathstrut \) \(521491127431162087307044631956063608\) \(\nu^{3}\mathstrut +\mathstrut \) \(737257168921916481180634357769803270549855680\) \(\nu^{2}\mathstrut -\mathstrut \) \(584969923936038046306057858489375490015648207868667392\) \(\nu\mathstrut -\mathstrut \) \(199882640008057964412149983364237685683878312304851746019344384\)\()/\)\(22\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(435832873032153699\) \(\nu^{5}\mathstrut +\mathstrut \) \(347491719717916643716757121\) \(\nu^{4}\mathstrut +\mathstrut \) \(2105163063868379798861919121276379736\) \(\nu^{3}\mathstrut -\mathstrut \) \(1082415425851397114846133417392816600222610240\) \(\nu^{2}\mathstrut -\mathstrut \) \(2323062035005819942853152802088639418336341889754955264\) \(\nu\mathstrut +\mathstrut \) \(471969830014766643367604179830241315488762618506341209216909312\)\()/\)\(30\!\cdots\!60\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(696122221\) \(\beta_{1}\mathstrut +\mathstrut \) \(51045408175403926710\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(902\) \(\beta_{5}\mathstrut +\mathstrut \) \(957\) \(\beta_{4}\mathstrut +\mathstrut \) \(1535423\) \(\beta_{3}\mathstrut +\mathstrut \) \(2102900233\) \(\beta_{2}\mathstrut +\mathstrut \) \(89489344159479276454\) \(\beta_{1}\mathstrut +\mathstrut \) \(35533843068204702904910754636\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(2253844874710\) \(\beta_{5}\mathstrut -\mathstrut \) \(3763900160107\) \(\beta_{4}\mathstrut +\mathstrut \) \(1637623494138631\) \(\beta_{3}\mathstrut +\mathstrut \) \(58456564662972687201\) \(\beta_{2}\mathstrut +\mathstrut \) \(88833175400517927855447807830\) \(\beta_{1}\mathstrut +\mathstrut \) \(2284010050042572295739582589377650637868\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(30830171529496779873862\) \(\beta_{5}\mathstrut +\mathstrut \) \(32599645253477211559773\) \(\beta_{4}\mathstrut +\mathstrut \) \(70664756610820794943662495\) \(\beta_{3}\mathstrut +\mathstrut \) \(97128090264824385005614260105\) \(\beta_{2}\mathstrut +\mathstrut \) \(2282417534443336051534027761857044439110\) \(\beta_{1}\mathstrut +\mathstrut \) \(2267262852465788443179627291856979571575121285132\)\()/1944\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.64208e9
−1.16886e9
−2.78112e8
2.86568e8
9.77853e8
1.82463e9
−8.81729e9 1.85302e15 4.08511e19 −7.15176e22 −1.63386e25 3.06094e27 −3.48954e28 3.43368e30 6.30591e32
1.2 −5.97799e9 1.85302e15 −1.15708e18 3.17694e22 −1.10773e25 6.82141e26 2.27466e29 3.43368e30 −1.89917e32
1.3 −6.33506e8 1.85302e15 −3.64922e19 −5.06462e22 −1.17390e24 −3.41514e27 4.64902e28 3.43368e30 3.20846e31
1.4 2.75457e9 1.85302e15 −2.93058e19 9.23075e22 5.10428e24 −1.32074e27 −1.82351e29 3.43368e30 2.54268e32
1.5 6.90228e9 1.85302e15 1.07480e19 −3.03033e22 1.27901e25 3.59710e27 −1.80464e29 3.43368e30 −2.09162e32
1.6 1.19829e10 1.85302e15 1.06697e20 6.35463e22 2.22046e25 −1.28649e27 8.36449e29 3.43368e30 7.61471e32
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 6210982962 T_{2}^{5} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!68\)\( T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!64\)\( T_{2}^{3} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!96\)\( T_{2}^{2} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!08\)\( T_{2} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!48\)\( \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(3))\).