Properties

Label 3.72.a.b
Level 3
Weight 72
Character orbit 3.a
Self dual Yes
Analytic conductor 95.774
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 72 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(95.7738481683\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{49}\cdot 3^{29}\cdot 5^{7}\cdot 7^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+(-12150609471 + \beta_{1}) q^{2}\) \(-50031545098999707 q^{3}\) \(+(\)\(17\!\cdots\!22\)\( - 17516044621 \beta_{1} + \beta_{2}) q^{4}\) \(+(\)\(16\!\cdots\!50\)\( + 28213031926853 \beta_{1} + 1134 \beta_{2} + \beta_{3}) q^{5}\) \(+(\)\(60\!\cdots\!97\)\( - 50031545098999707 \beta_{1}) q^{6}\) \(+(\)\(99\!\cdots\!48\)\( - 1082693092113113668 \beta_{1} + 104180280 \beta_{2} - 18415 \beta_{3} - 71 \beta_{4} + 38 \beta_{5}) q^{7}\) \(+(-\)\(61\!\cdots\!72\)\( + \)\(17\!\cdots\!98\)\( \beta_{1} - 18881241852 \beta_{2} - 6340812 \beta_{3} + 8666 \beta_{4} + 10241 \beta_{5}) q^{8}\) \(+\)\(25\!\cdots\!49\)\( q^{9}\) \(+O(q^{10})\) \( q\) \(+(-12150609471 + \beta_{1}) q^{2}\) \(-50031545098999707 q^{3}\) \(+(\)\(17\!\cdots\!22\)\( - 17516044621 \beta_{1} + \beta_{2}) q^{4}\) \(+(\)\(16\!\cdots\!50\)\( + 28213031926853 \beta_{1} + 1134 \beta_{2} + \beta_{3}) q^{5}\) \(+(\)\(60\!\cdots\!97\)\( - 50031545098999707 \beta_{1}) q^{6}\) \(+(\)\(99\!\cdots\!48\)\( - 1082693092113113668 \beta_{1} + 104180280 \beta_{2} - 18415 \beta_{3} - 71 \beta_{4} + 38 \beta_{5}) q^{7}\) \(+(-\)\(61\!\cdots\!72\)\( + \)\(17\!\cdots\!98\)\( \beta_{1} - 18881241852 \beta_{2} - 6340812 \beta_{3} + 8666 \beta_{4} + 10241 \beta_{5}) q^{8}\) \(+\)\(25\!\cdots\!49\)\( q^{9}\) \(+(\)\(90\!\cdots\!50\)\( + \)\(41\!\cdots\!78\)\( \beta_{1} - 7427576128556 \beta_{2} - 72797015944 \beta_{3} + 30812860 \beta_{4} + 9467030 \beta_{5}) q^{10}\) \(+(\)\(36\!\cdots\!44\)\( + \)\(51\!\cdots\!74\)\( \beta_{1} + 1059967454702844 \beta_{2} - 545209258238 \beta_{3} + 374353744 \beta_{4} + 443137504 \beta_{5}) q^{11}\) \(+(-\)\(86\!\cdots\!54\)\( + \)\(87\!\cdots\!47\)\( \beta_{1} - 50031545098999707 \beta_{2}) q^{12}\) \(+(-\)\(62\!\cdots\!46\)\( + \)\(13\!\cdots\!81\)\( \beta_{1} + 51014920186356326 \beta_{2} + 144291871839670 \beta_{3} + 17030753453 \beta_{4} + 243171997726 \beta_{5}) q^{13}\) \(+(-\)\(54\!\cdots\!72\)\( + \)\(34\!\cdots\!68\)\( \beta_{1} + 3195232706844438996 \beta_{2} - 2437317859969928 \beta_{3} - 55678808900 \beta_{4} + 3269326203286 \beta_{5}) q^{14}\) \(+(-\)\(84\!\cdots\!50\)\( - \)\(14\!\cdots\!71\)\( \beta_{1} - 56735772142265667738 \beta_{2} - 50031545098999707 \beta_{3}) q^{15}\) \(+(\)\(36\!\cdots\!80\)\( - \)\(73\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2} + 598606103381663032 \beta_{3} - 716962040898532 \beta_{4} - 128965651174778 \beta_{5}) q^{16}\) \(+(\)\(49\!\cdots\!66\)\( - \)\(60\!\cdots\!76\)\( \beta_{1} - \)\(21\!\cdots\!08\)\( \beta_{2} + 1119360920456075938 \beta_{3} + 2146481076140310 \beta_{4} - 218736202471836 \beta_{5}) q^{17}\) \(+(-\)\(30\!\cdots\!79\)\( + \)\(25\!\cdots\!49\)\( \beta_{1}) q^{18}\) \(+(-\)\(56\!\cdots\!12\)\( - \)\(34\!\cdots\!88\)\( \beta_{1} - \)\(35\!\cdots\!56\)\( \beta_{2} - 7564500504646046230 \beta_{3} + 327165097315901566 \beta_{4} + 58305069829914452 \beta_{5}) q^{19}\) \(+(\)\(11\!\cdots\!00\)\( - \)\(15\!\cdots\!46\)\( \beta_{1} + \)\(65\!\cdots\!42\)\( \beta_{2} + \)\(32\!\cdots\!08\)\( \beta_{3} - 2198822332160593920 \beta_{4} - 309308960437340160 \beta_{5}) q^{20}\) \(+(-\)\(49\!\cdots\!36\)\( + \)\(54\!\cdots\!76\)\( \beta_{1} - \)\(52\!\cdots\!60\)\( \beta_{2} + \)\(92\!\cdots\!05\)\( \beta_{3} + 3552239702028979197 \beta_{4} - 1901198713761988866 \beta_{5}) q^{21}\) \(+(\)\(15\!\cdots\!32\)\( + \)\(58\!\cdots\!16\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + \)\(31\!\cdots\!44\)\( \beta_{3} - 20435352797062709896 \beta_{4} + 14235034987616496940 \beta_{5}) q^{22}\) \(+(-\)\(40\!\cdots\!04\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(63\!\cdots\!16\)\( \beta_{2} + \)\(85\!\cdots\!98\)\( \beta_{3} + \)\(18\!\cdots\!62\)\( \beta_{4} + 58268384487909764828 \beta_{5}) q^{23}\) \(+(\)\(30\!\cdots\!04\)\( - \)\(88\!\cdots\!86\)\( \beta_{1} + \)\(94\!\cdots\!64\)\( \beta_{2} + \)\(31\!\cdots\!84\)\( \beta_{3} - \)\(43\!\cdots\!62\)\( \beta_{4} - \)\(51\!\cdots\!87\)\( \beta_{5}) q^{24}\) \(+(\)\(24\!\cdots\!75\)\( - \)\(56\!\cdots\!50\)\( \beta_{1} + \)\(98\!\cdots\!00\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3} + \)\(63\!\cdots\!50\)\( \beta_{4} - \)\(11\!\cdots\!00\)\( \beta_{5}) q^{25}\) \(+(\)\(60\!\cdots\!02\)\( - \)\(57\!\cdots\!02\)\( \beta_{1} + \)\(48\!\cdots\!88\)\( \beta_{2} - \)\(14\!\cdots\!68\)\( \beta_{3} - \)\(47\!\cdots\!64\)\( \beta_{4} + \)\(50\!\cdots\!88\)\( \beta_{5}) q^{26}\) \(-\)\(12\!\cdots\!43\)\( q^{27}\) \(+(\)\(11\!\cdots\!44\)\( + \)\(26\!\cdots\!52\)\( \beta_{1} + \)\(70\!\cdots\!20\)\( \beta_{2} + \)\(11\!\cdots\!68\)\( \beta_{3} + \)\(37\!\cdots\!44\)\( \beta_{4} + \)\(18\!\cdots\!52\)\( \beta_{5}) q^{28}\) \(+(\)\(23\!\cdots\!66\)\( + \)\(69\!\cdots\!01\)\( \beta_{1} + \)\(10\!\cdots\!62\)\( \beta_{2} - \)\(30\!\cdots\!65\)\( \beta_{3} + \)\(33\!\cdots\!06\)\( \beta_{4} - \)\(24\!\cdots\!08\)\( \beta_{5}) q^{29}\) \(+(-\)\(45\!\cdots\!50\)\( - \)\(20\!\cdots\!46\)\( \beta_{1} + \)\(37\!\cdots\!92\)\( \beta_{2} + \)\(36\!\cdots\!08\)\( \beta_{3} - \)\(15\!\cdots\!20\)\( \beta_{4} - \)\(47\!\cdots\!10\)\( \beta_{5}) q^{30}\) \(+(-\)\(45\!\cdots\!04\)\( + \)\(12\!\cdots\!44\)\( \beta_{1} - \)\(83\!\cdots\!44\)\( \beta_{2} + \)\(51\!\cdots\!89\)\( \beta_{3} - \)\(17\!\cdots\!03\)\( \beta_{4} + \)\(43\!\cdots\!26\)\( \beta_{5}) q^{31}\) \(+(-\)\(18\!\cdots\!52\)\( + \)\(35\!\cdots\!72\)\( \beta_{1} - \)\(95\!\cdots\!16\)\( \beta_{2} - \)\(54\!\cdots\!80\)\( \beta_{3} + \)\(16\!\cdots\!52\)\( \beta_{4} + \)\(17\!\cdots\!04\)\( \beta_{5}) q^{32}\) \(+(-\)\(18\!\cdots\!08\)\( - \)\(25\!\cdots\!18\)\( \beta_{1} - \)\(53\!\cdots\!08\)\( \beta_{2} + \)\(27\!\cdots\!66\)\( \beta_{3} - \)\(18\!\cdots\!08\)\( \beta_{4} - \)\(22\!\cdots\!28\)\( \beta_{5}) q^{33}\) \(+(-\)\(24\!\cdots\!86\)\( + \)\(32\!\cdots\!98\)\( \beta_{1} - \)\(60\!\cdots\!96\)\( \beta_{2} + \)\(11\!\cdots\!76\)\( \beta_{3} - \)\(81\!\cdots\!28\)\( \beta_{4} - \)\(69\!\cdots\!28\)\( \beta_{5}) q^{34}\) \(+(\)\(22\!\cdots\!00\)\( - \)\(13\!\cdots\!82\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2} + \)\(55\!\cdots\!26\)\( \beta_{3} - \)\(58\!\cdots\!60\)\( \beta_{4} + \)\(21\!\cdots\!20\)\( \beta_{5}) q^{35}\) \(+(\)\(43\!\cdots\!78\)\( - \)\(43\!\cdots\!29\)\( \beta_{1} + \)\(25\!\cdots\!49\)\( \beta_{2}) q^{36}\) \(+(-\)\(59\!\cdots\!42\)\( + \)\(22\!\cdots\!17\)\( \beta_{1} + \)\(90\!\cdots\!58\)\( \beta_{2} + \)\(83\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!91\)\( \beta_{4} + \)\(23\!\cdots\!78\)\( \beta_{5}) q^{37}\) \(+(-\)\(66\!\cdots\!64\)\( - \)\(13\!\cdots\!96\)\( \beta_{1} + \)\(17\!\cdots\!24\)\( \beta_{2} + \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(82\!\cdots\!36\)\( \beta_{4} - \)\(86\!\cdots\!92\)\( \beta_{5}) q^{38}\) \(+(\)\(31\!\cdots\!22\)\( - \)\(67\!\cdots\!67\)\( \beta_{1} - \)\(25\!\cdots\!82\)\( \beta_{2} - \)\(72\!\cdots\!90\)\( \beta_{3} - \)\(85\!\cdots\!71\)\( \beta_{4} - \)\(12\!\cdots\!82\)\( \beta_{5}) q^{39}\) \(+(-\)\(41\!\cdots\!00\)\( + \)\(16\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!80\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3} + \)\(82\!\cdots\!60\)\( \beta_{4} + \)\(72\!\cdots\!30\)\( \beta_{5}) q^{40}\) \(+(\)\(79\!\cdots\!86\)\( - \)\(12\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!44\)\( \beta_{2} + \)\(10\!\cdots\!38\)\( \beta_{3} - \)\(18\!\cdots\!06\)\( \beta_{4} - \)\(66\!\cdots\!48\)\( \beta_{5}) q^{41}\) \(+(\)\(27\!\cdots\!04\)\( - \)\(17\!\cdots\!76\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} - \)\(16\!\cdots\!02\)\( \beta_{5}) q^{42}\) \(+(\)\(19\!\cdots\!80\)\( + \)\(15\!\cdots\!40\)\( \beta_{1} - \)\(46\!\cdots\!48\)\( \beta_{2} + \)\(13\!\cdots\!74\)\( \beta_{3} + \)\(76\!\cdots\!58\)\( \beta_{4} - \)\(32\!\cdots\!12\)\( \beta_{5}) q^{43}\) \(+(\)\(12\!\cdots\!16\)\( + \)\(35\!\cdots\!20\)\( \beta_{1} + \)\(37\!\cdots\!52\)\( \beta_{2} - \)\(26\!\cdots\!68\)\( \beta_{3} + \)\(82\!\cdots\!80\)\( \beta_{4} + \)\(11\!\cdots\!16\)\( \beta_{5}) q^{44}\) \(+(\)\(42\!\cdots\!50\)\( + \)\(70\!\cdots\!97\)\( \beta_{1} + \)\(28\!\cdots\!66\)\( \beta_{2} + \)\(25\!\cdots\!49\)\( \beta_{3}) q^{45}\) \(+(-\)\(37\!\cdots\!64\)\( + \)\(11\!\cdots\!04\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} - \)\(13\!\cdots\!36\)\( \beta_{3} - \)\(13\!\cdots\!44\)\( \beta_{4} - \)\(18\!\cdots\!76\)\( \beta_{5}) q^{46}\) \(+(-\)\(12\!\cdots\!60\)\( - \)\(29\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!16\)\( \beta_{2} - \)\(80\!\cdots\!86\)\( \beta_{3} - \)\(15\!\cdots\!90\)\( \beta_{4} + \)\(20\!\cdots\!52\)\( \beta_{5}) q^{47}\) \(+(-\)\(18\!\cdots\!60\)\( + \)\(36\!\cdots\!96\)\( \beta_{1} - \)\(79\!\cdots\!40\)\( \beta_{2} - \)\(29\!\cdots\!24\)\( \beta_{3} + \)\(35\!\cdots\!24\)\( \beta_{4} + \)\(64\!\cdots\!46\)\( \beta_{5}) q^{48}\) \(+(-\)\(11\!\cdots\!15\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} - \)\(19\!\cdots\!68\)\( \beta_{2} + \)\(24\!\cdots\!36\)\( \beta_{3} + \)\(49\!\cdots\!66\)\( \beta_{4} - \)\(56\!\cdots\!40\)\( \beta_{5}) q^{49}\) \(+(-\)\(25\!\cdots\!25\)\( + \)\(50\!\cdots\!75\)\( \beta_{1} - \)\(43\!\cdots\!00\)\( \beta_{2} + \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(63\!\cdots\!00\)\( \beta_{4} + \)\(94\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(-\)\(24\!\cdots\!62\)\( + \)\(30\!\cdots\!32\)\( \beta_{1} + \)\(10\!\cdots\!56\)\( \beta_{2} - \)\(56\!\cdots\!66\)\( \beta_{3} - \)\(10\!\cdots\!70\)\( \beta_{4} + \)\(10\!\cdots\!52\)\( \beta_{5}) q^{51}\) \(+(-\)\(15\!\cdots\!24\)\( + \)\(14\!\cdots\!42\)\( \beta_{1} + \)\(37\!\cdots\!66\)\( \beta_{2} + \)\(35\!\cdots\!72\)\( \beta_{3} - \)\(40\!\cdots\!56\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5}) q^{52}\) \(+(\)\(90\!\cdots\!86\)\( + \)\(93\!\cdots\!65\)\( \beta_{1} + \)\(20\!\cdots\!42\)\( \beta_{2} - \)\(16\!\cdots\!21\)\( \beta_{3} + \)\(23\!\cdots\!02\)\( \beta_{4} - \)\(85\!\cdots\!84\)\( \beta_{5}) q^{53}\) \(+(\)\(15\!\cdots\!53\)\( - \)\(12\!\cdots\!43\)\( \beta_{1}) q^{54}\) \(+(-\)\(38\!\cdots\!00\)\( + \)\(10\!\cdots\!36\)\( \beta_{1} + \)\(48\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!32\)\( \beta_{3} - \)\(76\!\cdots\!60\)\( \beta_{4} + \)\(27\!\cdots\!20\)\( \beta_{5}) q^{55}\) \(+(\)\(87\!\cdots\!60\)\( + \)\(19\!\cdots\!04\)\( \beta_{1} - \)\(56\!\cdots\!00\)\( \beta_{2} - \)\(54\!\cdots\!84\)\( \beta_{3} + \)\(76\!\cdots\!80\)\( \beta_{4} - \)\(10\!\cdots\!92\)\( \beta_{5}) q^{56}\) \(+(\)\(28\!\cdots\!84\)\( + \)\(17\!\cdots\!16\)\( \beta_{1} + \)\(17\!\cdots\!92\)\( \beta_{2} + \)\(37\!\cdots\!10\)\( \beta_{3} - \)\(16\!\cdots\!62\)\( \beta_{4} - \)\(29\!\cdots\!64\)\( \beta_{5}) q^{57}\) \(+(-\)\(11\!\cdots\!18\)\( + \)\(46\!\cdots\!14\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(29\!\cdots\!04\)\( \beta_{3} + \)\(70\!\cdots\!72\)\( \beta_{4} - \)\(64\!\cdots\!74\)\( \beta_{5}) q^{58}\) \(+(-\)\(49\!\cdots\!08\)\( - \)\(33\!\cdots\!64\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2} + \)\(43\!\cdots\!16\)\( \beta_{3} - \)\(34\!\cdots\!56\)\( \beta_{4} - \)\(27\!\cdots\!44\)\( \beta_{5}) q^{59}\) \(+(-\)\(56\!\cdots\!00\)\( + \)\(79\!\cdots\!22\)\( \beta_{1} - \)\(32\!\cdots\!94\)\( \beta_{2} - \)\(16\!\cdots\!56\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(15\!\cdots\!20\)\( \beta_{5}) q^{60}\) \(+(\)\(33\!\cdots\!70\)\( - \)\(15\!\cdots\!71\)\( \beta_{1} + \)\(42\!\cdots\!74\)\( \beta_{2} + \)\(55\!\cdots\!00\)\( \beta_{3} + \)\(25\!\cdots\!27\)\( \beta_{4} - \)\(96\!\cdots\!86\)\( \beta_{5}) q^{61}\) \(+(\)\(10\!\cdots\!76\)\( - \)\(65\!\cdots\!28\)\( \beta_{1} + \)\(59\!\cdots\!96\)\( \beta_{2} - \)\(43\!\cdots\!96\)\( \beta_{3} - \)\(88\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!34\)\( \beta_{5}) q^{62}\) \(+(\)\(25\!\cdots\!52\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} + \)\(26\!\cdots\!20\)\( \beta_{2} - \)\(46\!\cdots\!35\)\( \beta_{3} - \)\(17\!\cdots\!79\)\( \beta_{4} + \)\(95\!\cdots\!62\)\( \beta_{5}) q^{63}\) \(+(\)\(75\!\cdots\!08\)\( - \)\(25\!\cdots\!92\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(32\!\cdots\!52\)\( \beta_{3} - \)\(48\!\cdots\!40\)\( \beta_{4} - \)\(84\!\cdots\!64\)\( \beta_{5}) q^{64}\) \(+(\)\(80\!\cdots\!00\)\( - \)\(15\!\cdots\!24\)\( \beta_{1} + \)\(44\!\cdots\!68\)\( \beta_{2} - \)\(24\!\cdots\!38\)\( \beta_{3} + \)\(10\!\cdots\!90\)\( \beta_{4} + \)\(14\!\cdots\!20\)\( \beta_{5}) q^{65}\) \(+(-\)\(78\!\cdots\!24\)\( - \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(76\!\cdots\!84\)\( \beta_{2} - \)\(15\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!72\)\( \beta_{4} - \)\(71\!\cdots\!80\)\( \beta_{5}) q^{66}\) \(+(-\)\(59\!\cdots\!64\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} - \)\(13\!\cdots\!96\)\( \beta_{2} + \)\(48\!\cdots\!04\)\( \beta_{3} + \)\(26\!\cdots\!88\)\( \beta_{4} + \)\(26\!\cdots\!88\)\( \beta_{5}) q^{67}\) \(+(\)\(30\!\cdots\!52\)\( - \)\(24\!\cdots\!54\)\( \beta_{1} - \)\(27\!\cdots\!06\)\( \beta_{2} + \)\(16\!\cdots\!84\)\( \beta_{3} - \)\(76\!\cdots\!36\)\( \beta_{4} - \)\(70\!\cdots\!60\)\( \beta_{5}) q^{68}\) \(+(\)\(20\!\cdots\!28\)\( + \)\(54\!\cdots\!60\)\( \beta_{1} - \)\(31\!\cdots\!12\)\( \beta_{2} - \)\(42\!\cdots\!86\)\( \beta_{3} - \)\(93\!\cdots\!34\)\( \beta_{4} - \)\(29\!\cdots\!96\)\( \beta_{5}) q^{69}\) \(+(-\)\(57\!\cdots\!00\)\( + \)\(38\!\cdots\!48\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} - \)\(19\!\cdots\!64\)\( \beta_{3} + \)\(78\!\cdots\!40\)\( \beta_{4} + \)\(21\!\cdots\!20\)\( \beta_{5}) q^{70}\) \(+(-\)\(85\!\cdots\!88\)\( + \)\(49\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(34\!\cdots\!38\)\( \beta_{3} + \)\(29\!\cdots\!58\)\( \beta_{4} - \)\(12\!\cdots\!20\)\( \beta_{5}) q^{71}\) \(+(-\)\(15\!\cdots\!28\)\( + \)\(44\!\cdots\!02\)\( \beta_{1} - \)\(47\!\cdots\!48\)\( \beta_{2} - \)\(15\!\cdots\!88\)\( \beta_{3} + \)\(21\!\cdots\!34\)\( \beta_{4} + \)\(25\!\cdots\!09\)\( \beta_{5}) q^{72}\) \(+(-\)\(32\!\cdots\!54\)\( - \)\(39\!\cdots\!86\)\( \beta_{1} - \)\(19\!\cdots\!24\)\( \beta_{2} + \)\(13\!\cdots\!16\)\( \beta_{3} - \)\(83\!\cdots\!42\)\( \beta_{4} - \)\(10\!\cdots\!36\)\( \beta_{5}) q^{73}\) \(+(\)\(95\!\cdots\!58\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + \)\(62\!\cdots\!08\)\( \beta_{2} - \)\(91\!\cdots\!48\)\( \beta_{3} - \)\(99\!\cdots\!12\)\( \beta_{4} + \)\(98\!\cdots\!72\)\( \beta_{5}) q^{74}\) \(+(-\)\(12\!\cdots\!25\)\( + \)\(28\!\cdots\!50\)\( \beta_{1} - \)\(49\!\cdots\!00\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} - \)\(31\!\cdots\!50\)\( \beta_{4} + \)\(57\!\cdots\!00\)\( \beta_{5}) q^{75}\) \(+(-\)\(39\!\cdots\!92\)\( + \)\(49\!\cdots\!56\)\( \beta_{1} - \)\(26\!\cdots\!48\)\( \beta_{2} - \)\(94\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4} - \)\(93\!\cdots\!56\)\( \beta_{5}) q^{76}\) \(+(\)\(31\!\cdots\!16\)\( + \)\(99\!\cdots\!16\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} - \)\(11\!\cdots\!72\)\( \beta_{3} + \)\(19\!\cdots\!36\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5}) q^{77}\) \(+(-\)\(30\!\cdots\!14\)\( + \)\(28\!\cdots\!14\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} + \)\(70\!\cdots\!76\)\( \beta_{3} + \)\(23\!\cdots\!48\)\( \beta_{4} - \)\(25\!\cdots\!16\)\( \beta_{5}) q^{78}\) \(+(\)\(80\!\cdots\!60\)\( - \)\(43\!\cdots\!12\)\( \beta_{1} - \)\(56\!\cdots\!16\)\( \beta_{2} + \)\(21\!\cdots\!85\)\( \beta_{3} + \)\(78\!\cdots\!17\)\( \beta_{4} + \)\(51\!\cdots\!54\)\( \beta_{5}) q^{79}\) \(+(\)\(43\!\cdots\!00\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(44\!\cdots\!56\)\( \beta_{3} - \)\(30\!\cdots\!20\)\( \beta_{4} - \)\(14\!\cdots\!60\)\( \beta_{5}) q^{80}\) \(+\)\(62\!\cdots\!01\)\( q^{81}\) \(+(-\)\(50\!\cdots\!94\)\( - \)\(20\!\cdots\!86\)\( \beta_{1} - \)\(22\!\cdots\!76\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3} + \)\(28\!\cdots\!76\)\( \beta_{4} + \)\(15\!\cdots\!32\)\( \beta_{5}) q^{82}\) \(+(\)\(38\!\cdots\!28\)\( - \)\(11\!\cdots\!50\)\( \beta_{1} + \)\(27\!\cdots\!08\)\( \beta_{2} + \)\(16\!\cdots\!78\)\( \beta_{3} + \)\(33\!\cdots\!64\)\( \beta_{4} + \)\(42\!\cdots\!52\)\( \beta_{5}) q^{83}\) \(+(-\)\(59\!\cdots\!08\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(35\!\cdots\!40\)\( \beta_{2} - \)\(57\!\cdots\!76\)\( \beta_{3} - \)\(18\!\cdots\!08\)\( \beta_{4} - \)\(92\!\cdots\!64\)\( \beta_{5}) q^{84}\) \(+(-\)\(25\!\cdots\!00\)\( - \)\(24\!\cdots\!42\)\( \beta_{1} - \)\(32\!\cdots\!76\)\( \beta_{2} + \)\(41\!\cdots\!86\)\( \beta_{3} - \)\(16\!\cdots\!00\)\( \beta_{4} - \)\(77\!\cdots\!00\)\( \beta_{5}) q^{85}\) \(+(\)\(38\!\cdots\!64\)\( + \)\(77\!\cdots\!40\)\( \beta_{1} - \)\(78\!\cdots\!28\)\( \beta_{2} - \)\(74\!\cdots\!40\)\( \beta_{3} + \)\(33\!\cdots\!52\)\( \beta_{4} - \)\(15\!\cdots\!96\)\( \beta_{5}) q^{86}\) \(+(-\)\(11\!\cdots\!62\)\( - \)\(34\!\cdots\!07\)\( \beta_{1} - \)\(50\!\cdots\!34\)\( \beta_{2} + \)\(15\!\cdots\!55\)\( \beta_{3} - \)\(16\!\cdots\!42\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5}) q^{87}\) \(+(\)\(88\!\cdots\!68\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!40\)\( \beta_{2} + \)\(80\!\cdots\!84\)\( \beta_{3} - \)\(21\!\cdots\!40\)\( \beta_{4} + \)\(18\!\cdots\!92\)\( \beta_{5}) q^{88}\) \(+(\)\(73\!\cdots\!18\)\( + \)\(78\!\cdots\!04\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2} + \)\(12\!\cdots\!88\)\( \beta_{3} - \)\(32\!\cdots\!72\)\( \beta_{4} + \)\(40\!\cdots\!80\)\( \beta_{5}) q^{89}\) \(+(\)\(22\!\cdots\!50\)\( + \)\(10\!\cdots\!22\)\( \beta_{1} - \)\(18\!\cdots\!44\)\( \beta_{2} - \)\(18\!\cdots\!56\)\( \beta_{3} + \)\(77\!\cdots\!40\)\( \beta_{4} + \)\(23\!\cdots\!70\)\( \beta_{5}) q^{90}\) \(+(\)\(17\!\cdots\!16\)\( + \)\(40\!\cdots\!68\)\( \beta_{1} + \)\(90\!\cdots\!12\)\( \beta_{2} + \)\(17\!\cdots\!06\)\( \beta_{3} + \)\(14\!\cdots\!62\)\( \beta_{4} - \)\(15\!\cdots\!88\)\( \beta_{5}) q^{91}\) \(+(\)\(58\!\cdots\!28\)\( + \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(35\!\cdots\!80\)\( \beta_{2} - \)\(18\!\cdots\!52\)\( \beta_{3} - \)\(29\!\cdots\!48\)\( \beta_{4} - \)\(31\!\cdots\!32\)\( \beta_{5}) q^{92}\) \(+(\)\(22\!\cdots\!28\)\( - \)\(61\!\cdots\!08\)\( \beta_{1} + \)\(41\!\cdots\!08\)\( \beta_{2} - \)\(25\!\cdots\!23\)\( \beta_{3} + \)\(89\!\cdots\!21\)\( \beta_{4} - \)\(21\!\cdots\!82\)\( \beta_{5}) q^{93}\) \(+(\)\(35\!\cdots\!52\)\( + \)\(39\!\cdots\!96\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3} + \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(59\!\cdots\!76\)\( \beta_{5}) q^{94}\) \(+(-\)\(53\!\cdots\!00\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} - \)\(61\!\cdots\!60\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!60\)\( \beta_{4} + \)\(19\!\cdots\!80\)\( \beta_{5}) q^{95}\) \(+(\)\(93\!\cdots\!64\)\( - \)\(17\!\cdots\!04\)\( \beta_{1} + \)\(47\!\cdots\!12\)\( \beta_{2} + \)\(27\!\cdots\!60\)\( \beta_{3} - \)\(84\!\cdots\!64\)\( \beta_{4} - \)\(88\!\cdots\!28\)\( \beta_{5}) q^{96}\) \(+(\)\(16\!\cdots\!66\)\( - \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(19\!\cdots\!04\)\( \beta_{2} - \)\(96\!\cdots\!48\)\( \beta_{3} - \)\(50\!\cdots\!20\)\( \beta_{4} - \)\(20\!\cdots\!24\)\( \beta_{5}) q^{97}\) \(+(\)\(43\!\cdots\!21\)\( - \)\(21\!\cdots\!19\)\( \beta_{1} + \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(21\!\cdots\!72\)\( \beta_{4} - \)\(15\!\cdots\!64\)\( \beta_{5}) q^{98}\) \(+(\)\(92\!\cdots\!56\)\( + \)\(12\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!56\)\( \beta_{2} - \)\(13\!\cdots\!62\)\( \beta_{3} + \)\(93\!\cdots\!56\)\( \beta_{4} + \)\(11\!\cdots\!96\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 72903656826q^{2} \) \(\mathstrut -\mathstrut 300189270593998242q^{3} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!32\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!82\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!88\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!32\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!94\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 72903656826q^{2} \) \(\mathstrut -\mathstrut 300189270593998242q^{3} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!32\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!82\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!88\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!32\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!94\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!64\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!24\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!76\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!32\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!80\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!96\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!74\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!72\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!16\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!92\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!24\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!24\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!12\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!58\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!64\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!96\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!24\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!12\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!48\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!16\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!68\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!52\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!84\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!32\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!16\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!24\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!96\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!84\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!60\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!90\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!50\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!72\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!44\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!16\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!18\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!04\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!08\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!48\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!56\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!12\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!48\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!44\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!84\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!12\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!68\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!28\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!68\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!24\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!48\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!50\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!52\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!96\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!84\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!60\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!06\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!64\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!68\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!48\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!84\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!72\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!08\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!08\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!68\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!68\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!12\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!84\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!96\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!26\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!36\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(327712249929578892882\) \(x^{4}\mathstrut -\mathstrut \) \(247064970776889197907293537206\) \(x^{3}\mathstrut +\mathstrut \) \(25021443441555202497465421340907628615965\) \(x^{2}\mathstrut +\mathstrut \) \(14077016837018585361473224499724917623325562275625\) \(x\mathstrut -\mathstrut \) \(279259064416593965218586682187488478036140359478345093745500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 3 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} - 40711045962 \nu - 3932546999134591191657 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(2647338101397\) \(\nu^{5}\mathstrut +\mathstrut \) \(15585983560336005406296\) \(\nu^{4}\mathstrut +\mathstrut \) \(665943825059194354529114527395714\) \(\nu^{3}\mathstrut -\mathstrut \) \(2709584662600088683935949720665253390974588\) \(\nu^{2}\mathstrut -\mathstrut \) \(28336308661367042176479914042787806437245868632255285\) \(\nu\mathstrut +\mathstrut \) \(63322920809782082624626193833747499036625134066418078569730780\)\()/\)\(13\!\cdots\!40\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(104778897869878551\) \(\nu^{5}\mathstrut -\mathstrut \) \(723744240429543687924148152\) \(\nu^{4}\mathstrut +\mathstrut \) \(34403186840616316215985349942118946422\) \(\nu^{3}\mathstrut +\mathstrut \) \(190818785626362696757958152258341409915806056556\) \(\nu^{2}\mathstrut -\mathstrut \) \(2182591086580538223721931001219022522622317728806750482935\) \(\nu\mathstrut -\mathstrut \) \(5417817678162782131119470382015236091722504014174630329062729721420\)\()/\)\(62\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(25300363094553231\) \(\nu^{5}\mathstrut +\mathstrut \) \(2119930948658032782322797432\) \(\nu^{4}\mathstrut +\mathstrut \) \(6541929024458501170586700888486179418\) \(\nu^{3}\mathstrut -\mathstrut \) \(556218784724563435725073421322083729718007982796\) \(\nu^{2}\mathstrut -\mathstrut \) \(2487365207004615067701905635488661412782409232072541917585\) \(\nu\mathstrut +\mathstrut \) \(18013763051749487640773267896799037138147661392064065749480539920300\)\()/\)\(12\!\cdots\!60\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(6785174327\) \(\beta_{1}\mathstrut +\mathstrut \) \(3932546999154946714638\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(10241\) \(\beta_{5}\mathstrut +\mathstrut \) \(8666\) \(\beta_{4}\mathstrut -\mathstrut \) \(6340812\) \(\beta_{3}\mathstrut +\mathstrut \) \(17570586570\) \(\beta_{2}\mathstrut +\mathstrut \) \(6300306283451987631960\) \(\beta_{1}\mathstrut +\mathstrut \) \(26683016950082802351171263312988\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(184385957659079\) \(\beta_{5}\mathstrut -\mathstrut \) \(147886657045898\) \(\beta_{4}\mathstrut +\mathstrut \) \(145213590970725740\) \(\beta_{3}\mathstrut +\mathstrut \) \(4318306282598743033064\) \(\beta_{2}\mathstrut +\mathstrut \) \(55109579314199972427421588418346\) \(\beta_{1}\mathstrut +\mathstrut \) \(12388125284412213438358600535730522935095644\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(13220994683194238297297297\) \(\beta_{5}\mathstrut +\mathstrut \) \(8503776992325990821009018\) \(\beta_{4}\mathstrut -\mathstrut \) \(10873323761890330947413991308\) \(\beta_{3}\mathstrut +\mathstrut \) \(30390299029534496433664184974350\) \(\beta_{2}\mathstrut +\mathstrut \) \(5697025877958147537301573458895105056666306\) \(\beta_{1}\mathstrut +\mathstrut \) \(54180252698368277670536905870375644096963350849575086\)\()/972\)

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
−1.38952e10
−1.03374e10
−3.99343e9
3.35353e9
9.35614e9
1.55163e10
−9.55218e10 −5.00315e16 6.76322e21 1.07436e25 4.77910e27 6.00629e29 −4.20491e32 2.50316e33 −1.02625e36
1.2 −7.41748e10 −5.00315e16 3.14071e21 −1.28474e25 3.71108e27 −1.80308e28 −5.78216e31 2.50316e33 9.52956e35
1.3 −3.61112e10 −5.00315e16 −1.05717e21 4.57686e24 1.80670e27 −2.20036e29 1.23441e32 2.50316e33 −1.65276e35
1.4 7.97054e9 −5.00315e16 −2.29765e21 −4.57364e24 −3.98779e26 9.29524e29 −3.71335e31 2.50316e33 −3.64544e34
1.5 4.39862e10 −5.00315e16 −4.26395e20 4.61068e24 −2.20070e27 −1.72926e30 −1.22615e32 2.50316e33 2.02806e35
1.6 8.09473e10 −5.00315e16 4.19128e21 7.59856e24 −4.04992e27 1.03665e30 1.48142e32 2.50316e33 6.15083e35
\(n\): 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 \(T_{2}^{6} \) \(\mathstrut +\mathstrut 72903656826 T_{2}^{5} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!72\)\( T_{2}^{4} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!88\)\( T_{2}^{3} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!36\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!44\)\( T_{2} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!72\)\( \) acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(3))\).