Properties

Label 3.68.a.b
Level 3
Weight 68
Character orbit 3.a
Self dual Yes
Analytic conductor 85.287
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 68 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(85.287105579\)
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 \( 2^{46}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11^{2}\cdot 13\cdot 17 \)
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.

\(f(q)\) \(=\) \( q\) \(+(2289225861 - \beta_{1}) q^{2}\) \(-5559060566555523 q^{3}\) \(+(76856221419709869142 - 871210566 \beta_{1} + \beta_{2}) q^{4}\) \(+(-\)\(30\!\cdots\!50\)\( - 5867760487466 \beta_{1} + 352 \beta_{2} - \beta_{3}) q^{5}\) \(+(-\)\(12\!\cdots\!03\)\( + 5559060566555523 \beta_{1}) q^{6}\) \(+(-\)\(46\!\cdots\!68\)\( + 291321334454968162 \beta_{1} - 80666795 \beta_{2} + 1598 \beta_{3} - \beta_{4} + 2 \beta_{5}) q^{7}\) \(+(\)\(29\!\cdots\!32\)\( - 43246795112688713255 \beta_{1} - 2546139725 \beta_{2} - 2166905 \beta_{3} - 110 \beta_{4} - 141 \beta_{5}) q^{8}\) \(+\)\(30\!\cdots\!29\)\( q^{9}\) \(+O(q^{10})\) \( q\) \(+(2289225861 - \beta_{1}) q^{2}\) \(-5559060566555523 q^{3}\) \(+(76856221419709869142 - 871210566 \beta_{1} + \beta_{2}) q^{4}\) \(+(-\)\(30\!\cdots\!50\)\( - 5867760487466 \beta_{1} + 352 \beta_{2} - \beta_{3}) q^{5}\) \(+(-\)\(12\!\cdots\!03\)\( + 5559060566555523 \beta_{1}) q^{6}\) \(+(-\)\(46\!\cdots\!68\)\( + 291321334454968162 \beta_{1} - 80666795 \beta_{2} + 1598 \beta_{3} - \beta_{4} + 2 \beta_{5}) q^{7}\) \(+(\)\(29\!\cdots\!32\)\( - 43246795112688713255 \beta_{1} - 2546139725 \beta_{2} - 2166905 \beta_{3} - 110 \beta_{4} - 141 \beta_{5}) q^{8}\) \(+\)\(30\!\cdots\!29\)\( q^{9}\) \(+(\)\(12\!\cdots\!50\)\( - \)\(21\!\cdots\!08\)\( \beta_{1} + 11275681472306 \beta_{2} - 11521777498 \beta_{3} - 286060 \beta_{4} + 16190 \beta_{5}) q^{10}\) \(+(\)\(20\!\cdots\!04\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} - 238526413697568 \beta_{2} + 101100377470 \beta_{3} - 489760 \beta_{4} - 2842176 \beta_{5}) q^{11}\) \(+(-\)\(42\!\cdots\!66\)\( + \)\(48\!\cdots\!18\)\( \beta_{1} - 5559060566555523 \beta_{2}) q^{12}\) \(+(-\)\(17\!\cdots\!94\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} - 5436148551484991 \beta_{2} - 18442904453723 \beta_{3} - 2441657309 \beta_{4} + 400101562 \beta_{5}) q^{13}\) \(+(-\)\(74\!\cdots\!92\)\( + \)\(13\!\cdots\!18\)\( \beta_{1} - 458625319886245086 \beta_{2} - 211620741738506 \beta_{3} + 11474357492 \beta_{4} + 28966144974 \beta_{5}) q^{14}\) \(+(\)\(16\!\cdots\!50\)\( + \)\(32\!\cdots\!18\)\( \beta_{1} - 1956789319427544096 \beta_{2} + 5559060566555523 \beta_{3}) q^{15}\) \(+(-\)\(17\!\cdots\!80\)\( + \)\(45\!\cdots\!78\)\( \beta_{1} + 42872511004161547578 \beta_{2} - 49623051005294894 \beta_{3} + 245849360668 \beta_{4} - 859755870086 \beta_{5}) q^{16}\) \(+(-\)\(37\!\cdots\!66\)\( - \)\(56\!\cdots\!24\)\( \beta_{1} + \)\(42\!\cdots\!86\)\( \beta_{2} - 484211423013981496 \beta_{3} - 2330416206678 \beta_{4} - 9069535208916 \beta_{5}) q^{17}\) \(+(\)\(70\!\cdots\!69\)\( - \)\(30\!\cdots\!29\)\( \beta_{1}) q^{18}\) \(+(-\)\(11\!\cdots\!92\)\( - \)\(21\!\cdots\!72\)\( \beta_{1} - \)\(17\!\cdots\!06\)\( \beta_{2} - 18473170732721507368 \beta_{3} + 311331020568626 \beta_{4} + 651351290112284 \beta_{5}) q^{19}\) \(+(\)\(76\!\cdots\!00\)\( - \)\(16\!\cdots\!16\)\( \beta_{1} + \)\(12\!\cdots\!62\)\( \beta_{2} - 96920911026180581696 \beta_{3} - 3060130678465920 \beta_{4} - 863367828793920 \beta_{5}) q^{20}\) \(+(\)\(26\!\cdots\!64\)\( - \)\(16\!\cdots\!26\)\( \beta_{1} + \)\(44\!\cdots\!85\)\( \beta_{2} - 8883378785355725754 \beta_{3} + 5559060566555523 \beta_{4} - 11118121133111046 \beta_{5}) q^{21}\) \(+(\)\(23\!\cdots\!28\)\( + \)\(28\!\cdots\!36\)\( \beta_{1} + \)\(28\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3} + 55067333963906728 \beta_{4} + 624153984764092 \beta_{5}) q^{22}\) \(+(\)\(20\!\cdots\!44\)\( + \)\(55\!\cdots\!76\)\( \beta_{1} + \)\(21\!\cdots\!82\)\( \beta_{2} + \)\(48\!\cdots\!96\)\( \beta_{3} - 301038046592161562 \beta_{4} + 120696402704387124 \beta_{5}) q^{23}\) \(+(-\)\(16\!\cdots\!36\)\( + \)\(24\!\cdots\!65\)\( \beta_{1} + \)\(14\!\cdots\!75\)\( \beta_{2} + \)\(12\!\cdots\!15\)\( \beta_{3} + 611496662321107530 \beta_{4} + 783827539884328743 \beta_{5}) q^{24}\) \(+(\)\(20\!\cdots\!75\)\( - \)\(34\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2} + \)\(12\!\cdots\!50\)\( \beta_{3} - 1651190398107421750 \beta_{4} - 4875320240304740500 \beta_{5}) q^{25}\) \(+(\)\(24\!\cdots\!82\)\( + \)\(96\!\cdots\!82\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!48\)\( \beta_{3} + 4639782283551053496 \beta_{4} + 4108393197068209236 \beta_{5}) q^{26}\) \(-\)\(17\!\cdots\!67\)\( q^{27}\) \(+(-\)\(24\!\cdots\!24\)\( + \)\(65\!\cdots\!96\)\( \beta_{1} - \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3} + 83109627162696689280 \beta_{4} + 56114506768901699520 \beta_{5}) q^{28}\) \(+(-\)\(31\!\cdots\!94\)\( - \)\(12\!\cdots\!30\)\( \beta_{1} - \)\(21\!\cdots\!98\)\( \beta_{2} - \)\(64\!\cdots\!17\)\( \beta_{3} - \)\(75\!\cdots\!46\)\( \beta_{4} - 3006089876479886988 \beta_{5}) q^{29}\) \(+(-\)\(67\!\cdots\!50\)\( + \)\(12\!\cdots\!84\)\( \beta_{1} - \)\(62\!\cdots\!38\)\( \beta_{2} + \)\(64\!\cdots\!54\)\( \beta_{3} + \)\(15\!\cdots\!80\)\( \beta_{4} - 90001190572533917370 \beta_{5}) q^{30}\) \(+(-\)\(37\!\cdots\!84\)\( - \)\(13\!\cdots\!18\)\( \beta_{1} + \)\(29\!\cdots\!97\)\( \beta_{2} + \)\(76\!\cdots\!74\)\( \beta_{3} + \)\(73\!\cdots\!27\)\( \beta_{4} - \)\(36\!\cdots\!18\)\( \beta_{5}) q^{31}\) \(+(-\)\(10\!\cdots\!08\)\( + \)\(25\!\cdots\!12\)\( \beta_{1} + \)\(38\!\cdots\!76\)\( \beta_{2} - \)\(19\!\cdots\!32\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!88\)\( \beta_{5}) q^{32}\) \(+(-\)\(11\!\cdots\!92\)\( + \)\(59\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!64\)\( \beta_{2} - \)\(56\!\cdots\!10\)\( \beta_{3} + \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(15\!\cdots\!48\)\( \beta_{5}) q^{33}\) \(+(\)\(11\!\cdots\!34\)\( - \)\(44\!\cdots\!86\)\( \beta_{1} + \)\(78\!\cdots\!88\)\( \beta_{2} - \)\(65\!\cdots\!76\)\( \beta_{3} - \)\(15\!\cdots\!88\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5}) q^{34}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(23\!\cdots\!80\)\( \beta_{1} - \)\(75\!\cdots\!40\)\( \beta_{2} + \)\(18\!\cdots\!90\)\( \beta_{3} + \)\(47\!\cdots\!60\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5}) q^{35}\) \(+(\)\(23\!\cdots\!18\)\( - \)\(26\!\cdots\!14\)\( \beta_{1} + \)\(30\!\cdots\!29\)\( \beta_{2}) q^{36}\) \(+(\)\(45\!\cdots\!42\)\( - \)\(71\!\cdots\!48\)\( \beta_{1} - \)\(17\!\cdots\!81\)\( \beta_{2} + \)\(41\!\cdots\!29\)\( \beta_{3} - \)\(50\!\cdots\!03\)\( \beta_{4} - \)\(24\!\cdots\!18\)\( \beta_{5}) q^{37}\) \(+(\)\(20\!\cdots\!84\)\( + \)\(11\!\cdots\!36\)\( \beta_{1} - \)\(32\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(61\!\cdots\!80\)\( \beta_{4} + \)\(74\!\cdots\!28\)\( \beta_{5}) q^{38}\) \(+(\)\(95\!\cdots\!62\)\( + \)\(63\!\cdots\!60\)\( \beta_{1} + \)\(30\!\cdots\!93\)\( \beta_{2} + \)\(10\!\cdots\!29\)\( \beta_{3} + \)\(13\!\cdots\!07\)\( \beta_{4} - \)\(22\!\cdots\!26\)\( \beta_{5}) q^{39}\) \(+(\)\(19\!\cdots\!00\)\( - \)\(19\!\cdots\!54\)\( \beta_{1} + \)\(21\!\cdots\!18\)\( \beta_{2} - \)\(61\!\cdots\!54\)\( \beta_{3} + \)\(15\!\cdots\!40\)\( \beta_{4} - \)\(23\!\cdots\!10\)\( \beta_{5}) q^{40}\) \(+(\)\(44\!\cdots\!66\)\( - \)\(44\!\cdots\!56\)\( \beta_{1} + \)\(66\!\cdots\!38\)\( \beta_{2} - \)\(64\!\cdots\!64\)\( \beta_{3} - \)\(46\!\cdots\!82\)\( \beta_{4} + \)\(55\!\cdots\!28\)\( \beta_{5}) q^{41}\) \(+(\)\(41\!\cdots\!16\)\( - \)\(75\!\cdots\!14\)\( \beta_{1} + \)\(25\!\cdots\!78\)\( \beta_{2} + \)\(11\!\cdots\!38\)\( \beta_{3} - \)\(63\!\cdots\!16\)\( \beta_{4} - \)\(16\!\cdots\!02\)\( \beta_{5}) q^{42}\) \(+(-\)\(18\!\cdots\!00\)\( - \)\(85\!\cdots\!64\)\( \beta_{1} + \)\(70\!\cdots\!82\)\( \beta_{2} + \)\(84\!\cdots\!64\)\( \beta_{3} + \)\(19\!\cdots\!82\)\( \beta_{4} + \)\(34\!\cdots\!64\)\( \beta_{5}) q^{43}\) \(+(-\)\(58\!\cdots\!84\)\( - \)\(44\!\cdots\!72\)\( \beta_{1} - \)\(42\!\cdots\!20\)\( \beta_{2} + \)\(13\!\cdots\!20\)\( \beta_{3} - \)\(57\!\cdots\!80\)\( \beta_{4} - \)\(39\!\cdots\!48\)\( \beta_{5}) q^{44}\) \(+(-\)\(92\!\cdots\!50\)\( - \)\(18\!\cdots\!14\)\( \beta_{1} + \)\(10\!\cdots\!08\)\( \beta_{2} - \)\(30\!\cdots\!29\)\( \beta_{3}) q^{45}\) \(+(-\)\(75\!\cdots\!04\)\( - \)\(46\!\cdots\!56\)\( \beta_{1} - \)\(63\!\cdots\!56\)\( \beta_{2} - \)\(99\!\cdots\!72\)\( \beta_{3} - \)\(19\!\cdots\!16\)\( \beta_{4} - \)\(24\!\cdots\!04\)\( \beta_{5}) q^{46}\) \(+(\)\(96\!\cdots\!60\)\( - \)\(80\!\cdots\!40\)\( \beta_{1} - \)\(40\!\cdots\!22\)\( \beta_{2} - \)\(11\!\cdots\!28\)\( \beta_{3} + \)\(86\!\cdots\!46\)\( \beta_{4} + \)\(11\!\cdots\!12\)\( \beta_{5}) q^{47}\) \(+(\)\(99\!\cdots\!40\)\( - \)\(25\!\cdots\!94\)\( \beta_{1} - \)\(23\!\cdots\!94\)\( \beta_{2} + \)\(27\!\cdots\!62\)\( \beta_{3} - \)\(13\!\cdots\!64\)\( \beta_{4} + \)\(47\!\cdots\!78\)\( \beta_{5}) q^{48}\) \(+(\)\(16\!\cdots\!45\)\( - \)\(18\!\cdots\!32\)\( \beta_{1} + \)\(23\!\cdots\!66\)\( \beta_{2} - \)\(90\!\cdots\!46\)\( \beta_{3} - \)\(10\!\cdots\!78\)\( \beta_{4} + \)\(23\!\cdots\!76\)\( \beta_{5}) q^{49}\) \(+(\)\(79\!\cdots\!75\)\( - \)\(29\!\cdots\!75\)\( \beta_{1} + \)\(53\!\cdots\!00\)\( \beta_{2} + \)\(73\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} - \)\(71\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(\)\(20\!\cdots\!18\)\( + \)\(31\!\cdots\!52\)\( \beta_{1} - \)\(23\!\cdots\!78\)\( \beta_{2} + \)\(26\!\cdots\!08\)\( \beta_{3} + \)\(12\!\cdots\!94\)\( \beta_{4} + \)\(50\!\cdots\!68\)\( \beta_{5}) q^{51}\) \(+(-\)\(12\!\cdots\!96\)\( - \)\(16\!\cdots\!44\)\( \beta_{1} - \)\(23\!\cdots\!34\)\( \beta_{2} - \)\(96\!\cdots\!88\)\( \beta_{3} - \)\(57\!\cdots\!44\)\( \beta_{4} - \)\(13\!\cdots\!28\)\( \beta_{5}) q^{52}\) \(+(-\)\(22\!\cdots\!06\)\( - \)\(53\!\cdots\!30\)\( \beta_{1} - \)\(21\!\cdots\!58\)\( \beta_{2} + \)\(25\!\cdots\!67\)\( \beta_{3} - \)\(59\!\cdots\!54\)\( \beta_{4} + \)\(31\!\cdots\!16\)\( \beta_{5}) q^{53}\) \(+(-\)\(39\!\cdots\!87\)\( + \)\(17\!\cdots\!67\)\( \beta_{1}) q^{54}\) \(+(-\)\(85\!\cdots\!00\)\( + \)\(53\!\cdots\!44\)\( \beta_{1} + \)\(22\!\cdots\!12\)\( \beta_{2} - \)\(35\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4} + \)\(41\!\cdots\!40\)\( \beta_{5}) q^{55}\) \(+(-\)\(88\!\cdots\!40\)\( + \)\(24\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!16\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(31\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5}) q^{56}\) \(+(\)\(62\!\cdots\!16\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(95\!\cdots\!38\)\( \beta_{2} + \)\(10\!\cdots\!64\)\( \beta_{3} - \)\(17\!\cdots\!98\)\( \beta_{4} - \)\(36\!\cdots\!32\)\( \beta_{5}) q^{57}\) \(+(-\)\(45\!\cdots\!02\)\( + \)\(56\!\cdots\!52\)\( \beta_{1} + \)\(52\!\cdots\!54\)\( \beta_{2} - \)\(27\!\cdots\!06\)\( \beta_{3} + \)\(35\!\cdots\!52\)\( \beta_{4} + \)\(29\!\cdots\!22\)\( \beta_{5}) q^{58}\) \(+(\)\(72\!\cdots\!52\)\( + \)\(64\!\cdots\!44\)\( \beta_{1} + \)\(23\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!12\)\( \beta_{3} - \)\(10\!\cdots\!36\)\( \beta_{4} + \)\(15\!\cdots\!92\)\( \beta_{5}) q^{59}\) \(+(-\)\(42\!\cdots\!00\)\( + \)\(91\!\cdots\!68\)\( \beta_{1} - \)\(71\!\cdots\!26\)\( \beta_{2} + \)\(53\!\cdots\!08\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4} + \)\(47\!\cdots\!60\)\( \beta_{5}) q^{60}\) \(+(-\)\(50\!\cdots\!90\)\( - \)\(93\!\cdots\!96\)\( \beta_{1} - \)\(20\!\cdots\!21\)\( \beta_{2} - \)\(14\!\cdots\!75\)\( \beta_{3} - \)\(72\!\cdots\!15\)\( \beta_{4} - \)\(45\!\cdots\!34\)\( \beta_{5}) q^{61}\) \(+(\)\(21\!\cdots\!64\)\( + \)\(35\!\cdots\!10\)\( \beta_{1} + \)\(21\!\cdots\!18\)\( \beta_{2} + \)\(11\!\cdots\!66\)\( \beta_{3} + \)\(15\!\cdots\!48\)\( \beta_{4} - \)\(42\!\cdots\!54\)\( \beta_{5}) q^{62}\) \(+(-\)\(14\!\cdots\!72\)\( + \)\(90\!\cdots\!98\)\( \beta_{1} - \)\(24\!\cdots\!55\)\( \beta_{2} + \)\(49\!\cdots\!42\)\( \beta_{3} - \)\(30\!\cdots\!29\)\( \beta_{4} + \)\(61\!\cdots\!58\)\( \beta_{5}) q^{63}\) \(+(-\)\(55\!\cdots\!52\)\( - \)\(70\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(36\!\cdots\!20\)\( \beta_{3} - \)\(13\!\cdots\!80\)\( \beta_{4} + \)\(17\!\cdots\!92\)\( \beta_{5}) q^{64}\) \(+(\)\(15\!\cdots\!00\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} + \)\(29\!\cdots\!38\)\( \beta_{2} + \)\(58\!\cdots\!76\)\( \beta_{3} + \)\(41\!\cdots\!10\)\( \beta_{4} - \)\(11\!\cdots\!40\)\( \beta_{5}) q^{65}\) \(+(-\)\(13\!\cdots\!44\)\( - \)\(15\!\cdots\!28\)\( \beta_{1} - \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(85\!\cdots\!88\)\( \beta_{3} - \)\(30\!\cdots\!44\)\( \beta_{4} - \)\(34\!\cdots\!16\)\( \beta_{5}) q^{66}\) \(+(-\)\(60\!\cdots\!16\)\( - \)\(75\!\cdots\!80\)\( \beta_{1} + \)\(34\!\cdots\!80\)\( \beta_{2} - \)\(89\!\cdots\!96\)\( \beta_{3} + \)\(21\!\cdots\!52\)\( \beta_{4} - \)\(20\!\cdots\!56\)\( \beta_{5}) q^{67}\) \(+(\)\(10\!\cdots\!48\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} + \)\(13\!\cdots\!70\)\( \beta_{2} - \)\(62\!\cdots\!08\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(45\!\cdots\!80\)\( \beta_{5}) q^{68}\) \(+(-\)\(11\!\cdots\!12\)\( - \)\(30\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!86\)\( \beta_{2} - \)\(27\!\cdots\!08\)\( \beta_{3} + \)\(16\!\cdots\!26\)\( \beta_{4} - \)\(67\!\cdots\!52\)\( \beta_{5}) q^{69}\) \(+(-\)\(29\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(45\!\cdots\!20\)\( \beta_{2} + \)\(36\!\cdots\!40\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4} + \)\(23\!\cdots\!40\)\( \beta_{5}) q^{70}\) \(+(\)\(42\!\cdots\!12\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!74\)\( \beta_{2} - \)\(12\!\cdots\!12\)\( \beta_{3} - \)\(42\!\cdots\!06\)\( \beta_{4} + \)\(59\!\cdots\!16\)\( \beta_{5}) q^{71}\) \(+(\)\(89\!\cdots\!28\)\( - \)\(13\!\cdots\!95\)\( \beta_{1} - \)\(78\!\cdots\!25\)\( \beta_{2} - \)\(66\!\cdots\!45\)\( \beta_{3} - \)\(33\!\cdots\!90\)\( \beta_{4} - \)\(43\!\cdots\!89\)\( \beta_{5}) q^{72}\) \(+(\)\(27\!\cdots\!54\)\( + \)\(39\!\cdots\!56\)\( \beta_{1} - \)\(20\!\cdots\!90\)\( \beta_{2} - \)\(54\!\cdots\!02\)\( \beta_{3} - \)\(11\!\cdots\!26\)\( \beta_{4} - \)\(18\!\cdots\!96\)\( \beta_{5}) q^{73}\) \(+(\)\(16\!\cdots\!78\)\( + \)\(16\!\cdots\!02\)\( \beta_{1} + \)\(24\!\cdots\!56\)\( \beta_{2} + \)\(74\!\cdots\!56\)\( \beta_{3} + \)\(33\!\cdots\!68\)\( \beta_{4} - \)\(45\!\cdots\!76\)\( \beta_{5}) q^{74}\) \(+(-\)\(11\!\cdots\!25\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(66\!\cdots\!50\)\( \beta_{2} - \)\(71\!\cdots\!50\)\( \beta_{3} + \)\(91\!\cdots\!50\)\( \beta_{4} + \)\(27\!\cdots\!00\)\( \beta_{5}) q^{75}\) \(+(-\)\(86\!\cdots\!52\)\( + \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(42\!\cdots\!88\)\( \beta_{2} - \)\(13\!\cdots\!84\)\( \beta_{3} - \)\(21\!\cdots\!52\)\( \beta_{4} + \)\(21\!\cdots\!40\)\( \beta_{5}) q^{76}\) \(+(\)\(58\!\cdots\!24\)\( + \)\(44\!\cdots\!16\)\( \beta_{1} + \)\(35\!\cdots\!84\)\( \beta_{2} - \)\(15\!\cdots\!52\)\( \beta_{3} - \)\(59\!\cdots\!76\)\( \beta_{4} - \)\(17\!\cdots\!32\)\( \beta_{5}) q^{77}\) \(+(-\)\(13\!\cdots\!86\)\( - \)\(53\!\cdots\!86\)\( \beta_{1} - \)\(74\!\cdots\!88\)\( \beta_{2} + \)\(56\!\cdots\!04\)\( \beta_{3} - \)\(25\!\cdots\!08\)\( \beta_{4} - \)\(22\!\cdots\!28\)\( \beta_{5}) q^{78}\) \(+(-\)\(79\!\cdots\!40\)\( - \)\(25\!\cdots\!46\)\( \beta_{1} - \)\(13\!\cdots\!51\)\( \beta_{2} - \)\(30\!\cdots\!38\)\( \beta_{3} + \)\(15\!\cdots\!71\)\( \beta_{4} + \)\(86\!\cdots\!22\)\( \beta_{5}) q^{79}\) \(+(\)\(36\!\cdots\!00\)\( - \)\(17\!\cdots\!64\)\( \beta_{1} + \)\(64\!\cdots\!68\)\( \beta_{2} + \)\(91\!\cdots\!76\)\( \beta_{3} + \)\(19\!\cdots\!80\)\( \beta_{4} - \)\(34\!\cdots\!20\)\( \beta_{5}) q^{80}\) \(+\)\(95\!\cdots\!41\)\( q^{81}\) \(+(\)\(10\!\cdots\!14\)\( - \)\(45\!\cdots\!02\)\( \beta_{1} + \)\(44\!\cdots\!36\)\( \beta_{2} - \)\(25\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!88\)\( \beta_{4} + \)\(42\!\cdots\!24\)\( \beta_{5}) q^{82}\) \(+(\)\(61\!\cdots\!72\)\( - \)\(38\!\cdots\!08\)\( \beta_{1} - \)\(61\!\cdots\!20\)\( \beta_{2} - \)\(17\!\cdots\!82\)\( \beta_{3} + \)\(39\!\cdots\!44\)\( \beta_{4} + \)\(10\!\cdots\!04\)\( \beta_{5}) q^{83}\) \(+(\)\(13\!\cdots\!52\)\( - \)\(36\!\cdots\!08\)\( \beta_{1} + \)\(10\!\cdots\!16\)\( \beta_{2} - \)\(72\!\cdots\!40\)\( \beta_{3} - \)\(46\!\cdots\!40\)\( \beta_{4} - \)\(31\!\cdots\!60\)\( \beta_{5}) q^{84}\) \(+(\)\(41\!\cdots\!00\)\( - \)\(24\!\cdots\!64\)\( \beta_{1} + \)\(12\!\cdots\!08\)\( \beta_{2} + \)\(29\!\cdots\!46\)\( \beta_{3} - \)\(44\!\cdots\!00\)\( \beta_{4} - \)\(35\!\cdots\!00\)\( \beta_{5}) q^{85}\) \(+(\)\(18\!\cdots\!84\)\( - \)\(50\!\cdots\!68\)\( \beta_{1} - \)\(25\!\cdots\!48\)\( \beta_{2} + \)\(12\!\cdots\!24\)\( \beta_{3} + \)\(37\!\cdots\!32\)\( \beta_{4} + \)\(17\!\cdots\!68\)\( \beta_{5}) q^{86}\) \(+(\)\(17\!\cdots\!62\)\( + \)\(67\!\cdots\!90\)\( \beta_{1} + \)\(11\!\cdots\!54\)\( \beta_{2} + \)\(35\!\cdots\!91\)\( \beta_{3} + \)\(42\!\cdots\!58\)\( \beta_{4} + \)\(16\!\cdots\!24\)\( \beta_{5}) q^{87}\) \(+(\)\(49\!\cdots\!32\)\( + \)\(70\!\cdots\!32\)\( \beta_{1} + \)\(12\!\cdots\!64\)\( \beta_{2} - \)\(40\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!20\)\( \beta_{4} + \)\(45\!\cdots\!16\)\( \beta_{5}) q^{88}\) \(+(-\)\(41\!\cdots\!82\)\( + \)\(57\!\cdots\!72\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} - \)\(37\!\cdots\!68\)\( \beta_{3} - \)\(65\!\cdots\!24\)\( \beta_{4} - \)\(20\!\cdots\!32\)\( \beta_{5}) q^{89}\) \(+(\)\(37\!\cdots\!50\)\( - \)\(67\!\cdots\!32\)\( \beta_{1} + \)\(34\!\cdots\!74\)\( \beta_{2} - \)\(35\!\cdots\!42\)\( \beta_{3} - \)\(88\!\cdots\!40\)\( \beta_{4} + \)\(50\!\cdots\!10\)\( \beta_{5}) q^{90}\) \(+(\)\(27\!\cdots\!96\)\( + \)\(15\!\cdots\!08\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3} + \)\(13\!\cdots\!86\)\( \beta_{4} + \)\(24\!\cdots\!88\)\( \beta_{5}) q^{91}\) \(+(\)\(69\!\cdots\!12\)\( + \)\(13\!\cdots\!64\)\( \beta_{1} + \)\(35\!\cdots\!28\)\( \beta_{2} - \)\(16\!\cdots\!28\)\( \beta_{3} + \)\(29\!\cdots\!16\)\( \beta_{4} - \)\(25\!\cdots\!28\)\( \beta_{5}) q^{92}\) \(+(\)\(20\!\cdots\!32\)\( + \)\(77\!\cdots\!14\)\( \beta_{1} - \)\(16\!\cdots\!31\)\( \beta_{2} - \)\(42\!\cdots\!02\)\( \beta_{3} - \)\(40\!\cdots\!21\)\( \beta_{4} + \)\(20\!\cdots\!14\)\( \beta_{5}) q^{93}\) \(+(\)\(17\!\cdots\!52\)\( + \)\(64\!\cdots\!84\)\( \beta_{1} + \)\(73\!\cdots\!12\)\( \beta_{2} + \)\(21\!\cdots\!32\)\( \beta_{3} - \)\(23\!\cdots\!84\)\( \beta_{4} + \)\(17\!\cdots\!52\)\( \beta_{5}) q^{94}\) \(+(\)\(14\!\cdots\!00\)\( - \)\(52\!\cdots\!16\)\( \beta_{1} - \)\(57\!\cdots\!28\)\( \beta_{2} + \)\(65\!\cdots\!84\)\( \beta_{3} - \)\(40\!\cdots\!40\)\( \beta_{4} - \)\(14\!\cdots\!40\)\( \beta_{5}) q^{95}\) \(+(\)\(60\!\cdots\!84\)\( - \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(21\!\cdots\!48\)\( \beta_{2} + \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(57\!\cdots\!88\)\( \beta_{4} - \)\(57\!\cdots\!24\)\( \beta_{5}) q^{96}\) \(+(\)\(64\!\cdots\!94\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(19\!\cdots\!04\)\( \beta_{2} - \)\(85\!\cdots\!60\)\( \beta_{3} + \)\(43\!\cdots\!60\)\( \beta_{4} + \)\(68\!\cdots\!68\)\( \beta_{5}) q^{97}\) \(+(\)\(43\!\cdots\!29\)\( - \)\(16\!\cdots\!01\)\( \beta_{1} + \)\(12\!\cdots\!04\)\( \beta_{2} - \)\(57\!\cdots\!88\)\( \beta_{3} - \)\(27\!\cdots\!44\)\( \beta_{4} + \)\(18\!\cdots\!56\)\( \beta_{5}) q^{98}\) \(+(\)\(64\!\cdots\!16\)\( - \)\(32\!\cdots\!52\)\( \beta_{1} - \)\(73\!\cdots\!72\)\( \beta_{2} + \)\(31\!\cdots\!30\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4} - \)\(87\!\cdots\!04\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 13735355166q^{2} \) \(\mathstrut -\mathstrut 33354363399333138q^{3} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!52\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!18\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!08\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!92\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!74\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 13735355166q^{2} \) \(\mathstrut -\mathstrut 33354363399333138q^{3} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!52\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!18\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!08\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!92\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!74\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!24\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!96\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!64\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!52\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!96\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!14\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!52\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!84\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!68\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!16\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!92\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!02\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!44\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!64\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!04\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!52\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!04\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!08\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!52\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!04\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!72\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!96\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!96\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!04\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!24\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!60\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!40\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!70\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!50\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!08\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!76\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!36\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!22\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!96\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!12\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!12\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!40\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!84\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!32\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!12\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!64\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!96\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!88\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!72\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!72\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!68\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!24\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!68\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!50\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!12\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!44\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!16\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!46\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!84\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!32\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!12\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!04\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!92\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!92\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!76\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!72\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!92\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!12\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!04\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!64\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!74\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!96\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(18265801580559590892\) \(x^{4}\mathstrut -\mathstrut \) \(7523970115455178716247853536\) \(x^{3}\mathstrut +\mathstrut \) \(78501261395396048023472482629898922880\) \(x^{2}\mathstrut +\mathstrut \) \(29605717530431234535889524616359650095893868800\) \(x\mathstrut -\mathstrut \) \(80637078962263575025979556823830075867035962985404800000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 3 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} - 22243446972 \nu - 219189618955593367272 \)
\(\beta_{3}\)\(=\)\((\)\(547656442731\) \(\nu^{5}\mathstrut -\mathstrut \) \(1948427421673813232157\) \(\nu^{4}\mathstrut -\mathstrut \) \(6610099066817607565987436485344\) \(\nu^{3}\mathstrut +\mathstrut \) \(22045821613742858828448779691123530008800\) \(\nu^{2}\mathstrut +\mathstrut \) \(15511387419718206739455612128355924712055616441600\) \(\nu\mathstrut -\mathstrut \) \(43849158917089635084556338654244679976800281181975924394240\)\()/\)\(50\!\cdots\!40\)
\(\beta_{4}\)\(=\)\((\)\(152285563749136221\) \(\nu^{5}\mathstrut -\mathstrut \) \(44198762943573650977329627\) \(\nu^{4}\mathstrut -\mathstrut \) \(2091774541352086674869707906974556704\) \(\nu^{3}\mathstrut -\mathstrut \) \(940502006414093841225016418875952277260352480\) \(\nu^{2}\mathstrut +\mathstrut \) \(3792145608704657568185822853913921837818945092995541760\) \(\nu\mathstrut +\mathstrut \) \(2514372192078802698287077358613077227357378667670079244064272640\)\()/\)\(35\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(35543898440648379\) \(\nu^{5}\mathstrut +\mathstrut \) \(48817395431103207716732493\) \(\nu^{4}\mathstrut +\mathstrut \) \(576416861397703537014097471318617312\) \(\nu^{3}\mathstrut -\mathstrut \) \(496748380676434680942250393566449513324067552\) \(\nu^{2}\mathstrut -\mathstrut \) \(1863568152173038225032573567221321416281990886847102208\) \(\nu\mathstrut +\mathstrut \) \(1175497631740102854763090187082864397921472619598164033538144512\)\()/\)\(70\!\cdots\!16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(3707241162\) \(\beta_{1}\mathstrut +\mathstrut \) \(219189618966715090758\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(141\) \(\beta_{5}\mathstrut +\mathstrut \) \(110\) \(\beta_{4}\mathstrut +\mathstrut \) \(2166905\) \(\beta_{3}\mathstrut +\mathstrut \) \(9413817317\) \(\beta_{2}\mathstrut +\mathstrut \) \(348133172179229341527\) \(\beta_{1}\mathstrut +\mathstrut \) \(812588778387279013456075631868\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(215683758605\) \(\beta_{5}\mathstrut +\mathstrut \) \(626554370414\) \(\beta_{4}\mathstrut -\mathstrut \) \(14890455560985607\) \(\beta_{3}\mathstrut +\mathstrut \) \(270176227379470727469\) \(\beta_{2}\mathstrut +\mathstrut \) \(1593848457539098681444566001255\) \(\beta_{1}\mathstrut +\mathstrut \) \(38153588681100594335049668773620993718764\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(17618615270372932220961\) \(\beta_{5}\mathstrut +\mathstrut \) \(18636474309890062085734\) \(\beta_{4}\mathstrut +\mathstrut \) \(254913665554799621734702653\) \(\beta_{3}\mathstrut +\mathstrut \) \(1732506467044683769447592070401\) \(\beta_{2}\mathstrut +\mathstrut \) \(37593243180169074975655494272242500516387\) \(\beta_{1}\mathstrut +\mathstrut \) \(174677518108355491808139991369890052791801708585372\)\()/1944\)

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
3.84072e9
2.02873e9
9.72648e8
−1.58092e9
−2.19503e9
−3.06614e9
−2.07551e10 −5.55906e15 2.83199e20 −8.85856e22 1.15379e26 −1.53665e28 −2.81491e30 3.09032e31 1.83860e33
1.2 −9.88316e9 −5.55906e15 −4.98972e19 −3.20918e23 5.49411e25 2.08311e28 1.95164e30 3.09032e31 3.17168e33
1.3 −3.54666e9 −5.55906e15 −1.34995e20 1.66367e23 1.97161e25 −1.26991e28 1.00218e30 3.09032e31 −5.90048e32
1.4 1.17747e10 −5.55906e15 −8.92992e18 8.66624e22 −6.54564e25 3.38994e28 −1.84279e30 3.09032e31 1.02043e33
1.5 1.54594e10 −5.55906e15 9.14202e19 −4.50800e23 −8.59399e25 −2.44076e28 −8.68106e29 3.09032e31 −6.96912e33
1.6 2.06861e10 −5.55906e15 2.80340e20 4.26985e23 −1.14995e26 −3.03250e28 2.74642e30 3.09032e31 8.83265e33
\(n\): 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 \(T_{2}^{6} \) \(\mathstrut -\mathstrut 13735355166 T_{2}^{5} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!32\)\( T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!48\)\( T_{2}^{3} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!96\)\( T_{2}^{2} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!44\)\( T_{2} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!52\)\( \) acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(3))\).