Properties

Label 1.80.a.a
Level 1
Weight 80
Character orbit 1.a
Self dual Yes
Analytic conductor 39.524
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 80 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(39.5237048722\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{54}\cdot 3^{24}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 13^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(+(-2681096220 + \beta_{1}) q^{2}\) \(+(323695821380196780 - 429230 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(25\!\cdots\!48\)\( + 103738873468 \beta_{1} - 8640 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(10\!\cdots\!90\)\( + 528731358938588 \beta_{1} - 135383940 \beta_{2} - 1928 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(37\!\cdots\!88\)\( + 933211626209604492 \beta_{1} - 58046406632 \beta_{2} + 1620233 \beta_{3} - 19 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(34\!\cdots\!00\)\( + \)\(36\!\cdots\!64\)\( \beta_{1} - 31355072821338 \beta_{2} + 386101744 \beta_{3} - 93324 \beta_{4} - 336 \beta_{5}) q^{7}\) \(+(\)\(90\!\cdots\!60\)\( + \)\(49\!\cdots\!04\)\( \beta_{1} - 11674271584003392 \beta_{2} - 128366288992 \beta_{3} - 83250968 \beta_{4} - 2952 \beta_{5}) q^{8}\) \(+(\)\(16\!\cdots\!37\)\( + \)\(40\!\cdots\!56\)\( \beta_{1} - 1337693128060671576 \beta_{2} - 6253994102256 \beta_{3} - 10041224742 \beta_{4} + 3460032 \beta_{5}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-2681096220 + \beta_{1}) q^{2}\) \(+(323695821380196780 - 429230 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(25\!\cdots\!48\)\( + 103738873468 \beta_{1} - 8640 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(10\!\cdots\!90\)\( + 528731358938588 \beta_{1} - 135383940 \beta_{2} - 1928 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(37\!\cdots\!88\)\( + 933211626209604492 \beta_{1} - 58046406632 \beta_{2} + 1620233 \beta_{3} - 19 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(34\!\cdots\!00\)\( + \)\(36\!\cdots\!64\)\( \beta_{1} - 31355072821338 \beta_{2} + 386101744 \beta_{3} - 93324 \beta_{4} - 336 \beta_{5}) q^{7}\) \(+(\)\(90\!\cdots\!60\)\( + \)\(49\!\cdots\!04\)\( \beta_{1} - 11674271584003392 \beta_{2} - 128366288992 \beta_{3} - 83250968 \beta_{4} - 2952 \beta_{5}) q^{8}\) \(+(\)\(16\!\cdots\!37\)\( + \)\(40\!\cdots\!56\)\( \beta_{1} - 1337693128060671576 \beta_{2} - 6253994102256 \beta_{3} - 10041224742 \beta_{4} + 3460032 \beta_{5}) q^{9}\) \(+(\)\(45\!\cdots\!40\)\( + \)\(42\!\cdots\!78\)\( \beta_{1} + 2598703832387364960 \beta_{2} + 5013150723351332 \beta_{3} + 407138302644 \beta_{4} - 276989700 \beta_{5}) q^{10}\) \(+(\)\(54\!\cdots\!92\)\( + \)\(72\!\cdots\!90\)\( \beta_{1} - \)\(15\!\cdots\!31\)\( \beta_{2} + 204943290007561952 \beta_{3} + 5908079984488 \beta_{4} + 12207853152 \beta_{5}) q^{11}\) \(+(\)\(61\!\cdots\!60\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} - \)\(24\!\cdots\!68\)\( \beta_{2} - 217351454451690132 \beta_{3} - 610139837478528 \beta_{4} - 369064017792 \beta_{5}) q^{12}\) \(+(-\)\(41\!\cdots\!90\)\( - \)\(32\!\cdots\!08\)\( \beta_{1} - \)\(47\!\cdots\!04\)\( \beta_{2} - \)\(10\!\cdots\!32\)\( \beta_{3} + 15125586094385247 \beta_{4} + 8354367585408 \beta_{5}) q^{13}\) \(+(\)\(31\!\cdots\!36\)\( + \)\(17\!\cdots\!44\)\( \beta_{1} - \)\(28\!\cdots\!40\)\( \beta_{2} + \)\(36\!\cdots\!98\)\( \beta_{3} - 170386606955801390 \beta_{4} - 148115161307610 \beta_{5}) q^{14}\) \(+(\)\(89\!\cdots\!80\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(34\!\cdots\!30\)\( \beta_{2} + \)\(24\!\cdots\!44\)\( \beta_{3} + 38413555280592748 \beta_{4} + 2105954551963600 \beta_{5}) q^{15}\) \(+(-\)\(11\!\cdots\!64\)\( - \)\(80\!\cdots\!36\)\( \beta_{1} + \)\(26\!\cdots\!68\)\( \beta_{2} - \)\(19\!\cdots\!08\)\( \beta_{3} + 29801977696108963776 \beta_{4} - 24218268871356096 \beta_{5}) q^{16}\) \(+(\)\(13\!\cdots\!90\)\( + \)\(60\!\cdots\!20\)\( \beta_{1} + \)\(35\!\cdots\!44\)\( \beta_{2} - \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(51\!\cdots\!26\)\( \beta_{4} + 223335882082950336 \beta_{5}) q^{17}\) \(+(\)\(34\!\cdots\!80\)\( + \)\(15\!\cdots\!65\)\( \beta_{1} + \)\(27\!\cdots\!92\)\( \beta_{2} + \)\(49\!\cdots\!24\)\( \beta_{3} + \)\(46\!\cdots\!96\)\( \beta_{4} - 1585614539297370456 \beta_{5}) q^{18}\) \(+(-\)\(84\!\cdots\!60\)\( + \)\(16\!\cdots\!22\)\( \beta_{1} - \)\(20\!\cdots\!53\)\( \beta_{2} - \)\(20\!\cdots\!80\)\( \beta_{3} - \)\(21\!\cdots\!36\)\( \beta_{4} + 7467464880301354656 \beta_{5}) q^{19}\) \(+(-\)\(24\!\cdots\!80\)\( + \)\(20\!\cdots\!24\)\( \beta_{1} - \)\(19\!\cdots\!20\)\( \beta_{2} - \)\(14\!\cdots\!94\)\( \beta_{3} - \)\(46\!\cdots\!48\)\( \beta_{4} - 3712084687919808000 \beta_{5}) q^{20}\) \(+(\)\(19\!\cdots\!12\)\( + \)\(21\!\cdots\!64\)\( \beta_{1} + \)\(61\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} + \)\(15\!\cdots\!80\)\( \beta_{4} - \)\(35\!\cdots\!80\)\( \beta_{5}) q^{21}\) \(+(\)\(61\!\cdots\!60\)\( + \)\(13\!\cdots\!92\)\( \beta_{1} + \)\(78\!\cdots\!56\)\( \beta_{2} - \)\(51\!\cdots\!17\)\( \beta_{3} - \)\(12\!\cdots\!93\)\( \beta_{4} + \)\(42\!\cdots\!73\)\( \beta_{5}) q^{22}\) \(+(-\)\(17\!\cdots\!60\)\( + \)\(45\!\cdots\!84\)\( \beta_{1} + \)\(21\!\cdots\!06\)\( \beta_{2} - \)\(28\!\cdots\!60\)\( \beta_{3} + \)\(51\!\cdots\!60\)\( \beta_{4} - \)\(31\!\cdots\!60\)\( \beta_{5}) q^{23}\) \(+(\)\(77\!\cdots\!20\)\( + \)\(19\!\cdots\!16\)\( \beta_{1} - \)\(28\!\cdots\!88\)\( \beta_{2} + \)\(21\!\cdots\!08\)\( \beta_{3} + \)\(27\!\cdots\!84\)\( \beta_{4} + \)\(16\!\cdots\!36\)\( \beta_{5}) q^{24}\) \(+(\)\(18\!\cdots\!75\)\( - \)\(97\!\cdots\!80\)\( \beta_{1} - \)\(22\!\cdots\!00\)\( \beta_{2} + \)\(18\!\cdots\!80\)\( \beta_{3} - \)\(13\!\cdots\!40\)\( \beta_{4} - \)\(46\!\cdots\!00\)\( \beta_{5}) q^{25}\) \(+(-\)\(27\!\cdots\!28\)\( - \)\(83\!\cdots\!94\)\( \beta_{1} + \)\(80\!\cdots\!84\)\( \beta_{2} - \)\(43\!\cdots\!76\)\( \beta_{3} + \)\(92\!\cdots\!28\)\( \beta_{4} - \)\(71\!\cdots\!88\)\( \beta_{5}) q^{26}\) \(+(\)\(75\!\cdots\!40\)\( - \)\(19\!\cdots\!44\)\( \beta_{1} - \)\(19\!\cdots\!98\)\( \beta_{2} + \)\(70\!\cdots\!32\)\( \beta_{3} - \)\(27\!\cdots\!72\)\( \beta_{4} + \)\(19\!\cdots\!92\)\( \beta_{5}) q^{27}\) \(+(\)\(16\!\cdots\!60\)\( + \)\(43\!\cdots\!56\)\( \beta_{1} - \)\(12\!\cdots\!72\)\( \beta_{2} + \)\(92\!\cdots\!52\)\( \beta_{3} - \)\(75\!\cdots\!92\)\( \beta_{4} - \)\(14\!\cdots\!88\)\( \beta_{5}) q^{28}\) \(+(-\)\(11\!\cdots\!90\)\( + \)\(32\!\cdots\!48\)\( \beta_{1} + \)\(70\!\cdots\!24\)\( \beta_{2} - \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(39\!\cdots\!03\)\( \beta_{4} + \)\(75\!\cdots\!12\)\( \beta_{5}) q^{29}\) \(+(-\)\(18\!\cdots\!20\)\( + \)\(20\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(27\!\cdots\!14\)\( \beta_{3} - \)\(10\!\cdots\!62\)\( \beta_{4} - \)\(26\!\cdots\!50\)\( \beta_{5}) q^{30}\) \(+(\)\(22\!\cdots\!72\)\( + \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(18\!\cdots\!32\)\( \beta_{2} + \)\(40\!\cdots\!56\)\( \beta_{3} - \)\(42\!\cdots\!36\)\( \beta_{4} + \)\(60\!\cdots\!56\)\( \beta_{5}) q^{31}\) \(+(-\)\(61\!\cdots\!20\)\( - \)\(21\!\cdots\!44\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - \)\(21\!\cdots\!84\)\( \beta_{3} + \)\(47\!\cdots\!64\)\( \beta_{4} - \)\(49\!\cdots\!04\)\( \beta_{5}) q^{32}\) \(+(\)\(11\!\cdots\!60\)\( - \)\(64\!\cdots\!60\)\( \beta_{1} - \)\(67\!\cdots\!60\)\( \beta_{2} - \)\(78\!\cdots\!24\)\( \beta_{3} - \)\(19\!\cdots\!46\)\( \beta_{4} - \)\(67\!\cdots\!44\)\( \beta_{5}) q^{33}\) \(+(\)\(52\!\cdots\!36\)\( - \)\(10\!\cdots\!74\)\( \beta_{1} + \)\(22\!\cdots\!72\)\( \beta_{2} + \)\(26\!\cdots\!08\)\( \beta_{3} + \)\(46\!\cdots\!04\)\( \beta_{4} + \)\(34\!\cdots\!16\)\( \beta_{5}) q^{34}\) \(+(\)\(15\!\cdots\!40\)\( + \)\(60\!\cdots\!28\)\( \beta_{1} + \)\(40\!\cdots\!60\)\( \beta_{2} + \)\(93\!\cdots\!32\)\( \beta_{3} - \)\(48\!\cdots\!56\)\( \beta_{4} - \)\(94\!\cdots\!00\)\( \beta_{5}) q^{35}\) \(+(\)\(33\!\cdots\!76\)\( + \)\(19\!\cdots\!44\)\( \beta_{1} - \)\(94\!\cdots\!12\)\( \beta_{2} - \)\(17\!\cdots\!63\)\( \beta_{3} + \)\(12\!\cdots\!16\)\( \beta_{4} + \)\(10\!\cdots\!64\)\( \beta_{5}) q^{36}\) \(+(\)\(33\!\cdots\!70\)\( + \)\(26\!\cdots\!52\)\( \beta_{1} - \)\(57\!\cdots\!44\)\( \beta_{2} - \)\(24\!\cdots\!36\)\( \beta_{3} - \)\(39\!\cdots\!69\)\( \beta_{4} + \)\(36\!\cdots\!84\)\( \beta_{5}) q^{37}\) \(+(\)\(14\!\cdots\!40\)\( - \)\(53\!\cdots\!44\)\( \beta_{1} + \)\(53\!\cdots\!80\)\( \beta_{2} + \)\(34\!\cdots\!61\)\( \beta_{3} + \)\(25\!\cdots\!69\)\( \beta_{4} - \)\(22\!\cdots\!09\)\( \beta_{5}) q^{38}\) \(+(\)\(30\!\cdots\!24\)\( - \)\(58\!\cdots\!76\)\( \beta_{1} + \)\(48\!\cdots\!54\)\( \beta_{2} - \)\(79\!\cdots\!20\)\( \beta_{3} - \)\(47\!\cdots\!52\)\( \beta_{4} + \)\(56\!\cdots\!92\)\( \beta_{5}) q^{39}\) \(+(\)\(15\!\cdots\!00\)\( - \)\(69\!\cdots\!20\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!80\)\( \beta_{3} - \)\(12\!\cdots\!60\)\( \beta_{4} - \)\(46\!\cdots\!00\)\( \beta_{5}) q^{40}\) \(+(\)\(32\!\cdots\!62\)\( + \)\(75\!\cdots\!20\)\( \beta_{1} - \)\(22\!\cdots\!88\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} + \)\(89\!\cdots\!24\)\( \beta_{4} - \)\(18\!\cdots\!04\)\( \beta_{5}) q^{41}\) \(+(\)\(18\!\cdots\!40\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} - \)\(62\!\cdots\!56\)\( \beta_{2} + \)\(96\!\cdots\!64\)\( \beta_{3} - \)\(20\!\cdots\!44\)\( \beta_{4} + \)\(81\!\cdots\!84\)\( \beta_{5}) q^{42}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(12\!\cdots\!78\)\( \beta_{1} + \)\(22\!\cdots\!53\)\( \beta_{2} - \)\(43\!\cdots\!20\)\( \beta_{3} + \)\(96\!\cdots\!20\)\( \beta_{4} - \)\(13\!\cdots\!20\)\( \beta_{5}) q^{43}\) \(+(\)\(82\!\cdots\!16\)\( - \)\(85\!\cdots\!24\)\( \beta_{1} + \)\(42\!\cdots\!92\)\( \beta_{2} + \)\(58\!\cdots\!68\)\( \beta_{3} + \)\(60\!\cdots\!44\)\( \beta_{4} - \)\(16\!\cdots\!24\)\( \beta_{5}) q^{44}\) \(+(\)\(18\!\cdots\!30\)\( - \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} - \)\(34\!\cdots\!36\)\( \beta_{3} - \)\(11\!\cdots\!37\)\( \beta_{4} + \)\(44\!\cdots\!00\)\( \beta_{5}) q^{45}\) \(+(\)\(38\!\cdots\!32\)\( - \)\(23\!\cdots\!20\)\( \beta_{1} - \)\(26\!\cdots\!08\)\( \beta_{2} + \)\(30\!\cdots\!06\)\( \beta_{3} - \)\(44\!\cdots\!66\)\( \beta_{4} - \)\(47\!\cdots\!14\)\( \beta_{5}) q^{46}\) \(+(\)\(12\!\cdots\!60\)\( - \)\(65\!\cdots\!72\)\( \beta_{1} + \)\(10\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(80\!\cdots\!60\)\( \beta_{4} - \)\(91\!\cdots\!40\)\( \beta_{5}) q^{47}\) \(+(-\)\(20\!\cdots\!60\)\( + \)\(14\!\cdots\!64\)\( \beta_{1} + \)\(23\!\cdots\!68\)\( \beta_{2} + \)\(53\!\cdots\!36\)\( \beta_{3} + \)\(26\!\cdots\!44\)\( \beta_{4} - \)\(29\!\cdots\!84\)\( \beta_{5}) q^{48}\) \(+(\)\(63\!\cdots\!93\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(17\!\cdots\!88\)\( \beta_{2} + \)\(33\!\cdots\!56\)\( \beta_{3} - \)\(74\!\cdots\!76\)\( \beta_{4} + \)\(19\!\cdots\!96\)\( \beta_{5}) q^{49}\) \(+(-\)\(84\!\cdots\!00\)\( + \)\(97\!\cdots\!95\)\( \beta_{1} - \)\(32\!\cdots\!00\)\( \beta_{2} - \)\(46\!\cdots\!20\)\( \beta_{3} + \)\(10\!\cdots\!60\)\( \beta_{4} - \)\(55\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(-\)\(23\!\cdots\!88\)\( - \)\(16\!\cdots\!32\)\( \beta_{1} - \)\(15\!\cdots\!42\)\( \beta_{2} + \)\(51\!\cdots\!00\)\( \beta_{3} + \)\(37\!\cdots\!96\)\( \beta_{4} + \)\(36\!\cdots\!84\)\( \beta_{5}) q^{51}\) \(+(-\)\(46\!\cdots\!00\)\( - \)\(38\!\cdots\!96\)\( \beta_{1} + \)\(39\!\cdots\!84\)\( \beta_{2} - \)\(57\!\cdots\!90\)\( \beta_{3} - \)\(79\!\cdots\!60\)\( \beta_{4} + \)\(37\!\cdots\!60\)\( \beta_{5}) q^{52}\) \(+(-\)\(52\!\cdots\!70\)\( - \)\(59\!\cdots\!20\)\( \beta_{1} - \)\(41\!\cdots\!80\)\( \beta_{2} + \)\(17\!\cdots\!88\)\( \beta_{3} + \)\(19\!\cdots\!27\)\( \beta_{4} - \)\(95\!\cdots\!72\)\( \beta_{5}) q^{53}\) \(+(-\)\(16\!\cdots\!60\)\( + \)\(81\!\cdots\!72\)\( \beta_{1} + \)\(17\!\cdots\!32\)\( \beta_{2} - \)\(89\!\cdots\!50\)\( \beta_{3} + \)\(10\!\cdots\!34\)\( \beta_{4} + \)\(29\!\cdots\!86\)\( \beta_{5}) q^{54}\) \(+(-\)\(14\!\cdots\!20\)\( + \)\(25\!\cdots\!96\)\( \beta_{1} - \)\(60\!\cdots\!30\)\( \beta_{2} - \)\(31\!\cdots\!76\)\( \beta_{3} + \)\(11\!\cdots\!08\)\( \beta_{4} + \)\(37\!\cdots\!00\)\( \beta_{5}) q^{55}\) \(+(\)\(18\!\cdots\!60\)\( + \)\(54\!\cdots\!48\)\( \beta_{1} + \)\(28\!\cdots\!64\)\( \beta_{2} + \)\(13\!\cdots\!88\)\( \beta_{3} - \)\(46\!\cdots\!92\)\( \beta_{4} - \)\(80\!\cdots\!68\)\( \beta_{5}) q^{56}\) \(+(\)\(12\!\cdots\!80\)\( - \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(95\!\cdots\!24\)\( \beta_{2} + \)\(35\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!30\)\( \beta_{4} - \)\(72\!\cdots\!20\)\( \beta_{5}) q^{57}\) \(+(\)\(27\!\cdots\!60\)\( - \)\(27\!\cdots\!26\)\( \beta_{1} + \)\(64\!\cdots\!84\)\( \beta_{2} + \)\(36\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!84\)\( \beta_{4} + \)\(26\!\cdots\!76\)\( \beta_{5}) q^{58}\) \(+(\)\(40\!\cdots\!20\)\( - \)\(52\!\cdots\!54\)\( \beta_{1} - \)\(63\!\cdots\!95\)\( \beta_{2} - \)\(11\!\cdots\!48\)\( \beta_{3} - \)\(58\!\cdots\!80\)\( \beta_{4} - \)\(19\!\cdots\!20\)\( \beta_{5}) q^{59}\) \(+(\)\(12\!\cdots\!40\)\( + \)\(23\!\cdots\!48\)\( \beta_{1} - \)\(22\!\cdots\!40\)\( \beta_{2} + \)\(40\!\cdots\!12\)\( \beta_{3} - \)\(12\!\cdots\!96\)\( \beta_{4} - \)\(97\!\cdots\!00\)\( \beta_{5}) q^{60}\) \(+(\)\(13\!\cdots\!42\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!00\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!75\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5}) q^{61}\) \(+(\)\(99\!\cdots\!60\)\( + \)\(37\!\cdots\!72\)\( \beta_{1} + \)\(45\!\cdots\!68\)\( \beta_{2} + \)\(19\!\cdots\!24\)\( \beta_{3} + \)\(65\!\cdots\!96\)\( \beta_{4} + \)\(34\!\cdots\!44\)\( \beta_{5}) q^{62}\) \(+(-\)\(46\!\cdots\!80\)\( + \)\(31\!\cdots\!76\)\( \beta_{1} + \)\(38\!\cdots\!94\)\( \beta_{2} - \)\(62\!\cdots\!24\)\( \beta_{3} + \)\(22\!\cdots\!04\)\( \beta_{4} - \)\(15\!\cdots\!44\)\( \beta_{5}) q^{63}\) \(+(-\)\(11\!\cdots\!52\)\( - \)\(80\!\cdots\!44\)\( \beta_{1} - \)\(32\!\cdots\!36\)\( \beta_{2} - \)\(25\!\cdots\!36\)\( \beta_{3} - \)\(28\!\cdots\!12\)\( \beta_{4} + \)\(10\!\cdots\!52\)\( \beta_{5}) q^{64}\) \(+(-\)\(16\!\cdots\!20\)\( - \)\(28\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(35\!\cdots\!64\)\( \beta_{3} + \)\(35\!\cdots\!88\)\( \beta_{4} + \)\(47\!\cdots\!00\)\( \beta_{5}) q^{65}\) \(+(-\)\(55\!\cdots\!96\)\( - \)\(20\!\cdots\!56\)\( \beta_{1} - \)\(46\!\cdots\!76\)\( \beta_{2} + \)\(36\!\cdots\!80\)\( \beta_{3} + \)\(81\!\cdots\!88\)\( \beta_{4} - \)\(11\!\cdots\!48\)\( \beta_{5}) q^{66}\) \(+(-\)\(30\!\cdots\!60\)\( + \)\(25\!\cdots\!50\)\( \beta_{1} + \)\(51\!\cdots\!39\)\( \beta_{2} - \)\(37\!\cdots\!04\)\( \beta_{3} - \)\(26\!\cdots\!16\)\( \beta_{4} + \)\(40\!\cdots\!76\)\( \beta_{5}) q^{67}\) \(+(-\)\(98\!\cdots\!20\)\( + \)\(91\!\cdots\!24\)\( \beta_{1} + \)\(65\!\cdots\!12\)\( \beta_{2} - \)\(19\!\cdots\!78\)\( \beta_{3} + \)\(14\!\cdots\!88\)\( \beta_{4} + \)\(27\!\cdots\!32\)\( \beta_{5}) q^{68}\) \(+(-\)\(23\!\cdots\!56\)\( + \)\(22\!\cdots\!12\)\( \beta_{1} + \)\(33\!\cdots\!88\)\( \beta_{2} + \)\(11\!\cdots\!08\)\( \beta_{3} + \)\(31\!\cdots\!96\)\( \beta_{4} - \)\(46\!\cdots\!16\)\( \beta_{5}) q^{69}\) \(+(\)\(51\!\cdots\!40\)\( + \)\(22\!\cdots\!68\)\( \beta_{1} - \)\(81\!\cdots\!40\)\( \beta_{2} + \)\(58\!\cdots\!92\)\( \beta_{3} - \)\(40\!\cdots\!36\)\( \beta_{4} + \)\(48\!\cdots\!00\)\( \beta_{5}) q^{70}\) \(+(\)\(64\!\cdots\!32\)\( - \)\(73\!\cdots\!00\)\( \beta_{1} - \)\(86\!\cdots\!50\)\( \beta_{2} - \)\(36\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4} + \)\(42\!\cdots\!00\)\( \beta_{5}) q^{71}\) \(+(\)\(14\!\cdots\!20\)\( - \)\(99\!\cdots\!32\)\( \beta_{1} - \)\(85\!\cdots\!20\)\( \beta_{2} - \)\(33\!\cdots\!72\)\( \beta_{3} - \)\(90\!\cdots\!88\)\( \beta_{4} + \)\(26\!\cdots\!68\)\( \beta_{5}) q^{72}\) \(+(\)\(10\!\cdots\!90\)\( - \)\(23\!\cdots\!56\)\( \beta_{1} + \)\(30\!\cdots\!48\)\( \beta_{2} - \)\(31\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!74\)\( \beta_{4} - \)\(32\!\cdots\!64\)\( \beta_{5}) q^{73}\) \(+(\)\(22\!\cdots\!36\)\( + \)\(30\!\cdots\!50\)\( \beta_{1} + \)\(26\!\cdots\!28\)\( \beta_{2} + \)\(62\!\cdots\!64\)\( \beta_{3} + \)\(46\!\cdots\!56\)\( \beta_{4} - \)\(52\!\cdots\!76\)\( \beta_{5}) q^{74}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(20\!\cdots\!90\)\( \beta_{1} - \)\(34\!\cdots\!75\)\( \beta_{2} + \)\(37\!\cdots\!60\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5}) q^{75}\) \(+(-\)\(41\!\cdots\!80\)\( + \)\(16\!\cdots\!56\)\( \beta_{1} - \)\(84\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!84\)\( \beta_{3} - \)\(32\!\cdots\!04\)\( \beta_{4} - \)\(96\!\cdots\!16\)\( \beta_{5}) q^{76}\) \(+(-\)\(19\!\cdots\!00\)\( - \)\(48\!\cdots\!12\)\( \beta_{1} - \)\(20\!\cdots\!60\)\( \beta_{2} + \)\(97\!\cdots\!96\)\( \beta_{3} + \)\(28\!\cdots\!84\)\( \beta_{4} - \)\(37\!\cdots\!24\)\( \beta_{5}) q^{77}\) \(+(-\)\(50\!\cdots\!00\)\( - \)\(87\!\cdots\!04\)\( \beta_{1} + \)\(62\!\cdots\!80\)\( \beta_{2} - \)\(25\!\cdots\!78\)\( \beta_{3} + \)\(10\!\cdots\!38\)\( \beta_{4} + \)\(56\!\cdots\!82\)\( \beta_{5}) q^{78}\) \(+(-\)\(31\!\cdots\!40\)\( - \)\(66\!\cdots\!92\)\( \beta_{1} - \)\(33\!\cdots\!08\)\( \beta_{2} + \)\(32\!\cdots\!72\)\( \beta_{3} - \)\(23\!\cdots\!36\)\( \beta_{4} - \)\(91\!\cdots\!44\)\( \beta_{5}) q^{79}\) \(+(-\)\(45\!\cdots\!60\)\( + \)\(19\!\cdots\!68\)\( \beta_{1} + \)\(86\!\cdots\!60\)\( \beta_{2} + \)\(62\!\cdots\!92\)\( \beta_{3} + \)\(36\!\cdots\!64\)\( \beta_{4} - \)\(48\!\cdots\!00\)\( \beta_{5}) q^{80}\) \(+(-\)\(70\!\cdots\!99\)\( - \)\(15\!\cdots\!28\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2} - \)\(34\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!30\)\( \beta_{4} + \)\(54\!\cdots\!20\)\( \beta_{5}) q^{81}\) \(+(\)\(63\!\cdots\!60\)\( + \)\(38\!\cdots\!62\)\( \beta_{1} - \)\(18\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!16\)\( \beta_{3} - \)\(42\!\cdots\!64\)\( \beta_{4} - \)\(15\!\cdots\!96\)\( \beta_{5}) q^{82}\) \(+(\)\(24\!\cdots\!20\)\( - \)\(93\!\cdots\!94\)\( \beta_{1} + \)\(34\!\cdots\!47\)\( \beta_{2} + \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(48\!\cdots\!00\)\( \beta_{4} + \)\(44\!\cdots\!00\)\( \beta_{5}) q^{83}\) \(+(\)\(81\!\cdots\!76\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(56\!\cdots\!04\)\( \beta_{2} + \)\(70\!\cdots\!08\)\( \beta_{3} - \)\(84\!\cdots\!12\)\( \beta_{4} - \)\(36\!\cdots\!48\)\( \beta_{5}) q^{84}\) \(+(\)\(64\!\cdots\!40\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} + \)\(12\!\cdots\!60\)\( \beta_{2} - \)\(57\!\cdots\!68\)\( \beta_{3} - \)\(22\!\cdots\!06\)\( \beta_{4} - \)\(62\!\cdots\!00\)\( \beta_{5}) q^{85}\) \(+(\)\(11\!\cdots\!52\)\( + \)\(14\!\cdots\!28\)\( \beta_{1} - \)\(59\!\cdots\!04\)\( \beta_{2} + \)\(14\!\cdots\!39\)\( \beta_{3} + \)\(28\!\cdots\!47\)\( \beta_{4} + \)\(17\!\cdots\!13\)\( \beta_{5}) q^{86}\) \(+(-\)\(20\!\cdots\!80\)\( - \)\(46\!\cdots\!72\)\( \beta_{1} - \)\(12\!\cdots\!38\)\( \beta_{2} - \)\(38\!\cdots\!36\)\( \beta_{3} - \)\(21\!\cdots\!44\)\( \beta_{4} - \)\(21\!\cdots\!16\)\( \beta_{5}) q^{87}\) \(+(-\)\(11\!\cdots\!80\)\( + \)\(19\!\cdots\!68\)\( \beta_{1} - \)\(50\!\cdots\!64\)\( \beta_{2} - \)\(23\!\cdots\!64\)\( \beta_{3} + \)\(13\!\cdots\!44\)\( \beta_{4} + \)\(27\!\cdots\!16\)\( \beta_{5}) q^{88}\) \(+(\)\(14\!\cdots\!30\)\( - \)\(83\!\cdots\!16\)\( \beta_{1} + \)\(96\!\cdots\!12\)\( \beta_{2} + \)\(13\!\cdots\!04\)\( \beta_{3} - \)\(10\!\cdots\!86\)\( \beta_{4} - \)\(85\!\cdots\!44\)\( \beta_{5}) q^{89}\) \(+(-\)\(11\!\cdots\!20\)\( + \)\(16\!\cdots\!86\)\( \beta_{1} + \)\(44\!\cdots\!20\)\( \beta_{2} - \)\(54\!\cdots\!16\)\( \beta_{3} + \)\(76\!\cdots\!28\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5}) q^{90}\) \(+(-\)\(18\!\cdots\!28\)\( - \)\(54\!\cdots\!92\)\( \beta_{1} + \)\(19\!\cdots\!64\)\( \beta_{2} - \)\(55\!\cdots\!92\)\( \beta_{3} + \)\(38\!\cdots\!08\)\( \beta_{4} + \)\(22\!\cdots\!32\)\( \beta_{5}) q^{91}\) \(+(-\)\(96\!\cdots\!60\)\( + \)\(22\!\cdots\!20\)\( \beta_{1} - \)\(67\!\cdots\!60\)\( \beta_{2} - \)\(71\!\cdots\!16\)\( \beta_{3} - \)\(57\!\cdots\!64\)\( \beta_{4} - \)\(80\!\cdots\!96\)\( \beta_{5}) q^{92}\) \(+(-\)\(12\!\cdots\!40\)\( - \)\(35\!\cdots\!60\)\( \beta_{1} - \)\(44\!\cdots\!76\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(37\!\cdots\!88\)\( \beta_{4} - \)\(19\!\cdots\!32\)\( \beta_{5}) q^{93}\) \(+(-\)\(57\!\cdots\!64\)\( - \)\(29\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!72\)\( \beta_{2} + \)\(20\!\cdots\!60\)\( \beta_{3} + \)\(87\!\cdots\!64\)\( \beta_{4} + \)\(28\!\cdots\!56\)\( \beta_{5}) q^{94}\) \(+(\)\(68\!\cdots\!00\)\( - \)\(59\!\cdots\!80\)\( \beta_{1} - \)\(52\!\cdots\!50\)\( \beta_{2} + \)\(16\!\cdots\!80\)\( \beta_{3} + \)\(62\!\cdots\!60\)\( \beta_{4} + \)\(33\!\cdots\!00\)\( \beta_{5}) q^{95}\) \(+(\)\(75\!\cdots\!72\)\( - \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(17\!\cdots\!44\)\( \beta_{2} - \)\(14\!\cdots\!64\)\( \beta_{3} - \)\(10\!\cdots\!92\)\( \beta_{4} + \)\(17\!\cdots\!32\)\( \beta_{5}) q^{96}\) \(+(\)\(34\!\cdots\!10\)\( + \)\(11\!\cdots\!08\)\( \beta_{1} + \)\(34\!\cdots\!40\)\( \beta_{2} - \)\(41\!\cdots\!88\)\( \beta_{3} - \)\(78\!\cdots\!02\)\( \beta_{4} - \)\(29\!\cdots\!28\)\( \beta_{5}) q^{97}\) \(+(\)\(35\!\cdots\!40\)\( + \)\(24\!\cdots\!33\)\( \beta_{1} + \)\(72\!\cdots\!68\)\( \beta_{2} + \)\(66\!\cdots\!64\)\( \beta_{3} - \)\(32\!\cdots\!44\)\( \beta_{4} - \)\(19\!\cdots\!16\)\( \beta_{5}) q^{98}\) \(+(\)\(20\!\cdots\!04\)\( + \)\(30\!\cdots\!82\)\( \beta_{1} - \)\(37\!\cdots\!19\)\( \beta_{2} - \)\(33\!\cdots\!68\)\( \beta_{3} - \)\(57\!\cdots\!68\)\( \beta_{4} + \)\(78\!\cdots\!28\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 16086577320q^{2} \) \(\mathstrut +\mathstrut 1942174928281180680q^{3} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!88\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!40\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!28\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!22\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 16086577320q^{2} \) \(\mathstrut +\mathstrut 1942174928281180680q^{3} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!88\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!40\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!28\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!22\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!52\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!60\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!40\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!16\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!80\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!84\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!40\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!80\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!72\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!60\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!68\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!40\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!40\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!32\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!20\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!60\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!16\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!40\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!56\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!44\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!72\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!96\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!92\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!60\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!60\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!58\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!28\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!20\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!60\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!20\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!80\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!60\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!40\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!52\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!60\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!80\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!12\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!76\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!60\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!20\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!36\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!40\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!92\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!40\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!16\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!80\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!40\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!60\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!94\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!60\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!20\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!56\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!40\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!12\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!80\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!20\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!68\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!60\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!40\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!84\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!32\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!60\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!40\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!24\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(4493465009931118088598\) \(x^{4}\mathstrut -\mathstrut \) \(13617827867154853888266546820774\) \(x^{3}\mathstrut +\mathstrut \) \(5183423914918797826459016270129188352477541\) \(x^{2}\mathstrut +\mathstrut \) \(36020237378286283561256608380952541686850032723239033\) \(x\mathstrut -\mathstrut \) \(768426424275848216319978876834525014212769442584294247793970688\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 12 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(597229958995063\) \(\nu^{5}\mathstrut +\mathstrut \) \(2471892005074829244563700\) \(\nu^{4}\mathstrut +\mathstrut \) \(1949419810644184872216322297437898662\) \(\nu^{3}\mathstrut -\mathstrut \) \(10584857959420731462820406471807880078864746348\) \(\nu^{2}\mathstrut -\mathstrut \) \(1133567706345786428042492708306981978734270096901781475239\) \(\nu\mathstrut +\mathstrut \) \(7013256411909823331435630200586455100629185588616638318485550887424\)\()/\)\(13\!\cdots\!68\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(26875348154777835\) \(\nu^{5}\mathstrut +\mathstrut \) \(111235140228367316005366500\) \(\nu^{4}\mathstrut +\mathstrut \) \(87723891478988319249734503384705439790\) \(\nu^{3}\mathstrut +\mathstrut \) \(3497416854160423168353384921769632575889662128644\) \(\nu^{2}\mathstrut -\mathstrut \) \(69074662396913184502283863666204694953251102850182445531227\) \(\nu\mathstrut -\mathstrut \) \(5636350547695653722650116311520020437553229011386033167054587936699904\)\()/\)\(68\!\cdots\!04\)
\(\beta_{4}\)\(=\)\((\)\(23563035749469300129\) \(\nu^{5}\mathstrut +\mathstrut \) \(66307116569785101256916660436\) \(\nu^{4}\mathstrut -\mathstrut \) \(132989037038325767256112053028336142484810\) \(\nu^{3}\mathstrut -\mathstrut \) \(324455418512060640190206526168398583786491956893836\) \(\nu^{2}\mathstrut +\mathstrut \) \(168407470197004289291595531243077017098147726427956207178762737\) \(\nu\mathstrut +\mathstrut \) \(680090936111077516374033709222384383888200213584841258707094776359461376\)\()/\)\(35\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(53467753519988644526221583\) \(\nu^{5}\mathstrut -\mathstrut \) \(11088337498782312511008436189797392148\) \(\nu^{4}\mathstrut -\mathstrut \) \(331758122929932654122303370460381914974231788950\) \(\nu^{3}\mathstrut +\mathstrut \) \(30732842646922767760470315074359454663705687372159107725388\) \(\nu^{2}\mathstrut +\mathstrut \) \(462040098753655047191834964855338460056320067077754414190014372854719\) \(\nu\mathstrut -\mathstrut \) \(8582922091606419036428140486631152566554588594377675331449370194280531978961408\)\()/\)\(82\!\cdots\!80\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(8640\) \(\beta_{2}\mathstrut +\mathstrut \) \(109101065932\) \(\beta_{1}\mathstrut +\mathstrut \) \(862745281906774673011680\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(369\) \(\beta_{5}\mathstrut -\mathstrut \) \(10406371\) \(\beta_{4}\mathstrut -\mathstrut \) \(15040375037\) \(\beta_{3}\mathstrut -\mathstrut \) \(1467970699792104\) \(\beta_{2}\mathstrut +\mathstrut \) \(157418686723624244836560\) \(\beta_{1}\mathstrut +\mathstrut \) \(11765803288868855065203754538802528\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(126301704456581\) \(\beta_{5}\mathstrut +\mathstrut \) \(150568553487751841\) \(\beta_{4}\mathstrut +\mathstrut \) \(8440249726986677178822\) \(\beta_{3}\mathstrut -\mathstrut \) \(68660126943966960119072584\) \(\beta_{2}\mathstrut +\mathstrut \) \(1008304140295319817672604183672860\) \(\beta_{1}\mathstrut +\mathstrut \) \(5658842860254359710247179919766744391201903008\)\()/1728\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(287867922024816448215201\) \(\beta_{5}\mathstrut -\mathstrut \) \(5557378214984384225007942259\) \(\beta_{4}\mathstrut -\mathstrut \) \(11221582939787767631829282270921\) \(\beta_{3}\mathstrut -\mathstrut \) \(1408146508860993468356104899623061864\) \(\beta_{2}\mathstrut +\mathstrut \) \(62590684798049532748462637503276446509637668\) \(\beta_{1}\mathstrut +\mathstrut \) \(6041038967774968863957714854327615343439459397006200464\)\()/288\)

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
−5.13988e10
−3.49960e10
−1.94020e10
9.55147e9
4.48729e10
5.13724e10
−1.23625e12 −2.24416e18 9.23858e23 −5.14707e27 2.77435e30 4.93307e32 −3.94854e35 −4.42334e37 6.36308e39
1.2 −8.42585e11 1.18388e19 1.05486e23 5.37236e27 −9.97521e30 −2.88361e33 4.20430e35 9.08882e37 −4.52667e39
1.3 −4.68329e11 −9.12223e18 −3.85131e23 5.54574e27 4.27221e30 1.91401e33 4.63456e35 3.39455e37 −2.59723e39
1.4 2.26554e11 2.57159e18 −5.53136e23 −2.99197e27 5.82605e29 −6.95219e31 −2.62259e35 −4.26565e37 −6.77842e38
1.5 1.07427e12 −9.45275e18 5.49591e23 −5.27758e26 −1.01548e31 −3.45278e33 −5.89470e34 4.00848e37 −5.66955e38
1.6 1.23026e12 8.35089e18 9.09069e23 3.84167e27 1.02737e31 3.79445e33 3.74744e35 2.04677e37 4.72625e39
\(n\): 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.

Hecke kernels

There are no other newforms in \(S_{80}^{\mathrm{new}}(\Gamma_0(1))\).