Properties

Label 1.82.a.a
Level 1
Weight 82
Character orbit 1.a
Self dual Yes
Analytic conductor 41.550
Analytic rank 1
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 82 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(41.5501285538\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 13 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+(-76812004440 - \beta_{1}) q^{2}\) \(+(-2608048654315521660 - 4130706 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(74\!\cdots\!92\)\( + 330453870896 \beta_{1} - 4668 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(30\!\cdots\!50\)\( + 3579253777224384 \beta_{1} + 21942656 \beta_{2} + 3247 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(13\!\cdots\!72\)\( + 4513221122897092521 \beta_{1} + 247691680988 \beta_{2} + 2224483 \beta_{3} - 108 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(52\!\cdots\!00\)\( - \)\(20\!\cdots\!44\)\( \beta_{1} - 102284077821438 \beta_{2} - 1096811324 \beta_{3} - 12780 \beta_{4} + 408 \beta_{5}) q^{7}\) \(+(-\)\(91\!\cdots\!20\)\( - \)\(69\!\cdots\!92\)\( \beta_{1} - 23354748483639488 \beta_{2} - 483724211632 \beta_{3} - 73855360 \beta_{4} + 87264 \beta_{5}) q^{8}\) \(+(\)\(19\!\cdots\!33\)\( - \)\(31\!\cdots\!88\)\( \beta_{1} + 2385861390791192256 \beta_{2} - 63041239844574 \beta_{3} - 22295404386 \beta_{4} - 11158752 \beta_{5}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-76812004440 - \beta_{1}) q^{2}\) \(+(-2608048654315521660 - 4130706 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(74\!\cdots\!92\)\( + 330453870896 \beta_{1} - 4668 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(30\!\cdots\!50\)\( + 3579253777224384 \beta_{1} + 21942656 \beta_{2} + 3247 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(13\!\cdots\!72\)\( + 4513221122897092521 \beta_{1} + 247691680988 \beta_{2} + 2224483 \beta_{3} - 108 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(52\!\cdots\!00\)\( - \)\(20\!\cdots\!44\)\( \beta_{1} - 102284077821438 \beta_{2} - 1096811324 \beta_{3} - 12780 \beta_{4} + 408 \beta_{5}) q^{7}\) \(+(-\)\(91\!\cdots\!20\)\( - \)\(69\!\cdots\!92\)\( \beta_{1} - 23354748483639488 \beta_{2} - 483724211632 \beta_{3} - 73855360 \beta_{4} + 87264 \beta_{5}) q^{8}\) \(+(\)\(19\!\cdots\!33\)\( - \)\(31\!\cdots\!88\)\( \beta_{1} + 2385861390791192256 \beta_{2} - 63041239844574 \beta_{3} - 22295404386 \beta_{4} - 11158752 \beta_{5}) q^{9}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(52\!\cdots\!42\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - 7823121005523436 \beta_{3} - 162776084688 \beta_{4} + 556990500 \beta_{5}) q^{10}\) \(+(-\)\(38\!\cdots\!88\)\( + \)\(75\!\cdots\!70\)\( \beta_{1} + \)\(60\!\cdots\!83\)\( \beta_{2} - 367894936402422424 \beta_{3} + 51221298912776 \beta_{4} - 15361340688 \beta_{5}) q^{11}\) \(+(-\)\(89\!\cdots\!80\)\( - \)\(92\!\cdots\!76\)\( \beta_{1} - \)\(27\!\cdots\!28\)\( \beta_{2} - 18961616389226821308 \beta_{3} - 1267640420797440 \beta_{4} + 218743785216 \beta_{5}) q^{12}\) \(+(\)\(21\!\cdots\!30\)\( + \)\(16\!\cdots\!44\)\( \beta_{1} - \)\(21\!\cdots\!76\)\( \beta_{2} + \)\(23\!\cdots\!63\)\( \beta_{3} + 768189821556105 \beta_{4} + 679320417984 \beta_{5}) q^{13}\) \(+(\)\(68\!\cdots\!04\)\( + \)\(81\!\cdots\!98\)\( \beta_{1} - \)\(86\!\cdots\!24\)\( \beta_{2} + \)\(81\!\cdots\!38\)\( \beta_{3} + 526793576543375720 \beta_{4} - 121314522929490 \beta_{5}) q^{14}\) \(+(-\)\(26\!\cdots\!00\)\( - \)\(80\!\cdots\!96\)\( \beta_{1} - \)\(20\!\cdots\!14\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} - 11969550314767844244 \beta_{4} + 3542243323729000 \beta_{5}) q^{15}\) \(+(\)\(46\!\cdots\!76\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(12\!\cdots\!32\)\( \beta_{4} - 67147365672685056 \beta_{5}) q^{16}\) \(+(\)\(39\!\cdots\!30\)\( - \)\(63\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!44\)\( \beta_{2} + \)\(13\!\cdots\!94\)\( \beta_{3} - \)\(34\!\cdots\!70\)\( \beta_{4} + 963043109681077152 \beta_{5}) q^{17}\) \(+(\)\(84\!\cdots\!40\)\( - \)\(44\!\cdots\!25\)\( \beta_{1} + \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(37\!\cdots\!96\)\( \beta_{3} - \)\(92\!\cdots\!20\)\( \beta_{4} - 10991883681013116168 \beta_{5}) q^{18}\) \(+(-\)\(29\!\cdots\!40\)\( + \)\(20\!\cdots\!74\)\( \beta_{1} + \)\(78\!\cdots\!49\)\( \beta_{2} - \)\(10\!\cdots\!84\)\( \beta_{3} + \)\(15\!\cdots\!12\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5}) q^{19}\) \(+(\)\(24\!\cdots\!00\)\( + \)\(21\!\cdots\!28\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(72\!\cdots\!74\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(75\!\cdots\!00\)\( \beta_{5}) q^{20}\) \(+(-\)\(19\!\cdots\!48\)\( + \)\(19\!\cdots\!72\)\( \beta_{1} - \)\(98\!\cdots\!36\)\( \beta_{2} + \)\(23\!\cdots\!52\)\( \beta_{3} + \)\(44\!\cdots\!80\)\( \beta_{4} + \)\(42\!\cdots\!20\)\( \beta_{5}) q^{21}\) \(+(-\)\(20\!\cdots\!80\)\( + \)\(12\!\cdots\!71\)\( \beta_{1} + \)\(39\!\cdots\!16\)\( \beta_{2} - \)\(43\!\cdots\!43\)\( \beta_{3} + \)\(14\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!99\)\( \beta_{5}) q^{22}\) \(+(\)\(17\!\cdots\!20\)\( + \)\(64\!\cdots\!60\)\( \beta_{1} + \)\(37\!\cdots\!54\)\( \beta_{2} + \)\(55\!\cdots\!40\)\( \beta_{3} - \)\(29\!\cdots\!00\)\( \beta_{4} - \)\(59\!\cdots\!80\)\( \beta_{5}) q^{23}\) \(+(-\)\(20\!\cdots\!20\)\( + \)\(43\!\cdots\!52\)\( \beta_{1} - \)\(64\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!64\)\( \beta_{3} + \)\(18\!\cdots\!72\)\( \beta_{4} + \)\(66\!\cdots\!44\)\( \beta_{5}) q^{24}\) \(+(\)\(68\!\cdots\!75\)\( + \)\(73\!\cdots\!00\)\( \beta_{1} - \)\(96\!\cdots\!00\)\( \beta_{2} - \)\(80\!\cdots\!00\)\( \beta_{3} - \)\(46\!\cdots\!00\)\( \beta_{4} - \)\(64\!\cdots\!00\)\( \beta_{5}) q^{25}\) \(+(-\)\(53\!\cdots\!28\)\( - \)\(81\!\cdots\!22\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} - \)\(14\!\cdots\!24\)\( \beta_{4} + \)\(41\!\cdots\!52\)\( \beta_{5}) q^{26}\) \(+(\)\(24\!\cdots\!80\)\( - \)\(32\!\cdots\!52\)\( \beta_{1} + \)\(29\!\cdots\!82\)\( \beta_{2} + \)\(28\!\cdots\!68\)\( \beta_{3} + \)\(17\!\cdots\!40\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5}) q^{27}\) \(+(-\)\(13\!\cdots\!20\)\( - \)\(23\!\cdots\!12\)\( \beta_{1} + \)\(80\!\cdots\!72\)\( \beta_{2} - \)\(88\!\cdots\!28\)\( \beta_{3} - \)\(57\!\cdots\!40\)\( \beta_{4} + \)\(76\!\cdots\!56\)\( \beta_{5}) q^{28}\) \(+(\)\(25\!\cdots\!90\)\( + \)\(15\!\cdots\!36\)\( \beta_{1} - \)\(34\!\cdots\!16\)\( \beta_{2} - \)\(16\!\cdots\!45\)\( \beta_{3} - \)\(73\!\cdots\!71\)\( \beta_{4} - \)\(21\!\cdots\!92\)\( \beta_{5}) q^{29}\) \(+(\)\(27\!\cdots\!00\)\( + \)\(32\!\cdots\!98\)\( \beta_{1} - \)\(88\!\cdots\!68\)\( \beta_{2} + \)\(16\!\cdots\!34\)\( \beta_{3} + \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(41\!\cdots\!50\)\( \beta_{5}) q^{30}\) \(+(\)\(20\!\cdots\!32\)\( + \)\(61\!\cdots\!80\)\( \beta_{1} + \)\(52\!\cdots\!84\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} - \)\(69\!\cdots\!32\)\( \beta_{4} - \)\(40\!\cdots\!44\)\( \beta_{5}) q^{31}\) \(+(-\)\(21\!\cdots\!40\)\( + \)\(96\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!36\)\( \beta_{3} + \)\(14\!\cdots\!80\)\( \beta_{4} + \)\(63\!\cdots\!12\)\( \beta_{5}) q^{32}\) \(+(\)\(35\!\cdots\!80\)\( - \)\(25\!\cdots\!88\)\( \beta_{1} - \)\(51\!\cdots\!80\)\( \beta_{2} + \)\(30\!\cdots\!66\)\( \beta_{3} - \)\(12\!\cdots\!10\)\( \beta_{4} - \)\(89\!\cdots\!92\)\( \beta_{5}) q^{33}\) \(+(\)\(19\!\cdots\!24\)\( - \)\(31\!\cdots\!38\)\( \beta_{1} - \)\(22\!\cdots\!60\)\( \beta_{2} + \)\(12\!\cdots\!84\)\( \beta_{3} - \)\(47\!\cdots\!48\)\( \beta_{4} + \)\(53\!\cdots\!24\)\( \beta_{5}) q^{34}\) \(+(-\)\(38\!\cdots\!00\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} + \)\(41\!\cdots\!52\)\( \beta_{2} - \)\(62\!\cdots\!76\)\( \beta_{3} - \)\(60\!\cdots\!08\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5}) q^{35}\) \(+(-\)\(34\!\cdots\!64\)\( + \)\(11\!\cdots\!72\)\( \beta_{1} + \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(31\!\cdots\!69\)\( \beta_{3} + \)\(41\!\cdots\!72\)\( \beta_{4} - \)\(24\!\cdots\!16\)\( \beta_{5}) q^{36}\) \(+(\)\(69\!\cdots\!90\)\( + \)\(78\!\cdots\!44\)\( \beta_{1} - \)\(28\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!91\)\( \beta_{3} - \)\(13\!\cdots\!95\)\( \beta_{4} + \)\(34\!\cdots\!68\)\( \beta_{5}) q^{37}\) \(+(-\)\(40\!\cdots\!80\)\( + \)\(52\!\cdots\!53\)\( \beta_{1} - \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(65\!\cdots\!31\)\( \beta_{3} + \)\(18\!\cdots\!00\)\( \beta_{4} - \)\(14\!\cdots\!57\)\( \beta_{5}) q^{38}\) \(+(-\)\(15\!\cdots\!64\)\( - \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} - \)\(32\!\cdots\!88\)\( \beta_{3} + \)\(15\!\cdots\!04\)\( \beta_{4} + \)\(29\!\cdots\!48\)\( \beta_{5}) q^{39}\) \(+(-\)\(43\!\cdots\!00\)\( - \)\(35\!\cdots\!80\)\( \beta_{1} + \)\(15\!\cdots\!80\)\( \beta_{2} - \)\(76\!\cdots\!40\)\( \beta_{3} - \)\(93\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5}) q^{40}\) \(+(-\)\(41\!\cdots\!58\)\( - \)\(43\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!64\)\( \beta_{2} + \)\(67\!\cdots\!08\)\( \beta_{3} - \)\(46\!\cdots\!52\)\( \beta_{4} - \)\(34\!\cdots\!44\)\( \beta_{5}) q^{41}\) \(+(-\)\(62\!\cdots\!20\)\( - \)\(92\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!64\)\( \beta_{3} + \)\(42\!\cdots\!20\)\( \beta_{4} + \)\(11\!\cdots\!88\)\( \beta_{5}) q^{42}\) \(+(-\)\(89\!\cdots\!00\)\( + \)\(56\!\cdots\!46\)\( \beta_{1} - \)\(20\!\cdots\!13\)\( \beta_{2} + \)\(11\!\cdots\!40\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(15\!\cdots\!60\)\( \beta_{5}) q^{43}\) \(+(-\)\(28\!\cdots\!96\)\( + \)\(73\!\cdots\!72\)\( \beta_{1} + \)\(43\!\cdots\!20\)\( \beta_{2} - \)\(59\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(28\!\cdots\!76\)\( \beta_{5}) q^{44}\) \(+(-\)\(10\!\cdots\!50\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} + \)\(12\!\cdots\!48\)\( \beta_{2} + \)\(33\!\cdots\!51\)\( \beta_{3} + \)\(29\!\cdots\!33\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5}) q^{45}\) \(+(-\)\(20\!\cdots\!28\)\( - \)\(43\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2} - \)\(62\!\cdots\!02\)\( \beta_{3} - \)\(11\!\cdots\!92\)\( \beta_{4} - \)\(42\!\cdots\!74\)\( \beta_{5}) q^{46}\) \(+(-\)\(38\!\cdots\!80\)\( - \)\(16\!\cdots\!04\)\( \beta_{1} - \)\(12\!\cdots\!24\)\( \beta_{2} - \)\(70\!\cdots\!80\)\( \beta_{3} - \)\(12\!\cdots\!60\)\( \beta_{4} + \)\(13\!\cdots\!20\)\( \beta_{5}) q^{47}\) \(+(-\)\(11\!\cdots\!80\)\( - \)\(21\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!16\)\( \beta_{3} + \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(17\!\cdots\!28\)\( \beta_{5}) q^{48}\) \(+(-\)\(24\!\cdots\!43\)\( + \)\(45\!\cdots\!60\)\( \beta_{1} + \)\(44\!\cdots\!16\)\( \beta_{2} + \)\(61\!\cdots\!72\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4} - \)\(47\!\cdots\!16\)\( \beta_{5}) q^{49}\) \(+(-\)\(23\!\cdots\!00\)\( + \)\(88\!\cdots\!25\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3} - \)\(19\!\cdots\!00\)\( \beta_{4} + \)\(35\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(\)\(12\!\cdots\!12\)\( + \)\(28\!\cdots\!24\)\( \beta_{1} - \)\(20\!\cdots\!46\)\( \beta_{2} - \)\(26\!\cdots\!24\)\( \beta_{3} - \)\(19\!\cdots\!28\)\( \beta_{4} + \)\(86\!\cdots\!64\)\( \beta_{5}) q^{51}\) \(+(\)\(20\!\cdots\!00\)\( + \)\(68\!\cdots\!92\)\( \beta_{1} - \)\(36\!\cdots\!16\)\( \beta_{2} + \)\(92\!\cdots\!70\)\( \beta_{3} + \)\(53\!\cdots\!40\)\( \beta_{4} - \)\(19\!\cdots\!80\)\( \beta_{5}) q^{52}\) \(+(\)\(43\!\cdots\!90\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(91\!\cdots\!20\)\( \beta_{2} - \)\(71\!\cdots\!57\)\( \beta_{3} + \)\(44\!\cdots\!65\)\( \beta_{4} - \)\(12\!\cdots\!36\)\( \beta_{5}) q^{53}\) \(+(\)\(10\!\cdots\!60\)\( - \)\(79\!\cdots\!86\)\( \beta_{1} + \)\(40\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!94\)\( \beta_{3} - \)\(44\!\cdots\!28\)\( \beta_{4} + \)\(62\!\cdots\!74\)\( \beta_{5}) q^{54}\) \(+(\)\(11\!\cdots\!00\)\( - \)\(34\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!22\)\( \beta_{2} - \)\(57\!\cdots\!36\)\( \beta_{3} + \)\(72\!\cdots\!12\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5}) q^{55}\) \(+(\)\(58\!\cdots\!60\)\( + \)\(20\!\cdots\!24\)\( \beta_{1} + \)\(65\!\cdots\!36\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(13\!\cdots\!96\)\( \beta_{4} - \)\(93\!\cdots\!68\)\( \beta_{5}) q^{56}\) \(+(\)\(50\!\cdots\!60\)\( + \)\(52\!\cdots\!52\)\( \beta_{1} - \)\(53\!\cdots\!24\)\( \beta_{2} - \)\(92\!\cdots\!90\)\( \beta_{3} - \)\(12\!\cdots\!30\)\( \beta_{4} + \)\(12\!\cdots\!60\)\( \beta_{5}) q^{57}\) \(+(-\)\(52\!\cdots\!20\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} + \)\(40\!\cdots\!56\)\( \beta_{2} - \)\(34\!\cdots\!24\)\( \beta_{3} - \)\(78\!\cdots\!40\)\( \beta_{4} + \)\(26\!\cdots\!68\)\( \beta_{5}) q^{58}\) \(+(-\)\(64\!\cdots\!20\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} + \)\(75\!\cdots\!59\)\( \beta_{2} - \)\(40\!\cdots\!88\)\( \beta_{3} + \)\(35\!\cdots\!80\)\( \beta_{4} - \)\(12\!\cdots\!80\)\( \beta_{5}) q^{59}\) \(+(-\)\(97\!\cdots\!00\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!12\)\( \beta_{2} + \)\(23\!\cdots\!44\)\( \beta_{3} + \)\(23\!\cdots\!52\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5}) q^{60}\) \(+(-\)\(76\!\cdots\!38\)\( - \)\(33\!\cdots\!60\)\( \beta_{1} - \)\(68\!\cdots\!40\)\( \beta_{2} + \)\(34\!\cdots\!95\)\( \beta_{3} - \)\(57\!\cdots\!15\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5}) q^{61}\) \(+(-\)\(19\!\cdots\!80\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} + \)\(34\!\cdots\!68\)\( \beta_{2} - \)\(48\!\cdots\!64\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4} - \)\(88\!\cdots\!12\)\( \beta_{5}) q^{62}\) \(+(-\)\(59\!\cdots\!40\)\( + \)\(23\!\cdots\!92\)\( \beta_{1} - \)\(21\!\cdots\!74\)\( \beta_{2} - \)\(93\!\cdots\!44\)\( \beta_{3} + \)\(18\!\cdots\!80\)\( \beta_{4} + \)\(14\!\cdots\!88\)\( \beta_{5}) q^{63}\) \(+(-\)\(40\!\cdots\!28\)\( + \)\(16\!\cdots\!32\)\( \beta_{1} + \)\(45\!\cdots\!16\)\( \beta_{2} - \)\(12\!\cdots\!24\)\( \beta_{3} - \)\(20\!\cdots\!96\)\( \beta_{4} - \)\(55\!\cdots\!12\)\( \beta_{5}) q^{64}\) \(+(\)\(66\!\cdots\!00\)\( + \)\(50\!\cdots\!04\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2} + \)\(44\!\cdots\!32\)\( \beta_{3} - \)\(59\!\cdots\!44\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5}) q^{65}\) \(+(\)\(77\!\cdots\!64\)\( - \)\(10\!\cdots\!28\)\( \beta_{1} - \)\(33\!\cdots\!68\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} - \)\(26\!\cdots\!44\)\( \beta_{4} + \)\(79\!\cdots\!72\)\( \beta_{5}) q^{66}\) \(+(\)\(46\!\cdots\!80\)\( - \)\(15\!\cdots\!66\)\( \beta_{1} - \)\(13\!\cdots\!91\)\( \beta_{2} - \)\(14\!\cdots\!56\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5}) q^{67}\) \(+(\)\(98\!\cdots\!40\)\( - \)\(23\!\cdots\!40\)\( \beta_{1} - \)\(26\!\cdots\!72\)\( \beta_{2} + \)\(80\!\cdots\!02\)\( \beta_{3} + \)\(69\!\cdots\!60\)\( \beta_{4} + \)\(82\!\cdots\!96\)\( \beta_{5}) q^{68}\) \(+(\)\(13\!\cdots\!36\)\( - \)\(65\!\cdots\!76\)\( \beta_{1} + \)\(45\!\cdots\!72\)\( \beta_{2} - \)\(31\!\cdots\!08\)\( \beta_{3} - \)\(61\!\cdots\!52\)\( \beta_{4} + \)\(34\!\cdots\!16\)\( \beta_{5}) q^{69}\) \(+(\)\(43\!\cdots\!00\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} + \)\(50\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!88\)\( \beta_{3} + \)\(22\!\cdots\!04\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5}) q^{70}\) \(+(-\)\(92\!\cdots\!28\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(18\!\cdots\!30\)\( \beta_{2} - \)\(84\!\cdots\!40\)\( \beta_{3} + \)\(11\!\cdots\!80\)\( \beta_{4} + \)\(82\!\cdots\!00\)\( \beta_{5}) q^{71}\) \(+(-\)\(52\!\cdots\!60\)\( + \)\(37\!\cdots\!76\)\( \beta_{1} - \)\(56\!\cdots\!40\)\( \beta_{2} - \)\(26\!\cdots\!48\)\( \beta_{3} + \)\(18\!\cdots\!40\)\( \beta_{4} + \)\(16\!\cdots\!16\)\( \beta_{5}) q^{72}\) \(+(-\)\(19\!\cdots\!30\)\( - \)\(55\!\cdots\!96\)\( \beta_{1} + \)\(89\!\cdots\!52\)\( \beta_{2} + \)\(54\!\cdots\!46\)\( \beta_{3} - \)\(94\!\cdots\!50\)\( \beta_{4} - \)\(36\!\cdots\!12\)\( \beta_{5}) q^{73}\) \(+(-\)\(24\!\cdots\!36\)\( - \)\(66\!\cdots\!70\)\( \beta_{1} - \)\(71\!\cdots\!36\)\( \beta_{2} + \)\(21\!\cdots\!88\)\( \beta_{3} + \)\(16\!\cdots\!28\)\( \beta_{4} - \)\(10\!\cdots\!04\)\( \beta_{5}) q^{74}\) \(+(-\)\(67\!\cdots\!00\)\( - \)\(21\!\cdots\!50\)\( \beta_{1} + \)\(23\!\cdots\!75\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!00\)\( \beta_{4} + \)\(63\!\cdots\!00\)\( \beta_{5}) q^{75}\) \(+(-\)\(93\!\cdots\!80\)\( - \)\(30\!\cdots\!12\)\( \beta_{1} + \)\(63\!\cdots\!52\)\( \beta_{2} - \)\(52\!\cdots\!80\)\( \beta_{3} - \)\(54\!\cdots\!08\)\( \beta_{4} + \)\(91\!\cdots\!64\)\( \beta_{5}) q^{76}\) \(+(-\)\(38\!\cdots\!00\)\( + \)\(43\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} - \)\(41\!\cdots\!76\)\( \beta_{3} + \)\(68\!\cdots\!80\)\( \beta_{4} - \)\(33\!\cdots\!08\)\( \beta_{5}) q^{77}\) \(+(\)\(49\!\cdots\!00\)\( + \)\(22\!\cdots\!30\)\( \beta_{1} - \)\(59\!\cdots\!40\)\( \beta_{2} + \)\(55\!\cdots\!22\)\( \beta_{3} + \)\(58\!\cdots\!00\)\( \beta_{4} - \)\(16\!\cdots\!34\)\( \beta_{5}) q^{78}\) \(+(\)\(12\!\cdots\!40\)\( - \)\(14\!\cdots\!44\)\( \beta_{1} - \)\(17\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!52\)\( \beta_{3} - \)\(13\!\cdots\!88\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5}) q^{79}\) \(+(\)\(54\!\cdots\!00\)\( + \)\(18\!\cdots\!84\)\( \beta_{1} + \)\(27\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!72\)\( \beta_{3} + \)\(35\!\cdots\!76\)\( \beta_{4} - \)\(75\!\cdots\!00\)\( \beta_{5}) q^{80}\) \(+(\)\(11\!\cdots\!81\)\( - \)\(90\!\cdots\!04\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} - \)\(46\!\cdots\!74\)\( \beta_{3} - \)\(26\!\cdots\!10\)\( \beta_{4} - \)\(33\!\cdots\!80\)\( \beta_{5}) q^{81}\) \(+(\)\(13\!\cdots\!20\)\( - \)\(99\!\cdots\!58\)\( \beta_{1} + \)\(62\!\cdots\!28\)\( \beta_{2} - \)\(21\!\cdots\!44\)\( \beta_{3} - \)\(90\!\cdots\!20\)\( \beta_{4} + \)\(38\!\cdots\!88\)\( \beta_{5}) q^{82}\) \(+(\)\(73\!\cdots\!60\)\( - \)\(87\!\cdots\!82\)\( \beta_{1} - \)\(15\!\cdots\!67\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4} + \)\(69\!\cdots\!00\)\( \beta_{5}) q^{83}\) \(+(\)\(38\!\cdots\!84\)\( + \)\(48\!\cdots\!96\)\( \beta_{1} - \)\(14\!\cdots\!36\)\( \beta_{2} + \)\(23\!\cdots\!80\)\( \beta_{3} + \)\(13\!\cdots\!24\)\( \beta_{4} - \)\(12\!\cdots\!72\)\( \beta_{5}) q^{84}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(29\!\cdots\!68\)\( \beta_{1} - \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(53\!\cdots\!06\)\( \beta_{3} - \)\(40\!\cdots\!98\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5}) q^{85}\) \(+(-\)\(17\!\cdots\!28\)\( + \)\(47\!\cdots\!79\)\( \beta_{1} + \)\(27\!\cdots\!80\)\( \beta_{2} - \)\(80\!\cdots\!67\)\( \beta_{3} - \)\(22\!\cdots\!16\)\( \beta_{4} + \)\(38\!\cdots\!53\)\( \beta_{5}) q^{86}\) \(+(-\)\(22\!\cdots\!60\)\( - \)\(72\!\cdots\!48\)\( \beta_{1} + \)\(64\!\cdots\!62\)\( \beta_{2} + \)\(61\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!40\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5}) q^{87}\) \(+(-\)\(16\!\cdots\!40\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2} + \)\(32\!\cdots\!96\)\( \beta_{3} - \)\(34\!\cdots\!80\)\( \beta_{4} - \)\(80\!\cdots\!32\)\( \beta_{5}) q^{88}\) \(+(-\)\(31\!\cdots\!30\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} - \)\(12\!\cdots\!48\)\( \beta_{2} + \)\(86\!\cdots\!90\)\( \beta_{3} - \)\(32\!\cdots\!78\)\( \beta_{4} - \)\(26\!\cdots\!36\)\( \beta_{5}) q^{89}\) \(+(-\)\(45\!\cdots\!00\)\( + \)\(23\!\cdots\!14\)\( \beta_{1} - \)\(47\!\cdots\!24\)\( \beta_{2} - \)\(81\!\cdots\!88\)\( \beta_{3} - \)\(32\!\cdots\!04\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5}) q^{90}\) \(+(-\)\(24\!\cdots\!48\)\( - \)\(96\!\cdots\!56\)\( \beta_{1} + \)\(23\!\cdots\!56\)\( \beta_{2} - \)\(25\!\cdots\!80\)\( \beta_{3} - \)\(23\!\cdots\!44\)\( \beta_{4} + \)\(81\!\cdots\!52\)\( \beta_{5}) q^{91}\) \(+(\)\(11\!\cdots\!80\)\( + \)\(20\!\cdots\!12\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2} - \)\(17\!\cdots\!64\)\( \beta_{3} + \)\(70\!\cdots\!80\)\( \beta_{4} - \)\(55\!\cdots\!72\)\( \beta_{5}) q^{92}\) \(+(\)\(21\!\cdots\!80\)\( - \)\(79\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!68\)\( \beta_{3} - \)\(30\!\cdots\!00\)\( \beta_{4} + \)\(18\!\cdots\!04\)\( \beta_{5}) q^{93}\) \(+(\)\(55\!\cdots\!84\)\( + \)\(43\!\cdots\!44\)\( \beta_{1} - \)\(59\!\cdots\!76\)\( \beta_{2} + \)\(25\!\cdots\!96\)\( \beta_{3} + \)\(23\!\cdots\!32\)\( \beta_{4} + \)\(78\!\cdots\!44\)\( \beta_{5}) q^{94}\) \(+(\)\(44\!\cdots\!00\)\( - \)\(45\!\cdots\!60\)\( \beta_{1} - \)\(96\!\cdots\!90\)\( \beta_{2} - \)\(43\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!40\)\( \beta_{4} - \)\(60\!\cdots\!00\)\( \beta_{5}) q^{95}\) \(+(\)\(82\!\cdots\!12\)\( + \)\(64\!\cdots\!16\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2} + \)\(38\!\cdots\!12\)\( \beta_{3} - \)\(24\!\cdots\!04\)\( \beta_{4} + \)\(23\!\cdots\!12\)\( \beta_{5}) q^{96}\) \(+(\)\(18\!\cdots\!70\)\( - \)\(51\!\cdots\!92\)\( \beta_{1} - \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(55\!\cdots\!18\)\( \beta_{3} + \)\(14\!\cdots\!90\)\( \beta_{4} - \)\(21\!\cdots\!36\)\( \beta_{5}) q^{97}\) \(+(-\)\(14\!\cdots\!80\)\( - \)\(12\!\cdots\!01\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(12\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4} + \)\(16\!\cdots\!72\)\( \beta_{5}) q^{98}\) \(+(-\)\(10\!\cdots\!04\)\( - \)\(56\!\cdots\!86\)\( \beta_{1} - \)\(14\!\cdots\!89\)\( \beta_{2} + \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(16\!\cdots\!64\)\( \beta_{4} + \)\(62\!\cdots\!12\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 460872026640q^{2} \) \(\mathstrut -\mathstrut 15648291925893129960q^{3} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!52\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!32\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!98\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 460872026640q^{2} \) \(\mathstrut -\mathstrut 15648291925893129960q^{3} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!52\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!32\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!98\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!28\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!80\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!24\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!56\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!80\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!40\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!40\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!88\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!68\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!20\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!40\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!92\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!40\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!80\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!44\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!84\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!40\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!80\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!84\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!48\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!20\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!76\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!68\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!80\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!58\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!72\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!40\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!60\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!20\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!28\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!40\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!68\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!84\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!80\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!40\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!16\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!68\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!60\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!16\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!80\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!86\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!20\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!60\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!04\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!68\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!40\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!88\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!80\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!04\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!72\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!80\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!24\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(457312410974129060102\) \(x^{4}\mathstrut -\mathstrut \) \(374381904318715551718938507366\) \(x^{3}\mathstrut +\mathstrut \) \(44188779498690494552956207183337235151045\) \(x^{2}\mathstrut +\mathstrut \) \(43570916530513802078515169165814719895908350666425\) \(x\mathstrut -\mathstrut \) \(274514886475906972833021199940770859283985201013623777350000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 144 \nu - 72 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(78099962708907\) \(\nu^{5}\mathstrut -\mathstrut \) \(1375691159918800801887660\) \(\nu^{4}\mathstrut +\mathstrut \) \(40226354593628598292993469800322382\) \(\nu^{3}\mathstrut +\mathstrut \) \(480737543221988187416410765264622639114508612\) \(\nu^{2}\mathstrut -\mathstrut \) \(4095995735119213827140200446492535473237363091214415291\) \(\nu\mathstrut -\mathstrut \) \(17130467806636669478709481975515934948439625934282286274145384368\)\()/\)\(69\!\cdots\!88\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(91142656481294469\) \(\nu^{5}\mathstrut -\mathstrut \) \(1605431583625240535802899220\) \(\nu^{4}\mathstrut +\mathstrut \) \(46944155810764574207923379256976219794\) \(\nu^{3}\mathstrut +\mathstrut \) \(4156478093089465029823565998569977702280963615996\) \(\nu^{2}\mathstrut -\mathstrut \) \(9195195307147830686701062611508965700721778119431024111477\) \(\nu\mathstrut -\mathstrut \) \(568073683618420740619803295994124603298936885106074825044960922999632\)\()/\)\(17\!\cdots\!72\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(606518390599808941384479\) \(\nu^{5}\mathstrut +\mathstrut \) \(1785096453037002977854674621991044\) \(\nu^{4}\mathstrut +\mathstrut \) \(272371810583872799198993532386144773328210262\) \(\nu^{3}\mathstrut -\mathstrut \) \(428159409320707302070751030867493812269669563678419532\) \(\nu^{2}\mathstrut -\mathstrut \) \(24830646082099940342148642768635140888295145794949529244263117551\) \(\nu\mathstrut +\mathstrut \) \(6940271488442406108725437805273034213041663446224288202723835834817750288\)\()/\)\(19\!\cdots\!28\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(289613468206311996503683251\) \(\nu^{5}\mathstrut +\mathstrut \) \(174972226505692037633509898090229300\) \(\nu^{4}\mathstrut +\mathstrut \) \(98117079324462934033155129479068624936612447902\) \(\nu^{3}\mathstrut +\mathstrut \) \(81654344557241561486631709673962807456824605977160938404\) \(\nu^{2}\mathstrut -\mathstrut \) \(3301073357707396020037591585583591704402760770904974187155182617667\) \(\nu\mathstrut -\mathstrut \) \(7051981094494430331918639309417795660138715723520929441699594261998477705264\)\()/\)\(99\!\cdots\!64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(72\)\()/144\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(4668\) \(\beta_{2}\mathstrut +\mathstrut \) \(176829862160\) \(\beta_{1}\mathstrut +\mathstrut \) \(3160943384653180063456128\)\()/20736\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(2727\) \(\beta_{5}\mathstrut +\mathstrut \) \(2307980\) \(\beta_{4}\mathstrut +\mathstrut \) \(7915256204\) \(\beta_{3}\mathstrut +\mathstrut \) \(763450743525280\) \(\beta_{2}\mathstrut +\mathstrut \) \(171109700008821821257987\) \(\beta_{1}\mathstrut +\mathstrut \) \(17467162191903096320225691284444736\)\()/93312\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(52520409734386\) \(\beta_{5}\mathstrut +\mathstrut \) \(133986338060164584\) \(\beta_{4}\mathstrut +\mathstrut \) \(9163979507037693318775\) \(\beta_{3}\mathstrut -\mathstrut \) \(269216097411919964130756292\) \(\beta_{2}\mathstrut +\mathstrut \) \(2712908864519921503572803358879354\) \(\beta_{1}\mathstrut +\mathstrut \) \(22536169715180661308714439099709591398970306176\)\()/559872\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(1875582683239936145189383\) \(\beta_{5}\mathstrut +\mathstrut \) \(1193105276567082329063209740\) \(\beta_{4}\mathstrut +\mathstrut \) \(7309604918011009905854873771614\) \(\beta_{3}\mathstrut +\mathstrut \) \(338424019486123722914639804333875304\) \(\beta_{2}\mathstrut +\mathstrut \) \(76621665860635844813220188031530036790903207\) \(\beta_{1}\mathstrut +\mathstrut \) \(14887763417556726787053963959092799432916570517581881184\)\()/139968\)

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
1.85696e10
1.12136e10
2.10202e9
−3.17952e9
−1.13941e10
−1.73117e10
−2.75084e12 −2.21125e19 5.14927e24 2.52982e28 6.08279e31 −2.62798e34 −7.51371e36 4.55346e37 −6.95912e40
1.2 −1.69157e12 1.74109e19 4.43572e23 −9.34342e27 −2.94518e31 1.25454e34 3.33964e36 −1.40288e38 1.58051e40
1.3 −3.79502e11 −3.74133e19 −2.27383e24 −3.57322e28 1.41984e31 −3.75122e33 1.78050e36 9.56330e38 1.35605e40
1.4 3.81039e11 −1.66183e18 −2.27266e24 2.26484e28 −6.33221e29 −1.01850e33 −1.78727e36 −4.40665e38 8.62992e39
1.5 1.56393e12 3.91464e19 2.80371e22 −1.92800e28 6.12224e31 −2.38829e34 −3.73751e36 1.08902e39 −3.01526e40
1.6 2.41607e12 −1.10180e19 3.41956e24 −1.95525e27 −2.66202e31 1.09563e34 2.42020e36 −3.22031e38 −4.72402e39
\(n\): 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 \(S_{82}^{\mathrm{new}}(\Gamma_0(1))\).