Properties

Label 1.86.a.a
Level 1
Weight 86
Character orbit 1.a
Self dual Yes
Analytic conductor 45.755
Analytic rank 1
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 86 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(45.7549576907\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{65}\cdot 3^{23}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 17^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(+(-599485114800 + \beta_{1}) q^{2}\) \(+(-26429057684102378700 - 6087800 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(23\!\cdots\!72\)\( + 1051215317901 \beta_{1} - 125655 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(15\!\cdots\!50\)\( - 1259712134775639 \beta_{1} + 458508359 \beta_{2} - 1638 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(36\!\cdots\!08\)\( - \)\(12\!\cdots\!63\)\( \beta_{1} - 492374310 \beta_{2} - 5743476 \beta_{3} - 174 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(62\!\cdots\!00\)\( + \)\(20\!\cdots\!92\)\( \beta_{1} - 6398682496962 \beta_{2} + 987599368 \beta_{3} + 14868 \beta_{4} - 168 \beta_{5}) q^{7}\) \(+(\)\(74\!\cdots\!00\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} - 23302112413172480 \beta_{2} + 1130791297824 \beta_{3} - 21143776 \beta_{4} - 183024 \beta_{5}) q^{8}\) \(+(\)\(95\!\cdots\!73\)\( + \)\(13\!\cdots\!94\)\( \beta_{1} - 15867554014098191070 \beta_{2} + 109812373643628 \beta_{3} - 36787298418 \beta_{4} - 10659168 \beta_{5}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-599485114800 + \beta_{1}) q^{2}\) \(+(-26429057684102378700 - 6087800 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(23\!\cdots\!72\)\( + 1051215317901 \beta_{1} - 125655 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(15\!\cdots\!50\)\( - 1259712134775639 \beta_{1} + 458508359 \beta_{2} - 1638 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(36\!\cdots\!08\)\( - \)\(12\!\cdots\!63\)\( \beta_{1} - 492374310 \beta_{2} - 5743476 \beta_{3} - 174 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(62\!\cdots\!00\)\( + \)\(20\!\cdots\!92\)\( \beta_{1} - 6398682496962 \beta_{2} + 987599368 \beta_{3} + 14868 \beta_{4} - 168 \beta_{5}) q^{7}\) \(+(\)\(74\!\cdots\!00\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} - 23302112413172480 \beta_{2} + 1130791297824 \beta_{3} - 21143776 \beta_{4} - 183024 \beta_{5}) q^{8}\) \(+(\)\(95\!\cdots\!73\)\( + \)\(13\!\cdots\!94\)\( \beta_{1} - 15867554014098191070 \beta_{2} + 109812373643628 \beta_{3} - 36787298418 \beta_{4} - 10659168 \beta_{5}) q^{9}\) \(+(\)\(15\!\cdots\!00\)\( - \)\(23\!\cdots\!78\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2} - 10509442468760176 \beta_{3} - 6263836481448 \beta_{4} + 1929895500 \beta_{5}) q^{10}\) \(+(-\)\(93\!\cdots\!48\)\( + \)\(95\!\cdots\!28\)\( \beta_{1} + \)\(24\!\cdots\!35\)\( \beta_{2} - 1432455142891590576 \beta_{3} - 7488625876792 \beta_{4} - 94807006992 \beta_{5}) q^{11}\) \(+(-\)\(63\!\cdots\!00\)\( - \)\(47\!\cdots\!36\)\( \beta_{1} + \)\(25\!\cdots\!84\)\( \beta_{2} - \)\(19\!\cdots\!24\)\( \beta_{3} + 9807228322834176 \beta_{4} + 2338324170624 \beta_{5}) q^{12}\) \(+(-\)\(63\!\cdots\!50\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(38\!\cdots\!47\)\( \beta_{2} - \)\(39\!\cdots\!06\)\( \beta_{3} - 187108684252781031 \beta_{4} - 23547471265344 \beta_{5}) q^{13}\) \(+(\)\(12\!\cdots\!84\)\( + \)\(12\!\cdots\!78\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(33\!\cdots\!48\)\( \beta_{3} - 1557622995397154140 \beta_{4} - 420407088200190 \beta_{5}) q^{14}\) \(+(\)\(24\!\cdots\!00\)\( + \)\(60\!\cdots\!56\)\( \beta_{1} - \)\(45\!\cdots\!86\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!96\)\( \beta_{4} + 22888906755689000 \beta_{5}) q^{15}\) \(+(\)\(43\!\cdots\!56\)\( + \)\(10\!\cdots\!88\)\( \beta_{1} - \)\(67\!\cdots\!40\)\( \beta_{2} - \)\(20\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!84\)\( \beta_{4} - 528917337722601984 \beta_{5}) q^{16}\) \(+(-\)\(85\!\cdots\!50\)\( - \)\(78\!\cdots\!78\)\( \beta_{1} - \)\(52\!\cdots\!22\)\( \beta_{2} - \)\(35\!\cdots\!88\)\( \beta_{3} + \)\(49\!\cdots\!62\)\( \beta_{4} + 8298680780279923488 \beta_{5}) q^{17}\) \(+(\)\(77\!\cdots\!00\)\( + \)\(15\!\cdots\!77\)\( \beta_{1} - \)\(61\!\cdots\!88\)\( \beta_{2} + \)\(24\!\cdots\!72\)\( \beta_{3} + \)\(13\!\cdots\!72\)\( \beta_{4} - 96811177722486238872 \beta_{5}) q^{18}\) \(+(\)\(44\!\cdots\!60\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(15\!\cdots\!25\)\( \beta_{2} + \)\(21\!\cdots\!92\)\( \beta_{3} - \)\(24\!\cdots\!64\)\( \beta_{4} + \)\(84\!\cdots\!36\)\( \beta_{5}) q^{19}\) \(+(-\)\(87\!\cdots\!00\)\( - \)\(45\!\cdots\!58\)\( \beta_{1} + \)\(11\!\cdots\!98\)\( \beta_{2} - \)\(26\!\cdots\!86\)\( \beta_{3} + \)\(17\!\cdots\!72\)\( \beta_{4} - \)\(52\!\cdots\!00\)\( \beta_{5}) q^{20}\) \(+(-\)\(97\!\cdots\!68\)\( - \)\(87\!\cdots\!08\)\( \beta_{1} + \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(43\!\cdots\!28\)\( \beta_{3} - \)\(24\!\cdots\!60\)\( \beta_{4} + \)\(15\!\cdots\!40\)\( \beta_{5}) q^{21}\) \(+(\)\(59\!\cdots\!00\)\( - \)\(50\!\cdots\!97\)\( \beta_{1} - \)\(59\!\cdots\!82\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3} - \)\(73\!\cdots\!74\)\( \beta_{4} + \)\(98\!\cdots\!99\)\( \beta_{5}) q^{22}\) \(+(-\)\(48\!\cdots\!00\)\( - \)\(52\!\cdots\!84\)\( \beta_{1} - \)\(78\!\cdots\!86\)\( \beta_{2} - \)\(41\!\cdots\!80\)\( \beta_{3} + \)\(75\!\cdots\!20\)\( \beta_{4} - \)\(19\!\cdots\!20\)\( \beta_{5}) q^{23}\) \(+(-\)\(11\!\cdots\!20\)\( - \)\(95\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(53\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!44\)\( \beta_{4} + \)\(16\!\cdots\!56\)\( \beta_{5}) q^{24}\) \(+(\)\(11\!\cdots\!75\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} + \)\(37\!\cdots\!00\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(31\!\cdots\!00\)\( \beta_{4} - \)\(91\!\cdots\!00\)\( \beta_{5}) q^{25}\) \(+(-\)\(24\!\cdots\!28\)\( - \)\(19\!\cdots\!74\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} + \)\(21\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!48\)\( \beta_{4} + \)\(30\!\cdots\!48\)\( \beta_{5}) q^{26}\) \(+(-\)\(48\!\cdots\!00\)\( - \)\(34\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} - \)\(80\!\cdots\!36\)\( \beta_{3} - \)\(55\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!36\)\( \beta_{5}) q^{27}\) \(+(\)\(43\!\cdots\!00\)\( + \)\(24\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!96\)\( \beta_{2} + \)\(13\!\cdots\!76\)\( \beta_{3} + \)\(60\!\cdots\!76\)\( \beta_{4} - \)\(91\!\cdots\!76\)\( \beta_{5}) q^{28}\) \(+(-\)\(21\!\cdots\!10\)\( + \)\(51\!\cdots\!33\)\( \beta_{1} + \)\(23\!\cdots\!35\)\( \beta_{2} + \)\(10\!\cdots\!22\)\( \beta_{3} + \)\(77\!\cdots\!97\)\( \beta_{4} + \)\(70\!\cdots\!72\)\( \beta_{5}) q^{29}\) \(+(\)\(22\!\cdots\!00\)\( + \)\(10\!\cdots\!62\)\( \beta_{1} + \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(20\!\cdots\!96\)\( \beta_{3} - \)\(42\!\cdots\!08\)\( \beta_{4} - \)\(34\!\cdots\!50\)\( \beta_{5}) q^{30}\) \(+(-\)\(86\!\cdots\!48\)\( + \)\(87\!\cdots\!64\)\( \beta_{1} - \)\(27\!\cdots\!20\)\( \beta_{2} - \)\(25\!\cdots\!88\)\( \beta_{3} + \)\(27\!\cdots\!04\)\( \beta_{4} + \)\(11\!\cdots\!04\)\( \beta_{5}) q^{31}\) \(+(\)\(31\!\cdots\!00\)\( + \)\(26\!\cdots\!28\)\( \beta_{1} - \)\(30\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(67\!\cdots\!52\)\( \beta_{4} - \)\(27\!\cdots\!52\)\( \beta_{5}) q^{32}\) \(+(\)\(68\!\cdots\!00\)\( - \)\(39\!\cdots\!42\)\( \beta_{1} + \)\(18\!\cdots\!46\)\( \beta_{2} + \)\(12\!\cdots\!08\)\( \beta_{3} - \)\(35\!\cdots\!42\)\( \beta_{4} + \)\(39\!\cdots\!92\)\( \beta_{5}) q^{33}\) \(+(-\)\(43\!\cdots\!56\)\( - \)\(14\!\cdots\!22\)\( \beta_{1} + \)\(52\!\cdots\!60\)\( \beta_{2} - \)\(15\!\cdots\!60\)\( \beta_{3} + \)\(64\!\cdots\!76\)\( \beta_{4} - \)\(68\!\cdots\!24\)\( \beta_{5}) q^{34}\) \(+(-\)\(91\!\cdots\!00\)\( - \)\(16\!\cdots\!88\)\( \beta_{1} - \)\(21\!\cdots\!72\)\( \beta_{2} + \)\(29\!\cdots\!04\)\( \beta_{3} + \)\(71\!\cdots\!92\)\( \beta_{4} + \)\(59\!\cdots\!00\)\( \beta_{5}) q^{35}\) \(+(\)\(56\!\cdots\!56\)\( + \)\(16\!\cdots\!33\)\( \beta_{1} - \)\(57\!\cdots\!15\)\( \beta_{2} + \)\(17\!\cdots\!45\)\( \beta_{3} - \)\(25\!\cdots\!24\)\( \beta_{4} - \)\(37\!\cdots\!24\)\( \beta_{5}) q^{36}\) \(+(-\)\(84\!\cdots\!50\)\( + \)\(28\!\cdots\!65\)\( \beta_{1} - \)\(14\!\cdots\!73\)\( \beta_{2} - \)\(24\!\cdots\!02\)\( \beta_{3} - \)\(29\!\cdots\!27\)\( \beta_{4} + \)\(12\!\cdots\!52\)\( \beta_{5}) q^{37}\) \(+(-\)\(81\!\cdots\!00\)\( + \)\(63\!\cdots\!77\)\( \beta_{1} + \)\(59\!\cdots\!86\)\( \beta_{2} - \)\(16\!\cdots\!92\)\( \beta_{3} + \)\(21\!\cdots\!58\)\( \beta_{4} - \)\(11\!\cdots\!83\)\( \beta_{5}) q^{38}\) \(+(\)\(18\!\cdots\!56\)\( + \)\(64\!\cdots\!60\)\( \beta_{1} - \)\(87\!\cdots\!50\)\( \beta_{2} + \)\(52\!\cdots\!44\)\( \beta_{3} - \)\(62\!\cdots\!68\)\( \beta_{4} - \)\(11\!\cdots\!68\)\( \beta_{5}) q^{39}\) \(+(-\)\(23\!\cdots\!00\)\( - \)\(16\!\cdots\!20\)\( \beta_{1} - \)\(25\!\cdots\!80\)\( \beta_{2} + \)\(19\!\cdots\!60\)\( \beta_{3} + \)\(79\!\cdots\!80\)\( \beta_{4} + \)\(67\!\cdots\!00\)\( \beta_{5}) q^{40}\) \(+(-\)\(62\!\cdots\!98\)\( - \)\(41\!\cdots\!16\)\( \beta_{1} - \)\(49\!\cdots\!20\)\( \beta_{2} - \)\(13\!\cdots\!28\)\( \beta_{3} + \)\(18\!\cdots\!24\)\( \beta_{4} - \)\(18\!\cdots\!76\)\( \beta_{5}) q^{41}\) \(+(-\)\(53\!\cdots\!00\)\( - \)\(44\!\cdots\!88\)\( \beta_{1} + \)\(18\!\cdots\!80\)\( \beta_{2} - \)\(97\!\cdots\!72\)\( \beta_{3} + \)\(28\!\cdots\!28\)\( \beta_{4} + \)\(13\!\cdots\!72\)\( \beta_{5}) q^{42}\) \(+(-\)\(20\!\cdots\!00\)\( + \)\(56\!\cdots\!32\)\( \beta_{1} + \)\(74\!\cdots\!27\)\( \beta_{2} + \)\(35\!\cdots\!80\)\( \beta_{3} - \)\(26\!\cdots\!20\)\( \beta_{4} + \)\(90\!\cdots\!20\)\( \beta_{5}) q^{43}\) \(+(-\)\(31\!\cdots\!56\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} + \)\(75\!\cdots\!60\)\( \beta_{2} + \)\(44\!\cdots\!00\)\( \beta_{3} + \)\(60\!\cdots\!16\)\( \beta_{4} - \)\(40\!\cdots\!84\)\( \beta_{5}) q^{44}\) \(+(-\)\(14\!\cdots\!50\)\( + \)\(88\!\cdots\!53\)\( \beta_{1} - \)\(29\!\cdots\!93\)\( \beta_{2} - \)\(10\!\cdots\!74\)\( \beta_{3} + \)\(54\!\cdots\!73\)\( \beta_{4} + \)\(72\!\cdots\!00\)\( \beta_{5}) q^{45}\) \(+(-\)\(31\!\cdots\!48\)\( - \)\(17\!\cdots\!66\)\( \beta_{1} - \)\(32\!\cdots\!20\)\( \beta_{2} - \)\(45\!\cdots\!88\)\( \beta_{3} - \)\(60\!\cdots\!56\)\( \beta_{4} + \)\(60\!\cdots\!94\)\( \beta_{5}) q^{46}\) \(+(-\)\(39\!\cdots\!00\)\( - \)\(40\!\cdots\!72\)\( \beta_{1} - \)\(93\!\cdots\!64\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} + \)\(12\!\cdots\!40\)\( \beta_{4} - \)\(34\!\cdots\!40\)\( \beta_{5}) q^{47}\) \(+(-\)\(33\!\cdots\!00\)\( - \)\(34\!\cdots\!36\)\( \beta_{1} + \)\(94\!\cdots\!28\)\( \beta_{2} - \)\(28\!\cdots\!52\)\( \beta_{3} - \)\(48\!\cdots\!52\)\( \beta_{4} + \)\(71\!\cdots\!52\)\( \beta_{5}) q^{48}\) \(+(-\)\(59\!\cdots\!43\)\( + \)\(77\!\cdots\!56\)\( \beta_{1} - \)\(30\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!28\)\( \beta_{3} - \)\(49\!\cdots\!44\)\( \beta_{4} - \)\(23\!\cdots\!44\)\( \beta_{5}) q^{49}\) \(+(-\)\(19\!\cdots\!00\)\( + \)\(11\!\cdots\!75\)\( \beta_{1} - \)\(85\!\cdots\!00\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(79\!\cdots\!00\)\( \beta_{4} - \)\(72\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(-\)\(17\!\cdots\!88\)\( + \)\(45\!\cdots\!40\)\( \beta_{1} - \)\(14\!\cdots\!50\)\( \beta_{2} + \)\(79\!\cdots\!12\)\( \beta_{3} + \)\(22\!\cdots\!56\)\( \beta_{4} - \)\(37\!\cdots\!44\)\( \beta_{5}) q^{51}\) \(+(-\)\(97\!\cdots\!00\)\( - \)\(16\!\cdots\!26\)\( \beta_{1} + \)\(23\!\cdots\!14\)\( \beta_{2} - \)\(77\!\cdots\!10\)\( \beta_{3} + \)\(31\!\cdots\!40\)\( \beta_{4} + \)\(18\!\cdots\!60\)\( \beta_{5}) q^{52}\) \(+(-\)\(84\!\cdots\!50\)\( - \)\(10\!\cdots\!91\)\( \beta_{1} + \)\(28\!\cdots\!63\)\( \beta_{2} + \)\(55\!\cdots\!94\)\( \beta_{3} - \)\(25\!\cdots\!31\)\( \beta_{4} - \)\(53\!\cdots\!44\)\( \beta_{5}) q^{53}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(20\!\cdots\!50\)\( \beta_{1} + \)\(65\!\cdots\!00\)\( \beta_{2} - \)\(32\!\cdots\!48\)\( \beta_{3} + \)\(41\!\cdots\!96\)\( \beta_{4} - \)\(15\!\cdots\!54\)\( \beta_{5}) q^{54}\) \(+(\)\(65\!\cdots\!00\)\( - \)\(79\!\cdots\!28\)\( \beta_{1} - \)\(35\!\cdots\!82\)\( \beta_{2} + \)\(64\!\cdots\!24\)\( \beta_{3} + \)\(53\!\cdots\!52\)\( \beta_{4} + \)\(45\!\cdots\!00\)\( \beta_{5}) q^{55}\) \(+(\)\(97\!\cdots\!60\)\( + \)\(90\!\cdots\!72\)\( \beta_{1} - \)\(31\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3} - \)\(56\!\cdots\!12\)\( \beta_{4} - \)\(28\!\cdots\!12\)\( \beta_{5}) q^{56}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(41\!\cdots\!62\)\( \beta_{1} + \)\(97\!\cdots\!66\)\( \beta_{2} - \)\(49\!\cdots\!80\)\( \beta_{3} - \)\(90\!\cdots\!30\)\( \beta_{4} - \)\(15\!\cdots\!20\)\( \beta_{5}) q^{57}\) \(+(\)\(32\!\cdots\!00\)\( - \)\(61\!\cdots\!26\)\( \beta_{1} + \)\(25\!\cdots\!32\)\( \beta_{2} - \)\(75\!\cdots\!72\)\( \beta_{3} + \)\(33\!\cdots\!28\)\( \beta_{4} + \)\(43\!\cdots\!72\)\( \beta_{5}) q^{58}\) \(+(\)\(11\!\cdots\!80\)\( + \)\(66\!\cdots\!32\)\( \beta_{1} - \)\(98\!\cdots\!85\)\( \beta_{2} - \)\(55\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!40\)\( \beta_{4} - \)\(24\!\cdots\!60\)\( \beta_{5}) q^{59}\) \(+(\)\(52\!\cdots\!00\)\( + \)\(63\!\cdots\!32\)\( \beta_{1} - \)\(12\!\cdots\!92\)\( \beta_{2} + \)\(74\!\cdots\!44\)\( \beta_{3} - \)\(43\!\cdots\!88\)\( \beta_{4} - \)\(87\!\cdots\!00\)\( \beta_{5}) q^{60}\) \(+(\)\(23\!\cdots\!02\)\( - \)\(70\!\cdots\!95\)\( \beta_{1} - \)\(10\!\cdots\!25\)\( \beta_{2} - \)\(10\!\cdots\!10\)\( \beta_{3} + \)\(81\!\cdots\!05\)\( \beta_{4} + \)\(13\!\cdots\!80\)\( \beta_{5}) q^{61}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(12\!\cdots\!60\)\( \beta_{1} + \)\(56\!\cdots\!84\)\( \beta_{2} - \)\(86\!\cdots\!12\)\( \beta_{3} + \)\(97\!\cdots\!88\)\( \beta_{4} + \)\(16\!\cdots\!12\)\( \beta_{5}) q^{62}\) \(+(\)\(59\!\cdots\!00\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} - \)\(10\!\cdots\!58\)\( \beta_{2} + \)\(90\!\cdots\!68\)\( \beta_{3} - \)\(59\!\cdots\!32\)\( \beta_{4} - \)\(36\!\cdots\!68\)\( \beta_{5}) q^{63}\) \(+(-\)\(25\!\cdots\!88\)\( + \)\(49\!\cdots\!64\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(48\!\cdots\!48\)\( \beta_{3} - \)\(81\!\cdots\!68\)\( \beta_{4} - \)\(12\!\cdots\!68\)\( \beta_{5}) q^{64}\) \(+(-\)\(34\!\cdots\!00\)\( + \)\(53\!\cdots\!96\)\( \beta_{1} - \)\(57\!\cdots\!76\)\( \beta_{2} + \)\(43\!\cdots\!32\)\( \beta_{3} - \)\(70\!\cdots\!64\)\( \beta_{4} + \)\(46\!\cdots\!00\)\( \beta_{5}) q^{65}\) \(+(-\)\(24\!\cdots\!16\)\( + \)\(18\!\cdots\!40\)\( \beta_{1} + \)\(72\!\cdots\!00\)\( \beta_{2} - \)\(39\!\cdots\!24\)\( \beta_{3} + \)\(24\!\cdots\!28\)\( \beta_{4} - \)\(27\!\cdots\!72\)\( \beta_{5}) q^{66}\) \(+(-\)\(11\!\cdots\!00\)\( + \)\(65\!\cdots\!92\)\( \beta_{1} - \)\(12\!\cdots\!87\)\( \beta_{2} + \)\(32\!\cdots\!52\)\( \beta_{3} - \)\(73\!\cdots\!48\)\( \beta_{4} - \)\(13\!\cdots\!52\)\( \beta_{5}) q^{67}\) \(+(-\)\(52\!\cdots\!00\)\( - \)\(12\!\cdots\!38\)\( \beta_{1} + \)\(30\!\cdots\!90\)\( \beta_{2} - \)\(52\!\cdots\!74\)\( \beta_{3} - \)\(67\!\cdots\!24\)\( \beta_{4} + \)\(36\!\cdots\!24\)\( \beta_{5}) q^{68}\) \(+(-\)\(12\!\cdots\!04\)\( + \)\(55\!\cdots\!48\)\( \beta_{1} - \)\(28\!\cdots\!40\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3} + \)\(80\!\cdots\!44\)\( \beta_{4} - \)\(37\!\cdots\!56\)\( \beta_{5}) q^{69}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(49\!\cdots\!56\)\( \beta_{2} - \)\(21\!\cdots\!92\)\( \beta_{3} + \)\(76\!\cdots\!84\)\( \beta_{4} - \)\(94\!\cdots\!00\)\( \beta_{5}) q^{70}\) \(+(-\)\(10\!\cdots\!48\)\( + \)\(21\!\cdots\!40\)\( \beta_{1} - \)\(97\!\cdots\!50\)\( \beta_{2} - \)\(55\!\cdots\!80\)\( \beta_{3} + \)\(21\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!40\)\( \beta_{5}) q^{71}\) \(+(\)\(70\!\cdots\!00\)\( + \)\(11\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(10\!\cdots\!76\)\( \beta_{3} - \)\(66\!\cdots\!24\)\( \beta_{4} - \)\(42\!\cdots\!76\)\( \beta_{5}) q^{72}\) \(+(\)\(59\!\cdots\!50\)\( - \)\(64\!\cdots\!66\)\( \beta_{1} + \)\(10\!\cdots\!38\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3} - \)\(43\!\cdots\!02\)\( \beta_{4} + \)\(84\!\cdots\!52\)\( \beta_{5}) q^{73}\) \(+(\)\(17\!\cdots\!64\)\( - \)\(87\!\cdots\!06\)\( \beta_{1} + \)\(30\!\cdots\!80\)\( \beta_{2} + \)\(20\!\cdots\!72\)\( \beta_{3} + \)\(30\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!56\)\( \beta_{5}) q^{74}\) \(+(\)\(24\!\cdots\!00\)\( - \)\(24\!\cdots\!00\)\( \beta_{1} + \)\(13\!\cdots\!75\)\( \beta_{2} - \)\(55\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5}) q^{75}\) \(+(\)\(26\!\cdots\!20\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} - \)\(26\!\cdots\!20\)\( \beta_{2} - \)\(29\!\cdots\!64\)\( \beta_{3} - \)\(51\!\cdots\!84\)\( \beta_{4} + \)\(62\!\cdots\!16\)\( \beta_{5}) q^{76}\) \(+(\)\(41\!\cdots\!00\)\( - \)\(19\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!76\)\( \beta_{2} + \)\(65\!\cdots\!72\)\( \beta_{3} - \)\(41\!\cdots\!28\)\( \beta_{4} + \)\(40\!\cdots\!28\)\( \beta_{5}) q^{77}\) \(+(\)\(28\!\cdots\!00\)\( + \)\(33\!\cdots\!10\)\( \beta_{1} - \)\(53\!\cdots\!68\)\( \beta_{2} + \)\(21\!\cdots\!56\)\( \beta_{3} + \)\(27\!\cdots\!56\)\( \beta_{4} - \)\(15\!\cdots\!06\)\( \beta_{5}) q^{78}\) \(+(-\)\(22\!\cdots\!60\)\( + \)\(11\!\cdots\!72\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2} + \)\(74\!\cdots\!64\)\( \beta_{3} - \)\(85\!\cdots\!84\)\( \beta_{4} + \)\(11\!\cdots\!16\)\( \beta_{5}) q^{79}\) \(+(-\)\(68\!\cdots\!00\)\( - \)\(74\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!04\)\( \beta_{2} - \)\(12\!\cdots\!28\)\( \beta_{3} - \)\(56\!\cdots\!44\)\( \beta_{4} + \)\(20\!\cdots\!00\)\( \beta_{5}) q^{80}\) \(+(-\)\(71\!\cdots\!59\)\( + \)\(14\!\cdots\!46\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} + \)\(56\!\cdots\!36\)\( \beta_{3} + \)\(74\!\cdots\!70\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5}) q^{81}\) \(+(-\)\(25\!\cdots\!00\)\( - \)\(34\!\cdots\!70\)\( \beta_{1} - \)\(78\!\cdots\!96\)\( \beta_{2} + \)\(92\!\cdots\!28\)\( \beta_{3} - \)\(14\!\cdots\!72\)\( \beta_{4} - \)\(82\!\cdots\!28\)\( \beta_{5}) q^{82}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(22\!\cdots\!56\)\( \beta_{1} - \)\(52\!\cdots\!07\)\( \beta_{2} + \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!00\)\( \beta_{4} + \)\(70\!\cdots\!00\)\( \beta_{5}) q^{83}\) \(+(-\)\(20\!\cdots\!96\)\( - \)\(66\!\cdots\!92\)\( \beta_{1} + \)\(60\!\cdots\!60\)\( \beta_{2} - \)\(41\!\cdots\!48\)\( \beta_{3} + \)\(41\!\cdots\!12\)\( \beta_{4} + \)\(24\!\cdots\!12\)\( \beta_{5}) q^{84}\) \(+(\)\(29\!\cdots\!00\)\( + \)\(73\!\cdots\!42\)\( \beta_{1} + \)\(20\!\cdots\!98\)\( \beta_{2} - \)\(51\!\cdots\!36\)\( \beta_{3} - \)\(18\!\cdots\!78\)\( \beta_{4} - \)\(40\!\cdots\!00\)\( \beta_{5}) q^{85}\) \(+(\)\(35\!\cdots\!12\)\( + \)\(72\!\cdots\!31\)\( \beta_{1} + \)\(39\!\cdots\!70\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(22\!\cdots\!22\)\( \beta_{4} - \)\(28\!\cdots\!53\)\( \beta_{5}) q^{86}\) \(+(\)\(84\!\cdots\!00\)\( + \)\(17\!\cdots\!56\)\( \beta_{1} - \)\(50\!\cdots\!42\)\( \beta_{2} - \)\(50\!\cdots\!72\)\( \beta_{3} + \)\(17\!\cdots\!28\)\( \beta_{4} + \)\(93\!\cdots\!72\)\( \beta_{5}) q^{87}\) \(+(\)\(45\!\cdots\!00\)\( + \)\(77\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(68\!\cdots\!68\)\( \beta_{3} - \)\(45\!\cdots\!32\)\( \beta_{4} + \)\(27\!\cdots\!32\)\( \beta_{5}) q^{88}\) \(+(\)\(26\!\cdots\!70\)\( + \)\(35\!\cdots\!74\)\( \beta_{1} - \)\(10\!\cdots\!70\)\( \beta_{2} - \)\(65\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!74\)\( \beta_{4} - \)\(18\!\cdots\!24\)\( \beta_{5}) q^{89}\) \(+(\)\(14\!\cdots\!00\)\( - \)\(14\!\cdots\!94\)\( \beta_{1} + \)\(35\!\cdots\!64\)\( \beta_{2} - \)\(53\!\cdots\!48\)\( \beta_{3} + \)\(37\!\cdots\!96\)\( \beta_{4} + \)\(58\!\cdots\!00\)\( \beta_{5}) q^{90}\) \(+(-\)\(43\!\cdots\!88\)\( - \)\(56\!\cdots\!88\)\( \beta_{1} + \)\(98\!\cdots\!40\)\( \beta_{2} - \)\(36\!\cdots\!52\)\( \beta_{3} - \)\(45\!\cdots\!72\)\( \beta_{4} + \)\(24\!\cdots\!28\)\( \beta_{5}) q^{91}\) \(+(-\)\(68\!\cdots\!00\)\( - \)\(23\!\cdots\!92\)\( \beta_{1} + \)\(64\!\cdots\!76\)\( \beta_{2} + \)\(34\!\cdots\!48\)\( \beta_{3} + \)\(11\!\cdots\!48\)\( \beta_{4} - \)\(13\!\cdots\!48\)\( \beta_{5}) q^{92}\) \(+(-\)\(98\!\cdots\!00\)\( + \)\(45\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!76\)\( \beta_{2} + \)\(57\!\cdots\!04\)\( \beta_{3} + \)\(20\!\cdots\!04\)\( \beta_{4} + \)\(51\!\cdots\!96\)\( \beta_{5}) q^{93}\) \(+(-\)\(22\!\cdots\!76\)\( + \)\(20\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} - \)\(23\!\cdots\!88\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4} - \)\(13\!\cdots\!04\)\( \beta_{5}) q^{94}\) \(+(-\)\(88\!\cdots\!00\)\( + \)\(15\!\cdots\!60\)\( \beta_{1} + \)\(36\!\cdots\!90\)\( \beta_{2} - \)\(75\!\cdots\!80\)\( \beta_{3} + \)\(75\!\cdots\!60\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5}) q^{95}\) \(+(-\)\(14\!\cdots\!08\)\( - \)\(17\!\cdots\!56\)\( \beta_{1} + \)\(17\!\cdots\!80\)\( \beta_{2} - \)\(18\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!28\)\( \beta_{4} + \)\(56\!\cdots\!28\)\( \beta_{5}) q^{96}\) \(+(\)\(33\!\cdots\!50\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} + \)\(35\!\cdots\!98\)\( \beta_{2} + \)\(36\!\cdots\!44\)\( \beta_{3} - \)\(26\!\cdots\!06\)\( \beta_{4} - \)\(15\!\cdots\!44\)\( \beta_{5}) q^{97}\) \(+(\)\(83\!\cdots\!00\)\( - \)\(51\!\cdots\!11\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} + \)\(10\!\cdots\!72\)\( \beta_{3} + \)\(89\!\cdots\!72\)\( \beta_{4} - \)\(51\!\cdots\!72\)\( \beta_{5}) q^{98}\) \(+(\)\(24\!\cdots\!96\)\( + \)\(27\!\cdots\!12\)\( \beta_{1} + \)\(60\!\cdots\!15\)\( \beta_{2} + \)\(16\!\cdots\!48\)\( \beta_{3} - \)\(29\!\cdots\!72\)\( \beta_{4} - \)\(13\!\cdots\!72\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 3596910688800q^{2} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!32\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!48\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!38\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 3596910688800q^{2} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!32\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!48\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!38\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!88\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!04\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!36\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!00\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!08\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!68\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!60\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!88\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!36\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!36\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!36\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!88\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!36\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!88\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!58\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!80\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!12\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!00\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!28\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!00\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!24\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!88\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!00\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!84\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!20\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!54\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!00\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!76\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!72\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!20\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!28\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!00\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!56\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!48\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!76\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(20125264230072359909772\) \(x^{4}\mathstrut -\mathstrut \) \(314483190102207151531480871396326\) \(x^{3}\mathstrut +\mathstrut \) \(90192008136736578291529572869233310390975625\) \(x^{2}\mathstrut +\mathstrut \) \(2091285328888251081155130451204419887619836339467659225\) \(x\mathstrut -\mathstrut \) \(17301527721628842869681120067092990846953613013306724567953408750\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 96 \nu - 48 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(22756186805176591\) \(\nu^{5}\mathstrut -\mathstrut \) \(483252804253965287758870110\) \(\nu^{4}\mathstrut +\mathstrut \) \(475875085781293238416161348550588875486\) \(\nu^{3}\mathstrut +\mathstrut \) \(7950249227554230137213770363263282294842780178256\) \(\nu^{2}\mathstrut -\mathstrut \) \(1957759152680685241303120220577649384815732101892511821184119\) \(\nu\mathstrut -\mathstrut \) \(11611706082054896493621404036255849164015328090393618969591323160096586\)\()/\)\(15\!\cdots\!84\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(953142884334821514035\) \(\nu^{5}\mathstrut -\mathstrut \) \(20241043706177336077780274557350\) \(\nu^{4}\mathstrut +\mathstrut \) \(19932027967949467291060918084041415049731110\) \(\nu^{3}\mathstrut +\mathstrut \) \(806458021859872956575172714137865191966760473745390608\) \(\nu^{2}\mathstrut -\mathstrut \) \(93098418921216404490093459616804899517367291279071188076514925051\) \(\nu\mathstrut -\mathstrut \) \(3662537806324698741181082066453925140126893958666522820402964954045709229666\)\()/\)\(51\!\cdots\!28\)
\(\beta_{4}\)\(=\)\((\)\(118867839963294866167023935\) \(\nu^{5}\mathstrut -\mathstrut \) \(6814015439217244141373784371057939778\) \(\nu^{4}\mathstrut -\mathstrut \) \(1971384792947708823283594916646760911941847080766\) \(\nu^{3}\mathstrut +\mathstrut \) \(63270266943438541747634348557772923345895522617105490245552\) \(\nu^{2}\mathstrut +\mathstrut \) \(6944448255893472511332738518423316583886904157463057265498212816888103\) \(\nu\mathstrut -\mathstrut \) \(24000351072165621607619955337066809163978940700200563952518285812797048159259798\)\()/\)\(15\!\cdots\!84\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(2375127850411434314498077925\) \(\nu^{5}\mathstrut +\mathstrut \) \(39461873411226592448116965379963705206\) \(\nu^{4}\mathstrut -\mathstrut \) \(17368620528250195743857681677050138748710176452086\) \(\nu^{3}\mathstrut +\mathstrut \) \(1715292838829365272842058111406248163340155080442297900162160\) \(\nu^{2}\mathstrut +\mathstrut \) \(475862781610197766965562800541539116096742085485443815429540457128310691\) \(\nu\mathstrut -\mathstrut \) \(3343063685479791914131038125580564492229149207958059873547432068107463230805345198\)\()/\)\(12\!\cdots\!32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(48\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(125655\) \(\beta_{2}\mathstrut +\mathstrut \) \(2250185547597\) \(\beta_{1}\mathstrut +\mathstrut \) \(61824811714782289642833408\)\()/9216\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(11439\) \(\beta_{5}\mathstrut -\mathstrut \) \(1321486\) \(\beta_{4}\mathstrut +\mathstrut \) \(183077915148\) \(\beta_{3}\mathstrut -\mathstrut \) \(15580438670740550\) \(\beta_{2}\mathstrut +\mathstrut \) \(6429427758825316187342585\) \(\beta_{1}\mathstrut +\mathstrut \) \(8694831241615093241841504952721998848\)\()/55296\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(630076817944247\) \(\beta_{5}\mathstrut -\mathstrut \) \(985408156577169566\) \(\beta_{4}\mathstrut +\mathstrut \) \(77418924907811888405478\) \(\beta_{3}\mathstrut -\mathstrut \) \(14086238524349083425115530700\) \(\beta_{2}\mathstrut +\mathstrut \) \(303220444217040214330536411781777379\) \(\beta_{1}\mathstrut +\mathstrut \) \(4140605842488921785528448110491099156793449755648\)\()/55296\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(18819236431109095703923252\) \(\beta_{5}\mathstrut -\mathstrut \) \(559045802632022974435261608\) \(\beta_{4}\mathstrut +\mathstrut \) \(356719113396061454336223392223977\) \(\beta_{3}\mathstrut -\mathstrut \) \(55381908678362670245119615831084812807\) \(\beta_{2}\mathstrut +\mathstrut \) \(6931229720847429771105310354580088090836717729\) \(\beta_{1}\mathstrut +\mathstrut \) \(16273044360112309560789075256857500126070557690400454466304\)\()/4608\)

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
−1.03394e11
−7.73032e10
−3.27167e10
6.50618e9
8.03621e10
1.26545e11
−1.05253e13 −2.03014e20 7.20957e25 −7.53919e29 2.13678e33 −3.48209e35 −3.51650e38 5.29713e39 7.93520e42
1.2 −8.02060e12 1.02906e20 2.56443e25 8.73356e29 −8.25371e32 1.26096e35 1.04599e38 −2.53278e40 −7.00484e42
1.3 −3.74029e12 2.81970e20 −2.46958e25 −8.27734e29 −1.05465e33 1.89091e35 2.37065e38 4.35893e40 3.09596e42
1.4 2.51077e10 −1.85192e20 −3.86850e25 5.79783e28 −4.64974e30 −1.94193e31 −1.94260e36 −1.62151e39 1.45570e39
1.5 7.11528e12 1.40385e20 1.19416e25 1.39389e29 9.98881e32 −1.69824e35 −1.90292e38 −1.62095e40 9.91791e41
1.6 1.15489e13 −2.95630e20 9.46906e25 −4.26442e29 −3.41419e33 5.79630e35 6.46794e38 5.14795e40 −4.92492e42
\(n\): 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 \(S_{86}^{\mathrm{new}}(\Gamma_0(1))\).