Properties

Label 1.64.a.a
Level 1
Weight 64
Character orbit 1.a
Self dual Yes
Analytic conductor 25.136
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 64 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(25.1360966918\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{37}\cdot 3^{17}\cdot 5^{3}\cdot 7^{2}\cdot 13 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(101463019 - \beta_{1}) q^{2}\) \(+(190649070219572 - 18394 \beta_{1} - \beta_{2}) q^{3}\) \(+(1354584576291079283 - 215090888 \beta_{1} + 669 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(10\!\cdots\!40\)\( - 659683719897 \beta_{1} + 145059 \beta_{2} + 245 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(21\!\cdots\!92\)\( - 114690955086372 \beta_{1} + 99399096 \beta_{2} + 192264 \beta_{3} + 120 \beta_{4}) q^{6}\) \(+(\)\(75\!\cdots\!48\)\( + 1747180379931712 \beta_{1} - 9692344086 \beta_{2} + 43833100 \beta_{3} - 19620 \beta_{4}) q^{7}\) \(+(\)\(14\!\cdots\!52\)\( - 881208717933160576 \beta_{1} - 12849034917768 \beta_{2} - 315813800 \beta_{3} + 1165760 \beta_{4}) q^{8}\) \(+(\)\(12\!\cdots\!77\)\( + 17846178675598245834 \beta_{1} - 1133383860802062 \beta_{2} - 72422061858 \beta_{3} - 42220890 \beta_{4}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(101463019 - \beta_{1}) q^{2}\) \(+(190649070219572 - 18394 \beta_{1} - \beta_{2}) q^{3}\) \(+(1354584576291079283 - 215090888 \beta_{1} + 669 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(10\!\cdots\!40\)\( - 659683719897 \beta_{1} + 145059 \beta_{2} + 245 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(21\!\cdots\!92\)\( - 114690955086372 \beta_{1} + 99399096 \beta_{2} + 192264 \beta_{3} + 120 \beta_{4}) q^{6}\) \(+(\)\(75\!\cdots\!48\)\( + 1747180379931712 \beta_{1} - 9692344086 \beta_{2} + 43833100 \beta_{3} - 19620 \beta_{4}) q^{7}\) \(+(\)\(14\!\cdots\!52\)\( - 881208717933160576 \beta_{1} - 12849034917768 \beta_{2} - 315813800 \beta_{3} + 1165760 \beta_{4}) q^{8}\) \(+(\)\(12\!\cdots\!77\)\( + 17846178675598245834 \beta_{1} - 1133383860802062 \beta_{2} - 72422061858 \beta_{3} - 42220890 \beta_{4}) q^{9}\) \(+(\)\(69\!\cdots\!10\)\( - \)\(21\!\cdots\!62\)\( \beta_{1} - 18004502499599136 \beta_{2} + 2663271065120 \beta_{3} + 1081776096 \beta_{4}) q^{10}\) \(+(-\)\(10\!\cdots\!68\)\( - \)\(10\!\cdots\!70\)\( \beta_{1} + 307238139008440365 \beta_{2} - 42229248861800 \beta_{3} - 21038877640 \beta_{4}) q^{11}\) \(+(-\)\(52\!\cdots\!00\)\( - \)\(17\!\cdots\!44\)\( \beta_{1} + 4972868297144221436 \beta_{2} + 273344406086700 \beta_{3} + 322909148160 \beta_{4}) q^{12}\) \(+(\)\(21\!\cdots\!88\)\( - \)\(34\!\cdots\!77\)\( \beta_{1} - 71084452195560207165 \beta_{2} + 1862677836522325 \beta_{3} - 3996043453215 \beta_{4}) q^{13}\) \(+(-\)\(10\!\cdots\!24\)\( - \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(26\!\cdots\!88\)\( \beta_{2} - 60032646663528752 \beta_{3} + 40204492977200 \beta_{4}) q^{14}\) \(+(-\)\(69\!\cdots\!80\)\( - \)\(22\!\cdots\!44\)\( \beta_{1} + \)\(74\!\cdots\!18\)\( \beta_{2} + 639401304412529940 \beta_{3} - 326717880482748 \beta_{4}) q^{15}\) \(+(-\)\(30\!\cdots\!64\)\( + \)\(41\!\cdots\!56\)\( \beta_{1} - \)\(18\!\cdots\!68\)\( \beta_{2} - 3493655345864697792 \beta_{3} + 2073868362385920 \beta_{4}) q^{16}\) \(+(\)\(46\!\cdots\!66\)\( + \)\(14\!\cdots\!98\)\( \beta_{1} - \)\(27\!\cdots\!78\)\( \beta_{2} + 2768074475558717150 \beta_{3} - 9153503316227930 \beta_{4}) q^{17}\) \(+(-\)\(17\!\cdots\!53\)\( + \)\(59\!\cdots\!11\)\( \beta_{1} + \)\(16\!\cdots\!84\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} + 12245970229980480 \beta_{4}) q^{18}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(18\!\cdots\!82\)\( \beta_{1} + \)\(28\!\cdots\!31\)\( \beta_{2} - \)\(10\!\cdots\!56\)\( \beta_{3} + 237205341825686280 \beta_{4}) q^{19}\) \(+(\)\(23\!\cdots\!30\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} - \)\(49\!\cdots\!78\)\( \beta_{2} + \)\(41\!\cdots\!10\)\( \beta_{3} - 2885463764230254592 \beta_{4}) q^{20}\) \(+(\)\(25\!\cdots\!12\)\( - \)\(39\!\cdots\!24\)\( \beta_{1} + \)\(51\!\cdots\!12\)\( \beta_{2} - \)\(21\!\cdots\!52\)\( \beta_{3} + 20454693249964733100 \beta_{4}) q^{21}\) \(+(\)\(10\!\cdots\!08\)\( + \)\(44\!\cdots\!48\)\( \beta_{1} + \)\(84\!\cdots\!40\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!40\)\( \beta_{4}) q^{22}\) \(+(\)\(31\!\cdots\!44\)\( + \)\(99\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!94\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3} + \)\(41\!\cdots\!00\)\( \beta_{4}) q^{23}\) \(+(\)\(16\!\cdots\!00\)\( - \)\(13\!\cdots\!56\)\( \beta_{1} - \)\(86\!\cdots\!32\)\( \beta_{2} - \)\(33\!\cdots\!08\)\( \beta_{3} - \)\(11\!\cdots\!20\)\( \beta_{4}) q^{24}\) \(+(\)\(58\!\cdots\!75\)\( - \)\(21\!\cdots\!40\)\( \beta_{1} + \)\(45\!\cdots\!80\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!20\)\( \beta_{4}) q^{25}\) \(+(\)\(36\!\cdots\!02\)\( - \)\(36\!\cdots\!86\)\( \beta_{1} + \)\(66\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!32\)\( \beta_{3} + \)\(76\!\cdots\!60\)\( \beta_{4}) q^{26}\) \(+(\)\(11\!\cdots\!08\)\( + \)\(24\!\cdots\!96\)\( \beta_{1} - \)\(70\!\cdots\!22\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(51\!\cdots\!60\)\( \beta_{4}) q^{27}\) \(+(\)\(41\!\cdots\!56\)\( + \)\(48\!\cdots\!80\)\( \beta_{1} + \)\(57\!\cdots\!24\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!40\)\( \beta_{4}) q^{28}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(36\!\cdots\!73\)\( \beta_{1} + \)\(43\!\cdots\!79\)\( \beta_{2} - \)\(89\!\cdots\!19\)\( \beta_{3} - \)\(27\!\cdots\!15\)\( \beta_{4}) q^{29}\) \(+(\)\(23\!\cdots\!20\)\( - \)\(63\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!72\)\( \beta_{2} + \)\(15\!\cdots\!40\)\( \beta_{3} - \)\(35\!\cdots\!08\)\( \beta_{4}) q^{30}\) \(+(\)\(31\!\cdots\!92\)\( - \)\(94\!\cdots\!60\)\( \beta_{1} - \)\(55\!\cdots\!80\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(36\!\cdots\!80\)\( \beta_{4}) q^{31}\) \(+(-\)\(57\!\cdots\!16\)\( + \)\(43\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!80\)\( \beta_{2} + \)\(77\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4}) q^{32}\) \(+(-\)\(39\!\cdots\!96\)\( + \)\(10\!\cdots\!02\)\( \beta_{1} + \)\(43\!\cdots\!98\)\( \beta_{2} + \)\(81\!\cdots\!50\)\( \beta_{3} + \)\(13\!\cdots\!70\)\( \beta_{4}) q^{33}\) \(+(-\)\(15\!\cdots\!54\)\( - \)\(32\!\cdots\!46\)\( \beta_{1} + \)\(28\!\cdots\!88\)\( \beta_{2} - \)\(12\!\cdots\!28\)\( \beta_{3} + \)\(28\!\cdots\!80\)\( \beta_{4}) q^{34}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(78\!\cdots\!32\)\( \beta_{1} - \)\(29\!\cdots\!96\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(17\!\cdots\!44\)\( \beta_{4}) q^{35}\) \(+(-\)\(74\!\cdots\!09\)\( - \)\(10\!\cdots\!84\)\( \beta_{1} + \)\(81\!\cdots\!77\)\( \beta_{2} - \)\(23\!\cdots\!87\)\( \beta_{3} + \)\(34\!\cdots\!20\)\( \beta_{4}) q^{36}\) \(+(-\)\(34\!\cdots\!08\)\( + \)\(26\!\cdots\!15\)\( \beta_{1} - \)\(55\!\cdots\!77\)\( \beta_{2} - \)\(94\!\cdots\!75\)\( \beta_{3} - \)\(52\!\cdots\!95\)\( \beta_{4}) q^{37}\) \(+(\)\(19\!\cdots\!88\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(10\!\cdots\!28\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!80\)\( \beta_{4}) q^{38}\) \(+(\)\(98\!\cdots\!24\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(94\!\cdots\!78\)\( \beta_{2} - \)\(23\!\cdots\!72\)\( \beta_{3} + \)\(34\!\cdots\!60\)\( \beta_{4}) q^{39}\) \(+(\)\(17\!\cdots\!00\)\( - \)\(39\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!00\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{40}\) \(+(\)\(31\!\cdots\!22\)\( - \)\(71\!\cdots\!60\)\( \beta_{1} + \)\(73\!\cdots\!20\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3} - \)\(84\!\cdots\!20\)\( \beta_{4}) q^{41}\) \(+(\)\(41\!\cdots\!24\)\( - \)\(16\!\cdots\!60\)\( \beta_{1} - \)\(26\!\cdots\!64\)\( \beta_{2} + \)\(74\!\cdots\!00\)\( \beta_{3} + \)\(73\!\cdots\!80\)\( \beta_{4}) q^{42}\) \(+(-\)\(59\!\cdots\!24\)\( + \)\(48\!\cdots\!34\)\( \beta_{1} - \)\(60\!\cdots\!27\)\( \beta_{2} + \)\(51\!\cdots\!00\)\( \beta_{3} + \)\(41\!\cdots\!00\)\( \beta_{4}) q^{43}\) \(+(-\)\(37\!\cdots\!44\)\( + \)\(58\!\cdots\!24\)\( \beta_{1} - \)\(68\!\cdots\!72\)\( \beta_{2} - \)\(40\!\cdots\!68\)\( \beta_{3} - \)\(78\!\cdots\!20\)\( \beta_{4}) q^{44}\) \(+(-\)\(85\!\cdots\!80\)\( - \)\(15\!\cdots\!69\)\( \beta_{1} + \)\(91\!\cdots\!43\)\( \beta_{2} + \)\(68\!\cdots\!65\)\( \beta_{3} - \)\(21\!\cdots\!23\)\( \beta_{4}) q^{45}\) \(+(-\)\(10\!\cdots\!88\)\( - \)\(60\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4}) q^{46}\) \(+(-\)\(97\!\cdots\!00\)\( + \)\(29\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!84\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{47}\) \(+(\)\(21\!\cdots\!84\)\( + \)\(30\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!24\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(32\!\cdots\!80\)\( \beta_{4}) q^{48}\) \(+(\)\(67\!\cdots\!93\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(88\!\cdots\!60\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} + \)\(12\!\cdots\!80\)\( \beta_{4}) q^{49}\) \(+(\)\(22\!\cdots\!25\)\( - \)\(14\!\cdots\!15\)\( \beta_{1} + \)\(67\!\cdots\!80\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4}) q^{50}\) \(+(\)\(31\!\cdots\!52\)\( + \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(36\!\cdots\!26\)\( \beta_{2} - \)\(52\!\cdots\!24\)\( \beta_{3} - \)\(24\!\cdots\!80\)\( \beta_{4}) q^{51}\) \(+(\)\(22\!\cdots\!98\)\( - \)\(38\!\cdots\!88\)\( \beta_{1} + \)\(69\!\cdots\!34\)\( \beta_{2} + \)\(82\!\cdots\!50\)\( \beta_{3} + \)\(81\!\cdots\!00\)\( \beta_{4}) q^{52}\) \(+(-\)\(13\!\cdots\!48\)\( + \)\(24\!\cdots\!71\)\( \beta_{1} - \)\(57\!\cdots\!81\)\( \beta_{2} + \)\(15\!\cdots\!25\)\( \beta_{3} - \)\(61\!\cdots\!15\)\( \beta_{4}) q^{53}\) \(+(-\)\(24\!\cdots\!00\)\( - \)\(58\!\cdots\!92\)\( \beta_{1} + \)\(22\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{54}\) \(+(-\)\(41\!\cdots\!80\)\( + \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(23\!\cdots\!62\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} + \)\(23\!\cdots\!32\)\( \beta_{4}) q^{55}\) \(+(-\)\(46\!\cdots\!00\)\( - \)\(17\!\cdots\!08\)\( \beta_{1} - \)\(31\!\cdots\!16\)\( \beta_{2} + \)\(24\!\cdots\!76\)\( \beta_{3} - \)\(46\!\cdots\!40\)\( \beta_{4}) q^{56}\) \(+(-\)\(34\!\cdots\!44\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} + \)\(29\!\cdots\!26\)\( \beta_{2} + \)\(48\!\cdots\!50\)\( \beta_{3} - \)\(46\!\cdots\!50\)\( \beta_{4}) q^{57}\) \(+(\)\(48\!\cdots\!62\)\( - \)\(97\!\cdots\!26\)\( \beta_{1} + \)\(52\!\cdots\!32\)\( \beta_{2} - \)\(93\!\cdots\!00\)\( \beta_{3} - \)\(84\!\cdots\!80\)\( \beta_{4}) q^{58}\) \(+(\)\(21\!\cdots\!00\)\( + \)\(55\!\cdots\!74\)\( \beta_{1} + \)\(31\!\cdots\!13\)\( \beta_{2} + \)\(29\!\cdots\!52\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4}) q^{59}\) \(+(\)\(69\!\cdots\!60\)\( - \)\(36\!\cdots\!52\)\( \beta_{1} - \)\(58\!\cdots\!56\)\( \beta_{2} + \)\(78\!\cdots\!20\)\( \beta_{3} + \)\(30\!\cdots\!16\)\( \beta_{4}) q^{60}\) \(+(\)\(78\!\cdots\!32\)\( + \)\(46\!\cdots\!75\)\( \beta_{1} - \)\(12\!\cdots\!25\)\( \beta_{2} - \)\(61\!\cdots\!75\)\( \beta_{3} - \)\(10\!\cdots\!75\)\( \beta_{4}) q^{61}\) \(+(\)\(10\!\cdots\!48\)\( - \)\(87\!\cdots\!52\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(67\!\cdots\!80\)\( \beta_{4}) q^{62}\) \(+(-\)\(13\!\cdots\!80\)\( - \)\(41\!\cdots\!68\)\( \beta_{1} + \)\(14\!\cdots\!22\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!20\)\( \beta_{4}) q^{63}\) \(+(-\)\(43\!\cdots\!12\)\( - \)\(53\!\cdots\!56\)\( \beta_{1} + \)\(19\!\cdots\!08\)\( \beta_{2} - \)\(61\!\cdots\!28\)\( \beta_{3} - \)\(34\!\cdots\!40\)\( \beta_{4}) q^{64}\) \(+(-\)\(35\!\cdots\!80\)\( - \)\(21\!\cdots\!64\)\( \beta_{1} - \)\(41\!\cdots\!92\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4}) q^{65}\) \(+(-\)\(11\!\cdots\!56\)\( + \)\(33\!\cdots\!56\)\( \beta_{1} - \)\(38\!\cdots\!48\)\( \beta_{2} - \)\(83\!\cdots\!52\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{66}\) \(+(-\)\(95\!\cdots\!44\)\( + \)\(77\!\cdots\!78\)\( \beta_{1} - \)\(67\!\cdots\!13\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(60\!\cdots\!20\)\( \beta_{4}) q^{67}\) \(+(-\)\(24\!\cdots\!10\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(36\!\cdots\!98\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} - \)\(47\!\cdots\!60\)\( \beta_{4}) q^{68}\) \(+(\)\(34\!\cdots\!44\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(12\!\cdots\!96\)\( \beta_{3} - \)\(33\!\cdots\!80\)\( \beta_{4}) q^{69}\) \(+(\)\(80\!\cdots\!60\)\( - \)\(69\!\cdots\!72\)\( \beta_{1} - \)\(96\!\cdots\!16\)\( \beta_{2} + \)\(79\!\cdots\!20\)\( \beta_{3} + \)\(64\!\cdots\!76\)\( \beta_{4}) q^{70}\) \(+(\)\(10\!\cdots\!12\)\( - \)\(44\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!50\)\( \beta_{2} - \)\(92\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{71}\) \(+(\)\(12\!\cdots\!04\)\( + \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!76\)\( \beta_{2} - \)\(59\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4}) q^{72}\) \(+(-\)\(59\!\cdots\!26\)\( + \)\(28\!\cdots\!30\)\( \beta_{1} + \)\(39\!\cdots\!66\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} + \)\(92\!\cdots\!70\)\( \beta_{4}) q^{73}\) \(+(-\)\(28\!\cdots\!14\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} - \)\(55\!\cdots\!80\)\( \beta_{4}) q^{74}\) \(+(-\)\(40\!\cdots\!00\)\( + \)\(30\!\cdots\!70\)\( \beta_{1} - \)\(37\!\cdots\!15\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3} + \)\(25\!\cdots\!40\)\( \beta_{4}) q^{75}\) \(+(-\)\(91\!\cdots\!00\)\( - \)\(21\!\cdots\!16\)\( \beta_{1} - \)\(41\!\cdots\!32\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!80\)\( \beta_{4}) q^{76}\) \(+(-\)\(11\!\cdots\!64\)\( - \)\(60\!\cdots\!36\)\( \beta_{1} + \)\(13\!\cdots\!88\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(79\!\cdots\!80\)\( \beta_{4}) q^{77}\) \(+(-\)\(44\!\cdots\!88\)\( - \)\(88\!\cdots\!72\)\( \beta_{1} - \)\(41\!\cdots\!64\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!40\)\( \beta_{4}) q^{78}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(89\!\cdots\!32\)\( \beta_{1} + \)\(23\!\cdots\!24\)\( \beta_{2} + \)\(27\!\cdots\!16\)\( \beta_{3} - \)\(25\!\cdots\!20\)\( \beta_{4}) q^{79}\) \(+(\)\(21\!\cdots\!60\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(49\!\cdots\!24\)\( \beta_{2} - \)\(31\!\cdots\!80\)\( \beta_{3} + \)\(33\!\cdots\!36\)\( \beta_{4}) q^{80}\) \(+(\)\(90\!\cdots\!21\)\( + \)\(29\!\cdots\!58\)\( \beta_{1} - \)\(16\!\cdots\!54\)\( \beta_{2} - \)\(63\!\cdots\!66\)\( \beta_{3} - \)\(20\!\cdots\!50\)\( \beta_{4}) q^{81}\) \(+(\)\(78\!\cdots\!18\)\( - \)\(11\!\cdots\!82\)\( \beta_{1} + \)\(48\!\cdots\!20\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3} - \)\(45\!\cdots\!20\)\( \beta_{4}) q^{82}\) \(+(\)\(55\!\cdots\!20\)\( - \)\(17\!\cdots\!58\)\( \beta_{1} + \)\(10\!\cdots\!87\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(-\)\(25\!\cdots\!04\)\( - \)\(68\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!36\)\( \beta_{2} + \)\(66\!\cdots\!96\)\( \beta_{3} - \)\(58\!\cdots\!40\)\( \beta_{4}) q^{84}\) \(+(-\)\(24\!\cdots\!40\)\( - \)\(22\!\cdots\!22\)\( \beta_{1} + \)\(40\!\cdots\!34\)\( \beta_{2} - \)\(14\!\cdots\!30\)\( \beta_{3} - \)\(27\!\cdots\!74\)\( \beta_{4}) q^{85}\) \(+(-\)\(51\!\cdots\!68\)\( + \)\(65\!\cdots\!92\)\( \beta_{1} - \)\(96\!\cdots\!76\)\( \beta_{2} - \)\(29\!\cdots\!44\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4}) q^{86}\) \(+(-\)\(33\!\cdots\!56\)\( + \)\(30\!\cdots\!08\)\( \beta_{1} - \)\(80\!\cdots\!06\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(34\!\cdots\!80\)\( \beta_{4}) q^{87}\) \(+(-\)\(75\!\cdots\!36\)\( + \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!76\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} + \)\(49\!\cdots\!20\)\( \beta_{4}) q^{88}\) \(+(\)\(65\!\cdots\!50\)\( - \)\(79\!\cdots\!54\)\( \beta_{1} + \)\(16\!\cdots\!42\)\( \beta_{2} + \)\(13\!\cdots\!38\)\( \beta_{3} - \)\(17\!\cdots\!70\)\( \beta_{4}) q^{89}\) \(+(\)\(79\!\cdots\!70\)\( + \)\(19\!\cdots\!26\)\( \beta_{1} - \)\(62\!\cdots\!72\)\( \beta_{2} - \)\(21\!\cdots\!60\)\( \beta_{3} - \)\(41\!\cdots\!08\)\( \beta_{4}) q^{90}\) \(+(\)\(21\!\cdots\!72\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} + \)\(79\!\cdots\!44\)\( \beta_{2} - \)\(30\!\cdots\!84\)\( \beta_{3} + \)\(40\!\cdots\!60\)\( \beta_{4}) q^{91}\) \(+(\)\(33\!\cdots\!16\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} + \)\(93\!\cdots\!12\)\( \beta_{2} + \)\(68\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4}) q^{92}\) \(+(\)\(75\!\cdots\!24\)\( - \)\(31\!\cdots\!68\)\( \beta_{1} - \)\(74\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!40\)\( \beta_{4}) q^{93}\) \(+(-\)\(40\!\cdots\!44\)\( + \)\(19\!\cdots\!48\)\( \beta_{1} + \)\(31\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!16\)\( \beta_{3} - \)\(17\!\cdots\!20\)\( \beta_{4}) q^{94}\) \(+(\)\(27\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} + \)\(32\!\cdots\!50\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(71\!\cdots\!00\)\( \beta_{4}) q^{95}\) \(+(-\)\(18\!\cdots\!88\)\( + \)\(99\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!96\)\( \beta_{3} + \)\(80\!\cdots\!60\)\( \beta_{4}) q^{96}\) \(+(-\)\(34\!\cdots\!10\)\( + \)\(37\!\cdots\!46\)\( \beta_{1} - \)\(27\!\cdots\!34\)\( \beta_{2} + \)\(13\!\cdots\!50\)\( \beta_{3} - \)\(21\!\cdots\!10\)\( \beta_{4}) q^{97}\) \(+(-\)\(38\!\cdots\!13\)\( - \)\(99\!\cdots\!13\)\( \beta_{1} - \)\(26\!\cdots\!60\)\( \beta_{2} - \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(33\!\cdots\!80\)\( \beta_{4}) q^{98}\) \(+(-\)\(25\!\cdots\!36\)\( - \)\(78\!\cdots\!02\)\( \beta_{1} + \)\(18\!\cdots\!21\)\( \beta_{2} + \)\(39\!\cdots\!44\)\( \beta_{3} + \)\(94\!\cdots\!40\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut +\mathstrut 507315096q^{2} \) \(\mathstrut +\mathstrut 953245351116252q^{3} \) \(\mathstrut +\mathstrut 6772922881670488640q^{4} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!30\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!56\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!85\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 507315096q^{2} \) \(\mathstrut +\mathstrut 953245351116252q^{3} \) \(\mathstrut +\mathstrut 6772922881670488640q^{4} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!30\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!56\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!85\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!20\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!40\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!84\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!62\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!20\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!60\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!26\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!08\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!72\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!48\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!50\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!60\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!44\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!36\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!20\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!20\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!34\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!20\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!10\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!52\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!08\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!10\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!40\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!64\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!65\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!60\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!96\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!98\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!20\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!10\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!32\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!88\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!60\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!60\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!80\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!24\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!92\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!20\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!60\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!78\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!08\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!96\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!05\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!12\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!32\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!40\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!50\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!40\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!76\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!84\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!20\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!40\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!14\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!72\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(5096287552528786\) \(x^{3}\mathstrut +\mathstrut \) \(574038763744383494840\) \(x^{2}\mathstrut +\mathstrut \) \(3502610791787684740809332695881\) \(x\mathstrut -\mathstrut \) \(35880030333954415007358004861309901934\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 29 \)
\(\beta_{2}\)\(=\)\((\)\(98361\) \(\nu^{4}\mathstrut +\mathstrut \) \(1875869661375\) \(\nu^{3}\mathstrut -\mathstrut \) \(478485724569513183099\) \(\nu^{2}\mathstrut -\mathstrut \) \(8652395815905765079896446187\) \(\nu\mathstrut +\mathstrut \) \(229804128899029201484728877488049518\)\()/\)\(24333542524587606016\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(65803509\) \(\nu^{4}\mathstrut -\mathstrut \) \(1254956803459875\) \(\nu^{3}\mathstrut +\mathstrut \) \(446252034184466469080175\) \(\nu^{2}\mathstrut +\mathstrut \) \(5809765699648259782584028883487\) \(\nu\mathstrut -\mathstrut \) \(410887611714893351947779231248849410166\)\()/\)\(24333542524587606016\)
\(\beta_{4}\)\(=\)\((\)\(2665776547737\) \(\nu^{4}\mathstrut +\mathstrut \) \(31362235844074290783\) \(\nu^{3}\mathstrut -\mathstrut \) \(12800126178842342648877747099\) \(\nu^{2}\mathstrut -\mathstrut \) \(161978831504363254446653476179804939\) \(\nu\mathstrut +\mathstrut \) \(5879417456202971889545707633040694674748078\)\()/\)\(60833856311469015040\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(29\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(669\) \(\beta_{2}\mathstrut -\mathstrut \) \(12164792\) \(\beta_{1}\mathstrut +\mathstrut \) \(10567661868921261571\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(18215\) \(\beta_{4}\mathstrut +\mathstrut \) \(9690671\) \(\beta_{3}\mathstrut +\mathstrut \) \(203947988361\) \(\beta_{2}\mathstrut +\mathstrut \) \(301458839229539738\) \(\beta_{1}\mathstrut -\mathstrut \) \(2008640715943852839028117\)\()/5832\)
\(\nu^{4}\)\(=\)\((\)\(2779066165000\) \(\beta_{4}\mathstrut +\mathstrut \) \(42302782965885731\) \(\beta_{3}\mathstrut +\mathstrut \) \(9715533155837233575\) \(\beta_{2}\mathstrut +\mathstrut \) \(10475558906127233571198464\) \(\beta_{1}\mathstrut +\mathstrut \) \(353968345455287156763638000502126737\)\()/46656\)

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
6.64596e7
1.73795e7
1.42670e7
−3.39250e7
−6.41810e7
−4.68363e9 −3.45751e14 1.27130e19 −3.68153e21 1.61937e24 6.76929e26 −1.63440e28 −1.02502e30 1.72429e31
1.2 −1.14986e9 2.06926e15 −7.90120e18 −9.56939e21 −2.37936e24 −1.15073e26 1.96908e28 3.13728e30 1.10034e31
1.3 −9.25761e8 −5.88048e14 −8.36634e18 1.60235e22 5.44392e23 −7.44490e26 1.62839e28 −7.98761e29 −1.48340e31
1.4 2.54406e9 −9.84533e14 −2.75111e18 −1.69176e22 −2.50471e24 1.59356e26 −3.04638e28 −1.75256e29 −4.30395e31
1.5 4.72250e9 8.02316e14 1.30786e19 1.36438e22 3.78894e24 4.00095e26 1.82063e28 −5.00850e29 6.44329e31
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{64}^{\mathrm{new}}(\Gamma_0(1))\).