Properties

Label 9.64.a.c
Level $9$
Weight $64$
Character orbit 9.a
Self dual yes
Analytic conductor $226.225$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 64 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(226.224870226\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5096287552528786 x^{3} + 574038763744383494840 x^{2} + 3502610791787684740809332695881 x - 35880030333954415007358004861309901934\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{37}\cdot 3^{22}\cdot 5^{3}\cdot 7^{2}\cdot 13 \)
Twist minimal: no (minimal twist has level 1)
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.

\(f(q)\) \(=\) \( q +(-101463019 + \beta_{1}) q^{2} +(1354584576291079553 - 215090880 \beta_{1} + 3 \beta_{2} + \beta_{3}) q^{4} +(\)\(10\!\cdots\!91\)\( + 659683718170 \beta_{1} - 657 \beta_{2} - 245 \beta_{3} + \beta_{4}) q^{5} +(\)\(75\!\cdots\!52\)\( + 1747180248915546 \beta_{1} - 29054482 \beta_{2} + 43833100 \beta_{3} + 19620 \beta_{4}) q^{7} +(-\)\(14\!\cdots\!04\)\( + 881208876481732224 \beta_{1} + 52955183736 \beta_{2} + 315813800 \beta_{3} + 1165760 \beta_{4}) q^{8} +O(q^{10})\) \( q +(-101463019 + \beta_{1}) q^{2} +(1354584576291079553 - 215090880 \beta_{1} + 3 \beta_{2} + \beta_{3}) q^{4} +(\)\(10\!\cdots\!91\)\( + 659683718170 \beta_{1} - 657 \beta_{2} - 245 \beta_{3} + \beta_{4}) q^{5} +(\)\(75\!\cdots\!52\)\( + 1747180248915546 \beta_{1} - 29054482 \beta_{2} + 43833100 \beta_{3} + 19620 \beta_{4}) q^{7} +(-\)\(14\!\cdots\!04\)\( + 881208876481732224 \beta_{1} + 52955183736 \beta_{2} + 315813800 \beta_{3} + 1165760 \beta_{4}) q^{8} +(\)\(69\!\cdots\!14\)\( - \)\(21\!\cdots\!70\)\( \beta_{1} - 73435485874528 \beta_{2} + 2663271065120 \beta_{3} - 1081776096 \beta_{4}) q^{10} +(\)\(10\!\cdots\!78\)\( + \)\(10\!\cdots\!25\)\( \beta_{1} - 1253936857100895 \beta_{2} + 42229248861800 \beta_{3} - 21038877640 \beta_{4}) q^{11} +(\)\(21\!\cdots\!73\)\( - \)\(34\!\cdots\!62\)\( \beta_{1} - 292066913385497615 \beta_{2} + 1862677836522325 \beta_{3} + 3996043453215 \beta_{4}) q^{13} +(\)\(10\!\cdots\!64\)\( + \)\(45\!\cdots\!40\)\( \beta_{1} + 1116929253406607856 \beta_{2} + 60032646663528752 \beta_{3} + 40204492977200 \beta_{4}) q^{14} +(-\)\(30\!\cdots\!24\)\( + \)\(41\!\cdots\!60\)\( \beta_{1} - 75642534947079523136 \beta_{2} - 3493655345864697792 \beta_{3} - 2073868362385920 \beta_{4}) q^{16} +(-\)\(46\!\cdots\!08\)\( - \)\(14\!\cdots\!20\)\( \beta_{1} + \)\(11\!\cdots\!66\)\( \beta_{2} - 2768074475558717150 \beta_{3} - 9153503316227930 \beta_{4}) q^{17} +(-\)\(15\!\cdots\!50\)\( - \)\(18\!\cdots\!95\)\( \beta_{1} + \)\(11\!\cdots\!17\)\( \beta_{2} - \)\(10\!\cdots\!56\)\( \beta_{3} - 237205341825686280 \beta_{4}) q^{19} +(-\)\(23\!\cdots\!22\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!94\)\( \beta_{2} - \)\(41\!\cdots\!10\)\( \beta_{3} - 2885463764230254592 \beta_{4}) q^{20} +(\)\(10\!\cdots\!48\)\( + \)\(44\!\cdots\!68\)\( \beta_{1} + \)\(34\!\cdots\!20\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!40\)\( \beta_{4}) q^{22} +(-\)\(31\!\cdots\!60\)\( - \)\(99\!\cdots\!86\)\( \beta_{1} + \)\(10\!\cdots\!58\)\( \beta_{2} - \)\(36\!\cdots\!00\)\( \beta_{3} + \)\(41\!\cdots\!00\)\( \beta_{4}) q^{23} +(\)\(58\!\cdots\!55\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(18\!\cdots\!40\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{25} +(-\)\(36\!\cdots\!82\)\( + \)\(36\!\cdots\!10\)\( \beta_{1} - \)\(28\!\cdots\!16\)\( \beta_{2} - \)\(37\!\cdots\!32\)\( \beta_{3} + \)\(76\!\cdots\!60\)\( \beta_{4}) q^{26} +(\)\(41\!\cdots\!20\)\( + \)\(48\!\cdots\!04\)\( \beta_{1} + \)\(24\!\cdots\!28\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!40\)\( \beta_{4}) q^{28} +(-\)\(10\!\cdots\!85\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(17\!\cdots\!13\)\( \beta_{2} + \)\(89\!\cdots\!19\)\( \beta_{3} - \)\(27\!\cdots\!15\)\( \beta_{4}) q^{29} +(\)\(31\!\cdots\!12\)\( - \)\(94\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!40\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!80\)\( \beta_{4}) q^{31} +(\)\(57\!\cdots\!36\)\( - \)\(43\!\cdots\!64\)\( \beta_{1} - \)\(54\!\cdots\!40\)\( \beta_{2} - \)\(77\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4}) q^{32} +(-\)\(15\!\cdots\!94\)\( - \)\(32\!\cdots\!10\)\( \beta_{1} + \)\(87\!\cdots\!76\)\( \beta_{2} - \)\(12\!\cdots\!28\)\( \beta_{3} - \)\(28\!\cdots\!80\)\( \beta_{4}) q^{34} +(\)\(21\!\cdots\!96\)\( + \)\(78\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!08\)\( \beta_{2} - \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(17\!\cdots\!44\)\( \beta_{4}) q^{35} +(-\)\(34\!\cdots\!55\)\( + \)\(26\!\cdots\!38\)\( \beta_{1} - \)\(23\!\cdots\!19\)\( \beta_{2} - \)\(94\!\cdots\!75\)\( \beta_{3} + \)\(52\!\cdots\!95\)\( \beta_{4}) q^{37} +(-\)\(19\!\cdots\!96\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} - \)\(45\!\cdots\!76\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!80\)\( \beta_{4}) q^{38} +(\)\(17\!\cdots\!00\)\( - \)\(39\!\cdots\!00\)\( \beta_{1} + \)\(72\!\cdots\!00\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{40} +(-\)\(31\!\cdots\!42\)\( + \)\(71\!\cdots\!00\)\( \beta_{1} - \)\(29\!\cdots\!60\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3} - \)\(84\!\cdots\!20\)\( \beta_{4}) q^{41} +(-\)\(59\!\cdots\!46\)\( + \)\(48\!\cdots\!67\)\( \beta_{1} - \)\(25\!\cdots\!89\)\( \beta_{2} + \)\(51\!\cdots\!00\)\( \beta_{3} - \)\(41\!\cdots\!00\)\( \beta_{4}) q^{43} +(\)\(37\!\cdots\!84\)\( - \)\(58\!\cdots\!40\)\( \beta_{1} + \)\(29\!\cdots\!44\)\( \beta_{2} + \)\(40\!\cdots\!68\)\( \beta_{3} - \)\(78\!\cdots\!20\)\( \beta_{4}) q^{44} +(-\)\(10\!\cdots\!68\)\( - \)\(60\!\cdots\!00\)\( \beta_{1} - \)\(47\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4}) q^{46} +(\)\(97\!\cdots\!24\)\( - \)\(29\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{47} +(\)\(67\!\cdots\!13\)\( + \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(36\!\cdots\!40\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4}) q^{49} +(-\)\(22\!\cdots\!05\)\( + \)\(14\!\cdots\!75\)\( \beta_{1} - \)\(28\!\cdots\!40\)\( \beta_{2} - \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4}) q^{50} +(\)\(22\!\cdots\!22\)\( - \)\(38\!\cdots\!24\)\( \beta_{1} + \)\(28\!\cdots\!38\)\( \beta_{2} + \)\(82\!\cdots\!50\)\( \beta_{3} - \)\(81\!\cdots\!00\)\( \beta_{4}) q^{52} +(\)\(13\!\cdots\!39\)\( - \)\(24\!\cdots\!50\)\( \beta_{1} + \)\(19\!\cdots\!27\)\( \beta_{2} - \)\(15\!\cdots\!25\)\( \beta_{3} - \)\(61\!\cdots\!15\)\( \beta_{4}) q^{53} +(-\)\(41\!\cdots\!12\)\( + \)\(19\!\cdots\!10\)\( \beta_{1} - \)\(95\!\cdots\!26\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} - \)\(23\!\cdots\!32\)\( \beta_{4}) q^{55} +(\)\(46\!\cdots\!40\)\( + \)\(17\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!52\)\( \beta_{2} - \)\(24\!\cdots\!76\)\( \beta_{3} - \)\(46\!\cdots\!40\)\( \beta_{4}) q^{56} +(\)\(48\!\cdots\!14\)\( - \)\(97\!\cdots\!94\)\( \beta_{1} + \)\(21\!\cdots\!04\)\( \beta_{2} - \)\(93\!\cdots\!00\)\( \beta_{3} + \)\(84\!\cdots\!80\)\( \beta_{4}) q^{58} +(-\)\(21\!\cdots\!90\)\( - \)\(55\!\cdots\!15\)\( \beta_{1} - \)\(13\!\cdots\!31\)\( \beta_{2} - \)\(29\!\cdots\!52\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4}) q^{59} +(\)\(78\!\cdots\!57\)\( + \)\(46\!\cdots\!50\)\( \beta_{1} - \)\(53\!\cdots\!75\)\( \beta_{2} - \)\(61\!\cdots\!75\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4}) q^{61} +(-\)\(10\!\cdots\!68\)\( + \)\(87\!\cdots\!92\)\( \beta_{1} + \)\(57\!\cdots\!40\)\( \beta_{2} - \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(67\!\cdots\!80\)\( \beta_{4}) q^{62} +(-\)\(43\!\cdots\!32\)\( - \)\(53\!\cdots\!60\)\( \beta_{1} + \)\(80\!\cdots\!36\)\( \beta_{2} - \)\(61\!\cdots\!28\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4}) q^{64} +(\)\(35\!\cdots\!92\)\( + \)\(21\!\cdots\!40\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4}) q^{65} +(-\)\(95\!\cdots\!62\)\( + \)\(77\!\cdots\!65\)\( \beta_{1} - \)\(27\!\cdots\!11\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} - \)\(60\!\cdots\!20\)\( \beta_{4}) q^{67} +(\)\(24\!\cdots\!82\)\( - \)\(12\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!46\)\( \beta_{2} - \)\(11\!\cdots\!50\)\( \beta_{3} - \)\(47\!\cdots\!60\)\( \beta_{4}) q^{68} +(\)\(80\!\cdots\!84\)\( - \)\(69\!\cdots\!20\)\( \beta_{1} - \)\(37\!\cdots\!68\)\( \beta_{2} + \)\(79\!\cdots\!20\)\( \beta_{3} - \)\(64\!\cdots\!76\)\( \beta_{4}) q^{70} +(-\)\(10\!\cdots\!12\)\( + \)\(44\!\cdots\!50\)\( \beta_{1} + \)\(94\!\cdots\!50\)\( \beta_{2} + \)\(92\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{71} +(-\)\(59\!\cdots\!00\)\( + \)\(28\!\cdots\!76\)\( \beta_{1} + \)\(16\!\cdots\!42\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} - \)\(92\!\cdots\!70\)\( \beta_{4}) q^{73} +(\)\(28\!\cdots\!34\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(47\!\cdots\!60\)\( \beta_{2} + \)\(14\!\cdots\!60\)\( \beta_{3} - \)\(55\!\cdots\!80\)\( \beta_{4}) q^{74} +(-\)\(91\!\cdots\!80\)\( - \)\(21\!\cdots\!60\)\( \beta_{1} - \)\(17\!\cdots\!04\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} + \)\(17\!\cdots\!80\)\( \beta_{4}) q^{76} +(\)\(11\!\cdots\!96\)\( + \)\(60\!\cdots\!28\)\( \beta_{1} - \)\(52\!\cdots\!96\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(79\!\cdots\!80\)\( \beta_{4}) q^{77} +(\)\(11\!\cdots\!40\)\( + \)\(89\!\cdots\!20\)\( \beta_{1} + \)\(99\!\cdots\!08\)\( \beta_{2} + \)\(27\!\cdots\!16\)\( \beta_{3} + \)\(25\!\cdots\!20\)\( \beta_{4}) q^{79} +(-\)\(21\!\cdots\!24\)\( - \)\(22\!\cdots\!80\)\( \beta_{1} - \)\(20\!\cdots\!52\)\( \beta_{2} + \)\(31\!\cdots\!80\)\( \beta_{3} + \)\(33\!\cdots\!36\)\( \beta_{4}) q^{80} +(\)\(78\!\cdots\!38\)\( - \)\(11\!\cdots\!22\)\( \beta_{1} + \)\(20\!\cdots\!60\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3} + \)\(45\!\cdots\!20\)\( \beta_{4}) q^{82} +(-\)\(55\!\cdots\!02\)\( + \)\(17\!\cdots\!31\)\( \beta_{1} - \)\(43\!\cdots\!09\)\( \beta_{2} - \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(24\!\cdots\!66\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!82\)\( \beta_{2} - \)\(14\!\cdots\!30\)\( \beta_{3} + \)\(27\!\cdots\!74\)\( \beta_{4}) q^{85} +(\)\(51\!\cdots\!88\)\( - \)\(65\!\cdots\!20\)\( \beta_{1} + \)\(40\!\cdots\!52\)\( \beta_{2} + \)\(29\!\cdots\!44\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4}) q^{86} +(-\)\(75\!\cdots\!72\)\( + \)\(32\!\cdots\!32\)\( \beta_{1} - \)\(70\!\cdots\!52\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(49\!\cdots\!20\)\( \beta_{4}) q^{88} +(-\)\(65\!\cdots\!80\)\( + \)\(79\!\cdots\!40\)\( \beta_{1} - \)\(66\!\cdots\!74\)\( \beta_{2} - \)\(13\!\cdots\!38\)\( \beta_{3} - \)\(17\!\cdots\!70\)\( \beta_{4}) q^{89} +(\)\(21\!\cdots\!32\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} + \)\(31\!\cdots\!68\)\( \beta_{2} - \)\(30\!\cdots\!84\)\( \beta_{3} - \)\(40\!\cdots\!60\)\( \beta_{4}) q^{91} +(-\)\(33\!\cdots\!48\)\( + \)\(19\!\cdots\!80\)\( \beta_{1} - \)\(40\!\cdots\!04\)\( \beta_{2} - \)\(68\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4}) q^{92} +(-\)\(40\!\cdots\!44\)\( + \)\(19\!\cdots\!80\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} - \)\(17\!\cdots\!16\)\( \beta_{3} + \)\(17\!\cdots\!20\)\( \beta_{4}) q^{94} +(-\)\(27\!\cdots\!00\)\( - \)\(23\!\cdots\!50\)\( \beta_{1} - \)\(12\!\cdots\!50\)\( \beta_{2} + \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(71\!\cdots\!00\)\( \beta_{4}) q^{95} +(-\)\(34\!\cdots\!84\)\( + \)\(37\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!78\)\( \beta_{2} + \)\(13\!\cdots\!50\)\( \beta_{3} + \)\(21\!\cdots\!10\)\( \beta_{4}) q^{97} +(\)\(38\!\cdots\!73\)\( + \)\(99\!\cdots\!33\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(33\!\cdots\!80\)\( \beta_{4}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 507315096q^{2} + 6772922881670488640q^{4} + \)\(50\!\cdots\!30\)\(q^{5} + \)\(37\!\cdots\!56\)\(q^{7} - \)\(73\!\cdots\!00\)\(q^{8} + O(q^{10}) \) \( 5q - 507315096q^{2} + 6772922881670488640q^{4} + \)\(50\!\cdots\!30\)\(q^{5} + \)\(37\!\cdots\!56\)\(q^{7} - \)\(73\!\cdots\!00\)\(q^{8} + \)\(34\!\cdots\!20\)\(q^{10} + \)\(54\!\cdots\!40\)\(q^{11} + \)\(10\!\cdots\!62\)\(q^{13} + \)\(54\!\cdots\!20\)\(q^{14} - \)\(15\!\cdots\!20\)\(q^{16} - \)\(23\!\cdots\!26\)\(q^{17} - \)\(78\!\cdots\!00\)\(q^{19} - \)\(11\!\cdots\!60\)\(q^{20} + \)\(52\!\cdots\!72\)\(q^{22} - \)\(15\!\cdots\!72\)\(q^{23} + \)\(29\!\cdots\!75\)\(q^{25} - \)\(18\!\cdots\!60\)\(q^{26} + \)\(20\!\cdots\!48\)\(q^{28} - \)\(50\!\cdots\!50\)\(q^{29} + \)\(15\!\cdots\!60\)\(q^{31} + \)\(28\!\cdots\!44\)\(q^{32} - \)\(77\!\cdots\!20\)\(q^{34} + \)\(10\!\cdots\!80\)\(q^{35} - \)\(17\!\cdots\!34\)\(q^{37} - \)\(96\!\cdots\!00\)\(q^{38} + \)\(89\!\cdots\!00\)\(q^{40} - \)\(15\!\cdots\!10\)\(q^{41} - \)\(29\!\cdots\!08\)\(q^{43} + \)\(18\!\cdots\!20\)\(q^{44} - \)\(51\!\cdots\!40\)\(q^{46} + \)\(48\!\cdots\!64\)\(q^{47} + \)\(33\!\cdots\!65\)\(q^{49} - \)\(11\!\cdots\!00\)\(q^{50} + \)\(11\!\cdots\!96\)\(q^{52} + \)\(69\!\cdots\!98\)\(q^{53} - \)\(20\!\cdots\!60\)\(q^{55} + \)\(23\!\cdots\!00\)\(q^{56} + \)\(24\!\cdots\!00\)\(q^{58} - \)\(10\!\cdots\!00\)\(q^{59} + \)\(39\!\cdots\!10\)\(q^{61} - \)\(51\!\cdots\!32\)\(q^{62} - \)\(21\!\cdots\!60\)\(q^{64} + \)\(17\!\cdots\!60\)\(q^{65} - \)\(47\!\cdots\!24\)\(q^{67} + \)\(12\!\cdots\!92\)\(q^{68} + \)\(40\!\cdots\!20\)\(q^{70} - \)\(50\!\cdots\!60\)\(q^{71} - \)\(29\!\cdots\!78\)\(q^{73} + \)\(14\!\cdots\!20\)\(q^{74} - \)\(45\!\cdots\!00\)\(q^{76} + \)\(58\!\cdots\!08\)\(q^{77} + \)\(58\!\cdots\!00\)\(q^{79} - \)\(10\!\cdots\!20\)\(q^{80} + \)\(39\!\cdots\!12\)\(q^{82} - \)\(27\!\cdots\!32\)\(q^{83} - \)\(12\!\cdots\!80\)\(q^{85} + \)\(25\!\cdots\!40\)\(q^{86} - \)\(37\!\cdots\!00\)\(q^{88} - \)\(32\!\cdots\!50\)\(q^{89} + \)\(10\!\cdots\!60\)\(q^{91} - \)\(16\!\cdots\!76\)\(q^{92} - \)\(20\!\cdots\!20\)\(q^{94} - \)\(13\!\cdots\!00\)\(q^{95} - \)\(17\!\cdots\!14\)\(q^{97} + \)\(19\!\cdots\!72\)\(q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 29 \)
\(\beta_{2}\)\(=\)\((\)\(23901723 \nu^{4} + 455836327714125 \nu^{3} - 116272031070391703493057 \nu^{2} - 2102532188521146099725759322897 \nu + 55842403322464095693120149459132366698\)\()/ 24333542524587606016 \)
\(\beta_{3}\)\(=\)\((\)\(-71705169 \nu^{4} - 1367508983142375 \nu^{3} + 474961177658637260066115 \nu^{2} + 6328909450354620749148123287859 \nu - 424675859448835104158530676521070411326\)\()/ 24333542524587606016 \)
\(\beta_{4}\)\(=\)\((\)\(-2665803105207 \nu^{4} - 31362742328882862033 \nu^{3} + 12800255369987976417437183829 \nu^{2} + 161981167651237929040879473989358309 \nu - 5879479503317774627429256835849255881907378\)\()/ 60833856311469015040 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 29\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} - 12164784 \beta_{1} + 10567661868921261841\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(18215 \beta_{4} + 9690671 \beta_{3} + 841692987 \beta_{2} + 301458841744909813 \beta_{1} - 2008640715943770586474740\)\()/5832\)
\(\nu^{4}\)\(=\)\((\)\(-25011595485000 \beta_{4} + 380725046692971579 \beta_{3} + 453829629431279905 \beta_{2} + 94280031135933955337139312 \beta_{1} + 3185715109097584446211720066759108427\)\()/419904\)

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
−6.41810e7
−3.39250e7
1.42670e7
1.73795e7
6.64596e7
−4.72250e9 0 1.30786e19 −1.36438e22 0 4.00095e26 −1.82063e28 0 6.44329e31
1.2 −2.54406e9 0 −2.75111e18 1.69176e22 0 1.59356e26 3.04638e28 0 −4.30395e31
1.3 9.25761e8 0 −8.36634e18 −1.60235e22 0 −7.44490e26 −1.62839e28 0 −1.48340e31
1.4 1.14986e9 0 −7.90120e18 9.56939e21 0 −1.15073e26 −1.96908e28 0 1.10034e31
1.5 4.68363e9 0 1.27130e19 3.68153e21 0 6.76929e26 1.63440e28 0 1.72429e31
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.64.a.c 5
3.b odd 2 1 1.64.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.64.a.a 5 3.b odd 2 1
9.64.a.c 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 507315096 T_{2}^{4} - \)\(26\!\cdots\!32\)\( T_{2}^{3} - \)\(78\!\cdots\!72\)\( T_{2}^{2} + \)\(93\!\cdots\!56\)\( T_{2} - \)\(59\!\cdots\!24\)\( \) acting on \(S_{64}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -\)\(59\!\cdots\!24\)\( + \)\(93\!\cdots\!56\)\( T - \)\(78\!\cdots\!72\)\( T^{2} - 26316207229657439232 T^{3} + 507315096 T^{4} + T^{5} \)
$3$ \( T^{5} \)
$5$ \( -\)\(13\!\cdots\!00\)\( + \)\(39\!\cdots\!00\)\( T + \)\(50\!\cdots\!00\)\( T^{2} - \)\(41\!\cdots\!00\)\( T^{3} - \)\(50\!\cdots\!30\)\( T^{4} + T^{5} \)
$7$ \( -\)\(36\!\cdots\!76\)\( + \)\(80\!\cdots\!96\)\( T + \)\(23\!\cdots\!32\)\( T^{2} - \)\(53\!\cdots\!72\)\( T^{3} - \)\(37\!\cdots\!56\)\( T^{4} + T^{5} \)
$11$ \( -\)\(11\!\cdots\!68\)\( - \)\(13\!\cdots\!20\)\( T + \)\(44\!\cdots\!80\)\( T^{2} - \)\(75\!\cdots\!60\)\( T^{3} - \)\(54\!\cdots\!40\)\( T^{4} + T^{5} \)
$13$ \( -\)\(46\!\cdots\!32\)\( + \)\(56\!\cdots\!36\)\( T + \)\(43\!\cdots\!56\)\( T^{2} - \)\(49\!\cdots\!88\)\( T^{3} - \)\(10\!\cdots\!62\)\( T^{4} + T^{5} \)
$17$ \( -\)\(31\!\cdots\!24\)\( + \)\(18\!\cdots\!76\)\( T - \)\(17\!\cdots\!52\)\( T^{2} - \)\(81\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!26\)\( T^{4} + T^{5} \)
$19$ \( \)\(28\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T - \)\(15\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(78\!\cdots\!00\)\( T^{4} + T^{5} \)
$23$ \( -\)\(19\!\cdots\!68\)\( - \)\(11\!\cdots\!44\)\( T - \)\(19\!\cdots\!96\)\( T^{2} - \)\(43\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!72\)\( T^{4} + T^{5} \)
$29$ \( \)\(48\!\cdots\!00\)\( + \)\(32\!\cdots\!00\)\( T + \)\(80\!\cdots\!00\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!50\)\( T^{4} + T^{5} \)
$31$ \( -\)\(48\!\cdots\!32\)\( + \)\(13\!\cdots\!80\)\( T + \)\(17\!\cdots\!20\)\( T^{2} - \)\(81\!\cdots\!60\)\( T^{3} - \)\(15\!\cdots\!60\)\( T^{4} + T^{5} \)
$37$ \( \)\(71\!\cdots\!24\)\( + \)\(16\!\cdots\!36\)\( T - \)\(51\!\cdots\!08\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!34\)\( T^{4} + T^{5} \)
$41$ \( -\)\(72\!\cdots\!68\)\( - \)\(49\!\cdots\!20\)\( T - \)\(47\!\cdots\!20\)\( T^{2} + \)\(65\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!10\)\( T^{4} + T^{5} \)
$43$ \( -\)\(13\!\cdots\!32\)\( + \)\(96\!\cdots\!96\)\( T - \)\(83\!\cdots\!24\)\( T^{2} - \)\(44\!\cdots\!28\)\( T^{3} + \)\(29\!\cdots\!08\)\( T^{4} + T^{5} \)
$47$ \( -\)\(18\!\cdots\!24\)\( - \)\(74\!\cdots\!84\)\( T + \)\(17\!\cdots\!88\)\( T^{2} + \)\(13\!\cdots\!08\)\( T^{3} - \)\(48\!\cdots\!64\)\( T^{4} + T^{5} \)
$53$ \( \)\(42\!\cdots\!32\)\( - \)\(26\!\cdots\!84\)\( T + \)\(93\!\cdots\!84\)\( T^{2} - \)\(83\!\cdots\!08\)\( T^{3} - \)\(69\!\cdots\!98\)\( T^{4} + T^{5} \)
$59$ \( \)\(31\!\cdots\!00\)\( - \)\(25\!\cdots\!00\)\( T - \)\(23\!\cdots\!00\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!00\)\( T^{4} + T^{5} \)
$61$ \( -\)\(11\!\cdots\!32\)\( - \)\(24\!\cdots\!20\)\( T + \)\(33\!\cdots\!20\)\( T^{2} - \)\(52\!\cdots\!60\)\( T^{3} - \)\(39\!\cdots\!10\)\( T^{4} + T^{5} \)
$67$ \( \)\(17\!\cdots\!24\)\( - \)\(74\!\cdots\!24\)\( T - \)\(16\!\cdots\!48\)\( T^{2} + \)\(14\!\cdots\!48\)\( T^{3} + \)\(47\!\cdots\!24\)\( T^{4} + T^{5} \)
$71$ \( -\)\(24\!\cdots\!68\)\( - \)\(45\!\cdots\!20\)\( T - \)\(23\!\cdots\!20\)\( T^{2} + \)\(26\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!60\)\( T^{4} + T^{5} \)
$73$ \( -\)\(26\!\cdots\!32\)\( + \)\(19\!\cdots\!56\)\( T + \)\(68\!\cdots\!96\)\( T^{2} - \)\(52\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!78\)\( T^{4} + T^{5} \)
$79$ \( -\)\(37\!\cdots\!00\)\( + \)\(63\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} - \)\(58\!\cdots\!00\)\( T^{4} + T^{5} \)
$83$ \( \)\(15\!\cdots\!32\)\( + \)\(31\!\cdots\!76\)\( T - \)\(30\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!48\)\( T^{3} + \)\(27\!\cdots\!32\)\( T^{4} + T^{5} \)
$89$ \( -\)\(42\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T + \)\(56\!\cdots\!00\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!50\)\( T^{4} + T^{5} \)
$97$ \( -\)\(82\!\cdots\!76\)\( + \)\(73\!\cdots\!16\)\( T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(20\!\cdots\!92\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} + T^{5} \)
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