Properties

Label 9.60.a.c
Level 9
Weight 60
Character orbit 9.a
Self dual yes
Analytic conductor 198.412
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 60 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(198.412204959\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 3976283494919360 x^{3} + 9065173660301515822976 x^{2} + 2677795447191606098169599438848 x - 23358185771581696169459363194340724736\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{19}\cdot 5^{3}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q +(89938373 + \beta_{1}) q^{2} +(347763875847469645 + 97803582 \beta_{1} - \beta_{2} + \beta_{3}) q^{4} +(-35992322464951840573 - 174673144666 \beta_{1} - 213 \beta_{2} - 12 \beta_{3} + \beta_{4}) q^{5} +(\)\(29\!\cdots\!76\)\( - 1403037836233582 \beta_{1} + 51206374 \beta_{2} - 1472240 \beta_{3} - 7020 \beta_{4}) q^{7} +(\)\(69\!\cdots\!28\)\( + 363735003386080208 \beta_{1} - 5658146712 \beta_{2} + 360632280 \beta_{3} - 266560 \beta_{4}) q^{8} +O(q^{10})\) \( q +(89938373 + \beta_{1}) q^{2} +(347763875847469645 + 97803582 \beta_{1} - \beta_{2} + \beta_{3}) q^{4} +(-35992322464951840573 - 174673144666 \beta_{1} - 213 \beta_{2} - 12 \beta_{3} + \beta_{4}) q^{5} +(\)\(29\!\cdots\!76\)\( - 1403037836233582 \beta_{1} + 51206374 \beta_{2} - 1472240 \beta_{3} - 7020 \beta_{4}) q^{7} +(\)\(69\!\cdots\!28\)\( + 363735003386080208 \beta_{1} - 5658146712 \beta_{2} + 360632280 \beta_{3} - 266560 \beta_{4}) q^{8} +(-\)\(16\!\cdots\!62\)\( - 43243016745318562654 \beta_{1} - 2508297240672 \beta_{2} - 323627897728 \beta_{3} + 158562144 \beta_{4}) q^{10} +(-\)\(85\!\cdots\!12\)\( + \)\(37\!\cdots\!85\)\( \beta_{1} + 10024580618085 \beta_{2} - 3717411735840 \beta_{3} + 2748847640 \beta_{4}) q^{11} +(-\)\(16\!\cdots\!31\)\( - \)\(58\!\cdots\!42\)\( \beta_{1} - 3097561149712155 \beta_{2} - 198523391408180 \beta_{3} - 476711357265 \beta_{4}) q^{13} +(-\)\(12\!\cdots\!12\)\( - \)\(91\!\cdots\!56\)\( \beta_{1} + 30788635066882608 \beta_{2} - 4908054819167808 \beta_{3} - 4910778324400 \beta_{4}) q^{14} +(\)\(13\!\cdots\!12\)\( + \)\(26\!\cdots\!96\)\( \beta_{1} - 1270560999623839808 \beta_{2} + 381700226166408768 \beta_{3} + 331129448133120 \beta_{4}) q^{16} +(\)\(68\!\cdots\!76\)\( + \)\(21\!\cdots\!84\)\( \beta_{1} + 7777130488502711538 \beta_{2} + 1778018042496512760 \beta_{3} + 2173189785854230 \beta_{4}) q^{17} +(\)\(12\!\cdots\!08\)\( + \)\(87\!\cdots\!93\)\( \beta_{1} + 2533851712950979781 \beta_{2} + 6757617129034581984 \beta_{3} - 57636170473930920 \beta_{4}) q^{19} +(-\)\(33\!\cdots\!54\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(14\!\cdots\!26\)\( \beta_{2} + 95991735237875522274 \beta_{3} - 220943909849546752 \beta_{4}) q^{20} +(\)\(33\!\cdots\!04\)\( - \)\(30\!\cdots\!52\)\( \beta_{1} + \)\(96\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!20\)\( \beta_{3} + 812126951793317160 \beta_{4}) q^{22} +(-\)\(53\!\cdots\!40\)\( + \)\(83\!\cdots\!06\)\( \beta_{1} + \)\(69\!\cdots\!94\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} - 3523640114226532300 \beta_{4}) q^{23} +(\)\(12\!\cdots\!95\)\( + \)\(38\!\cdots\!40\)\( \beta_{1} + \)\(13\!\cdots\!20\)\( \beta_{2} - \)\(22\!\cdots\!20\)\( \beta_{3} + \)\(12\!\cdots\!60\)\( \beta_{4}) q^{25} +(-\)\(54\!\cdots\!06\)\( - \)\(26\!\cdots\!74\)\( \beta_{1} + \)\(28\!\cdots\!72\)\( \beta_{2} - \)\(27\!\cdots\!52\)\( \beta_{3} + \)\(27\!\cdots\!40\)\( \beta_{4}) q^{26} +(-\)\(11\!\cdots\!00\)\( - \)\(35\!\cdots\!04\)\( \beta_{1} + \)\(13\!\cdots\!84\)\( \beta_{2} - \)\(27\!\cdots\!20\)\( \beta_{3} + \)\(24\!\cdots\!40\)\( \beta_{4}) q^{28} +(\)\(35\!\cdots\!63\)\( - \)\(46\!\cdots\!82\)\( \beta_{1} - \)\(77\!\cdots\!09\)\( \beta_{2} - \)\(30\!\cdots\!96\)\( \beta_{3} + \)\(11\!\cdots\!65\)\( \beta_{4}) q^{29} +(-\)\(71\!\cdots\!68\)\( - \)\(33\!\cdots\!80\)\( \beta_{1} + \)\(53\!\cdots\!20\)\( \beta_{2} + \)\(42\!\cdots\!20\)\( \beta_{3} - \)\(43\!\cdots\!20\)\( \beta_{4}) q^{31} +(\)\(21\!\cdots\!48\)\( + \)\(16\!\cdots\!76\)\( \beta_{1} - \)\(67\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(17\!\cdots\!80\)\( \beta_{4}) q^{32} +(\)\(26\!\cdots\!70\)\( + \)\(16\!\cdots\!14\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2} - \)\(49\!\cdots\!88\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{34} +(-\)\(17\!\cdots\!88\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(26\!\cdots\!28\)\( \beta_{2} + \)\(33\!\cdots\!28\)\( \beta_{3} + \)\(10\!\cdots\!56\)\( \beta_{4}) q^{35} +(\)\(24\!\cdots\!65\)\( - \)\(68\!\cdots\!58\)\( \beta_{1} - \)\(11\!\cdots\!27\)\( \beta_{2} - \)\(15\!\cdots\!40\)\( \beta_{3} - \)\(22\!\cdots\!45\)\( \beta_{4}) q^{37} +(\)\(91\!\cdots\!32\)\( + \)\(16\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(19\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{38} +(-\)\(12\!\cdots\!20\)\( + \)\(36\!\cdots\!60\)\( \beta_{1} + \)\(18\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3} - \)\(27\!\cdots\!60\)\( \beta_{4}) q^{40} +(\)\(25\!\cdots\!78\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} + \)\(27\!\cdots\!80\)\( \beta_{2} - \)\(42\!\cdots\!20\)\( \beta_{3} - \)\(28\!\cdots\!80\)\( \beta_{4}) q^{41} +(\)\(27\!\cdots\!52\)\( + \)\(29\!\cdots\!51\)\( \beta_{1} - \)\(94\!\cdots\!17\)\( \beta_{2} + \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(23\!\cdots\!00\)\( \beta_{4}) q^{43} +(-\)\(20\!\cdots\!00\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} - \)\(14\!\cdots\!28\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3} - \)\(29\!\cdots\!80\)\( \beta_{4}) q^{44} +(\)\(71\!\cdots\!32\)\( + \)\(47\!\cdots\!60\)\( \beta_{1} - \)\(94\!\cdots\!00\)\( \beta_{2} + \)\(60\!\cdots\!40\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4}) q^{46} +(-\)\(11\!\cdots\!08\)\( - \)\(13\!\cdots\!20\)\( \beta_{1} - \)\(68\!\cdots\!96\)\( \beta_{2} - \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(16\!\cdots\!00\)\( \beta_{4}) q^{47} +(-\)\(34\!\cdots\!27\)\( + \)\(25\!\cdots\!40\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!20\)\( \beta_{3} - \)\(32\!\cdots\!20\)\( \beta_{4}) q^{49} +(\)\(46\!\cdots\!55\)\( - \)\(17\!\cdots\!65\)\( \beta_{1} + \)\(94\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!80\)\( \beta_{3} - \)\(18\!\cdots\!60\)\( \beta_{4}) q^{50} +(-\)\(19\!\cdots\!34\)\( - \)\(39\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!34\)\( \beta_{2} - \)\(49\!\cdots\!50\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{52} +(-\)\(45\!\cdots\!93\)\( - \)\(11\!\cdots\!22\)\( \beta_{1} + \)\(71\!\cdots\!51\)\( \beta_{2} - \)\(64\!\cdots\!20\)\( \beta_{3} + \)\(41\!\cdots\!65\)\( \beta_{4}) q^{53} +(\)\(18\!\cdots\!16\)\( + \)\(59\!\cdots\!22\)\( \beta_{1} - \)\(16\!\cdots\!54\)\( \beta_{2} - \)\(23\!\cdots\!96\)\( \beta_{3} + \)\(21\!\cdots\!08\)\( \beta_{4}) q^{55} +(-\)\(26\!\cdots\!12\)\( - \)\(23\!\cdots\!32\)\( \beta_{1} - \)\(50\!\cdots\!84\)\( \beta_{2} - \)\(27\!\cdots\!96\)\( \beta_{3} + \)\(29\!\cdots\!40\)\( \beta_{4}) q^{56} +(-\)\(39\!\cdots\!98\)\( + \)\(22\!\cdots\!42\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(39\!\cdots\!60\)\( \beta_{3} + \)\(93\!\cdots\!20\)\( \beta_{4}) q^{58} +(\)\(43\!\cdots\!96\)\( + \)\(11\!\cdots\!21\)\( \beta_{1} + \)\(48\!\cdots\!97\)\( \beta_{2} - \)\(19\!\cdots\!72\)\( \beta_{3} + \)\(43\!\cdots\!00\)\( \beta_{4}) q^{59} +(-\)\(11\!\cdots\!83\)\( - \)\(35\!\cdots\!50\)\( \beta_{1} + \)\(18\!\cdots\!25\)\( \beta_{2} + \)\(65\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!25\)\( \beta_{4}) q^{61} +(-\)\(36\!\cdots\!04\)\( - \)\(73\!\cdots\!48\)\( \beta_{1} + \)\(42\!\cdots\!00\)\( \beta_{2} - \)\(77\!\cdots\!60\)\( \beta_{3} - \)\(49\!\cdots\!80\)\( \beta_{4}) q^{62} +(\)\(88\!\cdots\!32\)\( + \)\(16\!\cdots\!24\)\( \beta_{1} - \)\(89\!\cdots\!72\)\( \beta_{2} - \)\(48\!\cdots\!48\)\( \beta_{3} - \)\(22\!\cdots\!40\)\( \beta_{4}) q^{64} +(-\)\(21\!\cdots\!76\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(22\!\cdots\!44\)\( \beta_{2} + \)\(22\!\cdots\!56\)\( \beta_{3} - \)\(25\!\cdots\!88\)\( \beta_{4}) q^{65} +(\)\(50\!\cdots\!64\)\( + \)\(20\!\cdots\!81\)\( \beta_{1} - \)\(17\!\cdots\!23\)\( \beta_{2} + \)\(25\!\cdots\!40\)\( \beta_{3} - \)\(81\!\cdots\!80\)\( \beta_{4}) q^{67} +(\)\(11\!\cdots\!46\)\( - \)\(34\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} + \)\(20\!\cdots\!70\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{68} +(-\)\(11\!\cdots\!72\)\( + \)\(37\!\cdots\!76\)\( \beta_{1} - \)\(21\!\cdots\!32\)\( \beta_{2} + \)\(16\!\cdots\!32\)\( \beta_{3} + \)\(12\!\cdots\!64\)\( \beta_{4}) q^{70} +(\)\(12\!\cdots\!68\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(46\!\cdots\!50\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!00\)\( \beta_{4}) q^{71} +(\)\(24\!\cdots\!60\)\( - \)\(25\!\cdots\!96\)\( \beta_{1} - \)\(13\!\cdots\!14\)\( \beta_{2} - \)\(21\!\cdots\!60\)\( \beta_{3} + \)\(15\!\cdots\!70\)\( \beta_{4}) q^{73} +(-\)\(60\!\cdots\!06\)\( - \)\(65\!\cdots\!10\)\( \beta_{1} + \)\(15\!\cdots\!60\)\( \beta_{2} - \)\(49\!\cdots\!20\)\( \beta_{3} + \)\(34\!\cdots\!80\)\( \beta_{4}) q^{74} +(\)\(88\!\cdots\!64\)\( + \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(91\!\cdots\!52\)\( \beta_{2} + \)\(85\!\cdots\!12\)\( \beta_{3} + \)\(15\!\cdots\!20\)\( \beta_{4}) q^{76} +(-\)\(34\!\cdots\!12\)\( + \)\(72\!\cdots\!04\)\( \beta_{1} + \)\(52\!\cdots\!92\)\( \beta_{2} - \)\(37\!\cdots\!40\)\( \beta_{3} - \)\(33\!\cdots\!20\)\( \beta_{4}) q^{77} +(-\)\(39\!\cdots\!28\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} - \)\(84\!\cdots\!96\)\( \beta_{2} + \)\(26\!\cdots\!36\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4}) q^{79} +(\)\(41\!\cdots\!72\)\( - \)\(69\!\cdots\!76\)\( \beta_{1} + \)\(63\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!32\)\( \beta_{3} - \)\(27\!\cdots\!64\)\( \beta_{4}) q^{80} +(-\)\(18\!\cdots\!66\)\( + \)\(20\!\cdots\!58\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} - \)\(42\!\cdots\!40\)\( \beta_{3} - \)\(26\!\cdots\!20\)\( \beta_{4}) q^{82} +(-\)\(26\!\cdots\!96\)\( - \)\(15\!\cdots\!85\)\( \beta_{1} - \)\(13\!\cdots\!57\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4}) q^{83} +(\)\(49\!\cdots\!38\)\( - \)\(61\!\cdots\!04\)\( \beta_{1} - \)\(37\!\cdots\!22\)\( \beta_{2} - \)\(23\!\cdots\!28\)\( \beta_{3} + \)\(25\!\cdots\!94\)\( \beta_{4}) q^{85} +(\)\(29\!\cdots\!96\)\( + \)\(41\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!16\)\( \beta_{2} + \)\(80\!\cdots\!64\)\( \beta_{3} - \)\(31\!\cdots\!40\)\( \beta_{4}) q^{86} +(-\)\(15\!\cdots\!76\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} + \)\(84\!\cdots\!04\)\( \beta_{2} + \)\(25\!\cdots\!40\)\( \beta_{3} + \)\(94\!\cdots\!20\)\( \beta_{4}) q^{88} +(-\)\(82\!\cdots\!96\)\( - \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(15\!\cdots\!22\)\( \beta_{2} - \)\(84\!\cdots\!68\)\( \beta_{3} - \)\(92\!\cdots\!30\)\( \beta_{4}) q^{89} +(\)\(40\!\cdots\!64\)\( + \)\(11\!\cdots\!72\)\( \beta_{1} - \)\(26\!\cdots\!36\)\( \beta_{2} + \)\(24\!\cdots\!16\)\( \beta_{3} + \)\(45\!\cdots\!60\)\( \beta_{4}) q^{91} +(\)\(80\!\cdots\!56\)\( + \)\(65\!\cdots\!44\)\( \beta_{1} - \)\(46\!\cdots\!12\)\( \beta_{2} + \)\(57\!\cdots\!40\)\( \beta_{3} + \)\(64\!\cdots\!20\)\( \beta_{4}) q^{92} +(-\)\(22\!\cdots\!16\)\( - \)\(21\!\cdots\!52\)\( \beta_{1} + \)\(14\!\cdots\!16\)\( \beta_{2} + \)\(41\!\cdots\!24\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4}) q^{94} +(-\)\(17\!\cdots\!20\)\( - \)\(50\!\cdots\!90\)\( \beta_{1} - \)\(63\!\cdots\!70\)\( \beta_{2} + \)\(14\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!40\)\( \beta_{4}) q^{95} +(\)\(23\!\cdots\!28\)\( - \)\(85\!\cdots\!80\)\( \beta_{1} + \)\(49\!\cdots\!86\)\( \beta_{2} - \)\(41\!\cdots\!20\)\( \beta_{3} - \)\(53\!\cdots\!10\)\( \beta_{4}) q^{97} +(\)\(20\!\cdots\!09\)\( - \)\(43\!\cdots\!47\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(28\!\cdots\!80\)\( \beta_{4}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 449691864q^{2} + 1738819379139544640q^{4} - \)\(17\!\cdots\!90\)\(q^{5} + \)\(14\!\cdots\!56\)\(q^{7} + \)\(34\!\cdots\!80\)\(q^{8} + O(q^{10}) \) \( 5q + 449691864q^{2} + 1738819379139544640q^{4} - \)\(17\!\cdots\!90\)\(q^{5} + \)\(14\!\cdots\!56\)\(q^{7} + \)\(34\!\cdots\!80\)\(q^{8} - \)\(81\!\cdots\!60\)\(q^{10} - \)\(42\!\cdots\!60\)\(q^{11} - \)\(84\!\cdots\!78\)\(q^{13} - \)\(62\!\cdots\!80\)\(q^{14} + \)\(69\!\cdots\!80\)\(q^{16} + \)\(34\!\cdots\!54\)\(q^{17} + \)\(64\!\cdots\!00\)\(q^{19} - \)\(16\!\cdots\!20\)\(q^{20} + \)\(16\!\cdots\!12\)\(q^{22} - \)\(26\!\cdots\!12\)\(q^{23} + \)\(62\!\cdots\!75\)\(q^{25} - \)\(27\!\cdots\!60\)\(q^{26} - \)\(56\!\cdots\!52\)\(q^{28} + \)\(17\!\cdots\!50\)\(q^{29} - \)\(35\!\cdots\!40\)\(q^{31} + \)\(10\!\cdots\!84\)\(q^{32} + \)\(13\!\cdots\!80\)\(q^{34} - \)\(87\!\cdots\!40\)\(q^{35} + \)\(12\!\cdots\!26\)\(q^{37} + \)\(45\!\cdots\!20\)\(q^{38} - \)\(62\!\cdots\!00\)\(q^{40} + \)\(12\!\cdots\!90\)\(q^{41} + \)\(13\!\cdots\!92\)\(q^{43} - \)\(10\!\cdots\!80\)\(q^{44} + \)\(35\!\cdots\!60\)\(q^{46} - \)\(58\!\cdots\!16\)\(q^{47} - \)\(17\!\cdots\!35\)\(q^{49} + \)\(23\!\cdots\!00\)\(q^{50} - \)\(99\!\cdots\!24\)\(q^{52} - \)\(22\!\cdots\!82\)\(q^{53} + \)\(90\!\cdots\!80\)\(q^{55} - \)\(13\!\cdots\!00\)\(q^{56} - \)\(19\!\cdots\!80\)\(q^{58} + \)\(21\!\cdots\!00\)\(q^{59} - \)\(55\!\cdots\!90\)\(q^{61} - \)\(18\!\cdots\!92\)\(q^{62} + \)\(44\!\cdots\!40\)\(q^{64} - \)\(10\!\cdots\!80\)\(q^{65} + \)\(25\!\cdots\!96\)\(q^{67} + \)\(58\!\cdots\!32\)\(q^{68} - \)\(57\!\cdots\!60\)\(q^{70} + \)\(63\!\cdots\!40\)\(q^{71} + \)\(12\!\cdots\!62\)\(q^{73} - \)\(30\!\cdots\!80\)\(q^{74} + \)\(44\!\cdots\!00\)\(q^{76} - \)\(17\!\cdots\!52\)\(q^{77} - \)\(19\!\cdots\!00\)\(q^{79} + \)\(20\!\cdots\!60\)\(q^{80} - \)\(90\!\cdots\!68\)\(q^{82} - \)\(13\!\cdots\!52\)\(q^{83} + \)\(24\!\cdots\!40\)\(q^{85} + \)\(14\!\cdots\!40\)\(q^{86} - \)\(76\!\cdots\!60\)\(q^{88} - \)\(41\!\cdots\!50\)\(q^{89} + \)\(20\!\cdots\!60\)\(q^{91} + \)\(40\!\cdots\!04\)\(q^{92} - \)\(11\!\cdots\!20\)\(q^{94} - \)\(87\!\cdots\!00\)\(q^{95} + \)\(11\!\cdots\!66\)\(q^{97} + \)\(10\!\cdots\!52\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 3976283494919360 x^{3} + 9065173660301515822976 x^{2} + 2677795447191606098169599438848 x - 23358185771581696169459363194340724736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-567 \nu^{4} - 37846914417 \nu^{3} + 2141374975420432680 \nu^{2} + 102314614596804580479814464 \nu - 1240490506653641669866746605691392\)\()/ 12765437266362368 \)
\(\beta_{3}\)\(=\)\((\)\(-567 \nu^{4} - 37846914417 \nu^{3} + 9494266840845156648 \nu^{2} + 127459407364152524047862592 \nu - 12935363537448663253118554131751424\)\()/ 12765437266362368 \)
\(\beta_{4}\)\(=\)\((\)\(56341737 \nu^{4} + 450652208893551 \nu^{3} - 200258777797909324704216 \nu^{2} - 1420335560668568034930714332352 \nu + 85338870566699595484758853274423418368\)\()/ 63827186331811840 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - \beta_{2} - 82073154 \beta_{1} + 916135717213006029\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-33320 \beta_{4} + 11352147 \beta_{3} - 673541451 \beta_{2} + 189316794307441866 \beta_{1} - 9398767890526525072144529\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(6672270837960 \beta_{4} + 31716831694725369 \beta_{3} - 15827168625589793 \beta_{2} - 1723028863877764298597666 \beta_{1} + 21679984667196387398744450633344653\)\()/5184\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.61966e7
−3.26158e7
9.74166e6
2.35709e7
5.54999e7
−1.25878e9 0 1.00806e18 4.05126e20 0 2.66317e24 −5.43293e26 0 −5.09964e29
1.2 −6.92842e8 0 −9.64308e16 −3.87033e20 0 −7.47157e23 4.66207e26 0 2.68153e29
1.3 3.23738e8 0 −4.71654e17 7.71676e20 0 −4.97804e24 −3.39315e26 0 2.49821e29
1.4 6.55640e8 0 −1.46597e17 −7.23713e20 0 1.08717e25 −4.74066e26 0 −4.74495e29
1.5 1.42193e9 0 1.44544e18 −2.46018e20 0 −6.31094e24 1.23563e27 0 −3.49822e29
\(n\): 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 9.60.a.c 5
3.b odd 2 1 1.60.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.60.a.a 5 3.b odd 2 1
9.60.a.c 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 449691864 T_{2}^{4} - \)\(22\!\cdots\!92\)\( T_{2}^{3} + \)\(73\!\cdots\!88\)\( T_{2}^{2} + \)\(81\!\cdots\!16\)\( T_{2} - \)\(26\!\cdots\!24\)\( \) acting on \(S_{60}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 449691864 T + 672853577462683648 T^{2} - \)\(30\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!68\)\( T^{4} - \)\(31\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!84\)\( T^{6} - \)\(99\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!56\)\( T^{8} - \)\(49\!\cdots\!04\)\( T^{9} + \)\(63\!\cdots\!68\)\( T^{10} \)
$3$ 1
$5$ \( 1 + \)\(17\!\cdots\!90\)\( T + \)\(13\!\cdots\!25\)\( T^{2} - \)\(33\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} - \)\(82\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!50\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(71\!\cdots\!25\)\( T^{8} + \)\(16\!\cdots\!50\)\( T^{9} + \)\(15\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 - \)\(14\!\cdots\!56\)\( T + \)\(26\!\cdots\!43\)\( T^{2} - \)\(60\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!98\)\( T^{4} - \)\(70\!\cdots\!88\)\( T^{5} + \)\(23\!\cdots\!14\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} - \)\(41\!\cdots\!56\)\( T^{9} + \)\(20\!\cdots\!43\)\( T^{10} \)
$11$ \( 1 + \)\(42\!\cdots\!60\)\( T + \)\(88\!\cdots\!95\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!10\)\( T^{4} + \)\(13\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!10\)\( T^{6} + \)\(27\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!45\)\( T^{8} + \)\(24\!\cdots\!60\)\( T^{9} + \)\(16\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 + \)\(84\!\cdots\!78\)\( T + \)\(17\!\cdots\!17\)\( T^{2} + \)\(82\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(44\!\cdots\!24\)\( T^{5} + \)\(64\!\cdots\!26\)\( T^{6} + \)\(23\!\cdots\!80\)\( T^{7} + \)\(25\!\cdots\!61\)\( T^{8} + \)\(65\!\cdots\!98\)\( T^{9} + \)\(41\!\cdots\!57\)\( T^{10} \)
$17$ \( 1 - \)\(34\!\cdots\!54\)\( T + \)\(15\!\cdots\!33\)\( T^{2} - \)\(32\!\cdots\!20\)\( T^{3} + \)\(92\!\cdots\!58\)\( T^{4} - \)\(15\!\cdots\!72\)\( T^{5} + \)\(36\!\cdots\!74\)\( T^{6} - \)\(50\!\cdots\!80\)\( T^{7} + \)\(94\!\cdots\!41\)\( T^{8} - \)\(82\!\cdots\!74\)\( T^{9} + \)\(96\!\cdots\!93\)\( T^{10} \)
$19$ \( 1 - \)\(64\!\cdots\!00\)\( T + \)\(13\!\cdots\!95\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!10\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!90\)\( T^{6} - \)\(48\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!05\)\( T^{8} - \)\(39\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + \)\(26\!\cdots\!12\)\( T + \)\(67\!\cdots\!47\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!58\)\( T^{4} + \)\(37\!\cdots\!56\)\( T^{5} + \)\(56\!\cdots\!46\)\( T^{6} + \)\(70\!\cdots\!80\)\( T^{7} + \)\(71\!\cdots\!41\)\( T^{8} + \)\(62\!\cdots\!32\)\( T^{9} + \)\(51\!\cdots\!07\)\( T^{10} \)
$29$ \( 1 - \)\(17\!\cdots\!50\)\( T + \)\(80\!\cdots\!45\)\( T^{2} - \)\(97\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!10\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!90\)\( T^{6} - \)\(35\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!05\)\( T^{8} - \)\(23\!\cdots\!50\)\( T^{9} + \)\(25\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 + \)\(35\!\cdots\!40\)\( T + \)\(91\!\cdots\!95\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} + \)\(23\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!10\)\( T^{6} + \)\(15\!\cdots\!80\)\( T^{7} + \)\(85\!\cdots\!45\)\( T^{8} + \)\(32\!\cdots\!40\)\( T^{9} + \)\(89\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 - \)\(12\!\cdots\!26\)\( T + \)\(96\!\cdots\!13\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!78\)\( T^{4} - \)\(58\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!94\)\( T^{6} - \)\(14\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!21\)\( T^{8} - \)\(15\!\cdots\!66\)\( T^{9} + \)\(41\!\cdots\!93\)\( T^{10} \)
$41$ \( 1 - \)\(12\!\cdots\!90\)\( T + \)\(10\!\cdots\!45\)\( T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} - \)\(12\!\cdots\!48\)\( T^{5} + \)\(44\!\cdots\!10\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!45\)\( T^{8} - \)\(52\!\cdots\!90\)\( T^{9} + \)\(59\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(13\!\cdots\!92\)\( T + \)\(85\!\cdots\!07\)\( T^{2} - \)\(77\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!98\)\( T^{4} - \)\(23\!\cdots\!16\)\( T^{5} + \)\(78\!\cdots\!86\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} - \)\(42\!\cdots\!92\)\( T^{9} + \)\(74\!\cdots\!07\)\( T^{10} \)
$47$ \( 1 + \)\(58\!\cdots\!16\)\( T + \)\(28\!\cdots\!03\)\( T^{2} + \)\(92\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!38\)\( T^{4} + \)\(59\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!54\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!61\)\( T^{8} + \)\(23\!\cdots\!36\)\( T^{9} + \)\(18\!\cdots\!43\)\( T^{10} \)
$53$ \( 1 + \)\(22\!\cdots\!82\)\( T + \)\(40\!\cdots\!37\)\( T^{2} + \)\(48\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!18\)\( T^{4} + \)\(38\!\cdots\!96\)\( T^{5} + \)\(26\!\cdots\!06\)\( T^{6} + \)\(14\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!81\)\( T^{8} + \)\(19\!\cdots\!22\)\( T^{9} + \)\(45\!\cdots\!57\)\( T^{10} \)
$59$ \( 1 - \)\(21\!\cdots\!00\)\( T + \)\(79\!\cdots\!95\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!10\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{5} + \)\(99\!\cdots\!90\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(25\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + \)\(55\!\cdots\!90\)\( T + \)\(84\!\cdots\!45\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!10\)\( T^{4} + \)\(92\!\cdots\!48\)\( T^{5} + \)\(63\!\cdots\!10\)\( T^{6} + \)\(15\!\cdots\!80\)\( T^{7} + \)\(84\!\cdots\!45\)\( T^{8} + \)\(12\!\cdots\!90\)\( T^{9} + \)\(47\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 - \)\(25\!\cdots\!96\)\( T + \)\(20\!\cdots\!83\)\( T^{2} - \)\(32\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!58\)\( T^{4} - \)\(22\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!74\)\( T^{6} - \)\(96\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} - \)\(22\!\cdots\!76\)\( T^{9} + \)\(49\!\cdots\!43\)\( T^{10} \)
$71$ \( 1 - \)\(63\!\cdots\!40\)\( T + \)\(82\!\cdots\!95\)\( T^{2} - \)\(37\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!10\)\( T^{4} - \)\(89\!\cdots\!48\)\( T^{5} + \)\(44\!\cdots\!10\)\( T^{6} - \)\(10\!\cdots\!80\)\( T^{7} + \)\(38\!\cdots\!45\)\( T^{8} - \)\(50\!\cdots\!40\)\( T^{9} + \)\(13\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 - \)\(12\!\cdots\!62\)\( T + \)\(31\!\cdots\!97\)\( T^{2} - \)\(37\!\cdots\!20\)\( T^{3} + \)\(48\!\cdots\!58\)\( T^{4} - \)\(45\!\cdots\!56\)\( T^{5} + \)\(42\!\cdots\!46\)\( T^{6} - \)\(28\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!41\)\( T^{8} - \)\(69\!\cdots\!82\)\( T^{9} + \)\(47\!\cdots\!57\)\( T^{10} \)
$79$ \( 1 + \)\(19\!\cdots\!00\)\( T + \)\(33\!\cdots\!95\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!10\)\( T^{4} + \)\(30\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!90\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!05\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(63\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 + \)\(13\!\cdots\!52\)\( T + \)\(14\!\cdots\!27\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!78\)\( T^{4} + \)\(26\!\cdots\!36\)\( T^{5} + \)\(99\!\cdots\!66\)\( T^{6} + \)\(28\!\cdots\!40\)\( T^{7} + \)\(68\!\cdots\!21\)\( T^{8} + \)\(10\!\cdots\!12\)\( T^{9} + \)\(13\!\cdots\!07\)\( T^{10} \)
$89$ \( 1 + \)\(41\!\cdots\!50\)\( T + \)\(38\!\cdots\!45\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!10\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(65\!\cdots\!90\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!05\)\( T^{8} + \)\(47\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 - \)\(11\!\cdots\!66\)\( T + \)\(92\!\cdots\!53\)\( T^{2} - \)\(60\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!38\)\( T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(53\!\cdots\!54\)\( T^{6} - \)\(16\!\cdots\!80\)\( T^{7} + \)\(42\!\cdots\!61\)\( T^{8} - \)\(87\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \)
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