Newform orbit 531.2.i.c

• Label: 531.2.i.c
• Level: $531$
• Weight: $2$
• Character orbit: 531.i
• Analytic conductor: $4.240$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.i (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 140q + q^{2} - 9q^{4} + 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.31462	+	2.47964i		0		−3.29802	−	4.86421i	−1.27307	+	0.280224i		0		1.14406	−	0.869689i	10.8169	−	1.17641i		0		0.978753	−	3.52515i
19.2	−0.655840	+	1.23705i		0		0.0222185	+	0.0327698i	−2.32817	+	0.512469i		0		−1.36308	+	1.03619i	−2.83899	+	0.308758i		0		0.892959	−	3.21615i
19.3	−0.0292574	+	0.0551852i		0		1.12018	+	1.65215i	2.41343	−	0.531236i		0		−2.91820	+	2.21836i	−0.248138	+	0.0269866i		0		−0.0412942	+	0.148728i
19.4	0.454038	−	0.856407i		0		0.595092	+	0.877695i	−3.60206	+	0.792874i		0		1.56865	−	1.19245i	2.94914	−	0.320738i		0		−0.956451	+	3.44483i
19.5	1.07727	−	2.03195i		0		−1.84595	−	2.72257i	3.23491	−	0.712057i		0		1.31099	−	0.996592i	−2.94796	+	0.320610i		0		2.03802	−	7.34027i
28.1	−1.31462	−	2.47964i		0		−3.29802	+	4.86421i	−1.27307	−	0.280224i		0		1.14406	+	0.869689i	10.8169	+	1.17641i		0		0.978753	+	3.52515i
28.2	−0.655840	−	1.23705i		0		0.0222185	−	0.0327698i	−2.32817	−	0.512469i		0		−1.36308	−	1.03619i	−2.83899	−	0.308758i		0		0.892959	+	3.21615i
28.3	−0.0292574	−	0.0551852i		0		1.12018	−	1.65215i	2.41343	+	0.531236i		0		−2.91820	−	2.21836i	−0.248138	−	0.0269866i		0		−0.0412942	−	0.148728i
28.4	0.454038	+	0.856407i		0		0.595092	−	0.877695i	−3.60206	−	0.792874i		0		1.56865	+	1.19245i	2.94914	+	0.320738i		0		−0.956451	−	3.44483i
28.5	1.07727	+	2.03195i		0		−1.84595	+	2.72257i	3.23491	+	0.712057i		0		1.31099	+	0.996592i	−2.94796	−	0.320610i		0		2.03802	+	7.34027i
46.1	−1.65338	−	1.94651i		0		−0.731672	+	4.46300i	−0.865278	+	1.63209i		0		−2.96873	+	0.322869i	5.52030	−	3.32145i		0		4.60751	−	1.01419i
46.2	−1.15330	−	1.35777i		0		−0.189873	+	1.15817i	1.46043	−	2.75467i		0		1.72018	−	0.187081i	−1.26142	+	0.758972i		0		−5.42452	+	1.19403i
46.3	−0.398193	−	0.468789i		0		0.262358	−	1.60032i	−0.780962	+	1.47305i		0		−2.98728	+	0.324887i	−1.90875	+	1.14846i		0		1.00152	−	0.220452i
46.4	0.814427	+	0.958818i		0		0.0675241	−	0.411879i	−0.588484	+	1.11000i		0		2.16620	−	0.235588i	2.60580	−	1.56786i		0		−1.54356	+	0.339764i
46.5	1.74306	+	2.05209i		0		−0.849247	+	5.18018i	1.70562	−	3.21713i		0		3.51311	−	0.382074i	−7.49637	+	4.51042i		0		9.57483	−	2.10758i
64.1	−2.28036	−	0.768343i		0		3.01751	+	2.29385i	−0.739682	+	1.85646i		0		−0.807669	+	0.373667i	−2.41776	−	3.56592i		0		3.11314	−	3.66507i
64.2	−0.827908	−	0.278955i		0		−0.984570	−	0.748451i	−0.302390	+	0.758941i		0		3.54386	−	1.63956i	1.58690	+	2.34050i		0		0.462061	−	0.543981i
64.3	−0.288668	−	0.0972636i		0		−1.51832	−	1.15419i	1.30154	−	3.26662i		0		−3.34578	+	1.54792i	0.667919	+	0.985107i		0		−0.693437	+	0.816377i
64.4	1.73089	+	0.583204i		0		1.06366	+	0.808574i	0.835093	−	2.09593i		0		2.04844	−	0.947709i	−0.680501	−	1.00366i		0		2.66781	−	3.14078i
64.5	2.61370	+	0.880659i		0		4.46370	+	3.39321i	−0.422705	+	1.06091i		0		−3.24336	+	1.50054i	5.58292	+	8.23419i		0		−2.03912	+	2.40064i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.c	even	29	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	531.2.i.c		140
3.b	odd	2	1		177.2.e.a	✓	140
59.c	even	29	1	inner	531.2.i.c		140
177.h	odd	58	1		177.2.e.a	✓	140

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
177.2.e.a	✓	140	3.b	odd	2	1
177.2.e.a	✓	140	177.h	odd	58	1
531.2.i.c		140	1.a	even	1	1	trivial
531.2.i.c		140	59.c	even	29	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(17\!\cdots\!76\)\( T_{2}^{109} + \)\(15\!\cdots\!73\)\( T_{2}^{108} + \)\(54\!\cdots\!81\)\( T_{2}^{107} + \)\(26\!\cdots\!18\)\( T_{2}^{106} - \)\(19\!\cdots\!24\)\( T_{2}^{105} - \)\(44\!\cdots\!52\)\( T_{2}^{104} - \)\(95\!\cdots\!64\)\( T_{2}^{103} - \)\(52\!\cdots\!48\)\( T_{2}^{102} - \)\(85\!\cdots\!70\)\( T_{2}^{101} + \)\(21\!\cdots\!05\)\( T_{2}^{100} - \)\(40\!\cdots\!45\)\( T_{2}^{99} + \)\(13\!\cdots\!06\)\( T_{2}^{98} + \)\(26\!\cdots\!97\)\( T_{2}^{97} + \)\(21\!\cdots\!47\)\( T_{2}^{96} - \)\(57\!\cdots\!05\)\( T_{2}^{95} + \)\(84\!\cdots\!26\)\( T_{2}^{94} - \)\(28\!\cdots\!67\)\( T_{2}^{93} + \)\(35\!\cdots\!42\)\( T_{2}^{92} + \)\(33\!\cdots\!80\)\( T_{2}^{91} + \)\(43\!\cdots\!99\)\( T_{2}^{90} + \)\(18\!\cdots\!41\)\( T_{2}^{89} + \)\(11\!\cdots\!44\)\( T_{2}^{88} - \)\(66\!\cdots\!53\)\( T_{2}^{87} - \)\(48\!\cdots\!26\)\( T_{2}^{86} + \)\(46\!\cdots\!87\)\( T_{2}^{85} + \)\(48\!\cdots\!56\)\( T_{2}^{84} - \)\(83\!\cdots\!66\)\( T_{2}^{83} + \)\(62\!\cdots\!14\)\( T_{2}^{82} + \)\(44\!\cdots\!26\)\( T_{2}^{81} + \)\(91\!\cdots\!50\)\( T_{2}^{80} - \)\(49\!\cdots\!08\)\( T_{2}^{79} + \)\(13\!\cdots\!30\)\( T_{2}^{78} + \)\(89\!\cdots\!48\)\( T_{2}^{77} + \)\(16\!\cdots\!22\)\( T_{2}^{76} - \)\(16\!\cdots\!85\)\( T_{2}^{75} + \)\(71\!\cdots\!71\)\( T_{2}^{74} - \)\(86\!\cdots\!21\)\( T_{2}^{73} + \)\(13\!\cdots\!80\)\( T_{2}^{72} - \)\(23\!\cdots\!50\)\( T_{2}^{71} + \)\(11\!\cdots\!94\)\( T_{2}^{70} - \)\(19\!\cdots\!59\)\( T_{2}^{69} + \)\(43\!\cdots\!96\)\( T_{2}^{68} - \)\(99\!\cdots\!74\)\( T_{2}^{67} + \)\(16\!\cdots\!48\)\( T_{2}^{66} - \)\(30\!\cdots\!27\)\( T_{2}^{65} + \)\(55\!\cdots\!80\)\( T_{2}^{64} - \)\(90\!\cdots\!16\)\( T_{2}^{63} + \)\(14\!\cdots\!36\)\( T_{2}^{62} - \)\(22\!\cdots\!43\)\( T_{2}^{61} + \)\(31\!\cdots\!29\)\( T_{2}^{60} - \)\(39\!\cdots\!90\)\( T_{2}^{59} + \)\(49\!\cdots\!38\)\( T_{2}^{58} - \)\(53\!\cdots\!38\)\( T_{2}^{57} + \)\(51\!\cdots\!10\)\( T_{2}^{56} - \)\(47\!\cdots\!65\)\( T_{2}^{55} + \)\(48\!\cdots\!67\)\( T_{2}^{54} - \)\(38\!\cdots\!07\)\( T_{2}^{53} + \)\(46\!\cdots\!46\)\( T_{2}^{52} - \)\(53\!\cdots\!75\)\( T_{2}^{51} + \)\(57\!\cdots\!11\)\( T_{2}^{50} - \)\(56\!\cdots\!93\)\( T_{2}^{49} + \)\(59\!\cdots\!87\)\( T_{2}^{48} - \)\(42\!\cdots\!58\)\( T_{2}^{47} + \)\(25\!\cdots\!24\)\( T_{2}^{46} - \)\(13\!\cdots\!04\)\( T_{2}^{45} + \)\(10\!\cdots\!55\)\( T_{2}^{44} + \)\(16\!\cdots\!60\)\( T_{2}^{43} + \)\(38\!\cdots\!51\)\( T_{2}^{42} + \)\(40\!\cdots\!88\)\( T_{2}^{41} + \)\(98\!\cdots\!00\)\( T_{2}^{40} + \)\(97\!\cdots\!56\)\( T_{2}^{39} + \)\(16\!\cdots\!63\)\( T_{2}^{38} + \)\(19\!\cdots\!51\)\( T_{2}^{37} + \)\(25\!\cdots\!35\)\( T_{2}^{36} + \)\(31\!\cdots\!78\)\( T_{2}^{35} + \)\(34\!\cdots\!38\)\( T_{2}^{34} + \)\(35\!\cdots\!25\)\( T_{2}^{33} + \)\(33\!\cdots\!11\)\( T_{2}^{32} + \)\(29\!\cdots\!36\)\( T_{2}^{31} + \)\(24\!\cdots\!65\)\( T_{2}^{30} + \)\(19\!\cdots\!67\)\( T_{2}^{29} + \)\(15\!\cdots\!93\)\( T_{2}^{28} + \)\(12\!\cdots\!74\)\( T_{2}^{27} + \)\(93\!\cdots\!61\)\( T_{2}^{26} + \)\(71\!\cdots\!64\)\( T_{2}^{25} + \)\(53\!\cdots\!00\)\( T_{2}^{24} + \)\(39\!\cdots\!03\)\( T_{2}^{23} + \)\(28\!\cdots\!93\)\( T_{2}^{22} + \)\(19\!\cdots\!77\)\( T_{2}^{21} + \)\(12\!\cdots\!47\)\( T_{2}^{20} + \)\(72\!\cdots\!14\)\( T_{2}^{19} + \)\(40\!\cdots\!84\)\( T_{2}^{18} + \)\(20\!\cdots\!56\)\( T_{2}^{17} + \)\(93\!\cdots\!05\)\( T_{2}^{16} + \)\(38\!\cdots\!68\)\( T_{2}^{15} + \)\(14\!\cdots\!80\)\( T_{2}^{14} + \)\(46\!\cdots\!77\)\( T_{2}^{13} + \)\(13\!\cdots\!12\)\( T_{2}^{12} + \)\(35\!\cdots\!97\)\( T_{2}^{11} + \)\(79\!\cdots\!45\)\( T_{2}^{10} + \)\(15\!\cdots\!98\)\( T_{2}^{9} + \)\(24\!\cdots\!23\)\( T_{2}^{8} + \)\(33\!\cdots\!67\)\( T_{2}^{7} + \)\(37\!\cdots\!26\)\( T_{2}^{6} + \)\(33\!\cdots\!63\)\( T_{2}^{5} + \)\(24\!\cdots\!88\)\( T_{2}^{4} + \)\(13\!\cdots\!02\)\( T_{2}^{3} + \)\(57\!\cdots\!02\)\( T_{2}^{2} + \)\(16\!\cdots\!73\)\( T_{2} + 23694752761 \)">\(T_{2}^{140} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(531, [\chi])\).