Newform orbit 966.2.r.a

• Label: 966.2.r.a
• Level: $966$
• Weight: $2$
• Character orbit: 966.r
• Analytic conductor: $7.714$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.r (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.71354883526\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240q - 4q^{3} + 24q^{4} - 4q^{5} + 4q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
113.1	−0.540641	+	0.841254i	−1.72997	−	0.0847817i	−0.415415	−	0.909632i	−2.86323	−	0.840720i	1.00662	−	1.40951i	−0.755750	−	0.654861i	0.989821	+	0.142315i	2.98562	+	0.293340i	2.25524	−	1.95417i
113.2	−0.540641	+	0.841254i	−1.65958	−	0.495767i	−0.415415	−	0.909632i	1.08122	+	0.317476i	1.31430	−	1.12810i	−0.755750	−	0.654861i	0.989821	+	0.142315i	2.50843	+	1.64553i	−0.851632	+	0.737943i
113.3	−0.540641	+	0.841254i	−1.54004	+	0.792632i	−0.415415	−	0.909632i	−1.55379	−	0.456233i	0.165806	−	1.72410i	−0.755750	−	0.654861i	0.989821	+	0.142315i	1.74347	−	2.44138i	1.22385	−	1.06047i
113.4	−0.540641	+	0.841254i	−0.844377	+	1.51229i	−0.415415	−	0.909632i	2.57280	+	0.755442i	−0.815716	−	1.52794i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−1.57405	−	2.55389i	−2.02648	+	1.75595i
113.5	−0.540641	+	0.841254i	−0.618673	−	1.61779i	−0.415415	−	0.909632i	−0.285460	−	0.0838187i	1.69545	+	0.354183i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−2.23449	+	2.00176i	0.224844	−	0.194829i
113.6	−0.540641	+	0.841254i	0.0176252	+	1.73196i	−0.415415	−	0.909632i	0.959405	+	0.281707i	−1.46655	−	0.921542i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−2.99938	+	0.0610522i	−0.755680	+	0.654801i
113.7	−0.540641	+	0.841254i	0.822715	−	1.52418i	−0.415415	−	0.909632i	3.64992	+	1.07171i	0.837432	+	1.51615i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−1.64628	−	2.50794i	−2.87488	+	2.49110i
113.8	−0.540641	+	0.841254i	0.897526	+	1.48137i	−0.415415	−	0.909632i	1.68096	+	0.493574i	−1.73144	−	0.0458405i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−1.38889	+	2.65913i	−1.32402	+	1.14727i
113.9	−0.540641	+	0.841254i	0.899579	−	1.48012i	−0.415415	−	0.909632i	−0.854693	−	0.250960i	0.758807	+	1.55699i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−1.38151	−	2.66297i	0.673203	−	0.583334i
113.10	−0.540641	+	0.841254i	1.04501	+	1.38129i	−0.415415	−	0.909632i	−1.38055	−	0.405366i	−1.72699	+	0.132338i	−0.755750	−	0.654861i	0.989821	+	0.142315i	−0.815909	+	2.88692i	1.08740	−	0.942236i
113.11	−0.540641	+	0.841254i	1.32111	−	1.12013i	−0.415415	−	0.909632i	−1.47166	−	0.432119i	0.228066	+	1.71697i	−0.755750	−	0.654861i	0.989821	+	0.142315i	0.490639	−	2.95961i	1.15916	−	1.00442i
113.12	−0.540641	+	0.841254i	1.57548	+	0.719621i	−0.415415	−	0.909632i	−2.33211	−	0.684769i	−1.45715	+	0.936324i	−0.755750	−	0.654861i	0.989821	+	0.142315i	1.96429	+	2.26750i	1.83690	−	1.59168i
113.13	0.540641	−	0.841254i	−1.71466	+	0.244816i	−0.415415	−	0.909632i	−0.707825	−	0.207836i	−0.721063	+	1.57482i	0.755750	+	0.654861i	−0.989821	−	0.142315i	2.88013	−	0.839555i	−0.557522	+	0.483096i
113.14	0.540641	−	0.841254i	−1.70862	+	0.283914i	−0.415415	−	0.909632i	4.23656	+	1.24397i	−0.684908	+	1.59088i	0.755750	+	0.654861i	−0.989821	−	0.142315i	2.83879	−	0.970204i	3.33695	−	2.89148i
113.15	0.540641	−	0.841254i	−1.67713	−	0.432689i	−0.415415	−	0.909632i	−0.241809	−	0.0710016i	−1.27073	+	1.17697i	0.755750	+	0.654861i	−0.989821	−	0.142315i	2.62556	+	1.45135i	−0.190462	+	0.165036i
113.16	0.540641	−	0.841254i	−0.980063	+	1.42810i	−0.415415	−	0.909632i	−3.32190	−	0.975398i	0.671534	+	1.59657i	0.755750	+	0.654861i	−0.989821	−	0.142315i	−1.07895	−	2.79926i	−2.61651	+	2.26722i
113.17	0.540641	−	0.841254i	−0.628320	−	1.61407i	−0.415415	−	0.909632i	−2.58447	−	0.758869i	−1.69754	−	0.344054i	0.755750	+	0.654861i	−0.989821	−	0.142315i	−2.21043	+	2.02830i	−2.03567	+	1.76392i
113.18	0.540641	−	0.841254i	0.331308	+	1.70007i	−0.415415	−	0.909632i	2.40341	+	0.705705i	1.60931	+	0.640412i	0.755750	+	0.654861i	−0.989821	−	0.142315i	−2.78047	+	1.12649i	1.89306	−	1.64034i
113.19	0.540641	−	0.841254i	0.367161	−	1.69269i	−0.415415	−	0.909632i	−2.84093	−	0.834172i	−1.22548	−	1.22401i	0.755750	+	0.654861i	−0.989821	−	0.142315i	−2.73039	−	1.24298i	−2.23767	+	1.93895i
113.20	0.540641	−	0.841254i	0.677326	+	1.59412i	−0.415415	−	0.909632i	−2.64132	−	0.775563i	1.70725	+	0.292045i	0.755750	+	0.654861i	−0.989821	−	0.142315i	−2.08246	+	2.15948i	−2.08045	+	1.80272i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
69.g	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	966.2.r.a	✓	240
3.b	odd	2	1		966.2.r.b	yes	240
23.d	odd	22	1		966.2.r.b	yes	240
69.g	even	22	1	inner	966.2.r.a	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
966.2.r.a	✓	240	1.a	even	1	1	trivial
966.2.r.a	✓	240	69.g	even	22	1	inner
966.2.r.b	yes	240	3.b	odd	2	1
966.2.r.b	yes	240	23.d	odd	22	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!91\)\( T_{5}^{224} + \)\(75\!\cdots\!64\)\( T_{5}^{223} + \)\(37\!\cdots\!73\)\( T_{5}^{222} + \)\(17\!\cdots\!84\)\( T_{5}^{221} + \)\(81\!\cdots\!34\)\( T_{5}^{220} + \)\(37\!\cdots\!44\)\( T_{5}^{219} + \)\(17\!\cdots\!55\)\( T_{5}^{218} + \)\(78\!\cdots\!52\)\( T_{5}^{217} + \)\(35\!\cdots\!72\)\( T_{5}^{216} + \)\(15\!\cdots\!84\)\( T_{5}^{215} + \)\(71\!\cdots\!79\)\( T_{5}^{214} + \)\(31\!\cdots\!60\)\( T_{5}^{213} + \)\(13\!\cdots\!75\)\( T_{5}^{212} + \)\(59\!\cdots\!72\)\( T_{5}^{211} + \)\(25\!\cdots\!79\)\( T_{5}^{210} + \)\(11\!\cdots\!68\)\( T_{5}^{209} + \)\(46\!\cdots\!95\)\( T_{5}^{208} + \)\(19\!\cdots\!76\)\( T_{5}^{207} + \)\(81\!\cdots\!21\)\( T_{5}^{206} + \)\(33\!\cdots\!98\)\( T_{5}^{205} + \)\(13\!\cdots\!90\)\( T_{5}^{204} + \)\(56\!\cdots\!00\)\( T_{5}^{203} + \)\(22\!\cdots\!51\)\( T_{5}^{202} + \)\(91\!\cdots\!98\)\( T_{5}^{201} + \)\(36\!\cdots\!97\)\( T_{5}^{200} + \)\(14\!\cdots\!18\)\( T_{5}^{199} + \)\(56\!\cdots\!02\)\( T_{5}^{198} + \)\(22\!\cdots\!74\)\( T_{5}^{197} + \)\(86\!\cdots\!59\)\( T_{5}^{196} + \)\(33\!\cdots\!92\)\( T_{5}^{195} + \)\(12\!\cdots\!70\)\( T_{5}^{194} + \)\(49\!\cdots\!30\)\( T_{5}^{193} + \)\(18\!\cdots\!59\)\( T_{5}^{192} + \)\(70\!\cdots\!58\)\( T_{5}^{191} + \)\(26\!\cdots\!75\)\( T_{5}^{190} + \)\(96\!\cdots\!50\)\( T_{5}^{189} + \)\(35\!\cdots\!75\)\( T_{5}^{188} + \)\(13\!\cdots\!56\)\( T_{5}^{187} + \)\(47\!\cdots\!81\)\( T_{5}^{186} + \)\(17\!\cdots\!06\)\( T_{5}^{185} + \)\(61\!\cdots\!33\)\( T_{5}^{184} + \)\(21\!\cdots\!50\)\( T_{5}^{183} + \)\(76\!\cdots\!00\)\( T_{5}^{182} + \)\(27\!\cdots\!82\)\( T_{5}^{181} + \)\(94\!\cdots\!66\)\( T_{5}^{180} + \)\(32\!\cdots\!04\)\( T_{5}^{179} + \)\(11\!\cdots\!50\)\( T_{5}^{178} + \)\(37\!\cdots\!38\)\( T_{5}^{177} + \)\(12\!\cdots\!96\)\( T_{5}^{176} + \)\(42\!\cdots\!06\)\( T_{5}^{175} + \)\(14\!\cdots\!57\)\( T_{5}^{174} + \)\(45\!\cdots\!16\)\( T_{5}^{173} + \)\(14\!\cdots\!38\)\( T_{5}^{172} + \)\(47\!\cdots\!48\)\( T_{5}^{171} + \)\(14\!\cdots\!91\)\( T_{5}^{170} + \)\(46\!\cdots\!20\)\( T_{5}^{169} + \)\(14\!\cdots\!79\)\( T_{5}^{168} + \)\(43\!\cdots\!52\)\( T_{5}^{167} + \)\(13\!\cdots\!67\)\( T_{5}^{166} + \)\(38\!\cdots\!46\)\( T_{5}^{165} + \)\(11\!\cdots\!18\)\( T_{5}^{164} + \)\(33\!\cdots\!20\)\( T_{5}^{163} + \)\(96\!\cdots\!33\)\( T_{5}^{162} + \)\(27\!\cdots\!74\)\( T_{5}^{161} + \)\(79\!\cdots\!01\)\( T_{5}^{160} + \)\(22\!\cdots\!06\)\( T_{5}^{159} + \)\(63\!\cdots\!70\)\( T_{5}^{158} + \)\(17\!\cdots\!34\)\( T_{5}^{157} + \)\(49\!\cdots\!49\)\( T_{5}^{156} + \)\(13\!\cdots\!48\)\( T_{5}^{155} + \)\(37\!\cdots\!40\)\( T_{5}^{154} + \)\(10\!\cdots\!34\)\( T_{5}^{153} + \)\(27\!\cdots\!33\)\( T_{5}^{152} + \)\(73\!\cdots\!78\)\( T_{5}^{151} + \)\(19\!\cdots\!87\)\( T_{5}^{150} + \)\(51\!\cdots\!34\)\( T_{5}^{149} + \)\(13\!\cdots\!22\)\( T_{5}^{148} + \)\(35\!\cdots\!96\)\( T_{5}^{147} + \)\(92\!\cdots\!05\)\( T_{5}^{146} + \)\(23\!\cdots\!22\)\( T_{5}^{145} + \)\(60\!\cdots\!88\)\( T_{5}^{144} + \)\(15\!\cdots\!78\)\( T_{5}^{143} + \)\(38\!\cdots\!70\)\( T_{5}^{142} + \)\(97\!\cdots\!40\)\( T_{5}^{141} + \)\(24\!\cdots\!40\)\( T_{5}^{140} + \)\(60\!\cdots\!72\)\( T_{5}^{139} + \)\(15\!\cdots\!16\)\( T_{5}^{138} + \)\(36\!\cdots\!36\)\( T_{5}^{137} + \)\(89\!\cdots\!59\)\( T_{5}^{136} + \)\(21\!\cdots\!96\)\( T_{5}^{135} + \)\(52\!\cdots\!36\)\( T_{5}^{134} + \)\(12\!\cdots\!46\)\( T_{5}^{133} + \)\(29\!\cdots\!92\)\( T_{5}^{132} + \)\(69\!\cdots\!48\)\( T_{5}^{131} + \)\(16\!\cdots\!52\)\( T_{5}^{130} + \)\(37\!\cdots\!58\)\( T_{5}^{129} + \)\(86\!\cdots\!89\)\( T_{5}^{128} + \)\(19\!\cdots\!90\)\( T_{5}^{127} + \)\(45\!\cdots\!57\)\( T_{5}^{126} + \)\(10\!\cdots\!34\)\( T_{5}^{125} + \)\(22\!\cdots\!94\)\( T_{5}^{124} + \)\(50\!\cdots\!72\)\( T_{5}^{123} + \)\(11\!\cdots\!95\)\( T_{5}^{122} + \)\(24\!\cdots\!70\)\( T_{5}^{121} + \)\(53\!\cdots\!12\)\( T_{5}^{120} + \)\(11\!\cdots\!62\)\( T_{5}^{119} + \)\(24\!\cdots\!22\)\( T_{5}^{118} + \)\(51\!\cdots\!00\)\( T_{5}^{117} + \)\(10\!\cdots\!70\)\( T_{5}^{116} + \)\(21\!\cdots\!96\)\( T_{5}^{115} + \)\(44\!\cdots\!68\)\( T_{5}^{114} + \)\(88\!\cdots\!68\)\( T_{5}^{113} + \)\(17\!\cdots\!87\)\( T_{5}^{112} + \)\(33\!\cdots\!36\)\( T_{5}^{111} + \)\(64\!\cdots\!62\)\( T_{5}^{110} + \)\(12\!\cdots\!54\)\( T_{5}^{109} + \)\(22\!\cdots\!87\)\( T_{5}^{108} + \)\(40\!\cdots\!04\)\( T_{5}^{107} + \)\(71\!\cdots\!78\)\( T_{5}^{106} + \)\(12\!\cdots\!70\)\( T_{5}^{105} + \)\(21\!\cdots\!73\)\( T_{5}^{104} + \)\(37\!\cdots\!54\)\( T_{5}^{103} + \)\(64\!\cdots\!35\)\( T_{5}^{102} + \)\(11\!\cdots\!28\)\( T_{5}^{101} + \)\(18\!\cdots\!91\)\( T_{5}^{100} + \)\(31\!\cdots\!24\)\( T_{5}^{99} + \)\(53\!\cdots\!65\)\( T_{5}^{98} + \)\(90\!\cdots\!66\)\( T_{5}^{97} + \)\(15\!\cdots\!47\)\( T_{5}^{96} + \)\(24\!\cdots\!72\)\( T_{5}^{95} + \)\(39\!\cdots\!63\)\( T_{5}^{94} + \)\(61\!\cdots\!64\)\( T_{5}^{93} + \)\(92\!\cdots\!55\)\( T_{5}^{92} + \)\(13\!\cdots\!34\)\( T_{5}^{91} + \)\(18\!\cdots\!52\)\( T_{5}^{90} + \)\(24\!\cdots\!08\)\( T_{5}^{89} + \)\(31\!\cdots\!97\)\( T_{5}^{88} + \)\(38\!\cdots\!94\)\( T_{5}^{87} + \)\(44\!\cdots\!85\)\( T_{5}^{86} + \)\(52\!\cdots\!28\)\( T_{5}^{85} + \)\(60\!\cdots\!60\)\( T_{5}^{84} + \)\(73\!\cdots\!38\)\( T_{5}^{83} + \)\(90\!\cdots\!84\)\( T_{5}^{82} + \)\(11\!\cdots\!86\)\( T_{5}^{81} + \)\(14\!\cdots\!08\)\( T_{5}^{80} + \)\(17\!\cdots\!94\)\( T_{5}^{79} + \)\(20\!\cdots\!79\)\( T_{5}^{78} + \)\(23\!\cdots\!20\)\( T_{5}^{77} + \)\(25\!\cdots\!10\)\( T_{5}^{76} + \)\(29\!\cdots\!56\)\( T_{5}^{75} + \)\(33\!\cdots\!05\)\( T_{5}^{74} + \)\(38\!\cdots\!94\)\( T_{5}^{73} + \)\(43\!\cdots\!95\)\( T_{5}^{72} + \)\(46\!\cdots\!24\)\( T_{5}^{71} + \)\(49\!\cdots\!47\)\( T_{5}^{70} + \)\(51\!\cdots\!90\)\( T_{5}^{69} + \)\(53\!\cdots\!75\)\( T_{5}^{68} + \)\(57\!\cdots\!54\)\( T_{5}^{67} + \)\(61\!\cdots\!78\)\( T_{5}^{66} + \)\(66\!\cdots\!76\)\( T_{5}^{65} + \)\(71\!\cdots\!40\)\( T_{5}^{64} + \)\(77\!\cdots\!32\)\( T_{5}^{63} + \)\(83\!\cdots\!92\)\( T_{5}^{62} + \)\(86\!\cdots\!02\)\( T_{5}^{61} + \)\(87\!\cdots\!01\)\( T_{5}^{60} + \)\(83\!\cdots\!00\)\( T_{5}^{59} + \)\(76\!\cdots\!44\)\( T_{5}^{58} + \)\(69\!\cdots\!26\)\( T_{5}^{57} + \)\(66\!\cdots\!44\)\( T_{5}^{56} + \)\(66\!\cdots\!10\)\( T_{5}^{55} + \)\(68\!\cdots\!09\)\( T_{5}^{54} + \)\(67\!\cdots\!48\)\( T_{5}^{53} + \)\(64\!\cdots\!26\)\( T_{5}^{52} + \)\(59\!\cdots\!64\)\( T_{5}^{51} + \)\(54\!\cdots\!81\)\( T_{5}^{50} + \)\(49\!\cdots\!24\)\( T_{5}^{49} + \)\(45\!\cdots\!28\)\( T_{5}^{48} + \)\(41\!\cdots\!44\)\( T_{5}^{47} + \)\(37\!\cdots\!39\)\( T_{5}^{46} + \)\(31\!\cdots\!06\)\( T_{5}^{45} + \)\(26\!\cdots\!38\)\( T_{5}^{44} + \)\(20\!\cdots\!04\)\( T_{5}^{43} + \)\(16\!\cdots\!37\)\( T_{5}^{42} + \)\(12\!\cdots\!96\)\( T_{5}^{41} + \)\(90\!\cdots\!59\)\( T_{5}^{40} + \)\(65\!\cdots\!10\)\( T_{5}^{39} + \)\(45\!\cdots\!64\)\( T_{5}^{38} + \)\(30\!\cdots\!94\)\( T_{5}^{37} + \)\(19\!\cdots\!70\)\( T_{5}^{36} + \)\(11\!\cdots\!86\)\( T_{5}^{35} + \)\(64\!\cdots\!11\)\( T_{5}^{34} + \)\(33\!\cdots\!72\)\( T_{5}^{33} + \)\(16\!\cdots\!64\)\( T_{5}^{32} + \)\(76\!\cdots\!56\)\( T_{5}^{31} + \)\(34\!\cdots\!29\)\( T_{5}^{30} + \)\(13\!\cdots\!50\)\( T_{5}^{29} + \)\(55\!\cdots\!51\)\( T_{5}^{28} + \)\(18\!\cdots\!88\)\( T_{5}^{27} + \)\(73\!\cdots\!11\)\( T_{5}^{26} + \)\(19\!\cdots\!02\)\( T_{5}^{25} + \)\(81\!\cdots\!42\)\( T_{5}^{24} + \)\(17\!\cdots\!02\)\( T_{5}^{23} + \)\(56\!\cdots\!32\)\( T_{5}^{22} + \)\(20\!\cdots\!80\)\( T_{5}^{21} + \)\(16\!\cdots\!25\)\( T_{5}^{20} + \)\(14\!\cdots\!64\)\( T_{5}^{19} + \)\(38\!\cdots\!12\)\( T_{5}^{18} - \)\(28\!\cdots\!04\)\( T_{5}^{17} + \)\(34\!\cdots\!00\)\( T_{5}^{16} + \)\(29\!\cdots\!36\)\( T_{5}^{15} - \)\(82\!\cdots\!48\)\( T_{5}^{14} + \)\(38\!\cdots\!64\)\( T_{5}^{13} + \)\(68\!\cdots\!16\)\( T_{5}^{12} - \)\(60\!\cdots\!32\)\( T_{5}^{11} + \)\(26\!\cdots\!76\)\( T_{5}^{10} - \)\(32\!\cdots\!72\)\( T_{5}^{9} + \)\(47\!\cdots\!28\)\( T_{5}^{8} - \)\(20\!\cdots\!40\)\( T_{5}^{7} + \)\(13\!\cdots\!68\)\( T_{5}^{6} + \)\(14\!\cdots\!80\)\( T_{5}^{5} + \)\(49\!\cdots\!92\)\( T_{5}^{4} - \)\(39\!\cdots\!48\)\( T_{5}^{3} + \)\(27\!\cdots\!96\)\( T_{5}^{2} - \)\(71\!\cdots\!92\)\( T_{5} + \)\(11\!\cdots\!84\)\( \)">\(T_{5}^{240} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).