Properties

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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 240q - 4q^{3} + 24q^{4} - 4q^{5} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q - 4q^{3} + 24q^{4} - 4q^{5} + 4q^{9} + 4q^{12} - 8q^{13} + 24q^{14} + 18q^{15} - 24q^{16} + 32q^{17} - 4q^{18} + 4q^{20} - 8q^{23} + 12q^{25} - 148q^{27} + 40q^{30} + 16q^{31} + 42q^{33} - 4q^{36} + 22q^{37} + 8q^{39} + 154q^{41} + 4q^{42} + 22q^{43} + 24q^{45} + 4q^{46} - 4q^{48} + 24q^{49} + 88q^{50} + 24q^{51} + 8q^{52} - 108q^{53} + 12q^{54} - 16q^{55} - 24q^{56} - 62q^{57} - 4q^{58} + 22q^{59} - 18q^{60} - 4q^{63} + 24q^{64} - 100q^{66} - 44q^{67} - 32q^{68} - 86q^{69} + 4q^{70} + 4q^{72} - 12q^{73} - 16q^{74} - 26q^{75} - 78q^{78} - 4q^{80} + 52q^{81} + 8q^{82} + 16q^{83} - 28q^{85} + 16q^{86} - 196q^{87} + 24q^{89} + 126q^{90} + 8q^{92} - 16q^{93} - 8q^{94} + 132q^{97} + 172q^{99} + O(q^{100}) \)

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
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 953.24
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(966, [\chi])\).