Properties

Label 966.2.r.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 240q - 4q^{3} + 24q^{4} + 4q^{5} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q - 4q^{3} + 24q^{4} + 4q^{5} + 4q^{9} + 4q^{12} - 8q^{13} - 24q^{14} + 26q^{15} - 24q^{16} - 32q^{17} + 40q^{18} - 4q^{20} + 8q^{23} + 12q^{25} + 116q^{27} + 4q^{30} + 16q^{31} + 2q^{33} - 4q^{36} + 22q^{37} + 8q^{39} - 154q^{41} - 4q^{42} + 22q^{43} - 24q^{45} + 4q^{46} - 4q^{48} + 24q^{49} - 88q^{50} - 24q^{51} + 8q^{52} + 108q^{53} + 12q^{54} - 16q^{55} + 24q^{56} - 70q^{57} - 4q^{58} - 22q^{59} - 26q^{60} + 4q^{63} + 24q^{64} - 76q^{66} - 44q^{67} + 32q^{68} - 86q^{69} + 4q^{70} + 4q^{72} - 12q^{73} + 16q^{74} - 26q^{75} - 78q^{78} + 4q^{80} - 168q^{81} + 8q^{82} - 16q^{83} - 28q^{85} - 16q^{86} + 156q^{87} - 24q^{89} - 126q^{90} - 8q^{92} - 16q^{93} - 8q^{94} + 132q^{97} - 172q^{99} + O(q^{100}) \)

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 −1.66196 + 0.487739i −0.415415 0.909632i 2.08309 + 0.611649i 0.488211 1.66182i 0.755750 + 0.654861i 0.989821 + 0.142315i 2.52422 1.62121i −1.64075 + 1.42172i
113.2 −0.540641 + 0.841254i −1.65004 + 0.526650i −0.415415 0.909632i −3.65562 1.07339i 0.449034 1.67283i 0.755750 + 0.654861i 0.989821 + 0.142315i 2.44528 1.73799i 2.87937 2.49499i
113.3 −0.540641 + 0.841254i −1.24614 1.20297i −0.415415 0.909632i −2.28533 0.671033i 1.68572 0.397939i 0.755750 + 0.654861i 0.989821 + 0.142315i 0.105706 + 2.99814i 1.80005 1.55975i
113.4 −0.540641 + 0.841254i −0.829173 1.52068i −0.415415 0.909632i 2.84093 + 0.834172i 1.72756 + 0.124597i 0.755750 + 0.654861i 0.989821 + 0.142315i −1.62494 + 2.52182i −2.23767 + 1.93895i
113.5 −0.540641 + 0.841254i −0.798314 + 1.53711i −0.415415 0.909632i −1.04326 0.306329i −0.861494 1.50261i 0.755750 + 0.654861i 0.989821 + 0.142315i −1.72539 2.45419i 0.821728 0.712032i
113.6 −0.540641 + 0.841254i −0.200773 + 1.72037i −0.415415 0.909632i 2.64132 + 0.775563i −1.33873 1.09901i 0.755750 + 0.654861i 0.989821 + 0.142315i −2.91938 0.690811i −2.08045 + 1.80272i
113.7 −0.540641 + 0.841254i 0.148133 1.72570i −0.415415 0.909632i 2.58447 + 0.758869i 1.37167 + 1.05760i 0.755750 + 0.654861i 0.989821 + 0.142315i −2.95611 0.511269i −2.03567 + 1.76392i
113.8 −0.540641 + 0.841254i 0.161077 + 1.72454i −0.415415 0.909632i −2.40341 0.705705i −1.53786 0.796853i 0.755750 + 0.654861i 0.989821 + 0.142315i −2.94811 + 0.555568i 1.89306 1.64034i
113.9 −0.540641 + 0.841254i 1.34271 + 1.09414i −0.415415 0.909632i 3.32190 + 0.975398i −1.64637 + 0.538020i 0.755750 + 0.654861i 0.989821 + 0.142315i 0.605721 + 2.93821i −2.61651 + 2.26722i
113.10 −0.540641 + 0.841254i 1.48730 0.887665i −0.415415 0.909632i 0.241809 + 0.0710016i −0.0573418 + 1.73110i 0.755750 + 0.654861i 0.989821 + 0.142315i 1.42410 2.64044i −0.190462 + 0.165036i
113.11 −0.540641 + 0.841254i 1.71418 0.248176i −0.415415 0.909632i 0.707825 + 0.207836i −0.717976 + 1.57623i 0.755750 + 0.654861i 0.989821 + 0.142315i 2.87682 0.850837i −0.557522 + 0.483096i
113.12 −0.540641 + 0.841254i 1.71940 0.208961i −0.415415 0.909632i −4.23656 1.24397i −0.753788 + 1.55942i 0.755750 + 0.654861i 0.989821 + 0.142315i 2.91267 0.718576i 3.33695 2.89148i
113.13 0.540641 0.841254i −1.58317 0.702554i −0.415415 0.909632i 1.47166 + 0.432119i −1.44695 + 0.952016i −0.755750 0.654861i −0.989821 0.142315i 2.01284 + 2.22452i 1.15916 1.00442i
113.14 0.540641 0.841254i −1.30892 + 1.13434i −0.415415 0.909632i 2.33211 + 0.684769i 0.246607 + 1.71441i −0.755750 0.654861i −0.989821 0.142315i 0.426563 2.96952i 1.83690 1.59168i
113.15 0.540641 0.841254i −1.28014 1.16672i −0.415415 0.909632i 0.854693 + 0.250960i −1.67361 + 0.446142i −0.755750 0.654861i −0.989821 0.142315i 0.277508 + 2.98714i 0.673203 0.583334i
113.16 0.540641 0.841254i −1.21880 1.23066i −0.415415 0.909632i −3.64992 1.07171i −1.69423 + 0.359977i −0.755750 0.654861i −0.989821 0.142315i −0.0290431 + 2.99986i −2.87488 + 2.49110i
113.17 0.540641 0.841254i −0.613526 + 1.61975i −0.415415 0.909632i 1.38055 + 0.405366i 1.03092 + 1.39183i −0.755750 0.654861i −0.989821 0.142315i −2.24717 1.98752i 1.08740 0.942236i
113.18 0.540641 0.841254i −0.443821 + 1.67422i −0.415415 0.909632i −1.68096 0.493574i 1.16850 + 1.27852i −0.755750 0.654861i −0.989821 0.142315i −2.60605 1.48611i −1.32402 + 1.14727i
113.19 0.540641 0.841254i 0.137828 1.72656i −0.415415 0.909632i 0.285460 + 0.0838187i −1.37796 1.04940i −0.755750 0.654861i −0.989821 0.142315i −2.96201 0.475936i 0.224844 0.194829i
113.20 0.540641 0.841254i 0.471039 + 1.66677i −0.415415 0.909632i −0.959405 0.281707i 1.65684 + 0.504861i −0.755750 0.654861i −0.989821 0.142315i −2.55625 + 1.57023i −0.755680 + 0.654801i
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.b yes 240
3.b odd 2 1 966.2.r.a 240
23.d odd 22 1 966.2.r.a 240
69.g even 22 1 inner 966.2.r.b yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.r.a 240 3.b odd 2 1
966.2.r.a 240 23.d odd 22 1
966.2.r.b yes 240 1.a even 1 1 trivial
966.2.r.b yes 240 69.g even 22 1 inner

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