# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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