# 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 $2$

# 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.

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

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