# Properties

 Label 216.3.r.b Level $216$ Weight $3$ Character orbit 216.r Analytic conductor $5.886$ Analytic rank $0$ Dimension $408$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 216.r (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$408$$ Relative dimension: $$68$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$408q - 6q^{2} - 12q^{3} - 6q^{4} - 6q^{6} - 51q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$408q - 6q^{2} - 12q^{3} - 6q^{4} - 6q^{6} - 51q^{8} - 12q^{9} - 3q^{10} + 30q^{11} + 15q^{12} - 51q^{14} - 6q^{16} - 6q^{17} - 153q^{18} - 6q^{19} - 69q^{20} - 90q^{22} - 84q^{24} - 12q^{25} + 150q^{26} + 126q^{27} - 12q^{28} + 141q^{30} + 84q^{32} - 174q^{33} - 6q^{34} - 6q^{35} - 36q^{36} - 492q^{38} - 81q^{40} - 78q^{41} - 546q^{42} + 30q^{43} + 213q^{44} - 3q^{46} + 207q^{48} - 12q^{49} - 315q^{50} + 630q^{51} - 33q^{52} + 78q^{54} - 405q^{56} + 288q^{57} - 141q^{58} + 912q^{59} - 882q^{60} + 294q^{62} + 381q^{64} - 12q^{65} + 393q^{66} + 174q^{67} - 573q^{68} - 141q^{70} + 228q^{72} - 6q^{73} - 207q^{74} - 348q^{75} + 858q^{76} - 216q^{78} + 798q^{80} - 12q^{81} - 12q^{82} - 732q^{83} + 654q^{84} + 198q^{86} + 858q^{88} - 444q^{89} - 420q^{90} - 6q^{91} - 1077q^{92} + 345q^{94} - 1626q^{96} - 294q^{97} - 1104q^{98} - 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.99994 + 0.0149705i −2.98925 + 0.253717i 3.99955 0.0598802i 0.800962 2.20063i 5.97454 0.552171i −4.82501 0.850780i −7.99798 + 0.179632i 8.87125 1.51685i −1.56893 + 4.41312i
43.2 −1.99900 + 0.0631437i 2.99997 + 0.0137424i 3.99203 0.252449i 2.50897 6.89334i −5.99781 + 0.161958i 12.4678 + 2.19840i −7.96413 + 0.756718i 8.99962 + 0.0824533i −4.58017 + 13.9382i
43.3 −1.99064 0.193255i 0.606643 2.93802i 3.92530 + 0.769404i 2.98706 8.20689i −1.77540 + 5.73131i −11.0729 1.95245i −7.66518 2.29019i −8.26397 3.56466i −7.53219 + 15.7597i
43.4 −1.98421 0.250810i 2.36409 + 1.84692i 3.87419 + 0.995319i −0.923949 + 2.53853i −4.22762 4.25761i −9.76491 1.72182i −7.43757 2.94661i 2.17780 + 8.73254i 2.47000 4.80524i
43.5 −1.97753 + 0.298974i −1.80737 2.39446i 3.82123 1.18246i −0.497328 + 1.36640i 4.29000 + 4.19475i 4.84771 + 0.854783i −7.20306 + 3.48079i −2.46685 + 8.65533i 0.574963 2.85078i
43.6 −1.95702 + 0.412389i 1.01799 + 2.82200i 3.65987 1.61411i −2.81454 + 7.73289i −3.15598 5.10292i 11.3893 + 2.00825i −6.49681 + 4.66814i −6.92741 + 5.74552i 2.31916 16.2941i
43.7 −1.88567 + 0.666531i 1.57623 2.55255i 3.11147 2.51371i −2.77462 + 7.62322i −1.27090 + 5.86386i −4.46660 0.787582i −4.19173 + 6.81391i −4.03098 8.04681i 0.150905 16.2242i
43.8 −1.87801 0.687799i 2.64392 1.41764i 3.05387 + 2.58339i −1.12060 + 3.07881i −5.94036 + 0.843855i 1.91560 + 0.337772i −3.95835 6.95208i 4.98062 7.49623i 4.22210 5.01131i
43.9 −1.85647 0.743978i −0.763222 + 2.90129i 2.89299 + 2.76235i 2.17232 5.96840i 3.57540 4.81835i 1.76650 + 0.311482i −3.31564 7.28056i −7.83498 4.42866i −8.47321 + 9.46402i
43.10 −1.84655 + 0.768275i −1.75085 + 2.43608i 2.81951 2.83732i 0.197813 0.543487i 1.36146 5.84349i −0.709060 0.125026i −3.02652 + 7.40541i −2.86902 8.53046i 0.0522754 + 1.15555i
43.11 −1.82525 0.817590i −2.80001 + 1.07699i 2.66309 + 2.98461i −1.58292 + 4.34904i 5.99127 + 0.323475i 10.0819 + 1.77771i −2.42063 7.62499i 6.68016 6.03120i 6.44495 6.64391i
43.12 −1.74775 + 0.972303i 0.442507 + 2.96719i 2.10925 3.39868i 1.73504 4.76700i −3.65840 4.75564i −2.19570 0.387160i −0.381894 + 7.99088i −8.60837 + 2.62600i 1.60254 + 10.0185i
43.13 −1.72745 + 1.00793i 2.91257 + 0.718971i 1.96817 3.48228i 0.957213 2.62992i −5.75600 + 1.69367i −7.64494 1.34801i 0.109963 + 7.99924i 7.96616 + 4.18811i 0.997227 + 5.50786i
43.14 −1.66732 1.10455i −2.32187 1.89972i 1.55994 + 3.68328i −2.67920 + 7.36103i 1.77297 + 5.73206i −10.3449 1.82408i 1.46743 7.86426i 1.78214 + 8.82179i 12.5977 9.31393i
43.15 −1.53457 + 1.28261i −2.87953 + 0.841593i 0.709834 3.93651i −2.60916 + 7.16862i 3.33943 4.98480i −6.69780 1.18100i 3.95971 + 6.95131i 7.58344 4.84679i −5.19057 14.3473i
43.16 −1.51663 + 1.30378i 0.997347 2.82936i 0.600312 3.95470i 1.28525 3.53120i 2.17627 + 5.59141i 7.55489 + 1.33213i 4.24561 + 6.78047i −7.01060 5.64371i 2.65466 + 7.03119i
43.17 −1.39225 + 1.43584i −2.77266 1.14558i −0.123276 3.99810i 2.62649 7.21622i 5.50511 2.38616i −0.683553 0.120529i 5.91227 + 5.38935i 6.37529 + 6.35261i 6.70461 + 13.8180i
43.18 −1.37730 1.45019i −2.32187 1.89972i −0.206111 + 3.99469i 2.67920 7.36103i 0.442943 + 5.98363i 10.3449 + 1.82408i 6.07694 5.20297i 1.78214 + 8.82179i −14.3650 + 6.25298i
43.19 −1.18104 + 1.61405i 2.99216 + 0.216681i −1.21030 3.81250i −1.70614 + 4.68759i −3.88359 + 4.57359i 5.07601 + 0.895037i 7.58297 + 2.54923i 8.90610 + 1.29669i −5.55098 8.29002i
43.20 −1.12212 1.65555i −2.80001 + 1.07699i −1.48169 + 3.71545i 1.58292 4.34904i 4.92497 + 3.42705i −10.0819 1.77771i 7.81375 1.71617i 6.68016 6.03120i −8.97627 + 2.25954i
See next 80 embeddings (of 408 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.68 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
27.e even 9 1 inner
216.r odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.r.b 408
8.d odd 2 1 inner 216.3.r.b 408
27.e even 9 1 inner 216.3.r.b 408
216.r odd 18 1 inner 216.3.r.b 408

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.r.b 408 1.a even 1 1 trivial
216.3.r.b 408 8.d odd 2 1 inner
216.3.r.b 408 27.e even 9 1 inner
216.3.r.b 408 216.r odd 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$73\!\cdots\!03$$$$T_{5}^{396} +$$$$45\!\cdots\!97$$$$T_{5}^{394} +$$$$67\!\cdots\!98$$$$T_{5}^{392} -$$$$10\!\cdots\!22$$$$T_{5}^{390} -$$$$61\!\cdots\!75$$$$T_{5}^{388} -$$$$49\!\cdots\!20$$$$T_{5}^{386} +$$$$10\!\cdots\!85$$$$T_{5}^{384} +$$$$59\!\cdots\!94$$$$T_{5}^{382} -$$$$14\!\cdots\!89$$$$T_{5}^{380} -$$$$87\!\cdots\!70$$$$T_{5}^{378} -$$$$44\!\cdots\!04$$$$T_{5}^{376} +$$$$47\!\cdots\!88$$$$T_{5}^{374} +$$$$58\!\cdots\!47$$$$T_{5}^{372} +$$$$25\!\cdots\!59$$$$T_{5}^{370} -$$$$61\!\cdots\!96$$$$T_{5}^{368} -$$$$32\!\cdots\!20$$$$T_{5}^{366} -$$$$11\!\cdots\!71$$$$T_{5}^{364} +$$$$48\!\cdots\!13$$$$T_{5}^{362} +$$$$15\!\cdots\!18$$$$T_{5}^{360} +$$$$38\!\cdots\!99$$$$T_{5}^{358} -$$$$28\!\cdots\!77$$$$T_{5}^{356} -$$$$62\!\cdots\!70$$$$T_{5}^{354} -$$$$80\!\cdots\!88$$$$T_{5}^{352} +$$$$12\!\cdots\!68$$$$T_{5}^{350} +$$$$21\!\cdots\!87$$$$T_{5}^{348} +$$$$94\!\cdots\!06$$$$T_{5}^{346} -$$$$44\!\cdots\!09$$$$T_{5}^{344} -$$$$64\!\cdots\!73$$$$T_{5}^{342} +$$$$80\!\cdots\!73$$$$T_{5}^{340} +$$$$12\!\cdots\!23$$$$T_{5}^{338} +$$$$16\!\cdots\!03$$$$T_{5}^{336} -$$$$42\!\cdots\!21$$$$T_{5}^{334} -$$$$27\!\cdots\!96$$$$T_{5}^{332} -$$$$37\!\cdots\!12$$$$T_{5}^{330} +$$$$14\!\cdots\!46$$$$T_{5}^{328} +$$$$48\!\cdots\!32$$$$T_{5}^{326} +$$$$73\!\cdots\!22$$$$T_{5}^{324} -$$$$35\!\cdots\!81$$$$T_{5}^{322} -$$$$59\!\cdots\!06$$$$T_{5}^{320} -$$$$12\!\cdots\!23$$$$T_{5}^{318} +$$$$72\!\cdots\!67$$$$T_{5}^{316} +$$$$35\!\cdots\!05$$$$T_{5}^{314} +$$$$18\!\cdots\!25$$$$T_{5}^{312} -$$$$12\!\cdots\!15$$$$T_{5}^{310} +$$$$47\!\cdots\!63$$$$T_{5}^{308} -$$$$24\!\cdots\!74$$$$T_{5}^{306} +$$$$17\!\cdots\!52$$$$T_{5}^{304} -$$$$18\!\cdots\!81$$$$T_{5}^{302} +$$$$28\!\cdots\!36$$$$T_{5}^{300} -$$$$20\!\cdots\!17$$$$T_{5}^{298} +$$$$33\!\cdots\!54$$$$T_{5}^{296} -$$$$28\!\cdots\!17$$$$T_{5}^{294} +$$$$19\!\cdots\!24$$$$T_{5}^{292} -$$$$43\!\cdots\!04$$$$T_{5}^{290} +$$$$25\!\cdots\!04$$$$T_{5}^{288} -$$$$16\!\cdots\!65$$$$T_{5}^{286} +$$$$45\!\cdots\!67$$$$T_{5}^{284} -$$$$19\!\cdots\!40$$$$T_{5}^{282} +$$$$10\!\cdots\!94$$$$T_{5}^{280} -$$$$38\!\cdots\!15$$$$T_{5}^{278} +$$$$13\!\cdots\!07$$$$T_{5}^{276} -$$$$55\!\cdots\!33$$$$T_{5}^{274} +$$$$26\!\cdots\!89$$$$T_{5}^{272} -$$$$76\!\cdots\!70$$$$T_{5}^{270} +$$$$20\!\cdots\!83$$$$T_{5}^{268} -$$$$16\!\cdots\!03$$$$T_{5}^{266} +$$$$38\!\cdots\!51$$$$T_{5}^{264} -$$$$34\!\cdots\!12$$$$T_{5}^{262} +$$$$82\!\cdots\!81$$$$T_{5}^{260} -$$$$17\!\cdots\!36$$$$T_{5}^{258} -$$$$20\!\cdots\!25$$$$T_{5}^{256} -$$$$37\!\cdots\!09$$$$T_{5}^{254} +$$$$64\!\cdots\!64$$$$T_{5}^{252} +$$$$22\!\cdots\!27$$$$T_{5}^{250} +$$$$14\!\cdots\!93$$$$T_{5}^{248} -$$$$21\!\cdots\!10$$$$T_{5}^{246} -$$$$12\!\cdots\!64$$$$T_{5}^{244} -$$$$52\!\cdots\!70$$$$T_{5}^{242} +$$$$59\!\cdots\!29$$$$T_{5}^{240} +$$$$50\!\cdots\!62$$$$T_{5}^{238} +$$$$16\!\cdots\!38$$$$T_{5}^{236} -$$$$14\!\cdots\!05$$$$T_{5}^{234} -$$$$15\!\cdots\!81$$$$T_{5}^{232} -$$$$43\!\cdots\!96$$$$T_{5}^{230} +$$$$29\!\cdots\!51$$$$T_{5}^{228} +$$$$39\!\cdots\!87$$$$T_{5}^{226} +$$$$10\!\cdots\!16$$$$T_{5}^{224} -$$$$51\!\cdots\!06$$$$T_{5}^{222} -$$$$81\!\cdots\!14$$$$T_{5}^{220} -$$$$20\!\cdots\!83$$$$T_{5}^{218} +$$$$74\!\cdots\!91$$$$T_{5}^{216} +$$$$13\!\cdots\!76$$$$T_{5}^{214} +$$$$33\!\cdots\!98$$$$T_{5}^{212} -$$$$90\!\cdots\!43$$$$T_{5}^{210} -$$$$18\!\cdots\!09$$$$T_{5}^{208} -$$$$46\!\cdots\!83$$$$T_{5}^{206} +$$$$91\!\cdots\!83$$$$T_{5}^{204} +$$$$20\!\cdots\!76$$$$T_{5}^{202} +$$$$52\!\cdots\!33$$$$T_{5}^{200} -$$$$76\!\cdots\!59$$$$T_{5}^{198} -$$$$18\!\cdots\!49$$$$T_{5}^{196} -$$$$48\!\cdots\!34$$$$T_{5}^{194} +$$$$53\!\cdots\!71$$$$T_{5}^{192} +$$$$13\!\cdots\!38$$$$T_{5}^{190} +$$$$36\!\cdots\!31$$$$T_{5}^{188} -$$$$31\!\cdots\!09$$$$T_{5}^{186} -$$$$79\!\cdots\!24$$$$T_{5}^{184} -$$$$21\!\cdots\!33$$$$T_{5}^{182} +$$$$16\!\cdots\!14$$$$T_{5}^{180} +$$$$37\!\cdots\!82$$$$T_{5}^{178} +$$$$10\!\cdots\!25$$$$T_{5}^{176} -$$$$77\!\cdots\!93$$$$T_{5}^{174} -$$$$14\!\cdots\!10$$$$T_{5}^{172} -$$$$38\!\cdots\!77$$$$T_{5}^{170} +$$$$31\!\cdots\!54$$$$T_{5}^{168} +$$$$46\!\cdots\!10$$$$T_{5}^{166} +$$$$11\!\cdots\!41$$$$T_{5}^{164} -$$$$10\!\cdots\!87$$$$T_{5}^{162} -$$$$11\!\cdots\!58$$$$T_{5}^{160} -$$$$28\!\cdots\!47$$$$T_{5}^{158} +$$$$30\!\cdots\!27$$$$T_{5}^{156} +$$$$24\!\cdots\!42$$$$T_{5}^{154} +$$$$55\!\cdots\!14$$$$T_{5}^{152} -$$$$68\!\cdots\!44$$$$T_{5}^{150} -$$$$39\!\cdots\!89$$$$T_{5}^{148} -$$$$81\!\cdots\!30$$$$T_{5}^{146} +$$$$12\!\cdots\!92$$$$T_{5}^{144} +$$$$49\!\cdots\!07$$$$T_{5}^{142} +$$$$91\!\cdots\!82$$$$T_{5}^{140} -$$$$16\!\cdots\!44$$$$T_{5}^{138} -$$$$43\!\cdots\!10$$$$T_{5}^{136} -$$$$75\!\cdots\!00$$$$T_{5}^{134} +$$$$17\!\cdots\!89$$$$T_{5}^{132} +$$$$21\!\cdots\!51$$$$T_{5}^{130} +$$$$45\!\cdots\!35$$$$T_{5}^{128} -$$$$13\!\cdots\!96$$$$T_{5}^{126} -$$$$73\!\cdots\!48$$$$T_{5}^{124} -$$$$22\!\cdots\!66$$$$T_{5}^{122} +$$$$82\!\cdots\!02$$$$T_{5}^{120} -$$$$75\!\cdots\!42$$$$T_{5}^{118} +$$$$94\!\cdots\!05$$$$T_{5}^{116} -$$$$37\!\cdots\!50$$$$T_{5}^{114} +$$$$67\!\cdots\!47$$$$T_{5}^{112} -$$$$38\!\cdots\!13$$$$T_{5}^{110} +$$$$13\!\cdots\!41$$$$T_{5}^{108} -$$$$33\!\cdots\!81$$$$T_{5}^{106} +$$$$13\!\cdots\!85$$$$T_{5}^{104} -$$$$37\!\cdots\!53$$$$T_{5}^{102} +$$$$11\!\cdots\!02$$$$T_{5}^{100} -$$$$42\!\cdots\!50$$$$T_{5}^{98} +$$$$80\!\cdots\!55$$$$T_{5}^{96} -$$$$30\!\cdots\!72$$$$T_{5}^{94} +$$$$94\!\cdots\!61$$$$T_{5}^{92} -$$$$13\!\cdots\!94$$$$T_{5}^{90} +$$$$56\!\cdots\!55$$$$T_{5}^{88} -$$$$15\!\cdots\!04$$$$T_{5}^{86} +$$$$16\!\cdots\!44$$$$T_{5}^{84} -$$$$73\!\cdots\!70$$$$T_{5}^{82} +$$$$17\!\cdots\!37$$$$T_{5}^{80} -$$$$13\!\cdots\!91$$$$T_{5}^{78} +$$$$65\!\cdots\!04$$$$T_{5}^{76} -$$$$14\!\cdots\!06$$$$T_{5}^{74} +$$$$64\!\cdots\!98$$$$T_{5}^{72} -$$$$32\!\cdots\!55$$$$T_{5}^{70} +$$$$53\!\cdots\!14$$$$T_{5}^{68} -$$$$46\!\cdots\!40$$$$T_{5}^{66} +$$$$69\!\cdots\!23$$$$T_{5}^{64} -$$$$25\!\cdots\!76$$$$T_{5}^{62} +$$$$22\!\cdots\!92$$$$T_{5}^{60} -$$$$22\!\cdots\!06$$$$T_{5}^{58} +$$$$24\!\cdots\!68$$$$T_{5}^{56} -$$$$62\!\cdots\!75$$$$T_{5}^{54} +$$$$70\!\cdots\!89$$$$T_{5}^{52} -$$$$13\!\cdots\!95$$$$T_{5}^{50} +$$$$35\!\cdots\!28$$$$T_{5}^{48} -$$$$54\!\cdots\!60$$$$T_{5}^{46} +$$$$95\!\cdots\!52$$$$T_{5}^{44} -$$$$18\!\cdots\!84$$$$T_{5}^{42} +$$$$23\!\cdots\!57$$$$T_{5}^{40} -$$$$22\!\cdots\!27$$$$T_{5}^{38} +$$$$23\!\cdots\!97$$$$T_{5}^{36} -$$$$15\!\cdots\!76$$$$T_{5}^{34} +$$$$49\!\cdots\!84$$$$T_{5}^{32} -$$$$36\!\cdots\!04$$$$T_{5}^{30} +$$$$17\!\cdots\!68$$$$T_{5}^{28} +$$$$95\!\cdots\!76$$$$T_{5}^{26} +$$$$73\!\cdots\!08$$$$T_{5}^{24} +$$$$17\!\cdots\!60$$$$T_{5}^{22} +$$$$56\!\cdots\!28$$$$T_{5}^{20} +$$$$55\!\cdots\!12$$$$T_{5}^{18} +$$$$11\!\cdots\!60$$$$T_{5}^{16} -$$$$40\!\cdots\!16$$$$T_{5}^{14} +$$$$37\!\cdots\!04$$$$T_{5}^{12} -$$$$31\!\cdots\!92$$$$T_{5}^{10} +$$$$33\!\cdots\!20$$$$T_{5}^{8} +$$$$19\!\cdots\!72$$$$T_{5}^{6} +$$$$47\!\cdots\!20$$$$T_{5}^{4} +$$$$17\!\cdots\!16$$$$T_{5}^{2} +$$$$29\!\cdots\!44$$">$$T_{5}^{408} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.