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 $4$

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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:

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 \(S_{3}^{\mathrm{new}}(216, [\chi])\).