# Properties

 Label 216.2.v.b Level $216$ Weight $2$ Character orbit 216.v Analytic conductor $1.725$ Analytic rank $0$ Dimension $192$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$192$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$192 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 9 q^{8} - 12 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$192 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 9 q^{8} - 12 q^{9} - 3 q^{10} - 30 q^{11} - 3 q^{12} + 9 q^{14} - 6 q^{16} - 18 q^{17} + 63 q^{18} - 6 q^{19} - 27 q^{20} - 18 q^{22} - 30 q^{24} - 12 q^{25} + 18 q^{27} - 12 q^{28} - 21 q^{30} - 36 q^{32} - 30 q^{33} + 12 q^{34} - 18 q^{35} - 36 q^{36} - 102 q^{38} + 9 q^{40} + 18 q^{41} - 6 q^{42} - 42 q^{43} - 81 q^{44} - 3 q^{46} - 81 q^{48} - 12 q^{49} + 57 q^{50} - 18 q^{51} + 21 q^{52} + 78 q^{54} - 69 q^{56} - 36 q^{57} - 33 q^{58} - 84 q^{59} - 54 q^{60} + 90 q^{62} - 51 q^{64} - 12 q^{65} + 87 q^{66} + 30 q^{67} + 63 q^{68} - 33 q^{70} + 12 q^{72} - 6 q^{73} + 51 q^{74} - 96 q^{75} + 30 q^{76} + 90 q^{78} - 12 q^{81} - 12 q^{82} - 72 q^{83} - 48 q^{84} + 42 q^{86} - 78 q^{88} + 144 q^{89} + 120 q^{90} - 6 q^{91} - 3 q^{92} - 33 q^{94} - 78 q^{96} - 42 q^{97} + 162 q^{98} - 12 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41043 + 0.103314i 0.584277 + 1.63053i 1.97865 0.291436i −2.25103 0.819307i −0.992542 2.23939i −4.05746 0.715440i −2.76065 + 0.615475i −2.31724 + 1.90536i 3.25957 + 0.923015i
11.2 −1.40972 + 0.112665i 0.00307742 1.73205i 1.97461 0.317652i 2.47648 + 0.901367i 0.190803 + 2.44205i 2.55949 + 0.451307i −2.74786 + 0.670270i −2.99998 0.0106605i −3.59270 0.991660i
11.3 −1.38842 0.268854i −1.38044 1.04613i 1.85543 + 0.746566i −3.21396 1.16979i 1.63538 + 1.82361i 0.199218 + 0.0351275i −2.37541 1.53539i 0.811221 + 2.88824i 4.14784 + 2.48825i
11.4 −1.30304 + 0.549629i −1.04101 + 1.38431i 1.39582 1.43237i 0.437371 + 0.159190i 0.595617 2.37597i 3.46177 + 0.610403i −1.03153 + 2.63362i −0.832609 2.88215i −0.657406 + 0.0329610i
11.5 −1.27615 0.609466i 1.73063 0.0701536i 1.25710 + 1.55554i −0.426624 0.155279i −2.25129 0.965233i 1.28150 + 0.225962i −0.656202 2.75125i 2.99016 0.242820i 0.449799 + 0.458171i
11.6 −1.26354 0.635192i 0.0447181 + 1.73147i 1.19306 + 1.60518i 4.00434 + 1.45746i 1.04331 2.21619i −1.46723 0.258712i −0.487886 2.78603i −2.99600 + 0.154856i −4.13387 4.38508i
11.7 −1.21873 + 0.717416i −1.66645 0.472175i 0.970628 1.74868i 1.70638 + 0.621073i 2.36970 0.620081i −3.32069 0.585528i 0.0715925 + 2.82752i 2.55410 + 1.57371i −2.52520 + 0.467264i
11.8 −1.20377 + 0.742249i 1.24100 1.20827i 0.898133 1.78700i −3.07462 1.11907i −0.597038 + 2.37561i 1.33825 + 0.235969i 0.245250 + 2.81777i 0.0801499 2.99893i 4.53177 0.935029i
11.9 −1.03887 0.959554i 0.130423 1.72713i 0.158512 + 1.99371i 0.588722 + 0.214277i −1.79277 + 1.66912i −4.57427 0.806567i 1.74840 2.22331i −2.96598 0.450516i −0.405997 0.787518i
11.10 −0.955634 1.04248i −1.08798 + 1.34770i −0.173527 + 1.99246i −2.35728 0.857979i 2.44466 0.153705i 2.09025 + 0.368567i 2.24292 1.72316i −0.632581 2.93255i 1.35827 + 3.27733i
11.11 −0.521940 + 1.31437i 1.24100 1.20827i −1.45516 1.37205i 3.07462 + 1.11907i 0.940397 + 2.26178i −1.33825 0.235969i 2.56289 1.19649i 0.0801499 2.99893i −3.07564 + 3.45711i
11.12 −0.494886 + 1.32480i −1.66645 0.472175i −1.51018 1.31125i −1.70638 0.621073i 1.45024 1.97403i 3.32069 + 0.585528i 2.48450 1.35176i 2.55410 + 1.57371i 1.66726 1.95325i
11.13 −0.491528 1.32605i −1.72369 + 0.169964i −1.51680 + 1.30358i 0.715441 + 0.260399i 1.07262 + 2.20215i −3.32699 0.586638i 2.47416 + 1.37060i 2.94222 0.585931i −0.00635759 1.07670i
11.14 −0.322208 1.37702i 0.889428 + 1.48624i −1.79236 + 0.887374i 0.479395 + 0.174486i 1.76000 1.70364i 1.63176 + 0.287723i 1.79945 + 2.18220i −1.41783 + 2.64381i 0.0858049 0.716357i
11.15 −0.315008 + 1.37868i −1.04101 + 1.38431i −1.80154 0.868594i −0.437371 0.159190i −1.58060 1.87129i −3.46177 0.610403i 1.76502 2.21014i −0.832609 2.88215i 0.357248 0.552850i
11.16 −0.279321 1.38635i 1.40807 1.00863i −1.84396 + 0.774477i 2.25586 + 0.821067i −1.79163 1.67035i 1.95474 + 0.344674i 1.58876 + 2.34005i 0.965315 2.84045i 0.508180 3.35677i
11.17 0.133842 + 1.40787i 0.00307742 1.73205i −1.96417 + 0.376862i −2.47648 0.901367i 2.43890 0.227487i −2.55949 0.451307i −0.793459 2.71485i −2.99998 0.0106605i 0.937547 3.60720i
11.18 0.143175 + 1.40695i 0.584277 + 1.63053i −1.95900 + 0.402879i 2.25103 + 0.819307i −2.21041 + 1.05550i 4.05746 + 0.715440i −0.847309 2.69853i −2.31724 + 1.90536i −0.830431 + 3.28438i
11.19 0.160171 1.40511i −1.03533 1.38855i −1.94869 0.450118i −1.93709 0.705042i −2.11691 + 1.23236i 0.744451 + 0.131267i −0.944592 + 2.66604i −0.856167 + 2.87524i −1.30093 + 2.60890i
11.20 0.505867 + 1.32064i −1.38044 1.04613i −1.48820 + 1.33614i 3.21396 + 1.16979i 0.683248 2.35227i −0.199218 0.0351275i −2.51739 1.28947i 0.811221 + 2.88824i 0.0809654 + 4.83625i
See next 80 embeddings (of 192 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 203.32 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.f odd 18 1 inner
216.v even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.v.b 192
3.b odd 2 1 648.2.v.b 192
4.b odd 2 1 864.2.bh.b 192
8.b even 2 1 864.2.bh.b 192
8.d odd 2 1 inner 216.2.v.b 192
24.f even 2 1 648.2.v.b 192
27.e even 9 1 648.2.v.b 192
27.f odd 18 1 inner 216.2.v.b 192
108.l even 18 1 864.2.bh.b 192
216.r odd 18 1 648.2.v.b 192
216.v even 18 1 inner 216.2.v.b 192
216.x odd 18 1 864.2.bh.b 192

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.v.b 192 1.a even 1 1 trivial
216.2.v.b 192 8.d odd 2 1 inner
216.2.v.b 192 27.f odd 18 1 inner
216.2.v.b 192 216.v even 18 1 inner
648.2.v.b 192 3.b odd 2 1
648.2.v.b 192 24.f even 2 1
648.2.v.b 192 27.e even 9 1
648.2.v.b 192 216.r odd 18 1
864.2.bh.b 192 4.b odd 2 1
864.2.bh.b 192 8.b even 2 1
864.2.bh.b 192 108.l even 18 1
864.2.bh.b 192 216.x odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$22\!\cdots\!11$$$$T_{5}^{172} -$$$$26\!\cdots\!20$$$$T_{5}^{170} +$$$$19\!\cdots\!02$$$$T_{5}^{168} +$$$$69\!\cdots\!30$$$$T_{5}^{166} -$$$$10\!\cdots\!58$$$$T_{5}^{164} +$$$$50\!\cdots\!19$$$$T_{5}^{162} +$$$$16\!\cdots\!14$$$$T_{5}^{160} -$$$$30\!\cdots\!07$$$$T_{5}^{158} +$$$$95\!\cdots\!75$$$$T_{5}^{156} +$$$$31\!\cdots\!67$$$$T_{5}^{154} -$$$$56\!\cdots\!98$$$$T_{5}^{152} +$$$$13\!\cdots\!01$$$$T_{5}^{150} +$$$$46\!\cdots\!46$$$$T_{5}^{148} -$$$$71\!\cdots\!09$$$$T_{5}^{146} +$$$$14\!\cdots\!77$$$$T_{5}^{144} +$$$$50\!\cdots\!04$$$$T_{5}^{142} -$$$$64\!\cdots\!67$$$$T_{5}^{140} +$$$$10\!\cdots\!30$$$$T_{5}^{138} +$$$$42\!\cdots\!52$$$$T_{5}^{136} -$$$$41\!\cdots\!37$$$$T_{5}^{134} +$$$$64\!\cdots\!51$$$$T_{5}^{132} +$$$$26\!\cdots\!96$$$$T_{5}^{130} -$$$$18\!\cdots\!17$$$$T_{5}^{128} +$$$$28\!\cdots\!21$$$$T_{5}^{126} +$$$$12\!\cdots\!80$$$$T_{5}^{124} -$$$$57\!\cdots\!80$$$$T_{5}^{122} +$$$$95\!\cdots\!90$$$$T_{5}^{120} +$$$$43\!\cdots\!99$$$$T_{5}^{118} -$$$$11\!\cdots\!60$$$$T_{5}^{116} +$$$$25\!\cdots\!90$$$$T_{5}^{114} +$$$$11\!\cdots\!52$$$$T_{5}^{112} -$$$$11\!\cdots\!68$$$$T_{5}^{110} +$$$$53\!\cdots\!75$$$$T_{5}^{108} +$$$$23\!\cdots\!31$$$$T_{5}^{106} +$$$$93\!\cdots\!52$$$$T_{5}^{104} +$$$$90\!\cdots\!12$$$$T_{5}^{102} +$$$$38\!\cdots\!81$$$$T_{5}^{100} +$$$$58\!\cdots\!29$$$$T_{5}^{98} +$$$$12\!\cdots\!66$$$$T_{5}^{96} +$$$$48\!\cdots\!95$$$$T_{5}^{94} +$$$$11\!\cdots\!27$$$$T_{5}^{92} +$$$$13\!\cdots\!90$$$$T_{5}^{90} +$$$$48\!\cdots\!40$$$$T_{5}^{88} +$$$$14\!\cdots\!20$$$$T_{5}^{86} +$$$$11\!\cdots\!13$$$$T_{5}^{84} +$$$$36\!\cdots\!66$$$$T_{5}^{82} +$$$$12\!\cdots\!66$$$$T_{5}^{80} +$$$$70\!\cdots\!79$$$$T_{5}^{78} +$$$$20\!\cdots\!80$$$$T_{5}^{76} +$$$$71\!\cdots\!76$$$$T_{5}^{74} +$$$$31\!\cdots\!07$$$$T_{5}^{72} +$$$$84\!\cdots\!42$$$$T_{5}^{70} +$$$$28\!\cdots\!80$$$$T_{5}^{68} +$$$$94\!\cdots\!48$$$$T_{5}^{66} +$$$$24\!\cdots\!32$$$$T_{5}^{64} +$$$$69\!\cdots\!99$$$$T_{5}^{62} +$$$$19\!\cdots\!97$$$$T_{5}^{60} +$$$$44\!\cdots\!81$$$$T_{5}^{58} +$$$$10\!\cdots\!78$$$$T_{5}^{56} +$$$$21\!\cdots\!42$$$$T_{5}^{54} +$$$$36\!\cdots\!72$$$$T_{5}^{52} +$$$$55\!\cdots\!18$$$$T_{5}^{50} +$$$$72\!\cdots\!42$$$$T_{5}^{48} +$$$$74\!\cdots\!07$$$$T_{5}^{46} +$$$$50\!\cdots\!22$$$$T_{5}^{44} +$$$$26\!\cdots\!77$$$$T_{5}^{42} +$$$$16\!\cdots\!89$$$$T_{5}^{40} +$$$$65\!\cdots\!91$$$$T_{5}^{38} +$$$$31\!\cdots\!47$$$$T_{5}^{36} +$$$$50\!\cdots\!65$$$$T_{5}^{34} +$$$$43\!\cdots\!23$$$$T_{5}^{32} -$$$$12\!\cdots\!90$$$$T_{5}^{30} -$$$$78\!\cdots\!18$$$$T_{5}^{28} +$$$$67\!\cdots\!71$$$$T_{5}^{26} +$$$$79\!\cdots\!97$$$$T_{5}^{24} -$$$$12\!\cdots\!69$$$$T_{5}^{22} +$$$$30\!\cdots\!33$$$$T_{5}^{20} +$$$$66\!\cdots\!42$$$$T_{5}^{18} -$$$$49\!\cdots\!37$$$$T_{5}^{16} +$$$$10\!\cdots\!25$$$$T_{5}^{14} +$$$$14\!\cdots\!05$$$$T_{5}^{12} -$$$$13\!\cdots\!80$$$$T_{5}^{10} +$$$$39\!\cdots\!40$$$$T_{5}^{8} -$$$$64\!\cdots\!64$$$$T_{5}^{6} +$$$$69\!\cdots\!12$$$$T_{5}^{4} -$$$$47\!\cdots\!56$$$$T_{5}^{2} +$$$$23\!\cdots\!36$$">$$T_{5}^{192} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.