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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,2,Mod(11,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.v (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 11.32
Significant digits:
Format:

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 \( T_{5}^{192} + 6 T_{5}^{190} - 9 T_{5}^{188} + 13251 T_{5}^{186} + 71217 T_{5}^{184} + \cdots + 23\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display