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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(43,216)]
 
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, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.r (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: \(5.88557371018\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9} - 3 q^{10} + 30 q^{11} + 15 q^{12} - 51 q^{14} - 6 q^{16} - 6 q^{17} - 153 q^{18} - 6 q^{19} - 69 q^{20} - 90 q^{22} - 84 q^{24} - 12 q^{25} + 150 q^{26} + 126 q^{27} - 12 q^{28} + 141 q^{30} + 84 q^{32} - 174 q^{33} - 6 q^{34} - 6 q^{35} - 36 q^{36} - 492 q^{38} - 81 q^{40} - 78 q^{41} - 546 q^{42} + 30 q^{43} + 213 q^{44} - 3 q^{46} + 207 q^{48} - 12 q^{49} - 315 q^{50} + 630 q^{51} - 33 q^{52} + 78 q^{54} - 405 q^{56} + 288 q^{57} - 141 q^{58} + 912 q^{59} - 882 q^{60} + 294 q^{62} + 381 q^{64} - 12 q^{65} + 393 q^{66} + 174 q^{67} - 573 q^{68} - 141 q^{70} + 228 q^{72} - 6 q^{73} - 207 q^{74} - 348 q^{75} + 858 q^{76} - 216 q^{78} + 798 q^{80} - 12 q^{81} - 12 q^{82} - 732 q^{83} + 654 q^{84} + 198 q^{86} + 858 q^{88} - 444 q^{89} - 420 q^{90} - 6 q^{91} - 1077 q^{92} + 345 q^{94} - 1626 q^{96} - 294 q^{97} - 1104 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 43.68
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 \( T_{5}^{408} + 6 T_{5}^{406} + 171 T_{5}^{404} - 3620553 T_{5}^{402} - 22781187 T_{5}^{400} + \cdots + 29\!\cdots\!44 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display