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Properties

Label 891.2.o.b
Level $891$
Weight $2$
Character orbit 891.o
Analytic conductor $7.115$
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] = [891,2,Mod(98,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.98");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.o (of order \(18\), degree \(6\), not 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 297)
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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{4} + 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{4} + 18 q^{5} - 3 q^{11} + 12 q^{14} - 36 q^{20} - 15 q^{22} - 24 q^{23} - 6 q^{25} - 6 q^{31} - 18 q^{34} - 6 q^{37} + 102 q^{38} - 63 q^{44} - 12 q^{47} - 12 q^{49} - 12 q^{55} + 114 q^{56} - 12 q^{58} - 144 q^{59} - 114 q^{64} + 102 q^{67} - 42 q^{70} + 18 q^{71} + 9 q^{77} - 24 q^{82} + 18 q^{86} + 78 q^{88} - 6 q^{91} + 126 q^{92} - 186 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1 −2.58014 − 0.939096i 0 4.24315 + 3.56043i 0.597541 + 0.105363i 0 0.904555 + 1.07801i −4.85863 − 8.41540i 0 −1.44280 − 0.832999i
98.2 −2.30835 − 0.840172i 0 3.09051 + 2.59325i −3.84216 − 0.677476i 0 −1.90677 − 2.27241i −2.49872 − 4.32792i 0 8.29986 + 4.79193i
98.3 −2.30721 − 0.839756i 0 3.08595 + 2.58942i 1.10932 + 0.195604i 0 −2.12667 − 2.53447i −2.49017 − 4.31310i 0 −2.39519 − 1.38286i
98.4 −2.22314 − 0.809156i 0 2.75552 + 2.31215i 0.747912 + 0.131877i 0 −0.877752 − 1.04606i −1.88919 − 3.27217i 0 −1.55600 − 0.898358i
98.5 −2.11576 − 0.770073i 0 2.35133 + 1.97300i 3.28259 + 0.578809i 0 1.96006 + 2.33590i −1.20395 − 2.08530i 0 −6.49944 − 3.75245i
98.6 −1.77471 − 0.645941i 0 1.20026 + 1.00714i −0.520958 − 0.0918590i 0 2.50734 + 2.98813i 0.409045 + 0.708487i 0 0.865213 + 0.499531i
98.7 −1.73602 − 0.631860i 0 1.08243 + 0.908270i 3.14246 + 0.554101i 0 −2.29615 − 2.73645i 0.542206 + 0.939128i 0 −5.10527 − 2.94753i
98.8 −1.66410 − 0.605684i 0 0.870301 + 0.730269i −1.43335 − 0.252738i 0 1.86883 + 2.22718i 0.764944 + 1.32492i 0 2.23216 + 1.28874i
98.9 −1.42882 − 0.520048i 0 0.238990 + 0.200537i −2.24064 − 0.395085i 0 −0.982597 − 1.17101i 1.28333 + 2.22280i 0 2.99601 + 1.72975i
98.10 −1.13818 − 0.414264i 0 −0.408250 − 0.342562i 3.58460 + 0.632062i 0 −0.0196718 − 0.0234440i 1.53398 + 2.65692i 0 −3.81808 − 2.20437i
98.11 −1.12891 − 0.410890i 0 −0.426478 − 0.357857i −2.10891 − 0.371858i 0 −2.52158 − 3.00510i 1.53578 + 2.66005i 0 2.22799 + 1.28633i
98.12 −1.03946 − 0.378332i 0 −0.594748 − 0.499053i −0.474093 − 0.0835953i 0 −0.116143 − 0.138414i 1.53558 + 2.65970i 0 0.461173 + 0.266258i
98.13 −1.01597 − 0.369781i 0 −0.636641 − 0.534205i 2.91774 + 0.514477i 0 0.815076 + 0.971370i 1.53043 + 2.65079i 0 −2.77408 − 1.60162i
98.14 −0.543079 − 0.197664i 0 −1.27623 − 1.07088i −2.88423 − 0.508568i 0 3.04569 + 3.62971i 1.05935 + 1.83484i 0 1.46584 + 0.846302i
98.15 −0.327809 − 0.119313i 0 −1.43887 − 1.20735i −0.699723 − 0.123380i 0 1.00175 + 1.19384i 0.676467 + 1.17168i 0 0.214654 + 0.123931i
98.16 −0.107374 − 0.0390808i 0 −1.52209 − 1.27718i 2.02763 + 0.357525i 0 2.63324 + 3.13817i 0.227783 + 0.394532i 0 −0.203741 − 0.117630i
98.17 0.107374 + 0.0390808i 0 −1.52209 − 1.27718i 2.02763 + 0.357525i 0 −2.63324 − 3.13817i −0.227783 − 0.394532i 0 0.203741 + 0.117630i
98.18 0.327809 + 0.119313i 0 −1.43887 − 1.20735i −0.699723 − 0.123380i 0 −1.00175 − 1.19384i −0.676467 − 1.17168i 0 −0.214654 − 0.123931i
98.19 0.543079 + 0.197664i 0 −1.27623 − 1.07088i −2.88423 − 0.508568i 0 −3.04569 − 3.62971i −1.05935 − 1.83484i 0 −1.46584 − 0.846302i
98.20 1.01597 + 0.369781i 0 −0.636641 − 0.534205i 2.91774 + 0.514477i 0 −0.815076 − 0.971370i −1.53043 − 2.65079i 0 2.77408 + 1.60162i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 98.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
27.f odd 18 1 inner
297.o even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.o.b 192
3.b odd 2 1 297.2.o.b ✓ 192
11.b odd 2 1 inner 891.2.o.b 192
27.e even 9 1 297.2.o.b ✓ 192
27.f odd 18 1 inner 891.2.o.b 192
33.d even 2 1 297.2.o.b ✓ 192
297.o even 18 1 inner 891.2.o.b 192
297.q odd 18 1 297.2.o.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.o.b ✓ 192 3.b odd 2 1
297.2.o.b ✓ 192 27.e even 9 1
297.2.o.b ✓ 192 33.d even 2 1
297.2.o.b ✓ 192 297.q odd 18 1
891.2.o.b 192 1.a even 1 1 trivial
891.2.o.b 192 11.b odd 2 1 inner
891.2.o.b 192 27.f odd 18 1 inner
891.2.o.b 192 297.o even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} + 6 T_{2}^{190} + 9 T_{2}^{188} + 1389 T_{2}^{186} + 7974 T_{2}^{184} + \cdots + 48\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display