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Properties

Label 558.2.g.a
Level $558$
Weight $2$
Character orbit 558.g
Analytic conductor $4.456$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [558,2,Mod(211,558)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(558, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("558.211"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 558.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.45565243279\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{2} + 4 q^{3} - 16 q^{4} - q^{5} - 2 q^{6} + q^{7} + 32 q^{8} + 12 q^{9} - q^{10} + 4 q^{11} - 2 q^{12} - 10 q^{13} - 2 q^{14} - 13 q^{15} - 16 q^{16} - 4 q^{17} - 6 q^{18} + 4 q^{19} + 2 q^{20}+ \cdots + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1 −0.500000 − 0.866025i −1.69035 − 0.377782i −0.500000 + 0.866025i −1.17303 − 2.03175i 0.518006 + 1.65278i 1.52021 − 2.63308i 1.00000 2.71456 + 1.27717i −1.17303 + 2.03175i
211.2 −0.500000 − 0.866025i −1.64881 − 0.530487i −0.500000 + 0.866025i 1.26260 + 2.18689i 0.364991 + 1.69316i −1.93524 + 3.35193i 1.00000 2.43717 + 1.74935i 1.26260 − 2.18689i
211.3 −0.500000 − 0.866025i −1.59012 + 0.686665i −0.500000 + 0.866025i −1.38064 − 2.39134i 1.38973 + 1.03375i −0.281122 + 0.486917i 1.00000 2.05698 − 2.18376i −1.38064 + 2.39134i
211.4 −0.500000 − 0.866025i −1.35283 − 1.08159i −0.500000 + 0.866025i 0.281395 + 0.487391i −0.260271 + 1.71238i 0.502527 − 0.870402i 1.00000 0.660311 + 2.92643i 0.281395 − 0.487391i
211.5 −0.500000 − 0.866025i −1.06032 + 1.36957i −0.500000 + 0.866025i 1.73418 + 3.00368i 1.71624 + 0.233475i 0.0972558 − 0.168452i 1.00000 −0.751457 − 2.90436i 1.73418 − 3.00368i
211.6 −0.500000 − 0.866025i −0.735824 + 1.56798i −0.500000 + 0.866025i 0.112175 + 0.194293i 1.72582 − 0.146748i 1.36938 − 2.37184i 1.00000 −1.91712 − 2.30752i 0.112175 − 0.194293i
211.7 −0.500000 − 0.866025i −0.551268 − 1.64198i −0.500000 + 0.866025i −1.55594 − 2.69497i −1.14636 + 1.29840i −2.29219 + 3.97020i 1.00000 −2.39221 + 1.81034i −1.55594 + 2.69497i
211.8 −0.500000 − 0.866025i 0.236607 − 1.71581i −0.500000 + 0.866025i 0.111920 + 0.193852i −1.60424 + 0.652999i 0.637652 − 1.10445i 1.00000 −2.88803 − 0.811948i 0.111920 − 0.193852i
211.9 −0.500000 − 0.866025i 0.369655 + 1.69215i −0.500000 + 0.866025i 1.11634 + 1.93356i 1.28061 − 1.16620i −1.31227 + 2.27292i 1.00000 −2.72671 + 1.25102i 1.11634 − 1.93356i
211.10 −0.500000 − 0.866025i 1.06506 − 1.36589i −0.500000 + 0.866025i 0.473803 + 0.820651i −1.71542 − 0.239428i 0.356824 − 0.618038i 1.00000 −0.731286 − 2.90951i 0.473803 − 0.820651i
211.11 −0.500000 − 0.866025i 1.07682 + 1.35663i −0.500000 + 0.866025i 0.251592 + 0.435771i 0.636470 − 1.61087i 1.49612 − 2.59136i 1.00000 −0.680914 + 2.92170i 0.251592 − 0.435771i
211.12 −0.500000 − 0.866025i 1.21171 + 1.23764i −0.500000 + 0.866025i −2.00250 − 3.46843i 0.465974 − 1.66819i −0.535456 + 0.927438i 1.00000 −0.0635133 + 2.99933i −2.00250 + 3.46843i
211.13 −0.500000 − 0.866025i 1.54827 − 0.776432i −0.500000 + 0.866025i −1.91192 − 3.31155i −1.44655 − 0.952629i 1.95675 − 3.38919i 1.00000 1.79431 − 2.40426i −1.91192 + 3.31155i
211.14 −0.500000 − 0.866025i 1.68060 − 0.419045i −0.500000 + 0.866025i −0.487564 − 0.844486i −1.20320 − 1.24592i −0.761936 + 1.31971i 1.00000 2.64880 − 1.40849i −0.487564 + 0.844486i
211.15 −0.500000 − 0.866025i 1.72031 − 0.201361i −0.500000 + 0.866025i 2.15411 + 3.73103i −1.03454 − 1.38915i 1.90014 − 3.29114i 1.00000 2.91891 − 0.692806i 2.15411 − 3.73103i
211.16 −0.500000 − 0.866025i 1.72050 + 0.199742i −0.500000 + 0.866025i 0.513483 + 0.889379i −0.687266 − 1.58986i −2.21865 + 3.84281i 1.00000 2.92021 + 0.687310i 0.513483 − 0.889379i
439.1 −0.500000 + 0.866025i −1.69035 + 0.377782i −0.500000 − 0.866025i −1.17303 + 2.03175i 0.518006 − 1.65278i 1.52021 + 2.63308i 1.00000 2.71456 − 1.27717i −1.17303 − 2.03175i
439.2 −0.500000 + 0.866025i −1.64881 + 0.530487i −0.500000 − 0.866025i 1.26260 − 2.18689i 0.364991 − 1.69316i −1.93524 − 3.35193i 1.00000 2.43717 − 1.74935i 1.26260 + 2.18689i
439.3 −0.500000 + 0.866025i −1.59012 − 0.686665i −0.500000 − 0.866025i −1.38064 + 2.39134i 1.38973 − 1.03375i −0.281122 − 0.486917i 1.00000 2.05698 + 2.18376i −1.38064 − 2.39134i
439.4 −0.500000 + 0.866025i −1.35283 + 1.08159i −0.500000 − 0.866025i 0.281395 − 0.487391i −0.260271 − 1.71238i 0.502527 + 0.870402i 1.00000 0.660311 − 2.92643i 0.281395 + 0.487391i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.16
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
279.e even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 558.2.g.a ✓ 32
3.b odd 2 1 1674.2.g.b 32
9.c even 3 1 558.2.h.a yes 32
9.d odd 6 1 1674.2.h.b 32
31.c even 3 1 558.2.h.a yes 32
93.h odd 6 1 1674.2.h.b 32
279.e even 3 1 inner 558.2.g.a ✓ 32
279.q odd 6 1 1674.2.g.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
558.2.g.a ✓ 32 1.a even 1 1 trivial
558.2.g.a ✓ 32 279.e even 3 1 inner
558.2.h.a yes 32 9.c even 3 1
558.2.h.a yes 32 31.c even 3 1
1674.2.g.b 32 3.b odd 2 1
1674.2.g.b 32 279.q odd 6 1
1674.2.h.b 32 9.d odd 6 1
1674.2.h.b 32 93.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + T_{5}^{31} + 50 T_{5}^{30} + 27 T_{5}^{29} + 1530 T_{5}^{28} + 448 T_{5}^{27} + \cdots + 123201 \) acting on \(S_{2}^{\mathrm{new}}(558, [\chi])\). Copy content Toggle raw display