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Properties

Label 558.2.q.a
Level $558$
Weight $2$
Character orbit 558.q
Analytic conductor $4.456$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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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(185,558)] 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([5, 3])) 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("558.185"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 558.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.45565243279\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} + 12 q^{5} - 4 q^{7} - 4 q^{9} - 32 q^{16} + 8 q^{18} - 8 q^{19} + 12 q^{20} + 44 q^{25} - 8 q^{28} + 8 q^{31} - 36 q^{33} - 8 q^{36} + 36 q^{38} - 8 q^{39} + 24 q^{41} - 8 q^{45} - 48 q^{47}+ \cdots - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
185.1 −0.866025 + 0.500000i −1.71868 − 0.214796i 0.500000 − 0.866025i −1.16560 − 0.672958i 1.59582 − 0.673321i −2.59128 − 4.48824i 1.00000i 2.90773 + 0.738333i 1.34592
185.2 −0.866025 + 0.500000i −1.71705 + 0.227499i 0.500000 − 0.866025i 1.92707 + 1.11259i 1.37326 − 1.05554i 0.556441 + 0.963784i 1.00000i 2.89649 − 0.781251i −2.22519
185.3 −0.866025 + 0.500000i −1.62440 − 0.601111i 0.500000 − 0.866025i −2.48879 − 1.43691i 1.70732 − 0.291621i 1.86356 + 3.22779i 1.00000i 2.27733 + 1.95289i 2.87381
185.4 −0.866025 + 0.500000i −1.12280 + 1.31883i 0.500000 − 0.866025i 0.615038 + 0.355093i 0.312954 − 1.70354i 0.178545 + 0.309248i 1.00000i −0.478650 − 2.96157i −0.710185
185.5 −0.866025 + 0.500000i −0.841508 − 1.51389i 0.500000 − 0.866025i 2.53994 + 1.46644i 1.48571 + 0.890313i 2.03804 + 3.52999i 1.00000i −1.58373 + 2.54790i −2.93287
185.6 −0.866025 + 0.500000i −0.489221 + 1.66152i 0.500000 − 0.866025i 3.61004 + 2.08426i −0.407085 − 1.68353i −2.04301 − 3.53859i 1.00000i −2.52133 − 1.62570i −4.16852
185.7 −0.866025 + 0.500000i −0.296201 − 1.70654i 0.500000 − 0.866025i −2.42613 − 1.40073i 1.10979 + 1.32980i −0.855855 − 1.48238i 1.00000i −2.82453 + 1.01095i 2.80145
185.8 −0.866025 + 0.500000i −0.217529 + 1.71834i 0.500000 − 0.866025i −1.11157 − 0.641767i −0.670782 − 1.59689i 0.353559 + 0.612382i 1.00000i −2.90536 − 0.747578i 1.28353
185.9 −0.866025 + 0.500000i 0.217529 − 1.71834i 0.500000 − 0.866025i −1.11157 − 0.641767i 0.670782 + 1.59689i 0.353559 + 0.612382i 1.00000i −2.90536 − 0.747578i 1.28353
185.10 −0.866025 + 0.500000i 0.296201 + 1.70654i 0.500000 − 0.866025i −2.42613 − 1.40073i −1.10979 − 1.32980i −0.855855 − 1.48238i 1.00000i −2.82453 + 1.01095i 2.80145
185.11 −0.866025 + 0.500000i 0.489221 − 1.66152i 0.500000 − 0.866025i 3.61004 + 2.08426i 0.407085 + 1.68353i −2.04301 − 3.53859i 1.00000i −2.52133 − 1.62570i −4.16852
185.12 −0.866025 + 0.500000i 0.841508 + 1.51389i 0.500000 − 0.866025i 2.53994 + 1.46644i −1.48571 − 0.890313i 2.03804 + 3.52999i 1.00000i −1.58373 + 2.54790i −2.93287
185.13 −0.866025 + 0.500000i 1.12280 − 1.31883i 0.500000 − 0.866025i 0.615038 + 0.355093i −0.312954 + 1.70354i 0.178545 + 0.309248i 1.00000i −0.478650 − 2.96157i −0.710185
185.14 −0.866025 + 0.500000i 1.62440 + 0.601111i 0.500000 − 0.866025i −2.48879 − 1.43691i −1.70732 + 0.291621i 1.86356 + 3.22779i 1.00000i 2.27733 + 1.95289i 2.87381
185.15 −0.866025 + 0.500000i 1.71705 − 0.227499i 0.500000 − 0.866025i 1.92707 + 1.11259i −1.37326 + 1.05554i 0.556441 + 0.963784i 1.00000i 2.89649 − 0.781251i −2.22519
185.16 −0.866025 + 0.500000i 1.71868 + 0.214796i 0.500000 − 0.866025i −1.16560 − 0.672958i −1.59582 + 0.673321i −2.59128 − 4.48824i 1.00000i 2.90773 + 0.738333i 1.34592
185.17 0.866025 − 0.500000i −1.71519 + 0.241086i 0.500000 − 0.866025i −1.01733 − 0.587355i −1.36486 + 1.06638i −0.902070 − 1.56243i − 1.00000i 2.88375 − 0.827017i −1.17471
185.18 0.866025 − 0.500000i −1.58459 − 0.699349i 0.500000 − 0.866025i 0.254546 + 0.146962i −1.72197 + 0.186639i 1.66841 + 2.88977i − 1.00000i 2.02182 + 2.21636i 0.293925
185.19 0.866025 − 0.500000i −1.55011 − 0.772765i 0.500000 − 0.866025i 3.64602 + 2.10503i −1.72882 + 0.105819i −0.908070 − 1.57282i − 1.00000i 1.80567 + 2.39574i 4.21006
185.20 0.866025 − 0.500000i −1.37617 + 1.05175i 0.500000 − 0.866025i 1.74890 + 1.00973i −0.665921 + 1.59892i −1.62183 − 2.80910i − 1.00000i 0.787662 − 2.89475i 2.01945
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 185.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
31.b odd 2 1 inner
279.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 558.2.q.a ✓ 64
3.b odd 2 1 1674.2.q.a 64
9.c even 3 1 1674.2.q.a 64
9.d odd 6 1 inner 558.2.q.a ✓ 64
31.b odd 2 1 inner 558.2.q.a ✓ 64
93.c even 2 1 1674.2.q.a 64
279.m odd 6 1 1674.2.q.a 64
279.s even 6 1 inner 558.2.q.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
558.2.q.a ✓ 64 1.a even 1 1 trivial
558.2.q.a ✓ 64 9.d odd 6 1 inner
558.2.q.a ✓ 64 31.b odd 2 1 inner
558.2.q.a ✓ 64 279.s even 6 1 inner
1674.2.q.a 64 3.b odd 2 1
1674.2.q.a 64 9.c even 3 1
1674.2.q.a 64 93.c even 2 1
1674.2.q.a 64 279.m odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(558, [\chi])\).