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Properties

Label 247.2.z.a
Level $247$
Weight $2$
Character orbit 247.z
Analytic conductor $1.972$
Analytic rank $0$
Dimension $88$
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] = [247,2,Mod(141,247)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("247.141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(247, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([7, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 247 = 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 247.z (of order \(12\), degree \(4\), 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: \(1.97230492993\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 6 q^{2} - 2 q^{5} + 14 q^{6} - 12 q^{7} + 12 q^{8} - 84 q^{9} - 6 q^{10} - 4 q^{11} + 36 q^{12} - 6 q^{13} - 12 q^{14} + 30 q^{15} - 92 q^{16} - 6 q^{17} + 6 q^{18} - 14 q^{19} - 24 q^{20} - 18 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1 −1.86530 + 1.86530i − 2.56769i − 4.95868i −3.04891 + 0.816953i 4.78951 + 4.78951i −0.371832 − 1.38770i 5.51881 + 5.51881i −3.59305 4.16327 − 7.21099i
141.2 −1.81254 + 1.81254i 1.28516i − 4.57060i 2.20779 − 0.591576i −2.32940 − 2.32940i −0.454189 − 1.69506i 4.65931 + 4.65931i 1.34837 −2.92945 + 5.07396i
141.3 −1.71306 + 1.71306i 1.96487i − 3.86916i −2.94400 + 0.788843i −3.36594 − 3.36594i 1.07500 + 4.01196i 3.20198 + 3.20198i −0.860718 3.69192 − 6.39459i
141.4 −1.46122 + 1.46122i − 2.41824i − 2.27034i 3.58569 − 0.960783i 3.53359 + 3.53359i −0.590401 − 2.20341i 0.395022 + 0.395022i −2.84791 −3.83557 + 6.64341i
141.5 −1.30928 + 1.30928i 2.83019i − 1.42842i −0.575278 + 0.154145i −3.70551 − 3.70551i −1.04046 − 3.88303i −0.748356 − 0.748356i −5.00998 0.551380 − 0.955018i
141.6 −1.20848 + 1.20848i 0.308608i − 0.920861i 1.62953 − 0.436631i −0.372947 − 0.372947i 0.435093 + 1.62379i −1.30412 − 1.30412i 2.90476 −1.44160 + 2.49692i
141.7 −1.18233 + 1.18233i − 2.14610i − 0.795830i −0.374156 + 0.100255i 2.53741 + 2.53741i 0.889436 + 3.31942i −1.42373 − 1.42373i −1.60575 0.323843 − 0.560912i
141.8 −0.996021 + 0.996021i − 0.836119i 0.0158842i −2.41402 + 0.646834i 0.832792 + 0.832792i −0.702189 − 2.62060i −2.00786 − 2.00786i 2.30090 1.76015 − 3.04867i
141.9 −0.317768 + 0.317768i 1.19533i 1.79805i 2.57314 − 0.689472i −0.379836 − 0.379836i 0.544490 + 2.03206i −1.20690 − 1.20690i 1.57120 −0.598570 + 1.03675i
141.10 −0.273277 + 0.273277i 1.58138i 1.85064i −1.36285 + 0.365175i −0.432157 − 0.432157i 0.171253 + 0.639124i −1.05229 − 1.05229i 0.499222 0.272642 − 0.472230i
141.11 −0.0597405 + 0.0597405i − 0.623849i 1.99286i 0.894356 − 0.239642i 0.0372691 + 0.0372691i −1.18379 − 4.41795i −0.238536 − 0.238536i 2.61081 −0.0391129 + 0.0677456i
141.12 −0.0155569 + 0.0155569i − 2.63084i 1.99952i 3.09539 − 0.829408i 0.0409276 + 0.0409276i 0.629890 + 2.35078i −0.0622200 − 0.0622200i −3.92130 −0.0352517 + 0.0610577i
141.13 0.219425 − 0.219425i − 1.44626i 1.90371i −3.86150 + 1.03469i −0.317345 − 0.317345i 0.965483 + 3.60323i 0.856569 + 0.856569i 0.908331 −0.620272 + 1.07434i
141.14 0.557483 − 0.557483i 2.96213i 1.37843i 3.87110 − 1.03726i 1.65134 + 1.65134i −0.837739 − 3.12649i 1.88341 + 1.88341i −5.77421 1.57982 − 2.73632i
141.15 0.831895 − 0.831895i − 1.83299i 0.615900i 1.74214 − 0.466806i −1.52485 − 1.52485i −0.569588 − 2.12573i 2.17616 + 2.17616i −0.359836 1.06095 − 1.83761i
141.16 0.895490 − 0.895490i 0.516757i 0.396194i −1.13405 + 0.303867i 0.462751 + 0.462751i 0.349756 + 1.30531i 2.14577 + 2.14577i 2.73296 −0.743418 + 1.28764i
141.17 1.04858 − 1.04858i − 3.10378i − 0.199027i −2.60950 + 0.699214i −3.25455 − 3.25455i −0.699358 − 2.61004i 1.88846 + 1.88846i −6.63347 −2.00308 + 3.46944i
141.18 1.09743 − 1.09743i 2.75582i − 0.408698i −2.23116 + 0.597838i 3.02431 + 3.02431i −0.0404798 − 0.151073i 1.74634 + 1.74634i −4.59452 −1.79246 + 3.10462i
141.19 1.49300 − 1.49300i 0.393223i − 2.45810i 0.695308 − 0.186307i 0.587083 + 0.587083i −0.516731 − 1.92847i −0.683946 − 0.683946i 2.84538 0.759939 − 1.31625i
141.20 1.63590 − 1.63590i − 2.40603i − 3.35231i 0.659303 − 0.176660i −3.93602 − 3.93602i 1.09172 + 4.07434i −2.21224 − 2.21224i −2.78899 0.789554 − 1.36755i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
247.z even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 247.2.z.a ✓ 88
13.f odd 12 1 247.2.bf.a yes 88
19.d odd 6 1 247.2.bf.a yes 88
247.z even 12 1 inner 247.2.z.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
247.2.z.a ✓ 88 1.a even 1 1 trivial
247.2.z.a ✓ 88 247.z even 12 1 inner
247.2.bf.a yes 88 13.f odd 12 1
247.2.bf.a yes 88 19.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(247, [\chi])\).