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Properties

Label 627.2.s.a
Level $627$
Weight $2$
Character orbit 627.s
Analytic conductor $5.007$
Analytic rank $0$
Dimension $160$
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] = [627,2,Mod(94,627)] 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("627.94"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(627, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.s (of order \(10\), 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: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 40 q^{4} + 4 q^{5} - 10 q^{7} + 40 q^{9} + 10 q^{11} - 40 q^{16} - 50 q^{17} + 76 q^{20} + 16 q^{23} - 80 q^{25} - 68 q^{26} - 40 q^{35} + 40 q^{36} - 62 q^{38} + 8 q^{42} + 40 q^{44} - 4 q^{45}+ \cdots - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1 −0.867807 − 2.67084i −0.587785 + 0.809017i −4.76224 + 3.45997i 0.857404 − 2.63882i 2.67084 + 0.867807i −2.32024 − 3.19354i 8.82983 + 6.41525i −0.309017 − 0.951057i −7.79192
94.2 −0.820228 − 2.52440i 0.587785 − 0.809017i −4.08180 + 2.96560i −1.10292 + 3.39445i −2.52440 − 0.820228i 0.639656 + 0.880410i 6.53962 + 4.75131i −0.309017 − 0.951057i 9.47362
94.3 −0.757373 − 2.33095i 0.587785 − 0.809017i −3.24170 + 2.35523i −0.0812949 + 0.250200i −2.33095 − 0.757373i −0.777579 − 1.07025i 3.97945 + 2.89124i −0.309017 − 0.951057i 0.644775
94.4 −0.718483 − 2.21126i −0.587785 + 0.809017i −2.75543 + 2.00194i −0.792086 + 2.43779i 2.21126 + 0.718483i 0.238267 + 0.327946i 2.64453 + 1.92136i −0.309017 − 0.951057i 5.95969
94.5 −0.714800 − 2.19993i 0.587785 − 0.809017i −2.71071 + 1.96945i 1.17464 − 3.61516i −2.19993 − 0.714800i 2.08834 + 2.87435i 2.52753 + 1.83635i −0.309017 − 0.951057i −8.79273
94.6 −0.714061 − 2.19765i −0.587785 + 0.809017i −2.70176 + 1.96295i 0.312132 − 0.960644i 2.19765 + 0.714061i 0.825739 + 1.13653i 2.50423 + 1.81943i −0.309017 − 0.951057i −2.33404
94.7 −0.633940 − 1.95107i 0.587785 − 0.809017i −1.78674 + 1.29814i 1.01409 − 3.12106i −1.95107 − 0.633940i −1.52746 − 2.10237i 0.346100 + 0.251457i −0.309017 − 0.951057i −6.73226
94.8 −0.630524 − 1.94055i −0.587785 + 0.809017i −1.75015 + 1.27156i 0.436935 − 1.34475i 1.94055 + 0.630524i 1.37286 + 1.88957i 0.269578 + 0.195860i −0.309017 − 0.951057i −2.88506
94.9 −0.568233 − 1.74884i 0.587785 − 0.809017i −1.11753 + 0.811930i −0.691229 + 2.12738i −1.74884 − 0.568233i 1.54468 + 2.12607i −0.920356 − 0.668678i −0.309017 − 0.951057i 4.11324
94.10 −0.493829 − 1.51985i −0.587785 + 0.809017i −0.448040 + 0.325520i 0.704035 − 2.16680i 1.51985 + 0.493829i −2.84839 − 3.92047i −1.86973 − 1.35843i −0.309017 − 0.951057i −3.64088
94.11 −0.446228 − 1.37335i −0.587785 + 0.809017i −0.0689333 + 0.0500830i −1.08925 + 3.35238i 1.37335 + 0.446228i 2.43075 + 3.34564i −2.23694 − 1.62523i −0.309017 − 0.951057i 5.09004
94.12 −0.443338 − 1.36445i 0.587785 − 0.809017i −0.0471489 + 0.0342557i 0.197251 − 0.607076i −1.36445 − 0.443338i −1.69324 − 2.33055i −2.25370 − 1.63741i −0.309017 − 0.951057i −0.915775
94.13 −0.347788 − 1.07038i −0.587785 + 0.809017i 0.593275 − 0.431039i 1.10843 − 3.41140i 1.07038 + 0.347788i 0.927280 + 1.27629i −2.48875 − 1.80818i −0.309017 − 0.951057i −4.03699
94.14 −0.334649 − 1.02994i 0.587785 − 0.809017i 0.669241 − 0.486232i −1.27547 + 3.92550i −1.02994 − 0.334649i −2.44443 − 3.36447i −2.47700 − 1.79964i −0.309017 − 0.951057i 4.46987
94.15 −0.276003 − 0.849451i −0.587785 + 0.809017i 0.972645 − 0.706668i −0.519559 + 1.59904i 0.849451 + 0.276003i 0.570772 + 0.785601i −2.31390 − 1.68115i −0.309017 − 0.951057i 1.50170
94.16 −0.235532 − 0.724894i −0.587785 + 0.809017i 1.14804 − 0.834099i −0.896649 + 2.75960i 0.724894 + 0.235532i −1.44561 − 1.98971i −2.10830 − 1.53177i −0.309017 − 0.951057i 2.21161
94.17 −0.181729 − 0.559303i 0.587785 − 0.809017i 1.33824 − 0.972288i 0.281904 − 0.867612i −0.559303 − 0.181729i 1.33131 + 1.83239i −1.73854 − 1.26313i −0.309017 − 0.951057i −0.536488
94.18 −0.134089 − 0.412683i 0.587785 − 0.809017i 1.46571 − 1.06490i −0.0446214 + 0.137331i −0.412683 − 0.134089i 2.66082 + 3.66230i −1.33810 − 0.972186i −0.309017 − 0.951057i 0.0626572
94.19 −0.123092 − 0.378837i 0.587785 − 0.809017i 1.48967 − 1.08231i −0.0677455 + 0.208499i −0.378837 − 0.123092i −1.61387 − 2.22131i −1.23790 − 0.899388i −0.309017 − 0.951057i 0.0873262
94.20 −0.0187862 − 0.0578180i 0.587785 − 0.809017i 1.61504 − 1.17340i 0.974011 − 2.99770i −0.0578180 − 0.0187862i −0.650624 − 0.895507i −0.196550 − 0.142802i −0.309017 − 0.951057i −0.191619
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
19.b odd 2 1 inner
209.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 627.2.s.a ✓ 160
11.d odd 10 1 inner 627.2.s.a ✓ 160
19.b odd 2 1 inner 627.2.s.a ✓ 160
209.k even 10 1 inner 627.2.s.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.s.a ✓ 160 1.a even 1 1 trivial
627.2.s.a ✓ 160 11.d odd 10 1 inner
627.2.s.a ✓ 160 19.b odd 2 1 inner
627.2.s.a ✓ 160 209.k even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(627, [\chi])\).