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Properties

Label 820.2.bg.b
Level $820$
Weight $2$
Character orbit 820.bg
Analytic conductor $6.548$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [820,2,Mod(441,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) 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("820.441"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.bg (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.54773296574\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{5} - 5 q^{7} - 42 q^{9} + 5 q^{11} + 5 q^{13} - 5 q^{17} - 25 q^{19} - 6 q^{21} - 22 q^{23} - 8 q^{25} + 5 q^{29} - 9 q^{31} + 24 q^{33} + 5 q^{35} - 11 q^{37} + 20 q^{39} + 16 q^{41} + 26 q^{43}+ \cdots - 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
441.1 0 − 3.03671i 0 0.309017 + 0.951057i 0 −0.298250 − 0.410506i 0 −6.22161 0
441.2 0 − 2.34579i 0 0.309017 + 0.951057i 0 −0.869808 − 1.19719i 0 −2.50275 0
441.3 0 − 1.23652i 0 0.309017 + 0.951057i 0 0.581439 + 0.800283i 0 1.47101 0
441.4 0 0.0118557i 0 0.309017 + 0.951057i 0 2.84530 + 3.91622i 0 2.99986 0
441.5 0 0.804578i 0 0.309017 + 0.951057i 0 −1.73252 − 2.38461i 0 2.35265 0
441.6 0 2.10026i 0 0.309017 + 0.951057i 0 −2.62784 − 3.61690i 0 −1.41111 0
441.7 0 2.14480i 0 0.309017 + 0.951057i 0 0.606688 + 0.835034i 0 −1.60015 0
441.8 0 2.73311i 0 0.309017 + 0.951057i 0 1.92203 + 2.64545i 0 −4.46987 0
681.1 0 − 2.63835i 0 −0.809017 − 0.587785i 0 −4.34729 + 1.41252i 0 −3.96087 0
681.2 0 − 2.16903i 0 −0.809017 − 0.587785i 0 −1.30195 + 0.423028i 0 −1.70470 0
681.3 0 − 1.65530i 0 −0.809017 − 0.587785i 0 4.93296 − 1.60282i 0 0.259996 0
681.4 0 − 0.195916i 0 −0.809017 − 0.587785i 0 0.384497 − 0.124931i 0 2.96162 0
681.5 0 1.07205i 0 −0.809017 − 0.587785i 0 −2.50037 + 0.812419i 0 1.85070 0
681.6 0 1.94063i 0 −0.809017 − 0.587785i 0 −0.0642499 + 0.0208760i 0 −0.766036 0
681.7 0 2.11504i 0 −0.809017 − 0.587785i 0 3.27625 − 1.06452i 0 −1.47340 0
681.8 0 3.43298i 0 −0.809017 − 0.587785i 0 −3.30690 + 1.07448i 0 −8.78535 0
701.1 0 − 2.73311i 0 0.309017 − 0.951057i 0 1.92203 − 2.64545i 0 −4.46987 0
701.2 0 − 2.14480i 0 0.309017 − 0.951057i 0 0.606688 − 0.835034i 0 −1.60015 0
701.3 0 − 2.10026i 0 0.309017 − 0.951057i 0 −2.62784 + 3.61690i 0 −1.41111 0
701.4 0 − 0.804578i 0 0.309017 − 0.951057i 0 −1.73252 + 2.38461i 0 2.35265 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 441.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.bg.b ✓ 32
41.f even 10 1 inner 820.2.bg.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.bg.b ✓ 32 1.a even 1 1 trivial
820.2.bg.b ✓ 32 41.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 69 T_{3}^{30} + 2147 T_{3}^{28} + 39906 T_{3}^{26} + 494511 T_{3}^{24} + 4315995 T_{3}^{22} + \cdots + 841 \) acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\). Copy content Toggle raw display