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Properties

Label 820.2.bo.b
Level $820$
Weight $2$
Character orbit 820.bo
Analytic conductor $6.548$
Analytic rank $0$
Dimension $64$
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(21,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(20)) 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("820.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.bo (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] 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: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{3} - 10 q^{7} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 10 q^{19} - 22 q^{23} + 16 q^{25} + 20 q^{27} - 12 q^{29} + 22 q^{31} + 30 q^{33} + 10 q^{35} + 12 q^{37} + 20 q^{39} - 10 q^{41}+ \cdots - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 0 −2.39269 − 2.39269i 0 0.587785 − 0.809017i 0 0.802599 − 1.57519i 0 8.44996i 0
21.2 0 −1.65684 − 1.65684i 0 0.587785 − 0.809017i 0 −1.79341 + 3.51976i 0 2.49025i 0
21.3 0 −0.933435 − 0.933435i 0 0.587785 − 0.809017i 0 1.85288 − 3.63648i 0 − 1.25740i 0
21.4 0 0.180913 + 0.180913i 0 0.587785 − 0.809017i 0 0.338061 − 0.663482i 0 − 2.93454i 0
21.5 0 0.335549 + 0.335549i 0 0.587785 − 0.809017i 0 −1.60673 + 3.15338i 0 − 2.77481i 0
21.6 0 0.901610 + 0.901610i 0 0.587785 − 0.809017i 0 −4.93870e−6 0 9.69274e-6i 0 − 1.37420i 0
21.7 0 1.93460 + 1.93460i 0 0.587785 − 0.809017i 0 −1.78032 + 3.49408i 0 4.48538i 0
21.8 0 2.27233 + 2.27233i 0 0.587785 − 0.809017i 0 1.91674 − 3.76182i 0 7.32700i 0
61.1 0 −2.17362 + 2.17362i 0 −0.587785 + 0.809017i 0 3.05208 + 1.55511i 0 − 6.44925i 0
61.2 0 −1.90076 + 1.90076i 0 −0.587785 + 0.809017i 0 −4.30882 − 2.19546i 0 − 4.22575i 0
61.3 0 −1.41199 + 1.41199i 0 −0.587785 + 0.809017i 0 −0.268234 − 0.136672i 0 − 0.987434i 0
61.4 0 −0.654373 + 0.654373i 0 −0.587785 + 0.809017i 0 0.156007 + 0.0794893i 0 2.14359i 0
61.5 0 0.869863 − 0.869863i 0 −0.587785 + 0.809017i 0 −2.37416 − 1.20970i 0 1.48668i 0
61.6 0 0.882020 − 0.882020i 0 −0.587785 + 0.809017i 0 3.88017 + 1.97705i 0 1.44408i 0
61.7 0 0.915716 − 0.915716i 0 −0.587785 + 0.809017i 0 −3.85694 − 1.96521i 0 1.32293i 0
61.8 0 2.21307 − 2.21307i 0 −0.587785 + 0.809017i 0 0.372044 + 0.189566i 0 − 6.79534i 0
121.1 0 −2.17362 − 2.17362i 0 −0.587785 − 0.809017i 0 3.05208 − 1.55511i 0 6.44925i 0
121.2 0 −1.90076 − 1.90076i 0 −0.587785 − 0.809017i 0 −4.30882 + 2.19546i 0 4.22575i 0
121.3 0 −1.41199 − 1.41199i 0 −0.587785 − 0.809017i 0 −0.268234 + 0.136672i 0 0.987434i 0
121.4 0 −0.654373 − 0.654373i 0 −0.587785 − 0.809017i 0 0.156007 − 0.0794893i 0 − 2.14359i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.bo.b ✓ 64
41.g even 20 1 inner 820.2.bo.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.bo.b ✓ 64 1.a even 1 1 trivial
820.2.bo.b ✓ 64 41.g even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 2 T_{3}^{63} + 2 T_{3}^{62} - 4 T_{3}^{61} + 519 T_{3}^{60} - 1044 T_{3}^{59} + \cdots + 169300154521 \) acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\). Copy content Toggle raw display