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Properties

Label 820.2.u.b
Level $820$
Weight $2$
Character orbit 820.u
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(141,820)] 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, 4])) 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("820.141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.u (of order \(5\), 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 \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} + 8 q^{5} - 5 q^{7} + 46 q^{9} + q^{11} + q^{13} - 2 q^{15} + 7 q^{17} - 13 q^{19} - 6 q^{21} + 4 q^{23} - 8 q^{25} - 28 q^{27} + 3 q^{29} - q^{31} + 14 q^{33} + 5 q^{35} - 25 q^{37} + 26 q^{41}+ \cdots + 30 q^{99}+O(q^{100}) \) 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 0 −3.40143 0 0.809017 − 0.587785i 0 −1.56474 + 4.81578i 0 8.56970 0
141.2 0 −2.80651 0 0.809017 − 0.587785i 0 1.06034 − 3.26338i 0 4.87650 0
141.3 0 −1.37178 0 0.809017 − 0.587785i 0 −0.259586 + 0.798923i 0 −1.11821 0
141.4 0 −0.355750 0 0.809017 − 0.587785i 0 0.307592 − 0.946670i 0 −2.87344 0
141.5 0 0.300773 0 0.809017 − 0.587785i 0 −0.378452 + 1.16476i 0 −2.90954 0
141.6 0 1.58274 0 0.809017 − 0.587785i 0 −1.18200 + 3.63783i 0 −0.494940 0
141.7 0 2.57506 0 0.809017 − 0.587785i 0 1.08833 − 3.34952i 0 3.63096 0
141.8 0 2.85886 0 0.809017 − 0.587785i 0 0.237547 − 0.731093i 0 5.17308 0
201.1 0 −3.30240 0 −0.309017 + 0.951057i 0 −1.74674 − 1.26908i 0 7.90585 0
201.2 0 −1.57747 0 −0.309017 + 0.951057i 0 1.66472 + 1.20949i 0 −0.511595 0
201.3 0 −0.435978 0 −0.309017 + 0.951057i 0 0.796401 + 0.578619i 0 −2.80992 0
201.4 0 −0.279895 0 −0.309017 + 0.951057i 0 3.56351 + 2.58904i 0 −2.92166 0
201.5 0 0.0908517 0 −0.309017 + 0.951057i 0 −1.97649 − 1.43601i 0 −2.99175 0
201.6 0 1.59597 0 −0.309017 + 0.951057i 0 −3.74875 − 2.72363i 0 −0.452887 0
201.7 0 2.19153 0 −0.309017 + 0.951057i 0 1.32809 + 0.964915i 0 1.80282 0
201.8 0 3.33542 0 −0.309017 + 0.951057i 0 −1.68976 − 1.22768i 0 8.12504 0
221.1 0 −3.40143 0 0.809017 + 0.587785i 0 −1.56474 − 4.81578i 0 8.56970 0
221.2 0 −2.80651 0 0.809017 + 0.587785i 0 1.06034 + 3.26338i 0 4.87650 0
221.3 0 −1.37178 0 0.809017 + 0.587785i 0 −0.259586 − 0.798923i 0 −1.11821 0
221.4 0 −0.355750 0 0.809017 + 0.587785i 0 0.307592 + 0.946670i 0 −2.87344 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.u.b ✓ 32
41.d even 5 1 inner 820.2.u.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.u.b ✓ 32 1.a even 1 1 trivial
820.2.u.b ✓ 32 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - T_{3}^{15} - 35 T_{3}^{14} + 38 T_{3}^{13} + 461 T_{3}^{12} - 519 T_{3}^{11} - 2841 T_{3}^{10} + \cdots + 11 \) acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\). Copy content Toggle raw display