Properties

Label 1332.2.bg.b
Level $1332$
Weight $2$
Character orbit 1332.bg
Analytic conductor $10.636$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1332,2,Mod(517,1332)] 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("1332.517"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1332, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.bg (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] 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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{3} + 4 q^{7} + 14 q^{9} + 2 q^{11} + 24 q^{21} + 28 q^{25} + 22 q^{27} + 28 q^{33} - 8 q^{37} - 46 q^{41} + 18 q^{47} - 32 q^{49} - 24 q^{53} + 28 q^{63} - 30 q^{65} - 10 q^{67} - 20 q^{71}+ \cdots - 112 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
517.1 0 −1.71941 − 0.208899i 0 −2.46326 − 1.42216i 0 −1.98080 − 3.43084i 0 2.91272 + 0.718364i 0
517.2 0 −1.71941 − 0.208899i 0 2.46326 + 1.42216i 0 −1.98080 − 3.43084i 0 2.91272 + 0.718364i 0
517.3 0 −1.70307 + 0.315509i 0 −1.18476 − 0.684022i 0 −0.0762691 − 0.132102i 0 2.80091 − 1.07467i 0
517.4 0 −1.70307 + 0.315509i 0 1.18476 + 0.684022i 0 −0.0762691 − 0.132102i 0 2.80091 − 1.07467i 0
517.5 0 −1.69287 − 0.366340i 0 −2.65825 − 1.53474i 0 1.94338 + 3.36603i 0 2.73159 + 1.24033i 0
517.6 0 −1.69287 − 0.366340i 0 2.65825 + 1.53474i 0 1.94338 + 3.36603i 0 2.73159 + 1.24033i 0
517.7 0 −1.33924 + 1.09838i 0 −1.11412 − 0.643237i 0 1.51759 + 2.62854i 0 0.587120 − 2.94199i 0
517.8 0 −1.33924 + 1.09838i 0 1.11412 + 0.643237i 0 1.51759 + 2.62854i 0 0.587120 − 2.94199i 0
517.9 0 −1.28609 − 1.16016i 0 −0.937122 − 0.541048i 0 −1.11317 − 1.92807i 0 0.308047 + 2.98414i 0
517.10 0 −1.28609 − 1.16016i 0 0.937122 + 0.541048i 0 −1.11317 − 1.92807i 0 0.308047 + 2.98414i 0
517.11 0 −1.04628 + 1.38033i 0 −3.67589 − 2.12227i 0 −1.36321 − 2.36115i 0 −0.810600 − 2.88841i 0
517.12 0 −1.04628 + 1.38033i 0 3.67589 + 2.12227i 0 −1.36321 − 2.36115i 0 −0.810600 − 2.88841i 0
517.13 0 −0.803016 − 1.53465i 0 −0.760707 − 0.439194i 0 1.49800 + 2.59462i 0 −1.71033 + 2.46470i 0
517.14 0 −0.803016 − 1.53465i 0 0.760707 + 0.439194i 0 1.49800 + 2.59462i 0 −1.71033 + 2.46470i 0
517.15 0 −0.461158 + 1.66953i 0 −0.508828 − 0.293772i 0 −0.818763 − 1.41814i 0 −2.57467 − 1.53983i 0
517.16 0 −0.461158 + 1.66953i 0 0.508828 + 0.293772i 0 −0.818763 − 1.41814i 0 −2.57467 − 1.53983i 0
517.17 0 −0.432723 − 1.67713i 0 −3.66916 − 2.11839i 0 0.227714 + 0.394411i 0 −2.62550 + 1.45146i 0
517.18 0 −0.432723 − 1.67713i 0 3.66916 + 2.11839i 0 0.227714 + 0.394411i 0 −2.62550 + 1.45146i 0
517.19 0 −0.244688 + 1.71468i 0 −0.494028 − 0.285227i 0 1.72555 + 2.98874i 0 −2.88026 − 0.839123i 0
517.20 0 −0.244688 + 1.71468i 0 0.494028 + 0.285227i 0 1.72555 + 2.98874i 0 −2.88026 − 0.839123i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 517.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
37.b even 2 1 inner
333.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.bg.b ✓ 72
3.b odd 2 1 3996.2.bg.b 72
9.c even 3 1 inner 1332.2.bg.b ✓ 72
9.d odd 6 1 3996.2.bg.b 72
37.b even 2 1 inner 1332.2.bg.b ✓ 72
111.d odd 2 1 3996.2.bg.b 72
333.n odd 6 1 3996.2.bg.b 72
333.q even 6 1 inner 1332.2.bg.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.bg.b ✓ 72 1.a even 1 1 trivial
1332.2.bg.b ✓ 72 9.c even 3 1 inner
1332.2.bg.b ✓ 72 37.b even 2 1 inner
1332.2.bg.b ✓ 72 333.q even 6 1 inner
3996.2.bg.b 72 3.b odd 2 1
3996.2.bg.b 72 9.d odd 6 1
3996.2.bg.b 72 111.d odd 2 1
3996.2.bg.b 72 333.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 104 T_{5}^{70} + 5988 T_{5}^{68} - 236908 T_{5}^{66} + 7110650 T_{5}^{64} + \cdots + 12\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\). Copy content Toggle raw display