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Properties

Label 130.4.n.a
Level $130$
Weight $4$
Character orbit 130.n
Analytic conductor $7.670$
Analytic rank $0$
Dimension $44$
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] = [130,4,Mod(9,130)] 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("130.9"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(130, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 130.n (of order \(6\), degree \(2\), 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: \(7.67024830075\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 88 q^{4} - 12 q^{5} + 266 q^{9} + 28 q^{10} + 2 q^{11} - 216 q^{14} - 64 q^{15} - 352 q^{16} + 14 q^{19} - 24 q^{20} - 32 q^{21} + 504 q^{25} + 336 q^{26} + 308 q^{29} + 104 q^{30} + 792 q^{31} - 352 q^{34}+ \cdots + 2916 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.73205 − 1.00000i −8.47005 − 4.89018i 2.00000 + 3.46410i 1.89034 + 11.0194i 9.78037 + 16.9401i 15.2112 − 8.78218i − 8.00000i 34.3278 + 59.4575i 7.74521 − 20.9765i
9.2 −1.73205 − 1.00000i −6.45441 − 3.72645i 2.00000 + 3.46410i −9.31537 − 6.18255i 7.45291 + 12.9088i −23.1745 + 13.3798i − 8.00000i 14.2729 + 24.7214i 9.95214 + 20.0239i
9.3 −1.73205 − 1.00000i −5.30007 − 3.06000i 2.00000 + 3.46410i 6.73291 − 8.92569i 6.11999 + 10.6001i 12.2028 − 7.04529i − 8.00000i 5.22716 + 9.05371i −20.5874 + 8.72683i
9.4 −1.73205 − 1.00000i −3.44569 − 1.98937i 2.00000 + 3.46410i 11.1790 − 0.173268i 3.97874 + 6.89138i −16.2727 + 9.39504i − 8.00000i −5.58482 − 9.67320i −19.5359 − 10.8789i
9.5 −1.73205 − 1.00000i −2.39309 − 1.38165i 2.00000 + 3.46410i −10.8544 − 2.68005i 2.76331 + 4.78619i 29.2023 − 16.8600i − 8.00000i −9.68207 − 16.7698i 16.1203 + 15.4964i
9.6 −1.73205 − 1.00000i 0.162462 + 0.0937974i 2.00000 + 3.46410i 2.25511 + 10.9505i −0.187595 − 0.324924i −16.1121 + 9.30232i − 8.00000i −13.4824 − 23.3522i 7.04459 − 21.2220i
9.7 −1.73205 − 1.00000i 2.13838 + 1.23459i 2.00000 + 3.46410i −10.7802 + 2.96444i −2.46918 − 4.27675i 0.660166 − 0.381147i − 8.00000i −10.4516 − 18.1026i 21.6362 + 5.64562i
9.8 −1.73205 − 1.00000i 3.43924 + 1.98565i 2.00000 + 3.46410i 2.28471 − 10.9444i −3.97130 − 6.87849i 5.96841 − 3.44587i − 8.00000i −5.61440 − 9.72443i −14.9016 + 16.6716i
9.9 −1.73205 − 1.00000i 4.53157 + 2.61630i 2.00000 + 3.46410i 3.51208 + 10.6144i −5.23260 − 9.06313i 21.0781 − 12.1694i − 8.00000i 0.190066 + 0.329205i 4.53129 − 21.8967i
9.10 −1.73205 − 1.00000i 6.95834 + 4.01740i 2.00000 + 3.46410i 10.8454 − 2.71598i −8.03480 − 13.9167i 8.29606 − 4.78973i − 8.00000i 18.7790 + 32.5262i −21.5008 − 6.14122i
9.11 −1.73205 − 1.00000i 8.83332 + 5.09992i 2.00000 + 3.46410i −10.7497 + 3.07320i −10.1998 − 17.6666i −13.6770 + 7.89644i − 8.00000i 38.5183 + 66.7157i 21.6922 + 5.42674i
9.12 1.73205 + 1.00000i −8.83332 − 5.09992i 2.00000 + 3.46410i −10.7497 − 3.07320i −10.1998 − 17.6666i 13.6770 − 7.89644i 8.00000i 38.5183 + 66.7157i −15.5458 − 16.0726i
9.13 1.73205 + 1.00000i −6.95834 − 4.01740i 2.00000 + 3.46410i 10.8454 + 2.71598i −8.03480 − 13.9167i −8.29606 + 4.78973i 8.00000i 18.7790 + 32.5262i 16.0689 + 15.5496i
9.14 1.73205 + 1.00000i −4.53157 − 2.61630i 2.00000 + 3.46410i 3.51208 − 10.6144i −5.23260 − 9.06313i −21.0781 + 12.1694i 8.00000i 0.190066 + 0.329205i 16.6975 − 14.8726i
9.15 1.73205 + 1.00000i −3.43924 − 1.98565i 2.00000 + 3.46410i 2.28471 + 10.9444i −3.97130 − 6.87849i −5.96841 + 3.44587i 8.00000i −5.61440 − 9.72443i −6.98717 + 21.2410i
9.16 1.73205 + 1.00000i −2.13838 − 1.23459i 2.00000 + 3.46410i −10.7802 − 2.96444i −2.46918 − 4.27675i −0.660166 + 0.381147i 8.00000i −10.4516 − 18.1026i −15.7074 − 15.9147i
9.17 1.73205 + 1.00000i −0.162462 − 0.0937974i 2.00000 + 3.46410i 2.25511 − 10.9505i −0.187595 − 0.324924i 16.1121 − 9.30232i 8.00000i −13.4824 − 23.3522i 14.8565 − 16.7118i
9.18 1.73205 + 1.00000i 2.39309 + 1.38165i 2.00000 + 3.46410i −10.8544 + 2.68005i 2.76331 + 4.78619i −29.2023 + 16.8600i 8.00000i −9.68207 − 16.7698i −21.4804 − 6.21238i
9.19 1.73205 + 1.00000i 3.44569 + 1.98937i 2.00000 + 3.46410i 11.1790 + 0.173268i 3.97874 + 6.89138i 16.2727 − 9.39504i 8.00000i −5.58482 − 9.67320i 19.1893 + 11.4791i
9.20 1.73205 + 1.00000i 5.30007 + 3.06000i 2.00000 + 3.46410i 6.73291 + 8.92569i 6.11999 + 10.6001i −12.2028 + 7.04529i 8.00000i 5.22716 + 9.05371i 2.73606 + 22.1927i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.4.n.a ✓ 44
5.b even 2 1 inner 130.4.n.a ✓ 44
13.c even 3 1 inner 130.4.n.a ✓ 44
65.n even 6 1 inner 130.4.n.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.4.n.a ✓ 44 1.a even 1 1 trivial
130.4.n.a ✓ 44 5.b even 2 1 inner
130.4.n.a ✓ 44 13.c even 3 1 inner
130.4.n.a ✓ 44 65.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(130, [\chi])\).