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Properties

Label 3129.2.a.k
Level $3129$
Weight $2$
Character orbit 3129.a
Self dual yes
Analytic conductor $24.985$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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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] = [3129,2,Mod(1,3129)] 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("3129.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3129, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3129 = 3 \cdot 7 \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3129.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [21,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.9851907925\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 3 q^{2} - 21 q^{3} + 27 q^{4} + 3 q^{6} + 21 q^{7} - 9 q^{8} + 21 q^{9} + 11 q^{10} - 3 q^{11} - 27 q^{12} + 15 q^{13} - 3 q^{14} + 39 q^{16} + 3 q^{17} - 3 q^{18} + 25 q^{19} + 10 q^{20} - 21 q^{21}+ \cdots - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80022 −1.00000 5.84122 −2.52647 2.80022 1.00000 −10.7563 1.00000 7.07468
1.2 −2.67639 −1.00000 5.16306 3.51294 2.67639 1.00000 −8.46559 1.00000 −9.40200
1.3 −2.46640 −1.00000 4.08311 −2.04482 2.46640 1.00000 −5.13778 1.00000 5.04333
1.4 −2.21547 −1.00000 2.90829 0.455706 2.21547 1.00000 −2.01228 1.00000 −1.00960
1.5 −2.07853 −1.00000 2.32028 −3.59676 2.07853 1.00000 −0.665706 1.00000 7.47596
1.6 −1.94531 −1.00000 1.78424 2.67298 1.94531 1.00000 0.419722 1.00000 −5.19978
1.7 −1.33431 −1.00000 −0.219620 2.42431 1.33431 1.00000 2.96166 1.00000 −3.23478
1.8 −1.27133 −1.00000 −0.383714 −1.92256 1.27133 1.00000 3.03049 1.00000 2.44421
1.9 −0.857878 −1.00000 −1.26405 −4.04478 0.857878 1.00000 2.80015 1.00000 3.46993
1.10 −0.389194 −1.00000 −1.84853 3.65335 0.389194 1.00000 1.49783 1.00000 −1.42186
1.11 −0.168762 −1.00000 −1.97152 2.18214 0.168762 1.00000 0.670242 1.00000 −0.368262
1.12 0.236458 −1.00000 −1.94409 0.319681 −0.236458 1.00000 −0.932612 1.00000 0.0755912
1.13 0.288776 −1.00000 −1.91661 −0.604244 −0.288776 1.00000 −1.13102 1.00000 −0.174491
1.14 0.518346 −1.00000 −1.73132 −2.75013 −0.518346 1.00000 −1.93411 1.00000 −1.42552
1.15 1.46863 −1.00000 0.156872 −1.48971 −1.46863 1.00000 −2.70687 1.00000 −2.18782
1.16 1.50777 −1.00000 0.273362 −1.78236 −1.50777 1.00000 −2.60337 1.00000 −2.68738
1.17 1.56508 −1.00000 0.449485 1.86124 −1.56508 1.00000 −2.42668 1.00000 2.91300
1.18 2.17169 −1.00000 2.71624 −3.23440 −2.17169 1.00000 1.55544 1.00000 −7.02412
1.19 2.35080 −1.00000 3.52628 1.87463 −2.35080 1.00000 3.58799 1.00000 4.40690
1.20 2.35941 −1.00000 3.56684 4.13207 −2.35941 1.00000 3.69682 1.00000 9.74927
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( -1 \)
\(149\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3129.2.a.k ✓ 21
3.b odd 2 1 9387.2.a.r 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3129.2.a.k ✓ 21 1.a even 1 1 trivial
9387.2.a.r 21 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} + 3 T_{2}^{20} - 30 T_{2}^{19} - 92 T_{2}^{18} + 372 T_{2}^{17} + 1177 T_{2}^{16} - 2461 T_{2}^{15} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3129))\). Copy content Toggle raw display