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Properties

Label 729.6.a.b
Level $729$
Weight $6$
Character orbit 729.a
Self dual yes
Analytic conductor $116.920$
Analytic rank $0$
Dimension $24$
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] = [729,6,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,12] 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: \(116.919804644\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} + 432 q^{4} + 33 q^{5} + 294 q^{7} + 843 q^{8} + 600 q^{10} + 30 q^{11} + 1014 q^{13} - 3720 q^{14} + 8448 q^{16} - 2709 q^{17} + 4332 q^{19} + 861 q^{20} - 2193 q^{22} + 4020 q^{23} + 15417 q^{25}+ \cdots + 944667 q^{98}+O(q^{100}) \) 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 −10.5190 0 78.6494 8.61028 0 93.6853 −490.705 0 −90.5716
1.2 −9.81893 0 64.4115 −77.7846 0 224.654 −318.246 0 763.762
1.3 −9.36178 0 55.6429 −51.5051 0 9.78880 −221.340 0 482.179
1.4 −8.16395 0 34.6500 49.1579 0 −112.628 −21.6346 0 −401.322
1.5 −7.67489 0 26.9040 88.7585 0 168.615 39.1112 0 −681.212
1.6 −6.48467 0 10.0509 −57.2659 0 −99.4053 142.332 0 371.350
1.7 −5.35036 0 −3.37370 94.5899 0 9.88816 189.262 0 −506.089
1.8 −4.86323 0 −8.34897 −8.85823 0 −189.829 196.226 0 43.0796
1.9 −2.25556 0 −26.9125 −51.7887 0 −26.8498 132.881 0 116.812
1.10 −2.05892 0 −27.7608 22.2951 0 85.7048 123.043 0 −45.9038
1.11 −1.69057 0 −29.1420 −41.0011 0 164.613 103.365 0 69.3154
1.12 −1.09514 0 −30.8007 27.6575 0 −27.4634 68.7757 0 −30.2890
1.13 0.908243 0 −31.1751 −83.9961 0 204.577 −57.3784 0 −76.2889
1.14 2.60656 0 −25.2058 22.7076 0 −124.035 −149.110 0 59.1888
1.15 3.74508 0 −17.9744 104.284 0 30.6711 −187.158 0 390.551
1.16 5.30646 0 −3.84147 −47.6126 0 −184.697 −190.191 0 −252.654
1.17 6.03324 0 4.39994 75.3717 0 64.8318 −166.518 0 454.735
1.18 6.30275 0 7.72471 −40.2114 0 163.750 −153.001 0 −253.443
1.19 6.92211 0 15.9156 −42.9376 0 −174.162 −111.338 0 −297.218
1.20 7.76070 0 28.2284 −33.6629 0 205.544 −29.2700 0 −261.248
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)

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 729.6.a.b yes 24
3.b odd 2 1 729.6.a.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.6.a.a ✓ 24 3.b odd 2 1
729.6.a.b yes 24 1.a even 1 1 trivial