Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [729,6,Mod(1,729)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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")
//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);
Level: | \( N \) | \(=\) | \( 729 = 3^{6} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 729.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −11.2521 | 0 | 94.6093 | 8.52580 | 0 | 152.526 | −704.485 | 0 | −95.9329 | ||||||||||||||||||
1.2 | −11.0172 | 0 | 89.3790 | −58.3688 | 0 | −214.792 | −632.156 | 0 | 643.061 | ||||||||||||||||||
1.3 | −10.0448 | 0 | 68.8976 | 83.2630 | 0 | −225.519 | −370.628 | 0 | −836.358 | ||||||||||||||||||
1.4 | −9.43780 | 0 | 57.0721 | −109.612 | 0 | 94.5333 | −236.626 | 0 | 1034.50 | ||||||||||||||||||
1.5 | −7.76070 | 0 | 28.2284 | 33.6629 | 0 | 205.544 | 29.2700 | 0 | −261.248 | ||||||||||||||||||
1.6 | −6.92211 | 0 | 15.9156 | 42.9376 | 0 | −174.162 | 111.338 | 0 | −297.218 | ||||||||||||||||||
1.7 | −6.30275 | 0 | 7.72471 | 40.2114 | 0 | 163.750 | 153.001 | 0 | −253.443 | ||||||||||||||||||
1.8 | −6.03324 | 0 | 4.39994 | −75.3717 | 0 | 64.8318 | 166.518 | 0 | 454.735 | ||||||||||||||||||
1.9 | −5.30646 | 0 | −3.84147 | 47.6126 | 0 | −184.697 | 190.191 | 0 | −252.654 | ||||||||||||||||||
1.10 | −3.74508 | 0 | −17.9744 | −104.284 | 0 | 30.6711 | 187.158 | 0 | 390.551 | ||||||||||||||||||
1.11 | −2.60656 | 0 | −25.2058 | −22.7076 | 0 | −124.035 | 149.110 | 0 | 59.1888 | ||||||||||||||||||
1.12 | −0.908243 | 0 | −31.1751 | 83.9961 | 0 | 204.577 | 57.3784 | 0 | −76.2889 | ||||||||||||||||||
1.13 | 1.09514 | 0 | −30.8007 | −27.6575 | 0 | −27.4634 | −68.7757 | 0 | −30.2890 | ||||||||||||||||||
1.14 | 1.69057 | 0 | −29.1420 | 41.0011 | 0 | 164.613 | −103.365 | 0 | 69.3154 | ||||||||||||||||||
1.15 | 2.05892 | 0 | −27.7608 | −22.2951 | 0 | 85.7048 | −123.043 | 0 | −45.9038 | ||||||||||||||||||
1.16 | 2.25556 | 0 | −26.9125 | 51.7887 | 0 | −26.8498 | −132.881 | 0 | 116.812 | ||||||||||||||||||
1.17 | 4.86323 | 0 | −8.34897 | 8.85823 | 0 | −189.829 | −196.226 | 0 | 43.0796 | ||||||||||||||||||
1.18 | 5.35036 | 0 | −3.37370 | −94.5899 | 0 | 9.88816 | −189.262 | 0 | −506.089 | ||||||||||||||||||
1.19 | 6.48467 | 0 | 10.0509 | 57.2659 | 0 | −99.4053 | −142.332 | 0 | 371.350 | ||||||||||||||||||
1.20 | 7.67489 | 0 | 26.9040 | −88.7585 | 0 | 168.615 | −39.1112 | 0 | −681.212 | ||||||||||||||||||
See all 24 embeddings |
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.a | ✓ | 24 |
3.b | odd | 2 | 1 | 729.6.a.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
729.6.a.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
729.6.a.b | yes | 24 | 3.b | odd | 2 | 1 |