Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,12,Mod(4,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.4");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.e (of order \(6\), degree \(2\), minimal) |
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: | no |
Analytic conductor: | \(9.98846134727\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −74.2981 | − | 42.8960i | −148.593 | + | 257.371i | 2656.14 | + | 4600.57i | − | 10410.3i | 22080.4 | − | 12748.1i | 18278.8 | − | 10553.3i | − | 280049.i | 44413.7 | + | 76926.7i | −446559. | + | 773464.i | ||
4.2 | −57.1912 | − | 33.0193i | 212.159 | − | 367.469i | 1156.55 | + | 2003.21i | 3712.50i | −24267.2 | + | 14010.7i | −35175.6 | + | 20308.7i | − | 17507.1i | −1448.99 | − | 2509.72i | 122584. | − | 212322.i | |||
4.3 | −47.1587 | − | 27.2271i | −219.942 | + | 380.951i | 458.627 | + | 794.365i | 8334.97i | 20744.3 | − | 11976.8i | 19275.1 | − | 11128.5i | 61573.8i | −8175.47 | − | 14160.3i | 226937. | − | 393066.i | ||||
4.4 | −28.4687 | − | 16.4364i | 286.164 | − | 495.650i | −483.689 | − | 837.774i | − | 6962.98i | −16293.4 | + | 9407.01i | 68090.0 | − | 39311.8i | 99124.0i | −75206.1 | − | 130261.i | −114446. | + | 198227.i | |||
4.5 | −18.1006 | − | 10.4504i | −200.570 | + | 347.397i | −805.579 | − | 1395.30i | − | 5339.78i | 7260.86 | − | 4192.06i | −15062.6 | + | 8696.38i | 76479.2i | 8117.03 | + | 14059.1i | −55802.7 | + | 96653.1i | |||
4.6 | 8.35978 | + | 4.82652i | 155.366 | − | 269.102i | −977.409 | − | 1692.92i | − | 71.9990i | 2597.66 | − | 1499.76i | −53916.5 | + | 31128.7i | − | 38639.4i | 40296.2 | + | 69795.0i | 347.505 | − | 601.896i | ||
4.7 | 15.7596 | + | 9.09880i | 80.7581 | − | 139.877i | −858.424 | − | 1486.83i | 13894.4i | 2545.43 | − | 1469.60i | 55086.8 | − | 31804.4i | − | 68511.2i | 75529.7 | + | 130821.i | −126422. | + | 218969.i | |||
4.8 | 35.3488 | + | 20.4086i | −366.998 | + | 635.659i | −190.977 | − | 330.782i | 3052.06i | −25945.9 | + | 14979.8i | −6894.91 | + | 3980.78i | − | 99184.0i | −180802. | − | 313158.i | −62288.3 | + | 107887.i | |||
4.9 | 42.3913 | + | 24.4746i | −27.6563 | + | 47.9021i | 174.016 | + | 301.405i | − | 10452.4i | −2344.78 | + | 1353.76i | 39866.5 | − | 23016.9i | − | 83212.2i | 87043.8 | + | 150764.i | 255820. | − | 443093.i | ||
4.10 | 53.9750 | + | 31.1625i | 401.709 | − | 695.781i | 918.197 | + | 1590.36i | − | 673.285i | 43364.5 | − | 25036.5i | −9262.47 | + | 5347.69i | − | 13188.3i | −234167. | − | 405589.i | 20981.2 | − | 36340.5i | ||
4.11 | 67.8828 | + | 39.1921i | −51.3966 | + | 89.0216i | 2048.05 | + | 3547.32i | 2746.61i | −6977.89 | + | 4028.69i | −16037.2 | + | 9259.07i | 160539.i | 83290.3 | + | 144263.i | −107645. | + | 186447.i | ||||
10.1 | −74.2981 | + | 42.8960i | −148.593 | − | 257.371i | 2656.14 | − | 4600.57i | 10410.3i | 22080.4 | + | 12748.1i | 18278.8 | + | 10553.3i | 280049.i | 44413.7 | − | 76926.7i | −446559. | − | 773464.i | ||||
10.2 | −57.1912 | + | 33.0193i | 212.159 | + | 367.469i | 1156.55 | − | 2003.21i | − | 3712.50i | −24267.2 | − | 14010.7i | −35175.6 | − | 20308.7i | 17507.1i | −1448.99 | + | 2509.72i | 122584. | + | 212322.i | |||
10.3 | −47.1587 | + | 27.2271i | −219.942 | − | 380.951i | 458.627 | − | 794.365i | − | 8334.97i | 20744.3 | + | 11976.8i | 19275.1 | + | 11128.5i | − | 61573.8i | −8175.47 | + | 14160.3i | 226937. | + | 393066.i | ||
10.4 | −28.4687 | + | 16.4364i | 286.164 | + | 495.650i | −483.689 | + | 837.774i | 6962.98i | −16293.4 | − | 9407.01i | 68090.0 | + | 39311.8i | − | 99124.0i | −75206.1 | + | 130261.i | −114446. | − | 198227.i | |||
10.5 | −18.1006 | + | 10.4504i | −200.570 | − | 347.397i | −805.579 | + | 1395.30i | 5339.78i | 7260.86 | + | 4192.06i | −15062.6 | − | 8696.38i | − | 76479.2i | 8117.03 | − | 14059.1i | −55802.7 | − | 96653.1i | |||
10.6 | 8.35978 | − | 4.82652i | 155.366 | + | 269.102i | −977.409 | + | 1692.92i | 71.9990i | 2597.66 | + | 1499.76i | −53916.5 | − | 31128.7i | 38639.4i | 40296.2 | − | 69795.0i | 347.505 | + | 601.896i | ||||
10.7 | 15.7596 | − | 9.09880i | 80.7581 | + | 139.877i | −858.424 | + | 1486.83i | − | 13894.4i | 2545.43 | + | 1469.60i | 55086.8 | + | 31804.4i | 68511.2i | 75529.7 | − | 130821.i | −126422. | − | 218969.i | |||
10.8 | 35.3488 | − | 20.4086i | −366.998 | − | 635.659i | −190.977 | + | 330.782i | − | 3052.06i | −25945.9 | − | 14979.8i | −6894.91 | − | 3980.78i | 99184.0i | −180802. | + | 313158.i | −62288.3 | − | 107887.i | |||
10.9 | 42.3913 | − | 24.4746i | −27.6563 | − | 47.9021i | 174.016 | − | 301.405i | 10452.4i | −2344.78 | − | 1353.76i | 39866.5 | + | 23016.9i | 83212.2i | 87043.8 | − | 150764.i | 255820. | + | 443093.i | ||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.12.e.a | ✓ | 22 |
13.e | even | 6 | 1 | inner | 13.12.e.a | ✓ | 22 |
13.f | odd | 12 | 2 | 169.12.a.g | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.12.e.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
13.12.e.a | ✓ | 22 | 13.e | even | 6 | 1 | inner |
169.12.a.g | 22 | 13.f | odd | 12 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(13, [\chi])\).