Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,12,Mod(3,13)]
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([2]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.3");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.c (of order \(3\), 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: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −39.1473 | − | 67.8051i | −245.228 | − | 424.747i | −2041.02 | + | 3535.16i | 3256.36 | −19200.0 | + | 33255.4i | −27428.1 | + | 47506.9i | 159255. | −31700.0 | + | 54905.9i | −127478. | − | 220798.i | ||||
3.2 | −37.0263 | − | 64.1314i | 226.124 | + | 391.658i | −1717.89 | + | 2975.47i | 250.396 | 16745.0 | − | 29003.3i | 22976.7 | − | 39796.8i | 102768. | −13690.5 | + | 23712.6i | −9271.21 | − | 16058.2i | ||||
3.3 | −22.8013 | − | 39.4930i | −242.260 | − | 419.607i | −15.7983 | + | 27.3635i | −12226.1 | −11047.7 | + | 19135.2i | 35068.0 | − | 60739.5i | −91953.2 | −28806.7 | + | 49894.7i | 278772. | + | 482847.i | ||||
3.4 | −21.6665 | − | 37.5275i | 147.320 | + | 255.166i | 85.1270 | − | 147.444i | −1410.79 | 6383.83 | − | 11057.1i | −33309.3 | + | 57693.3i | −96123.5 | 45166.9 | − | 78231.3i | 30566.8 | + | 52943.2i | ||||
3.5 | −12.8467 | − | 22.2512i | −179.645 | − | 311.153i | 693.923 | − | 1201.91i | 11234.1 | −4615.69 | + | 7994.60i | 14221.5 | − | 24632.4i | −88278.7 | 24029.2 | − | 41619.8i | −144322. | − | 249972.i | ||||
3.6 | 0.177445 | + | 0.307344i | 85.3755 | + | 147.875i | 1023.94 | − | 1773.51i | −6278.25 | −30.2989 | + | 52.4793i | −7753.50 | + | 13429.5i | 1453.59 | 73995.6 | − | 128164.i | −1114.05 | − | 1929.58i | ||||
3.7 | 2.17841 | + | 3.77312i | 374.885 | + | 649.321i | 1014.51 | − | 1757.18i | 7470.18 | −1633.31 | + | 2828.98i | 19231.4 | − | 33309.7i | 17762.8 | −192505. | + | 333428.i | 16273.1 | + | 28185.9i | ||||
3.8 | 12.6130 | + | 21.8464i | −378.076 | − | 654.847i | 705.823 | − | 1222.52i | −3120.93 | 9537.36 | − | 16519.2i | −34271.3 | + | 59359.7i | 87273.2 | −197309. | + | 341750.i | −39364.3 | − | 68181.0i | ||||
3.9 | 22.3480 | + | 38.7079i | −79.2583 | − | 137.279i | 25.1334 | − | 43.5322i | 2002.50 | 3542.53 | − | 6135.84i | 12414.0 | − | 21501.7i | 93784.1 | 76009.7 | − | 131653.i | 44751.9 | + | 77512.5i | ||||
3.10 | 31.0971 | + | 53.8618i | 289.297 | + | 501.076i | −910.060 | + | 1576.27i | −11626.3 | −17992.6 | + | 31164.1i | 6582.83 | − | 11401.8i | 14172.8 | −78811.5 | + | 136506.i | −361545. | − | 626214.i | ||||
3.11 | 36.6814 | + | 63.5340i | 151.750 | + | 262.839i | −1667.05 | + | 2887.41i | 10169.3 | −11132.8 | + | 19282.6i | −33907.8 | + | 58730.1i | −94351.1 | 42517.2 | − | 73641.9i | 373023. | + | 646094.i | ||||
3.12 | 43.8927 | + | 76.0244i | −272.285 | − | 471.612i | −2829.14 | + | 4900.21i | −4938.39 | 23902.7 | − | 41400.6i | 24711.6 | − | 42801.8i | −316930. | −59704.8 | + | 103412.i | −216759. | − | 375438.i | ||||
9.1 | −39.1473 | + | 67.8051i | −245.228 | + | 424.747i | −2041.02 | − | 3535.16i | 3256.36 | −19200.0 | − | 33255.4i | −27428.1 | − | 47506.9i | 159255. | −31700.0 | − | 54905.9i | −127478. | + | 220798.i | ||||
9.2 | −37.0263 | + | 64.1314i | 226.124 | − | 391.658i | −1717.89 | − | 2975.47i | 250.396 | 16745.0 | + | 29003.3i | 22976.7 | + | 39796.8i | 102768. | −13690.5 | − | 23712.6i | −9271.21 | + | 16058.2i | ||||
9.3 | −22.8013 | + | 39.4930i | −242.260 | + | 419.607i | −15.7983 | − | 27.3635i | −12226.1 | −11047.7 | − | 19135.2i | 35068.0 | + | 60739.5i | −91953.2 | −28806.7 | − | 49894.7i | 278772. | − | 482847.i | ||||
9.4 | −21.6665 | + | 37.5275i | 147.320 | − | 255.166i | 85.1270 | + | 147.444i | −1410.79 | 6383.83 | + | 11057.1i | −33309.3 | − | 57693.3i | −96123.5 | 45166.9 | + | 78231.3i | 30566.8 | − | 52943.2i | ||||
9.5 | −12.8467 | + | 22.2512i | −179.645 | + | 311.153i | 693.923 | + | 1201.91i | 11234.1 | −4615.69 | − | 7994.60i | 14221.5 | + | 24632.4i | −88278.7 | 24029.2 | + | 41619.8i | −144322. | + | 249972.i | ||||
9.6 | 0.177445 | − | 0.307344i | 85.3755 | − | 147.875i | 1023.94 | + | 1773.51i | −6278.25 | −30.2989 | − | 52.4793i | −7753.50 | − | 13429.5i | 1453.59 | 73995.6 | + | 128164.i | −1114.05 | + | 1929.58i | ||||
9.7 | 2.17841 | − | 3.77312i | 374.885 | − | 649.321i | 1014.51 | + | 1757.18i | 7470.18 | −1633.31 | − | 2828.98i | 19231.4 | + | 33309.7i | 17762.8 | −192505. | − | 333428.i | 16273.1 | − | 28185.9i | ||||
9.8 | 12.6130 | − | 21.8464i | −378.076 | + | 654.847i | 705.823 | + | 1222.52i | −3120.93 | 9537.36 | + | 16519.2i | −34271.3 | − | 59359.7i | 87273.2 | −197309. | − | 341750.i | −39364.3 | + | 68181.0i | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.12.c.a | ✓ | 24 |
13.c | even | 3 | 1 | inner | 13.12.c.a | ✓ | 24 |
13.c | even | 3 | 1 | 169.12.a.d | 12 | ||
13.e | even | 6 | 1 | 169.12.a.f | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.12.c.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
13.12.c.a | ✓ | 24 | 13.c | even | 3 | 1 | inner |
169.12.a.d | 12 | 13.c | even | 3 | 1 | ||
169.12.a.f | 12 | 13.e | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(13, [\chi])\).