Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [690,3,Mod(229,690)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("690.229");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.f (of order \(2\), degree \(1\), 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: | \(18.8011382409\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
229.1 | − | 1.41421i | 1.73205i | −2.00000 | −1.71908 | + | 4.69518i | 2.44949 | −12.9397 | 2.82843i | −3.00000 | 6.63999 | + | 2.43115i | |||||||||||||
229.2 | − | 1.41421i | 1.73205i | −2.00000 | 1.71908 | − | 4.69518i | 2.44949 | 12.9397 | 2.82843i | −3.00000 | −6.63999 | − | 2.43115i | |||||||||||||
229.3 | 1.41421i | − | 1.73205i | −2.00000 | −1.71908 | − | 4.69518i | 2.44949 | −12.9397 | − | 2.82843i | −3.00000 | 6.63999 | − | 2.43115i | ||||||||||||
229.4 | 1.41421i | − | 1.73205i | −2.00000 | 1.71908 | + | 4.69518i | 2.44949 | 12.9397 | − | 2.82843i | −3.00000 | −6.63999 | + | 2.43115i | ||||||||||||
229.5 | − | 1.41421i | − | 1.73205i | −2.00000 | −3.10921 | − | 3.91571i | −2.44949 | 3.09410 | 2.82843i | −3.00000 | −5.53765 | + | 4.39709i | ||||||||||||
229.6 | − | 1.41421i | − | 1.73205i | −2.00000 | 3.10921 | + | 3.91571i | −2.44949 | −3.09410 | 2.82843i | −3.00000 | 5.53765 | − | 4.39709i | ||||||||||||
229.7 | 1.41421i | 1.73205i | −2.00000 | −3.10921 | + | 3.91571i | −2.44949 | 3.09410 | − | 2.82843i | −3.00000 | −5.53765 | − | 4.39709i | |||||||||||||
229.8 | 1.41421i | 1.73205i | −2.00000 | 3.10921 | − | 3.91571i | −2.44949 | −3.09410 | − | 2.82843i | −3.00000 | 5.53765 | + | 4.39709i | |||||||||||||
229.9 | − | 1.41421i | − | 1.73205i | −2.00000 | −4.91938 | + | 0.894267i | −2.44949 | 6.65471 | 2.82843i | −3.00000 | 1.26468 | + | 6.95705i | ||||||||||||
229.10 | − | 1.41421i | − | 1.73205i | −2.00000 | 4.91938 | − | 0.894267i | −2.44949 | −6.65471 | 2.82843i | −3.00000 | −1.26468 | − | 6.95705i | ||||||||||||
229.11 | 1.41421i | 1.73205i | −2.00000 | −4.91938 | − | 0.894267i | −2.44949 | 6.65471 | − | 2.82843i | −3.00000 | 1.26468 | − | 6.95705i | |||||||||||||
229.12 | 1.41421i | 1.73205i | −2.00000 | 4.91938 | + | 0.894267i | −2.44949 | −6.65471 | − | 2.82843i | −3.00000 | −1.26468 | + | 6.95705i | |||||||||||||
229.13 | − | 1.41421i | 1.73205i | −2.00000 | −4.27075 | − | 2.60014i | 2.44949 | −3.19570 | 2.82843i | −3.00000 | −3.67715 | + | 6.03975i | |||||||||||||
229.14 | − | 1.41421i | 1.73205i | −2.00000 | 4.27075 | + | 2.60014i | 2.44949 | 3.19570 | 2.82843i | −3.00000 | 3.67715 | − | 6.03975i | |||||||||||||
229.15 | 1.41421i | − | 1.73205i | −2.00000 | −4.27075 | + | 2.60014i | 2.44949 | −3.19570 | − | 2.82843i | −3.00000 | −3.67715 | − | 6.03975i | ||||||||||||
229.16 | 1.41421i | − | 1.73205i | −2.00000 | 4.27075 | − | 2.60014i | 2.44949 | 3.19570 | − | 2.82843i | −3.00000 | 3.67715 | + | 6.03975i | ||||||||||||
229.17 | − | 1.41421i | 1.73205i | −2.00000 | −1.15789 | + | 4.86408i | 2.44949 | 1.79343 | 2.82843i | −3.00000 | 6.87885 | + | 1.63751i | |||||||||||||
229.18 | − | 1.41421i | 1.73205i | −2.00000 | 1.15789 | − | 4.86408i | 2.44949 | −1.79343 | 2.82843i | −3.00000 | −6.87885 | − | 1.63751i | |||||||||||||
229.19 | 1.41421i | − | 1.73205i | −2.00000 | −1.15789 | − | 4.86408i | 2.44949 | 1.79343 | − | 2.82843i | −3.00000 | 6.87885 | − | 1.63751i | ||||||||||||
229.20 | 1.41421i | − | 1.73205i | −2.00000 | 1.15789 | + | 4.86408i | 2.44949 | −1.79343 | − | 2.82843i | −3.00000 | −6.87885 | + | 1.63751i | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
115.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.3.f.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 690.3.f.a | ✓ | 48 |
23.b | odd | 2 | 1 | inner | 690.3.f.a | ✓ | 48 |
115.c | odd | 2 | 1 | inner | 690.3.f.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.3.f.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
690.3.f.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
690.3.f.a | ✓ | 48 | 23.b | odd | 2 | 1 | inner |
690.3.f.a | ✓ | 48 | 115.c | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(690, [\chi])\).