Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3420,2,Mod(341,3420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3420, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3420.341");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3420.j (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: | \(27.3088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
341.1 | 0 | 0 | 0 | − | 1.00000i | 0 | −1.79230 | 0 | 0 | 0 | |||||||||||||||||
341.2 | 0 | 0 | 0 | 1.00000i | 0 | −1.79230 | 0 | 0 | 0 | ||||||||||||||||||
341.3 | 0 | 0 | 0 | − | 1.00000i | 0 | 1.16916 | 0 | 0 | 0 | |||||||||||||||||
341.4 | 0 | 0 | 0 | 1.00000i | 0 | 1.16916 | 0 | 0 | 0 | ||||||||||||||||||
341.5 | 0 | 0 | 0 | − | 1.00000i | 0 | 3.59503 | 0 | 0 | 0 | |||||||||||||||||
341.6 | 0 | 0 | 0 | 1.00000i | 0 | 3.59503 | 0 | 0 | 0 | ||||||||||||||||||
341.7 | 0 | 0 | 0 | − | 1.00000i | 0 | −1.89214 | 0 | 0 | 0 | |||||||||||||||||
341.8 | 0 | 0 | 0 | 1.00000i | 0 | −1.89214 | 0 | 0 | 0 | ||||||||||||||||||
341.9 | 0 | 0 | 0 | − | 1.00000i | 0 | 3.59503 | 0 | 0 | 0 | |||||||||||||||||
341.10 | 0 | 0 | 0 | 1.00000i | 0 | 3.59503 | 0 | 0 | 0 | ||||||||||||||||||
341.11 | 0 | 0 | 0 | − | 1.00000i | 0 | −1.89214 | 0 | 0 | 0 | |||||||||||||||||
341.12 | 0 | 0 | 0 | 1.00000i | 0 | −1.89214 | 0 | 0 | 0 | ||||||||||||||||||
341.13 | 0 | 0 | 0 | − | 1.00000i | 0 | −1.79230 | 0 | 0 | 0 | |||||||||||||||||
341.14 | 0 | 0 | 0 | 1.00000i | 0 | −1.79230 | 0 | 0 | 0 | ||||||||||||||||||
341.15 | 0 | 0 | 0 | − | 1.00000i | 0 | 1.16916 | 0 | 0 | 0 | |||||||||||||||||
341.16 | 0 | 0 | 0 | 1.00000i | 0 | 1.16916 | 0 | 0 | 0 | ||||||||||||||||||
341.17 | 0 | 0 | 0 | − | 1.00000i | 0 | −3.21694 | 0 | 0 | 0 | |||||||||||||||||
341.18 | 0 | 0 | 0 | 1.00000i | 0 | −3.21694 | 0 | 0 | 0 | ||||||||||||||||||
341.19 | 0 | 0 | 0 | − | 1.00000i | 0 | 2.13719 | 0 | 0 | 0 | |||||||||||||||||
341.20 | 0 | 0 | 0 | 1.00000i | 0 | 2.13719 | 0 | 0 | 0 | ||||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
57.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3420.2.j.a | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 3420.2.j.a | ✓ | 24 |
19.b | odd | 2 | 1 | inner | 3420.2.j.a | ✓ | 24 |
57.d | even | 2 | 1 | inner | 3420.2.j.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3420.2.j.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
3420.2.j.a | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
3420.2.j.a | ✓ | 24 | 19.b | odd | 2 | 1 | inner |
3420.2.j.a | ✓ | 24 | 57.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(3420, [\chi])\).