Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4410,2,Mod(4409,4410)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4410.4409");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4410.d (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: | \(35.2140272914\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4409.1 | 1.00000 | 0 | 1.00000 | −2.12190 | − | 0.705358i | 0 | 0 | 1.00000 | 0 | −2.12190 | − | 0.705358i | ||||||||||||||
4409.2 | 1.00000 | 0 | 1.00000 | −2.12190 | + | 0.705358i | 0 | 0 | 1.00000 | 0 | −2.12190 | + | 0.705358i | ||||||||||||||
4409.3 | 1.00000 | 0 | 1.00000 | −2.00118 | − | 0.997634i | 0 | 0 | 1.00000 | 0 | −2.00118 | − | 0.997634i | ||||||||||||||
4409.4 | 1.00000 | 0 | 1.00000 | −2.00118 | + | 0.997634i | 0 | 0 | 1.00000 | 0 | −2.00118 | + | 0.997634i | ||||||||||||||
4409.5 | 1.00000 | 0 | 1.00000 | −1.60367 | − | 1.55828i | 0 | 0 | 1.00000 | 0 | −1.60367 | − | 1.55828i | ||||||||||||||
4409.6 | 1.00000 | 0 | 1.00000 | −1.60367 | + | 1.55828i | 0 | 0 | 1.00000 | 0 | −1.60367 | + | 1.55828i | ||||||||||||||
4409.7 | 1.00000 | 0 | 1.00000 | −1.12606 | − | 1.93184i | 0 | 0 | 1.00000 | 0 | −1.12606 | − | 1.93184i | ||||||||||||||
4409.8 | 1.00000 | 0 | 1.00000 | −1.12606 | + | 1.93184i | 0 | 0 | 1.00000 | 0 | −1.12606 | + | 1.93184i | ||||||||||||||
4409.9 | 1.00000 | 0 | 1.00000 | −0.613159 | − | 2.15036i | 0 | 0 | 1.00000 | 0 | −0.613159 | − | 2.15036i | ||||||||||||||
4409.10 | 1.00000 | 0 | 1.00000 | −0.613159 | + | 2.15036i | 0 | 0 | 1.00000 | 0 | −0.613159 | + | 2.15036i | ||||||||||||||
4409.11 | 1.00000 | 0 | 1.00000 | −0.526370 | − | 2.17323i | 0 | 0 | 1.00000 | 0 | −0.526370 | − | 2.17323i | ||||||||||||||
4409.12 | 1.00000 | 0 | 1.00000 | −0.526370 | + | 2.17323i | 0 | 0 | 1.00000 | 0 | −0.526370 | + | 2.17323i | ||||||||||||||
4409.13 | 1.00000 | 0 | 1.00000 | 0.526370 | − | 2.17323i | 0 | 0 | 1.00000 | 0 | 0.526370 | − | 2.17323i | ||||||||||||||
4409.14 | 1.00000 | 0 | 1.00000 | 0.526370 | + | 2.17323i | 0 | 0 | 1.00000 | 0 | 0.526370 | + | 2.17323i | ||||||||||||||
4409.15 | 1.00000 | 0 | 1.00000 | 0.613159 | − | 2.15036i | 0 | 0 | 1.00000 | 0 | 0.613159 | − | 2.15036i | ||||||||||||||
4409.16 | 1.00000 | 0 | 1.00000 | 0.613159 | + | 2.15036i | 0 | 0 | 1.00000 | 0 | 0.613159 | + | 2.15036i | ||||||||||||||
4409.17 | 1.00000 | 0 | 1.00000 | 1.12606 | − | 1.93184i | 0 | 0 | 1.00000 | 0 | 1.12606 | − | 1.93184i | ||||||||||||||
4409.18 | 1.00000 | 0 | 1.00000 | 1.12606 | + | 1.93184i | 0 | 0 | 1.00000 | 0 | 1.12606 | + | 1.93184i | ||||||||||||||
4409.19 | 1.00000 | 0 | 1.00000 | 1.60367 | − | 1.55828i | 0 | 0 | 1.00000 | 0 | 1.60367 | − | 1.55828i | ||||||||||||||
4409.20 | 1.00000 | 0 | 1.00000 | 1.60367 | + | 1.55828i | 0 | 0 | 1.00000 | 0 | 1.60367 | + | 1.55828i | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
105.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4410.2.d.d | yes | 24 |
3.b | odd | 2 | 1 | 4410.2.d.c | ✓ | 24 | |
5.b | even | 2 | 1 | 4410.2.d.c | ✓ | 24 | |
7.b | odd | 2 | 1 | inner | 4410.2.d.d | yes | 24 |
15.d | odd | 2 | 1 | inner | 4410.2.d.d | yes | 24 |
21.c | even | 2 | 1 | 4410.2.d.c | ✓ | 24 | |
35.c | odd | 2 | 1 | 4410.2.d.c | ✓ | 24 | |
105.g | even | 2 | 1 | inner | 4410.2.d.d | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4410.2.d.c | ✓ | 24 | 3.b | odd | 2 | 1 | |
4410.2.d.c | ✓ | 24 | 5.b | even | 2 | 1 | |
4410.2.d.c | ✓ | 24 | 21.c | even | 2 | 1 | |
4410.2.d.c | ✓ | 24 | 35.c | odd | 2 | 1 | |
4410.2.d.d | yes | 24 | 1.a | even | 1 | 1 | trivial |
4410.2.d.d | yes | 24 | 7.b | odd | 2 | 1 | inner |
4410.2.d.d | yes | 24 | 15.d | odd | 2 | 1 | inner |
4410.2.d.d | yes | 24 | 105.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4410, [\chi])\):
\( T_{11}^{12} + 56T_{11}^{10} + 1140T_{11}^{8} + 10032T_{11}^{6} + 33476T_{11}^{4} + 13696T_{11}^{2} + 1024 \) |
\( T_{23}^{6} + 8T_{23}^{5} - 20T_{23}^{4} - 192T_{23}^{3} - 68T_{23}^{2} + 832T_{23} + 896 \) |