Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2160,2,Mod(593,2160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2160.593");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2160.w (of order \(4\), degree \(2\), not 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: | \(17.2476868366\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 1080) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
593.1 | 0 | 0 | 0 | −2.23600 | − | 0.0178032i | 0 | 0.600231 | − | 0.600231i | 0 | 0 | 0 | ||||||||||||||
593.2 | 0 | 0 | 0 | −2.04677 | + | 0.900415i | 0 | 1.47577 | − | 1.47577i | 0 | 0 | 0 | ||||||||||||||
593.3 | 0 | 0 | 0 | −1.95707 | − | 1.08160i | 0 | 0.0505342 | − | 0.0505342i | 0 | 0 | 0 | ||||||||||||||
593.4 | 0 | 0 | 0 | −1.69619 | + | 1.45703i | 0 | −2.91751 | + | 2.91751i | 0 | 0 | 0 | ||||||||||||||
593.5 | 0 | 0 | 0 | −0.260885 | − | 2.22080i | 0 | 1.86697 | − | 1.86697i | 0 | 0 | 0 | ||||||||||||||
593.6 | 0 | 0 | 0 | −0.189230 | + | 2.22805i | 0 | −3.07601 | + | 3.07601i | 0 | 0 | 0 | ||||||||||||||
593.7 | 0 | 0 | 0 | 0.189230 | − | 2.22805i | 0 | −3.07601 | + | 3.07601i | 0 | 0 | 0 | ||||||||||||||
593.8 | 0 | 0 | 0 | 0.260885 | + | 2.22080i | 0 | 1.86697 | − | 1.86697i | 0 | 0 | 0 | ||||||||||||||
593.9 | 0 | 0 | 0 | 1.69619 | − | 1.45703i | 0 | −2.91751 | + | 2.91751i | 0 | 0 | 0 | ||||||||||||||
593.10 | 0 | 0 | 0 | 1.95707 | + | 1.08160i | 0 | 0.0505342 | − | 0.0505342i | 0 | 0 | 0 | ||||||||||||||
593.11 | 0 | 0 | 0 | 2.04677 | − | 0.900415i | 0 | 1.47577 | − | 1.47577i | 0 | 0 | 0 | ||||||||||||||
593.12 | 0 | 0 | 0 | 2.23600 | + | 0.0178032i | 0 | 0.600231 | − | 0.600231i | 0 | 0 | 0 | ||||||||||||||
1457.1 | 0 | 0 | 0 | −2.23600 | + | 0.0178032i | 0 | 0.600231 | + | 0.600231i | 0 | 0 | 0 | ||||||||||||||
1457.2 | 0 | 0 | 0 | −2.04677 | − | 0.900415i | 0 | 1.47577 | + | 1.47577i | 0 | 0 | 0 | ||||||||||||||
1457.3 | 0 | 0 | 0 | −1.95707 | + | 1.08160i | 0 | 0.0505342 | + | 0.0505342i | 0 | 0 | 0 | ||||||||||||||
1457.4 | 0 | 0 | 0 | −1.69619 | − | 1.45703i | 0 | −2.91751 | − | 2.91751i | 0 | 0 | 0 | ||||||||||||||
1457.5 | 0 | 0 | 0 | −0.260885 | + | 2.22080i | 0 | 1.86697 | + | 1.86697i | 0 | 0 | 0 | ||||||||||||||
1457.6 | 0 | 0 | 0 | −0.189230 | − | 2.22805i | 0 | −3.07601 | − | 3.07601i | 0 | 0 | 0 | ||||||||||||||
1457.7 | 0 | 0 | 0 | 0.189230 | + | 2.22805i | 0 | −3.07601 | − | 3.07601i | 0 | 0 | 0 | ||||||||||||||
1457.8 | 0 | 0 | 0 | 0.260885 | − | 2.22080i | 0 | 1.86697 | + | 1.86697i | 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 |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2160.2.w.g | 24 | |
3.b | odd | 2 | 1 | inner | 2160.2.w.g | 24 | |
4.b | odd | 2 | 1 | 1080.2.s.b | ✓ | 24 | |
5.c | odd | 4 | 1 | inner | 2160.2.w.g | 24 | |
12.b | even | 2 | 1 | 1080.2.s.b | ✓ | 24 | |
15.e | even | 4 | 1 | inner | 2160.2.w.g | 24 | |
20.e | even | 4 | 1 | 1080.2.s.b | ✓ | 24 | |
60.l | odd | 4 | 1 | 1080.2.s.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1080.2.s.b | ✓ | 24 | 4.b | odd | 2 | 1 | |
1080.2.s.b | ✓ | 24 | 12.b | even | 2 | 1 | |
1080.2.s.b | ✓ | 24 | 20.e | even | 4 | 1 | |
1080.2.s.b | ✓ | 24 | 60.l | odd | 4 | 1 | |
2160.2.w.g | 24 | 1.a | even | 1 | 1 | trivial | |
2160.2.w.g | 24 | 3.b | odd | 2 | 1 | inner | |
2160.2.w.g | 24 | 5.c | odd | 4 | 1 | inner | |
2160.2.w.g | 24 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 4 T_{7}^{11} + 8 T_{7}^{10} - 48 T_{7}^{9} + 134 T_{7}^{8} + 240 T_{7}^{7} + 1040 T_{7}^{6} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(2160, [\chi])\).