Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,4,Mod(289,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.289");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.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: | \(56.6418336055\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 | 0 | 3.00000 | 0 | −11.1502 | − | 0.819892i | 0 | − | 22.5310i | 0 | 9.00000 | 0 | |||||||||||||||
289.2 | 0 | 3.00000 | 0 | −11.1502 | + | 0.819892i | 0 | 22.5310i | 0 | 9.00000 | 0 | ||||||||||||||||
289.3 | 0 | 3.00000 | 0 | −9.56341 | − | 5.79148i | 0 | 10.3120i | 0 | 9.00000 | 0 | ||||||||||||||||
289.4 | 0 | 3.00000 | 0 | −9.56341 | + | 5.79148i | 0 | − | 10.3120i | 0 | 9.00000 | 0 | |||||||||||||||
289.5 | 0 | 3.00000 | 0 | −8.55031 | − | 7.20362i | 0 | − | 33.9552i | 0 | 9.00000 | 0 | |||||||||||||||
289.6 | 0 | 3.00000 | 0 | −8.55031 | + | 7.20362i | 0 | 33.9552i | 0 | 9.00000 | 0 | ||||||||||||||||
289.7 | 0 | 3.00000 | 0 | −8.18789 | − | 7.61305i | 0 | − | 16.3920i | 0 | 9.00000 | 0 | |||||||||||||||
289.8 | 0 | 3.00000 | 0 | −8.18789 | + | 7.61305i | 0 | 16.3920i | 0 | 9.00000 | 0 | ||||||||||||||||
289.9 | 0 | 3.00000 | 0 | −2.24733 | − | 10.9521i | 0 | 9.55408i | 0 | 9.00000 | 0 | ||||||||||||||||
289.10 | 0 | 3.00000 | 0 | −2.24733 | + | 10.9521i | 0 | − | 9.55408i | 0 | 9.00000 | 0 | |||||||||||||||
289.11 | 0 | 3.00000 | 0 | −1.73599 | − | 11.0447i | 0 | 1.04186i | 0 | 9.00000 | 0 | ||||||||||||||||
289.12 | 0 | 3.00000 | 0 | −1.73599 | + | 11.0447i | 0 | − | 1.04186i | 0 | 9.00000 | 0 | |||||||||||||||
289.13 | 0 | 3.00000 | 0 | 1.73599 | − | 11.0447i | 0 | 1.04186i | 0 | 9.00000 | 0 | ||||||||||||||||
289.14 | 0 | 3.00000 | 0 | 1.73599 | + | 11.0447i | 0 | − | 1.04186i | 0 | 9.00000 | 0 | |||||||||||||||
289.15 | 0 | 3.00000 | 0 | 2.24733 | − | 10.9521i | 0 | 9.55408i | 0 | 9.00000 | 0 | ||||||||||||||||
289.16 | 0 | 3.00000 | 0 | 2.24733 | + | 10.9521i | 0 | − | 9.55408i | 0 | 9.00000 | 0 | |||||||||||||||
289.17 | 0 | 3.00000 | 0 | 8.18789 | − | 7.61305i | 0 | − | 16.3920i | 0 | 9.00000 | 0 | |||||||||||||||
289.18 | 0 | 3.00000 | 0 | 8.18789 | + | 7.61305i | 0 | 16.3920i | 0 | 9.00000 | 0 | ||||||||||||||||
289.19 | 0 | 3.00000 | 0 | 8.55031 | − | 7.20362i | 0 | − | 33.9552i | 0 | 9.00000 | 0 | |||||||||||||||
289.20 | 0 | 3.00000 | 0 | 8.55031 | + | 7.20362i | 0 | 33.9552i | 0 | 9.00000 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.4.d.d | yes | 24 |
4.b | odd | 2 | 1 | 960.4.d.c | ✓ | 24 | |
5.b | even | 2 | 1 | 960.4.d.c | ✓ | 24 | |
8.b | even | 2 | 1 | 960.4.d.c | ✓ | 24 | |
8.d | odd | 2 | 1 | inner | 960.4.d.d | yes | 24 |
20.d | odd | 2 | 1 | inner | 960.4.d.d | yes | 24 |
40.e | odd | 2 | 1 | 960.4.d.c | ✓ | 24 | |
40.f | even | 2 | 1 | inner | 960.4.d.d | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
960.4.d.c | ✓ | 24 | 4.b | odd | 2 | 1 | |
960.4.d.c | ✓ | 24 | 5.b | even | 2 | 1 | |
960.4.d.c | ✓ | 24 | 8.b | even | 2 | 1 | |
960.4.d.c | ✓ | 24 | 40.e | odd | 2 | 1 | |
960.4.d.d | yes | 24 | 1.a | even | 1 | 1 | trivial |
960.4.d.d | yes | 24 | 8.d | odd | 2 | 1 | inner |
960.4.d.d | yes | 24 | 20.d | odd | 2 | 1 | inner |
960.4.d.d | yes | 24 | 40.f | 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 \(S_{4}^{\mathrm{new}}(960, [\chi])\):
\( T_{7}^{12} + 2128 T_{7}^{10} + 1424768 T_{7}^{8} + 381377152 T_{7}^{6} + 41502918656 T_{7}^{4} + \cdots + 1656966221824 \) |
\( T_{43}^{6} - 368 T_{43}^{5} - 55616 T_{43}^{4} + 14683008 T_{43}^{3} + 2310125568 T_{43}^{2} + \cdots - 1275295150080 \) |