Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2880,2,Mod(721,2880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2880.721");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2880.t (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: | \(22.9969157821\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 720) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
721.1 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 4.30899i | 0 | 0 | 0 | ||||||||||||||||
721.2 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 1.47784i | 0 | 0 | 0 | ||||||||||||||||
721.3 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | − | 2.05446i | 0 | 0 | 0 | |||||||||||||||
721.4 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | − | 0.511707i | 0 | 0 | 0 | |||||||||||||||
721.5 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 1.69880i | 0 | 0 | 0 | ||||||||||||||||
721.6 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | − | 0.635963i | 0 | 0 | 0 | |||||||||||||||
721.7 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | − | 4.35099i | 0 | 0 | 0 | |||||||||||||||
721.8 | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 4.06749i | 0 | 0 | 0 | ||||||||||||||||
721.9 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | − | 2.05446i | 0 | 0 | 0 | |||||||||||||||
721.10 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 1.47784i | 0 | 0 | 0 | ||||||||||||||||
721.11 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | − | 0.511707i | 0 | 0 | 0 | |||||||||||||||
721.12 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | − | 0.635963i | 0 | 0 | 0 | |||||||||||||||
721.13 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 1.69880i | 0 | 0 | 0 | ||||||||||||||||
721.14 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 4.30899i | 0 | 0 | 0 | ||||||||||||||||
721.15 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 4.06749i | 0 | 0 | 0 | ||||||||||||||||
721.16 | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | − | 4.35099i | 0 | 0 | 0 | |||||||||||||||
2161.1 | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | − | 4.30899i | 0 | 0 | 0 | |||||||||||||||
2161.2 | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | − | 1.47784i | 0 | 0 | 0 | |||||||||||||||
2161.3 | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 2.05446i | 0 | 0 | 0 | ||||||||||||||||
2161.4 | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 0.511707i | 0 | 0 | 0 | ||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2880.2.t.e | 32 | |
3.b | odd | 2 | 1 | inner | 2880.2.t.e | 32 | |
4.b | odd | 2 | 1 | 720.2.t.e | ✓ | 32 | |
12.b | even | 2 | 1 | 720.2.t.e | ✓ | 32 | |
16.e | even | 4 | 1 | inner | 2880.2.t.e | 32 | |
16.f | odd | 4 | 1 | 720.2.t.e | ✓ | 32 | |
48.i | odd | 4 | 1 | inner | 2880.2.t.e | 32 | |
48.k | even | 4 | 1 | 720.2.t.e | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.2.t.e | ✓ | 32 | 4.b | odd | 2 | 1 | |
720.2.t.e | ✓ | 32 | 12.b | even | 2 | 1 | |
720.2.t.e | ✓ | 32 | 16.f | odd | 4 | 1 | |
720.2.t.e | ✓ | 32 | 48.k | even | 4 | 1 | |
2880.2.t.e | 32 | 1.a | even | 1 | 1 | trivial | |
2880.2.t.e | 32 | 3.b | odd | 2 | 1 | inner | |
2880.2.t.e | 32 | 16.e | even | 4 | 1 | inner | |
2880.2.t.e | 32 | 48.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 64 T_{7}^{14} + 1544 T_{7}^{12} + 17376 T_{7}^{10} + 93456 T_{7}^{8} + 243584 T_{7}^{6} + 288000 T_{7}^{4} + 122880 T_{7}^{2} + 16384 \)
acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\).