Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1248,2,Mod(463,1248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1248, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 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("1248.463");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1248.bb (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: | \(9.96533017226\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 312) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
463.1 | 0 | 1.00000 | 0 | −3.08779 | + | 3.08779i | 0 | 2.85816 | + | 2.85816i | 0 | 1.00000 | 0 | ||||||||||||||
463.2 | 0 | 1.00000 | 0 | −1.98058 | + | 1.98058i | 0 | −3.05943 | − | 3.05943i | 0 | 1.00000 | 0 | ||||||||||||||
463.3 | 0 | 1.00000 | 0 | −1.67211 | + | 1.67211i | 0 | 1.16012 | + | 1.16012i | 0 | 1.00000 | 0 | ||||||||||||||
463.4 | 0 | 1.00000 | 0 | −1.61967 | + | 1.61967i | 0 | −2.08158 | − | 2.08158i | 0 | 1.00000 | 0 | ||||||||||||||
463.5 | 0 | 1.00000 | 0 | −0.273741 | + | 0.273741i | 0 | 1.75949 | + | 1.75949i | 0 | 1.00000 | 0 | ||||||||||||||
463.6 | 0 | 1.00000 | 0 | −0.220561 | + | 0.220561i | 0 | 0.834354 | + | 0.834354i | 0 | 1.00000 | 0 | ||||||||||||||
463.7 | 0 | 1.00000 | 0 | 0.220561 | − | 0.220561i | 0 | −0.834354 | − | 0.834354i | 0 | 1.00000 | 0 | ||||||||||||||
463.8 | 0 | 1.00000 | 0 | 0.273741 | − | 0.273741i | 0 | −1.75949 | − | 1.75949i | 0 | 1.00000 | 0 | ||||||||||||||
463.9 | 0 | 1.00000 | 0 | 1.61967 | − | 1.61967i | 0 | 2.08158 | + | 2.08158i | 0 | 1.00000 | 0 | ||||||||||||||
463.10 | 0 | 1.00000 | 0 | 1.67211 | − | 1.67211i | 0 | −1.16012 | − | 1.16012i | 0 | 1.00000 | 0 | ||||||||||||||
463.11 | 0 | 1.00000 | 0 | 1.98058 | − | 1.98058i | 0 | 3.05943 | + | 3.05943i | 0 | 1.00000 | 0 | ||||||||||||||
463.12 | 0 | 1.00000 | 0 | 3.08779 | − | 3.08779i | 0 | −2.85816 | − | 2.85816i | 0 | 1.00000 | 0 | ||||||||||||||
655.1 | 0 | 1.00000 | 0 | −3.08779 | − | 3.08779i | 0 | 2.85816 | − | 2.85816i | 0 | 1.00000 | 0 | ||||||||||||||
655.2 | 0 | 1.00000 | 0 | −1.98058 | − | 1.98058i | 0 | −3.05943 | + | 3.05943i | 0 | 1.00000 | 0 | ||||||||||||||
655.3 | 0 | 1.00000 | 0 | −1.67211 | − | 1.67211i | 0 | 1.16012 | − | 1.16012i | 0 | 1.00000 | 0 | ||||||||||||||
655.4 | 0 | 1.00000 | 0 | −1.61967 | − | 1.61967i | 0 | −2.08158 | + | 2.08158i | 0 | 1.00000 | 0 | ||||||||||||||
655.5 | 0 | 1.00000 | 0 | −0.273741 | − | 0.273741i | 0 | 1.75949 | − | 1.75949i | 0 | 1.00000 | 0 | ||||||||||||||
655.6 | 0 | 1.00000 | 0 | −0.220561 | − | 0.220561i | 0 | 0.834354 | − | 0.834354i | 0 | 1.00000 | 0 | ||||||||||||||
655.7 | 0 | 1.00000 | 0 | 0.220561 | + | 0.220561i | 0 | −0.834354 | + | 0.834354i | 0 | 1.00000 | 0 | ||||||||||||||
655.8 | 0 | 1.00000 | 0 | 0.273741 | + | 0.273741i | 0 | −1.75949 | + | 1.75949i | 0 | 1.00000 | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
13.d | odd | 4 | 1 | inner |
104.m | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1248.2.bb.f | 24 | |
4.b | odd | 2 | 1 | 312.2.t.e | ✓ | 24 | |
8.b | even | 2 | 1 | 312.2.t.e | ✓ | 24 | |
8.d | odd | 2 | 1 | inner | 1248.2.bb.f | 24 | |
12.b | even | 2 | 1 | 936.2.w.j | 24 | ||
13.d | odd | 4 | 1 | inner | 1248.2.bb.f | 24 | |
24.h | odd | 2 | 1 | 936.2.w.j | 24 | ||
52.f | even | 4 | 1 | 312.2.t.e | ✓ | 24 | |
104.j | odd | 4 | 1 | 312.2.t.e | ✓ | 24 | |
104.m | even | 4 | 1 | inner | 1248.2.bb.f | 24 | |
156.l | odd | 4 | 1 | 936.2.w.j | 24 | ||
312.y | even | 4 | 1 | 936.2.w.j | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
312.2.t.e | ✓ | 24 | 4.b | odd | 2 | 1 | |
312.2.t.e | ✓ | 24 | 8.b | even | 2 | 1 | |
312.2.t.e | ✓ | 24 | 52.f | even | 4 | 1 | |
312.2.t.e | ✓ | 24 | 104.j | odd | 4 | 1 | |
936.2.w.j | 24 | 12.b | even | 2 | 1 | ||
936.2.w.j | 24 | 24.h | odd | 2 | 1 | ||
936.2.w.j | 24 | 156.l | odd | 4 | 1 | ||
936.2.w.j | 24 | 312.y | even | 4 | 1 | ||
1248.2.bb.f | 24 | 1.a | even | 1 | 1 | trivial | |
1248.2.bb.f | 24 | 8.d | odd | 2 | 1 | inner | |
1248.2.bb.f | 24 | 13.d | odd | 4 | 1 | inner | |
1248.2.bb.f | 24 | 104.m | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\):
\( T_{5}^{24} + 484T_{5}^{20} + 48256T_{5}^{16} + 1683456T_{5}^{12} + 19318784T_{5}^{8} + 615424T_{5}^{4} + 4096 \) |
\( T_{7}^{24} + 740T_{7}^{20} + 173184T_{7}^{16} + 13927936T_{7}^{12} + 385439744T_{7}^{8} + 2647475200T_{7}^{4} + 3782742016 \) |