Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,2,Mod(47,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 9, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.z (of order \(12\), degree \(4\), 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: | \(3.57729801055\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 112) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | 0 | −0.743580 | + | 2.77508i | 0 | −3.38680 | + | 0.907491i | 0 | −0.0791552 | − | 2.64457i | 0 | −4.55008 | − | 2.62699i | 0 | ||||||||||
47.2 | 0 | −0.615336 | + | 2.29647i | 0 | −0.835825 | + | 0.223959i | 0 | 1.25761 | + | 2.32775i | 0 | −2.29704 | − | 1.32620i | 0 | ||||||||||
47.3 | 0 | −0.524583 | + | 1.95777i | 0 | 2.48890 | − | 0.666898i | 0 | 2.38027 | + | 1.15512i | 0 | −0.959602 | − | 0.554026i | 0 | ||||||||||
47.4 | 0 | −0.449868 | + | 1.67893i | 0 | 0.731029 | − | 0.195879i | 0 | −2.52163 | − | 0.800849i | 0 | −0.0183525 | − | 0.0105958i | 0 | ||||||||||
47.5 | 0 | −0.350301 | + | 1.30734i | 0 | 1.09872 | − | 0.294400i | 0 | 1.56831 | − | 2.13082i | 0 | 1.01165 | + | 0.584076i | 0 | ||||||||||
47.6 | 0 | −0.0530773 | + | 0.198087i | 0 | −1.82029 | + | 0.487744i | 0 | −1.84933 | + | 1.89208i | 0 | 2.56165 | + | 1.47897i | 0 | ||||||||||
47.7 | 0 | 0.0661591 | − | 0.246909i | 0 | −0.499339 | + | 0.133797i | 0 | −2.45011 | − | 0.998472i | 0 | 2.54149 | + | 1.46733i | 0 | ||||||||||
47.8 | 0 | 0.204601 | − | 0.763582i | 0 | −3.81370 | + | 1.02188i | 0 | 2.64575 | + | 0.00379639i | 0 | 2.05688 | + | 1.18754i | 0 | ||||||||||
47.9 | 0 | 0.237312 | − | 0.885661i | 0 | 3.05579 | − | 0.818796i | 0 | 0.640837 | + | 2.56697i | 0 | 1.87000 | + | 1.07964i | 0 | ||||||||||
47.10 | 0 | 0.282507 | − | 1.05433i | 0 | −1.39922 | + | 0.374919i | 0 | 0.298614 | + | 2.62885i | 0 | 1.56627 | + | 0.904287i | 0 | ||||||||||
47.11 | 0 | 0.388227 | − | 1.44888i | 0 | 2.81809 | − | 0.755106i | 0 | 1.47726 | − | 2.19493i | 0 | 0.649537 | + | 0.375010i | 0 | ||||||||||
47.12 | 0 | 0.665801 | − | 2.48480i | 0 | −3.12401 | + | 0.837076i | 0 | −1.56215 | − | 2.13534i | 0 | −3.13287 | − | 1.80877i | 0 | ||||||||||
47.13 | 0 | 0.700587 | − | 2.61463i | 0 | 0.450647 | − | 0.120751i | 0 | 2.37326 | − | 1.16946i | 0 | −3.74737 | − | 2.16355i | 0 | ||||||||||
47.14 | 0 | 0.825526 | − | 3.08091i | 0 | 1.86998 | − | 0.501060i | 0 | −2.17953 | + | 1.49988i | 0 | −6.21241 | − | 3.58674i | 0 | ||||||||||
143.1 | 0 | −0.743580 | − | 2.77508i | 0 | −3.38680 | − | 0.907491i | 0 | −0.0791552 | + | 2.64457i | 0 | −4.55008 | + | 2.62699i | 0 | ||||||||||
143.2 | 0 | −0.615336 | − | 2.29647i | 0 | −0.835825 | − | 0.223959i | 0 | 1.25761 | − | 2.32775i | 0 | −2.29704 | + | 1.32620i | 0 | ||||||||||
143.3 | 0 | −0.524583 | − | 1.95777i | 0 | 2.48890 | + | 0.666898i | 0 | 2.38027 | − | 1.15512i | 0 | −0.959602 | + | 0.554026i | 0 | ||||||||||
143.4 | 0 | −0.449868 | − | 1.67893i | 0 | 0.731029 | + | 0.195879i | 0 | −2.52163 | + | 0.800849i | 0 | −0.0183525 | + | 0.0105958i | 0 | ||||||||||
143.5 | 0 | −0.350301 | − | 1.30734i | 0 | 1.09872 | + | 0.294400i | 0 | 1.56831 | + | 2.13082i | 0 | 1.01165 | − | 0.584076i | 0 | ||||||||||
143.6 | 0 | −0.0530773 | − | 0.198087i | 0 | −1.82029 | − | 0.487744i | 0 | −1.84933 | − | 1.89208i | 0 | 2.56165 | − | 1.47897i | 0 | ||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
16.f | odd | 4 | 1 | inner |
112.v | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 448.2.z.a | 56 | |
4.b | odd | 2 | 1 | 112.2.v.a | ✓ | 56 | |
7.d | odd | 6 | 1 | inner | 448.2.z.a | 56 | |
8.b | even | 2 | 1 | 896.2.z.a | 56 | ||
8.d | odd | 2 | 1 | 896.2.z.b | 56 | ||
16.e | even | 4 | 1 | 112.2.v.a | ✓ | 56 | |
16.e | even | 4 | 1 | 896.2.z.b | 56 | ||
16.f | odd | 4 | 1 | inner | 448.2.z.a | 56 | |
16.f | odd | 4 | 1 | 896.2.z.a | 56 | ||
28.d | even | 2 | 1 | 784.2.w.f | 56 | ||
28.f | even | 6 | 1 | 112.2.v.a | ✓ | 56 | |
28.f | even | 6 | 1 | 784.2.j.a | 56 | ||
28.g | odd | 6 | 1 | 784.2.j.a | 56 | ||
28.g | odd | 6 | 1 | 784.2.w.f | 56 | ||
56.j | odd | 6 | 1 | 896.2.z.a | 56 | ||
56.m | even | 6 | 1 | 896.2.z.b | 56 | ||
112.l | odd | 4 | 1 | 784.2.w.f | 56 | ||
112.v | even | 12 | 1 | inner | 448.2.z.a | 56 | |
112.v | even | 12 | 1 | 896.2.z.a | 56 | ||
112.w | even | 12 | 1 | 784.2.j.a | 56 | ||
112.w | even | 12 | 1 | 784.2.w.f | 56 | ||
112.x | odd | 12 | 1 | 112.2.v.a | ✓ | 56 | |
112.x | odd | 12 | 1 | 784.2.j.a | 56 | ||
112.x | odd | 12 | 1 | 896.2.z.b | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.2.v.a | ✓ | 56 | 4.b | odd | 2 | 1 | |
112.2.v.a | ✓ | 56 | 16.e | even | 4 | 1 | |
112.2.v.a | ✓ | 56 | 28.f | even | 6 | 1 | |
112.2.v.a | ✓ | 56 | 112.x | odd | 12 | 1 | |
448.2.z.a | 56 | 1.a | even | 1 | 1 | trivial | |
448.2.z.a | 56 | 7.d | odd | 6 | 1 | inner | |
448.2.z.a | 56 | 16.f | odd | 4 | 1 | inner | |
448.2.z.a | 56 | 112.v | even | 12 | 1 | inner | |
784.2.j.a | 56 | 28.f | even | 6 | 1 | ||
784.2.j.a | 56 | 28.g | odd | 6 | 1 | ||
784.2.j.a | 56 | 112.w | even | 12 | 1 | ||
784.2.j.a | 56 | 112.x | odd | 12 | 1 | ||
784.2.w.f | 56 | 28.d | even | 2 | 1 | ||
784.2.w.f | 56 | 28.g | odd | 6 | 1 | ||
784.2.w.f | 56 | 112.l | odd | 4 | 1 | ||
784.2.w.f | 56 | 112.w | even | 12 | 1 | ||
896.2.z.a | 56 | 8.b | even | 2 | 1 | ||
896.2.z.a | 56 | 16.f | odd | 4 | 1 | ||
896.2.z.a | 56 | 56.j | odd | 6 | 1 | ||
896.2.z.a | 56 | 112.v | even | 12 | 1 | ||
896.2.z.b | 56 | 8.d | odd | 2 | 1 | ||
896.2.z.b | 56 | 16.e | even | 4 | 1 | ||
896.2.z.b | 56 | 56.m | even | 6 | 1 | ||
896.2.z.b | 56 | 112.x | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(448, [\chi])\).