Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [224,3,Mod(17,224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("224.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.n (of order \(6\), 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: | \(6.10355792167\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 56) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −2.78005 | + | 4.81519i | 0 | −1.52921 | − | 2.64866i | 0 | −0.608243 | − | 6.97352i | 0 | −10.9574 | − | 18.9787i | 0 | ||||||||||
17.2 | 0 | −1.94818 | + | 3.37434i | 0 | 4.42985 | + | 7.67272i | 0 | 6.92329 | + | 1.03347i | 0 | −3.09078 | − | 5.35338i | 0 | ||||||||||
17.3 | 0 | −1.93494 | + | 3.35141i | 0 | 2.33882 | + | 4.05096i | 0 | −6.95505 | + | 0.792023i | 0 | −2.98798 | − | 5.17534i | 0 | ||||||||||
17.4 | 0 | −1.70138 | + | 2.94687i | 0 | −2.15858 | − | 3.73877i | 0 | 1.43197 | + | 6.85197i | 0 | −1.28938 | − | 2.23327i | 0 | ||||||||||
17.5 | 0 | −1.16781 | + | 2.02271i | 0 | −1.55055 | − | 2.68563i | 0 | 6.89374 | − | 1.21502i | 0 | 1.77242 | + | 3.06992i | 0 | ||||||||||
17.6 | 0 | −0.455431 | + | 0.788830i | 0 | −3.17251 | − | 5.49495i | 0 | −3.79106 | + | 5.88455i | 0 | 4.08516 | + | 7.07571i | 0 | ||||||||||
17.7 | 0 | −0.126628 | + | 0.219326i | 0 | −1.78589 | − | 3.09325i | 0 | −2.89466 | − | 6.37346i | 0 | 4.46793 | + | 7.73868i | 0 | ||||||||||
17.8 | 0 | 0.126628 | − | 0.219326i | 0 | 1.78589 | + | 3.09325i | 0 | −2.89466 | − | 6.37346i | 0 | 4.46793 | + | 7.73868i | 0 | ||||||||||
17.9 | 0 | 0.455431 | − | 0.788830i | 0 | 3.17251 | + | 5.49495i | 0 | −3.79106 | + | 5.88455i | 0 | 4.08516 | + | 7.07571i | 0 | ||||||||||
17.10 | 0 | 1.16781 | − | 2.02271i | 0 | 1.55055 | + | 2.68563i | 0 | 6.89374 | − | 1.21502i | 0 | 1.77242 | + | 3.06992i | 0 | ||||||||||
17.11 | 0 | 1.70138 | − | 2.94687i | 0 | 2.15858 | + | 3.73877i | 0 | 1.43197 | + | 6.85197i | 0 | −1.28938 | − | 2.23327i | 0 | ||||||||||
17.12 | 0 | 1.93494 | − | 3.35141i | 0 | −2.33882 | − | 4.05096i | 0 | −6.95505 | + | 0.792023i | 0 | −2.98798 | − | 5.17534i | 0 | ||||||||||
17.13 | 0 | 1.94818 | − | 3.37434i | 0 | −4.42985 | − | 7.67272i | 0 | 6.92329 | + | 1.03347i | 0 | −3.09078 | − | 5.35338i | 0 | ||||||||||
17.14 | 0 | 2.78005 | − | 4.81519i | 0 | 1.52921 | + | 2.64866i | 0 | −0.608243 | − | 6.97352i | 0 | −10.9574 | − | 18.9787i | 0 | ||||||||||
145.1 | 0 | −2.78005 | − | 4.81519i | 0 | −1.52921 | + | 2.64866i | 0 | −0.608243 | + | 6.97352i | 0 | −10.9574 | + | 18.9787i | 0 | ||||||||||
145.2 | 0 | −1.94818 | − | 3.37434i | 0 | 4.42985 | − | 7.67272i | 0 | 6.92329 | − | 1.03347i | 0 | −3.09078 | + | 5.35338i | 0 | ||||||||||
145.3 | 0 | −1.93494 | − | 3.35141i | 0 | 2.33882 | − | 4.05096i | 0 | −6.95505 | − | 0.792023i | 0 | −2.98798 | + | 5.17534i | 0 | ||||||||||
145.4 | 0 | −1.70138 | − | 2.94687i | 0 | −2.15858 | + | 3.73877i | 0 | 1.43197 | − | 6.85197i | 0 | −1.28938 | + | 2.23327i | 0 | ||||||||||
145.5 | 0 | −1.16781 | − | 2.02271i | 0 | −1.55055 | + | 2.68563i | 0 | 6.89374 | + | 1.21502i | 0 | 1.77242 | − | 3.06992i | 0 | ||||||||||
145.6 | 0 | −0.455431 | − | 0.788830i | 0 | −3.17251 | + | 5.49495i | 0 | −3.79106 | − | 5.88455i | 0 | 4.08516 | − | 7.07571i | 0 | ||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
8.b | even | 2 | 1 | inner |
56.j | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.3.n.a | 28 | |
4.b | odd | 2 | 1 | 56.3.j.a | ✓ | 28 | |
7.c | even | 3 | 1 | 1568.3.h.a | 28 | ||
7.d | odd | 6 | 1 | inner | 224.3.n.a | 28 | |
7.d | odd | 6 | 1 | 1568.3.h.a | 28 | ||
8.b | even | 2 | 1 | inner | 224.3.n.a | 28 | |
8.d | odd | 2 | 1 | 56.3.j.a | ✓ | 28 | |
28.d | even | 2 | 1 | 392.3.j.e | 28 | ||
28.f | even | 6 | 1 | 56.3.j.a | ✓ | 28 | |
28.f | even | 6 | 1 | 392.3.h.a | 28 | ||
28.g | odd | 6 | 1 | 392.3.h.a | 28 | ||
28.g | odd | 6 | 1 | 392.3.j.e | 28 | ||
56.e | even | 2 | 1 | 392.3.j.e | 28 | ||
56.j | odd | 6 | 1 | inner | 224.3.n.a | 28 | |
56.j | odd | 6 | 1 | 1568.3.h.a | 28 | ||
56.k | odd | 6 | 1 | 392.3.h.a | 28 | ||
56.k | odd | 6 | 1 | 392.3.j.e | 28 | ||
56.m | even | 6 | 1 | 56.3.j.a | ✓ | 28 | |
56.m | even | 6 | 1 | 392.3.h.a | 28 | ||
56.p | even | 6 | 1 | 1568.3.h.a | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.3.j.a | ✓ | 28 | 4.b | odd | 2 | 1 | |
56.3.j.a | ✓ | 28 | 8.d | odd | 2 | 1 | |
56.3.j.a | ✓ | 28 | 28.f | even | 6 | 1 | |
56.3.j.a | ✓ | 28 | 56.m | even | 6 | 1 | |
224.3.n.a | 28 | 1.a | even | 1 | 1 | trivial | |
224.3.n.a | 28 | 7.d | odd | 6 | 1 | inner | |
224.3.n.a | 28 | 8.b | even | 2 | 1 | inner | |
224.3.n.a | 28 | 56.j | odd | 6 | 1 | inner | |
392.3.h.a | 28 | 28.f | even | 6 | 1 | ||
392.3.h.a | 28 | 28.g | odd | 6 | 1 | ||
392.3.h.a | 28 | 56.k | odd | 6 | 1 | ||
392.3.h.a | 28 | 56.m | even | 6 | 1 | ||
392.3.j.e | 28 | 28.d | even | 2 | 1 | ||
392.3.j.e | 28 | 28.g | odd | 6 | 1 | ||
392.3.j.e | 28 | 56.e | even | 2 | 1 | ||
392.3.j.e | 28 | 56.k | odd | 6 | 1 | ||
1568.3.h.a | 28 | 7.c | even | 3 | 1 | ||
1568.3.h.a | 28 | 7.d | odd | 6 | 1 | ||
1568.3.h.a | 28 | 56.j | odd | 6 | 1 | ||
1568.3.h.a | 28 | 56.p | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(224, [\chi])\).