Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [280,2,Mod(97,280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(280, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("280.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 280.x (of order \(4\), degree \(2\), 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: | \(2.23581125660\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 | 0 | −2.16993 | − | 2.16993i | 0 | 0.272751 | + | 2.21937i | 0 | −1.63589 | + | 2.07939i | 0 | 6.41716i | 0 | ||||||||||||
97.2 | 0 | −2.16341 | − | 2.16341i | 0 | 1.91827 | − | 1.14901i | 0 | −0.409966 | − | 2.61380i | 0 | 6.36066i | 0 | ||||||||||||
97.3 | 0 | −1.41195 | − | 1.41195i | 0 | −1.94983 | − | 1.09461i | 0 | 0.0496428 | + | 2.64529i | 0 | 0.987218i | 0 | ||||||||||||
97.4 | 0 | −1.03893 | − | 1.03893i | 0 | 1.21973 | − | 1.87411i | 0 | 2.11557 | + | 1.58883i | 0 | − | 0.841261i | 0 | |||||||||||
97.5 | 0 | −0.730185 | − | 0.730185i | 0 | −1.51201 | + | 1.64737i | 0 | 2.41329 | − | 1.08445i | 0 | − | 1.93366i | 0 | |||||||||||
97.6 | 0 | −0.0703127 | − | 0.0703127i | 0 | −2.16104 | − | 0.574360i | 0 | −2.58523 | − | 0.562657i | 0 | − | 2.99011i | 0 | |||||||||||
97.7 | 0 | 0.0703127 | + | 0.0703127i | 0 | 2.16104 | + | 0.574360i | 0 | −0.562657 | − | 2.58523i | 0 | − | 2.99011i | 0 | |||||||||||
97.8 | 0 | 0.730185 | + | 0.730185i | 0 | 1.51201 | − | 1.64737i | 0 | −1.08445 | + | 2.41329i | 0 | − | 1.93366i | 0 | |||||||||||
97.9 | 0 | 1.03893 | + | 1.03893i | 0 | −1.21973 | + | 1.87411i | 0 | 1.58883 | + | 2.11557i | 0 | − | 0.841261i | 0 | |||||||||||
97.10 | 0 | 1.41195 | + | 1.41195i | 0 | 1.94983 | + | 1.09461i | 0 | 2.64529 | + | 0.0496428i | 0 | 0.987218i | 0 | ||||||||||||
97.11 | 0 | 2.16341 | + | 2.16341i | 0 | −1.91827 | + | 1.14901i | 0 | −2.61380 | − | 0.409966i | 0 | 6.36066i | 0 | ||||||||||||
97.12 | 0 | 2.16993 | + | 2.16993i | 0 | −0.272751 | − | 2.21937i | 0 | 2.07939 | − | 1.63589i | 0 | 6.41716i | 0 | ||||||||||||
153.1 | 0 | −2.16993 | + | 2.16993i | 0 | 0.272751 | − | 2.21937i | 0 | −1.63589 | − | 2.07939i | 0 | − | 6.41716i | 0 | |||||||||||
153.2 | 0 | −2.16341 | + | 2.16341i | 0 | 1.91827 | + | 1.14901i | 0 | −0.409966 | + | 2.61380i | 0 | − | 6.36066i | 0 | |||||||||||
153.3 | 0 | −1.41195 | + | 1.41195i | 0 | −1.94983 | + | 1.09461i | 0 | 0.0496428 | − | 2.64529i | 0 | − | 0.987218i | 0 | |||||||||||
153.4 | 0 | −1.03893 | + | 1.03893i | 0 | 1.21973 | + | 1.87411i | 0 | 2.11557 | − | 1.58883i | 0 | 0.841261i | 0 | ||||||||||||
153.5 | 0 | −0.730185 | + | 0.730185i | 0 | −1.51201 | − | 1.64737i | 0 | 2.41329 | + | 1.08445i | 0 | 1.93366i | 0 | ||||||||||||
153.6 | 0 | −0.0703127 | + | 0.0703127i | 0 | −2.16104 | + | 0.574360i | 0 | −2.58523 | + | 0.562657i | 0 | 2.99011i | 0 | ||||||||||||
153.7 | 0 | 0.0703127 | − | 0.0703127i | 0 | 2.16104 | − | 0.574360i | 0 | −0.562657 | + | 2.58523i | 0 | 2.99011i | 0 | ||||||||||||
153.8 | 0 | 0.730185 | − | 0.730185i | 0 | 1.51201 | + | 1.64737i | 0 | −1.08445 | − | 2.41329i | 0 | 1.93366i | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 280.2.x.a | ✓ | 24 |
4.b | odd | 2 | 1 | 560.2.bj.d | 24 | ||
5.b | even | 2 | 1 | 1400.2.x.b | 24 | ||
5.c | odd | 4 | 1 | inner | 280.2.x.a | ✓ | 24 |
5.c | odd | 4 | 1 | 1400.2.x.b | 24 | ||
7.b | odd | 2 | 1 | inner | 280.2.x.a | ✓ | 24 |
20.e | even | 4 | 1 | 560.2.bj.d | 24 | ||
28.d | even | 2 | 1 | 560.2.bj.d | 24 | ||
35.c | odd | 2 | 1 | 1400.2.x.b | 24 | ||
35.f | even | 4 | 1 | inner | 280.2.x.a | ✓ | 24 |
35.f | even | 4 | 1 | 1400.2.x.b | 24 | ||
140.j | odd | 4 | 1 | 560.2.bj.d | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
280.2.x.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
280.2.x.a | ✓ | 24 | 5.c | odd | 4 | 1 | inner |
280.2.x.a | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
280.2.x.a | ✓ | 24 | 35.f | even | 4 | 1 | inner |
560.2.bj.d | 24 | 4.b | odd | 2 | 1 | ||
560.2.bj.d | 24 | 20.e | even | 4 | 1 | ||
560.2.bj.d | 24 | 28.d | even | 2 | 1 | ||
560.2.bj.d | 24 | 140.j | odd | 4 | 1 | ||
1400.2.x.b | 24 | 5.b | even | 2 | 1 | ||
1400.2.x.b | 24 | 5.c | odd | 4 | 1 | ||
1400.2.x.b | 24 | 35.c | odd | 2 | 1 | ||
1400.2.x.b | 24 | 35.f | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(280, [\chi])\).