Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1120,2,Mod(1119,1120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1120.1119");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1120 = 2^{5} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1120.e (of order \(2\), degree \(1\), 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: | \(8.94324502638\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1119.1 | 0 | − | 2.21376i | 0 | −1.81856 | + | 1.30110i | 0 | −1.09155 | + | 2.41009i | 0 | −1.90072 | 0 | |||||||||||||
1119.2 | 0 | 2.21376i | 0 | −1.81856 | − | 1.30110i | 0 | −1.09155 | − | 2.41009i | 0 | −1.90072 | 0 | ||||||||||||||
1119.3 | 0 | − | 3.06386i | 0 | −1.39003 | − | 1.75152i | 0 | −2.46063 | + | 0.972270i | 0 | −6.38723 | 0 | |||||||||||||
1119.4 | 0 | 3.06386i | 0 | −1.39003 | + | 1.75152i | 0 | −2.46063 | − | 0.972270i | 0 | −6.38723 | 0 | ||||||||||||||
1119.5 | 0 | − | 1.86733i | 0 | 2.07366 | + | 0.836614i | 0 | −2.64040 | − | 0.168201i | 0 | −0.486908 | 0 | |||||||||||||
1119.6 | 0 | 1.86733i | 0 | 2.07366 | − | 0.836614i | 0 | −2.64040 | + | 0.168201i | 0 | −0.486908 | 0 | ||||||||||||||
1119.7 | 0 | − | 3.06386i | 0 | −1.39003 | + | 1.75152i | 0 | 2.46063 | + | 0.972270i | 0 | −6.38723 | 0 | |||||||||||||
1119.8 | 0 | 3.06386i | 0 | −1.39003 | − | 1.75152i | 0 | 2.46063 | − | 0.972270i | 0 | −6.38723 | 0 | ||||||||||||||
1119.9 | 0 | − | 1.86733i | 0 | −2.07366 | − | 0.836614i | 0 | 2.64040 | − | 0.168201i | 0 | −0.486908 | 0 | |||||||||||||
1119.10 | 0 | 1.86733i | 0 | −2.07366 | + | 0.836614i | 0 | 2.64040 | + | 0.168201i | 0 | −0.486908 | 0 | ||||||||||||||
1119.11 | 0 | − | 0.326439i | 0 | −0.278349 | + | 2.21868i | 0 | 2.25431 | + | 1.38495i | 0 | 2.89344 | 0 | |||||||||||||
1119.12 | 0 | 0.326439i | 0 | −0.278349 | − | 2.21868i | 0 | 2.25431 | − | 1.38495i | 0 | 2.89344 | 0 | ||||||||||||||
1119.13 | 0 | − | 0.326439i | 0 | 0.278349 | + | 2.21868i | 0 | 2.25431 | + | 1.38495i | 0 | 2.89344 | 0 | |||||||||||||
1119.14 | 0 | 0.326439i | 0 | 0.278349 | − | 2.21868i | 0 | 2.25431 | − | 1.38495i | 0 | 2.89344 | 0 | ||||||||||||||
1119.15 | 0 | − | 3.06386i | 0 | 1.39003 | − | 1.75152i | 0 | −2.46063 | + | 0.972270i | 0 | −6.38723 | 0 | |||||||||||||
1119.16 | 0 | 3.06386i | 0 | 1.39003 | + | 1.75152i | 0 | −2.46063 | − | 0.972270i | 0 | −6.38723 | 0 | ||||||||||||||
1119.17 | 0 | − | 2.40740i | 0 | 0.475379 | − | 2.18495i | 0 | 0.283298 | − | 2.63054i | 0 | −2.79558 | 0 | |||||||||||||
1119.18 | 0 | 2.40740i | 0 | 0.475379 | + | 2.18495i | 0 | 0.283298 | + | 2.63054i | 0 | −2.79558 | 0 | ||||||||||||||
1119.19 | 0 | − | 3.06386i | 0 | 1.39003 | + | 1.75152i | 0 | 2.46063 | + | 0.972270i | 0 | −6.38723 | 0 | |||||||||||||
1119.20 | 0 | 3.06386i | 0 | 1.39003 | − | 1.75152i | 0 | 2.46063 | − | 0.972270i | 0 | −6.38723 | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
140.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1120.2.e.a | ✓ | 48 |
4.b | odd | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
7.b | odd | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
8.b | even | 2 | 1 | 2240.2.e.g | 48 | ||
8.d | odd | 2 | 1 | 2240.2.e.g | 48 | ||
20.d | odd | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
28.d | even | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
35.c | odd | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
40.e | odd | 2 | 1 | 2240.2.e.g | 48 | ||
40.f | even | 2 | 1 | 2240.2.e.g | 48 | ||
56.e | even | 2 | 1 | 2240.2.e.g | 48 | ||
56.h | odd | 2 | 1 | 2240.2.e.g | 48 | ||
140.c | even | 2 | 1 | inner | 1120.2.e.a | ✓ | 48 |
280.c | odd | 2 | 1 | 2240.2.e.g | 48 | ||
280.n | even | 2 | 1 | 2240.2.e.g | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1120.2.e.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
1120.2.e.a | ✓ | 48 | 4.b | odd | 2 | 1 | inner |
1120.2.e.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
1120.2.e.a | ✓ | 48 | 7.b | odd | 2 | 1 | inner |
1120.2.e.a | ✓ | 48 | 20.d | odd | 2 | 1 | inner |
1120.2.e.a | ✓ | 48 | 28.d | even | 2 | 1 | inner |
1120.2.e.a | ✓ | 48 | 35.c | odd | 2 | 1 | inner |
1120.2.e.a | ✓ | 48 | 140.c | even | 2 | 1 | inner |
2240.2.e.g | 48 | 8.b | even | 2 | 1 | ||
2240.2.e.g | 48 | 8.d | odd | 2 | 1 | ||
2240.2.e.g | 48 | 40.e | odd | 2 | 1 | ||
2240.2.e.g | 48 | 40.f | even | 2 | 1 | ||
2240.2.e.g | 48 | 56.e | even | 2 | 1 | ||
2240.2.e.g | 48 | 56.h | odd | 2 | 1 | ||
2240.2.e.g | 48 | 280.c | odd | 2 | 1 | ||
2240.2.e.g | 48 | 280.n | even | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1120, [\chi])\).