Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1440,2,Mod(17,1440)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1440.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1440.bj (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: | \(11.4984578911\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 360) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | 0 | 0 | −2.23604 | − | 0.0113158i | 0 | −0.471963 | − | 0.471963i | 0 | 0 | 0 | ||||||||||||||
17.2 | 0 | 0 | 0 | −2.23604 | − | 0.0113158i | 0 | −0.471963 | − | 0.471963i | 0 | 0 | 0 | ||||||||||||||
17.3 | 0 | 0 | 0 | −2.03368 | + | 0.929591i | 0 | 2.49469 | + | 2.49469i | 0 | 0 | 0 | ||||||||||||||
17.4 | 0 | 0 | 0 | −2.03368 | + | 0.929591i | 0 | 2.49469 | + | 2.49469i | 0 | 0 | 0 | ||||||||||||||
17.5 | 0 | 0 | 0 | −1.45381 | + | 1.69895i | 0 | −1.53029 | − | 1.53029i | 0 | 0 | 0 | ||||||||||||||
17.6 | 0 | 0 | 0 | −1.45381 | + | 1.69895i | 0 | −1.53029 | − | 1.53029i | 0 | 0 | 0 | ||||||||||||||
17.7 | 0 | 0 | 0 | −1.42597 | − | 1.72238i | 0 | −3.11972 | − | 3.11972i | 0 | 0 | 0 | ||||||||||||||
17.8 | 0 | 0 | 0 | −1.42597 | − | 1.72238i | 0 | −3.11972 | − | 3.11972i | 0 | 0 | 0 | ||||||||||||||
17.9 | 0 | 0 | 0 | −1.29263 | − | 1.82458i | 0 | 0.306649 | + | 0.306649i | 0 | 0 | 0 | ||||||||||||||
17.10 | 0 | 0 | 0 | −1.29263 | − | 1.82458i | 0 | 0.306649 | + | 0.306649i | 0 | 0 | 0 | ||||||||||||||
17.11 | 0 | 0 | 0 | −0.215413 | + | 2.22567i | 0 | 2.32063 | + | 2.32063i | 0 | 0 | 0 | ||||||||||||||
17.12 | 0 | 0 | 0 | −0.215413 | + | 2.22567i | 0 | 2.32063 | + | 2.32063i | 0 | 0 | 0 | ||||||||||||||
17.13 | 0 | 0 | 0 | 0.215413 | − | 2.22567i | 0 | 2.32063 | + | 2.32063i | 0 | 0 | 0 | ||||||||||||||
17.14 | 0 | 0 | 0 | 0.215413 | − | 2.22567i | 0 | 2.32063 | + | 2.32063i | 0 | 0 | 0 | ||||||||||||||
17.15 | 0 | 0 | 0 | 1.29263 | + | 1.82458i | 0 | 0.306649 | + | 0.306649i | 0 | 0 | 0 | ||||||||||||||
17.16 | 0 | 0 | 0 | 1.29263 | + | 1.82458i | 0 | 0.306649 | + | 0.306649i | 0 | 0 | 0 | ||||||||||||||
17.17 | 0 | 0 | 0 | 1.42597 | + | 1.72238i | 0 | −3.11972 | − | 3.11972i | 0 | 0 | 0 | ||||||||||||||
17.18 | 0 | 0 | 0 | 1.42597 | + | 1.72238i | 0 | −3.11972 | − | 3.11972i | 0 | 0 | 0 | ||||||||||||||
17.19 | 0 | 0 | 0 | 1.45381 | − | 1.69895i | 0 | −1.53029 | − | 1.53029i | 0 | 0 | 0 | ||||||||||||||
17.20 | 0 | 0 | 0 | 1.45381 | − | 1.69895i | 0 | −1.53029 | − | 1.53029i | 0 | 0 | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
8.b | even | 2 | 1 | inner |
15.e | even | 4 | 1 | inner |
24.h | odd | 2 | 1 | inner |
40.i | odd | 4 | 1 | inner |
120.w | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1440.2.bj.a | 48 | |
3.b | odd | 2 | 1 | inner | 1440.2.bj.a | 48 | |
4.b | odd | 2 | 1 | 360.2.x.a | ✓ | 48 | |
5.c | odd | 4 | 1 | inner | 1440.2.bj.a | 48 | |
8.b | even | 2 | 1 | inner | 1440.2.bj.a | 48 | |
8.d | odd | 2 | 1 | 360.2.x.a | ✓ | 48 | |
12.b | even | 2 | 1 | 360.2.x.a | ✓ | 48 | |
15.e | even | 4 | 1 | inner | 1440.2.bj.a | 48 | |
20.e | even | 4 | 1 | 360.2.x.a | ✓ | 48 | |
24.f | even | 2 | 1 | 360.2.x.a | ✓ | 48 | |
24.h | odd | 2 | 1 | inner | 1440.2.bj.a | 48 | |
40.i | odd | 4 | 1 | inner | 1440.2.bj.a | 48 | |
40.k | even | 4 | 1 | 360.2.x.a | ✓ | 48 | |
60.l | odd | 4 | 1 | 360.2.x.a | ✓ | 48 | |
120.q | odd | 4 | 1 | 360.2.x.a | ✓ | 48 | |
120.w | even | 4 | 1 | inner | 1440.2.bj.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.2.x.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
360.2.x.a | ✓ | 48 | 8.d | odd | 2 | 1 | |
360.2.x.a | ✓ | 48 | 12.b | even | 2 | 1 | |
360.2.x.a | ✓ | 48 | 20.e | even | 4 | 1 | |
360.2.x.a | ✓ | 48 | 24.f | even | 2 | 1 | |
360.2.x.a | ✓ | 48 | 40.k | even | 4 | 1 | |
360.2.x.a | ✓ | 48 | 60.l | odd | 4 | 1 | |
360.2.x.a | ✓ | 48 | 120.q | odd | 4 | 1 | |
1440.2.bj.a | 48 | 1.a | even | 1 | 1 | trivial | |
1440.2.bj.a | 48 | 3.b | odd | 2 | 1 | inner | |
1440.2.bj.a | 48 | 5.c | odd | 4 | 1 | inner | |
1440.2.bj.a | 48 | 8.b | even | 2 | 1 | inner | |
1440.2.bj.a | 48 | 15.e | even | 4 | 1 | inner | |
1440.2.bj.a | 48 | 24.h | odd | 2 | 1 | inner | |
1440.2.bj.a | 48 | 40.i | odd | 4 | 1 | inner | |
1440.2.bj.a | 48 | 120.w | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1440, [\chi])\).