Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2640,2,Mod(1121,2640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2640, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2640.1121");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2640.f (of order \(2\), degree \(1\), 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: | \(21.0805061336\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 1320) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1121.1 | 0 | −1.64700 | − | 0.536081i | 0 | 1.00000i | 0 | − | 1.15647i | 0 | 2.42523 | + | 1.76585i | 0 | |||||||||||||
1121.2 | 0 | −1.64700 | + | 0.536081i | 0 | − | 1.00000i | 0 | 1.15647i | 0 | 2.42523 | − | 1.76585i | 0 | |||||||||||||
1121.3 | 0 | −1.42733 | − | 0.981193i | 0 | − | 1.00000i | 0 | 1.59862i | 0 | 1.07452 | + | 2.80096i | 0 | |||||||||||||
1121.4 | 0 | −1.42733 | + | 0.981193i | 0 | 1.00000i | 0 | − | 1.59862i | 0 | 1.07452 | − | 2.80096i | 0 | |||||||||||||
1121.5 | 0 | −1.32197 | − | 1.11911i | 0 | 1.00000i | 0 | 5.22520i | 0 | 0.495201 | + | 2.95885i | 0 | ||||||||||||||
1121.6 | 0 | −1.32197 | + | 1.11911i | 0 | − | 1.00000i | 0 | − | 5.22520i | 0 | 0.495201 | − | 2.95885i | 0 | ||||||||||||
1121.7 | 0 | −0.875092 | − | 1.49473i | 0 | − | 1.00000i | 0 | − | 0.174070i | 0 | −1.46843 | + | 2.61605i | 0 | ||||||||||||
1121.8 | 0 | −0.875092 | + | 1.49473i | 0 | 1.00000i | 0 | 0.174070i | 0 | −1.46843 | − | 2.61605i | 0 | ||||||||||||||
1121.9 | 0 | −0.488478 | − | 1.66174i | 0 | 1.00000i | 0 | − | 3.72161i | 0 | −2.52278 | + | 1.62345i | 0 | |||||||||||||
1121.10 | 0 | −0.488478 | + | 1.66174i | 0 | − | 1.00000i | 0 | 3.72161i | 0 | −2.52278 | − | 1.62345i | 0 | |||||||||||||
1121.11 | 0 | −0.100991 | − | 1.72910i | 0 | − | 1.00000i | 0 | 3.28518i | 0 | −2.97960 | + | 0.349247i | 0 | |||||||||||||
1121.12 | 0 | −0.100991 | + | 1.72910i | 0 | 1.00000i | 0 | − | 3.28518i | 0 | −2.97960 | − | 0.349247i | 0 | |||||||||||||
1121.13 | 0 | 0.566021 | − | 1.63695i | 0 | 1.00000i | 0 | − | 1.58806i | 0 | −2.35924 | − | 1.85310i | 0 | |||||||||||||
1121.14 | 0 | 0.566021 | + | 1.63695i | 0 | − | 1.00000i | 0 | 1.58806i | 0 | −2.35924 | + | 1.85310i | 0 | |||||||||||||
1121.15 | 0 | 0.804860 | − | 1.53369i | 0 | − | 1.00000i | 0 | − | 1.44652i | 0 | −1.70440 | − | 2.46881i | 0 | ||||||||||||
1121.16 | 0 | 0.804860 | + | 1.53369i | 0 | 1.00000i | 0 | 1.44652i | 0 | −1.70440 | + | 2.46881i | 0 | ||||||||||||||
1121.17 | 0 | 0.861793 | − | 1.50244i | 0 | 1.00000i | 0 | 3.29695i | 0 | −1.51462 | − | 2.58958i | 0 | ||||||||||||||
1121.18 | 0 | 0.861793 | + | 1.50244i | 0 | − | 1.00000i | 0 | − | 3.29695i | 0 | −1.51462 | + | 2.58958i | 0 | ||||||||||||
1121.19 | 0 | 1.24397 | − | 1.20522i | 0 | − | 1.00000i | 0 | − | 2.71015i | 0 | 0.0949023 | − | 2.99850i | 0 | ||||||||||||
1121.20 | 0 | 1.24397 | + | 1.20522i | 0 | 1.00000i | 0 | 2.71015i | 0 | 0.0949023 | + | 2.99850i | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
33.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2640.2.f.f | 24 | |
3.b | odd | 2 | 1 | 2640.2.f.g | 24 | ||
4.b | odd | 2 | 1 | 1320.2.f.b | yes | 24 | |
11.b | odd | 2 | 1 | 2640.2.f.g | 24 | ||
12.b | even | 2 | 1 | 1320.2.f.a | ✓ | 24 | |
33.d | even | 2 | 1 | inner | 2640.2.f.f | 24 | |
44.c | even | 2 | 1 | 1320.2.f.a | ✓ | 24 | |
132.d | odd | 2 | 1 | 1320.2.f.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1320.2.f.a | ✓ | 24 | 12.b | even | 2 | 1 | |
1320.2.f.a | ✓ | 24 | 44.c | even | 2 | 1 | |
1320.2.f.b | yes | 24 | 4.b | odd | 2 | 1 | |
1320.2.f.b | yes | 24 | 132.d | odd | 2 | 1 | |
2640.2.f.f | 24 | 1.a | even | 1 | 1 | trivial | |
2640.2.f.f | 24 | 33.d | even | 2 | 1 | inner | |
2640.2.f.g | 24 | 3.b | odd | 2 | 1 | ||
2640.2.f.g | 24 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2640, [\chi])\):
\( T_{7}^{24} + 94 T_{7}^{22} + 3721 T_{7}^{20} + 82120 T_{7}^{18} + 1121760 T_{7}^{16} + 9930912 T_{7}^{14} + \cdots + 7573504 \) |
\( T_{17}^{12} - 6 T_{17}^{11} - 75 T_{17}^{10} + 404 T_{17}^{9} + 1672 T_{17}^{8} - 7520 T_{17}^{7} + \cdots + 18624 \) |