Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6048,2,Mod(5615,6048)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6048.5615");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6048.j (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: | \(48.2935231425\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | no (minimal twist has level 1512) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5615.1 | 0 | 0 | 0 | −4.18263 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.2 | 0 | 0 | 0 | −4.18263 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.3 | 0 | 0 | 0 | −3.69622 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.4 | 0 | 0 | 0 | −3.69622 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.5 | 0 | 0 | 0 | −2.99711 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.6 | 0 | 0 | 0 | −2.99711 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.7 | 0 | 0 | 0 | −2.90802 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.8 | 0 | 0 | 0 | −2.90802 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.9 | 0 | 0 | 0 | −2.85878 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.10 | 0 | 0 | 0 | −2.85878 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.11 | 0 | 0 | 0 | −2.35025 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.12 | 0 | 0 | 0 | −2.35025 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.13 | 0 | 0 | 0 | −2.29653 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.14 | 0 | 0 | 0 | −2.29653 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.15 | 0 | 0 | 0 | −1.27819 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.16 | 0 | 0 | 0 | −1.27819 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.17 | 0 | 0 | 0 | −1.27600 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.18 | 0 | 0 | 0 | −1.27600 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||
5615.19 | 0 | 0 | 0 | −0.863507 | 0 | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||
5615.20 | 0 | 0 | 0 | −0.863507 | 0 | − | 1.00000i | 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 |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6048.2.j.d | 48 | |
3.b | odd | 2 | 1 | inner | 6048.2.j.d | 48 | |
4.b | odd | 2 | 1 | 1512.2.j.d | ✓ | 48 | |
8.b | even | 2 | 1 | 1512.2.j.d | ✓ | 48 | |
8.d | odd | 2 | 1 | inner | 6048.2.j.d | 48 | |
12.b | even | 2 | 1 | 1512.2.j.d | ✓ | 48 | |
24.f | even | 2 | 1 | inner | 6048.2.j.d | 48 | |
24.h | odd | 2 | 1 | 1512.2.j.d | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1512.2.j.d | ✓ | 48 | 4.b | odd | 2 | 1 | |
1512.2.j.d | ✓ | 48 | 8.b | even | 2 | 1 | |
1512.2.j.d | ✓ | 48 | 12.b | even | 2 | 1 | |
1512.2.j.d | ✓ | 48 | 24.h | odd | 2 | 1 | |
6048.2.j.d | 48 | 1.a | even | 1 | 1 | trivial | |
6048.2.j.d | 48 | 3.b | odd | 2 | 1 | inner | |
6048.2.j.d | 48 | 8.d | odd | 2 | 1 | inner | |
6048.2.j.d | 48 | 24.f | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 2204 T_{5}^{20} - 37640 T_{5}^{18} + 395238 T_{5}^{16} - 2649064 T_{5}^{14} + \cdots + 350464 \) acting on \(S_{2}^{\mathrm{new}}(6048, [\chi])\).