Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8001,2,Mod(1,8001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8001 = 3^{2} \cdot 7 \cdot 127 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8001.a (trivial) |
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: | yes |
Analytic conductor: | \(63.8883066572\) |
Analytic rank: | \(1\) |
Dimension: | \(22\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.37730 | 0 | 3.65154 | 2.39021 | 0 | 1.00000 | −3.92621 | 0 | −5.68223 | ||||||||||||||||||
1.2 | −2.32723 | 0 | 3.41602 | 1.82757 | 0 | 1.00000 | −3.29542 | 0 | −4.25318 | ||||||||||||||||||
1.3 | −2.21969 | 0 | 2.92701 | 0.739015 | 0 | 1.00000 | −2.05767 | 0 | −1.64038 | ||||||||||||||||||
1.4 | −1.94436 | 0 | 1.78053 | −1.34248 | 0 | 1.00000 | 0.426737 | 0 | 2.61025 | ||||||||||||||||||
1.5 | −1.69586 | 0 | 0.875937 | −2.42277 | 0 | 1.00000 | 1.90625 | 0 | 4.10867 | ||||||||||||||||||
1.6 | −1.56609 | 0 | 0.452631 | −2.71234 | 0 | 1.00000 | 2.42332 | 0 | 4.24776 | ||||||||||||||||||
1.7 | −0.775659 | 0 | −1.39835 | −0.833166 | 0 | 1.00000 | 2.63596 | 0 | 0.646253 | ||||||||||||||||||
1.8 | −0.691771 | 0 | −1.52145 | −0.236679 | 0 | 1.00000 | 2.43604 | 0 | 0.163728 | ||||||||||||||||||
1.9 | −0.684852 | 0 | −1.53098 | −0.182709 | 0 | 1.00000 | 2.41820 | 0 | 0.125129 | ||||||||||||||||||
1.10 | −0.446366 | 0 | −1.80076 | 2.33773 | 0 | 1.00000 | 1.69653 | 0 | −1.04348 | ||||||||||||||||||
1.11 | −0.384540 | 0 | −1.85213 | 3.33517 | 0 | 1.00000 | 1.48130 | 0 | −1.28251 | ||||||||||||||||||
1.12 | 0.384540 | 0 | −1.85213 | −3.33517 | 0 | 1.00000 | −1.48130 | 0 | −1.28251 | ||||||||||||||||||
1.13 | 0.446366 | 0 | −1.80076 | −2.33773 | 0 | 1.00000 | −1.69653 | 0 | −1.04348 | ||||||||||||||||||
1.14 | 0.684852 | 0 | −1.53098 | 0.182709 | 0 | 1.00000 | −2.41820 | 0 | 0.125129 | ||||||||||||||||||
1.15 | 0.691771 | 0 | −1.52145 | 0.236679 | 0 | 1.00000 | −2.43604 | 0 | 0.163728 | ||||||||||||||||||
1.16 | 0.775659 | 0 | −1.39835 | 0.833166 | 0 | 1.00000 | −2.63596 | 0 | 0.646253 | ||||||||||||||||||
1.17 | 1.56609 | 0 | 0.452631 | 2.71234 | 0 | 1.00000 | −2.42332 | 0 | 4.24776 | ||||||||||||||||||
1.18 | 1.69586 | 0 | 0.875937 | 2.42277 | 0 | 1.00000 | −1.90625 | 0 | 4.10867 | ||||||||||||||||||
1.19 | 1.94436 | 0 | 1.78053 | 1.34248 | 0 | 1.00000 | −0.426737 | 0 | 2.61025 | ||||||||||||||||||
1.20 | 2.21969 | 0 | 2.92701 | −0.739015 | 0 | 1.00000 | 2.05767 | 0 | −1.64038 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(-1\) |
\(127\) | \(-1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8001.2.a.x | ✓ | 22 |
3.b | odd | 2 | 1 | inner | 8001.2.a.x | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8001.2.a.x | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
8001.2.a.x | ✓ | 22 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
\( T_{2}^{22} - 27 T_{2}^{20} + 307 T_{2}^{18} - 1912 T_{2}^{16} + 7111 T_{2}^{14} - 16187 T_{2}^{12} + \cdots - 16 \) |
\( T_{5}^{22} - 42 T_{5}^{20} + 737 T_{5}^{18} - 7048 T_{5}^{16} + 40077 T_{5}^{14} - 138490 T_{5}^{12} + \cdots - 64 \) |