Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8043,2,Mod(1,8043)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8043, 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("8043.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8043 = 3 \cdot 7 \cdot 383 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8043.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: | \(64.2236783457\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
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 | −2.75477 | −1.00000 | 5.58878 | −0.0622071 | 2.75477 | 1.00000 | −9.88627 | 1.00000 | 0.171367 | ||||||||||||||||||
1.2 | −2.71076 | −1.00000 | 5.34821 | 0.130231 | 2.71076 | 1.00000 | −9.07620 | 1.00000 | −0.353025 | ||||||||||||||||||
1.3 | −2.56328 | −1.00000 | 4.57042 | −4.02293 | 2.56328 | 1.00000 | −6.58873 | 1.00000 | 10.3119 | ||||||||||||||||||
1.4 | −2.55848 | −1.00000 | 4.54580 | 3.55238 | 2.55848 | 1.00000 | −6.51338 | 1.00000 | −9.08869 | ||||||||||||||||||
1.5 | −2.45692 | −1.00000 | 4.03647 | −2.79922 | 2.45692 | 1.00000 | −5.00345 | 1.00000 | 6.87748 | ||||||||||||||||||
1.6 | −2.35415 | −1.00000 | 3.54201 | −1.85354 | 2.35415 | 1.00000 | −3.63012 | 1.00000 | 4.36349 | ||||||||||||||||||
1.7 | −2.28128 | −1.00000 | 3.20426 | 2.19375 | 2.28128 | 1.00000 | −2.74725 | 1.00000 | −5.00457 | ||||||||||||||||||
1.8 | −1.95494 | −1.00000 | 1.82179 | −1.46647 | 1.95494 | 1.00000 | 0.348396 | 1.00000 | 2.86686 | ||||||||||||||||||
1.9 | −1.95089 | −1.00000 | 1.80596 | 3.26605 | 1.95089 | 1.00000 | 0.378554 | 1.00000 | −6.37169 | ||||||||||||||||||
1.10 | −1.90192 | −1.00000 | 1.61732 | 0.767188 | 1.90192 | 1.00000 | 0.727834 | 1.00000 | −1.45913 | ||||||||||||||||||
1.11 | −1.85290 | −1.00000 | 1.43325 | −3.21014 | 1.85290 | 1.00000 | 1.05014 | 1.00000 | 5.94808 | ||||||||||||||||||
1.12 | −1.77779 | −1.00000 | 1.16053 | 3.56820 | 1.77779 | 1.00000 | 1.49240 | 1.00000 | −6.34350 | ||||||||||||||||||
1.13 | −1.64173 | −1.00000 | 0.695293 | 0.237832 | 1.64173 | 1.00000 | 2.14198 | 1.00000 | −0.390457 | ||||||||||||||||||
1.14 | −1.61741 | −1.00000 | 0.616031 | −3.81830 | 1.61741 | 1.00000 | 2.23845 | 1.00000 | 6.17577 | ||||||||||||||||||
1.15 | −1.58348 | −1.00000 | 0.507395 | −2.03406 | 1.58348 | 1.00000 | 2.36350 | 1.00000 | 3.22089 | ||||||||||||||||||
1.16 | −1.33568 | −1.00000 | −0.215968 | 1.69750 | 1.33568 | 1.00000 | 2.95982 | 1.00000 | −2.26731 | ||||||||||||||||||
1.17 | −1.10272 | −1.00000 | −0.784010 | 0.760743 | 1.10272 | 1.00000 | 3.06998 | 1.00000 | −0.838886 | ||||||||||||||||||
1.18 | −0.890431 | −1.00000 | −1.20713 | −2.94587 | 0.890431 | 1.00000 | 2.85573 | 1.00000 | 2.62309 | ||||||||||||||||||
1.19 | −0.869058 | −1.00000 | −1.24474 | −3.05181 | 0.869058 | 1.00000 | 2.81987 | 1.00000 | 2.65220 | ||||||||||||||||||
1.20 | −0.837626 | −1.00000 | −1.29838 | 2.34855 | 0.837626 | 1.00000 | 2.76281 | 1.00000 | −1.96721 | ||||||||||||||||||
See all 52 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(-1\) |
\(383\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8043.2.a.t | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8043.2.a.t | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):
\( T_{2}^{52} - 3 T_{2}^{51} - 78 T_{2}^{50} + 231 T_{2}^{49} + 2852 T_{2}^{48} - 8311 T_{2}^{47} + \cdots - 384 \) |
\( T_{5}^{52} + 7 T_{5}^{51} - 147 T_{5}^{50} - 1121 T_{5}^{49} + 9710 T_{5}^{48} + 82735 T_{5}^{47} + \cdots - 10636333056 \) |
\( T_{11}^{52} - 9 T_{11}^{51} - 300 T_{11}^{50} + 2922 T_{11}^{49} + 40661 T_{11}^{48} + \cdots + 12\!\cdots\!72 \) |