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: | \(32\) |
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 | −2.69601 | 0 | 5.26849 | 3.35934 | 0 | −1.00000 | −8.81189 | 0 | −9.05682 | ||||||||||||||||||
1.2 | −2.49464 | 0 | 4.22324 | −0.373762 | 0 | −1.00000 | −5.54618 | 0 | 0.932401 | ||||||||||||||||||
1.3 | −2.44637 | 0 | 3.98471 | 1.74324 | 0 | −1.00000 | −4.85534 | 0 | −4.26460 | ||||||||||||||||||
1.4 | −2.41110 | 0 | 3.81340 | −2.91376 | 0 | −1.00000 | −4.37229 | 0 | 7.02536 | ||||||||||||||||||
1.5 | −2.23843 | 0 | 3.01059 | −0.715002 | 0 | −1.00000 | −2.26213 | 0 | 1.60049 | ||||||||||||||||||
1.6 | −1.96743 | 0 | 1.87078 | 2.35101 | 0 | −1.00000 | 0.254227 | 0 | −4.62544 | ||||||||||||||||||
1.7 | −1.66842 | 0 | 0.783626 | −3.76494 | 0 | −1.00000 | 2.02942 | 0 | 6.28150 | ||||||||||||||||||
1.8 | −1.63487 | 0 | 0.672794 | 0.107826 | 0 | −1.00000 | 2.16981 | 0 | −0.176281 | ||||||||||||||||||
1.9 | −1.56027 | 0 | 0.434445 | 1.69832 | 0 | −1.00000 | 2.44269 | 0 | −2.64985 | ||||||||||||||||||
1.10 | −1.18013 | 0 | −0.607291 | 4.43826 | 0 | −1.00000 | 3.07694 | 0 | −5.23773 | ||||||||||||||||||
1.11 | −0.959867 | 0 | −1.07866 | −2.40466 | 0 | −1.00000 | 2.95510 | 0 | 2.30815 | ||||||||||||||||||
1.12 | −0.942656 | 0 | −1.11140 | −2.02754 | 0 | −1.00000 | 2.93298 | 0 | 1.91127 | ||||||||||||||||||
1.13 | −0.904420 | 0 | −1.18202 | −0.974726 | 0 | −1.00000 | 2.87789 | 0 | 0.881562 | ||||||||||||||||||
1.14 | −0.903971 | 0 | −1.18284 | 2.97865 | 0 | −1.00000 | 2.87719 | 0 | −2.69261 | ||||||||||||||||||
1.15 | −0.242500 | 0 | −1.94119 | 2.79671 | 0 | −1.00000 | 0.955741 | 0 | −0.678204 | ||||||||||||||||||
1.16 | −0.203292 | 0 | −1.95867 | −2.16833 | 0 | −1.00000 | 0.804765 | 0 | 0.440803 | ||||||||||||||||||
1.17 | 0.203292 | 0 | −1.95867 | 2.16833 | 0 | −1.00000 | −0.804765 | 0 | 0.440803 | ||||||||||||||||||
1.18 | 0.242500 | 0 | −1.94119 | −2.79671 | 0 | −1.00000 | −0.955741 | 0 | −0.678204 | ||||||||||||||||||
1.19 | 0.903971 | 0 | −1.18284 | −2.97865 | 0 | −1.00000 | −2.87719 | 0 | −2.69261 | ||||||||||||||||||
1.20 | 0.904420 | 0 | −1.18202 | 0.974726 | 0 | −1.00000 | −2.87789 | 0 | 0.881562 | ||||||||||||||||||
See all 32 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.z | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 8001.2.a.z | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8001.2.a.z | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
8001.2.a.z | ✓ | 32 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
\( T_{2}^{32} - 47 T_{2}^{30} + 991 T_{2}^{28} - 12396 T_{2}^{26} + 102523 T_{2}^{24} - 591710 T_{2}^{22} + \cdots + 1024 \) |
\( T_{5}^{32} - 98 T_{5}^{30} + 4277 T_{5}^{28} - 110184 T_{5}^{26} + 1870977 T_{5}^{24} - 22127702 T_{5}^{22} + \cdots + 7929856 \) |