Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8002,2,Mod(1,8002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8002, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8002 = 2 \cdot 4001 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8002.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: | \(63.8962916974\) |
Analytic rank: | \(1\) |
Dimension: | \(89\) |
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 | −1.00000 | −3.34684 | 1.00000 | −3.62971 | 3.34684 | −3.29144 | −1.00000 | 8.20136 | 3.62971 | ||||||||||||||||||
1.2 | −1.00000 | −3.32456 | 1.00000 | 2.37351 | 3.32456 | −1.54900 | −1.00000 | 8.05269 | −2.37351 | ||||||||||||||||||
1.3 | −1.00000 | −3.29759 | 1.00000 | −2.40937 | 3.29759 | 1.21018 | −1.00000 | 7.87411 | 2.40937 | ||||||||||||||||||
1.4 | −1.00000 | −3.26711 | 1.00000 | 0.417272 | 3.26711 | −4.66212 | −1.00000 | 7.67400 | −0.417272 | ||||||||||||||||||
1.5 | −1.00000 | −3.23014 | 1.00000 | 4.30971 | 3.23014 | −1.93951 | −1.00000 | 7.43383 | −4.30971 | ||||||||||||||||||
1.6 | −1.00000 | −3.19543 | 1.00000 | −0.753590 | 3.19543 | −0.614331 | −1.00000 | 7.21079 | 0.753590 | ||||||||||||||||||
1.7 | −1.00000 | −2.99019 | 1.00000 | 2.92742 | 2.99019 | 4.68595 | −1.00000 | 5.94125 | −2.92742 | ||||||||||||||||||
1.8 | −1.00000 | −2.96590 | 1.00000 | −3.55325 | 2.96590 | 2.76315 | −1.00000 | 5.79657 | 3.55325 | ||||||||||||||||||
1.9 | −1.00000 | −2.88148 | 1.00000 | 0.529257 | 2.88148 | −3.48565 | −1.00000 | 5.30294 | −0.529257 | ||||||||||||||||||
1.10 | −1.00000 | −2.87972 | 1.00000 | −2.30779 | 2.87972 | −0.146606 | −1.00000 | 5.29280 | 2.30779 | ||||||||||||||||||
1.11 | −1.00000 | −2.78190 | 1.00000 | 1.30300 | 2.78190 | 4.15781 | −1.00000 | 4.73899 | −1.30300 | ||||||||||||||||||
1.12 | −1.00000 | −2.76801 | 1.00000 | −2.71168 | 2.76801 | −4.48343 | −1.00000 | 4.66186 | 2.71168 | ||||||||||||||||||
1.13 | −1.00000 | −2.61902 | 1.00000 | 1.94954 | 2.61902 | −2.60943 | −1.00000 | 3.85928 | −1.94954 | ||||||||||||||||||
1.14 | −1.00000 | −2.61462 | 1.00000 | 2.63805 | 2.61462 | 1.49750 | −1.00000 | 3.83623 | −2.63805 | ||||||||||||||||||
1.15 | −1.00000 | −2.39406 | 1.00000 | 3.62374 | 2.39406 | −4.52806 | −1.00000 | 2.73150 | −3.62374 | ||||||||||||||||||
1.16 | −1.00000 | −2.39329 | 1.00000 | −1.79652 | 2.39329 | −1.61915 | −1.00000 | 2.72784 | 1.79652 | ||||||||||||||||||
1.17 | −1.00000 | −2.23021 | 1.00000 | −2.99176 | 2.23021 | 3.17674 | −1.00000 | 1.97381 | 2.99176 | ||||||||||||||||||
1.18 | −1.00000 | −2.19996 | 1.00000 | −2.22300 | 2.19996 | 1.97846 | −1.00000 | 1.83980 | 2.22300 | ||||||||||||||||||
1.19 | −1.00000 | −2.15966 | 1.00000 | 1.78112 | 2.15966 | 1.44604 | −1.00000 | 1.66414 | −1.78112 | ||||||||||||||||||
1.20 | −1.00000 | −2.15321 | 1.00000 | 0.345955 | 2.15321 | 4.65391 | −1.00000 | 1.63633 | −0.345955 | ||||||||||||||||||
See all 89 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(4001\) | \(1\) |
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 | 8002.2.a.f | ✓ | 89 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8002.2.a.f | ✓ | 89 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{89} + 12 T_{3}^{88} - 109 T_{3}^{87} - 1845 T_{3}^{86} + 4038 T_{3}^{85} + 134790 T_{3}^{84} + \cdots + 1997131777424 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).