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: | \(0\) |
Dimension: | \(77\) |
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 | −3.18344 | 1.00000 | −0.604370 | 3.18344 | 1.50072 | −1.00000 | 7.13428 | 0.604370 | ||||||||||||||||||
1.2 | −1.00000 | −3.14716 | 1.00000 | −2.17339 | 3.14716 | 4.77090 | −1.00000 | 6.90459 | 2.17339 | ||||||||||||||||||
1.3 | −1.00000 | −3.14624 | 1.00000 | 3.08369 | 3.14624 | 2.61077 | −1.00000 | 6.89883 | −3.08369 | ||||||||||||||||||
1.4 | −1.00000 | −2.93461 | 1.00000 | 1.73622 | 2.93461 | −0.969734 | −1.00000 | 5.61192 | −1.73622 | ||||||||||||||||||
1.5 | −1.00000 | −2.91842 | 1.00000 | 3.04954 | 2.91842 | 3.39338 | −1.00000 | 5.51716 | −3.04954 | ||||||||||||||||||
1.6 | −1.00000 | −2.69277 | 1.00000 | 0.930791 | 2.69277 | −2.75733 | −1.00000 | 4.25101 | −0.930791 | ||||||||||||||||||
1.7 | −1.00000 | −2.62668 | 1.00000 | 0.854809 | 2.62668 | 1.79286 | −1.00000 | 3.89945 | −0.854809 | ||||||||||||||||||
1.8 | −1.00000 | −2.61328 | 1.00000 | −3.49569 | 2.61328 | −0.478818 | −1.00000 | 3.82924 | 3.49569 | ||||||||||||||||||
1.9 | −1.00000 | −2.50919 | 1.00000 | −3.69214 | 2.50919 | −2.32427 | −1.00000 | 3.29603 | 3.69214 | ||||||||||||||||||
1.10 | −1.00000 | −2.45617 | 1.00000 | 0.358646 | 2.45617 | −1.91182 | −1.00000 | 3.03279 | −0.358646 | ||||||||||||||||||
1.11 | −1.00000 | −2.42114 | 1.00000 | −1.19106 | 2.42114 | −3.77401 | −1.00000 | 2.86190 | 1.19106 | ||||||||||||||||||
1.12 | −1.00000 | −2.37610 | 1.00000 | −3.32970 | 2.37610 | 3.18362 | −1.00000 | 2.64585 | 3.32970 | ||||||||||||||||||
1.13 | −1.00000 | −2.19428 | 1.00000 | 0.0495220 | 2.19428 | 1.71879 | −1.00000 | 1.81487 | −0.0495220 | ||||||||||||||||||
1.14 | −1.00000 | −2.17233 | 1.00000 | −1.31184 | 2.17233 | −2.50290 | −1.00000 | 1.71901 | 1.31184 | ||||||||||||||||||
1.15 | −1.00000 | −2.13252 | 1.00000 | 2.05525 | 2.13252 | −1.57763 | −1.00000 | 1.54765 | −2.05525 | ||||||||||||||||||
1.16 | −1.00000 | −1.85869 | 1.00000 | −0.435894 | 1.85869 | 1.03466 | −1.00000 | 0.454718 | 0.435894 | ||||||||||||||||||
1.17 | −1.00000 | −1.73843 | 1.00000 | 4.07186 | 1.73843 | −2.38793 | −1.00000 | 0.0221381 | −4.07186 | ||||||||||||||||||
1.18 | −1.00000 | −1.70235 | 1.00000 | 4.07630 | 1.70235 | 1.03752 | −1.00000 | −0.101993 | −4.07630 | ||||||||||||||||||
1.19 | −1.00000 | −1.65847 | 1.00000 | −0.802626 | 1.65847 | −0.0977002 | −1.00000 | −0.249472 | 0.802626 | ||||||||||||||||||
1.20 | −1.00000 | −1.59889 | 1.00000 | −0.0810696 | 1.59889 | 5.11538 | −1.00000 | −0.443564 | 0.0810696 | ||||||||||||||||||
See all 77 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.e | ✓ | 77 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8002.2.a.e | ✓ | 77 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{77} - 10 T_{3}^{76} - 101 T_{3}^{75} + 1309 T_{3}^{74} + 4091 T_{3}^{73} - 81236 T_{3}^{72} + \cdots - 1136453792 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).