Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8007,2,Mod(1,8007)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, 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("8007.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8007 = 3 \cdot 17 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8007.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: | \(63.9362168984\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
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 | −2.81598 | −1.00000 | 5.92975 | −4.24598 | 2.81598 | 0.0516498 | −11.0661 | 1.00000 | 11.9566 | ||||||||||||||||||
1.2 | −2.69942 | −1.00000 | 5.28690 | 3.29410 | 2.69942 | 4.13717 | −8.87273 | 1.00000 | −8.89217 | ||||||||||||||||||
1.3 | −2.69904 | −1.00000 | 5.28480 | −1.39519 | 2.69904 | −4.02432 | −8.86580 | 1.00000 | 3.76567 | ||||||||||||||||||
1.4 | −2.54273 | −1.00000 | 4.46545 | −2.67798 | 2.54273 | 2.94599 | −6.26898 | 1.00000 | 6.80937 | ||||||||||||||||||
1.5 | −2.53596 | −1.00000 | 4.43108 | −0.923448 | 2.53596 | −0.313569 | −6.16510 | 1.00000 | 2.34182 | ||||||||||||||||||
1.6 | −2.38697 | −1.00000 | 3.69761 | 2.25593 | 2.38697 | −4.86795 | −4.05215 | 1.00000 | −5.38482 | ||||||||||||||||||
1.7 | −2.37978 | −1.00000 | 3.66334 | 0.734525 | 2.37978 | 2.72642 | −3.95838 | 1.00000 | −1.74801 | ||||||||||||||||||
1.8 | −2.37098 | −1.00000 | 3.62156 | −2.47104 | 2.37098 | 5.22837 | −3.84468 | 1.00000 | 5.85879 | ||||||||||||||||||
1.9 | −2.32739 | −1.00000 | 3.41675 | 3.32274 | 2.32739 | 1.34387 | −3.29733 | 1.00000 | −7.73332 | ||||||||||||||||||
1.10 | −2.16822 | −1.00000 | 2.70116 | −4.31017 | 2.16822 | −1.36362 | −1.52027 | 1.00000 | 9.34539 | ||||||||||||||||||
1.11 | −2.11517 | −1.00000 | 2.47395 | 2.21169 | 2.11517 | −2.99289 | −1.00248 | 1.00000 | −4.67809 | ||||||||||||||||||
1.12 | −1.82641 | −1.00000 | 1.33579 | 0.0479756 | 1.82641 | −0.416922 | 1.21313 | 1.00000 | −0.0876232 | ||||||||||||||||||
1.13 | −1.78864 | −1.00000 | 1.19923 | −3.68881 | 1.78864 | 4.99843 | 1.43229 | 1.00000 | 6.59796 | ||||||||||||||||||
1.14 | −1.70394 | −1.00000 | 0.903397 | 3.92360 | 1.70394 | −2.84859 | 1.86854 | 1.00000 | −6.68557 | ||||||||||||||||||
1.15 | −1.65285 | −1.00000 | 0.731922 | 2.84674 | 1.65285 | −0.210210 | 2.09595 | 1.00000 | −4.70524 | ||||||||||||||||||
1.16 | −1.62615 | −1.00000 | 0.644366 | 0.536374 | 1.62615 | 4.53678 | 2.20447 | 1.00000 | −0.872225 | ||||||||||||||||||
1.17 | −1.58531 | −1.00000 | 0.513198 | −0.0940015 | 1.58531 | −1.19806 | 2.35704 | 1.00000 | 0.149021 | ||||||||||||||||||
1.18 | −1.55552 | −1.00000 | 0.419651 | −2.03554 | 1.55552 | −4.57317 | 2.45827 | 1.00000 | 3.16632 | ||||||||||||||||||
1.19 | −1.43835 | −1.00000 | 0.0688642 | −3.82222 | 1.43835 | 1.52533 | 2.77766 | 1.00000 | 5.49771 | ||||||||||||||||||
1.20 | −1.28460 | −1.00000 | −0.349799 | −0.580730 | 1.28460 | −1.93204 | 3.01856 | 1.00000 | 0.746007 | ||||||||||||||||||
See all 64 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(17\) | \(-1\) |
\(157\) | \(1\) |
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 | 8007.2.a.j | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8007.2.a.j | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 5 T_{2}^{63} - 90 T_{2}^{62} + 479 T_{2}^{61} + 3792 T_{2}^{60} - 21725 T_{2}^{59} + \cdots + 2527 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).