Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8019,2,Mod(1,8019)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8019, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8019.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8019 = 3^{6} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8019.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: | \(64.0320373809\) |
Analytic rank: | \(1\) |
Dimension: | \(36\) |
Twist minimal: | no (minimal twist has level 297) |
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 | −2.57896 | 0 | 4.65103 | 1.81453 | 0 | 0.597625 | −6.83689 | 0 | −4.67961 | ||||||||||||||||||
1.2 | −2.57869 | 0 | 4.64963 | 0.549951 | 0 | −2.53498 | −6.83256 | 0 | −1.41815 | ||||||||||||||||||
1.3 | −2.51998 | 0 | 4.35029 | −1.98987 | 0 | −2.89486 | −5.92267 | 0 | 5.01444 | ||||||||||||||||||
1.4 | −2.21310 | 0 | 2.89780 | 1.25911 | 0 | 0.887911 | −1.98693 | 0 | −2.78652 | ||||||||||||||||||
1.5 | −2.16415 | 0 | 2.68354 | 3.54207 | 0 | 1.71775 | −1.47929 | 0 | −7.66558 | ||||||||||||||||||
1.6 | −2.10597 | 0 | 2.43509 | −0.0342445 | 0 | 2.90025 | −0.916291 | 0 | 0.0721177 | ||||||||||||||||||
1.7 | −2.04863 | 0 | 2.19689 | −0.556265 | 0 | 2.53749 | −0.403346 | 0 | 1.13958 | ||||||||||||||||||
1.8 | −2.01026 | 0 | 2.04114 | −1.99399 | 0 | −3.74353 | −0.0827043 | 0 | 4.00844 | ||||||||||||||||||
1.9 | −1.56100 | 0 | 0.436708 | −1.98729 | 0 | −2.82745 | 2.44029 | 0 | 3.10215 | ||||||||||||||||||
1.10 | −1.54922 | 0 | 0.400087 | −4.30670 | 0 | 0.529623 | 2.47862 | 0 | 6.67204 | ||||||||||||||||||
1.11 | −1.11300 | 0 | −0.761231 | −0.296230 | 0 | 3.44297 | 3.07325 | 0 | 0.329705 | ||||||||||||||||||
1.12 | −1.09746 | 0 | −0.795590 | −0.844390 | 0 | −0.470010 | 3.06804 | 0 | 0.926681 | ||||||||||||||||||
1.13 | −1.04649 | 0 | −0.904867 | 3.42777 | 0 | −2.37542 | 3.03990 | 0 | −3.58711 | ||||||||||||||||||
1.14 | −0.834408 | 0 | −1.30376 | −0.396805 | 0 | 1.16343 | 2.75669 | 0 | 0.331097 | ||||||||||||||||||
1.15 | −0.759333 | 0 | −1.42341 | −2.62463 | 0 | 1.90497 | 2.59951 | 0 | 1.99297 | ||||||||||||||||||
1.16 | −0.272778 | 0 | −1.92559 | 2.10562 | 0 | −2.25775 | 1.07081 | 0 | −0.574367 | ||||||||||||||||||
1.17 | −0.271934 | 0 | −1.92605 | 1.50806 | 0 | −2.46200 | 1.06763 | 0 | −0.410092 | ||||||||||||||||||
1.18 | 0.0197513 | 0 | −1.99961 | 3.39148 | 0 | 3.53205 | −0.0789976 | 0 | 0.0669862 | ||||||||||||||||||
1.19 | 0.403238 | 0 | −1.83740 | 1.10652 | 0 | 3.55888 | −1.54739 | 0 | 0.446190 | ||||||||||||||||||
1.20 | 0.431171 | 0 | −1.81409 | −1.32759 | 0 | 2.32692 | −1.64453 | 0 | −0.572419 | ||||||||||||||||||
See all 36 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(11\) | \(-1\) |
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 | 8019.2.a.g | 36 | |
3.b | odd | 2 | 1 | 8019.2.a.h | 36 | ||
27.e | even | 9 | 2 | 891.2.j.b | 72 | ||
27.f | odd | 18 | 2 | 297.2.j.b | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
297.2.j.b | ✓ | 72 | 27.f | odd | 18 | 2 | |
891.2.j.b | 72 | 27.e | even | 9 | 2 | ||
8019.2.a.g | 36 | 1.a | even | 1 | 1 | trivial | |
8019.2.a.h | 36 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 51 T_{2}^{34} + T_{2}^{33} + 1179 T_{2}^{32} - 51 T_{2}^{31} - 16358 T_{2}^{30} + \cdots - 111 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8019))\).