Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4006,2,Mod(1,4006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4006, 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("4006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4006 = 2 \cdot 2003 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4006.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: | \(31.9880710497\) |
Analytic rank: | \(1\) |
Dimension: | \(40\) |
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 | −3.34571 | 1.00000 | −3.64453 | 3.34571 | 5.09237 | −1.00000 | 8.19376 | 3.64453 | ||||||||||||||||||
1.2 | −1.00000 | −3.11880 | 1.00000 | 1.57138 | 3.11880 | 2.57640 | −1.00000 | 6.72691 | −1.57138 | ||||||||||||||||||
1.3 | −1.00000 | −2.96542 | 1.00000 | 0.539090 | 2.96542 | −0.220082 | −1.00000 | 5.79371 | −0.539090 | ||||||||||||||||||
1.4 | −1.00000 | −2.86035 | 1.00000 | −2.90727 | 2.86035 | 0.924767 | −1.00000 | 5.18162 | 2.90727 | ||||||||||||||||||
1.5 | −1.00000 | −2.39522 | 1.00000 | 0.491091 | 2.39522 | 3.07111 | −1.00000 | 2.73707 | −0.491091 | ||||||||||||||||||
1.6 | −1.00000 | −2.30611 | 1.00000 | 3.88128 | 2.30611 | 0.245423 | −1.00000 | 2.31816 | −3.88128 | ||||||||||||||||||
1.7 | −1.00000 | −2.23215 | 1.00000 | −2.75227 | 2.23215 | −2.11016 | −1.00000 | 1.98248 | 2.75227 | ||||||||||||||||||
1.8 | −1.00000 | −1.94007 | 1.00000 | −0.275091 | 1.94007 | −2.80268 | −1.00000 | 0.763873 | 0.275091 | ||||||||||||||||||
1.9 | −1.00000 | −1.91121 | 1.00000 | −1.39712 | 1.91121 | 3.46874 | −1.00000 | 0.652717 | 1.39712 | ||||||||||||||||||
1.10 | −1.00000 | −1.84543 | 1.00000 | 2.20410 | 1.84543 | −0.412853 | −1.00000 | 0.405623 | −2.20410 | ||||||||||||||||||
1.11 | −1.00000 | −1.82093 | 1.00000 | −3.80048 | 1.82093 | −3.54475 | −1.00000 | 0.315788 | 3.80048 | ||||||||||||||||||
1.12 | −1.00000 | −1.69686 | 1.00000 | 3.10338 | 1.69686 | 1.10094 | −1.00000 | −0.120661 | −3.10338 | ||||||||||||||||||
1.13 | −1.00000 | −1.01040 | 1.00000 | −1.91784 | 1.01040 | 4.92355 | −1.00000 | −1.97910 | 1.91784 | ||||||||||||||||||
1.14 | −1.00000 | −0.956879 | 1.00000 | −0.0953014 | 0.956879 | −4.20577 | −1.00000 | −2.08438 | 0.0953014 | ||||||||||||||||||
1.15 | −1.00000 | −0.941592 | 1.00000 | −0.129713 | 0.941592 | 0.350515 | −1.00000 | −2.11340 | 0.129713 | ||||||||||||||||||
1.16 | −1.00000 | −0.934245 | 1.00000 | −4.17073 | 0.934245 | 1.30479 | −1.00000 | −2.12719 | 4.17073 | ||||||||||||||||||
1.17 | −1.00000 | −0.779235 | 1.00000 | −0.867411 | 0.779235 | −3.45139 | −1.00000 | −2.39279 | 0.867411 | ||||||||||||||||||
1.18 | −1.00000 | −0.712075 | 1.00000 | −4.09209 | 0.712075 | 2.33011 | −1.00000 | −2.49295 | 4.09209 | ||||||||||||||||||
1.19 | −1.00000 | −0.535052 | 1.00000 | 2.84024 | 0.535052 | 2.46812 | −1.00000 | −2.71372 | −2.84024 | ||||||||||||||||||
1.20 | −1.00000 | −0.0866100 | 1.00000 | 2.47499 | 0.0866100 | 0.719708 | −1.00000 | −2.99250 | −2.47499 | ||||||||||||||||||
See all 40 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(2003\) | \(1\) |
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 | 4006.2.a.g | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4006.2.a.g | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + T_{3}^{39} - 74 T_{3}^{38} - 70 T_{3}^{37} + 2498 T_{3}^{36} + 2231 T_{3}^{35} + \cdots - 6140 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\).