Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8006,2,Mod(1,8006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8006, 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("8006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8006 = 2 \cdot 4003 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8006.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: | \(63.9282318582\) |
Analytic rank: | \(1\) |
Dimension: | \(75\) |
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 | −3.43129 | 1.00000 | 0.837927 | 3.43129 | 3.88626 | −1.00000 | 8.77378 | −0.837927 | ||||||||||||||||||
1.2 | −1.00000 | −3.12981 | 1.00000 | −3.93566 | 3.12981 | 0.0773402 | −1.00000 | 6.79570 | 3.93566 | ||||||||||||||||||
1.3 | −1.00000 | −2.99270 | 1.00000 | −0.0677361 | 2.99270 | −2.03477 | −1.00000 | 5.95625 | 0.0677361 | ||||||||||||||||||
1.4 | −1.00000 | −2.91158 | 1.00000 | 1.02232 | 2.91158 | −2.06566 | −1.00000 | 5.47731 | −1.02232 | ||||||||||||||||||
1.5 | −1.00000 | −2.90089 | 1.00000 | −2.73153 | 2.90089 | 4.86656 | −1.00000 | 5.41514 | 2.73153 | ||||||||||||||||||
1.6 | −1.00000 | −2.82604 | 1.00000 | 3.06326 | 2.82604 | 0.692372 | −1.00000 | 4.98649 | −3.06326 | ||||||||||||||||||
1.7 | −1.00000 | −2.82362 | 1.00000 | −2.07564 | 2.82362 | 0.568941 | −1.00000 | 4.97283 | 2.07564 | ||||||||||||||||||
1.8 | −1.00000 | −2.77545 | 1.00000 | 4.25856 | 2.77545 | 2.96891 | −1.00000 | 4.70314 | −4.25856 | ||||||||||||||||||
1.9 | −1.00000 | −2.68989 | 1.00000 | −1.33052 | 2.68989 | 1.26652 | −1.00000 | 4.23551 | 1.33052 | ||||||||||||||||||
1.10 | −1.00000 | −2.63254 | 1.00000 | −2.25670 | 2.63254 | −0.475543 | −1.00000 | 3.93027 | 2.25670 | ||||||||||||||||||
1.11 | −1.00000 | −2.43887 | 1.00000 | 2.67337 | 2.43887 | 3.98930 | −1.00000 | 2.94808 | −2.67337 | ||||||||||||||||||
1.12 | −1.00000 | −2.34883 | 1.00000 | 2.23352 | 2.34883 | 0.995037 | −1.00000 | 2.51700 | −2.23352 | ||||||||||||||||||
1.13 | −1.00000 | −2.29820 | 1.00000 | −2.18058 | 2.29820 | −2.19565 | −1.00000 | 2.28173 | 2.18058 | ||||||||||||||||||
1.14 | −1.00000 | −2.22415 | 1.00000 | 2.52556 | 2.22415 | 1.21093 | −1.00000 | 1.94686 | −2.52556 | ||||||||||||||||||
1.15 | −1.00000 | −2.19035 | 1.00000 | 2.34635 | 2.19035 | −2.32589 | −1.00000 | 1.79762 | −2.34635 | ||||||||||||||||||
1.16 | −1.00000 | −2.05716 | 1.00000 | −1.30027 | 2.05716 | −1.55677 | −1.00000 | 1.23189 | 1.30027 | ||||||||||||||||||
1.17 | −1.00000 | −1.79620 | 1.00000 | −0.413445 | 1.79620 | −3.61034 | −1.00000 | 0.226347 | 0.413445 | ||||||||||||||||||
1.18 | −1.00000 | −1.74787 | 1.00000 | −1.73571 | 1.74787 | 2.21934 | −1.00000 | 0.0550411 | 1.73571 | ||||||||||||||||||
1.19 | −1.00000 | −1.72100 | 1.00000 | 1.49175 | 1.72100 | −1.07018 | −1.00000 | −0.0381718 | −1.49175 | ||||||||||||||||||
1.20 | −1.00000 | −1.69760 | 1.00000 | 2.70478 | 1.69760 | −3.93964 | −1.00000 | −0.118138 | −2.70478 | ||||||||||||||||||
See all 75 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(4003\) | \(1\) |
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 | 8006.2.a.b | ✓ | 75 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8006.2.a.b | ✓ | 75 | 1.a | even | 1 | 1 | trivial |