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: | \(0\) |
Dimension: | \(98\) |
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 | −3.34117 | 1.00000 | −2.63189 | −3.34117 | −3.23365 | 1.00000 | 8.16344 | −2.63189 | ||||||||||||||||||
1.2 | 1.00000 | −3.30357 | 1.00000 | −2.71504 | −3.30357 | 0.802195 | 1.00000 | 7.91355 | −2.71504 | ||||||||||||||||||
1.3 | 1.00000 | −3.23335 | 1.00000 | 1.72220 | −3.23335 | 4.52664 | 1.00000 | 7.45455 | 1.72220 | ||||||||||||||||||
1.4 | 1.00000 | −3.15680 | 1.00000 | −2.35136 | −3.15680 | −0.0333758 | 1.00000 | 6.96536 | −2.35136 | ||||||||||||||||||
1.5 | 1.00000 | −3.07582 | 1.00000 | 1.83927 | −3.07582 | −1.83961 | 1.00000 | 6.46070 | 1.83927 | ||||||||||||||||||
1.6 | 1.00000 | −3.05574 | 1.00000 | 2.27846 | −3.05574 | −0.374830 | 1.00000 | 6.33757 | 2.27846 | ||||||||||||||||||
1.7 | 1.00000 | −3.04444 | 1.00000 | −4.16552 | −3.04444 | −3.08322 | 1.00000 | 6.26861 | −4.16552 | ||||||||||||||||||
1.8 | 1.00000 | −2.80568 | 1.00000 | 1.82812 | −2.80568 | 4.55893 | 1.00000 | 4.87183 | 1.82812 | ||||||||||||||||||
1.9 | 1.00000 | −2.77826 | 1.00000 | −3.83157 | −2.77826 | 3.52049 | 1.00000 | 4.71872 | −3.83157 | ||||||||||||||||||
1.10 | 1.00000 | −2.73443 | 1.00000 | −0.0905939 | −2.73443 | −4.21224 | 1.00000 | 4.47709 | −0.0905939 | ||||||||||||||||||
1.11 | 1.00000 | −2.69452 | 1.00000 | −4.09763 | −2.69452 | 4.00132 | 1.00000 | 4.26044 | −4.09763 | ||||||||||||||||||
1.12 | 1.00000 | −2.67018 | 1.00000 | 0.0387933 | −2.67018 | −0.245209 | 1.00000 | 4.12985 | 0.0387933 | ||||||||||||||||||
1.13 | 1.00000 | −2.57565 | 1.00000 | −1.28503 | −2.57565 | −4.37657 | 1.00000 | 3.63399 | −1.28503 | ||||||||||||||||||
1.14 | 1.00000 | −2.52562 | 1.00000 | 3.79609 | −2.52562 | −3.83950 | 1.00000 | 3.37877 | 3.79609 | ||||||||||||||||||
1.15 | 1.00000 | −2.49687 | 1.00000 | 3.81192 | −2.49687 | 2.89844 | 1.00000 | 3.23436 | 3.81192 | ||||||||||||||||||
1.16 | 1.00000 | −2.29904 | 1.00000 | −0.379667 | −2.29904 | −2.21994 | 1.00000 | 2.28559 | −0.379667 | ||||||||||||||||||
1.17 | 1.00000 | −2.28004 | 1.00000 | 0.593845 | −2.28004 | −0.744586 | 1.00000 | 2.19860 | 0.593845 | ||||||||||||||||||
1.18 | 1.00000 | −2.25105 | 1.00000 | −0.876874 | −2.25105 | −0.763073 | 1.00000 | 2.06722 | −0.876874 | ||||||||||||||||||
1.19 | 1.00000 | −2.20795 | 1.00000 | −3.75545 | −2.20795 | −1.87926 | 1.00000 | 1.87502 | −3.75545 | ||||||||||||||||||
1.20 | 1.00000 | −2.20682 | 1.00000 | 0.892162 | −2.20682 | −0.549139 | 1.00000 | 1.87006 | 0.892162 | ||||||||||||||||||
See all 98 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.d | ✓ | 98 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8006.2.a.d | ✓ | 98 | 1.a | even | 1 | 1 | trivial |