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: | \(69\) |
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 | −3.36322 | 1.00000 | 3.90848 | −3.36322 | −1.71915 | 1.00000 | 8.31124 | 3.90848 | ||||||||||||||||||
1.2 | 1.00000 | −3.28301 | 1.00000 | −1.18461 | −3.28301 | 0.314789 | 1.00000 | 7.77813 | −1.18461 | ||||||||||||||||||
1.3 | 1.00000 | −3.15967 | 1.00000 | −0.278823 | −3.15967 | 1.77562 | 1.00000 | 6.98353 | −0.278823 | ||||||||||||||||||
1.4 | 1.00000 | −3.08249 | 1.00000 | 1.32186 | −3.08249 | 1.26521 | 1.00000 | 6.50177 | 1.32186 | ||||||||||||||||||
1.5 | 1.00000 | −2.86585 | 1.00000 | 2.47496 | −2.86585 | −3.03619 | 1.00000 | 5.21309 | 2.47496 | ||||||||||||||||||
1.6 | 1.00000 | −2.78283 | 1.00000 | 2.27257 | −2.78283 | −2.57189 | 1.00000 | 4.74415 | 2.27257 | ||||||||||||||||||
1.7 | 1.00000 | −2.75233 | 1.00000 | −1.92549 | −2.75233 | 3.54771 | 1.00000 | 4.57534 | −1.92549 | ||||||||||||||||||
1.8 | 1.00000 | −2.73399 | 1.00000 | 1.40012 | −2.73399 | 2.82233 | 1.00000 | 4.47471 | 1.40012 | ||||||||||||||||||
1.9 | 1.00000 | −2.72077 | 1.00000 | 0.316882 | −2.72077 | −4.75223 | 1.00000 | 4.40260 | 0.316882 | ||||||||||||||||||
1.10 | 1.00000 | −2.65418 | 1.00000 | −3.04088 | −2.65418 | −4.15627 | 1.00000 | 4.04466 | −3.04088 | ||||||||||||||||||
1.11 | 1.00000 | −2.51689 | 1.00000 | −2.29473 | −2.51689 | 0.132144 | 1.00000 | 3.33476 | −2.29473 | ||||||||||||||||||
1.12 | 1.00000 | −2.38200 | 1.00000 | 2.27463 | −2.38200 | −1.51830 | 1.00000 | 2.67393 | 2.27463 | ||||||||||||||||||
1.13 | 1.00000 | −2.26149 | 1.00000 | −2.88813 | −2.26149 | 2.25047 | 1.00000 | 2.11432 | −2.88813 | ||||||||||||||||||
1.14 | 1.00000 | −2.16991 | 1.00000 | −2.35323 | −2.16991 | −2.98482 | 1.00000 | 1.70849 | −2.35323 | ||||||||||||||||||
1.15 | 1.00000 | −2.10040 | 1.00000 | −0.571471 | −2.10040 | 3.01756 | 1.00000 | 1.41167 | −0.571471 | ||||||||||||||||||
1.16 | 1.00000 | −2.04988 | 1.00000 | 4.28371 | −2.04988 | 3.78593 | 1.00000 | 1.20199 | 4.28371 | ||||||||||||||||||
1.17 | 1.00000 | −1.97198 | 1.00000 | 2.38225 | −1.97198 | 2.34177 | 1.00000 | 0.888724 | 2.38225 | ||||||||||||||||||
1.18 | 1.00000 | −1.76933 | 1.00000 | 3.70543 | −1.76933 | −4.46804 | 1.00000 | 0.130514 | 3.70543 | ||||||||||||||||||
1.19 | 1.00000 | −1.67001 | 1.00000 | 1.85681 | −1.67001 | 0.410918 | 1.00000 | −0.211058 | 1.85681 | ||||||||||||||||||
1.20 | 1.00000 | −1.63641 | 1.00000 | −3.89748 | −1.63641 | 0.627952 | 1.00000 | −0.322178 | −3.89748 | ||||||||||||||||||
See all 69 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.a | ✓ | 69 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8006.2.a.a | ✓ | 69 | 1.a | even | 1 | 1 | trivial |