Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8038,2,Mod(1,8038)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8038, 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("8038.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8038 = 2 \cdot 4019 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8038.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: | \(64.1837531447\) |
Analytic rank: | \(0\) |
Dimension: | \(92\) |
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 | −3.24576 | 1.00000 | −1.41185 | −3.24576 | −1.89864 | 1.00000 | 7.53496 | −1.41185 | ||||||||||||||||||
1.2 | 1.00000 | −3.23664 | 1.00000 | 3.20472 | −3.23664 | 2.34255 | 1.00000 | 7.47583 | 3.20472 | ||||||||||||||||||
1.3 | 1.00000 | −3.14562 | 1.00000 | −0.303255 | −3.14562 | 2.95430 | 1.00000 | 6.89495 | −0.303255 | ||||||||||||||||||
1.4 | 1.00000 | −2.82708 | 1.00000 | −2.48763 | −2.82708 | −1.73566 | 1.00000 | 4.99240 | −2.48763 | ||||||||||||||||||
1.5 | 1.00000 | −2.74439 | 1.00000 | 2.72118 | −2.74439 | 4.78598 | 1.00000 | 4.53169 | 2.72118 | ||||||||||||||||||
1.6 | 1.00000 | −2.70157 | 1.00000 | 0.143003 | −2.70157 | −2.92792 | 1.00000 | 4.29850 | 0.143003 | ||||||||||||||||||
1.7 | 1.00000 | −2.67211 | 1.00000 | 1.54258 | −2.67211 | 1.53058 | 1.00000 | 4.14019 | 1.54258 | ||||||||||||||||||
1.8 | 1.00000 | −2.65852 | 1.00000 | 2.22462 | −2.65852 | 0.870094 | 1.00000 | 4.06774 | 2.22462 | ||||||||||||||||||
1.9 | 1.00000 | −2.58627 | 1.00000 | −0.666816 | −2.58627 | −1.51739 | 1.00000 | 3.68880 | −0.666816 | ||||||||||||||||||
1.10 | 1.00000 | −2.54065 | 1.00000 | −2.68248 | −2.54065 | 5.07030 | 1.00000 | 3.45490 | −2.68248 | ||||||||||||||||||
1.11 | 1.00000 | −2.45077 | 1.00000 | −1.39013 | −2.45077 | 4.27652 | 1.00000 | 3.00630 | −1.39013 | ||||||||||||||||||
1.12 | 1.00000 | −2.41917 | 1.00000 | 2.79360 | −2.41917 | 0.842715 | 1.00000 | 2.85238 | 2.79360 | ||||||||||||||||||
1.13 | 1.00000 | −2.40015 | 1.00000 | −4.19507 | −2.40015 | 1.33387 | 1.00000 | 2.76072 | −4.19507 | ||||||||||||||||||
1.14 | 1.00000 | −2.39141 | 1.00000 | 3.31334 | −2.39141 | −3.74056 | 1.00000 | 2.71882 | 3.31334 | ||||||||||||||||||
1.15 | 1.00000 | −2.38521 | 1.00000 | −2.77606 | −2.38521 | −1.22030 | 1.00000 | 2.68923 | −2.77606 | ||||||||||||||||||
1.16 | 1.00000 | −2.15753 | 1.00000 | −0.384990 | −2.15753 | −2.87024 | 1.00000 | 1.65493 | −0.384990 | ||||||||||||||||||
1.17 | 1.00000 | −1.94937 | 1.00000 | 2.21034 | −1.94937 | 3.83725 | 1.00000 | 0.800030 | 2.21034 | ||||||||||||||||||
1.18 | 1.00000 | −1.90955 | 1.00000 | 4.18412 | −1.90955 | −2.40151 | 1.00000 | 0.646387 | 4.18412 | ||||||||||||||||||
1.19 | 1.00000 | −1.78655 | 1.00000 | 0.381834 | −1.78655 | 1.78538 | 1.00000 | 0.191778 | 0.381834 | ||||||||||||||||||
1.20 | 1.00000 | −1.75958 | 1.00000 | 2.55886 | −1.75958 | −0.837744 | 1.00000 | 0.0961261 | 2.55886 | ||||||||||||||||||
See all 92 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(4019\) | \(1\) |
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 | 8038.2.a.d | ✓ | 92 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8038.2.a.d | ✓ | 92 | 1.a | even | 1 | 1 | trivial |