Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8026,2,Mod(1,8026)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8026, 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("8026.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8026 = 2 \cdot 4013 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8026.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: | \(64.0879326623\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
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 | −3.38260 | 1.00000 | 3.86577 | −3.38260 | 1.67092 | 1.00000 | 8.44195 | 3.86577 | ||||||||||||||||||
1.2 | 1.00000 | −3.34006 | 1.00000 | 0.570499 | −3.34006 | −3.09347 | 1.00000 | 8.15597 | 0.570499 | ||||||||||||||||||
1.3 | 1.00000 | −3.28199 | 1.00000 | 4.44756 | −3.28199 | −4.87209 | 1.00000 | 7.77147 | 4.44756 | ||||||||||||||||||
1.4 | 1.00000 | −3.27254 | 1.00000 | −0.667760 | −3.27254 | −3.79895 | 1.00000 | 7.70950 | −0.667760 | ||||||||||||||||||
1.5 | 1.00000 | −3.25028 | 1.00000 | −3.55308 | −3.25028 | 1.03732 | 1.00000 | 7.56432 | −3.55308 | ||||||||||||||||||
1.6 | 1.00000 | −3.15603 | 1.00000 | 0.126707 | −3.15603 | 5.16404 | 1.00000 | 6.96052 | 0.126707 | ||||||||||||||||||
1.7 | 1.00000 | −2.94262 | 1.00000 | −2.75708 | −2.94262 | −0.162948 | 1.00000 | 5.65901 | −2.75708 | ||||||||||||||||||
1.8 | 1.00000 | −2.86733 | 1.00000 | −3.62078 | −2.86733 | −3.60760 | 1.00000 | 5.22157 | −3.62078 | ||||||||||||||||||
1.9 | 1.00000 | −2.84755 | 1.00000 | 2.66905 | −2.84755 | 0.0156637 | 1.00000 | 5.10852 | 2.66905 | ||||||||||||||||||
1.10 | 1.00000 | −2.83728 | 1.00000 | 3.92961 | −2.83728 | 4.49024 | 1.00000 | 5.05014 | 3.92961 | ||||||||||||||||||
1.11 | 1.00000 | −2.81326 | 1.00000 | −0.734328 | −2.81326 | −3.92682 | 1.00000 | 4.91446 | −0.734328 | ||||||||||||||||||
1.12 | 1.00000 | −2.75991 | 1.00000 | 1.94869 | −2.75991 | −1.07781 | 1.00000 | 4.61711 | 1.94869 | ||||||||||||||||||
1.13 | 1.00000 | −2.59431 | 1.00000 | −1.49267 | −2.59431 | 1.08103 | 1.00000 | 3.73042 | −1.49267 | ||||||||||||||||||
1.14 | 1.00000 | −2.44392 | 1.00000 | 2.01179 | −2.44392 | 1.54759 | 1.00000 | 2.97273 | 2.01179 | ||||||||||||||||||
1.15 | 1.00000 | −2.41544 | 1.00000 | 3.01333 | −2.41544 | −2.26454 | 1.00000 | 2.83435 | 3.01333 | ||||||||||||||||||
1.16 | 1.00000 | −2.39250 | 1.00000 | 1.27700 | −2.39250 | 3.72505 | 1.00000 | 2.72406 | 1.27700 | ||||||||||||||||||
1.17 | 1.00000 | −2.32778 | 1.00000 | 2.23848 | −2.32778 | 3.76217 | 1.00000 | 2.41856 | 2.23848 | ||||||||||||||||||
1.18 | 1.00000 | −2.32502 | 1.00000 | −1.28151 | −2.32502 | −2.20923 | 1.00000 | 2.40572 | −1.28151 | ||||||||||||||||||
1.19 | 1.00000 | −2.26664 | 1.00000 | −3.59818 | −2.26664 | 5.23582 | 1.00000 | 2.13767 | −3.59818 | ||||||||||||||||||
1.20 | 1.00000 | −2.21370 | 1.00000 | 0.840207 | −2.21370 | −5.05822 | 1.00000 | 1.90049 | 0.840207 | ||||||||||||||||||
See all 96 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(4013\) | \(1\) |
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 | 8026.2.a.d | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8026.2.a.d | ✓ | 96 | 1.a | even | 1 | 1 | trivial |