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: | \(86\) |
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 | −3.25486 | 1.00000 | 3.01406 | 3.25486 | −0.942473 | −1.00000 | 7.59411 | −3.01406 | ||||||||||||||||||
1.2 | −1.00000 | −3.25191 | 1.00000 | −1.61920 | 3.25191 | −4.93737 | −1.00000 | 7.57490 | 1.61920 | ||||||||||||||||||
1.3 | −1.00000 | −3.20966 | 1.00000 | 0.782750 | 3.20966 | 3.68309 | −1.00000 | 7.30189 | −0.782750 | ||||||||||||||||||
1.4 | −1.00000 | −3.19050 | 1.00000 | −0.434778 | 3.19050 | 2.23469 | −1.00000 | 7.17932 | 0.434778 | ||||||||||||||||||
1.5 | −1.00000 | −3.14403 | 1.00000 | 0.849411 | 3.14403 | 0.337791 | −1.00000 | 6.88495 | −0.849411 | ||||||||||||||||||
1.6 | −1.00000 | −3.04565 | 1.00000 | −0.405483 | 3.04565 | −0.211812 | −1.00000 | 6.27596 | 0.405483 | ||||||||||||||||||
1.7 | −1.00000 | −2.97454 | 1.00000 | −1.53544 | 2.97454 | 2.95859 | −1.00000 | 5.84792 | 1.53544 | ||||||||||||||||||
1.8 | −1.00000 | −2.68740 | 1.00000 | 3.50149 | 2.68740 | −3.96925 | −1.00000 | 4.22214 | −3.50149 | ||||||||||||||||||
1.9 | −1.00000 | −2.64301 | 1.00000 | −0.786430 | 2.64301 | −4.08167 | −1.00000 | 3.98548 | 0.786430 | ||||||||||||||||||
1.10 | −1.00000 | −2.62565 | 1.00000 | −4.13736 | 2.62565 | −3.98462 | −1.00000 | 3.89402 | 4.13736 | ||||||||||||||||||
1.11 | −1.00000 | −2.61892 | 1.00000 | 3.93925 | 2.61892 | 4.49129 | −1.00000 | 3.85875 | −3.93925 | ||||||||||||||||||
1.12 | −1.00000 | −2.55894 | 1.00000 | −3.60377 | 2.55894 | −1.20803 | −1.00000 | 3.54817 | 3.60377 | ||||||||||||||||||
1.13 | −1.00000 | −2.34399 | 1.00000 | 1.04947 | 2.34399 | −2.18495 | −1.00000 | 2.49429 | −1.04947 | ||||||||||||||||||
1.14 | −1.00000 | −2.31833 | 1.00000 | 1.68969 | 2.31833 | −0.814652 | −1.00000 | 2.37465 | −1.68969 | ||||||||||||||||||
1.15 | −1.00000 | −2.09967 | 1.00000 | 3.00804 | 2.09967 | 1.34654 | −1.00000 | 1.40862 | −3.00804 | ||||||||||||||||||
1.16 | −1.00000 | −2.07592 | 1.00000 | 1.44155 | 2.07592 | −4.11205 | −1.00000 | 1.30942 | −1.44155 | ||||||||||||||||||
1.17 | −1.00000 | −2.06398 | 1.00000 | −1.63652 | 2.06398 | 0.580095 | −1.00000 | 1.26000 | 1.63652 | ||||||||||||||||||
1.18 | −1.00000 | −1.90629 | 1.00000 | −3.44864 | 1.90629 | 0.576337 | −1.00000 | 0.633932 | 3.44864 | ||||||||||||||||||
1.19 | −1.00000 | −1.83924 | 1.00000 | 4.20278 | 1.83924 | −0.870895 | −1.00000 | 0.382816 | −4.20278 | ||||||||||||||||||
1.20 | −1.00000 | −1.77113 | 1.00000 | −2.88416 | 1.77113 | −2.34784 | −1.00000 | 0.136917 | 2.88416 | ||||||||||||||||||
See all 86 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.c | ✓ | 86 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8026.2.a.c | ✓ | 86 | 1.a | even | 1 | 1 | trivial |