Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8012,2,Mod(1,8012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8012.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8012 = 2^{2} \cdot 2003 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8012.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: | \(63.9761420994\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
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 | 0 | −3.23075 | 0 | 0.167791 | 0 | 3.73844 | 0 | 7.43772 | 0 | ||||||||||||||||||
1.2 | 0 | −3.15616 | 0 | −3.31880 | 0 | 0.188587 | 0 | 6.96136 | 0 | ||||||||||||||||||
1.3 | 0 | −3.04318 | 0 | 2.19188 | 0 | 3.84643 | 0 | 6.26097 | 0 | ||||||||||||||||||
1.4 | 0 | −2.94029 | 0 | −0.651594 | 0 | 3.16105 | 0 | 5.64533 | 0 | ||||||||||||||||||
1.5 | 0 | −2.81005 | 0 | 4.18979 | 0 | 3.58597 | 0 | 4.89636 | 0 | ||||||||||||||||||
1.6 | 0 | −2.80891 | 0 | −0.455894 | 0 | −2.00885 | 0 | 4.88996 | 0 | ||||||||||||||||||
1.7 | 0 | −2.80739 | 0 | 2.13337 | 0 | −1.30534 | 0 | 4.88143 | 0 | ||||||||||||||||||
1.8 | 0 | −2.71734 | 0 | −0.520933 | 0 | −1.18624 | 0 | 4.38393 | 0 | ||||||||||||||||||
1.9 | 0 | −2.71377 | 0 | −3.80725 | 0 | 4.98615 | 0 | 4.36453 | 0 | ||||||||||||||||||
1.10 | 0 | −2.59974 | 0 | −0.601241 | 0 | −1.05390 | 0 | 3.75864 | 0 | ||||||||||||||||||
1.11 | 0 | −2.56711 | 0 | −4.07865 | 0 | 0.492202 | 0 | 3.59005 | 0 | ||||||||||||||||||
1.12 | 0 | −2.52878 | 0 | 2.30231 | 0 | 0.0873376 | 0 | 3.39472 | 0 | ||||||||||||||||||
1.13 | 0 | −2.48630 | 0 | −2.95359 | 0 | −3.19667 | 0 | 3.18170 | 0 | ||||||||||||||||||
1.14 | 0 | −2.24743 | 0 | −3.82088 | 0 | −1.51483 | 0 | 2.05096 | 0 | ||||||||||||||||||
1.15 | 0 | −2.08516 | 0 | 1.63979 | 0 | 1.21456 | 0 | 1.34791 | 0 | ||||||||||||||||||
1.16 | 0 | −2.06140 | 0 | 1.88031 | 0 | −1.75026 | 0 | 1.24936 | 0 | ||||||||||||||||||
1.17 | 0 | −2.04818 | 0 | 1.65769 | 0 | −0.109887 | 0 | 1.19506 | 0 | ||||||||||||||||||
1.18 | 0 | −1.98040 | 0 | 1.50325 | 0 | 0.615748 | 0 | 0.921997 | 0 | ||||||||||||||||||
1.19 | 0 | −1.83871 | 0 | −1.10593 | 0 | 4.51431 | 0 | 0.380865 | 0 | ||||||||||||||||||
1.20 | 0 | −1.72366 | 0 | 3.80404 | 0 | −2.57590 | 0 | −0.0289913 | 0 | ||||||||||||||||||
See all 88 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(2003\) | \(1\) |
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 | 8012.2.a.b | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8012.2.a.b | ✓ | 88 | 1.a | even | 1 | 1 | trivial |