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: | \(1\) |
Dimension: | \(79\) |
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 | −3.41908 | 0 | −3.90530 | 0 | −4.72288 | 0 | 8.69013 | 0 | ||||||||||||||||||
1.2 | 0 | −3.40885 | 0 | −1.83301 | 0 | 1.65667 | 0 | 8.62026 | 0 | ||||||||||||||||||
1.3 | 0 | −3.26026 | 0 | 0.973052 | 0 | −2.06988 | 0 | 7.62930 | 0 | ||||||||||||||||||
1.4 | 0 | −3.24921 | 0 | 4.44468 | 0 | −2.04361 | 0 | 7.55734 | 0 | ||||||||||||||||||
1.5 | 0 | −3.22159 | 0 | 3.40512 | 0 | −1.04725 | 0 | 7.37863 | 0 | ||||||||||||||||||
1.6 | 0 | −3.17871 | 0 | −1.27908 | 0 | 2.10162 | 0 | 7.10419 | 0 | ||||||||||||||||||
1.7 | 0 | −3.11170 | 0 | 0.355389 | 0 | −3.83214 | 0 | 6.68266 | 0 | ||||||||||||||||||
1.8 | 0 | −2.87938 | 0 | 2.41591 | 0 | 2.96745 | 0 | 5.29083 | 0 | ||||||||||||||||||
1.9 | 0 | −2.79728 | 0 | 3.00547 | 0 | −2.16180 | 0 | 4.82477 | 0 | ||||||||||||||||||
1.10 | 0 | −2.71048 | 0 | −2.47152 | 0 | 2.42570 | 0 | 4.34670 | 0 | ||||||||||||||||||
1.11 | 0 | −2.68647 | 0 | −0.588575 | 0 | −1.13984 | 0 | 4.21715 | 0 | ||||||||||||||||||
1.12 | 0 | −2.62866 | 0 | 2.26034 | 0 | −4.56855 | 0 | 3.90985 | 0 | ||||||||||||||||||
1.13 | 0 | −2.59561 | 0 | −1.81816 | 0 | −1.23436 | 0 | 3.73718 | 0 | ||||||||||||||||||
1.14 | 0 | −2.46774 | 0 | −2.83803 | 0 | 1.20715 | 0 | 3.08972 | 0 | ||||||||||||||||||
1.15 | 0 | −2.30440 | 0 | 3.02168 | 0 | 4.18657 | 0 | 2.31024 | 0 | ||||||||||||||||||
1.16 | 0 | −2.27913 | 0 | −0.601984 | 0 | 3.53846 | 0 | 2.19442 | 0 | ||||||||||||||||||
1.17 | 0 | −2.21676 | 0 | 1.73805 | 0 | −5.05782 | 0 | 1.91402 | 0 | ||||||||||||||||||
1.18 | 0 | −2.02583 | 0 | 1.86007 | 0 | 1.58288 | 0 | 1.10398 | 0 | ||||||||||||||||||
1.19 | 0 | −1.92727 | 0 | −2.52260 | 0 | −3.66775 | 0 | 0.714366 | 0 | ||||||||||||||||||
1.20 | 0 | −1.87872 | 0 | 3.33381 | 0 | 1.62469 | 0 | 0.529574 | 0 | ||||||||||||||||||
See all 79 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.a | ✓ | 79 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8012.2.a.a | ✓ | 79 | 1.a | even | 1 | 1 | trivial |