Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6002,2,Mod(1,6002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6002, 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("6002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6002 = 2 \cdot 3001 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6002.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: | \(47.9262112932\) |
Analytic rank: | \(1\) |
Dimension: | \(56\) |
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 | −3.33546 | 1.00000 | 0.467319 | 3.33546 | 1.63589 | −1.00000 | 8.12527 | −0.467319 | ||||||||||||||||||
1.2 | −1.00000 | −3.32670 | 1.00000 | −1.32100 | 3.32670 | −1.80854 | −1.00000 | 8.06690 | 1.32100 | ||||||||||||||||||
1.3 | −1.00000 | −3.25643 | 1.00000 | 4.04919 | 3.25643 | −2.73149 | −1.00000 | 7.60432 | −4.04919 | ||||||||||||||||||
1.4 | −1.00000 | −3.13871 | 1.00000 | −2.43791 | 3.13871 | −2.06269 | −1.00000 | 6.85148 | 2.43791 | ||||||||||||||||||
1.5 | −1.00000 | −3.00129 | 1.00000 | −0.204934 | 3.00129 | −4.79366 | −1.00000 | 6.00776 | 0.204934 | ||||||||||||||||||
1.6 | −1.00000 | −2.97926 | 1.00000 | −2.47117 | 2.97926 | 1.46435 | −1.00000 | 5.87597 | 2.47117 | ||||||||||||||||||
1.7 | −1.00000 | −2.77449 | 1.00000 | 2.09468 | 2.77449 | −4.11800 | −1.00000 | 4.69780 | −2.09468 | ||||||||||||||||||
1.8 | −1.00000 | −2.57676 | 1.00000 | 1.81538 | 2.57676 | 4.12880 | −1.00000 | 3.63972 | −1.81538 | ||||||||||||||||||
1.9 | −1.00000 | −2.53853 | 1.00000 | 3.20594 | 2.53853 | 2.08261 | −1.00000 | 3.44415 | −3.20594 | ||||||||||||||||||
1.10 | −1.00000 | −2.29725 | 1.00000 | −0.0563132 | 2.29725 | 2.55969 | −1.00000 | 2.27735 | 0.0563132 | ||||||||||||||||||
1.11 | −1.00000 | −2.23559 | 1.00000 | 2.58834 | 2.23559 | 1.55053 | −1.00000 | 1.99788 | −2.58834 | ||||||||||||||||||
1.12 | −1.00000 | −2.14369 | 1.00000 | −1.86034 | 2.14369 | 1.91377 | −1.00000 | 1.59541 | 1.86034 | ||||||||||||||||||
1.13 | −1.00000 | −2.11800 | 1.00000 | −3.39608 | 2.11800 | −4.21736 | −1.00000 | 1.48593 | 3.39608 | ||||||||||||||||||
1.14 | −1.00000 | −2.09636 | 1.00000 | −1.63895 | 2.09636 | 0.351798 | −1.00000 | 1.39473 | 1.63895 | ||||||||||||||||||
1.15 | −1.00000 | −2.01653 | 1.00000 | −1.99229 | 2.01653 | −2.17859 | −1.00000 | 1.06638 | 1.99229 | ||||||||||||||||||
1.16 | −1.00000 | −1.91820 | 1.00000 | 2.08437 | 1.91820 | −4.09357 | −1.00000 | 0.679480 | −2.08437 | ||||||||||||||||||
1.17 | −1.00000 | −1.79165 | 1.00000 | 2.16923 | 1.79165 | −2.84770 | −1.00000 | 0.210015 | −2.16923 | ||||||||||||||||||
1.18 | −1.00000 | −1.64063 | 1.00000 | −2.79431 | 1.64063 | 2.69140 | −1.00000 | −0.308341 | 2.79431 | ||||||||||||||||||
1.19 | −1.00000 | −1.59777 | 1.00000 | 0.373803 | 1.59777 | 0.398412 | −1.00000 | −0.447134 | −0.373803 | ||||||||||||||||||
1.20 | −1.00000 | −1.48075 | 1.00000 | 3.81565 | 1.48075 | 2.08581 | −1.00000 | −0.807375 | −3.81565 | ||||||||||||||||||
See all 56 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(3001\) | \(1\) |
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 | 6002.2.a.b | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6002.2.a.b | ✓ | 56 | 1.a | even | 1 | 1 | trivial |