Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6022,2,Mod(1,6022)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6022, 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("6022.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6022 = 2 \cdot 3011 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6022.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: | \(48.0859120972\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
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 | −3.35476 | 1.00000 | −3.57894 | −3.35476 | 1.79257 | 1.00000 | 8.25439 | −3.57894 | ||||||||||||||||||
1.2 | 1.00000 | −3.25636 | 1.00000 | 2.14369 | −3.25636 | −2.21695 | 1.00000 | 7.60391 | 2.14369 | ||||||||||||||||||
1.3 | 1.00000 | −3.01193 | 1.00000 | 3.00375 | −3.01193 | 5.03538 | 1.00000 | 6.07172 | 3.00375 | ||||||||||||||||||
1.4 | 1.00000 | −2.72056 | 1.00000 | −0.689023 | −2.72056 | 0.199511 | 1.00000 | 4.40143 | −0.689023 | ||||||||||||||||||
1.5 | 1.00000 | −2.64273 | 1.00000 | −1.57154 | −2.64273 | −2.07612 | 1.00000 | 3.98402 | −1.57154 | ||||||||||||||||||
1.6 | 1.00000 | −2.61602 | 1.00000 | −0.861265 | −2.61602 | 0.881327 | 1.00000 | 3.84357 | −0.861265 | ||||||||||||||||||
1.7 | 1.00000 | −2.49368 | 1.00000 | −0.272983 | −2.49368 | −4.45268 | 1.00000 | 3.21842 | −0.272983 | ||||||||||||||||||
1.8 | 1.00000 | −2.49125 | 1.00000 | 3.68938 | −2.49125 | 1.92510 | 1.00000 | 3.20631 | 3.68938 | ||||||||||||||||||
1.9 | 1.00000 | −2.43949 | 1.00000 | −1.96660 | −2.43949 | 4.60253 | 1.00000 | 2.95113 | −1.96660 | ||||||||||||||||||
1.10 | 1.00000 | −2.32468 | 1.00000 | 2.75364 | −2.32468 | 1.28652 | 1.00000 | 2.40413 | 2.75364 | ||||||||||||||||||
1.11 | 1.00000 | −2.31895 | 1.00000 | 2.61531 | −2.31895 | 1.26807 | 1.00000 | 2.37754 | 2.61531 | ||||||||||||||||||
1.12 | 1.00000 | −2.31495 | 1.00000 | −4.12889 | −2.31495 | −1.35948 | 1.00000 | 2.35897 | −4.12889 | ||||||||||||||||||
1.13 | 1.00000 | −2.26646 | 1.00000 | 2.42379 | −2.26646 | −1.89494 | 1.00000 | 2.13682 | 2.42379 | ||||||||||||||||||
1.14 | 1.00000 | −2.24505 | 1.00000 | −1.17279 | −2.24505 | 1.15296 | 1.00000 | 2.04026 | −1.17279 | ||||||||||||||||||
1.15 | 1.00000 | −1.47177 | 1.00000 | 0.895866 | −1.47177 | 0.963361 | 1.00000 | −0.833884 | 0.895866 | ||||||||||||||||||
1.16 | 1.00000 | −1.21963 | 1.00000 | −1.41374 | −1.21963 | −2.00566 | 1.00000 | −1.51249 | −1.41374 | ||||||||||||||||||
1.17 | 1.00000 | −1.15681 | 1.00000 | 1.22743 | −1.15681 | 3.85871 | 1.00000 | −1.66179 | 1.22743 | ||||||||||||||||||
1.18 | 1.00000 | −1.15100 | 1.00000 | 1.71186 | −1.15100 | 3.72563 | 1.00000 | −1.67520 | 1.71186 | ||||||||||||||||||
1.19 | 1.00000 | −1.02270 | 1.00000 | 3.27073 | −1.02270 | 4.67919 | 1.00000 | −1.95408 | 3.27073 | ||||||||||||||||||
1.20 | 1.00000 | −0.927049 | 1.00000 | −3.29749 | −0.927049 | 4.05370 | 1.00000 | −2.14058 | −3.29749 | ||||||||||||||||||
See all 68 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3011\) | \(1\) |
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 | 6022.2.a.e | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6022.2.a.e | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{68} - 25 T_{3}^{67} + 167 T_{3}^{66} + 961 T_{3}^{65} - 17331 T_{3}^{64} + 33713 T_{3}^{63} + \cdots - 87273344 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\).