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: | \(1\) |
Dimension: | \(64\) |
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 | −3.41719 | 1.00000 | 0.0151085 | 3.41719 | −2.13229 | −1.00000 | 8.67716 | −0.0151085 | ||||||||||||||||||
1.2 | −1.00000 | −3.28572 | 1.00000 | −3.45147 | 3.28572 | −1.04970 | −1.00000 | 7.79593 | 3.45147 | ||||||||||||||||||
1.3 | −1.00000 | −3.15898 | 1.00000 | −0.473594 | 3.15898 | −3.23851 | −1.00000 | 6.97917 | 0.473594 | ||||||||||||||||||
1.4 | −1.00000 | −3.12891 | 1.00000 | 3.39803 | 3.12891 | 2.97458 | −1.00000 | 6.79008 | −3.39803 | ||||||||||||||||||
1.5 | −1.00000 | −3.12816 | 1.00000 | −3.78389 | 3.12816 | 3.19899 | −1.00000 | 6.78536 | 3.78389 | ||||||||||||||||||
1.6 | −1.00000 | −3.07409 | 1.00000 | 0.701582 | 3.07409 | 1.60448 | −1.00000 | 6.45002 | −0.701582 | ||||||||||||||||||
1.7 | −1.00000 | −2.84165 | 1.00000 | −2.41894 | 2.84165 | 4.66695 | −1.00000 | 5.07496 | 2.41894 | ||||||||||||||||||
1.8 | −1.00000 | −2.65762 | 1.00000 | −0.0671999 | 2.65762 | 1.45309 | −1.00000 | 4.06295 | 0.0671999 | ||||||||||||||||||
1.9 | −1.00000 | −2.46766 | 1.00000 | 0.852564 | 2.46766 | −2.40231 | −1.00000 | 3.08936 | −0.852564 | ||||||||||||||||||
1.10 | −1.00000 | −2.38767 | 1.00000 | 3.65260 | 2.38767 | −5.18773 | −1.00000 | 2.70099 | −3.65260 | ||||||||||||||||||
1.11 | −1.00000 | −2.35063 | 1.00000 | −2.93009 | 2.35063 | −2.60832 | −1.00000 | 2.52545 | 2.93009 | ||||||||||||||||||
1.12 | −1.00000 | −2.23814 | 1.00000 | 3.11624 | 2.23814 | 2.78730 | −1.00000 | 2.00926 | −3.11624 | ||||||||||||||||||
1.13 | −1.00000 | −2.21674 | 1.00000 | 1.41824 | 2.21674 | −0.343724 | −1.00000 | 1.91395 | −1.41824 | ||||||||||||||||||
1.14 | −1.00000 | −2.19043 | 1.00000 | −0.477734 | 2.19043 | −4.48401 | −1.00000 | 1.79797 | 0.477734 | ||||||||||||||||||
1.15 | −1.00000 | −2.16331 | 1.00000 | −3.33265 | 2.16331 | 3.92579 | −1.00000 | 1.67989 | 3.33265 | ||||||||||||||||||
1.16 | −1.00000 | −2.12241 | 1.00000 | −2.62309 | 2.12241 | −2.27034 | −1.00000 | 1.50461 | 2.62309 | ||||||||||||||||||
1.17 | −1.00000 | −2.10760 | 1.00000 | 3.51120 | 2.10760 | 0.432948 | −1.00000 | 1.44198 | −3.51120 | ||||||||||||||||||
1.18 | −1.00000 | −1.92893 | 1.00000 | 1.84484 | 1.92893 | 2.06972 | −1.00000 | 0.720763 | −1.84484 | ||||||||||||||||||
1.19 | −1.00000 | −1.85756 | 1.00000 | −0.233219 | 1.85756 | 0.276932 | −1.00000 | 0.450535 | 0.233219 | ||||||||||||||||||
1.20 | −1.00000 | −1.43853 | 1.00000 | −4.29281 | 1.43853 | 1.20415 | −1.00000 | −0.930640 | 4.29281 | ||||||||||||||||||
See all 64 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.d | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6022.2.a.d | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 9 T_{3}^{63} - 86 T_{3}^{62} - 987 T_{3}^{61} + 2983 T_{3}^{60} + 50801 T_{3}^{59} + \cdots - 90066107 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\).