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: | \(54\) |
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 | −3.41232 | 1.00000 | −1.85663 | −3.41232 | 3.02838 | 1.00000 | 8.64394 | −1.85663 | ||||||||||||||||||
1.2 | 1.00000 | −3.40089 | 1.00000 | 3.63611 | −3.40089 | −2.39111 | 1.00000 | 8.56603 | 3.63611 | ||||||||||||||||||
1.3 | 1.00000 | −3.18590 | 1.00000 | −1.36832 | −3.18590 | −3.72265 | 1.00000 | 7.14997 | −1.36832 | ||||||||||||||||||
1.4 | 1.00000 | −3.17082 | 1.00000 | 0.413863 | −3.17082 | −1.62952 | 1.00000 | 7.05407 | 0.413863 | ||||||||||||||||||
1.5 | 1.00000 | −3.06406 | 1.00000 | 3.64593 | −3.06406 | 0.427453 | 1.00000 | 6.38849 | 3.64593 | ||||||||||||||||||
1.6 | 1.00000 | −2.86198 | 1.00000 | −0.548157 | −2.86198 | 2.20032 | 1.00000 | 5.19094 | −0.548157 | ||||||||||||||||||
1.7 | 1.00000 | −2.75819 | 1.00000 | 0.593272 | −2.75819 | 4.20000 | 1.00000 | 4.60761 | 0.593272 | ||||||||||||||||||
1.8 | 1.00000 | −2.72056 | 1.00000 | 1.67549 | −2.72056 | −1.89176 | 1.00000 | 4.40146 | 1.67549 | ||||||||||||||||||
1.9 | 1.00000 | −2.71927 | 1.00000 | −3.84841 | −2.71927 | −3.35161 | 1.00000 | 4.39444 | −3.84841 | ||||||||||||||||||
1.10 | 1.00000 | −2.46224 | 1.00000 | −1.05561 | −2.46224 | 2.87810 | 1.00000 | 3.06261 | −1.05561 | ||||||||||||||||||
1.11 | 1.00000 | −2.44368 | 1.00000 | 2.22438 | −2.44368 | −3.70978 | 1.00000 | 2.97155 | 2.22438 | ||||||||||||||||||
1.12 | 1.00000 | −2.44169 | 1.00000 | −4.13245 | −2.44169 | 3.61005 | 1.00000 | 2.96185 | −4.13245 | ||||||||||||||||||
1.13 | 1.00000 | −1.92077 | 1.00000 | 3.09656 | −1.92077 | −3.07737 | 1.00000 | 0.689347 | 3.09656 | ||||||||||||||||||
1.14 | 1.00000 | −1.82742 | 1.00000 | −0.665814 | −1.82742 | 1.40743 | 1.00000 | 0.339448 | −0.665814 | ||||||||||||||||||
1.15 | 1.00000 | −1.65622 | 1.00000 | 3.86194 | −1.65622 | 0.0747047 | 1.00000 | −0.256946 | 3.86194 | ||||||||||||||||||
1.16 | 1.00000 | −1.62746 | 1.00000 | 2.10353 | −1.62746 | −4.91169 | 1.00000 | −0.351369 | 2.10353 | ||||||||||||||||||
1.17 | 1.00000 | −1.61281 | 1.00000 | 1.01065 | −1.61281 | 0.302486 | 1.00000 | −0.398851 | 1.01065 | ||||||||||||||||||
1.18 | 1.00000 | −1.59103 | 1.00000 | −2.29870 | −1.59103 | −3.52840 | 1.00000 | −0.468610 | −2.29870 | ||||||||||||||||||
1.19 | 1.00000 | −1.52904 | 1.00000 | −3.77354 | −1.52904 | 0.908453 | 1.00000 | −0.662042 | −3.77354 | ||||||||||||||||||
1.20 | 1.00000 | −1.50713 | 1.00000 | −3.56652 | −1.50713 | −0.922198 | 1.00000 | −0.728548 | −3.56652 | ||||||||||||||||||
See all 54 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.b | ✓ | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6022.2.a.b | ✓ | 54 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{54} + 22 T_{3}^{53} + 145 T_{3}^{52} - 288 T_{3}^{51} - 7760 T_{3}^{50} - 21976 T_{3}^{49} + \cdots - 50985 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\).