Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4022,2,Mod(1,4022)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4022, 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("4022.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4022 = 2 \cdot 2011 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4022.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: | \(32.1158316930\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
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 | −3.20664 | 1.00000 | −0.511117 | −3.20664 | 0.304730 | 1.00000 | 7.28252 | −0.511117 | ||||||||||||||||||
1.2 | 1.00000 | −2.97777 | 1.00000 | −2.48115 | −2.97777 | 4.88773 | 1.00000 | 5.86711 | −2.48115 | ||||||||||||||||||
1.3 | 1.00000 | −2.74563 | 1.00000 | −2.60075 | −2.74563 | 0.636977 | 1.00000 | 4.53848 | −2.60075 | ||||||||||||||||||
1.4 | 1.00000 | −2.61133 | 1.00000 | −0.740567 | −2.61133 | −2.94518 | 1.00000 | 3.81902 | −0.740567 | ||||||||||||||||||
1.5 | 1.00000 | −2.59841 | 1.00000 | 4.27842 | −2.59841 | 3.34363 | 1.00000 | 3.75173 | 4.27842 | ||||||||||||||||||
1.6 | 1.00000 | −2.50306 | 1.00000 | 0.0732871 | −2.50306 | −1.25699 | 1.00000 | 3.26529 | 0.0732871 | ||||||||||||||||||
1.7 | 1.00000 | −2.49143 | 1.00000 | 2.52331 | −2.49143 | 1.07230 | 1.00000 | 3.20721 | 2.52331 | ||||||||||||||||||
1.8 | 1.00000 | −2.46190 | 1.00000 | 2.99591 | −2.46190 | 1.50567 | 1.00000 | 3.06093 | 2.99591 | ||||||||||||||||||
1.9 | 1.00000 | −2.20305 | 1.00000 | 2.72503 | −2.20305 | 3.76461 | 1.00000 | 1.85345 | 2.72503 | ||||||||||||||||||
1.10 | 1.00000 | −1.73177 | 1.00000 | 0.638996 | −1.73177 | 0.168919 | 1.00000 | −0.000964633 | 0 | 0.638996 | |||||||||||||||||
1.11 | 1.00000 | −1.60405 | 1.00000 | 1.46390 | −1.60405 | 4.50208 | 1.00000 | −0.427040 | 1.46390 | ||||||||||||||||||
1.12 | 1.00000 | −1.54111 | 1.00000 | −4.38749 | −1.54111 | 4.20037 | 1.00000 | −0.624992 | −4.38749 | ||||||||||||||||||
1.13 | 1.00000 | −1.49263 | 1.00000 | −2.25527 | −1.49263 | 0.477234 | 1.00000 | −0.772062 | −2.25527 | ||||||||||||||||||
1.14 | 1.00000 | −1.45431 | 1.00000 | −2.49956 | −1.45431 | −3.69872 | 1.00000 | −0.884987 | −2.49956 | ||||||||||||||||||
1.15 | 1.00000 | −1.41980 | 1.00000 | 2.76587 | −1.41980 | −4.92374 | 1.00000 | −0.984179 | 2.76587 | ||||||||||||||||||
1.16 | 1.00000 | −1.05832 | 1.00000 | −2.35340 | −1.05832 | −2.98821 | 1.00000 | −1.87996 | −2.35340 | ||||||||||||||||||
1.17 | 1.00000 | −0.537568 | 1.00000 | 2.09045 | −0.537568 | 4.78863 | 1.00000 | −2.71102 | 2.09045 | ||||||||||||||||||
1.18 | 1.00000 | −0.445408 | 1.00000 | −2.87764 | −0.445408 | −1.74390 | 1.00000 | −2.80161 | −2.87764 | ||||||||||||||||||
1.19 | 1.00000 | −0.424610 | 1.00000 | −2.05502 | −0.424610 | 3.71032 | 1.00000 | −2.81971 | −2.05502 | ||||||||||||||||||
1.20 | 1.00000 | −0.367906 | 1.00000 | 2.32640 | −0.367906 | 1.76735 | 1.00000 | −2.86465 | 2.32640 | ||||||||||||||||||
See all 50 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(2011\) | \(1\) |
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 | 4022.2.a.f | ✓ | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4022.2.a.f | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} - 18 T_{3}^{49} + 54 T_{3}^{48} + 914 T_{3}^{47} - 6636 T_{3}^{46} - 12202 T_{3}^{45} + \cdots - 7247872 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).