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: | \(1\) |
Dimension: | \(37\) |
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 | −3.25505 | 1.00000 | −0.797463 | 3.25505 | −4.64387 | −1.00000 | 7.59538 | 0.797463 | ||||||||||||||||||
1.2 | −1.00000 | −3.25277 | 1.00000 | −3.41922 | 3.25277 | 1.95122 | −1.00000 | 7.58049 | 3.41922 | ||||||||||||||||||
1.3 | −1.00000 | −3.07138 | 1.00000 | 0.455356 | 3.07138 | 0.869394 | −1.00000 | 6.43339 | −0.455356 | ||||||||||||||||||
1.4 | −1.00000 | −2.82634 | 1.00000 | 2.27969 | 2.82634 | −0.755653 | −1.00000 | 4.98821 | −2.27969 | ||||||||||||||||||
1.5 | −1.00000 | −2.78933 | 1.00000 | 0.171910 | 2.78933 | 1.94661 | −1.00000 | 4.78038 | −0.171910 | ||||||||||||||||||
1.6 | −1.00000 | −2.37393 | 1.00000 | 1.41638 | 2.37393 | 3.16627 | −1.00000 | 2.63555 | −1.41638 | ||||||||||||||||||
1.7 | −1.00000 | −2.35142 | 1.00000 | −3.94455 | 2.35142 | −0.732372 | −1.00000 | 2.52919 | 3.94455 | ||||||||||||||||||
1.8 | −1.00000 | −1.96616 | 1.00000 | −1.21238 | 1.96616 | −3.03603 | −1.00000 | 0.865786 | 1.21238 | ||||||||||||||||||
1.9 | −1.00000 | −1.94782 | 1.00000 | 3.45647 | 1.94782 | −2.39115 | −1.00000 | 0.794011 | −3.45647 | ||||||||||||||||||
1.10 | −1.00000 | −1.92131 | 1.00000 | −0.749362 | 1.92131 | −4.27002 | −1.00000 | 0.691416 | 0.749362 | ||||||||||||||||||
1.11 | −1.00000 | −1.46057 | 1.00000 | −0.684995 | 1.46057 | −3.06021 | −1.00000 | −0.866744 | 0.684995 | ||||||||||||||||||
1.12 | −1.00000 | −1.41095 | 1.00000 | −3.55049 | 1.41095 | −3.54920 | −1.00000 | −1.00921 | 3.55049 | ||||||||||||||||||
1.13 | −1.00000 | −1.40012 | 1.00000 | −1.65031 | 1.40012 | 2.90659 | −1.00000 | −1.03967 | 1.65031 | ||||||||||||||||||
1.14 | −1.00000 | −1.24109 | 1.00000 | −1.11151 | 1.24109 | 1.91739 | −1.00000 | −1.45969 | 1.11151 | ||||||||||||||||||
1.15 | −1.00000 | −1.16293 | 1.00000 | 3.05784 | 1.16293 | 2.13691 | −1.00000 | −1.64760 | −3.05784 | ||||||||||||||||||
1.16 | −1.00000 | −1.10960 | 1.00000 | 2.36799 | 1.10960 | 0.505980 | −1.00000 | −1.76879 | −2.36799 | ||||||||||||||||||
1.17 | −1.00000 | −0.586559 | 1.00000 | 2.99859 | 0.586559 | 1.29059 | −1.00000 | −2.65595 | −2.99859 | ||||||||||||||||||
1.18 | −1.00000 | −0.0611935 | 1.00000 | −3.49914 | 0.0611935 | −5.08120 | −1.00000 | −2.99626 | 3.49914 | ||||||||||||||||||
1.19 | −1.00000 | −0.00489128 | 1.00000 | −4.12652 | 0.00489128 | −1.34401 | −1.00000 | −2.99998 | 4.12652 | ||||||||||||||||||
1.20 | −1.00000 | 0.0519256 | 1.00000 | −2.57858 | −0.0519256 | 3.17465 | −1.00000 | −2.99730 | 2.57858 | ||||||||||||||||||
See all 37 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.d | ✓ | 37 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4022.2.a.d | ✓ | 37 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{37} + 5 T_{3}^{36} - 59 T_{3}^{35} - 320 T_{3}^{34} + 1532 T_{3}^{33} + 9243 T_{3}^{32} + \cdots + 136 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).