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: | \(31\) |
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 | −3.32689 | 1.00000 | −3.79290 | −3.32689 | −3.92698 | 1.00000 | 8.06821 | −3.79290 | ||||||||||||||||||
1.2 | 1.00000 | −3.17886 | 1.00000 | 2.94122 | −3.17886 | −2.37061 | 1.00000 | 7.10517 | 2.94122 | ||||||||||||||||||
1.3 | 1.00000 | −3.04963 | 1.00000 | −0.453463 | −3.04963 | 2.87005 | 1.00000 | 6.30022 | −0.453463 | ||||||||||||||||||
1.4 | 1.00000 | −2.91271 | 1.00000 | 3.14140 | −2.91271 | −3.38046 | 1.00000 | 5.48388 | 3.14140 | ||||||||||||||||||
1.5 | 1.00000 | −2.85499 | 1.00000 | 1.11825 | −2.85499 | −3.01039 | 1.00000 | 5.15094 | 1.11825 | ||||||||||||||||||
1.6 | 1.00000 | −2.62663 | 1.00000 | −2.01799 | −2.62663 | 0.869262 | 1.00000 | 3.89919 | −2.01799 | ||||||||||||||||||
1.7 | 1.00000 | −2.46221 | 1.00000 | −2.36990 | −2.46221 | 2.92726 | 1.00000 | 3.06248 | −2.36990 | ||||||||||||||||||
1.8 | 1.00000 | −2.43557 | 1.00000 | 0.846052 | −2.43557 | 3.32195 | 1.00000 | 2.93199 | 0.846052 | ||||||||||||||||||
1.9 | 1.00000 | −2.37527 | 1.00000 | −1.97700 | −2.37527 | −4.20609 | 1.00000 | 2.64192 | −1.97700 | ||||||||||||||||||
1.10 | 1.00000 | −1.80853 | 1.00000 | −3.10189 | −1.80853 | −2.67852 | 1.00000 | 0.270780 | −3.10189 | ||||||||||||||||||
1.11 | 1.00000 | −1.75046 | 1.00000 | 1.28935 | −1.75046 | −0.876800 | 1.00000 | 0.0641056 | 1.28935 | ||||||||||||||||||
1.12 | 1.00000 | −1.49586 | 1.00000 | 3.62778 | −1.49586 | −1.50310 | 1.00000 | −0.762393 | 3.62778 | ||||||||||||||||||
1.13 | 1.00000 | −1.32273 | 1.00000 | −4.14420 | −1.32273 | −0.0466766 | 1.00000 | −1.25038 | −4.14420 | ||||||||||||||||||
1.14 | 1.00000 | −0.917670 | 1.00000 | −0.382051 | −0.917670 | 1.33651 | 1.00000 | −2.15788 | −0.382051 | ||||||||||||||||||
1.15 | 1.00000 | −0.691780 | 1.00000 | −1.18527 | −0.691780 | 3.22162 | 1.00000 | −2.52144 | −1.18527 | ||||||||||||||||||
1.16 | 1.00000 | −0.653164 | 1.00000 | 1.42104 | −0.653164 | −3.96008 | 1.00000 | −2.57338 | 1.42104 | ||||||||||||||||||
1.17 | 1.00000 | −0.0253042 | 1.00000 | 0.720155 | −0.0253042 | −3.77805 | 1.00000 | −2.99936 | 0.720155 | ||||||||||||||||||
1.18 | 1.00000 | −0.0246012 | 1.00000 | 2.47730 | −0.0246012 | 1.25791 | 1.00000 | −2.99939 | 2.47730 | ||||||||||||||||||
1.19 | 1.00000 | 0.524728 | 1.00000 | 0.579848 | 0.524728 | 1.74824 | 1.00000 | −2.72466 | 0.579848 | ||||||||||||||||||
1.20 | 1.00000 | 0.689428 | 1.00000 | −0.213014 | 0.689428 | −0.113005 | 1.00000 | −2.52469 | −0.213014 | ||||||||||||||||||
See all 31 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.c | ✓ | 31 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4022.2.a.c | ✓ | 31 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{31} + 14 T_{3}^{30} + 38 T_{3}^{29} - 333 T_{3}^{28} - 1985 T_{3}^{27} + 1593 T_{3}^{26} + \cdots - 1093 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\).