Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8020,2,Mod(1,8020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8020, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8020.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8020 = 2^{2} \cdot 5 \cdot 401 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8020.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: | \(64.0400224211\) |
Analytic rank: | \(1\) |
Dimension: | \(29\) |
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 | 0 | −3.11347 | 0 | 1.00000 | 0 | 3.18726 | 0 | 6.69370 | 0 | ||||||||||||||||||
1.2 | 0 | −2.75932 | 0 | 1.00000 | 0 | −2.48714 | 0 | 4.61387 | 0 | ||||||||||||||||||
1.3 | 0 | −2.73282 | 0 | 1.00000 | 0 | 1.81499 | 0 | 4.46832 | 0 | ||||||||||||||||||
1.4 | 0 | −2.62437 | 0 | 1.00000 | 0 | −4.07774 | 0 | 3.88732 | 0 | ||||||||||||||||||
1.5 | 0 | −2.50668 | 0 | 1.00000 | 0 | −0.362320 | 0 | 3.28346 | 0 | ||||||||||||||||||
1.6 | 0 | −2.25625 | 0 | 1.00000 | 0 | 0.417578 | 0 | 2.09065 | 0 | ||||||||||||||||||
1.7 | 0 | −1.73579 | 0 | 1.00000 | 0 | 2.72114 | 0 | 0.0129592 | 0 | ||||||||||||||||||
1.8 | 0 | −1.59991 | 0 | 1.00000 | 0 | −1.91437 | 0 | −0.440296 | 0 | ||||||||||||||||||
1.9 | 0 | −1.35123 | 0 | 1.00000 | 0 | 0.606242 | 0 | −1.17417 | 0 | ||||||||||||||||||
1.10 | 0 | −1.12115 | 0 | 1.00000 | 0 | 0.722262 | 0 | −1.74302 | 0 | ||||||||||||||||||
1.11 | 0 | −0.848679 | 0 | 1.00000 | 0 | −3.71403 | 0 | −2.27974 | 0 | ||||||||||||||||||
1.12 | 0 | −0.697718 | 0 | 1.00000 | 0 | −4.99015 | 0 | −2.51319 | 0 | ||||||||||||||||||
1.13 | 0 | −0.537747 | 0 | 1.00000 | 0 | −1.46052 | 0 | −2.71083 | 0 | ||||||||||||||||||
1.14 | 0 | −0.202645 | 0 | 1.00000 | 0 | 1.27188 | 0 | −2.95893 | 0 | ||||||||||||||||||
1.15 | 0 | −0.107530 | 0 | 1.00000 | 0 | 2.68164 | 0 | −2.98844 | 0 | ||||||||||||||||||
1.16 | 0 | 0.0219398 | 0 | 1.00000 | 0 | 2.04593 | 0 | −2.99952 | 0 | ||||||||||||||||||
1.17 | 0 | 0.121745 | 0 | 1.00000 | 0 | 5.25775 | 0 | −2.98518 | 0 | ||||||||||||||||||
1.18 | 0 | 0.358058 | 0 | 1.00000 | 0 | 1.78556 | 0 | −2.87179 | 0 | ||||||||||||||||||
1.19 | 0 | 0.540664 | 0 | 1.00000 | 0 | −1.71282 | 0 | −2.70768 | 0 | ||||||||||||||||||
1.20 | 0 | 1.03867 | 0 | 1.00000 | 0 | −0.0217287 | 0 | −1.92117 | 0 | ||||||||||||||||||
See all 29 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(-1\) |
\(401\) | \(1\) |
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 | 8020.2.a.d | ✓ | 29 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8020.2.a.d | ✓ | 29 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{29} + 3 T_{3}^{28} - 44 T_{3}^{27} - 130 T_{3}^{26} + 851 T_{3}^{25} + 2470 T_{3}^{24} - 9512 T_{3}^{23} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\).