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: | \(0\) |
Dimension: | \(35\) |
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 | −3.41155 | 0 | −1.00000 | 0 | 0.311908 | 0 | 8.63864 | 0 | ||||||||||||||||||
1.2 | 0 | −3.20562 | 0 | −1.00000 | 0 | −2.96453 | 0 | 7.27598 | 0 | ||||||||||||||||||
1.3 | 0 | −3.08956 | 0 | −1.00000 | 0 | 4.75064 | 0 | 6.54541 | 0 | ||||||||||||||||||
1.4 | 0 | −2.93370 | 0 | −1.00000 | 0 | 4.42731 | 0 | 5.60662 | 0 | ||||||||||||||||||
1.5 | 0 | −2.91827 | 0 | −1.00000 | 0 | −4.31238 | 0 | 5.51629 | 0 | ||||||||||||||||||
1.6 | 0 | −2.89929 | 0 | −1.00000 | 0 | 2.68176 | 0 | 5.40586 | 0 | ||||||||||||||||||
1.7 | 0 | −2.36832 | 0 | −1.00000 | 0 | −4.12967 | 0 | 2.60896 | 0 | ||||||||||||||||||
1.8 | 0 | −2.25358 | 0 | −1.00000 | 0 | −0.859031 | 0 | 2.07862 | 0 | ||||||||||||||||||
1.9 | 0 | −1.64682 | 0 | −1.00000 | 0 | 1.19631 | 0 | −0.287991 | 0 | ||||||||||||||||||
1.10 | 0 | −1.51742 | 0 | −1.00000 | 0 | −1.44284 | 0 | −0.697439 | 0 | ||||||||||||||||||
1.11 | 0 | −1.37216 | 0 | −1.00000 | 0 | −1.82698 | 0 | −1.11719 | 0 | ||||||||||||||||||
1.12 | 0 | −1.36773 | 0 | −1.00000 | 0 | 2.86190 | 0 | −1.12932 | 0 | ||||||||||||||||||
1.13 | 0 | −1.28894 | 0 | −1.00000 | 0 | 0.0476003 | 0 | −1.33863 | 0 | ||||||||||||||||||
1.14 | 0 | −0.839936 | 0 | −1.00000 | 0 | 3.45440 | 0 | −2.29451 | 0 | ||||||||||||||||||
1.15 | 0 | −0.713769 | 0 | −1.00000 | 0 | −1.26919 | 0 | −2.49053 | 0 | ||||||||||||||||||
1.16 | 0 | −0.505617 | 0 | −1.00000 | 0 | −0.615205 | 0 | −2.74435 | 0 | ||||||||||||||||||
1.17 | 0 | −0.478839 | 0 | −1.00000 | 0 | 1.59155 | 0 | −2.77071 | 0 | ||||||||||||||||||
1.18 | 0 | −0.167962 | 0 | −1.00000 | 0 | 2.31084 | 0 | −2.97179 | 0 | ||||||||||||||||||
1.19 | 0 | 0.0559400 | 0 | −1.00000 | 0 | −0.736794 | 0 | −2.99687 | 0 | ||||||||||||||||||
1.20 | 0 | 0.309116 | 0 | −1.00000 | 0 | −4.23374 | 0 | −2.90445 | 0 | ||||||||||||||||||
See all 35 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.e | ✓ | 35 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8020.2.a.e | ✓ | 35 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{35} + T_{3}^{34} - 78 T_{3}^{33} - 74 T_{3}^{32} + 2745 T_{3}^{31} + 2470 T_{3}^{30} + \cdots - 168832 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\).