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: | \(28\) |
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 | −3.10189 | 0 | −1.00000 | 0 | 1.97262 | 0 | 6.62170 | 0 | ||||||||||||||||||
1.2 | 0 | −2.61261 | 0 | −1.00000 | 0 | −1.36796 | 0 | 3.82575 | 0 | ||||||||||||||||||
1.3 | 0 | −2.59496 | 0 | −1.00000 | 0 | 1.42420 | 0 | 3.73381 | 0 | ||||||||||||||||||
1.4 | 0 | −2.32143 | 0 | −1.00000 | 0 | 0.515797 | 0 | 2.38904 | 0 | ||||||||||||||||||
1.5 | 0 | −2.26002 | 0 | −1.00000 | 0 | −3.39221 | 0 | 2.10771 | 0 | ||||||||||||||||||
1.6 | 0 | −1.62538 | 0 | −1.00000 | 0 | 4.42157 | 0 | −0.358130 | 0 | ||||||||||||||||||
1.7 | 0 | −1.58112 | 0 | −1.00000 | 0 | 3.33363 | 0 | −0.500055 | 0 | ||||||||||||||||||
1.8 | 0 | −1.53415 | 0 | −1.00000 | 0 | 0.759833 | 0 | −0.646394 | 0 | ||||||||||||||||||
1.9 | 0 | −1.48695 | 0 | −1.00000 | 0 | −3.53428 | 0 | −0.788974 | 0 | ||||||||||||||||||
1.10 | 0 | −1.42561 | 0 | −1.00000 | 0 | −3.82644 | 0 | −0.967639 | 0 | ||||||||||||||||||
1.11 | 0 | −0.750059 | 0 | −1.00000 | 0 | −4.31732 | 0 | −2.43741 | 0 | ||||||||||||||||||
1.12 | 0 | −0.324068 | 0 | −1.00000 | 0 | −1.35293 | 0 | −2.89498 | 0 | ||||||||||||||||||
1.13 | 0 | −0.172341 | 0 | −1.00000 | 0 | 3.94901 | 0 | −2.97030 | 0 | ||||||||||||||||||
1.14 | 0 | 0.0547853 | 0 | −1.00000 | 0 | 3.87235 | 0 | −2.99700 | 0 | ||||||||||||||||||
1.15 | 0 | 0.167528 | 0 | −1.00000 | 0 | −3.89143 | 0 | −2.97193 | 0 | ||||||||||||||||||
1.16 | 0 | 0.356039 | 0 | −1.00000 | 0 | −0.400497 | 0 | −2.87324 | 0 | ||||||||||||||||||
1.17 | 0 | 0.910150 | 0 | −1.00000 | 0 | 1.40834 | 0 | −2.17163 | 0 | ||||||||||||||||||
1.18 | 0 | 1.05681 | 0 | −1.00000 | 0 | 3.74444 | 0 | −1.88316 | 0 | ||||||||||||||||||
1.19 | 0 | 1.28191 | 0 | −1.00000 | 0 | −0.791593 | 0 | −1.35670 | 0 | ||||||||||||||||||
1.20 | 0 | 1.67955 | 0 | −1.00000 | 0 | −2.58133 | 0 | −0.179108 | 0 | ||||||||||||||||||
See all 28 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.c | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8020.2.a.c | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 3 T_{3}^{27} - 46 T_{3}^{26} + 138 T_{3}^{25} + 923 T_{3}^{24} - 2776 T_{3}^{23} + \cdots - 256 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\).