Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6010,2,Mod(1,6010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("6010.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6010 = 2 \cdot 5 \cdot 601 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6010.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: | \(47.9900916148\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
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 | −2.89060 | 1.00000 | −1.00000 | 2.89060 | 1.07596 | −1.00000 | 5.35558 | 1.00000 | ||||||||||||||||||
1.2 | −1.00000 | −2.14296 | 1.00000 | −1.00000 | 2.14296 | 0.736597 | −1.00000 | 1.59228 | 1.00000 | ||||||||||||||||||
1.3 | −1.00000 | −1.93787 | 1.00000 | −1.00000 | 1.93787 | −0.661252 | −1.00000 | 0.755329 | 1.00000 | ||||||||||||||||||
1.4 | −1.00000 | −1.82478 | 1.00000 | −1.00000 | 1.82478 | −1.84575 | −1.00000 | 0.329836 | 1.00000 | ||||||||||||||||||
1.5 | −1.00000 | −1.73819 | 1.00000 | −1.00000 | 1.73819 | 4.80844 | −1.00000 | 0.0213193 | 1.00000 | ||||||||||||||||||
1.6 | −1.00000 | −1.41773 | 1.00000 | −1.00000 | 1.41773 | −1.60089 | −1.00000 | −0.990039 | 1.00000 | ||||||||||||||||||
1.7 | −1.00000 | −1.41757 | 1.00000 | −1.00000 | 1.41757 | −2.14903 | −1.00000 | −0.990504 | 1.00000 | ||||||||||||||||||
1.8 | −1.00000 | −0.220793 | 1.00000 | −1.00000 | 0.220793 | 1.03342 | −1.00000 | −2.95125 | 1.00000 | ||||||||||||||||||
1.9 | −1.00000 | 0.0461180 | 1.00000 | −1.00000 | −0.0461180 | 2.74163 | −1.00000 | −2.99787 | 1.00000 | ||||||||||||||||||
1.10 | −1.00000 | 0.464276 | 1.00000 | −1.00000 | −0.464276 | −1.37823 | −1.00000 | −2.78445 | 1.00000 | ||||||||||||||||||
1.11 | −1.00000 | 0.474795 | 1.00000 | −1.00000 | −0.474795 | 0.427918 | −1.00000 | −2.77457 | 1.00000 | ||||||||||||||||||
1.12 | −1.00000 | 0.478752 | 1.00000 | −1.00000 | −0.478752 | 0.579883 | −1.00000 | −2.77080 | 1.00000 | ||||||||||||||||||
1.13 | −1.00000 | 1.19881 | 1.00000 | −1.00000 | −1.19881 | 2.82809 | −1.00000 | −1.56285 | 1.00000 | ||||||||||||||||||
1.14 | −1.00000 | 1.26768 | 1.00000 | −1.00000 | −1.26768 | −4.76657 | −1.00000 | −1.39299 | 1.00000 | ||||||||||||||||||
1.15 | −1.00000 | 1.46860 | 1.00000 | −1.00000 | −1.46860 | −3.37454 | −1.00000 | −0.843201 | 1.00000 | ||||||||||||||||||
1.16 | −1.00000 | 1.81728 | 1.00000 | −1.00000 | −1.81728 | 0.464151 | −1.00000 | 0.302521 | 1.00000 | ||||||||||||||||||
1.17 | −1.00000 | 2.50288 | 1.00000 | −1.00000 | −2.50288 | 1.45988 | −1.00000 | 3.26441 | 1.00000 | ||||||||||||||||||
1.18 | −1.00000 | 2.66896 | 1.00000 | −1.00000 | −2.66896 | 3.89028 | −1.00000 | 4.12335 | 1.00000 | ||||||||||||||||||
1.19 | −1.00000 | 2.82986 | 1.00000 | −1.00000 | −2.82986 | −4.43096 | −1.00000 | 5.00809 | 1.00000 | ||||||||||||||||||
1.20 | −1.00000 | 3.15846 | 1.00000 | −1.00000 | −3.15846 | −1.63952 | −1.00000 | 6.97584 | 1.00000 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(5\) | \(1\) |
\(601\) | \(-1\) |
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 | 6010.2.a.e | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6010.2.a.e | ✓ | 21 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{21} - 8 T_{3}^{20} - 7 T_{3}^{19} + 201 T_{3}^{18} - 240 T_{3}^{17} - 1978 T_{3}^{16} + 4141 T_{3}^{15} + \cdots - 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).