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: | \(27\) |
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 | −3.12246 | 1.00000 | 1.00000 | 3.12246 | −0.227875 | −1.00000 | 6.74974 | −1.00000 | ||||||||||||||||||
1.2 | −1.00000 | −2.89176 | 1.00000 | 1.00000 | 2.89176 | 4.20794 | −1.00000 | 5.36230 | −1.00000 | ||||||||||||||||||
1.3 | −1.00000 | −2.88499 | 1.00000 | 1.00000 | 2.88499 | 0.210491 | −1.00000 | 5.32318 | −1.00000 | ||||||||||||||||||
1.4 | −1.00000 | −2.52918 | 1.00000 | 1.00000 | 2.52918 | −2.70859 | −1.00000 | 3.39676 | −1.00000 | ||||||||||||||||||
1.5 | −1.00000 | −2.12852 | 1.00000 | 1.00000 | 2.12852 | −4.55968 | −1.00000 | 1.53060 | −1.00000 | ||||||||||||||||||
1.6 | −1.00000 | −2.00989 | 1.00000 | 1.00000 | 2.00989 | −0.566476 | −1.00000 | 1.03966 | −1.00000 | ||||||||||||||||||
1.7 | −1.00000 | −1.92694 | 1.00000 | 1.00000 | 1.92694 | −3.44591 | −1.00000 | 0.713113 | −1.00000 | ||||||||||||||||||
1.8 | −1.00000 | −1.21224 | 1.00000 | 1.00000 | 1.21224 | 2.32820 | −1.00000 | −1.53047 | −1.00000 | ||||||||||||||||||
1.9 | −1.00000 | −1.11003 | 1.00000 | 1.00000 | 1.11003 | 1.51458 | −1.00000 | −1.76783 | −1.00000 | ||||||||||||||||||
1.10 | −1.00000 | −0.805731 | 1.00000 | 1.00000 | 0.805731 | 2.46129 | −1.00000 | −2.35080 | −1.00000 | ||||||||||||||||||
1.11 | −1.00000 | −0.716010 | 1.00000 | 1.00000 | 0.716010 | 3.53027 | −1.00000 | −2.48733 | −1.00000 | ||||||||||||||||||
1.12 | −1.00000 | −0.409284 | 1.00000 | 1.00000 | 0.409284 | −1.22599 | −1.00000 | −2.83249 | −1.00000 | ||||||||||||||||||
1.13 | −1.00000 | −0.114121 | 1.00000 | 1.00000 | 0.114121 | −3.33992 | −1.00000 | −2.98698 | −1.00000 | ||||||||||||||||||
1.14 | −1.00000 | −0.0770650 | 1.00000 | 1.00000 | 0.0770650 | −1.82853 | −1.00000 | −2.99406 | −1.00000 | ||||||||||||||||||
1.15 | −1.00000 | 1.03782 | 1.00000 | 1.00000 | −1.03782 | −3.65688 | −1.00000 | −1.92294 | −1.00000 | ||||||||||||||||||
1.16 | −1.00000 | 1.08273 | 1.00000 | 1.00000 | −1.08273 | 3.52082 | −1.00000 | −1.82769 | −1.00000 | ||||||||||||||||||
1.17 | −1.00000 | 1.23890 | 1.00000 | 1.00000 | −1.23890 | −4.00661 | −1.00000 | −1.46513 | −1.00000 | ||||||||||||||||||
1.18 | −1.00000 | 1.45178 | 1.00000 | 1.00000 | −1.45178 | 3.96566 | −1.00000 | −0.892348 | −1.00000 | ||||||||||||||||||
1.19 | −1.00000 | 1.60467 | 1.00000 | 1.00000 | −1.60467 | −3.11642 | −1.00000 | −0.425041 | −1.00000 | ||||||||||||||||||
1.20 | −1.00000 | 1.83794 | 1.00000 | 1.00000 | −1.83794 | 2.63564 | −1.00000 | 0.378009 | −1.00000 | ||||||||||||||||||
See all 27 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.g | ✓ | 27 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6010.2.a.g | ✓ | 27 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{27} - 6 T_{3}^{26} - 41 T_{3}^{25} + 297 T_{3}^{24} + 641 T_{3}^{23} - 6360 T_{3}^{22} + \cdots - 11264 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).