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: | \(28\) |
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 | 1.00000 | −3.20632 | 1.00000 | −1.00000 | −3.20632 | −2.43054 | 1.00000 | 7.28047 | −1.00000 | ||||||||||||||||||
1.2 | 1.00000 | −2.95613 | 1.00000 | −1.00000 | −2.95613 | −3.28957 | 1.00000 | 5.73868 | −1.00000 | ||||||||||||||||||
1.3 | 1.00000 | −2.81740 | 1.00000 | −1.00000 | −2.81740 | −2.15395 | 1.00000 | 4.93776 | −1.00000 | ||||||||||||||||||
1.4 | 1.00000 | −2.80565 | 1.00000 | −1.00000 | −2.80565 | 2.92877 | 1.00000 | 4.87166 | −1.00000 | ||||||||||||||||||
1.5 | 1.00000 | −2.45664 | 1.00000 | −1.00000 | −2.45664 | 3.60299 | 1.00000 | 3.03507 | −1.00000 | ||||||||||||||||||
1.6 | 1.00000 | −2.33584 | 1.00000 | −1.00000 | −2.33584 | 4.65545 | 1.00000 | 2.45616 | −1.00000 | ||||||||||||||||||
1.7 | 1.00000 | −1.85629 | 1.00000 | −1.00000 | −1.85629 | 2.08107 | 1.00000 | 0.445825 | −1.00000 | ||||||||||||||||||
1.8 | 1.00000 | −1.73502 | 1.00000 | −1.00000 | −1.73502 | −0.456244 | 1.00000 | 0.0103032 | −1.00000 | ||||||||||||||||||
1.9 | 1.00000 | −1.14577 | 1.00000 | −1.00000 | −1.14577 | 0.345550 | 1.00000 | −1.68722 | −1.00000 | ||||||||||||||||||
1.10 | 1.00000 | −0.869813 | 1.00000 | −1.00000 | −0.869813 | −1.24307 | 1.00000 | −2.24343 | −1.00000 | ||||||||||||||||||
1.11 | 1.00000 | −0.734514 | 1.00000 | −1.00000 | −0.734514 | 0.223072 | 1.00000 | −2.46049 | −1.00000 | ||||||||||||||||||
1.12 | 1.00000 | −0.503837 | 1.00000 | −1.00000 | −0.503837 | −4.37336 | 1.00000 | −2.74615 | −1.00000 | ||||||||||||||||||
1.13 | 1.00000 | −0.134464 | 1.00000 | −1.00000 | −0.134464 | 4.82249 | 1.00000 | −2.98192 | −1.00000 | ||||||||||||||||||
1.14 | 1.00000 | 0.184156 | 1.00000 | −1.00000 | 0.184156 | −4.92010 | 1.00000 | −2.96609 | −1.00000 | ||||||||||||||||||
1.15 | 1.00000 | 0.321033 | 1.00000 | −1.00000 | 0.321033 | −2.34471 | 1.00000 | −2.89694 | −1.00000 | ||||||||||||||||||
1.16 | 1.00000 | 0.519061 | 1.00000 | −1.00000 | 0.519061 | −3.03480 | 1.00000 | −2.73058 | −1.00000 | ||||||||||||||||||
1.17 | 1.00000 | 0.657748 | 1.00000 | −1.00000 | 0.657748 | 4.85947 | 1.00000 | −2.56737 | −1.00000 | ||||||||||||||||||
1.18 | 1.00000 | 0.983549 | 1.00000 | −1.00000 | 0.983549 | 0.242326 | 1.00000 | −2.03263 | −1.00000 | ||||||||||||||||||
1.19 | 1.00000 | 1.25977 | 1.00000 | −1.00000 | 1.25977 | 3.22153 | 1.00000 | −1.41297 | −1.00000 | ||||||||||||||||||
1.20 | 1.00000 | 1.41587 | 1.00000 | −1.00000 | 1.41587 | 2.06613 | 1.00000 | −0.995326 | −1.00000 | ||||||||||||||||||
See all 28 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.h | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6010.2.a.h | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 4 T_{3}^{27} - 54 T_{3}^{26} + 223 T_{3}^{25} + 1259 T_{3}^{24} - 5421 T_{3}^{23} + \cdots - 7936 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).