Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6046,2,Mod(1,6046)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6046.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6046 = 2 \cdot 3023 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6046.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: | \(48.2775530621\) |
Analytic rank: | \(1\) |
Dimension: | \(55\) |
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 | −3.14242 | 1.00000 | 1.46177 | 3.14242 | −0.391407 | −1.00000 | 6.87480 | −1.46177 | ||||||||||||||||||
1.2 | −1.00000 | −3.11598 | 1.00000 | −1.88193 | 3.11598 | −1.15467 | −1.00000 | 6.70936 | 1.88193 | ||||||||||||||||||
1.3 | −1.00000 | −2.90391 | 1.00000 | −2.93195 | 2.90391 | 1.22051 | −1.00000 | 5.43272 | 2.93195 | ||||||||||||||||||
1.4 | −1.00000 | −2.84928 | 1.00000 | 3.68221 | 2.84928 | 3.75882 | −1.00000 | 5.11842 | −3.68221 | ||||||||||||||||||
1.5 | −1.00000 | −2.76458 | 1.00000 | −3.35839 | 2.76458 | 2.59254 | −1.00000 | 4.64289 | 3.35839 | ||||||||||||||||||
1.6 | −1.00000 | −2.68677 | 1.00000 | 2.41964 | 2.68677 | 0.726544 | −1.00000 | 4.21874 | −2.41964 | ||||||||||||||||||
1.7 | −1.00000 | −2.59616 | 1.00000 | 0.628601 | 2.59616 | 3.52393 | −1.00000 | 3.74004 | −0.628601 | ||||||||||||||||||
1.8 | −1.00000 | −2.51732 | 1.00000 | 1.76331 | 2.51732 | 1.11201 | −1.00000 | 3.33688 | −1.76331 | ||||||||||||||||||
1.9 | −1.00000 | −2.34064 | 1.00000 | 0.439606 | 2.34064 | −1.53111 | −1.00000 | 2.47858 | −0.439606 | ||||||||||||||||||
1.10 | −1.00000 | −2.07899 | 1.00000 | −2.29890 | 2.07899 | −3.17260 | −1.00000 | 1.32220 | 2.29890 | ||||||||||||||||||
1.11 | −1.00000 | −2.05889 | 1.00000 | −2.17707 | 2.05889 | −2.39868 | −1.00000 | 1.23901 | 2.17707 | ||||||||||||||||||
1.12 | −1.00000 | −1.88890 | 1.00000 | −2.81426 | 1.88890 | 4.34333 | −1.00000 | 0.567942 | 2.81426 | ||||||||||||||||||
1.13 | −1.00000 | −1.82825 | 1.00000 | −1.60206 | 1.82825 | 1.75337 | −1.00000 | 0.342490 | 1.60206 | ||||||||||||||||||
1.14 | −1.00000 | −1.81717 | 1.00000 | 2.05072 | 1.81717 | 1.90646 | −1.00000 | 0.302089 | −2.05072 | ||||||||||||||||||
1.15 | −1.00000 | −1.56981 | 1.00000 | 3.21985 | 1.56981 | −0.392480 | −1.00000 | −0.535687 | −3.21985 | ||||||||||||||||||
1.16 | −1.00000 | −1.49688 | 1.00000 | 2.53302 | 1.49688 | −3.95759 | −1.00000 | −0.759363 | −2.53302 | ||||||||||||||||||
1.17 | −1.00000 | −1.42034 | 1.00000 | 1.09471 | 1.42034 | 4.85520 | −1.00000 | −0.982644 | −1.09471 | ||||||||||||||||||
1.18 | −1.00000 | −1.39612 | 1.00000 | −2.68054 | 1.39612 | −4.55830 | −1.00000 | −1.05084 | 2.68054 | ||||||||||||||||||
1.19 | −1.00000 | −0.961298 | 1.00000 | 0.560610 | 0.961298 | −3.48762 | −1.00000 | −2.07591 | −0.560610 | ||||||||||||||||||
1.20 | −1.00000 | −0.859002 | 1.00000 | 0.293314 | 0.859002 | 1.77335 | −1.00000 | −2.26211 | −0.293314 | ||||||||||||||||||
See all 55 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(3023\) | \(1\) |
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 | 6046.2.a.d | ✓ | 55 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6046.2.a.d | ✓ | 55 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):
\( T_{3}^{55} + 4 T_{3}^{54} - 89 T_{3}^{53} - 369 T_{3}^{52} + 3668 T_{3}^{51} + 15881 T_{3}^{50} + \cdots + 14201 \) |
\( T_{11}^{55} + 28 T_{11}^{54} + 98 T_{11}^{53} - 4338 T_{11}^{52} - 43087 T_{11}^{51} + \cdots - 12\!\cdots\!39 \) |