Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6003,2,Mod(1,6003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, 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("6003.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6003 = 3^{2} \cdot 23 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6003.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: | \(47.9341963334\) |
Analytic rank: | \(1\) |
Dimension: | \(22\) |
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 | −2.68779 | 0 | 5.22422 | −0.687677 | 0 | −3.46312 | −8.66603 | 0 | 1.84833 | ||||||||||||||||||
1.2 | −2.33095 | 0 | 3.43331 | 2.03702 | 0 | 4.28183 | −3.34096 | 0 | −4.74817 | ||||||||||||||||||
1.3 | −2.24483 | 0 | 3.03928 | 2.70888 | 0 | −1.81542 | −2.33301 | 0 | −6.08099 | ||||||||||||||||||
1.4 | −2.18238 | 0 | 2.76278 | 0.541552 | 0 | 0.936842 | −1.66467 | 0 | −1.18187 | ||||||||||||||||||
1.5 | −1.96070 | 0 | 1.84436 | −2.84151 | 0 | −2.21458 | 0.305158 | 0 | 5.57136 | ||||||||||||||||||
1.6 | −1.78641 | 0 | 1.19125 | −0.326188 | 0 | 1.17242 | 1.44475 | 0 | 0.582705 | ||||||||||||||||||
1.7 | −1.35991 | 0 | −0.150655 | 4.20064 | 0 | 0.870260 | 2.92469 | 0 | −5.71248 | ||||||||||||||||||
1.8 | −1.24737 | 0 | −0.444068 | −1.99313 | 0 | −4.26394 | 3.04866 | 0 | 2.48617 | ||||||||||||||||||
1.9 | −1.06922 | 0 | −0.856774 | 0.420836 | 0 | 1.13415 | 3.05451 | 0 | −0.449966 | ||||||||||||||||||
1.10 | −0.424349 | 0 | −1.81993 | −2.43761 | 0 | 3.41293 | 1.62098 | 0 | 1.03440 | ||||||||||||||||||
1.11 | −0.230149 | 0 | −1.94703 | 2.59479 | 0 | −4.77655 | 0.908406 | 0 | −0.597188 | ||||||||||||||||||
1.12 | 0.144466 | 0 | −1.97913 | −4.08805 | 0 | −0.828594 | −0.574849 | 0 | −0.590584 | ||||||||||||||||||
1.13 | 0.183492 | 0 | −1.96633 | 0.926834 | 0 | −0.385808 | −0.727789 | 0 | 0.170066 | ||||||||||||||||||
1.14 | 0.289055 | 0 | −1.91645 | −0.894239 | 0 | 3.31166 | −1.13207 | 0 | −0.258484 | ||||||||||||||||||
1.15 | 1.05077 | 0 | −0.895876 | 3.10634 | 0 | 3.07400 | −3.04291 | 0 | 3.26406 | ||||||||||||||||||
1.16 | 1.14621 | 0 | −0.686192 | 0.461250 | 0 | −3.03232 | −3.07895 | 0 | 0.528691 | ||||||||||||||||||
1.17 | 1.16689 | 0 | −0.638362 | −1.66410 | 0 | −0.970636 | −3.07868 | 0 | −1.94183 | ||||||||||||||||||
1.18 | 1.60427 | 0 | 0.573689 | −1.49095 | 0 | 2.64378 | −2.28819 | 0 | −2.39188 | ||||||||||||||||||
1.19 | 1.93234 | 0 | 1.73393 | 2.82056 | 0 | −3.97268 | −0.514145 | 0 | 5.45026 | ||||||||||||||||||
1.20 | 2.08661 | 0 | 2.35393 | 1.43041 | 0 | −0.274571 | 0.738503 | 0 | 2.98470 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(23\) | \(-1\) |
\(29\) | \(-1\) |
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 | 6003.2.a.t | ✓ | 22 |
3.b | odd | 2 | 1 | 6003.2.a.u | yes | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6003.2.a.t | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
6003.2.a.u | yes | 22 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):
\( T_{2}^{22} + 3 T_{2}^{21} - 26 T_{2}^{20} - 81 T_{2}^{19} + 279 T_{2}^{18} + 914 T_{2}^{17} - 1603 T_{2}^{16} + \cdots - 8 \) |
\( T_{5}^{22} - 55 T_{5}^{20} - 8 T_{5}^{19} + 1244 T_{5}^{18} + 356 T_{5}^{17} - 15211 T_{5}^{16} + \cdots - 11776 \) |