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: | \(0\) |
Dimension: | \(30\) |
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 | −2.64113 | 0 | 4.97557 | −1.42245 | 0 | 2.00224 | −7.85888 | 0 | 3.75688 | ||||||||||||||||||
1.2 | −2.60906 | 0 | 4.80718 | 3.69173 | 0 | 1.51238 | −7.32410 | 0 | −9.63193 | ||||||||||||||||||
1.3 | −2.60849 | 0 | 4.80424 | −1.19758 | 0 | −4.49655 | −7.31485 | 0 | 3.12388 | ||||||||||||||||||
1.4 | −2.55489 | 0 | 4.52748 | −3.88186 | 0 | 2.95370 | −6.45744 | 0 | 9.91773 | ||||||||||||||||||
1.5 | −2.23657 | 0 | 3.00224 | −0.453557 | 0 | 2.23602 | −2.24159 | 0 | 1.01441 | ||||||||||||||||||
1.6 | −1.79065 | 0 | 1.20642 | 3.95479 | 0 | −4.84424 | 1.42102 | 0 | −7.08163 | ||||||||||||||||||
1.7 | −1.59749 | 0 | 0.551985 | 2.02263 | 0 | −1.90895 | 2.31319 | 0 | −3.23113 | ||||||||||||||||||
1.8 | −1.59580 | 0 | 0.546582 | 1.23722 | 0 | 2.46504 | 2.31937 | 0 | −1.97436 | ||||||||||||||||||
1.9 | −1.56095 | 0 | 0.436554 | −3.73116 | 0 | −1.31733 | 2.44046 | 0 | 5.82414 | ||||||||||||||||||
1.10 | −1.20001 | 0 | −0.559985 | 3.40098 | 0 | 4.71373 | 3.07200 | 0 | −4.08120 | ||||||||||||||||||
1.11 | −1.12036 | 0 | −0.744798 | −4.16877 | 0 | 2.81319 | 3.07516 | 0 | 4.67051 | ||||||||||||||||||
1.12 | −0.619961 | 0 | −1.61565 | −0.522930 | 0 | −1.16804 | 2.24156 | 0 | 0.324196 | ||||||||||||||||||
1.13 | −0.534050 | 0 | −1.71479 | −2.64563 | 0 | −5.04348 | 1.98388 | 0 | 1.41290 | ||||||||||||||||||
1.14 | −0.480943 | 0 | −1.76869 | −1.51137 | 0 | 3.82960 | 1.81253 | 0 | 0.726880 | ||||||||||||||||||
1.15 | −0.395940 | 0 | −1.84323 | 0.579308 | 0 | −2.59556 | 1.52169 | 0 | −0.229371 | ||||||||||||||||||
1.16 | 0.0795257 | 0 | −1.99368 | 3.50033 | 0 | 0.586662 | −0.317600 | 0 | 0.278366 | ||||||||||||||||||
1.17 | 0.473233 | 0 | −1.77605 | −1.04483 | 0 | 3.83278 | −1.78695 | 0 | −0.494446 | ||||||||||||||||||
1.18 | 0.784780 | 0 | −1.38412 | 2.15464 | 0 | −2.00724 | −2.65579 | 0 | 1.69092 | ||||||||||||||||||
1.19 | 0.809520 | 0 | −1.34468 | 2.87436 | 0 | 0.883619 | −2.70758 | 0 | 2.32685 | ||||||||||||||||||
1.20 | 0.959410 | 0 | −1.07953 | −2.76105 | 0 | 2.95320 | −2.95453 | 0 | −2.64898 | ||||||||||||||||||
See all 30 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.w | yes | 30 |
3.b | odd | 2 | 1 | 6003.2.a.v | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6003.2.a.v | ✓ | 30 | 3.b | odd | 2 | 1 | |
6003.2.a.w | yes | 30 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):
\( T_{2}^{30} - T_{2}^{29} - 48 T_{2}^{28} + 45 T_{2}^{27} + 1028 T_{2}^{26} - 893 T_{2}^{25} - 12975 T_{2}^{24} + \cdots - 2304 \) |
\( T_{5}^{30} - 103 T_{5}^{28} - 12 T_{5}^{27} + 4692 T_{5}^{26} + 1088 T_{5}^{25} - 124647 T_{5}^{24} + \cdots + 1081344 \) |