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 | −2.78139 | 0 | 5.73612 | −0.00312667 | 0 | 2.66771 | −10.3916 | 0 | 0.00869647 | ||||||||||||||||||
1.2 | −2.66815 | 0 | 5.11905 | −1.38302 | 0 | 0.919018 | −8.32211 | 0 | 3.69010 | ||||||||||||||||||
1.3 | −2.53344 | 0 | 4.41832 | −4.44002 | 0 | −2.94217 | −6.12666 | 0 | 11.2485 | ||||||||||||||||||
1.4 | −2.40738 | 0 | 3.79549 | 3.50117 | 0 | 5.09845 | −4.32244 | 0 | −8.42865 | ||||||||||||||||||
1.5 | −2.27733 | 0 | 3.18622 | 2.98462 | 0 | −2.44710 | −2.70140 | 0 | −6.79696 | ||||||||||||||||||
1.6 | −1.99322 | 0 | 1.97291 | 0.735105 | 0 | −3.83374 | 0.0540029 | 0 | −1.46522 | ||||||||||||||||||
1.7 | −1.96558 | 0 | 1.86349 | −2.31356 | 0 | 5.13967 | 0.268317 | 0 | 4.54748 | ||||||||||||||||||
1.8 | −1.89039 | 0 | 1.57357 | −2.79891 | 0 | 1.18730 | 0.806117 | 0 | 5.29102 | ||||||||||||||||||
1.9 | −1.58386 | 0 | 0.508607 | 2.90489 | 0 | 0.0968084 | 2.36216 | 0 | −4.60094 | ||||||||||||||||||
1.10 | −1.33909 | 0 | −0.206831 | 0.887651 | 0 | −3.28671 | 2.95515 | 0 | −1.18865 | ||||||||||||||||||
1.11 | −0.959410 | 0 | −1.07953 | 2.76105 | 0 | 2.95320 | 2.95453 | 0 | −2.64898 | ||||||||||||||||||
1.12 | −0.809520 | 0 | −1.34468 | −2.87436 | 0 | 0.883619 | 2.70758 | 0 | 2.32685 | ||||||||||||||||||
1.13 | −0.784780 | 0 | −1.38412 | −2.15464 | 0 | −2.00724 | 2.65579 | 0 | 1.69092 | ||||||||||||||||||
1.14 | −0.473233 | 0 | −1.77605 | 1.04483 | 0 | 3.83278 | 1.78695 | 0 | −0.494446 | ||||||||||||||||||
1.15 | −0.0795257 | 0 | −1.99368 | −3.50033 | 0 | 0.586662 | 0.317600 | 0 | 0.278366 | ||||||||||||||||||
1.16 | 0.395940 | 0 | −1.84323 | −0.579308 | 0 | −2.59556 | −1.52169 | 0 | −0.229371 | ||||||||||||||||||
1.17 | 0.480943 | 0 | −1.76869 | 1.51137 | 0 | 3.82960 | −1.81253 | 0 | 0.726880 | ||||||||||||||||||
1.18 | 0.534050 | 0 | −1.71479 | 2.64563 | 0 | −5.04348 | −1.98388 | 0 | 1.41290 | ||||||||||||||||||
1.19 | 0.619961 | 0 | −1.61565 | 0.522930 | 0 | −1.16804 | −2.24156 | 0 | 0.324196 | ||||||||||||||||||
1.20 | 1.12036 | 0 | −0.744798 | 4.16877 | 0 | 2.81319 | −3.07516 | 0 | 4.67051 | ||||||||||||||||||
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.v | ✓ | 30 |
3.b | odd | 2 | 1 | 6003.2.a.w | yes | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6003.2.a.v | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
6003.2.a.w | yes | 30 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):
\( T_{2}^{30} + T_{2}^{29} - 48 T_{2}^{28} - 45 T_{2}^{27} + 1028 T_{2}^{26} + 893 T_{2}^{25} - 12975 T_{2}^{24} + \cdots - 2304 \) |
\( T_{5}^{30} - 103 T_{5}^{28} + 12 T_{5}^{27} + 4692 T_{5}^{26} - 1088 T_{5}^{25} - 124647 T_{5}^{24} + \cdots + 1081344 \) |