Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6019,2,Mod(1,6019)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, 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("6019.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6019 = 13 \cdot 463 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6019.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: | \(48.0619569766\) |
Analytic rank: | \(0\) |
Dimension: | \(130\) |
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 | −2.80453 | 3.24972 | 5.86540 | 2.53460 | −9.11395 | 2.92678 | −10.8406 | 7.56069 | −7.10835 | ||||||||||||||||||
1.2 | −2.76072 | 1.85217 | 5.62158 | −1.74367 | −5.11334 | −4.49860 | −9.99819 | 0.430551 | 4.81380 | ||||||||||||||||||
1.3 | −2.70229 | −1.73872 | 5.30239 | −0.220244 | 4.69853 | −4.26508 | −8.92402 | 0.0231491 | 0.595165 | ||||||||||||||||||
1.4 | −2.65690 | −1.17656 | 5.05912 | −3.97344 | 3.12601 | −1.98444 | −8.12779 | −1.61571 | 10.5570 | ||||||||||||||||||
1.5 | −2.57909 | 0.0546080 | 4.65172 | 2.10708 | −0.140839 | −1.58025 | −6.83904 | −2.99702 | −5.43435 | ||||||||||||||||||
1.6 | −2.55222 | −2.23917 | 4.51381 | 1.89858 | 5.71484 | −3.17598 | −6.41579 | 2.01387 | −4.84558 | ||||||||||||||||||
1.7 | −2.52323 | 3.28941 | 4.36670 | 3.66616 | −8.29994 | −5.12423 | −5.97175 | 7.82019 | −9.25058 | ||||||||||||||||||
1.8 | −2.52101 | −1.65021 | 4.35552 | 2.33988 | 4.16020 | 4.41588 | −5.93829 | −0.276814 | −5.89887 | ||||||||||||||||||
1.9 | −2.51967 | −3.32656 | 4.34873 | −0.493062 | 8.38182 | −0.741150 | −5.91802 | 8.06597 | 1.24235 | ||||||||||||||||||
1.10 | −2.50771 | −1.46096 | 4.28860 | 1.53494 | 3.66367 | 0.422474 | −5.73914 | −0.865591 | −3.84918 | ||||||||||||||||||
1.11 | −2.49834 | 2.94692 | 4.24169 | −1.72770 | −7.36239 | −2.26517 | −5.60049 | 5.68431 | 4.31639 | ||||||||||||||||||
1.12 | −2.49278 | −3.38832 | 4.21397 | 3.98474 | 8.44636 | 1.44276 | −5.51896 | 8.48073 | −9.93310 | ||||||||||||||||||
1.13 | −2.48853 | 0.814378 | 4.19279 | −1.02272 | −2.02661 | 0.518159 | −5.45682 | −2.33679 | 2.54508 | ||||||||||||||||||
1.14 | −2.40942 | 1.96845 | 3.80532 | 3.79888 | −4.74282 | 2.98166 | −4.34977 | 0.874791 | −9.15310 | ||||||||||||||||||
1.15 | −2.39514 | 1.83196 | 3.73671 | 3.10935 | −4.38781 | 4.25584 | −4.15967 | 0.356083 | −7.44733 | ||||||||||||||||||
1.16 | −2.34374 | 2.49578 | 3.49312 | −2.30500 | −5.84945 | 0.790647 | −3.49949 | 3.22890 | 5.40232 | ||||||||||||||||||
1.17 | −2.33535 | −1.90450 | 3.45386 | −1.81837 | 4.44767 | 2.51906 | −3.39528 | 0.627108 | 4.24653 | ||||||||||||||||||
1.18 | −2.28421 | 0.124993 | 3.21761 | −2.44620 | −0.285511 | −2.39008 | −2.78127 | −2.98438 | 5.58762 | ||||||||||||||||||
1.19 | −2.17988 | −0.326610 | 2.75186 | 0.291618 | 0.711970 | 2.70764 | −1.63895 | −2.89333 | −0.635691 | ||||||||||||||||||
1.20 | −2.07170 | 0.913939 | 2.29195 | −0.284252 | −1.89341 | −4.46498 | −0.604826 | −2.16472 | 0.588884 | ||||||||||||||||||
See next 80 embeddings (of 130 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(-1\) |
\(463\) | \(1\) |
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 | 6019.2.a.e | ✓ | 130 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6019.2.a.e | ✓ | 130 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{130} - 10 T_{2}^{129} - 153 T_{2}^{128} + 1842 T_{2}^{127} + 10563 T_{2}^{126} - 164654 T_{2}^{125} + \cdots - 8624128 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6019))\).