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: | \(123\) |
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 | −2.75439 | −0.861643 | 5.58668 | 3.86069 | 2.37330 | 2.91861 | −9.87912 | −2.25757 | −10.6339 | ||||||||||||||||||
1.2 | −2.74057 | −2.54852 | 5.51070 | −0.432732 | 6.98439 | 3.36424 | −9.62130 | 3.49496 | 1.18593 | ||||||||||||||||||
1.3 | −2.71192 | −2.81974 | 5.35451 | 4.11778 | 7.64691 | −3.85646 | −9.09715 | 4.95095 | −11.1671 | ||||||||||||||||||
1.4 | −2.69112 | 1.73469 | 5.24211 | 2.53592 | −4.66825 | −4.16450 | −8.72491 | 0.00913932 | −6.82446 | ||||||||||||||||||
1.5 | −2.68053 | 2.57455 | 5.18522 | 0.534092 | −6.90114 | −0.399106 | −8.53805 | 3.62830 | −1.43165 | ||||||||||||||||||
1.6 | −2.65517 | −0.380089 | 5.04992 | 1.30385 | 1.00920 | 0.922271 | −8.09806 | −2.85553 | −3.46193 | ||||||||||||||||||
1.7 | −2.60439 | −3.42443 | 4.78284 | −1.92258 | 8.91854 | −3.26289 | −7.24759 | 8.72671 | 5.00714 | ||||||||||||||||||
1.8 | −2.52129 | 0.498790 | 4.35689 | −0.537697 | −1.25759 | −2.34122 | −5.94240 | −2.75121 | 1.35569 | ||||||||||||||||||
1.9 | −2.52093 | 1.18363 | 4.35509 | −0.839879 | −2.98385 | 2.35424 | −5.93701 | −1.59902 | 2.11728 | ||||||||||||||||||
1.10 | −2.49040 | −2.13188 | 4.20211 | −1.49942 | 5.30925 | −0.951507 | −5.48415 | 1.54493 | 3.73417 | ||||||||||||||||||
1.11 | −2.47080 | 2.36731 | 4.10488 | −1.23232 | −5.84916 | 4.18566 | −5.20074 | 2.60415 | 3.04483 | ||||||||||||||||||
1.12 | −2.26767 | −1.90418 | 3.14232 | −3.16055 | 4.31804 | −3.88251 | −2.59040 | 0.625886 | 7.16707 | ||||||||||||||||||
1.13 | −2.26442 | 2.63903 | 3.12758 | 3.87429 | −5.97587 | 0.802453 | −2.55330 | 3.96450 | −8.77301 | ||||||||||||||||||
1.14 | −2.25898 | −0.131578 | 3.10301 | 3.21477 | 0.297232 | 0.192917 | −2.49169 | −2.98269 | −7.26212 | ||||||||||||||||||
1.15 | −2.22237 | 3.41646 | 2.93895 | 1.79804 | −7.59266 | 3.28634 | −2.08670 | 8.67221 | −3.99592 | ||||||||||||||||||
1.16 | −2.19076 | 0.906147 | 2.79943 | −2.56920 | −1.98515 | −1.27799 | −1.75137 | −2.17890 | 5.62849 | ||||||||||||||||||
1.17 | −2.17075 | −1.42264 | 2.71217 | −2.44924 | 3.08820 | 1.96298 | −1.54596 | −0.976102 | 5.31670 | ||||||||||||||||||
1.18 | −2.16481 | 1.72504 | 2.68638 | −0.175994 | −3.73438 | −2.81780 | −1.48588 | −0.0242310 | 0.380992 | ||||||||||||||||||
1.19 | −2.14591 | −1.26575 | 2.60492 | 3.62169 | 2.71619 | 0.00950737 | −1.29811 | −1.39787 | −7.77182 | ||||||||||||||||||
1.20 | −2.11210 | −0.313775 | 2.46095 | −4.01398 | 0.662722 | 2.82593 | −0.973578 | −2.90155 | 8.47792 | ||||||||||||||||||
See next 80 embeddings (of 123 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.d | ✓ | 123 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6019.2.a.d | ✓ | 123 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{123} - 10 T_{2}^{122} - 141 T_{2}^{121} + 1720 T_{2}^{120} + 8849 T_{2}^{119} - 143302 T_{2}^{118} + \cdots + 586577920 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6019))\).