Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6021,2,Mod(1,6021)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6021, 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("6021.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6021 = 3^{3} \cdot 223 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6021.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: | \(48.0779270570\) |
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 | −2.62208 | 0 | 4.87530 | −2.85914 | 0 | −3.15839 | −7.53926 | 0 | 7.49690 | ||||||||||||||||||
1.2 | −2.61208 | 0 | 4.82295 | −0.781189 | 0 | −0.254267 | −7.37378 | 0 | 2.04053 | ||||||||||||||||||
1.3 | −2.48811 | 0 | 4.19069 | 2.67155 | 0 | 3.81403 | −5.45068 | 0 | −6.64711 | ||||||||||||||||||
1.4 | −2.32514 | 0 | 3.40627 | −3.10806 | 0 | 4.20004 | −3.26977 | 0 | 7.22668 | ||||||||||||||||||
1.5 | −2.02272 | 0 | 2.09142 | 0.405475 | 0 | −4.34576 | −0.184909 | 0 | −0.820164 | ||||||||||||||||||
1.6 | −2.01253 | 0 | 2.05027 | −3.97367 | 0 | 1.72063 | −0.101176 | 0 | 7.99712 | ||||||||||||||||||
1.7 | −1.75221 | 0 | 1.07024 | 1.33036 | 0 | −3.64016 | 1.62913 | 0 | −2.33106 | ||||||||||||||||||
1.8 | −1.71011 | 0 | 0.924479 | 3.76518 | 0 | −2.83710 | 1.83926 | 0 | −6.43887 | ||||||||||||||||||
1.9 | −1.44814 | 0 | 0.0971206 | 1.39837 | 0 | 2.53252 | 2.75564 | 0 | −2.02504 | ||||||||||||||||||
1.10 | −1.31983 | 0 | −0.258046 | 0.212807 | 0 | 4.34948 | 2.98024 | 0 | −0.280869 | ||||||||||||||||||
1.11 | −0.869700 | 0 | −1.24362 | −3.21655 | 0 | 1.50813 | 2.82098 | 0 | 2.79743 | ||||||||||||||||||
1.12 | −0.795724 | 0 | −1.36682 | 0.986468 | 0 | −3.41517 | 2.67906 | 0 | −0.784957 | ||||||||||||||||||
1.13 | −0.556736 | 0 | −1.69004 | −4.00675 | 0 | 4.91682 | 2.05438 | 0 | 2.23070 | ||||||||||||||||||
1.14 | −0.125963 | 0 | −1.98413 | 1.92011 | 0 | 2.00612 | 0.501853 | 0 | −0.241862 | ||||||||||||||||||
1.15 | −0.118028 | 0 | −1.98607 | 1.85904 | 0 | −0.396926 | 0.470468 | 0 | −0.219419 | ||||||||||||||||||
1.16 | 0.118028 | 0 | −1.98607 | −1.85904 | 0 | −0.396926 | −0.470468 | 0 | −0.219419 | ||||||||||||||||||
1.17 | 0.125963 | 0 | −1.98413 | −1.92011 | 0 | 2.00612 | −0.501853 | 0 | −0.241862 | ||||||||||||||||||
1.18 | 0.556736 | 0 | −1.69004 | 4.00675 | 0 | 4.91682 | −2.05438 | 0 | 2.23070 | ||||||||||||||||||
1.19 | 0.795724 | 0 | −1.36682 | −0.986468 | 0 | −3.41517 | −2.67906 | 0 | −0.784957 | ||||||||||||||||||
1.20 | 0.869700 | 0 | −1.24362 | 3.21655 | 0 | 1.50813 | −2.82098 | 0 | 2.79743 | ||||||||||||||||||
See all 30 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(223\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6021.2.a.n | ✓ | 30 |
3.b | odd | 2 | 1 | inner | 6021.2.a.n | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6021.2.a.n | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
6021.2.a.n | ✓ | 30 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):
\( T_{2}^{30} - 45 T_{2}^{28} + 902 T_{2}^{26} - 10629 T_{2}^{24} + 81847 T_{2}^{22} - 433021 T_{2}^{20} + 1610983 T_{2}^{18} - 4240555 T_{2}^{16} + 7837897 T_{2}^{14} - 9955497 T_{2}^{12} + 8359905 T_{2}^{10} + \cdots - 28 \) |
\( T_{5}^{30} - 94 T_{5}^{28} + 3888 T_{5}^{26} - 93260 T_{5}^{24} + 1439928 T_{5}^{22} - 15026061 T_{5}^{20} + 108358761 T_{5}^{18} - 543324836 T_{5}^{16} + 1883425344 T_{5}^{14} - 4440409747 T_{5}^{12} + \cdots - 4085872 \) |