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: | \(40\) |
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 | −2.70299 | 0 | 5.30613 | −3.42111 | 0 | −1.82484 | −8.93642 | 0 | 9.24720 | ||||||||||||||||||
1.2 | −2.69102 | 0 | 5.24161 | −2.40662 | 0 | 2.96509 | −8.72327 | 0 | 6.47626 | ||||||||||||||||||
1.3 | −2.65109 | 0 | 5.02828 | 1.58126 | 0 | −3.54711 | −8.02824 | 0 | −4.19207 | ||||||||||||||||||
1.4 | −2.63563 | 0 | 4.94652 | 3.15326 | 0 | 2.40096 | −7.76593 | 0 | −8.31081 | ||||||||||||||||||
1.5 | −2.32144 | 0 | 3.38909 | −1.08423 | 0 | 4.03069 | −3.22470 | 0 | 2.51697 | ||||||||||||||||||
1.6 | −2.16137 | 0 | 2.67154 | −3.69668 | 0 | 0.733541 | −1.45144 | 0 | 7.98990 | ||||||||||||||||||
1.7 | −2.08210 | 0 | 2.33515 | 0.145427 | 0 | 4.41479 | −0.697822 | 0 | −0.302795 | ||||||||||||||||||
1.8 | −2.00467 | 0 | 2.01871 | 4.10087 | 0 | 1.86508 | −0.0375024 | 0 | −8.22090 | ||||||||||||||||||
1.9 | −1.74618 | 0 | 1.04913 | −3.84450 | 0 | 0.0845192 | 1.66038 | 0 | 6.71318 | ||||||||||||||||||
1.10 | −1.69867 | 0 | 0.885473 | −0.550686 | 0 | 0.164494 | 1.89321 | 0 | 0.935432 | ||||||||||||||||||
1.11 | −1.47084 | 0 | 0.163365 | −1.43270 | 0 | −5.26244 | 2.70139 | 0 | 2.10726 | ||||||||||||||||||
1.12 | −1.37704 | 0 | −0.103765 | 2.60531 | 0 | −2.14080 | 2.89697 | 0 | −3.58761 | ||||||||||||||||||
1.13 | −1.35612 | 0 | −0.160951 | −1.12934 | 0 | −1.90121 | 2.93050 | 0 | 1.53151 | ||||||||||||||||||
1.14 | −1.23529 | 0 | −0.474050 | 1.76412 | 0 | −1.12531 | 3.05618 | 0 | −2.17920 | ||||||||||||||||||
1.15 | −1.01187 | 0 | −0.976110 | −1.54012 | 0 | 4.83269 | 3.01145 | 0 | 1.55841 | ||||||||||||||||||
1.16 | −0.968595 | 0 | −1.06182 | 2.59691 | 0 | 3.83305 | 2.96567 | 0 | −2.51535 | ||||||||||||||||||
1.17 | −0.538428 | 0 | −1.71010 | −1.52796 | 0 | 0.425996 | 1.99762 | 0 | 0.822698 | ||||||||||||||||||
1.18 | −0.536475 | 0 | −1.71219 | −2.69435 | 0 | −1.96571 | 1.99150 | 0 | 1.44545 | ||||||||||||||||||
1.19 | −0.323417 | 0 | −1.89540 | 3.36365 | 0 | 3.36190 | 1.25984 | 0 | −1.08786 | ||||||||||||||||||
1.20 | −0.243699 | 0 | −1.94061 | −0.214655 | 0 | −3.34538 | 0.960324 | 0 | 0.0523112 | ||||||||||||||||||
See all 40 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.t | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 6021.2.a.t | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6021.2.a.t | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
6021.2.a.t | ✓ | 40 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):
\( T_{2}^{40} - 63 T_{2}^{38} + 1823 T_{2}^{36} - 32146 T_{2}^{34} + 386488 T_{2}^{32} - 3358868 T_{2}^{30} + 21829705 T_{2}^{28} - 108281137 T_{2}^{26} + 414740672 T_{2}^{24} - 1233202486 T_{2}^{22} + \cdots + 57132 \) |
\( T_{5}^{40} - 120 T_{5}^{38} + 6568 T_{5}^{36} - 217444 T_{5}^{34} + 4869438 T_{5}^{32} - 78157240 T_{5}^{30} + 929658705 T_{5}^{28} - 8359000882 T_{5}^{26} + 57453294656 T_{5}^{24} + \cdots + 337334448 \) |