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: | \(1\) |
Dimension: | \(35\) |
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 | −2.76954 | 0 | 5.67037 | 0.397024 | 0 | 1.86856 | −10.1653 | 0 | −1.09958 | ||||||||||||||||||
1.2 | −2.71512 | 0 | 5.37185 | 2.73826 | 0 | 0.227565 | −9.15497 | 0 | −7.43468 | ||||||||||||||||||
1.3 | −2.57522 | 0 | 4.63175 | −3.44083 | 0 | −2.38378 | −6.77735 | 0 | 8.86089 | ||||||||||||||||||
1.4 | −2.51287 | 0 | 4.31451 | −2.91287 | 0 | 3.68266 | −5.81606 | 0 | 7.31966 | ||||||||||||||||||
1.5 | −2.50224 | 0 | 4.26123 | −0.804329 | 0 | 3.78595 | −5.65814 | 0 | 2.01263 | ||||||||||||||||||
1.6 | −2.29192 | 0 | 3.25289 | −2.24227 | 0 | −1.95732 | −2.87153 | 0 | 5.13910 | ||||||||||||||||||
1.7 | −1.77853 | 0 | 1.16318 | 0.209108 | 0 | 1.24629 | 1.48830 | 0 | −0.371906 | ||||||||||||||||||
1.8 | −1.74295 | 0 | 1.03787 | 3.36855 | 0 | −3.59458 | 1.67694 | 0 | −5.87121 | ||||||||||||||||||
1.9 | −1.72263 | 0 | 0.967467 | −3.66365 | 0 | 0.816918 | 1.77868 | 0 | 6.31112 | ||||||||||||||||||
1.10 | −1.58898 | 0 | 0.524854 | −1.96674 | 0 | 3.69318 | 2.34398 | 0 | 3.12510 | ||||||||||||||||||
1.11 | −1.32840 | 0 | −0.235355 | 0.0596677 | 0 | −3.41560 | 2.96944 | 0 | −0.0792625 | ||||||||||||||||||
1.12 | −1.29059 | 0 | −0.334371 | −0.472881 | 0 | −4.11430 | 3.01272 | 0 | 0.610296 | ||||||||||||||||||
1.13 | −1.20962 | 0 | −0.536819 | −3.36438 | 0 | −4.87159 | 3.06859 | 0 | 4.06962 | ||||||||||||||||||
1.14 | −1.17334 | 0 | −0.623271 | 4.01472 | 0 | 1.71701 | 3.07799 | 0 | −4.71064 | ||||||||||||||||||
1.15 | −0.890971 | 0 | −1.20617 | 1.02214 | 0 | 1.73110 | 2.85661 | 0 | −0.910699 | ||||||||||||||||||
1.16 | −0.444304 | 0 | −1.80259 | 1.53364 | 0 | −0.0368149 | 1.68951 | 0 | −0.681403 | ||||||||||||||||||
1.17 | −0.238471 | 0 | −1.94313 | 0.830062 | 0 | 1.83678 | 0.940324 | 0 | −0.197946 | ||||||||||||||||||
1.18 | −0.184073 | 0 | −1.96612 | −3.73214 | 0 | 4.37279 | 0.730057 | 0 | 0.686988 | ||||||||||||||||||
1.19 | 0.0361660 | 0 | −1.99869 | 2.14054 | 0 | 4.70710 | −0.144617 | 0 | 0.0774147 | ||||||||||||||||||
1.20 | 0.306610 | 0 | −1.90599 | −2.77514 | 0 | −1.39648 | −1.19761 | 0 | −0.850885 | ||||||||||||||||||
See all 35 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(223\) | \(1\) |
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 | 6021.2.a.p | ✓ | 35 |
3.b | odd | 2 | 1 | 6021.2.a.s | yes | 35 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6021.2.a.p | ✓ | 35 | 1.a | even | 1 | 1 | trivial |
6021.2.a.s | yes | 35 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):
\( T_{2}^{35} + 4 T_{2}^{34} - 45 T_{2}^{33} - 191 T_{2}^{32} + 900 T_{2}^{31} + 4116 T_{2}^{30} - 10519 T_{2}^{29} - 52982 T_{2}^{28} + 79265 T_{2}^{27} + 454465 T_{2}^{26} - 398432 T_{2}^{25} - 2744239 T_{2}^{24} + 1314611 T_{2}^{23} + \cdots - 336 \) |
\( T_{5}^{35} + 14 T_{5}^{34} - 3 T_{5}^{33} - 887 T_{5}^{32} - 2828 T_{5}^{31} + 21605 T_{5}^{30} + 117281 T_{5}^{29} - 229098 T_{5}^{28} - 2245013 T_{5}^{27} + 238516 T_{5}^{26} + 24887450 T_{5}^{25} + 20420308 T_{5}^{24} + \cdots + 81 \) |