Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [639,3,Mod(496,639)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(639, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("639.496");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 639 = 3^{2} \cdot 71 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 639.d (of order \(2\), degree \(1\), minimal) |
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: | no |
Analytic conductor: | \(17.4114888926\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
496.1 | −3.83989 | 0 | 10.7448 | −2.25297 | 0 | 11.1817i | −25.8992 | 0 | 8.65118 | ||||||||||||||||||
496.2 | −3.83989 | 0 | 10.7448 | −2.25297 | 0 | − | 11.1817i | −25.8992 | 0 | 8.65118 | |||||||||||||||||
496.3 | −3.14297 | 0 | 5.87828 | 6.38552 | 0 | − | 5.89299i | −5.90338 | 0 | −20.0695 | |||||||||||||||||
496.4 | −3.14297 | 0 | 5.87828 | 6.38552 | 0 | 5.89299i | −5.90338 | 0 | −20.0695 | ||||||||||||||||||
496.5 | −2.96982 | 0 | 4.81983 | −6.96665 | 0 | 1.70133i | −2.43476 | 0 | 20.6897 | ||||||||||||||||||
496.6 | −2.96982 | 0 | 4.81983 | −6.96665 | 0 | − | 1.70133i | −2.43476 | 0 | 20.6897 | |||||||||||||||||
496.7 | −2.02401 | 0 | 0.0966251 | −0.798722 | 0 | − | 4.78394i | 7.90048 | 0 | 1.61662 | |||||||||||||||||
496.8 | −2.02401 | 0 | 0.0966251 | −0.798722 | 0 | 4.78394i | 7.90048 | 0 | 1.61662 | ||||||||||||||||||
496.9 | −1.07799 | 0 | −2.83793 | 3.69839 | 0 | − | 9.66006i | 7.37125 | 0 | −3.98684 | |||||||||||||||||
496.10 | −1.07799 | 0 | −2.83793 | 3.69839 | 0 | 9.66006i | 7.37125 | 0 | −3.98684 | ||||||||||||||||||
496.11 | −0.546276 | 0 | −3.70158 | −7.50327 | 0 | − | 5.39869i | 4.20719 | 0 | 4.09886 | |||||||||||||||||
496.12 | −0.546276 | 0 | −3.70158 | −7.50327 | 0 | 5.39869i | 4.20719 | 0 | 4.09886 | ||||||||||||||||||
496.13 | 0.546276 | 0 | −3.70158 | 7.50327 | 0 | − | 5.39869i | −4.20719 | 0 | 4.09886 | |||||||||||||||||
496.14 | 0.546276 | 0 | −3.70158 | 7.50327 | 0 | 5.39869i | −4.20719 | 0 | 4.09886 | ||||||||||||||||||
496.15 | 1.07799 | 0 | −2.83793 | −3.69839 | 0 | − | 9.66006i | −7.37125 | 0 | −3.98684 | |||||||||||||||||
496.16 | 1.07799 | 0 | −2.83793 | −3.69839 | 0 | 9.66006i | −7.37125 | 0 | −3.98684 | ||||||||||||||||||
496.17 | 2.02401 | 0 | 0.0966251 | 0.798722 | 0 | − | 4.78394i | −7.90048 | 0 | 1.61662 | |||||||||||||||||
496.18 | 2.02401 | 0 | 0.0966251 | 0.798722 | 0 | 4.78394i | −7.90048 | 0 | 1.61662 | ||||||||||||||||||
496.19 | 2.96982 | 0 | 4.81983 | 6.96665 | 0 | 1.70133i | 2.43476 | 0 | 20.6897 | ||||||||||||||||||
496.20 | 2.96982 | 0 | 4.81983 | 6.96665 | 0 | − | 1.70133i | 2.43476 | 0 | 20.6897 | |||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
71.b | odd | 2 | 1 | inner |
213.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 639.3.d.d | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 639.3.d.d | ✓ | 24 |
71.b | odd | 2 | 1 | inner | 639.3.d.d | ✓ | 24 |
213.b | even | 2 | 1 | inner | 639.3.d.d | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
639.3.d.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
639.3.d.d | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
639.3.d.d | ✓ | 24 | 71.b | odd | 2 | 1 | inner |
639.3.d.d | ✓ | 24 | 213.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 39T_{2}^{10} + 555T_{2}^{8} - 3514T_{2}^{6} + 9483T_{2}^{4} - 8647T_{2}^{2} + 1825 \)
acting on \(S_{3}^{\mathrm{new}}(639, [\chi])\).