Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [429,2,Mod(166,429)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(429, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("429.166");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 429.s (of order \(6\), degree \(2\), 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: | \(3.42558224671\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
166.1 | −2.14836 | + | 1.24036i | −0.500000 | − | 0.866025i | 2.07696 | − | 3.59741i | − | 0.0418579i | 2.14836 | + | 1.24036i | −3.59366 | − | 2.07480i | 5.34327i | −0.500000 | + | 0.866025i | 0.0519187 | + | 0.0899258i | |||
166.2 | −2.12238 | + | 1.22536i | −0.500000 | − | 0.866025i | 2.00301 | − | 3.46932i | 4.42067i | 2.12238 | + | 1.22536i | 0.902680 | + | 0.521163i | 4.91619i | −0.500000 | + | 0.866025i | −5.41690 | − | 9.38235i | ||||
166.3 | −2.08148 | + | 1.20174i | −0.500000 | − | 0.866025i | 1.88836 | − | 3.27074i | − | 3.20365i | 2.08148 | + | 1.20174i | 2.13033 | + | 1.22995i | 4.27033i | −0.500000 | + | 0.866025i | 3.84996 | + | 6.66833i | |||
166.4 | −1.57303 | + | 0.908192i | −0.500000 | − | 0.866025i | 0.649624 | − | 1.12518i | − | 0.705953i | 1.57303 | + | 0.908192i | 1.35880 | + | 0.784504i | − | 1.27283i | −0.500000 | + | 0.866025i | 0.641141 | + | 1.11049i | ||
166.5 | −0.945010 | + | 0.545602i | −0.500000 | − | 0.866025i | −0.404638 | + | 0.700853i | 1.08345i | 0.945010 | + | 0.545602i | −2.79805 | − | 1.61546i | − | 3.06549i | −0.500000 | + | 0.866025i | −0.591129 | − | 1.02387i | |||
166.6 | −0.489701 | + | 0.282729i | −0.500000 | − | 0.866025i | −0.840128 | + | 1.45515i | − | 1.14515i | 0.489701 | + | 0.282729i | 2.75529 | + | 1.59077i | − | 2.08103i | −0.500000 | + | 0.866025i | 0.323767 | + | 0.560780i | ||
166.7 | −0.458122 | + | 0.264497i | −0.500000 | − | 0.866025i | −0.860083 | + | 1.48971i | 3.29610i | 0.458122 | + | 0.264497i | 0.609605 | + | 0.351955i | − | 1.96794i | −0.500000 | + | 0.866025i | −0.871809 | − | 1.51002i | |||
166.8 | 0.252478 | − | 0.145768i | −0.500000 | − | 0.866025i | −0.957503 | + | 1.65844i | − | 3.62473i | −0.252478 | − | 0.145768i | −1.17162 | − | 0.676437i | 1.14137i | −0.500000 | + | 0.866025i | −0.528371 | − | 0.915165i | |||
166.9 | 0.813989 | − | 0.469957i | −0.500000 | − | 0.866025i | −0.558281 | + | 0.966972i | − | 0.163878i | −0.813989 | − | 0.469957i | 2.40248 | + | 1.38708i | 2.92930i | −0.500000 | + | 0.866025i | −0.0770155 | − | 0.133395i | |||
166.10 | 0.962812 | − | 0.555880i | −0.500000 | − | 0.866025i | −0.381995 | + | 0.661635i | 2.79143i | −0.962812 | − | 0.555880i | −3.71222 | − | 2.14325i | 3.07289i | −0.500000 | + | 0.866025i | 1.55170 | + | 2.68762i | ||||
166.11 | 1.56752 | − | 0.905006i | −0.500000 | − | 0.866025i | 0.638073 | − | 1.10517i | 2.99723i | −1.56752 | − | 0.905006i | 3.93315 | + | 2.27081i | 1.31019i | −0.500000 | + | 0.866025i | 2.71252 | + | 4.69821i | ||||
166.12 | 1.80882 | − | 1.04432i | −0.500000 | − | 0.866025i | 1.18121 | − | 2.04591i | − | 2.92121i | −1.80882 | − | 1.04432i | 0.684943 | + | 0.395452i | − | 0.756961i | −0.500000 | + | 0.866025i | −3.05068 | − | 5.28394i | ||
166.13 | 1.96842 | − | 1.13647i | −0.500000 | − | 0.866025i | 1.58313 | − | 2.74206i | − | 1.61338i | −1.96842 | − | 1.13647i | −2.77524 | − | 1.60229i | − | 2.65083i | −0.500000 | + | 0.866025i | −1.83356 | − | 3.17581i | ||
166.14 | 2.44405 | − | 1.41107i | −0.500000 | − | 0.866025i | 2.98226 | − | 5.16542i | 2.29504i | −2.44405 | − | 1.41107i | −0.726492 | − | 0.419441i | − | 11.1884i | −0.500000 | + | 0.866025i | 3.23847 | + | 5.60919i | |||
199.1 | −2.14836 | − | 1.24036i | −0.500000 | + | 0.866025i | 2.07696 | + | 3.59741i | 0.0418579i | 2.14836 | − | 1.24036i | −3.59366 | + | 2.07480i | − | 5.34327i | −0.500000 | − | 0.866025i | 0.0519187 | − | 0.0899258i | |||
199.2 | −2.12238 | − | 1.22536i | −0.500000 | + | 0.866025i | 2.00301 | + | 3.46932i | − | 4.42067i | 2.12238 | − | 1.22536i | 0.902680 | − | 0.521163i | − | 4.91619i | −0.500000 | − | 0.866025i | −5.41690 | + | 9.38235i | ||
199.3 | −2.08148 | − | 1.20174i | −0.500000 | + | 0.866025i | 1.88836 | + | 3.27074i | 3.20365i | 2.08148 | − | 1.20174i | 2.13033 | − | 1.22995i | − | 4.27033i | −0.500000 | − | 0.866025i | 3.84996 | − | 6.66833i | |||
199.4 | −1.57303 | − | 0.908192i | −0.500000 | + | 0.866025i | 0.649624 | + | 1.12518i | 0.705953i | 1.57303 | − | 0.908192i | 1.35880 | − | 0.784504i | 1.27283i | −0.500000 | − | 0.866025i | 0.641141 | − | 1.11049i | ||||
199.5 | −0.945010 | − | 0.545602i | −0.500000 | + | 0.866025i | −0.404638 | − | 0.700853i | − | 1.08345i | 0.945010 | − | 0.545602i | −2.79805 | + | 1.61546i | 3.06549i | −0.500000 | − | 0.866025i | −0.591129 | + | 1.02387i | |||
199.6 | −0.489701 | − | 0.282729i | −0.500000 | + | 0.866025i | −0.840128 | − | 1.45515i | 1.14515i | 0.489701 | − | 0.282729i | 2.75529 | − | 1.59077i | 2.08103i | −0.500000 | − | 0.866025i | 0.323767 | − | 0.560780i | ||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 429.2.s.b | ✓ | 28 |
13.e | even | 6 | 1 | inner | 429.2.s.b | ✓ | 28 |
13.f | odd | 12 | 1 | 5577.2.a.bf | 14 | ||
13.f | odd | 12 | 1 | 5577.2.a.bg | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
429.2.s.b | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
429.2.s.b | ✓ | 28 | 13.e | even | 6 | 1 | inner |
5577.2.a.bf | 14 | 13.f | odd | 12 | 1 | ||
5577.2.a.bg | 14 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 23 T_{2}^{26} + 325 T_{2}^{24} + 6 T_{2}^{23} - 2938 T_{2}^{22} - 54 T_{2}^{21} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).