Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(101,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.bn (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: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(34\) |
Relative dimension: | \(17\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 | 0.500000 | − | 0.866025i | −1.71486 | + | 0.243407i | −0.500000 | − | 0.866025i | −2.88000 | + | 1.66277i | −0.646635 | + | 1.60682i | −0.187179 | − | 2.63912i | −1.00000 | 2.88151 | − | 0.834819i | 3.32554i | ||||
101.2 | 0.500000 | − | 0.866025i | −1.70046 | + | 0.329307i | −0.500000 | − | 0.866025i | 1.26448 | − | 0.730045i | −0.565041 | + | 1.63729i | −2.63261 | + | 0.263392i | −1.00000 | 2.78311 | − | 1.11994i | − | 1.46009i | |||
101.3 | 0.500000 | − | 0.866025i | −1.60671 | − | 0.646913i | −0.500000 | − | 0.866025i | 2.60318 | − | 1.50295i | −1.36360 | + | 1.06799i | −0.916975 | − | 2.48177i | −1.00000 | 2.16301 | + | 2.07880i | − | 3.00590i | |||
101.4 | 0.500000 | − | 0.866025i | −1.57643 | − | 0.717543i | −0.500000 | − | 0.866025i | −1.41302 | + | 0.815806i | −1.40963 | + | 1.00646i | 1.06124 | + | 2.42359i | −1.00000 | 1.97026 | + | 2.26231i | 1.63161i | ||||
101.5 | 0.500000 | − | 0.866025i | −1.51226 | + | 0.844435i | −0.500000 | − | 0.866025i | −0.511132 | + | 0.295102i | −0.0248275 | + | 1.73187i | 1.57814 | + | 2.12355i | −1.00000 | 1.57386 | − | 2.55401i | 0.590205i | ||||
101.6 | 0.500000 | − | 0.866025i | −0.925467 | + | 1.46407i | −0.500000 | − | 0.866025i | 2.64914 | − | 1.52948i | 0.805192 | + | 1.53351i | 2.61668 | − | 0.391106i | −1.00000 | −1.28702 | − | 2.70990i | − | 3.05896i | |||
101.7 | 0.500000 | − | 0.866025i | −0.801045 | + | 1.53568i | −0.500000 | − | 0.866025i | 1.09866 | − | 0.634311i | 0.929420 | + | 1.46157i | −2.21179 | + | 1.45190i | −1.00000 | −1.71665 | − | 2.46030i | − | 1.26862i | |||
101.8 | 0.500000 | − | 0.866025i | −0.458130 | + | 1.67036i | −0.500000 | − | 0.866025i | −1.57344 | + | 0.908426i | 1.21751 | + | 1.23193i | −0.414054 | − | 2.61315i | −1.00000 | −2.58023 | − | 1.53049i | 1.81685i | ||||
101.9 | 0.500000 | − | 0.866025i | −0.428009 | − | 1.67834i | −0.500000 | − | 0.866025i | −1.58996 | + | 0.917964i | −1.66749 | − | 0.468501i | 2.08732 | − | 1.62576i | −1.00000 | −2.63362 | + | 1.43668i | 1.83593i | ||||
101.10 | 0.500000 | − | 0.866025i | 0.320103 | − | 1.70221i | −0.500000 | − | 0.866025i | 1.62172 | − | 0.936303i | −1.31411 | − | 1.12832i | 2.47749 | + | 0.928453i | −1.00000 | −2.79507 | − | 1.08977i | − | 1.87261i | |||
101.11 | 0.500000 | − | 0.866025i | 0.786858 | + | 1.54300i | −0.500000 | − | 0.866025i | −1.98183 | + | 1.14421i | 1.72971 | + | 0.0900624i | −2.60041 | + | 0.487738i | −1.00000 | −1.76171 | + | 2.42825i | 2.28842i | ||||
101.12 | 0.500000 | − | 0.866025i | 0.991815 | − | 1.41997i | −0.500000 | − | 0.866025i | −3.72094 | + | 2.14828i | −0.733819 | − | 1.56892i | −1.08595 | + | 2.41262i | −1.00000 | −1.03261 | − | 2.81669i | 4.29657i | ||||
101.13 | 0.500000 | − | 0.866025i | 1.17271 | + | 1.27466i | −0.500000 | − | 0.866025i | −0.870413 | + | 0.502533i | 1.69024 | − | 0.378264i | 1.33597 | + | 2.28368i | −1.00000 | −0.249516 | + | 2.98961i | 1.00507i | ||||
101.14 | 0.500000 | − | 0.866025i | 1.23327 | − | 1.21616i | −0.500000 | − | 0.866025i | −0.567570 | + | 0.327687i | −0.436593 | − | 1.67612i | −2.19987 | − | 1.46989i | −1.00000 | 0.0419009 | − | 2.99971i | 0.655374i | ||||
101.15 | 0.500000 | − | 0.866025i | 1.25705 | + | 1.19156i | −0.500000 | − | 0.866025i | 1.80315 | − | 1.04105i | 1.66045 | − | 0.492861i | 1.78353 | − | 1.95424i | −1.00000 | 0.160371 | + | 2.99571i | − | 2.08210i | |||
101.16 | 0.500000 | − | 0.866025i | 1.72973 | − | 0.0896134i | −0.500000 | − | 0.866025i | −3.27919 | + | 1.89324i | 0.787258 | − | 1.54280i | 2.49160 | − | 0.889895i | −1.00000 | 2.98394 | − | 0.310014i | 3.78649i | ||||
101.17 | 0.500000 | − | 0.866025i | 1.73183 | − | 0.0276700i | −0.500000 | − | 0.866025i | 2.84717 | − | 1.64381i | 0.841952 | − | 1.51364i | −0.683149 | + | 2.55603i | −1.00000 | 2.99847 | − | 0.0958395i | − | 3.28763i | |||
173.1 | 0.500000 | + | 0.866025i | −1.71486 | − | 0.243407i | −0.500000 | + | 0.866025i | −2.88000 | − | 1.66277i | −0.646635 | − | 1.60682i | −0.187179 | + | 2.63912i | −1.00000 | 2.88151 | + | 0.834819i | − | 3.32554i | |||
173.2 | 0.500000 | + | 0.866025i | −1.70046 | − | 0.329307i | −0.500000 | + | 0.866025i | 1.26448 | + | 0.730045i | −0.565041 | − | 1.63729i | −2.63261 | − | 0.263392i | −1.00000 | 2.78311 | + | 1.11994i | 1.46009i | ||||
173.3 | 0.500000 | + | 0.866025i | −1.60671 | + | 0.646913i | −0.500000 | + | 0.866025i | 2.60318 | + | 1.50295i | −1.36360 | − | 1.06799i | −0.916975 | + | 2.48177i | −1.00000 | 2.16301 | − | 2.07880i | 3.00590i | ||||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
273.y | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.bn.f | yes | 34 |
3.b | odd | 2 | 1 | 546.2.bn.e | yes | 34 | |
7.d | odd | 6 | 1 | 546.2.bi.e | ✓ | 34 | |
13.e | even | 6 | 1 | 546.2.bi.f | yes | 34 | |
21.g | even | 6 | 1 | 546.2.bi.f | yes | 34 | |
39.h | odd | 6 | 1 | 546.2.bi.e | ✓ | 34 | |
91.p | odd | 6 | 1 | 546.2.bn.e | yes | 34 | |
273.y | even | 6 | 1 | inner | 546.2.bn.f | yes | 34 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.bi.e | ✓ | 34 | 7.d | odd | 6 | 1 | |
546.2.bi.e | ✓ | 34 | 39.h | odd | 6 | 1 | |
546.2.bi.f | yes | 34 | 13.e | even | 6 | 1 | |
546.2.bi.f | yes | 34 | 21.g | even | 6 | 1 | |
546.2.bn.e | yes | 34 | 3.b | odd | 2 | 1 | |
546.2.bn.e | yes | 34 | 91.p | odd | 6 | 1 | |
546.2.bn.f | yes | 34 | 1.a | even | 1 | 1 | trivial |
546.2.bn.f | yes | 34 | 273.y | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{34} + 9 T_{5}^{33} - 10 T_{5}^{32} - 333 T_{5}^{31} - 234 T_{5}^{30} + 7575 T_{5}^{29} + \cdots + 3270763083 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).