Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,3,Mod(353,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.353");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 684.m (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: | \(18.6376500822\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(40\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
353.1 | 0 | −2.99548 | − | 0.164631i | 0 | 2.57048i | 0 | 1.88687 | − | 3.26816i | 0 | 8.94579 | + | 0.986299i | 0 | ||||||||||||
353.2 | 0 | −2.99460 | − | 0.179992i | 0 | − | 0.944063i | 0 | −5.01764 | + | 8.69081i | 0 | 8.93521 | + | 1.07801i | 0 | |||||||||||
353.3 | 0 | −2.88193 | + | 0.833354i | 0 | − | 7.26979i | 0 | 3.75776 | − | 6.50863i | 0 | 7.61104 | − | 4.80334i | 0 | |||||||||||
353.4 | 0 | −2.84492 | − | 0.952067i | 0 | − | 6.24143i | 0 | 4.14502 | − | 7.17938i | 0 | 7.18714 | + | 5.41711i | 0 | |||||||||||
353.5 | 0 | −2.80552 | + | 1.06257i | 0 | 5.14812i | 0 | 2.38165 | − | 4.12514i | 0 | 6.74191 | − | 5.96210i | 0 | ||||||||||||
353.6 | 0 | −2.74110 | − | 1.21916i | 0 | − | 1.38038i | 0 | −1.29258 | + | 2.23881i | 0 | 6.02728 | + | 6.68370i | 0 | |||||||||||
353.7 | 0 | −2.64568 | + | 1.41435i | 0 | − | 6.34957i | 0 | −3.73312 | + | 6.46596i | 0 | 4.99921 | − | 7.48384i | 0 | |||||||||||
353.8 | 0 | −2.55262 | − | 1.57611i | 0 | 9.08186i | 0 | 3.71777 | − | 6.43936i | 0 | 4.03172 | + | 8.04644i | 0 | ||||||||||||
353.9 | 0 | −2.38029 | + | 1.82598i | 0 | 5.89395i | 0 | −0.351537 | + | 0.608880i | 0 | 2.33159 | − | 8.69274i | 0 | ||||||||||||
353.10 | 0 | −2.11779 | − | 2.12485i | 0 | 8.55202i | 0 | −5.68709 | + | 9.85033i | 0 | −0.0299448 | + | 8.99995i | 0 | ||||||||||||
353.11 | 0 | −1.79577 | + | 2.40317i | 0 | − | 4.55087i | 0 | 5.85150 | − | 10.1351i | 0 | −2.55043 | − | 8.63107i | 0 | |||||||||||
353.12 | 0 | −1.70356 | − | 2.46939i | 0 | − | 9.28650i | 0 | −2.28067 | + | 3.95023i | 0 | −3.19575 | + | 8.41351i | 0 | |||||||||||
353.13 | 0 | −1.61882 | − | 2.52575i | 0 | − | 0.497329i | 0 | 5.73504 | − | 9.93338i | 0 | −3.75887 | + | 8.17746i | 0 | |||||||||||
353.14 | 0 | −1.60029 | − | 2.53754i | 0 | 1.56237i | 0 | −4.59594 | + | 7.96040i | 0 | −3.87817 | + | 8.12156i | 0 | ||||||||||||
353.15 | 0 | −1.55115 | + | 2.56787i | 0 | 5.60337i | 0 | −5.05578 | + | 8.75686i | 0 | −4.18789 | − | 7.96628i | 0 | ||||||||||||
353.16 | 0 | −1.21870 | + | 2.74131i | 0 | − | 2.92593i | 0 | −2.71227 | + | 4.69779i | 0 | −6.02956 | − | 6.68165i | 0 | |||||||||||
353.17 | 0 | −1.06149 | + | 2.80593i | 0 | − | 1.84834i | 0 | 2.46982 | − | 4.27785i | 0 | −6.74649 | − | 5.95692i | 0 | |||||||||||
353.18 | 0 | −1.03515 | − | 2.81575i | 0 | 0.329507i | 0 | 2.81673 | − | 4.87871i | 0 | −6.85692 | + | 5.82947i | 0 | ||||||||||||
353.19 | 0 | −0.358884 | − | 2.97846i | 0 | 0.408400i | 0 | −0.674294 | + | 1.16791i | 0 | −8.74240 | + | 2.13784i | 0 | ||||||||||||
353.20 | 0 | −0.0106609 | + | 2.99998i | 0 | 4.51007i | 0 | 1.48435 | − | 2.57096i | 0 | −8.99977 | − | 0.0639652i | 0 | ||||||||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.n | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 684.3.m.a | ✓ | 80 |
3.b | odd | 2 | 1 | 2052.3.m.a | 80 | ||
9.c | even | 3 | 1 | 2052.3.be.a | 80 | ||
9.d | odd | 6 | 1 | 684.3.be.a | yes | 80 | |
19.c | even | 3 | 1 | 684.3.be.a | yes | 80 | |
57.h | odd | 6 | 1 | 2052.3.be.a | 80 | ||
171.g | even | 3 | 1 | 2052.3.m.a | 80 | ||
171.n | odd | 6 | 1 | inner | 684.3.m.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
684.3.m.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
684.3.m.a | ✓ | 80 | 171.n | odd | 6 | 1 | inner |
684.3.be.a | yes | 80 | 9.d | odd | 6 | 1 | |
684.3.be.a | yes | 80 | 19.c | even | 3 | 1 | |
2052.3.m.a | 80 | 3.b | odd | 2 | 1 | ||
2052.3.m.a | 80 | 171.g | even | 3 | 1 | ||
2052.3.be.a | 80 | 9.c | even | 3 | 1 | ||
2052.3.be.a | 80 | 57.h | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).