Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [644,2,Mod(5,644)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([0, 55, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("644.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 644 = 2^{2} \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 644.bc (of order \(66\), degree \(20\), 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: | \(5.14236589017\) |
Analytic rank: | \(0\) |
Dimension: | \(320\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{66})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{66}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | 0 | −0.583248 | + | 3.02618i | 0 | 0.619931 | + | 0.487519i | 0 | −2.04497 | − | 1.67872i | 0 | −6.03248 | − | 2.41504i | 0 | ||||||||||
5.2 | 0 | −0.558795 | + | 2.89930i | 0 | −1.41352 | − | 1.11160i | 0 | 2.23525 | + | 1.41551i | 0 | −5.30859 | − | 2.12524i | 0 | ||||||||||
5.3 | 0 | −0.409381 | + | 2.12407i | 0 | −2.83055 | − | 2.22597i | 0 | −1.98760 | − | 1.74627i | 0 | −1.55898 | − | 0.624120i | 0 | ||||||||||
5.4 | 0 | −0.401668 | + | 2.08405i | 0 | 0.909054 | + | 0.714888i | 0 | 2.64571 | + | 0.0144750i | 0 | −1.39684 | − | 0.559210i | 0 | ||||||||||
5.5 | 0 | −0.277301 | + | 1.43878i | 0 | 2.43721 | + | 1.91664i | 0 | −1.23059 | − | 2.34215i | 0 | 0.791924 | + | 0.317039i | 0 | ||||||||||
5.6 | 0 | −0.248197 | + | 1.28777i | 0 | −1.66823 | − | 1.31191i | 0 | −1.56473 | + | 2.13345i | 0 | 1.18836 | + | 0.475746i | 0 | ||||||||||
5.7 | 0 | −0.206510 | + | 1.07147i | 0 | 2.27508 | + | 1.78914i | 0 | −1.75579 | + | 1.97920i | 0 | 1.67969 | + | 0.672448i | 0 | ||||||||||
5.8 | 0 | −0.0944134 | + | 0.489863i | 0 | −0.697492 | − | 0.548514i | 0 | 1.89346 | − | 1.84792i | 0 | 2.55405 | + | 1.02249i | 0 | ||||||||||
5.9 | 0 | 0.0419878 | − | 0.217853i | 0 | 2.18268 | + | 1.71648i | 0 | 2.01215 | + | 1.71792i | 0 | 2.73941 | + | 1.09669i | 0 | ||||||||||
5.10 | 0 | 0.134558 | − | 0.698152i | 0 | −1.09153 | − | 0.858389i | 0 | −1.45127 | + | 2.21220i | 0 | 2.31579 | + | 0.927104i | 0 | ||||||||||
5.11 | 0 | 0.280147 | − | 1.45354i | 0 | −1.86247 | − | 1.46466i | 0 | −1.88051 | − | 1.86110i | 0 | 0.750811 | + | 0.300579i | 0 | ||||||||||
5.12 | 0 | 0.281126 | − | 1.45862i | 0 | −1.67022 | − | 1.31348i | 0 | 1.72584 | + | 2.00536i | 0 | 0.736559 | + | 0.294874i | 0 | ||||||||||
5.13 | 0 | 0.402820 | − | 2.09003i | 0 | 1.75200 | + | 1.37779i | 0 | 1.01319 | − | 2.44406i | 0 | −1.42084 | − | 0.568820i | 0 | ||||||||||
5.14 | 0 | 0.507936 | − | 2.63542i | 0 | 2.11879 | + | 1.66623i | 0 | 0.210927 | + | 2.63733i | 0 | −3.90236 | − | 1.56227i | 0 | ||||||||||
5.15 | 0 | 0.519798 | − | 2.69697i | 0 | −3.05496 | − | 2.40245i | 0 | 2.63650 | − | 0.221078i | 0 | −4.21833 | − | 1.68877i | 0 | ||||||||||
5.16 | 0 | 0.611142 | − | 3.17090i | 0 | −0.0636489 | − | 0.0500540i | 0 | −2.64575 | − | 0.00346821i | 0 | −6.89603 | − | 2.76075i | 0 | ||||||||||
17.1 | 0 | −2.08208 | − | 2.18362i | 0 | −3.79402 | − | 1.95596i | 0 | −1.98214 | + | 1.75246i | 0 | −0.290404 | + | 6.09632i | 0 | ||||||||||
17.2 | 0 | −1.98732 | − | 2.08424i | 0 | 0.595169 | + | 0.306831i | 0 | −0.315825 | − | 2.62683i | 0 | −0.251873 | + | 5.28747i | 0 | ||||||||||
17.3 | 0 | −1.59849 | − | 1.67645i | 0 | 2.22824 | + | 1.14874i | 0 | −2.60800 | − | 0.445329i | 0 | −0.112562 | + | 2.36298i | 0 | ||||||||||
17.4 | 0 | −1.52748 | − | 1.60197i | 0 | 0.141790 | + | 0.0730976i | 0 | 1.67447 | + | 2.04845i | 0 | −0.0903819 | + | 1.89735i | 0 | ||||||||||
See next 80 embeddings (of 320 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
23.d | odd | 22 | 1 | inner |
161.o | even | 66 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 644.2.bc.a | ✓ | 320 |
7.d | odd | 6 | 1 | inner | 644.2.bc.a | ✓ | 320 |
23.d | odd | 22 | 1 | inner | 644.2.bc.a | ✓ | 320 |
161.o | even | 66 | 1 | inner | 644.2.bc.a | ✓ | 320 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
644.2.bc.a | ✓ | 320 | 1.a | even | 1 | 1 | trivial |
644.2.bc.a | ✓ | 320 | 7.d | odd | 6 | 1 | inner |
644.2.bc.a | ✓ | 320 | 23.d | odd | 22 | 1 | inner |
644.2.bc.a | ✓ | 320 | 161.o | even | 66 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).