Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [667,2,Mod(10,667)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(667, base_ring=CyclotomicField(308))
chi = DirichletCharacter(H, H._module([42, 253]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("667.10");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 667 = 23 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 667.x (of order \(308\), degree \(120\), 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: | \(5.32602181482\) |
Analytic rank: | \(0\) |
Dimension: | \(6960\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{308})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{308}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −2.65729 | − | 0.809781i | −0.342570 | − | 0.438674i | 4.74534 | + | 3.18827i | 1.70314 | + | 1.30227i | 0.555077 | + | 1.44309i | −0.549665 | − | 3.33810i | −6.51978 | − | 8.00663i | 0.652004 | − | 2.61000i | −3.47119 | − | 4.83969i |
10.2 | −2.54833 | − | 0.776576i | 0.0554222 | + | 0.0709704i | 4.23079 | + | 2.84256i | −2.60951 | − | 1.99531i | −0.0861200 | − | 0.223895i | 0.0248235 | + | 0.150752i | −5.20967 | − | 6.39775i | 0.725120 | − | 2.90269i | 5.10038 | + | 7.11118i |
10.3 | −2.48306 | − | 0.756687i | 1.06978 | + | 1.36989i | 3.93292 | + | 2.64243i | 1.15510 | + | 0.883222i | −1.61974 | − | 4.21101i | 0.221686 | + | 1.34629i | −4.48807 | − | 5.51158i | −0.00509350 | + | 0.0203895i | −2.19986 | − | 3.06714i |
10.4 | −2.45646 | − | 0.748582i | 2.01386 | + | 2.57883i | 3.81374 | + | 2.56235i | −0.155463 | − | 0.118871i | −3.01651 | − | 7.84233i | −0.639256 | − | 3.88218i | −4.20716 | − | 5.16661i | −1.86763 | + | 7.47623i | 0.292903 | + | 0.408379i |
10.5 | −2.43184 | − | 0.741079i | −1.78101 | − | 2.28065i | 3.70457 | + | 2.48900i | −0.499891 | − | 0.382232i | 2.64099 | + | 6.86606i | 0.676915 | + | 4.11088i | −3.95387 | − | 4.85556i | −1.30230 | + | 5.21316i | 0.932394 | + | 1.29999i |
10.6 | −2.26057 | − | 0.688884i | −0.279898 | − | 0.358420i | 2.97550 | + | 1.99916i | 3.34647 | + | 2.55881i | 0.385818 | + | 1.00305i | 0.738312 | + | 4.48375i | −2.36473 | − | 2.90401i | 0.676963 | − | 2.70992i | −5.80220 | − | 8.08969i |
10.7 | −2.18547 | − | 0.666000i | −1.42722 | − | 1.82761i | 2.67263 | + | 1.79567i | 1.89766 | + | 1.45101i | 1.90195 | + | 4.94471i | −0.270958 | − | 1.64552i | −1.75979 | − | 2.16111i | −0.576116 | + | 2.30622i | −3.18091 | − | 4.43498i |
10.8 | −2.15111 | − | 0.655530i | 1.73583 | + | 2.22280i | 2.53747 | + | 1.70486i | −3.19826 | − | 2.44548i | −2.27686 | − | 5.91938i | 0.777456 | + | 4.72146i | −1.50091 | − | 1.84319i | −1.20064 | + | 4.80623i | 5.27673 | + | 7.35706i |
10.9 | −2.09159 | − | 0.637391i | −0.992867 | − | 1.27141i | 2.30839 | + | 1.55094i | −1.59950 | − | 1.22303i | 1.26629 | + | 3.29210i | 0.0668614 | + | 0.406047i | −1.07832 | − | 1.32424i | 0.0963981 | − | 0.385886i | 2.56596 | + | 3.57758i |
10.10 | −1.98974 | − | 0.606353i | 0.437664 | + | 0.560446i | 1.93131 | + | 1.29760i | −0.227511 | − | 0.173961i | −0.531010 | − | 1.38052i | 0.186489 | + | 1.13254i | −0.429147 | − | 0.527015i | 0.604535 | − | 2.41998i | 0.347206 | + | 0.484090i |
10.11 | −1.98560 | − | 0.605090i | −0.775169 | − | 0.992635i | 1.91637 | + | 1.28756i | −1.79715 | − | 1.37415i | 0.938541 | + | 2.44002i | −0.486271 | − | 2.95311i | −0.404658 | − | 0.496941i | 0.342648 | − | 1.37164i | 2.73693 | + | 3.81595i |
10.12 | −1.88604 | − | 0.574752i | 0.762299 | + | 0.976154i | 1.56672 | + | 1.05264i | 2.50001 | + | 1.91158i | −0.876682 | − | 2.27920i | −0.517569 | − | 3.14318i | 0.140056 | + | 0.171996i | 0.355308 | − | 1.42232i | −3.61644 | − | 5.04221i |
10.13 | −1.66696 | − | 0.507988i | 1.27446 | + | 1.63200i | 0.860600 | + | 0.578215i | −2.62311 | − | 2.00571i | −1.29544 | − | 3.36789i | −0.605300 | − | 3.67597i | 1.05986 | + | 1.30157i | −0.312085 | + | 1.24929i | 3.35374 | + | 4.67594i |
10.14 | −1.64944 | − | 0.502649i | 1.73011 | + | 2.21548i | 0.807886 | + | 0.542797i | 2.52603 | + | 1.93147i | −1.74011 | − | 4.52394i | −0.0283849 | − | 0.172381i | 1.11787 | + | 1.37280i | −1.18797 | + | 4.75552i | −3.19567 | − | 4.45555i |
10.15 | −1.62394 | − | 0.494880i | −1.90243 | − | 2.43613i | 0.732189 | + | 0.491938i | 1.54627 | + | 1.18232i | 1.88384 | + | 4.89762i | −0.262699 | − | 1.59536i | 1.19835 | + | 1.47164i | −1.58843 | + | 6.35858i | −1.92595 | − | 2.68525i |
10.16 | −1.37254 | − | 0.418269i | −0.575339 | − | 0.736745i | 0.0488291 | + | 0.0328070i | −0.314379 | − | 0.240383i | 0.481522 | + | 1.25186i | −0.752034 | − | 4.56708i | 1.75874 | + | 2.15982i | 0.515308 | − | 2.06280i | 0.330954 | + | 0.461432i |
10.17 | −1.35040 | − | 0.411519i | −1.53960 | − | 1.97151i | −0.00587983 | − | 0.00395050i | −2.62859 | − | 2.00990i | 1.26775 | + | 3.29590i | 0.536640 | + | 3.25900i | 1.78911 | + | 2.19712i | −0.789426 | + | 3.16011i | 2.72252 | + | 3.79587i |
10.18 | −1.29910 | − | 0.395886i | −0.468338 | − | 0.599725i | −0.129175 | − | 0.0867889i | −0.0558674 | − | 0.0427178i | 0.370993 | + | 0.964509i | 0.673018 | + | 4.08722i | 1.84852 | + | 2.27008i | 0.586755 | − | 2.34881i | 0.0556657 | + | 0.0776117i |
10.19 | −1.29780 | − | 0.395490i | 1.78122 | + | 2.28092i | −0.132239 | − | 0.0888481i | 1.04153 | + | 0.796385i | −1.40958 | − | 3.66463i | 0.540900 | + | 3.28487i | 1.84983 | + | 2.27169i | −1.30278 | + | 5.21509i | −1.03673 | − | 1.44546i |
10.20 | −1.09968 | − | 0.335116i | 1.00277 | + | 1.28409i | −0.563109 | − | 0.378338i | −1.56158 | − | 1.19403i | −0.672411 | − | 1.74814i | 0.126397 | + | 0.767606i | 1.94425 | + | 2.38764i | 0.0837475 | − | 0.335245i | 1.31710 | + | 1.83636i |
See next 80 embeddings (of 6960 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.d | odd | 22 | 1 | inner |
29.f | odd | 28 | 1 | inner |
667.x | even | 308 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 667.2.x.a | ✓ | 6960 |
23.d | odd | 22 | 1 | inner | 667.2.x.a | ✓ | 6960 |
29.f | odd | 28 | 1 | inner | 667.2.x.a | ✓ | 6960 |
667.x | even | 308 | 1 | inner | 667.2.x.a | ✓ | 6960 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
667.2.x.a | ✓ | 6960 | 1.a | even | 1 | 1 | trivial |
667.2.x.a | ✓ | 6960 | 23.d | odd | 22 | 1 | inner |
667.2.x.a | ✓ | 6960 | 29.f | odd | 28 | 1 | inner |
667.2.x.a | ✓ | 6960 | 667.x | even | 308 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(667, [\chi])\).