Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(474,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.474");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.e (of order \(3\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
474.1 | −1.39981 | − | 2.42454i | 2.03714 | −2.91894 | + | 5.05576i | −0.880908 | − | 1.52578i | −2.85161 | − | 4.93914i | −2.33090 | − | 4.03724i | 10.7446 | 1.14995 | −2.46621 | + | 4.27160i | ||||||
474.2 | −1.28918 | − | 2.23293i | −3.07366 | −2.32398 | + | 4.02525i | 0.692759 | + | 1.19989i | 3.96250 | + | 6.86325i | −0.857264 | − | 1.48482i | 6.82738 | 6.44736 | 1.78618 | − | 3.09376i | ||||||
474.3 | −1.21768 | − | 2.10909i | −0.119390 | −1.96550 | + | 3.40434i | 0.0110850 | + | 0.0191998i | 0.145379 | + | 0.251804i | 0.177168 | + | 0.306863i | 4.70267 | −2.98575 | 0.0269960 | − | 0.0467585i | ||||||
474.4 | −1.10662 | − | 1.91672i | 2.91652 | −1.44921 | + | 2.51010i | 1.12816 | + | 1.95402i | −3.22747 | − | 5.59015i | 0.991463 | + | 1.71726i | 1.98840 | 5.50610 | 2.49688 | − | 4.32472i | ||||||
474.5 | −1.07290 | − | 1.85831i | −1.75555 | −1.30222 | + | 2.25550i | −0.834005 | − | 1.44454i | 1.88352 | + | 3.26235i | 2.55493 | + | 4.42527i | 1.29698 | 0.0819409 | −1.78960 | + | 3.09968i | ||||||
474.6 | −0.968326 | − | 1.67719i | −1.30850 | −0.875312 | + | 1.51608i | −1.56023 | − | 2.70239i | 1.26706 | + | 2.19461i | −1.74364 | − | 3.02008i | −0.482955 | −1.28782 | −3.02161 | + | 5.23359i | ||||||
474.7 | −0.894433 | − | 1.54920i | −0.575368 | −0.600022 | + | 1.03927i | 2.08743 | + | 3.61553i | 0.514629 | + | 0.891363i | 0.371978 | + | 0.644284i | −1.43101 | −2.66895 | 3.73413 | − | 6.46770i | ||||||
474.8 | −0.738467 | − | 1.27906i | 2.69547 | −0.0906660 | + | 0.157038i | −1.97155 | − | 3.41482i | −1.99052 | − | 3.44767i | −0.812537 | − | 1.40736i | −2.68605 | 4.26557 | −2.91185 | + | 5.04347i | ||||||
474.9 | −0.602444 | − | 1.04346i | 2.88926 | 0.274123 | − | 0.474794i | 1.22195 | + | 2.11647i | −1.74062 | − | 3.01484i | −1.97926 | − | 3.42819i | −3.07035 | 5.34781 | 1.47231 | − | 2.55011i | ||||||
474.10 | −0.536790 | − | 0.929747i | −2.63709 | 0.423714 | − | 0.733894i | 1.18192 | + | 2.04715i | 1.41556 | + | 2.45183i | −0.752816 | − | 1.30392i | −3.05694 | 3.95425 | 1.26889 | − | 2.19778i | ||||||
474.11 | −0.372490 | − | 0.645171i | −1.26258 | 0.722503 | − | 1.25141i | −0.0314823 | − | 0.0545290i | 0.470297 | + | 0.814579i | 1.61404 | + | 2.79560i | −2.56646 | −1.40590 | −0.0234537 | + | 0.0406230i | ||||||
474.12 | −0.320705 | − | 0.555478i | 0.649053 | 0.794296 | − | 1.37576i | 0.442759 | + | 0.766881i | −0.208155 | − | 0.360534i | −1.50175 | − | 2.60110i | −2.30176 | −2.57873 | 0.283990 | − | 0.491885i | ||||||
474.13 | −0.310878 | − | 0.538456i | 0.974785 | 0.806710 | − | 1.39726i | −0.223659 | − | 0.387389i | −0.303039 | − | 0.524879i | 1.06871 | + | 1.85105i | −2.24666 | −2.04979 | −0.139061 | + | 0.240861i | ||||||
474.14 | 0.131418 | + | 0.227623i | 2.69076 | 0.965459 | − | 1.67222i | −1.45219 | − | 2.51526i | 0.353614 | + | 0.612478i | 1.92643 | + | 3.33667i | 1.03319 | 4.24017 | 0.381687 | − | 0.661101i | ||||||
474.15 | 0.172443 | + | 0.298681i | 1.48381 | 0.940527 | − | 1.62904i | 1.21544 | + | 2.10521i | 0.255873 | + | 0.443185i | 0.147655 | + | 0.255746i | 1.33852 | −0.798307 | −0.419190 | + | 0.726058i | ||||||
474.16 | 0.196695 | + | 0.340685i | −1.57016 | 0.922622 | − | 1.59803i | −1.67457 | − | 2.90045i | −0.308843 | − | 0.534931i | −1.04200 | − | 1.80479i | 1.51268 | −0.534592 | 0.658760 | − | 1.14101i | ||||||
474.17 | 0.408544 | + | 0.707618i | −2.11340 | 0.666184 | − | 1.15387i | −0.472681 | − | 0.818707i | −0.863417 | − | 1.49548i | 1.09515 | + | 1.89685i | 2.72284 | 1.46647 | 0.386222 | − | 0.668955i | ||||||
474.18 | 0.566722 | + | 0.981591i | 2.34130 | 0.357653 | − | 0.619474i | −0.833325 | − | 1.44336i | 1.32687 | + | 2.29820i | −0.538269 | − | 0.932309i | 3.07765 | 2.48171 | 0.944526 | − | 1.63597i | ||||||
474.19 | 0.716721 | + | 1.24140i | −0.188236 | −0.0273779 | + | 0.0474198i | 1.37339 | + | 2.37878i | −0.134913 | − | 0.233676i | 1.70991 | + | 2.96164i | 2.78839 | −2.96457 | −1.96867 | + | 3.40984i | ||||||
474.20 | 0.772502 | + | 1.33801i | −0.117047 | −0.193519 | + | 0.335184i | −0.0594005 | − | 0.102885i | −0.0904189 | − | 0.156610i | −2.59858 | − | 4.50087i | 2.49203 | −2.98630 | 0.0917739 | − | 0.158957i | ||||||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.e.a | ✓ | 52 |
61.c | even | 3 | 1 | inner | 671.2.e.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.e.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
671.2.e.a | ✓ | 52 | 61.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 2 T_{2}^{51} + 43 T_{2}^{50} + 78 T_{2}^{49} + 1018 T_{2}^{48} + 1724 T_{2}^{47} + \cdots + 2531281 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\).