Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(4,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([14, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.z (of order \(21\), degree \(12\), 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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(648\) |
Relative dimension: | \(54\) over \(\Q(\zeta_{21})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −0.209537 | + | 2.79608i | −0.318381 | − | 1.70254i | −5.79651 | − | 0.873684i | −0.429093 | + | 1.87998i | 4.82715 | − | 0.533474i | 0.156954 | − | 2.64109i | 2.40961 | − | 10.5572i | −2.79727 | + | 1.08411i | −5.16666 | − | 1.59370i |
4.2 | −0.193400 | + | 2.58074i | 1.69682 | + | 0.347567i | −4.64515 | − | 0.700143i | 0.742208 | − | 3.25183i | −1.22514 | + | 4.31183i | 2.07445 | − | 1.64215i | 1.55350 | − | 6.80632i | 2.75839 | + | 1.17952i | 8.24857 | + | 2.54435i |
4.3 | −0.192530 | + | 2.56913i | 1.55341 | − | 0.766101i | −4.58569 | − | 0.691182i | −0.144697 | + | 0.633958i | 1.66913 | + | 4.13841i | −1.19412 | + | 2.36095i | 1.51204 | − | 6.62468i | 1.82618 | − | 2.38014i | −1.60086 | − | 0.493800i |
4.4 | −0.187690 | + | 2.50455i | −1.73007 | + | 0.0827751i | −4.25986 | − | 0.642071i | 0.223906 | − | 0.980997i | 0.117403 | − | 4.34858i | −2.61535 | + | 0.399936i | 1.28987 | − | 5.65131i | 2.98630 | − | 0.286414i | 2.41493 | + | 0.744907i |
4.5 | −0.173812 | + | 2.31936i | 0.977861 | + | 1.42961i | −3.37158 | − | 0.508183i | −0.186808 | + | 0.818458i | −3.48575 | + | 2.01953i | −2.56874 | − | 0.633687i | 0.729575 | − | 3.19648i | −1.08757 | + | 2.79592i | −1.86583 | − | 0.575533i |
4.6 | −0.173112 | + | 2.31002i | −0.732646 | + | 1.56947i | −3.32857 | − | 0.501701i | −0.445172 | + | 1.95043i | −3.49867 | − | 1.96412i | 0.693731 | − | 2.55318i | 0.704215 | − | 3.08537i | −1.92646 | − | 2.29973i | −4.42846 | − | 1.36600i |
4.7 | −0.168873 | + | 2.25346i | −1.65465 | + | 0.511977i | −3.07188 | − | 0.463011i | 0.626084 | − | 2.74306i | −0.874290 | − | 3.81515i | 2.34287 | − | 1.22921i | 0.556439 | − | 2.43792i | 2.47576 | − | 1.69429i | 6.07562 | + | 1.87408i |
4.8 | −0.163627 | + | 2.18345i | −1.55326 | − | 0.766401i | −2.76302 | − | 0.416458i | −0.523133 | + | 2.29200i | 1.92755 | − | 3.26607i | 1.85748 | + | 1.88409i | 0.386968 | − | 1.69542i | 1.82526 | + | 2.38085i | −4.91886 | − | 1.51727i |
4.9 | −0.162086 | + | 2.16289i | 0.451822 | − | 1.67208i | −2.67415 | − | 0.403063i | 0.397052 | − | 1.73960i | 3.54329 | + | 1.24826i | 1.01797 | + | 2.44208i | 0.339946 | − | 1.48940i | −2.59171 | − | 1.51097i | 3.69820 | + | 1.14074i |
4.10 | −0.152487 | + | 2.03479i | −0.743876 | − | 1.56418i | −2.13947 | − | 0.322473i | 0.723985 | − | 3.17199i | 3.29621 | − | 1.27512i | −2.58955 | − | 0.542443i | 0.0742997 | − | 0.325528i | −1.89330 | + | 2.32711i | 6.34394 | + | 1.95685i |
4.11 | −0.132809 | + | 1.77221i | 1.41906 | − | 0.993117i | −1.14543 | − | 0.172646i | −0.788881 | + | 3.45631i | 1.57155 | + | 2.64676i | 2.57823 | − | 0.593925i | −0.332830 | + | 1.45822i | 1.02744 | − | 2.81858i | −6.02055 | − | 1.85709i |
4.12 | −0.129339 | + | 1.72591i | −1.02279 | + | 1.39782i | −0.984361 | − | 0.148369i | −0.845939 | + | 3.70630i | −2.28022 | − | 1.94603i | −1.47863 | + | 2.19400i | −0.386868 | + | 1.69498i | −0.907804 | − | 2.85935i | −6.28731 | − | 1.93938i |
4.13 | −0.122748 | + | 1.63795i | 0.324451 | + | 1.70139i | −0.690165 | − | 0.104026i | 0.354390 | − | 1.55268i | −2.82663 | + | 0.322594i | 2.24116 | + | 1.40613i | −0.475897 | + | 2.08504i | −2.78946 | + | 1.10404i | 2.49973 | + | 0.771063i |
4.14 | −0.109094 | + | 1.45576i | 1.61060 | + | 0.637154i | −0.129671 | − | 0.0195447i | −0.0506126 | + | 0.221748i | −1.10325 | + | 2.27514i | 1.03301 | − | 2.43575i | −0.607092 | + | 2.65984i | 2.18807 | + | 2.05240i | −0.317290 | − | 0.0978711i |
4.15 | −0.107721 | + | 1.43743i | 0.0199394 | − | 1.73194i | −0.0769418 | − | 0.0115971i | −0.762536 | + | 3.34089i | 2.48739 | + | 0.215227i | −2.60405 | − | 0.467908i | −0.616552 | + | 2.70129i | −2.99920 | − | 0.0690675i | −4.72015 | − | 1.45597i |
4.16 | −0.105772 | + | 1.41143i | 1.42810 | − | 0.980073i | −0.00328183 | 0.000494657i | 0.365770 | − | 1.60254i | 1.23225 | + | 2.11932i | −2.11011 | − | 1.59607i | −0.628861 | + | 2.75522i | 1.07891 | − | 2.79928i | 2.22319 | + | 0.685763i | |
4.17 | −0.101230 | + | 1.35082i | 1.66963 | + | 0.460807i | 0.163204 | + | 0.0245990i | −0.157203 | + | 0.688750i | −0.791482 | + | 2.20871i | −1.05230 | + | 2.42748i | −0.652605 | + | 2.85925i | 2.57531 | + | 1.53875i | −0.914461 | − | 0.282074i |
4.18 | −0.0832951 | + | 1.11150i | −0.959106 | + | 1.44226i | 0.749175 | + | 0.112920i | 0.757564 | − | 3.31911i | −1.52318 | − | 1.18618i | −1.18524 | − | 2.36542i | −0.683962 | + | 2.99663i | −1.16023 | − | 2.76656i | 3.62607 | + | 1.11850i |
4.19 | −0.0806131 | + | 1.07571i | −1.11699 | − | 1.32376i | 0.827015 | + | 0.124652i | 0.478314 | − | 2.09563i | 1.51402 | − | 1.09484i | 2.41117 | − | 1.08917i | −0.680835 | + | 2.98293i | −0.504687 | + | 2.95724i | 2.21572 | + | 0.683461i |
4.20 | −0.0535758 | + | 0.714919i | 1.52049 | − | 0.829523i | 1.46942 | + | 0.221480i | 0.990551 | − | 4.33989i | 0.511580 | + | 1.13147i | 1.86250 | + | 1.87912i | −0.556127 | + | 2.43655i | 1.62378 | − | 2.52256i | 3.04960 | + | 0.940676i |
See next 80 embeddings (of 648 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
441.z | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.z.a | yes | 648 |
9.c | even | 3 | 1 | 441.2.y.a | ✓ | 648 | |
49.g | even | 21 | 1 | 441.2.y.a | ✓ | 648 | |
441.z | even | 21 | 1 | inner | 441.2.z.a | yes | 648 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.y.a | ✓ | 648 | 9.c | even | 3 | 1 | |
441.2.y.a | ✓ | 648 | 49.g | even | 21 | 1 | |
441.2.z.a | yes | 648 | 1.a | even | 1 | 1 | trivial |
441.2.z.a | yes | 648 | 441.z | even | 21 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).