Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [418,2,Mod(13,418)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(418, base_ring=CyclotomicField(90))
chi = DirichletCharacter(H, H._module([9, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("418.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 418.v (of order \(90\), degree \(24\), 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: | \(3.33774680449\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{90})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{90}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −0.374607 | − | 0.927184i | −0.873028 | − | 3.04461i | −0.719340 | + | 0.694658i | 2.51211 | − | 1.33571i | −2.49587 | + | 1.94999i | −1.97806 | − | 4.44279i | 0.913545 | + | 0.406737i | −5.96333 | + | 3.72630i | −2.17951 | − | 1.82882i |
13.2 | −0.374607 | − | 0.927184i | −0.582769 | − | 2.03236i | −0.719340 | + | 0.694658i | −2.69676 | + | 1.43389i | −1.66606 | + | 1.30167i | 0.507726 | + | 1.14037i | 0.913545 | + | 0.406737i | −1.24671 | + | 0.779029i | 2.33971 | + | 1.96325i |
13.3 | −0.374607 | − | 0.927184i | −0.419952 | − | 1.46455i | −0.719340 | + | 0.694658i | −1.24254 | + | 0.660669i | −1.20059 | + | 0.938002i | −1.61772 | − | 3.63346i | 0.913545 | + | 0.406737i | 0.575603 | − | 0.359677i | 1.07802 | + | 0.904570i |
13.4 | −0.374607 | − | 0.927184i | −0.414539 | − | 1.44567i | −0.719340 | + | 0.694658i | 1.07843 | − | 0.573409i | −1.18511 | + | 0.925911i | 1.15655 | + | 2.59766i | 0.913545 | + | 0.406737i | 0.626029 | − | 0.391186i | −0.935641 | − | 0.785096i |
13.5 | −0.374607 | − | 0.927184i | −0.270648 | − | 0.943861i | −0.719340 | + | 0.694658i | 1.29541 | − | 0.688784i | −0.773746 | + | 0.604517i | 1.11507 | + | 2.50449i | 0.913545 | + | 0.406737i | 1.72652 | − | 1.07885i | −1.12390 | − | 0.943064i |
13.6 | −0.374607 | − | 0.927184i | 0.116237 | + | 0.405365i | −0.719340 | + | 0.694658i | 3.30844 | − | 1.75913i | 0.332305 | − | 0.259625i | −0.632649 | − | 1.42095i | 0.913545 | + | 0.406737i | 2.39333 | − | 1.49552i | −2.87040 | − | 2.40855i |
13.7 | −0.374607 | − | 0.927184i | 0.222302 | + | 0.775258i | −0.719340 | + | 0.694658i | −2.70784 | + | 1.43978i | 0.635531 | − | 0.496531i | −1.35747 | − | 3.04893i | 0.913545 | + | 0.406737i | 1.99254 | − | 1.24508i | 2.34932 | + | 1.97131i |
13.8 | −0.374607 | − | 0.927184i | 0.411485 | + | 1.43502i | −0.719340 | + | 0.694658i | −0.764836 | + | 0.406670i | 1.17638 | − | 0.919089i | 0.622864 | + | 1.39897i | 0.913545 | + | 0.406737i | 0.654189 | − | 0.408783i | 0.663571 | + | 0.556802i |
13.9 | −0.374607 | − | 0.927184i | 0.581094 | + | 2.02651i | −0.719340 | + | 0.694658i | 0.405909 | − | 0.215825i | 1.66127 | − | 1.29793i | 0.326823 | + | 0.734057i | 0.913545 | + | 0.406737i | −1.22494 | + | 0.765430i | −0.352166 | − | 0.295502i |
13.10 | −0.374607 | − | 0.927184i | 0.937688 | + | 3.27011i | −0.719340 | + | 0.694658i | −2.61697 | + | 1.39147i | 2.68073 | − | 2.09441i | −0.352914 | − | 0.792659i | 0.913545 | + | 0.406737i | −7.27020 | + | 4.54292i | 2.27048 | + | 1.90516i |
29.1 | 0.438371 | − | 0.898794i | −2.63027 | + | 1.06270i | −0.615661 | − | 0.788011i | 1.03456 | − | 0.296655i | −0.197888 | + | 2.82992i | 0.0323920 | − | 0.152392i | −0.978148 | + | 0.207912i | 3.63096 | − | 3.50638i | 0.186889 | − | 1.05990i |
29.2 | 0.438371 | − | 0.898794i | −2.41147 | + | 0.974297i | −0.615661 | − | 0.788011i | −2.76916 | + | 0.794045i | −0.181426 | + | 2.59452i | 0.709936 | − | 3.33998i | −0.978148 | + | 0.207912i | 2.70791 | − | 2.61500i | −0.500239 | + | 2.83699i |
29.3 | 0.438371 | − | 0.898794i | −1.31507 | + | 0.531324i | −0.615661 | − | 0.788011i | 2.95966 | − | 0.848670i | −0.0989393 | + | 1.41490i | 0.318233 | − | 1.49717i | −0.978148 | + | 0.207912i | −0.710907 | + | 0.686515i | 0.534652 | − | 3.03216i |
29.4 | 0.438371 | − | 0.898794i | −1.03102 | + | 0.416557i | −0.615661 | − | 0.788011i | 1.04375 | − | 0.299291i | −0.0775682 | + | 1.10928i | −0.760853 | + | 3.57953i | −0.978148 | + | 0.207912i | −1.26855 | + | 1.22502i | 0.188549 | − | 1.06932i |
29.5 | 0.438371 | − | 0.898794i | 0.264611 | − | 0.106910i | −0.615661 | − | 0.788011i | −2.79611 | + | 0.801772i | 0.0199080 | − | 0.284697i | 0.419245 | − | 1.97239i | −0.978148 | + | 0.207912i | −2.09943 | + | 2.02740i | −0.505107 | + | 2.86460i |
29.6 | 0.438371 | − | 0.898794i | 0.332827 | − | 0.134471i | −0.615661 | − | 0.788011i | −2.40319 | + | 0.689104i | 0.0250402 | − | 0.358091i | −0.957149 | + | 4.50303i | −0.978148 | + | 0.207912i | −2.06533 | + | 1.99446i | −0.434127 | + | 2.46206i |
29.7 | 0.438371 | − | 0.898794i | 0.856586 | − | 0.346083i | −0.615661 | − | 0.788011i | 2.52606 | − | 0.724335i | 0.0644450 | − | 0.921607i | 1.07271 | − | 5.04669i | −0.978148 | + | 0.207912i | −1.54405 | + | 1.49108i | 0.456322 | − | 2.58793i |
29.8 | 0.438371 | − | 0.898794i | 1.88473 | − | 0.761481i | −0.615661 | − | 0.788011i | 0.993014 | − | 0.284742i | 0.141797 | − | 2.02780i | 0.0468569 | − | 0.220445i | −0.978148 | + | 0.207912i | 0.814339 | − | 0.786398i | 0.179384 | − | 1.01734i |
29.9 | 0.438371 | − | 0.898794i | 2.45510 | − | 0.991923i | −0.615661 | − | 0.788011i | 2.52718 | − | 0.724656i | 0.184709 | − | 2.64146i | −0.616865 | + | 2.90212i | −0.978148 | + | 0.207912i | 2.88556 | − | 2.78656i | 0.456524 | − | 2.58908i |
29.10 | 0.438371 | − | 0.898794i | 2.88213 | − | 1.16445i | −0.615661 | − | 0.788011i | −3.70984 | + | 1.06378i | 0.216836 | − | 3.10090i | 0.625748 | − | 2.94391i | −0.978148 | + | 0.207912i | 4.79268 | − | 4.62824i | −0.670169 | + | 3.80072i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
209.w | even | 90 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 418.2.v.b | yes | 240 |
11.d | odd | 10 | 1 | 418.2.v.a | ✓ | 240 | |
19.f | odd | 18 | 1 | 418.2.v.a | ✓ | 240 | |
209.w | even | 90 | 1 | inner | 418.2.v.b | yes | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
418.2.v.a | ✓ | 240 | 11.d | odd | 10 | 1 | |
418.2.v.a | ✓ | 240 | 19.f | odd | 18 | 1 | |
418.2.v.b | yes | 240 | 1.a | even | 1 | 1 | trivial |
418.2.v.b | yes | 240 | 209.w | even | 90 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{240} - 3 T_{3}^{239} + 3 T_{3}^{238} + 3 T_{3}^{237} + 72 T_{3}^{236} - 267 T_{3}^{235} + \cdots + 14\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).