[N,k,chi] = [48,9,Mod(17,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.17");
S:= CuspForms(chi, 9);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
\(n\)
\(17\)
\(31\)
\(37\)
\(\chi(n)\)
\(-1\)
\(1\)
\(1\)
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.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 1852352T_{5}^{6} + 1055033650176T_{5}^{4} + 183162750181376000T_{5}^{2} + 9126569679992258560000 \)
T5^8 + 1852352*T5^6 + 1055033650176*T5^4 + 183162750181376000*T5^2 + 9126569679992258560000
acting on \(S_{9}^{\mathrm{new}}(48, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{8} \)
T^8
$3$
\( T^{8} + 56 T^{7} + \cdots + 18\!\cdots\!41 \)
T^8 + 56*T^7 + 1404*T^6 - 71928*T^5 - 53661690*T^4 - 471919608*T^3 + 60437596284*T^2 + 15816054042936*T + 1853020188851841
$5$
\( T^{8} + 1852352 T^{6} + \cdots + 91\!\cdots\!00 \)
T^8 + 1852352*T^6 + 1055033650176*T^4 + 183162750181376000*T^2 + 9126569679992258560000
$7$
\( (T^{4} + 792 T^{3} + \cdots - 1739819024496)^{2} \)
(T^4 + 792*T^3 - 9744840*T^2 + 9117353952*T - 1739819024496)^2
$11$
\( T^{8} + 1024591808 T^{6} + \cdots + 54\!\cdots\!24 \)
T^8 + 1024591808*T^6 + 267212609442917376*T^4 + 22935703719705182389207040*T^2 + 548428109383318170560741663309824
$13$
\( (T^{4} - 12616 T^{3} + \cdots + 50\!\cdots\!00)^{2} \)
(T^4 - 12616*T^3 - 2223361128*T^2 + 23076612243680*T + 509880818254723600)^2
$17$
\( T^{8} + 38975642624 T^{6} + \cdots + 68\!\cdots\!36 \)
T^8 + 38975642624*T^6 + 434176286220872318976*T^4 + 1572750383496135979865552715776*T^2 + 685230731379581462331564410672603201536
$19$
\( (T^{4} + 78968 T^{3} + \cdots + 14\!\cdots\!56)^{2} \)
(T^4 + 78968*T^3 - 31227532680*T^2 - 2019603063660448*T + 14506207797099776656)^2
$23$
\( T^{8} + 230963324672 T^{6} + \cdots + 10\!\cdots\!56 \)
T^8 + 230963324672*T^6 + 19830959358501280727040*T^4 + 751165150830372873275634329059328*T^2 + 10601385423663910529329469523447082601414656
$29$
\( T^{8} + 3384568626624 T^{6} + \cdots + 20\!\cdots\!76 \)
T^8 + 3384568626624*T^6 + 3509467670550572750416896*T^4 + 1040613272305389779775115505590665216*T^2 + 20857441821648821046900723767401639880898379776
$31$
\( (T^{4} + 402776 T^{3} + \cdots - 24\!\cdots\!88)^{2} \)
(T^4 + 402776*T^3 - 2619620276808*T^2 - 1596354934488024352*T - 240382985784557123954288)^2
$37$
\( (T^{4} + 1992504 T^{3} + \cdots + 13\!\cdots\!76)^{2} \)
(T^4 + 1992504*T^3 - 2588448481128*T^2 - 4033743309893818656*T + 1360511691545601218639376)^2
$41$
\( T^{8} + 27732111683328 T^{6} + \cdots + 71\!\cdots\!76 \)
T^8 + 27732111683328*T^6 + 237395687480759224147820544*T^4 + 654324728793706594432197371615376310272*T^2 + 71837514773739495084407324627773586151902571134976
$43$
\( (T^{4} + 3481336 T^{3} + \cdots + 40\!\cdots\!44)^{2} \)
(T^4 + 3481336*T^3 - 16301672021640*T^2 - 33190984130304000416*T + 40330467464906581677145744)^2
$47$
\( T^{8} + 106160522738688 T^{6} + \cdots + 14\!\cdots\!56 \)
T^8 + 106160522738688*T^6 + 3500838560234764803949461504*T^4 + 40808933846199213034266392706342602145792*T^2 + 148216394999136190871310639090995438982682461975085056
$53$
\( T^{8} + 195356643589568 T^{6} + \cdots + 12\!\cdots\!44 \)
T^8 + 195356643589568*T^6 + 11775592427949142224873925632*T^4 + 229695883672715927258679535051178997776384*T^2 + 1276517276644834886638113043231301453571759906403385344
$59$
\( T^{8} + 756536765756864 T^{6} + \cdots + 36\!\cdots\!44 \)
T^8 + 756536765756864*T^6 + 183411974981110046772520578048*T^4 + 15112662537982409975957562209114482318573568*T^2 + 360963826714479136961773512558661334084194604123809644544
$61$
\( (T^{4} + 25791800 T^{3} + \cdots - 77\!\cdots\!96)^{2} \)
(T^4 + 25791800*T^3 + 9290106686616*T^2 - 1966132158874808406304*T - 7759447059091582118197197296)^2
$67$
\( (T^{4} + 29100344 T^{3} + \cdots + 23\!\cdots\!36)^{2} \)
(T^4 + 29100344*T^3 - 233974991467272*T^2 - 4287230580311881834144*T + 23781988970439615422912161936)^2
$71$
\( T^{8} + \cdots + 11\!\cdots\!96 \)
T^8 + 2068706599685888*T^6 + 859768763441409875715262955520*T^4 + 100433411947712684479600949047591497579364352*T^2 + 110636933249338066437295469630097953679162675387323908096
$73$
\( (T^{4} + 58427384 T^{3} + \cdots + 58\!\cdots\!04)^{2} \)
(T^4 + 58427384*T^3 - 4916145475560*T^2 - 13020214976326471196704*T + 58628764526610638988731301904)^2
$79$
\( (T^{4} + 86227288 T^{3} + \cdots - 61\!\cdots\!00)^{2} \)
(T^4 + 86227288*T^3 - 315064273304136*T^2 - 110378173479910161216800*T - 610712968497649393592477951600)^2
$83$
\( T^{8} + \cdots + 98\!\cdots\!96 \)
T^8 + 4125247868766656*T^6 + 5044423619805193162091101940736*T^4 + 1868908950894805027788732263820153289178415104*T^2 + 98182080083086176609032637669848757931711894470177137360896
$89$
\( T^{8} + \cdots + 25\!\cdots\!36 \)
T^8 + 12837983045103360*T^6 + 53089088259433606721936183181312*T^4 + 75591240233907023853864883021127872208028303360*T^2 + 25174057551267902071908494322331918861543413169795603403636736
$97$
\( (T^{4} + 168663352 T^{3} + \cdots - 90\!\cdots\!04)^{2} \)
(T^4 + 168663352*T^3 + 2091321027589272*T^2 - 383438742644222500536608*T - 9071556103171275826079087761904)^2
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