[N,k,chi] = [336,6,Mod(193,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.193");
S:= CuspForms(chi, 6);
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 \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
\(n\)
\(85\)
\(113\)
\(127\)
\(241\)
\(\chi(n)\)
\(1\)
\(1\)
\(1\)
\(-\beta_{1}\)
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
\( T_{5}^{8} + 7657 T_{5}^{6} + 605400 T_{5}^{5} + 61817893 T_{5}^{4} + 2317773900 T_{5}^{3} + 67214905692 T_{5}^{2} + 965081458800 T_{5} + 10164899803536 \)
T5^8 + 7657*T5^6 + 605400*T5^5 + 61817893*T5^4 + 2317773900*T5^3 + 67214905692*T5^2 + 965081458800*T5 + 10164899803536
acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{8} \)
T^8
$3$
\( (T^{2} - 9 T + 81)^{4} \)
(T^2 - 9*T + 81)^4
$5$
\( T^{8} + 7657 T^{6} + \cdots + 10164899803536 \)
T^8 + 7657*T^6 + 605400*T^5 + 61817893*T^4 + 2317773900*T^3 + 67214905692*T^2 + 965081458800*T + 10164899803536
$7$
\( T^{8} + 258 T^{7} + \cdots + 79\!\cdots\!01 \)
T^8 + 258*T^7 + 65408*T^6 + 12311544*T^5 + 1643194665*T^4 + 206920120008*T^3 + 18476141086592*T^2 + 1224870869565294*T + 79792266297612001
$11$
\( T^{8} - 402 T^{7} + \cdots + 28\!\cdots\!76 \)
T^8 - 402*T^7 + 399967*T^6 - 105799218*T^5 + 99024901837*T^4 - 25381944601332*T^3 + 9761813367494172*T^2 - 169576701600014928*T + 2829568482592751376
$13$
\( (T^{4} - 462 T^{3} + \cdots + 149501563456)^{2} \)
(T^4 - 462*T^3 - 1148423*T^2 + 515112852*T + 149501563456)^2
$17$
\( T^{8} + 276 T^{7} + \cdots + 25\!\cdots\!24 \)
T^8 + 276*T^7 + 1670928*T^6 + 1349604864*T^5 + 2840324474880*T^4 + 1454766301642752*T^3 + 720903316973027328*T^2 + 44837110146970681344*T + 2510425643599238529024
$19$
\( T^{8} - 510 T^{7} + \cdots + 54\!\cdots\!76 \)
T^8 - 510*T^7 + 6156971*T^6 + 1411756650*T^5 + 27788839301865*T^4 + 2834297273515140*T^3 + 44221112569626422096*T^2 + 5896825990015878977280*T + 54628927092985580511563776
$23$
\( T^{8} - 6900 T^{7} + \cdots + 90\!\cdots\!76 \)
T^8 - 6900*T^7 + 40488144*T^6 - 83386679040*T^5 + 165861153884160*T^4 + 163456653366558720*T^3 + 271772835469800505344*T^2 + 51504758372052573880320*T + 9047700688840340092747776
$29$
\( (T^{4} - 540 T^{3} + \cdots + 408027025117872)^{2} \)
(T^4 - 540*T^3 - 52650397*T^2 + 62527747272*T + 408027025117872)^2
$31$
\( T^{8} + 6410 T^{7} + \cdots + 75\!\cdots\!09 \)
T^8 + 6410*T^7 + 58801968*T^6 + 3691164188*T^5 + 602809801419817*T^4 - 73339378003949028*T^3 + 4972209303844040870952*T^2 - 5083169593044764929487598*T + 7519680127798113647275945209
$37$
\( T^{8} + 15250 T^{7} + \cdots + 25\!\cdots\!36 \)
T^8 + 15250*T^7 + 279418707*T^6 + 2279577843610*T^5 + 30068709801011305*T^4 + 223956156491010025260*T^3 + 2004918702019590959929008*T^2 + 7549601052824889027271102080*T + 25431105554668088404743342375936
$41$
\( (T^{4} - 4308 T^{3} + \cdots - 18\!\cdots\!52)^{2} \)
(T^4 - 4308*T^3 - 192741244*T^2 - 1101575496480*T - 1856858915261952)^2
$43$
\( (T^{4} + 29198 T^{3} + \cdots - 991662745581932)^{2} \)
(T^4 + 29198*T^3 + 199961493*T^2 + 199921376588*T - 991662745581932)^2
$47$
\( T^{8} + 15060 T^{7} + \cdots + 73\!\cdots\!36 \)
T^8 + 15060*T^7 + 176153856*T^6 + 852661119456*T^5 + 3512848312989072*T^4 + 5876954381324343168*T^3 + 15729581860689981040128*T^2 + 12164068572051486246766848*T + 73270724017161642212281211136
$53$
\( T^{8} + 13692 T^{7} + \cdots + 72\!\cdots\!84 \)
T^8 + 13692*T^7 + 685378293*T^6 + 9260410268772*T^5 + 366485627229994953*T^4 + 4235531909662997427528*T^3 + 60388638865198905797772912*T^2 + 68374549794314156759450087040*T + 72343236355801451678844053270784
$59$
\( T^{8} - 34830 T^{7} + \cdots + 66\!\cdots\!96 \)
T^8 - 34830*T^7 + 1024872459*T^6 - 7480212648750*T^5 + 54097578476747217*T^4 - 92739830872274379000*T^3 + 698578937563895542379376*T^2 - 1190450802227713314066840960*T + 6650478746304621097786096548096
$61$
\( T^{8} - 5364 T^{7} + \cdots + 32\!\cdots\!76 \)
T^8 - 5364*T^7 + 561368672*T^6 - 3902598174816*T^5 + 283845325040842512*T^4 - 1607543208856194807168*T^3 + 20978463126071642702311424*T^2 + 60639616949719778744166178560*T + 321922204446252136379290155141376
$67$
\( T^{8} + 5994 T^{7} + \cdots + 30\!\cdots\!56 \)
T^8 + 5994*T^7 + 2881922927*T^6 - 26942404854654*T^5 + 7519978747196869197*T^4 - 20658658000321909656792*T^3 + 1589966562463725863769911444*T^2 + 2718397014904060046722099476000*T + 302596024971115875614695445835555856
$71$
\( (T^{4} + 89268 T^{3} + \cdots + 21\!\cdots\!68)^{2} \)
(T^4 + 89268*T^3 + 1521744768*T^2 - 16377596837712*T + 21932335650275568)^2
$73$
\( T^{8} + 59638 T^{7} + \cdots + 14\!\cdots\!00 \)
T^8 + 59638*T^7 + 3193750863*T^6 + 62299772588518*T^5 + 1466139740779982221*T^4 + 7189592332501483112580*T^3 + 457528085093028893259077100*T^2 + 2482588017082558125236593314000*T + 14915808373949680059326036318490000
$79$
\( T^{8} + 44062 T^{7} + \cdots + 27\!\cdots\!81 \)
T^8 + 44062*T^7 + 5723264752*T^6 - 196348017119564*T^5 + 13482185316532207897*T^4 - 70747342212365881335788*T^3 + 845604017658862099686665128*T^2 + 2454848594648640401391441575206*T + 27301304458234581168518344089781081
$83$
\( (T^{4} - 208446 T^{3} + \cdots - 41\!\cdots\!32)^{2} \)
(T^4 - 208446*T^3 + 7607249829*T^2 + 738604511000820*T - 41533908097096407132)^2
$89$
\( T^{8} - 77520 T^{7} + \cdots + 22\!\cdots\!24 \)
T^8 - 77520*T^7 + 22815563908*T^6 - 2429594520301248*T^5 + 474439153312967172112*T^4 - 38700667842298647676388352*T^3 + 2687420811627864787647658831872*T^2 - 88312687527564464010904384415465472*T + 2239375876373079606378438995149165953024
$97$
\( (T^{4} + 188630 T^{3} + \cdots - 11\!\cdots\!44)^{2} \)
(T^4 + 188630*T^3 + 9271508101*T^2 - 99054118022220*T - 11638556269792123644)^2
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