[N,k,chi] = [168,6,Mod(25,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, 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("168.25");
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 \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
\(n\)
\(73\)
\(85\)
\(113\)
\(127\)
\(\chi(n)\)
\(\beta_{1}\)
\(1\)
\(1\)
\(1\)
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
\( T_{5}^{10} + 75 T_{5}^{9} + 10726 T_{5}^{8} + 1823 T_{5}^{7} + 29874586 T_{5}^{6} - 712256737 T_{5}^{5} + 98953482841 T_{5}^{4} - 3137487320332 T_{5}^{3} + 90655492595236 T_{5}^{2} + \cdots + 11\!\cdots\!96 \)
T5^10 + 75*T5^9 + 10726*T5^8 + 1823*T5^7 + 29874586*T5^6 - 712256737*T5^5 + 98953482841*T5^4 - 3137487320332*T5^3 + 90655492595236*T5^2 - 1146693382891440*T5 + 11790232273802896
acting on \(S_{6}^{\mathrm{new}}(168, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} \)
T^10
$3$
\( (T^{2} + 9 T + 81)^{5} \)
(T^2 + 9*T + 81)^5
$5$
\( T^{10} + 75 T^{9} + \cdots + 11\!\cdots\!96 \)
T^10 + 75*T^9 + 10726*T^8 + 1823*T^7 + 29874586*T^6 - 712256737*T^5 + 98953482841*T^4 - 3137487320332*T^3 + 90655492595236*T^2 - 1146693382891440*T + 11790232273802896
$7$
\( T^{10} + 113 T^{9} + \cdots + 13\!\cdots\!07 \)
T^10 + 113*T^9 - 8183*T^8 + 548898*T^7 + 166288801*T^6 - 10823316637*T^5 + 2794815878407*T^4 + 155050099225602*T^3 - 38849295835863569*T^2 + 9016526091630156113*T + 1341068619663964900807
$11$
\( T^{10} - 91 T^{9} + \cdots + 22\!\cdots\!04 \)
T^10 - 91*T^9 + 598118*T^8 - 147419143*T^7 + 314162633234*T^6 - 56232542141803*T^5 + 34979996326030837*T^4 - 1270408926714507552*T^3 + 2315854505203874622540*T^2 - 203020079068861630017120*T + 22401019845965451939557904
$13$
\( (T^{5} - 290 T^{4} + \cdots + 20518255200000)^{2} \)
(T^5 - 290*T^4 - 822543*T^3 - 1314880*T^2 + 129431486000*T + 20518255200000)^2
$17$
\( T^{10} + 1128 T^{9} + \cdots + 30\!\cdots\!16 \)
T^10 + 1128*T^9 + 4228032*T^8 + 120899200*T^7 + 7995359820288*T^6 - 1134851765624832*T^5 + 11128236651032055808*T^4 - 5674083075801728286720*T^3 + 6930264674916821879488512*T^2 - 468062555866699764357660672*T + 30297906104775059027332694016
$19$
\( T^{10} - 282 T^{9} + \cdots + 26\!\cdots\!00 \)
T^10 - 282*T^9 + 6469687*T^8 + 4832903774*T^7 + 31446759892201*T^6 + 19868267265428072*T^5 + 60999769669640391856*T^4 + 51995963622426776988800*T^3 + 88058390448428664842563840*T^2 + 45973099227642267862129766400*T + 26325995787283939237919668633600
$23$
\( T^{10} + 1808 T^{9} + \cdots + 34\!\cdots\!56 \)
T^10 + 1808*T^9 + 3851200*T^8 + 1153909376*T^7 + 1795187825152*T^6 - 1364820947390464*T^5 + 1637709837323210752*T^4 - 663136628732208381952*T^3 + 225391146621407638847488*T^2 - 31727905849348144811212800*T + 3466944934810669929889529856
$29$
\( (T^{5} - 2513 T^{4} + \cdots + 21\!\cdots\!12)^{2} \)
(T^5 - 2513*T^4 - 65834889*T^3 + 66337446345*T^2 + 637402568448672*T + 212603520563312112)^2
$31$
\( T^{10} + 5069 T^{9} + \cdots + 53\!\cdots\!41 \)
T^10 + 5069*T^9 + 69293283*T^8 - 129728017742*T^7 + 1833186154644245*T^6 - 1273373590145295153*T^5 + 16293788822959793335085*T^4 - 33877006033913292230652542*T^3 + 78797442212970283080556856523*T^2 - 69373034857119480923812969388131*T + 53840418208910850296652953599977241
$37$
\( T^{10} + 5010 T^{9} + \cdots + 51\!\cdots\!24 \)
T^10 + 5010*T^9 + 170610283*T^8 + 349227482682*T^7 + 23302028391341353*T^6 + 74972518933584196260*T^5 + 362586452947582475245584*T^4 + 348832975953938013910734336*T^3 + 1545007412611625919830676753408*T^2 + 1292271172487373305982530928771072*T + 5101201830296435221410209606822068224
$41$
\( (T^{5} - 6436 T^{4} + \cdots - 16\!\cdots\!08)^{2} \)
(T^5 - 6436*T^4 - 426979180*T^3 + 2630201702848*T^2 + 26759753361703680*T - 167764481435970859008)^2
$43$
\( (T^{5} - 8664 T^{4} + \cdots - 36\!\cdots\!32)^{2} \)
(T^5 - 8664*T^4 - 337167515*T^3 + 647782557486*T^2 + 13623853960189804*T - 3696481843416753432)^2
$47$
\( T^{10} + 50 T^{9} + \cdots + 19\!\cdots\!00 \)
T^10 + 50*T^9 + 630902972*T^8 - 9496545108880*T^7 + 347141292007792208*T^6 - 3435353676548826004000*T^5 + 54378617667214775689713472*T^4 - 321198830922037030896279901440*T^3 + 4670020747346920922589865873272576*T^2 - 22519858323559892080251562571379064320*T + 197625962402265536283355126562394012902400
$53$
\( T^{10} - 1167 T^{9} + \cdots + 10\!\cdots\!00 \)
T^10 - 1167*T^9 + 788480670*T^8 - 2187038543659*T^7 + 501318365630242206*T^6 - 1043497748766758768619*T^5 + 96786199242701247867871645*T^4 + 26701913051943006085399305828*T^3 + 14569474984412085530002270657238976*T^2 - 12179447728048617024793658602110125760*T + 10292782825865968339871876855174892345600
$59$
\( T^{10} + 42797 T^{9} + \cdots + 82\!\cdots\!00 \)
T^10 + 42797*T^9 + 2827722986*T^8 + 58693508157413*T^7 + 3300358188381351434*T^6 + 65304497769509321566049*T^5 + 2304702353467609253072625805*T^4 + 12796716399990371516364377390388*T^3 + 164795393725162258460923060174486944*T^2 - 400924494985142154780378127602926326080*T + 8218145370484083650568779085171701847609600
$61$
\( T^{10} + 26546 T^{9} + \cdots + 58\!\cdots\!00 \)
T^10 + 26546*T^9 + 2169544972*T^8 + 74307636848656*T^7 + 4184353079521638992*T^6 + 110603379519467782455136*T^5 + 2438209517545353000651017536*T^4 + 28095850455804916952585323147520*T^3 + 244019258610458220610778537369056000*T^2 + 411312199870264288081204427748571968000*T + 588568443553397074498063142235044434560000
$67$
\( T^{10} + 13440 T^{9} + \cdots + 20\!\cdots\!00 \)
T^10 + 13440*T^9 + 5415749235*T^8 - 59429067623932*T^7 + 22693335380157061197*T^6 - 114194640596243210295570*T^5 + 25278143305066631202480718936*T^4 - 174254173607694419280000142979592*T^3 + 22833812706196864041196898209180949664*T^2 - 67702222729104011289601716503393159139360*T + 200059687474523540553277468148947259900558400
$71$
\( (T^{5} + 19678 T^{4} + \cdots - 45\!\cdots\!04)^{2} \)
(T^5 + 19678*T^4 - 4643310264*T^3 - 160406593539088*T^2 - 1647280639679857456*T - 4524864345060734169504)^2
$73$
\( T^{10} + 27768 T^{9} + \cdots + 34\!\cdots\!64 \)
T^10 + 27768*T^9 + 7750629247*T^8 + 47033508353884*T^7 + 42978457204032063253*T^6 + 277433819120838583202806*T^5 + 79506758535671802603775697704*T^4 - 1914188206715160853409862002367560*T^3 + 75361242995670678043497649965776680672*T^2 - 533887797432488379461564839266389167080864*T + 3457415640445227100999353986690052871025395264
$79$
\( T^{10} + 123369 T^{9} + \cdots + 11\!\cdots\!21 \)
T^10 + 123369*T^9 + 16584994255*T^8 + 943114521620254*T^7 + 85371765175129509853*T^6 + 4340353267581732734607847*T^5 + 301509837399308084796902136613*T^4 + 8017041982186445195381796484510558*T^3 + 164599082025500977238284580371571261239*T^2 + 1579529723630038967500985156555069050716393*T + 11170968742201288257051824605946045327707664521
$83$
\( (T^{5} - 167125 T^{4} + \cdots - 27\!\cdots\!68)^{2} \)
(T^5 - 167125*T^4 + 2488807167*T^3 + 465845520194669*T^2 - 5011793218604974844*T - 27805556580679380734868)^2
$89$
\( T^{10} + 59350 T^{9} + \cdots + 71\!\cdots\!00 \)
T^10 + 59350*T^9 + 24568942360*T^8 - 499415155041096*T^7 + 341055808001851796832*T^6 - 9525926712059234451908640*T^5 + 2912373027106527940526318462784*T^4 - 159298285290750809311230534175796736*T^3 + 14408410419793174316478600023666404245504*T^2 - 332575095926942075332433295815395795259719680*T + 7176358095095821977252021807938105543101258137600
$97$
\( (T^{5} - 262641 T^{4} + \cdots - 39\!\cdots\!96)^{2} \)
(T^5 - 262641*T^4 + 647569699*T^3 + 3206085171622545*T^2 - 92503484133556600976*T - 3952483184464369374633996)^2
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