[N,k,chi] = [63,6,Mod(37,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.37");
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 \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
\(n\)
\(10\)
\(29\)
\(\chi(n)\)
\(-1 - \beta_{2}\)
\(1\)
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
\( T_{2}^{12} + 187T_{2}^{10} + 25399T_{2}^{8} + 1518438T_{2}^{6} + 66232188T_{2}^{4} + 1297462320T_{2}^{2} + 18380851776 \)
T2^12 + 187*T2^10 + 25399*T2^8 + 1518438*T2^6 + 66232188*T2^4 + 1297462320*T2^2 + 18380851776
acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} + 187 T^{10} + \cdots + 18380851776 \)
T^12 + 187*T^10 + 25399*T^8 + 1518438*T^6 + 66232188*T^4 + 1297462320*T^2 + 18380851776
$3$
\( T^{12} \)
T^12
$5$
\( T^{12} + 13138 T^{10} + \cdots + 31\!\cdots\!96 \)
T^12 + 13138*T^10 + 121169971*T^8 + 563323769202*T^6 + 1907045757424161*T^4 + 2892216493746131328*T^2 + 3161615865952288260096
$7$
\( (T^{6} - 71 T^{5} + \cdots + 4747561509943)^{2} \)
(T^6 - 71*T^5 - 5068*T^4 + 2295013*T^3 - 85177876*T^2 - 20055742679*T + 4747561509943)^2
$11$
\( T^{12} + 902578 T^{10} + \cdots + 29\!\cdots\!76 \)
T^12 + 902578*T^10 + 580016478163*T^8 + 177518781588298386*T^6 + 39593227288205595046113*T^4 + 4018471648852121112018640896*T^2 + 293327400605894153205095728152576
$13$
\( (T^{3} - 77 T^{2} - 783976 T + 60080636)^{4} \)
(T^3 - 77*T^2 - 783976*T + 60080636)^4
$17$
\( T^{12} + 2284392 T^{10} + \cdots + 20\!\cdots\!76 \)
T^12 + 2284392*T^10 + 4045590496944*T^8 + 2392812592337973888*T^6 + 1048408760351235505713408*T^4 + 167982923366516356509136158720*T^2 + 20513541786240598579940809796222976
$19$
\( (T^{6} - 4711 T^{5} + \cdots + 13\!\cdots\!64)^{2} \)
(T^6 - 4711*T^5 + 14897117*T^4 - 26945608228*T^3 + 35741443813028*T^2 - 27097936112073232*T + 13792871288922258064)^2
$23$
\( T^{12} + 10191432 T^{10} + \cdots + 12\!\cdots\!56 \)
T^12 + 10191432*T^10 + 83626347598128*T^8 + 183755707807315250304*T^6 + 294919960732369541578268928*T^4 + 227769610312631991943770483032064*T^2 + 126653177894882167920928387679970656256
$29$
\( (T^{6} - 66189634 T^{4} + \cdots - 75\!\cdots\!56)^{2} \)
(T^6 - 66189634*T^4 + 1273417115193153*T^2 - 7589524345822376326656)^2
$31$
\( (T^{6} - 11711 T^{5} + \cdots + 73\!\cdots\!01)^{2} \)
(T^6 - 11711*T^5 + 131877810*T^4 - 233327355023*T^3 + 1032654281342482*T^2 + 452177484448076961*T + 7362821470671396332001)^2
$37$
\( (T^{6} + 9091 T^{5} + \cdots + 10\!\cdots\!16)^{2} \)
(T^6 + 9091*T^5 + 159187641*T^4 - 631138256552*T^3 + 6152670218833564*T^2 + 2476084186138261440*T + 1046498147320473948816)^2
$41$
\( (T^{6} - 727900480 T^{4} + \cdots - 13\!\cdots\!64)^{2} \)
(T^6 - 727900480*T^4 + 172110236322742272*T^2 - 13307641909409328308158464)^2
$43$
\( (T^{3} + 21843 T^{2} + \cdots - 1643621929904)^{4} \)
(T^3 + 21843*T^2 - 6827376*T - 1643621929904)^4
$47$
\( T^{12} + 613454880 T^{10} + \cdots + 39\!\cdots\!96 \)
T^12 + 613454880*T^10 + 297422431977302784*T^8 + 44438107878839472361316352*T^6 + 5009365930820787007740231915995136*T^4 + 156476094031589807545805039079872972980224*T^2 + 3932718973111821984149528959773083809868987498496
$53$
\( T^{12} + 666461298 T^{10} + \cdots + 26\!\cdots\!96 \)
T^12 + 666461298*T^10 + 430059152067367059*T^8 + 9404450812290231583222482*T^6 + 199026659825498225610975762592353*T^4 + 2287725752460133794695315839129399680*T^2 + 26282154676099336766998866726254204829696
$59$
\( T^{12} + 4695322386 T^{10} + \cdots + 98\!\cdots\!16 \)
T^12 + 4695322386*T^10 + 15087337798234200435*T^8 + 26407162578639271263178281138*T^6 + 33712686391894075799246490990681154977*T^4 + 21802506530774863144984648578127556882892226944*T^2 + 9816457975656170484737818991927008245099333024845807616
$61$
\( (T^{6} + 8078 T^{5} + \cdots + 28\!\cdots\!04)^{2} \)
(T^6 + 8078*T^5 + 561262580*T^4 + 6685509994016*T^3 + 289210493001361472*T^2 + 2651727543675213742592*T + 28581141393439767195771904)^2
$67$
\( (T^{6} - 72325 T^{5} + \cdots + 50\!\cdots\!76)^{2} \)
(T^6 - 72325*T^5 + 3726770693*T^4 - 94561461641452*T^3 + 1748006811992756324*T^2 - 10698232891632380014768*T + 50588348408491474104860176)^2
$71$
\( (T^{6} - 3481391808 T^{4} + \cdots - 30\!\cdots\!44)^{2} \)
(T^6 - 3481391808*T^4 + 2282421134301143040*T^2 - 305717063785244517721964544)^2
$73$
\( (T^{6} + 50029 T^{5} + \cdots + 87\!\cdots\!44)^{2} \)
(T^6 + 50029*T^5 + 2019673425*T^4 + 22299735073840*T^3 + 186590305628279308*T^2 + 453182587408613738592*T + 879515094052012984214544)^2
$79$
\( (T^{6} - 50997 T^{5} + \cdots + 12\!\cdots\!89)^{2} \)
(T^6 - 50997*T^5 + 6165068082*T^4 - 41346706088653*T^3 + 18393964666219142178*T^2 - 397639951022377122322341*T + 12445532157519968466359810089)^2
$83$
\( (T^{6} - 7727019426 T^{4} + \cdots - 11\!\cdots\!64)^{2} \)
(T^6 - 7727019426*T^4 + 2503294783326306081*T^2 - 116392317447085148489134464)^2
$89$
\( T^{12} + 2820380296 T^{10} + \cdots + 36\!\cdots\!16 \)
T^12 + 2820380296*T^10 + 5520669130180094512*T^8 + 5651417432352960936431294592*T^6 + 4213137364991256452323773335959814400*T^4 + 1476192153976305420832483405113703082537385984*T^2 + 367865390440895915206491569666825019578046539121033216
$97$
\( (T^{3} + 216524 T^{2} + \cdots - 546802680855102)^{4} \)
(T^3 + 216524*T^2 + 6003490045*T - 546802680855102)^4
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