[N,k,chi] = [216,4,Mod(73,216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.73");
S:= CuspForms(chi, 4);
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 \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
\(n\)
\(55\)
\(109\)
\(137\)
\(\chi(n)\)
\(1\)
\(1\)
\(-1 - \beta_{1}\)
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
\( T_{5}^{8} - 5T_{5}^{7} + 214T_{5}^{6} + 1951T_{5}^{5} + 31798T_{5}^{4} + 109147T_{5}^{3} + 519121T_{5}^{2} - 708224T_{5} + 1982464 \)
T5^8 - 5*T5^7 + 214*T5^6 + 1951*T5^5 + 31798*T5^4 + 109147*T5^3 + 519121*T5^2 - 708224*T5 + 1982464
acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{8} \)
T^8
$3$
\( T^{8} \)
T^8
$5$
\( T^{8} - 5 T^{7} + 214 T^{6} + \cdots + 1982464 \)
T^8 - 5*T^7 + 214*T^6 + 1951*T^5 + 31798*T^4 + 109147*T^3 + 519121*T^2 - 708224*T + 1982464
$7$
\( T^{8} - 3 T^{7} + 276 T^{6} + \cdots + 352836 \)
T^8 - 3*T^7 + 276*T^6 + 27*T^5 + 71856*T^4 - 99765*T^3 + 308367*T^2 + 229878*T + 352836
$11$
\( T^{8} + 16 T^{7} + \cdots + 1515467041 \)
T^8 + 16*T^7 + 2122*T^6 - 33968*T^5 + 3410131*T^4 - 5082224*T^3 + 76868650*T^2 + 80038024*T + 1515467041
$13$
\( T^{8} - 29 T^{7} + \cdots + 19954952644 \)
T^8 - 29*T^7 + 4612*T^6 + 176029*T^5 + 13112464*T^4 + 133899481*T^3 + 1643921227*T^2 - 4708968770*T + 19954952644
$17$
\( (T^{4} - 17 T^{3} - 14562 T^{2} + \cdots + 2567224)^{2} \)
(T^4 - 17*T^3 - 14562*T^2 + 723268*T + 2567224)^2
$19$
\( (T^{4} + 109 T^{3} - 11772 T^{2} + \cdots - 5081408)^{2} \)
(T^4 + 109*T^3 - 11772*T^2 - 1246544*T - 5081408)^2
$23$
\( T^{8} + 37 T^{7} + \cdots + 14\!\cdots\!84 \)
T^8 + 37*T^7 + 20224*T^6 - 146297*T^5 + 327759100*T^4 + 2389314823*T^3 + 791572286251*T^2 - 10462101122582*T + 1440329863015684
$29$
\( T^{8} - 3 T^{7} + \cdots + 24\!\cdots\!84 \)
T^8 - 3*T^7 + 94416*T^6 + 6136119*T^5 + 7331160780*T^4 + 285713719875*T^3 + 157041912617055*T^2 - 4602547854315378*T + 2473515895022876484
$31$
\( T^{8} - 331 T^{7} + \cdots + 10\!\cdots\!16 \)
T^8 - 331*T^7 + 176710*T^6 - 47011399*T^5 + 19150528126*T^4 - 4431567571543*T^3 + 984750437588077*T^2 - 110181359315562964*T + 10129563987584179216
$37$
\( (T^{4} + 366 T^{3} - 18492 T^{2} + \cdots - 215981856)^{2} \)
(T^4 + 366*T^3 - 18492*T^2 - 7645176*T - 215981856)^2
$41$
\( T^{8} - 378 T^{7} + \cdots + 13\!\cdots\!89 \)
T^8 - 378*T^7 + 278880*T^6 - 24717276*T^5 + 33243122745*T^4 - 5449043120220*T^3 + 1399636875371256*T^2 - 13733462875964706*T + 130190591066564889
$43$
\( T^{8} - 506 T^{7} + \cdots + 32\!\cdots\!89 \)
T^8 - 506*T^7 + 324364*T^6 - 55804340*T^5 + 28101716725*T^4 - 3661781202116*T^3 + 2003297901616540*T^2 - 25635319789552418*T + 321817055724473089
$47$
\( T^{8} + 171 T^{7} + \cdots + 96\!\cdots\!84 \)
T^8 + 171*T^7 + 66300*T^6 + 8583669*T^5 + 2959788312*T^4 + 382731540885*T^3 + 44143245062343*T^2 + 2317895157154338*T + 96530775306536484
$53$
\( (T^{4} + 410 T^{3} + \cdots - 15963093536)^{2} \)
(T^4 + 410*T^3 - 239580*T^2 - 137282536*T - 15963093536)^2
$59$
\( T^{8} + 616 T^{7} + \cdots + 20\!\cdots\!89 \)
T^8 + 616*T^7 + 865450*T^6 + 181440880*T^5 + 382865497075*T^4 + 115092719100592*T^3 + 58483074959902666*T^2 - 340206973120924736*T + 2002597538717246689
$61$
\( T^{8} - 1331 T^{7} + \cdots + 23\!\cdots\!16 \)
T^8 - 1331*T^7 + 1421902*T^6 - 732776447*T^5 + 348318833614*T^4 - 81338334098267*T^3 + 34697186616690217*T^2 - 6432611051882456936*T + 2315131720861293015616
$67$
\( T^{8} - 1162 T^{7} + \cdots + 18\!\cdots\!41 \)
T^8 - 1162*T^7 + 1680184*T^6 - 933812188*T^5 + 1011475237129*T^4 - 536431035065164*T^3 + 388448158763370016*T^2 - 90439252923516932314*T + 18856837223951288543641
$71$
\( (T^{4} - 344 T^{3} - 654384 T^{2} + \cdots + 9762389248)^{2} \)
(T^4 - 344*T^3 - 654384*T^2 + 196817728*T + 9762389248)^2
$73$
\( (T^{4} + 1307 T^{3} + \cdots - 88005243128)^{2} \)
(T^4 + 1307*T^3 - 217110*T^2 - 613828588*T - 88005243128)^2
$79$
\( T^{8} - 1853 T^{7} + \cdots + 97\!\cdots\!36 \)
T^8 - 1853*T^7 + 2558350*T^6 - 1804104137*T^5 + 1033603053190*T^4 - 285916513673369*T^3 + 94666237048254421*T^2 - 8991341597401497620*T + 9735935267013683185936
$83$
\( T^{8} + 1421 T^{7} + \cdots + 46\!\cdots\!84 \)
T^8 + 1421*T^7 + 2300014*T^6 + 591084425*T^5 + 714384390310*T^4 - 53948361912559*T^3 + 264117443091693085*T^2 - 33607063850315486212*T + 4608882081653774717584
$89$
\( (T^{4} + 816 T^{3} + \cdots + 22003976592)^{2} \)
(T^4 + 816*T^3 - 307656*T^2 - 86780160*T + 22003976592)^2
$97$
\( T^{8} - 2506 T^{7} + \cdots + 67\!\cdots\!49 \)
T^8 - 2506*T^7 + 5414884*T^6 - 5227083700*T^5 + 5400080946325*T^4 - 2779817982781396*T^3 + 3047646442054046500*T^2 - 1252130317458996945658*T + 670187724027869717180449
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