[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 \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
\(n\)
\(55\)
\(109\)
\(137\)
\(\chi(n)\)
\(1\)
\(1\)
\(-\beta_{1}\)
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
\( T_{5}^{10} - 5 T_{5}^{9} + 486 T_{5}^{8} + 111 T_{5}^{7} + 181830 T_{5}^{6} - 57429 T_{5}^{5} + 18313185 T_{5}^{4} + 119463888 T_{5}^{3} + 1213781760 T_{5}^{2} + 3130236928 T_{5} + 7487094784 \)
T5^10 - 5*T5^9 + 486*T5^8 + 111*T5^7 + 181830*T5^6 - 57429*T5^5 + 18313185*T5^4 + 119463888*T5^3 + 1213781760*T5^2 + 3130236928*T5 + 7487094784
acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} \)
T^10
$3$
\( T^{10} \)
T^10
$5$
\( T^{10} - 5 T^{9} + \cdots + 7487094784 \)
T^10 - 5*T^9 + 486*T^8 + 111*T^7 + 181830*T^6 - 57429*T^5 + 18313185*T^4 + 119463888*T^3 + 1213781760*T^2 + 3130236928*T + 7487094784
$7$
\( T^{10} + 3 T^{9} + \cdots + 15465374220816 \)
T^10 + 3*T^9 + 1398*T^8 + 9063*T^7 + 1514322*T^6 + 10511775*T^5 + 635958729*T^4 + 8048280852*T^3 + 215103479796*T^2 + 1710069253776*T + 15465374220816
$11$
\( T^{10} + 25 T^{9} + \cdots + 6004376846689 \)
T^10 + 25*T^9 + 4515*T^8 + 57714*T^7 + 14397453*T^6 + 165369747*T^5 + 16457621229*T^4 - 226072406694*T^3 + 6948104284203*T^2 - 6546680909951*T + 6004376846689
$13$
\( T^{10} + \cdots + 137434637455504 \)
T^10 + 29*T^9 + 6246*T^8 - 243615*T^7 + 25934178*T^6 - 363662883*T^5 + 13145927493*T^4 - 38979577200*T^3 + 4590536784444*T^2 - 23683688834464*T + 137434637455504
$17$
\( (T^{5} + 28 T^{4} - 8927 T^{3} + \cdots + 465026264)^{2} \)
(T^5 + 28*T^4 - 8927*T^3 - 387766*T^2 + 9589868*T + 465026264)^2
$19$
\( (T^{5} - 64 T^{4} - 10133 T^{3} + \cdots - 219796928)^{2} \)
(T^5 - 64*T^4 - 10133*T^3 + 365212*T^2 + 25870544*T - 219796928)^2
$23$
\( T^{10} - 89 T^{9} + \cdots + 29\!\cdots\!04 \)
T^10 - 89*T^9 + 56286*T^8 - 4560837*T^7 + 2152773786*T^6 - 165642245613*T^5 + 42879461020113*T^4 - 2712099273970812*T^3 + 579732087296018196*T^2 - 31751123706799541936*T + 2987487349884166850704
$29$
\( T^{10} - 129 T^{9} + \cdots + 57\!\cdots\!76 \)
T^10 - 129*T^9 + 52734*T^8 - 469125*T^7 + 1428418314*T^6 - 37232108217*T^5 + 14270905441989*T^4 + 698592295583208*T^3 + 35802345615537852*T^2 + 492757465614276096*T + 5785835260059752976
$31$
\( T^{10} + 241 T^{9} + \cdots + 49\!\cdots\!16 \)
T^10 + 241*T^9 + 91746*T^8 - 222495*T^7 + 1926106698*T^6 + 58860019737*T^5 + 20351003023569*T^4 - 473801264251440*T^3 + 33763084411916304*T^2 + 351714593870877952*T + 4951292949728809216
$37$
\( (T^{5} - 366 T^{4} + \cdots - 525481299072)^{2} \)
(T^5 - 366*T^4 - 163524*T^3 + 69985944*T^2 - 3133248768*T - 525481299072)^2
$41$
\( T^{10} - 171 T^{9} + \cdots + 66\!\cdots\!25 \)
T^10 - 171*T^9 + 29019*T^8 - 2230326*T^7 + 243875637*T^6 - 18222196833*T^5 + 1353692434221*T^4 - 62164267803702*T^3 + 2283971198818131*T^2 - 46013783373068715*T + 666097174314495225
$43$
\( T^{10} + 803 T^{9} + \cdots + 55\!\cdots\!89 \)
T^10 + 803*T^9 + 501723*T^8 + 151510182*T^7 + 40675613565*T^6 + 5981093063649*T^5 + 1312047873503733*T^4 + 168308777538459870*T^3 + 25951383201888717435*T^2 + 1299448165545601486811*T + 55774859501607337757689
$47$
\( T^{10} + 477 T^{9} + \cdots + 18\!\cdots\!24 \)
T^10 + 477*T^9 + 412374*T^8 + 101308473*T^7 + 89097069042*T^6 + 26668160191089*T^5 + 7239693860556345*T^4 + 872878125415640796*T^3 + 82090173090149512308*T^2 + 1310469068599892983728*T + 18107861499577577566224
$53$
\( (T^{5} + 374 T^{4} + \cdots + 758189779072)^{2} \)
(T^5 + 374*T^4 - 242948*T^3 - 95073272*T^2 - 3898947712*T + 758189779072)^2
$59$
\( T^{10} + 607 T^{9} + \cdots + 64\!\cdots\!41 \)
T^10 + 607*T^9 + 462363*T^8 + 167720406*T^7 + 96505949709*T^6 + 33398465342493*T^5 + 11284372673673165*T^4 + 2037861889875036510*T^3 + 289107570489720707859*T^2 + 15678796313383118194279*T + 647696281501992184856641
$61$
\( T^{10} + 1349 T^{9} + \cdots + 19\!\cdots\!44 \)
T^10 + 1349*T^9 + 1709622*T^8 + 845729601*T^7 + 546493241046*T^6 + 139156143495237*T^5 + 122549677800944817*T^4 + 21382233233661825504*T^3 + 5668453132159375183680*T^2 - 285656582847338453923840*T + 19821856393930856864518144
$67$
\( T^{10} + 1549 T^{9} + \cdots + 36\!\cdots\!25 \)
T^10 + 1549*T^9 + 1971243*T^8 + 958326762*T^7 + 451407981693*T^6 + 65340412922367*T^5 + 29300040116699637*T^4 + 638078828063777490*T^3 + 2457333144060875971371*T^2 - 239753968061577526486475*T + 36793060410231280567725625
$71$
\( (T^{5} - 812 T^{4} + \cdots - 24420797440)^{2} \)
(T^5 - 812*T^4 - 329744*T^3 + 243239552*T^2 + 6494698496*T - 24420797440)^2
$73$
\( (T^{5} - 1928 T^{4} + \cdots + 28702127905256)^{2} \)
(T^5 - 1928*T^4 + 1119481*T^3 - 56743786*T^2 - 130341479668*T + 28702127905256)^2
$79$
\( T^{10} + 1727 T^{9} + \cdots + 72\!\cdots\!76 \)
T^10 + 1727*T^9 + 3208626*T^8 + 3061867839*T^7 + 3893703895002*T^6 + 3280523650333479*T^5 + 2932271936355536769*T^4 + 1448484963876433224384*T^3 + 594843135402160336023504*T^2 + 73525564407095363237143040*T + 7284093531849455479580498176
$83$
\( T^{10} + 1025 T^{9} + \cdots + 49\!\cdots\!00 \)
T^10 + 1025*T^9 + 1786482*T^8 + 556890513*T^7 + 1056762343578*T^6 + 231358942706193*T^5 + 473163243212398233*T^4 + 529928803884609816*T^3 + 70541563811913185759856*T^2 + 10992172607322839063653760*T + 4921701710201467143475513600
$89$
\( (T^{5} + 2310 T^{4} + \cdots - 887800297007520)^{2} \)
(T^5 + 2310*T^4 - 277992*T^3 - 4312041264*T^2 - 3596560713072*T - 887800297007520)^2
$97$
\( T^{10} + 2875 T^{9} + \cdots + 76\!\cdots\!09 \)
T^10 + 2875*T^9 + 6377931*T^8 + 6173318982*T^7 + 4922800050261*T^6 + 916843862454849*T^5 + 760067644702093581*T^4 + 211472008979783513286*T^3 + 71900298347866731208323*T^2 + 7935950291259918312818875*T + 765929500375186252292263609
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