[N,k,chi] = [62,4,Mod(5,62)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(62, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("62.5");
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 \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).
\(n\)
\(3\)
\(\chi(n)\)
\(-1 - \beta_{1}\)
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
\( T_{3}^{8} - 10T_{3}^{7} + 124T_{3}^{6} - 284T_{3}^{5} + 3569T_{3}^{4} - 13748T_{3}^{3} + 59692T_{3}^{2} - 97726T_{3} + 139129 \)
T3^8 - 10*T3^7 + 124*T3^6 - 284*T3^5 + 3569*T3^4 - 13748*T3^3 + 59692*T3^2 - 97726*T3 + 139129
acting on \(S_{4}^{\mathrm{new}}(62, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T - 2)^{8} \)
(T - 2)^8
$3$
\( T^{8} - 10 T^{7} + 124 T^{6} + \cdots + 139129 \)
T^8 - 10*T^7 + 124*T^6 - 284*T^5 + 3569*T^4 - 13748*T^3 + 59692*T^2 - 97726*T + 139129
$5$
\( T^{8} + 2 T^{7} + 304 T^{6} + \cdots + 25130169 \)
T^8 + 2*T^7 + 304*T^6 - 5268*T^5 + 90345*T^4 - 680148*T^3 + 3943656*T^2 - 11700342*T + 25130169
$7$
\( T^{8} + 16 T^{7} + \cdots + 6214641889 \)
T^8 + 16*T^7 + 910*T^6 + 344*T^5 + 435347*T^4 + 1011560*T^3 + 80759998*T^2 - 426013532*T + 6214641889
$11$
\( T^{8} - 22 T^{7} + \cdots + 118985673249 \)
T^8 - 22*T^7 + 2200*T^6 + 20628*T^5 + 2788077*T^4 + 485100*T^3 + 665230032*T^2 + 2953401966*T + 118985673249
$13$
\( T^{8} - 42 T^{7} + \cdots + 731107521 \)
T^8 - 42*T^7 + 4020*T^6 + 85484*T^5 + 5257125*T^4 - 8183028*T^3 + 82473940*T^2 + 125298726*T + 731107521
$17$
\( T^{8} - 60 T^{7} + \cdots + 35162065517121 \)
T^8 - 60*T^7 + 18058*T^6 - 1038096*T^5 + 260271283*T^4 - 13063837584*T^3 + 993537457482*T^2 + 5649805123668*T + 35162065517121
$19$
\( T^{8} + 30 T^{7} + \cdots + 6026450543689 \)
T^8 + 30*T^7 + 6452*T^6 + 54180*T^5 + 31680921*T^4 + 465481260*T^3 + 25811047316*T^2 - 270945436710*T + 6026450543689
$23$
\( (T^{4} + 56 T^{3} - 10176 T^{2} + \cdots + 8626176)^{2} \)
(T^4 + 56*T^3 - 10176*T^2 - 216576*T + 8626176)^2
$29$
\( (T^{4} + 148 T^{3} - 5096 T^{2} + \cdots - 36022608)^{2} \)
(T^4 + 148*T^3 - 5096*T^2 - 1322112*T - 36022608)^2
$31$
\( T^{8} + 788 T^{7} + \cdots + 78\!\cdots\!61 \)
T^8 + 788*T^7 + 296980*T^6 + 72932212*T^5 + 13835714966*T^4 + 2172723527692*T^3 + 263570843183380*T^2 + 20834422262608748*T + 787662783788549761
$37$
\( T^{8} + 186 T^{7} + \cdots + 68\!\cdots\!49 \)
T^8 + 186*T^7 + 89712*T^6 - 2056996*T^5 + 3538446489*T^4 + 128576914236*T^3 + 31196370286312*T^2 - 1071125462622030*T + 68341936304330649
$41$
\( T^{8} - 380 T^{7} + \cdots + 15\!\cdots\!21 \)
T^8 - 380*T^7 + 192010*T^6 - 33247056*T^5 + 13270797651*T^4 - 2171895679440*T^3 + 599421178254474*T^2 - 32079158895304908*T + 1561757365761873921
$43$
\( T^{8} - 366 T^{7} + \cdots + 20\!\cdots\!69 \)
T^8 - 366*T^7 + 86736*T^6 - 12279244*T^5 + 1268810205*T^4 - 84954067476*T^3 + 4118244282904*T^2 - 113370949509306*T + 2053783391403969
$47$
\( (T^{4} - 396 T^{3} - 135904 T^{2} + \cdots - 172657152)^{2} \)
(T^4 - 396*T^3 - 135904*T^2 - 9810816*T - 172657152)^2
$53$
\( T^{8} + 250 T^{7} + \cdots + 27\!\cdots\!49 \)
T^8 + 250*T^7 + 173392*T^6 + 25805820*T^5 + 19154488521*T^4 + 3051134132220*T^3 + 697886718859656*T^2 + 4452269158394370*T + 27672441274027449
$59$
\( T^{8} - 982 T^{7} + \cdots + 16\!\cdots\!21 \)
T^8 - 982*T^7 + 1116676*T^6 - 448582452*T^5 + 358050984321*T^4 - 126342338705820*T^3 + 83192591456809092*T^2 - 12301080142487616738*T + 1691473186035105418521
$61$
\( (T^{4} + 748 T^{3} + \cdots - 26488681104)^{2} \)
(T^4 + 748*T^3 - 303688*T^2 - 232790656*T - 26488681104)^2
$67$
\( T^{8} + 418 T^{7} + \cdots + 51\!\cdots\!69 \)
T^8 + 418*T^7 + 561624*T^6 - 48094076*T^5 + 150797458429*T^4 + 3052326988668*T^3 + 11988468962784144*T^2 - 1286454783830449194*T + 512698542319542215169
$71$
\( T^{8} - 860 T^{7} + \cdots + 19\!\cdots\!09 \)
T^8 - 860*T^7 + 1108030*T^6 - 480084840*T^5 + 522183488403*T^4 - 222076111388760*T^3 + 142653366293308110*T^2 - 17437299821608557960*T + 1915018845940306459809
$73$
\( T^{8} + 1100 T^{7} + \cdots + 10\!\cdots\!89 \)
T^8 + 1100*T^7 + 2029138*T^6 + 1515236816*T^5 + 2322182115211*T^4 + 1698556801573904*T^3 + 1195656659693981218*T^2 + 389307967825245365036*T + 103836241652355422876689
$79$
\( T^{8} - 608 T^{7} + \cdots + 14\!\cdots\!29 \)
T^8 - 608*T^7 + 952366*T^6 + 75674360*T^5 + 386330536451*T^4 - 35076657412024*T^3 + 41494767389883838*T^2 + 5280752333552542756*T + 1437020632139254470529
$83$
\( T^{8} - 306 T^{7} + \cdots + 20\!\cdots\!01 \)
T^8 - 306*T^7 + 2008168*T^6 - 510684420*T^5 + 3379336527061*T^4 - 771906344418204*T^3 + 1169535250624809504*T^2 + 248838858421922338794*T + 205993916961172419705201
$89$
\( (T^{4} + 1272 T^{3} + \cdots - 1251333571248)^{2} \)
(T^4 + 1272*T^3 - 1835136*T^2 - 3344193648*T - 1251333571248)^2
$97$
\( (T^{4} + 2220 T^{3} + \cdots - 203751443472)^{2} \)
(T^4 + 2220*T^3 + 148584*T^2 - 1545398144*T - 203751443472)^2
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