[N,k,chi] = [976,2,Mod(113,976)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(976, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("976.113");
S:= CuspForms(chi, 2);
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 \(\left(\mathbb{Z}/976\mathbb{Z}\right)^\times\).
\(n\)
\(245\)
\(367\)
\(673\)
\(\chi(n)\)
\(1\)
\(1\)
\(-\beta_{1}\)
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
\( T_{3}^{16} - T_{3}^{15} + 6 T_{3}^{14} + T_{3}^{13} + 43 T_{3}^{12} - 131 T_{3}^{11} + 504 T_{3}^{10} - 652 T_{3}^{9} + 1948 T_{3}^{8} - 2091 T_{3}^{7} + 2466 T_{3}^{6} - 1352 T_{3}^{5} + 633 T_{3}^{4} - 52 T_{3}^{3} - 16 T_{3}^{2} + 8 T_{3} + 16 \)
T3^16 - T3^15 + 6*T3^14 + T3^13 + 43*T3^12 - 131*T3^11 + 504*T3^10 - 652*T3^9 + 1948*T3^8 - 2091*T3^7 + 2466*T3^6 - 1352*T3^5 + 633*T3^4 - 52*T3^3 - 16*T3^2 + 8*T3 + 16
acting on \(S_{2}^{\mathrm{new}}(976, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} \)
T^16
$3$
\( T^{16} - T^{15} + 6 T^{14} + T^{13} + 43 T^{12} + \cdots + 16 \)
T^16 - T^15 + 6*T^14 + T^13 + 43*T^12 - 131*T^11 + 504*T^10 - 652*T^9 + 1948*T^8 - 2091*T^7 + 2466*T^6 - 1352*T^5 + 633*T^4 - 52*T^3 - 16*T^2 + 8*T + 16
$5$
\( T^{16} + 11 T^{14} - 15 T^{13} + 112 T^{12} + \cdots + 1 \)
T^16 + 11*T^14 - 15*T^13 + 112*T^12 + 255*T^11 + 1363*T^10 + 2440*T^9 + 6385*T^8 + 7020*T^7 + 14037*T^6 + 860*T^5 - 2483*T^4 - 50*T^3 + 1629*T^2 - 25*T + 1
$7$
\( T^{16} + 10 T^{15} + 51 T^{14} + \cdots + 1936 \)
T^16 + 10*T^15 + 51*T^14 + 165*T^13 + 704*T^12 + 3945*T^11 + 12657*T^10 + 13920*T^9 - 14751*T^8 - 57050*T^7 + 9194*T^6 + 84060*T^5 - 22619*T^4 - 43920*T^3 + 54848*T^2 - 18040*T + 1936
$11$
\( T^{16} + 87 T^{14} + 2756 T^{12} + \cdots + 1936 \)
T^16 + 87*T^14 + 2756*T^12 + 39221*T^10 + 255109*T^8 + 661998*T^6 + 477629*T^4 + 78724*T^2 + 1936
$13$
\( (T^{8} + 6 T^{7} - 38 T^{6} - 242 T^{5} + \cdots + 10261)^{2} \)
(T^8 + 6*T^7 - 38*T^6 - 242*T^5 + 430*T^4 + 2772*T^3 - 2638*T^2 - 9881*T + 10261)^2
$17$
\( T^{16} - 25 T^{14} - 210 T^{13} + \cdots + 4879681 \)
T^16 - 25*T^14 - 210*T^13 + 2208*T^12 + 5250*T^11 - 105200*T^10 + 93460*T^9 + 3004179*T^8 - 2114915*T^7 - 3305715*T^6 + 60532010*T^5 + 76201662*T^4 - 37873305*T^3 + 46720350*T^2 + 32185130*T + 4879681
$19$
\( T^{16} + 3 T^{15} + 36 T^{14} + \cdots + 7311616 \)
T^16 + 3*T^15 + 36*T^14 + 327*T^13 + 2696*T^12 - 13786*T^11 + 70329*T^10 + 261136*T^9 + 1051517*T^8 + 966106*T^7 + 14007439*T^6 - 4969806*T^5 + 35081121*T^4 - 38471108*T^3 + 42179696*T^2 - 23762752*T + 7311616
$23$
\( T^{16} - 15 T^{15} + 115 T^{14} + \cdots + 30976 \)
T^16 - 15*T^15 + 115*T^14 - 660*T^13 + 3637*T^12 - 19830*T^11 + 93360*T^10 - 344880*T^9 + 968974*T^8 - 2054040*T^7 + 3267730*T^6 - 3868080*T^5 + 3359253*T^4 - 2090900*T^3 + 893760*T^2 - 239360*T + 30976
$29$
\( T^{16} + 185 T^{14} + 11273 T^{12} + \cdots + 383161 \)
T^16 + 185*T^14 + 11273*T^12 + 304345*T^10 + 3899509*T^8 + 22386825*T^6 + 45705437*T^4 + 22311765*T^2 + 383161
$31$
\( T^{16} - 15 T^{15} + 105 T^{14} + \cdots + 1008016 \)
T^16 - 15*T^15 + 105*T^14 - 240*T^13 - 1258*T^12 + 10545*T^11 - 18875*T^10 - 98620*T^9 + 666689*T^8 - 1846210*T^7 + 3667805*T^6 - 9349335*T^5 + 27726233*T^4 - 49968300*T^3 + 41931520*T^2 - 11274920*T + 1008016
$37$
\( T^{16} + 5 T^{15} + \cdots + 46787420416 \)
T^16 + 5*T^15 - 115*T^14 + 70*T^13 + 11927*T^12 - 31290*T^11 - 720670*T^10 - 560520*T^9 + 54558584*T^8 + 164222510*T^7 - 364807040*T^6 + 3607503380*T^5 + 29232507733*T^4 + 21744165380*T^3 - 4662352160*T^2 + 33925119360*T + 46787420416
$41$
\( T^{16} - 12 T^{15} + 159 T^{14} + \cdots + 1437601 \)
T^16 - 12*T^15 + 159*T^14 - 1812*T^13 + 16218*T^12 - 91142*T^11 + 444796*T^10 - 1709906*T^9 + 5755703*T^8 - 14094927*T^7 + 29581129*T^6 - 28184306*T^5 + 29631578*T^4 + 47844491*T^3 + 27151796*T^2 - 10831766*T + 1437601
$43$
\( T^{16} - 25 T^{15} + \cdots + 591267856 \)
T^16 - 25*T^15 + 303*T^14 - 3180*T^13 + 42551*T^12 - 525840*T^11 + 4588559*T^10 - 31929235*T^9 + 254995504*T^8 - 2077327020*T^7 + 11969319757*T^6 - 42328629475*T^5 + 86510714899*T^4 - 91683240430*T^3 + 35723310576*T^2 + 4995965360*T + 591267856
$47$
\( (T^{8} + 3 T^{7} - 124 T^{6} - 716 T^{5} + \cdots - 3524)^{2} \)
(T^8 + 3*T^7 - 124*T^6 - 716*T^5 + 2409*T^4 + 27367*T^3 + 68369*T^2 + 49326*T - 3524)^2
$53$
\( T^{16} + 20 T^{15} + \cdots + 162537001 \)
T^16 + 20*T^15 - 13*T^14 - 2425*T^13 - 9469*T^12 + 64765*T^11 + 1245631*T^10 + 2488705*T^9 + 3961189*T^8 - 42058290*T^7 + 99124723*T^6 - 113980200*T^5 + 337276979*T^4 - 1309557500*T^3 + 2087323354*T^2 - 1014820400*T + 162537001
$59$
\( T^{16} + 5 T^{15} + \cdots + 138529862416 \)
T^16 + 5*T^15 - 180*T^14 - 1515*T^13 + 9607*T^12 + 173485*T^11 + 1027865*T^10 + 9220145*T^9 + 142003914*T^8 + 1421450690*T^7 + 8574249190*T^6 + 31973478985*T^5 + 70357909053*T^4 + 75581348920*T^3 + 29476272760*T^2 + 66202502520*T + 138529862416
$61$
\( T^{16} + \cdots + 191707312997281 \)
T^16 + 53*T^15 + 1264*T^14 + 17700*T^13 + 163718*T^12 + 1199030*T^11 + 10144001*T^10 + 106364807*T^9 + 963246013*T^8 + 6488253227*T^7 + 37745827721*T^6 + 272157028430*T^5 + 2266813396838*T^4 + 14949354527700*T^3 + 65121753192304*T^2 + 166565370309113*T + 191707312997281
$67$
\( T^{16} - 55 T^{15} + \cdots + 395837272336 \)
T^16 - 55*T^15 + 1355*T^14 - 20475*T^13 + 232053*T^12 - 2369540*T^11 + 23395915*T^10 - 207458760*T^9 + 1513828394*T^8 - 8725004425*T^7 + 39178639680*T^6 - 136043784405*T^5 + 360845437077*T^4 - 713102272280*T^3 + 1000050118920*T^2 - 899246379240*T + 395837272336
$71$
\( T^{16} - 50 T^{15} + \cdots + 121101216016 \)
T^16 - 50*T^15 + 1026*T^14 - 10900*T^13 + 61464*T^12 - 122800*T^11 - 293448*T^10 - 1005760*T^9 + 22086204*T^8 + 526455*T^7 - 558306726*T^6 + 1269000190*T^5 + 12074523441*T^4 + 7842296090*T^3 - 5005860392*T^2 + 70872865360*T + 121101216016
$73$
\( T^{16} + 11 T^{15} + \cdots + 687101735056 \)
T^16 + 11*T^15 + 74*T^14 + 491*T^13 + 14331*T^12 + 18513*T^11 + 672131*T^10 + 5912003*T^9 + 45173152*T^8 + 241017122*T^7 + 1656349196*T^6 + 6784966787*T^5 + 26600133711*T^4 + 81283937854*T^3 + 238029559164*T^2 + 406677794424*T + 687101735056
$79$
\( T^{16} + 40 T^{15} + \cdots + 2515192996096 \)
T^16 + 40*T^15 + 688*T^14 + 5195*T^13 - 6454*T^12 - 377760*T^11 - 1194136*T^10 + 16499840*T^9 + 95303404*T^8 - 491020160*T^7 - 207839798*T^6 + 66003267600*T^5 + 318703374109*T^4 - 103531566980*T^3 + 663609099456*T^2 + 3152460143360*T + 2515192996096
$83$
\( T^{16} + 31 T^{15} + \cdots + 5678526736 \)
T^16 + 31*T^15 + 524*T^14 + 6096*T^13 + 75326*T^12 + 648143*T^11 + 4786831*T^10 + 29607513*T^9 + 157205207*T^8 + 691901457*T^7 + 2469560586*T^6 + 6792072547*T^5 + 14127820391*T^4 + 21652351954*T^3 + 23659165664*T^2 + 16357074784*T + 5678526736
$89$
\( T^{16} - 60 T^{15} + \cdots + 10\!\cdots\!01 \)
T^16 - 60*T^15 + 1515*T^14 - 16245*T^13 - 63203*T^12 + 3656505*T^11 - 27789715*T^10 - 234436195*T^9 + 5298727629*T^8 - 29624245130*T^7 + 122099493905*T^6 - 3264182062050*T^5 + 38426648249173*T^4 - 141132542698590*T^3 + 49109649674910*T^2 - 1360843528250550*T + 10571397384924601
$97$
\( T^{16} - 45 T^{15} + \cdots + 394856641 \)
T^16 - 45*T^15 + 1244*T^14 - 23500*T^13 + 332912*T^12 - 3624925*T^11 + 30813742*T^10 - 203926145*T^9 + 1048592185*T^8 - 4106238865*T^7 + 11873488188*T^6 - 24138326165*T^5 + 32766093282*T^4 - 28230829385*T^3 + 17322855371*T^2 - 3991984545*T + 394856641
show more
show less