[N,k,chi] = [882,2,Mod(127,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.127");
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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
\(n\)
\(199\)
\(785\)
\(\chi(n)\)
\(-\beta_{10}\)
\(1\)
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
\( T_{5}^{18} - 6 T_{5}^{17} + 37 T_{5}^{16} - 166 T_{5}^{15} + 780 T_{5}^{14} - 3448 T_{5}^{13} + 13353 T_{5}^{12} - 39036 T_{5}^{11} + 92243 T_{5}^{10} - 170517 T_{5}^{9} + 267705 T_{5}^{8} - 260614 T_{5}^{7} + 136735 T_{5}^{6} + \cdots + 729 \)
T5^18 - 6*T5^17 + 37*T5^16 - 166*T5^15 + 780*T5^14 - 3448*T5^13 + 13353*T5^12 - 39036*T5^11 + 92243*T5^10 - 170517*T5^9 + 267705*T5^8 - 260614*T5^7 + 136735*T5^6 + 118758*T5^5 + 234*T5^4 - 14418*T5^3 + 2025*T5^2 - 972*T5 + 729
acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1)^{3} \)
(T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^3
$3$
\( T^{18} \)
T^18
$5$
\( T^{18} - 6 T^{17} + 37 T^{16} - 166 T^{15} + \cdots + 729 \)
T^18 - 6*T^17 + 37*T^16 - 166*T^15 + 780*T^14 - 3448*T^13 + 13353*T^12 - 39036*T^11 + 92243*T^10 - 170517*T^9 + 267705*T^8 - 260614*T^7 + 136735*T^6 + 118758*T^5 + 234*T^4 - 14418*T^3 + 2025*T^2 - 972*T + 729
$7$
\( T^{18} + 7 T^{17} - 154 T^{15} + \cdots + 40353607 \)
T^18 + 7*T^17 - 154*T^15 - 490*T^14 + 630*T^13 + 7259*T^12 + 13034*T^11 - 29008*T^10 - 172186*T^9 - 203056*T^8 + 638666*T^7 + 2489837*T^6 + 1512630*T^5 - 8235430*T^4 - 18117946*T^3 + 40353607*T + 40353607
$11$
\( T^{18} - T^{17} + 38 T^{16} + \cdots + 6126662529 \)
T^18 - T^17 + 38*T^16 - 46*T^15 + 730*T^14 - 364*T^13 + 24514*T^12 - 182543*T^11 + 1307443*T^10 - 6555946*T^9 + 30425033*T^8 - 144474632*T^7 + 680531887*T^6 - 2436327558*T^5 + 6448182129*T^4 - 12046757454*T^3 + 16194243168*T^2 - 14117318280*T + 6126662529
$13$
\( T^{18} + 14 T^{16} + 721 T^{14} + \cdots + 19882681 \)
T^18 + 14*T^16 + 721*T^14 + 224*T^13 + 3101*T^12 - 9702*T^11 + 34447*T^10 + 41062*T^9 + 1483867*T^8 - 2067065*T^7 + 13551636*T^6 - 11249028*T^5 + 53604726*T^4 - 2156098*T^3 + 75934026*T^2 + 71883539*T + 19882681
$17$
\( T^{18} - 11 T^{17} + \cdots + 438986304 \)
T^18 - 11*T^17 + 53*T^16 - 261*T^15 + 2175*T^14 - 9834*T^13 + 59742*T^12 - 167585*T^11 + 469218*T^10 - 2547177*T^9 + 12656674*T^8 - 907451*T^7 + 3342763*T^6 + 119862402*T^5 + 663591312*T^4 + 1421224488*T^3 + 3259471104*T^2 - 349982208*T + 438986304
$19$
\( (T^{9} - 18 T^{8} + 49 T^{7} + \cdots + 500387)^{2} \)
(T^9 - 18*T^8 + 49*T^7 + 762*T^6 - 4437*T^5 - 4789*T^4 + 70778*T^3 - 60732*T^2 - 323830*T + 500387)^2
$23$
\( T^{18} + 5 T^{17} + 89 T^{16} + \cdots + 1225449 \)
T^18 + 5*T^17 + 89*T^16 + 101*T^15 + 6353*T^14 + 7259*T^13 + 229502*T^12 + 186577*T^11 + 2514647*T^10 + 3627899*T^9 + 3694646*T^8 - 838324*T^7 - 7271873*T^6 - 656250*T^5 + 19414530*T^4 + 28602234*T^3 + 20043936*T^2 + 7422435*T + 1225449
$29$
\( T^{18} - 13 T^{17} + 101 T^{16} + \cdots + 1347921 \)
T^18 - 13*T^17 + 101*T^16 + 18*T^15 - 1720*T^14 + 9856*T^13 + 40663*T^12 - 498172*T^11 + 2177102*T^10 - 1545147*T^9 - 10910296*T^8 + 32428621*T^7 + 89139589*T^6 - 148049349*T^5 + 727402176*T^4 - 283596498*T^3 + 254290671*T^2 - 23196780*T + 1347921
$31$
\( (T^{9} + 2 T^{8} - 42 T^{7} - 62 T^{6} + \cdots - 13)^{2} \)
(T^9 + 2*T^8 - 42*T^7 - 62*T^6 + 515*T^5 + 282*T^4 - 2134*T^3 + 973*T^2 - 55*T - 13)^2
$37$
\( T^{18} - 33 T^{17} + \cdots + 1703523546481 \)
T^18 - 33*T^17 + 705*T^16 - 10496*T^15 + 119060*T^14 - 1074633*T^13 + 8065218*T^12 - 51855229*T^11 + 292835902*T^10 - 1475117058*T^9 + 6714838141*T^8 - 27768570742*T^7 + 104555523254*T^6 - 353102661625*T^5 + 1032144676369*T^4 - 2414654901424*T^3 + 4005241149640*T^2 - 3913889303610*T + 1703523546481
$41$
\( T^{18} + \cdots + 661949588881161 \)
T^18 - 28*T^17 + 462*T^16 - 4585*T^15 + 23891*T^14 - 17969*T^13 + 496237*T^12 - 25932319*T^11 + 430054625*T^10 - 4290001989*T^9 + 32867102555*T^8 - 230242501293*T^7 + 1673024849230*T^6 - 11313620160591*T^5 + 60848696518911*T^4 - 224691590458629*T^3 + 522541700179947*T^2 - 573823529046054*T + 661949588881161
$43$
\( T^{18} + 20 T^{17} + \cdots + 63645702961 \)
T^18 + 20*T^17 + 206*T^16 + 969*T^15 + 800*T^14 - 17913*T^13 - 56252*T^12 + 218200*T^11 + 3051654*T^10 + 8321924*T^9 + 13573814*T^8 - 131717720*T^7 + 162855672*T^6 - 393598233*T^5 + 6914508668*T^4 + 8246715781*T^3 + 91620147169*T^2 + 36996504088*T + 63645702961
$47$
\( T^{18} + 36 T^{17} + 625 T^{16} + \cdots + 9308601 \)
T^18 + 36*T^17 + 625*T^16 + 6883*T^15 + 53868*T^14 + 318202*T^13 + 1475562*T^12 + 5505573*T^11 + 16799087*T^10 + 42499887*T^9 + 90813342*T^8 + 163288450*T^7 + 248123026*T^6 + 312502578*T^5 + 316017243*T^4 + 244880631*T^3 + 137054997*T^2 + 49865544*T + 9308601
$53$
\( T^{18} + 48 T^{17} + \cdots + 97031627001 \)
T^18 + 48*T^17 + 1186*T^16 + 19498*T^15 + 235776*T^14 + 2202277*T^13 + 16323216*T^12 + 97612140*T^11 + 478409327*T^10 + 1957291065*T^9 + 6815152143*T^8 + 20432519512*T^7 + 52573806922*T^6 + 114516133515*T^5 + 209866774842*T^4 + 318061519128*T^3 + 363412003302*T^2 + 250155501930*T + 97031627001
$59$
\( T^{18} - 25 T^{17} + \cdots + 83822409441 \)
T^18 - 25*T^17 + 459*T^16 - 5553*T^15 + 50902*T^14 - 351252*T^13 + 1848536*T^12 - 7087407*T^11 + 17997078*T^10 - 18212764*T^9 - 39885921*T^8 + 55625270*T^7 + 1039563109*T^6 - 5161602423*T^5 + 11144614347*T^4 - 8766030870*T^3 + 13727918394*T^2 - 60194021589*T + 83822409441
$61$
\( T^{18} - T^{17} + \cdots + 4441105256449 \)
T^18 - T^17 + 79*T^16 + 425*T^15 + 6723*T^14 + 11512*T^13 + 516555*T^12 + 801280*T^11 + 22211774*T^10 - 52127225*T^9 + 1598762239*T^8 - 6773297730*T^7 + 46190613326*T^6 - 132250224046*T^5 + 377509347491*T^4 - 871786428368*T^3 + 568918787579*T^2 + 4894835398921*T + 4441105256449
$67$
\( (T^{9} - 17 T^{8} - 297 T^{7} + \cdots - 149884139)^{2} \)
(T^9 - 17*T^8 - 297*T^7 + 5425*T^6 + 26355*T^5 - 557340*T^4 - 574637*T^3 + 19662029*T^2 - 5099891*T - 149884139)^2
$71$
\( T^{18} - 6 T^{17} + 3 T^{16} + \cdots + 14085009 \)
T^18 - 6*T^17 + 3*T^16 + 242*T^15 + 2312*T^14 - 12845*T^13 - 40502*T^12 + 617747*T^11 + 3822025*T^10 + 5942270*T^9 + 48132912*T^8 + 315771751*T^7 + 1589624470*T^6 + 5180474523*T^5 + 8210233665*T^4 - 221862861*T^3 + 1173382929*T^2 + 46848699*T + 14085009
$73$
\( T^{18} + \cdots + 351446147128609 \)
T^18 + 39*T^17 + 816*T^16 + 13673*T^15 + 220281*T^14 + 3432274*T^13 + 49881395*T^12 + 590504220*T^11 + 5385235815*T^10 + 41554709115*T^9 + 316126001746*T^8 + 2297916849822*T^7 + 14410271880603*T^6 + 72777579463937*T^5 + 266975232821443*T^4 + 596512099485276*T^3 + 1115835500278486*T^2 + 802493508191986*T + 351446147128609
$79$
\( (T^{9} + T^{8} - 243 T^{7} - 742 T^{6} + \cdots + 5529329)^{2} \)
(T^9 + T^8 - 243*T^7 - 742*T^6 + 18382*T^5 + 88277*T^4 - 354690*T^3 - 2431936*T^2 - 1532338*T + 5529329)^2
$83$
\( T^{18} + 35 T^{17} + \cdots + 25\!\cdots\!84 \)
T^18 + 35*T^17 + 728*T^16 + 8638*T^15 + 70420*T^14 + 320691*T^13 + 2036601*T^12 + 45652369*T^11 + 1037478127*T^10 + 15203882043*T^9 + 172583943161*T^8 + 1520431490312*T^7 + 10902799991377*T^6 + 58616229537012*T^5 + 272171250033984*T^4 + 1010912235036480*T^3 + 2515561263290880*T^2 + 3482210671220736*T + 2589575516688384
$89$
\( T^{18} - 6 T^{17} + \cdots + 10124977628529 \)
T^18 - 6*T^17 + 30*T^16 + 1794*T^15 - 1712*T^14 - 97822*T^13 + 4649684*T^12 - 26779092*T^11 + 265358356*T^10 + 1657855643*T^9 + 16977804372*T^8 + 177534013002*T^7 + 1844764504678*T^6 + 10970387846526*T^5 + 42420666698406*T^4 + 103730375826639*T^3 + 176617362555720*T^2 + 53963567963964*T + 10124977628529
$97$
\( (T^{9} - 28 T^{8} + 238 T^{7} - 224 T^{6} + \cdots + 8869)^{2} \)
(T^9 - 28*T^8 + 238*T^7 - 224*T^6 - 5159*T^5 + 16170*T^4 + 2310*T^3 - 36603*T^2 + 4263*T + 8869)^2
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