[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
\( T_{5}^{18} + 7 T_{5}^{16} + 28 T_{5}^{15} + 42 T_{5}^{14} - 42 T_{5}^{13} + 2415 T_{5}^{12} - 1372 T_{5}^{11} + 30331 T_{5}^{10} + 10143 T_{5}^{9} + 224861 T_{5}^{8} + 383866 T_{5}^{7} + 1522969 T_{5}^{6} + \cdots + 1750329 \)
T5^18 + 7*T5^16 + 28*T5^15 + 42*T5^14 - 42*T5^13 + 2415*T5^12 - 1372*T5^11 + 30331*T5^10 + 10143*T5^9 + 224861*T5^8 + 383866*T5^7 + 1522969*T5^6 + 2364642*T5^5 + 5130594*T5^4 + 6982794*T5^3 + 3306177*T5^2 - 2333772*T5 + 1750329
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
$5$
\( T^{18} + 7 T^{16} + 28 T^{15} + \cdots + 1750329 \)
T^18 + 7*T^16 + 28*T^15 + 42*T^14 - 42*T^13 + 2415*T^12 - 1372*T^11 + 30331*T^10 + 10143*T^9 + 224861*T^8 + 383866*T^7 + 1522969*T^6 + 2364642*T^5 + 5130594*T^4 + 6982794*T^3 + 3306177*T^2 - 2333772*T + 1750329
$7$
\( T^{18} + T^{17} + 22 T^{16} + \cdots + 40353607 \)
T^18 + T^17 + 22*T^16 + 8*T^15 + 274*T^14 + 36*T^13 + 2633*T^12 + 21728*T^10 + 294*T^9 + 152096*T^8 + 903119*T^6 + 86436*T^5 + 4605118*T^4 + 941192*T^3 + 18117946*T^2 + 5764801*T + 40353607
$11$
\( T^{18} + 7 T^{17} + 28 T^{16} + \cdots + 1750329 \)
T^18 + 7*T^17 + 28*T^16 + 84*T^15 + 392*T^14 + 1512*T^13 + 4032*T^12 + 8673*T^11 + 26509*T^10 + 36456*T^9 + 85407*T^8 + 462658*T^7 + 1973671*T^6 + 5525730*T^5 + 10869327*T^4 + 11428074*T^3 + 7779240*T^2 + 1166886*T + 1750329
$13$
\( T^{18} + 10 T^{17} + 60 T^{16} + \cdots + 253009 \)
T^18 + 10*T^17 + 60*T^16 + 12*T^15 - 1017*T^14 - 3268*T^13 + 46967*T^12 + 529496*T^11 + 2635977*T^10 + 8336402*T^9 + 20946011*T^8 + 42480775*T^7 + 62115862*T^6 + 52048418*T^5 + 13737954*T^4 - 9071428*T^3 + 13645304*T^2 - 981353*T + 253009
$17$
\( T^{18} + T^{17} + 9 T^{16} + \cdots + 78428736 \)
T^18 + T^17 + 9*T^16 - 5*T^15 + 115*T^14 - 298*T^13 + 34542*T^12 - 195913*T^11 + 1182262*T^10 - 4010969*T^9 + 18592130*T^8 - 21913427*T^7 + 42736579*T^6 + 58593762*T^5 + 158596056*T^4 + 177948360*T^3 + 268207200*T^2 + 225721728*T + 78428736
$19$
\( (T^{9} + 22 T^{8} + 189 T^{7} + 778 T^{6} + \cdots + 1597)^{2} \)
(T^9 + 22*T^8 + 189*T^7 + 778*T^6 + 1381*T^5 - 171*T^4 - 3884*T^3 - 3612*T^2 + 1010*T + 1597)^2
$23$
\( T^{18} + 21 T^{17} + \cdots + 371981669409 \)
T^18 + 21*T^17 + 189*T^16 + 1113*T^15 + 7525*T^14 + 39165*T^13 + 161420*T^12 + 735147*T^11 + 3602921*T^10 + 10527601*T^9 + 52204796*T^8 + 55030332*T^7 + 818884423*T^6 + 3390939846*T^5 + 21961368702*T^4 + 48862814112*T^3 + 134469997830*T^2 + 146587136535*T + 371981669409
$29$
\( T^{18} + 11 T^{17} + \cdots + 86066940809529 \)
T^18 + 11*T^17 + 139*T^16 + 536*T^15 + 5188*T^14 + 1190*T^13 + 150129*T^12 - 188256*T^11 + 6845378*T^10 - 36441183*T^9 + 491445270*T^8 - 3143094995*T^7 + 21975520063*T^6 - 102481655745*T^5 + 549604784772*T^4 - 1679264344278*T^3 + 5293169066331*T^2 - 25121320577496*T + 86066940809529
$31$
\( (T^{9} + 12 T^{8} - 56 T^{7} - 1240 T^{6} + \cdots - 24347)^{2} \)
(T^9 + 12*T^8 - 56*T^7 - 1240*T^6 - 3377*T^5 + 17400*T^4 + 98988*T^3 + 130109*T^2 - 1161*T - 24347)^2
$37$
\( T^{18} + 13 T^{17} + \cdots + 1254429120169 \)
T^18 + 13*T^17 + 101*T^16 + 1228*T^15 + 15472*T^14 + 68537*T^13 + 595812*T^12 + 5133075*T^11 + 39695086*T^10 - 172176114*T^9 - 555566401*T^8 + 11051482070*T^7 + 84471288972*T^6 + 265028734817*T^5 + 605499153109*T^4 + 902032887234*T^3 + 1468869870774*T^2 + 1012518632312*T + 1254429120169
$41$
\( T^{18} + 8 T^{17} - 26 T^{16} + \cdots + 463067361 \)
T^18 + 8*T^17 - 26*T^16 - 1293*T^15 - 613*T^14 + 89295*T^13 + 544597*T^12 - 1356555*T^11 - 19269603*T^10 - 16537329*T^9 + 461219791*T^8 + 1948621279*T^7 + 5210596822*T^6 + 7911800865*T^5 + 10185015627*T^4 + 6839377911*T^3 + 4818611835*T^2 + 1223613378*T + 463067361
$43$
\( T^{18} + 24 T^{17} + \cdots + 2751859630129 \)
T^18 + 24*T^17 + 300*T^16 + 2173*T^15 + 7764*T^14 - 6139*T^13 + 196238*T^12 + 8859982*T^11 + 135246868*T^10 + 1330651744*T^9 + 9881024770*T^8 + 54046154636*T^7 + 203088943106*T^6 + 437423920521*T^5 + 387592774626*T^4 - 106853537249*T^3 + 2978953524455*T^2 - 1716542060972*T + 2751859630129
$47$
\( T^{18} + 40 T^{17} + \cdots + 13724825409 \)
T^18 + 40*T^17 + 785*T^16 + 9441*T^15 + 78784*T^14 + 538582*T^13 + 3682158*T^12 + 24722755*T^11 + 140514817*T^10 + 621770769*T^9 + 2107900992*T^8 + 5496161624*T^7 + 11131940992*T^6 + 17548401858*T^5 + 22523807745*T^4 + 25715323233*T^3 + 24526723617*T^2 + 11747868534*T + 13724825409
$53$
\( T^{18} + 10 T^{17} + \cdots + 9696939948081 \)
T^18 + 10*T^17 + 188*T^16 + 864*T^15 + 2080*T^14 - 48937*T^13 + 728112*T^12 - 16936922*T^11 + 193342451*T^10 - 656118713*T^9 + 11463793909*T^8 - 114073595330*T^7 + 631392395788*T^6 - 2479980287421*T^5 + 9170586885132*T^4 - 22823575343370*T^3 + 35911426470156*T^2 + 21760500600198*T + 9696939948081
$59$
\( T^{18} + 13 T^{17} + \cdots + 3058927542441 \)
T^18 + 13*T^17 + 9*T^16 + 999*T^15 + 33764*T^14 + 135720*T^13 + 422832*T^12 + 2964231*T^11 + 9413212*T^10 - 40058896*T^9 + 444392575*T^8 - 1623231088*T^7 + 9423894757*T^6 - 22187218707*T^5 + 106979389821*T^4 - 195676081722*T^3 + 805137414804*T^2 - 1116133685577*T + 3058927542441
$61$
\( T^{18} - 27 T^{17} + \cdots + 18702537077449 \)
T^18 - 27*T^17 + 541*T^16 - 9105*T^15 + 132317*T^14 - 1639726*T^13 + 18449617*T^12 - 190188464*T^11 + 1783324330*T^10 - 14912838273*T^9 + 113420415035*T^8 - 760416751840*T^7 + 4268179720662*T^6 - 19173934130166*T^5 + 65128660023409*T^4 - 148687498946362*T^3 + 186973597546143*T^2 - 77520363156109*T + 18702537077449
$67$
\( (T^{9} + 43 T^{8} + 653 T^{7} + \cdots - 73289)^{2} \)
(T^9 + 43*T^8 + 653*T^7 + 3255*T^6 - 14189*T^5 - 198982*T^4 - 547855*T^3 + 10175*T^2 + 279521*T - 73289)^2
$71$
\( T^{18} + \cdots + 700061588590329 \)
T^18 - 77*T^16 - 448*T^15 + 43610*T^14 + 90013*T^13 - 2384830*T^12 - 1084223*T^11 + 954662639*T^10 + 3657147634*T^9 + 64705211676*T^8 + 2158875957*T^7 + 2235915465826*T^6 - 10032653250333*T^5 + 47748587519367*T^4 - 259540028474265*T^3 + 808814383217127*T^2 - 547650560202795*T + 700061588590329
$73$
\( T^{18} - 5 T^{17} + \cdots + 10720627932169 \)
T^18 - 5*T^17 - 104*T^16 + 205*T^15 + 12137*T^14 + 3358*T^13 + 242127*T^12 - 3738820*T^11 + 20298539*T^10 - 220055049*T^9 + 2662931230*T^8 + 7869492426*T^7 + 131359311339*T^6 + 421846588365*T^5 + 2635189696963*T^4 + 5868592458080*T^3 + 20494138900218*T^2 + 18371632482942*T + 10720627932169
$79$
\( (T^{9} + 33 T^{8} + 225 T^{7} + \cdots + 10692919)^{2} \)
(T^9 + 33*T^8 + 225*T^7 - 3234*T^6 - 48300*T^5 - 93387*T^4 + 1413622*T^3 + 8634534*T^2 + 17020054*T + 10692919)^2
$83$
\( T^{18} + \cdots + 115077625675776 \)
T^18 + 55*T^17 + 1556*T^16 + 30084*T^15 + 445294*T^14 + 5297933*T^13 + 52526751*T^12 + 444252729*T^11 + 3226311939*T^10 + 19971804283*T^9 + 105636504439*T^8 + 474680480628*T^7 + 1799871526345*T^6 + 5377697633424*T^5 + 14189527077120*T^4 + 44555584868352*T^3 + 127519029264384*T^2 + 157762131591168*T + 115077625675776
$89$
\( T^{18} + 62 T^{17} + \cdots + 60940353321 \)
T^18 + 62*T^17 + 1962*T^16 + 39198*T^15 + 536840*T^14 + 5128414*T^13 + 33608320*T^12 + 140228540*T^11 + 305329812*T^10 + 57820123*T^9 - 549648704*T^8 - 2129269138*T^7 + 11567434834*T^6 + 40402211322*T^5 + 67987862946*T^4 + 93551803875*T^3 + 120306613716*T^2 + 67443412644*T + 60940353321
$97$
\( (T^{9} + 16 T^{8} - 238 T^{7} + \cdots + 1259329)^{2} \)
(T^9 + 16*T^8 - 238*T^7 - 4836*T^6 - 8123*T^5 + 128690*T^4 + 251546*T^3 - 961219*T^2 - 1444057*T + 1259329)^2
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