[N,k,chi] = [880,2,Mod(81,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("880.81");
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 \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).
\(n\)
\(111\)
\(177\)
\(321\)
\(661\)
\(\chi(n)\)
\(1\)
\(1\)
\(-\beta_{12}\)
\(1\)
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
\( T_{3}^{16} - 3 T_{3}^{15} + 14 T_{3}^{14} - 32 T_{3}^{13} + 141 T_{3}^{12} - 220 T_{3}^{11} + 1105 T_{3}^{10} - 1935 T_{3}^{9} + 9865 T_{3}^{8} - 18475 T_{3}^{7} + 34075 T_{3}^{6} - 18400 T_{3}^{5} + 23025 T_{3}^{4} + \cdots + 10000 \)
T3^16 - 3*T3^15 + 14*T3^14 - 32*T3^13 + 141*T3^12 - 220*T3^11 + 1105*T3^10 - 1935*T3^9 + 9865*T3^8 - 18475*T3^7 + 34075*T3^6 - 18400*T3^5 + 23025*T3^4 + 26000*T3^3 + 41000*T3^2 + 25000*T3 + 10000
acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} \)
T^16
$3$
\( T^{16} - 3 T^{15} + 14 T^{14} + \cdots + 10000 \)
T^16 - 3*T^15 + 14*T^14 - 32*T^13 + 141*T^12 - 220*T^11 + 1105*T^10 - 1935*T^9 + 9865*T^8 - 18475*T^7 + 34075*T^6 - 18400*T^5 + 23025*T^4 + 26000*T^3 + 41000*T^2 + 25000*T + 10000
$5$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \)
(T^4 + T^3 + T^2 + T + 1)^4
$7$
\( T^{16} + 8 T^{15} + 44 T^{14} + 194 T^{13} + \cdots + 256 \)
T^16 + 8*T^15 + 44*T^14 + 194*T^13 + 964*T^12 + 2810*T^11 + 9107*T^10 + 30363*T^9 + 147278*T^8 + 587948*T^7 + 1554637*T^6 + 2348205*T^5 + 1916969*T^4 + 601184*T^3 + 188384*T^2 + 4928*T + 256
$11$
\( T^{16} - 7 T^{15} + 35 T^{14} + \cdots + 214358881 \)
T^16 - 7*T^15 + 35*T^14 - 74*T^13 - 24*T^12 + 603*T^11 + 738*T^10 - 16446*T^9 + 88553*T^8 - 180906*T^7 + 89298*T^6 + 802593*T^5 - 351384*T^4 - 11917774*T^3 + 62004635*T^2 - 136410197*T + 214358881
$13$
\( T^{16} + 11 T^{15} + 93 T^{14} + \cdots + 36772096 \)
T^16 + 11*T^15 + 93*T^14 + 649*T^13 + 3918*T^12 + 17021*T^11 + 57090*T^10 + 141685*T^9 + 353726*T^8 + 1088405*T^7 + 3826465*T^6 + 9377454*T^5 + 21632193*T^4 + 46121096*T^3 + 83794048*T^2 + 49943104*T + 36772096
$17$
\( T^{16} - 9 T^{15} + 52 T^{14} + \cdots + 9759376 \)
T^16 - 9*T^15 + 52*T^14 - 36*T^13 + 1249*T^12 - 20834*T^11 + 247763*T^10 - 1056019*T^9 + 3484653*T^8 - 8571961*T^7 + 35514633*T^6 + 6618782*T^5 + 68199785*T^4 + 129377168*T^3 + 124744376*T^2 + 30796392*T + 9759376
$19$
\( T^{16} - 2 T^{15} + \cdots + 5603271025 \)
T^16 - 2*T^15 + 6*T^14 + 135*T^13 + 2385*T^12 - 21152*T^11 + 156029*T^10 + 90738*T^9 + 585255*T^8 - 6380595*T^7 + 70465926*T^6 + 24632973*T^5 + 1037562406*T^4 + 131538385*T^3 + 4817025235*T^2 + 2079097625*T + 5603271025
$23$
\( (T^{8} + 10 T^{7} + 8 T^{6} - 89 T^{5} + \cdots + 4)^{2} \)
(T^8 + 10*T^7 + 8*T^6 - 89*T^5 - 55*T^4 + 239*T^3 + 15*T^2 - 108*T + 4)^2
$29$
\( T^{16} - T^{15} + 109 T^{14} + \cdots + 4833030400 \)
T^16 - T^15 + 109*T^14 + 45*T^13 + 5990*T^12 - 12041*T^11 + 258481*T^10 - 1461484*T^9 + 13205365*T^8 - 69122490*T^7 + 292252864*T^6 - 283582584*T^5 - 355722704*T^4 + 771472800*T^3 + 5200988160*T^2 + 4446499200*T + 4833030400
$31$
\( T^{16} - 2 T^{15} + 13 T^{14} + \cdots + 6150400 \)
T^16 - 2*T^15 + 13*T^14 - 86*T^13 + 2373*T^12 + 8066*T^11 + 89417*T^10 - 47606*T^9 + 1122873*T^8 - 1303634*T^7 + 34025688*T^6 + 51811592*T^5 + 338599696*T^4 + 110108000*T^3 + 48242560*T^2 + 19740800*T + 6150400
$37$
\( T^{16} + 16 T^{15} + \cdots + 1041256976400 \)
T^16 + 16*T^15 + 237*T^14 + 2742*T^13 + 28843*T^12 + 203872*T^11 + 1298433*T^10 + 6725372*T^9 + 50568468*T^8 + 390493212*T^7 + 2721063592*T^6 + 13612584416*T^5 + 53465513881*T^4 + 147672620130*T^3 + 370260480060*T^2 + 748508682600*T + 1041256976400
$41$
\( T^{16} + 11 T^{15} + \cdots + 16066830025 \)
T^16 + 11*T^15 + 31*T^14 - 461*T^13 + 4622*T^12 - 5401*T^11 + 181136*T^10 - 242863*T^9 + 2237749*T^8 + 355442*T^7 + 55251263*T^6 + 135244810*T^5 + 879516221*T^4 + 1664305145*T^3 + 7222228610*T^2 + 7483615200*T + 16066830025
$43$
\( (T^{8} - 8 T^{7} - 195 T^{6} + 1955 T^{5} + \cdots - 76780)^{2} \)
(T^8 - 8*T^7 - 195*T^6 + 1955*T^5 + 6195*T^4 - 106368*T^3 + 217169*T^2 + 278850*T - 76780)^2
$47$
\( T^{16} - 10 T^{15} + \cdots + 786279452176 \)
T^16 - 10*T^15 + 21*T^14 + 367*T^13 + 16484*T^12 - 42381*T^11 + 1003986*T^10 - 4670437*T^9 + 81623486*T^8 + 134106017*T^7 + 2630917509*T^6 + 16085031373*T^5 + 94435332441*T^4 + 150261641672*T^3 + 474283017024*T^2 - 24159682104*T + 786279452176
$53$
\( T^{16} + 9 T^{15} + 301 T^{14} + \cdots + 23232400 \)
T^16 + 9*T^15 + 301*T^14 + 2856*T^13 + 45912*T^12 + 388321*T^11 + 4136691*T^10 + 26730088*T^9 + 239745024*T^8 + 950632228*T^7 + 6580017778*T^6 + 22227191900*T^5 + 38513550961*T^4 + 27051795470*T^3 + 7485471460*T^2 - 699141000*T + 23232400
$59$
\( T^{16} - 60 T^{15} + \cdots + 138039001 \)
T^16 - 60*T^15 + 1716*T^14 - 30107*T^13 + 360429*T^12 - 3117884*T^11 + 20298541*T^10 - 102145953*T^9 + 403949096*T^8 - 1258731377*T^7 + 3076804589*T^6 - 5784010048*T^5 + 8117152481*T^4 - 7839355697*T^3 + 5041824594*T^2 - 694060426*T + 138039001
$61$
\( T^{16} - 30 T^{15} + \cdots + 2548544045056 \)
T^16 - 30*T^15 + 494*T^14 - 4736*T^13 + 34004*T^12 - 182708*T^11 + 1986869*T^10 - 23159024*T^9 + 250313801*T^8 - 1541339604*T^7 + 8096422336*T^6 - 11248436264*T^5 + 46975189776*T^4 + 286993610496*T^3 + 1214472114176*T^2 + 1871050637312*T + 2548544045056
$67$
\( (T^{8} - 2 T^{7} - 266 T^{6} + \cdots + 2034124)^{2} \)
(T^8 - 2*T^7 - 266*T^6 + 435*T^5 + 21219*T^4 - 40229*T^3 - 567145*T^2 + 1278334*T + 2034124)^2
$71$
\( T^{16} - 10 T^{15} + \cdots + 240100000000 \)
T^16 - 10*T^15 + 255*T^14 - 2375*T^13 + 43590*T^12 - 82475*T^11 + 4005975*T^10 + 15228375*T^9 + 332458275*T^8 + 559968250*T^7 + 8404514000*T^6 - 26600285000*T^5 + 165185410000*T^4 - 354220300000*T^3 + 4329444000000*T^2 - 1642970000000*T + 240100000000
$73$
\( T^{16} - T^{15} + 242 T^{14} + \cdots + 2853696400 \)
T^16 - T^15 + 242*T^14 + 2048*T^13 + 20163*T^12 + 215618*T^11 + 2156633*T^10 + 2231653*T^9 - 13391387*T^8 + 85367973*T^7 + 1154082267*T^6 + 3067806084*T^5 + 3693842761*T^4 + 2737158540*T^3 + 10979068560*T^2 - 2579651800*T + 2853696400
$79$
\( T^{16} + \cdots + 153593389158400 \)
T^16 + 19*T^15 + 188*T^14 + 1915*T^13 + 47581*T^12 + 510445*T^11 + 6123582*T^10 + 90717143*T^9 + 1200301411*T^8 + 8886052590*T^7 + 75927048600*T^6 + 524088023840*T^5 + 2742753021840*T^4 + 7019508056000*T^3 + 30632480505600*T^2 + 32261195033600*T + 153593389158400
$83$
\( T^{16} + 64 T^{15} + \cdots + 61931296686736 \)
T^16 + 64*T^15 + 2174*T^14 + 49604*T^13 + 860910*T^12 + 11934014*T^11 + 137486207*T^10 + 1311911977*T^9 + 10007117498*T^8 + 54615898422*T^7 + 188903938737*T^6 + 317759585223*T^5 + 622749674729*T^4 - 802182022672*T^3 + 15798407980048*T^2 + 15509195348728*T + 61931296686736
$89$
\( (T^{8} - 12 T^{7} - 277 T^{6} + \cdots + 2898841)^{2} \)
(T^8 - 12*T^7 - 277*T^6 + 2296*T^5 + 24520*T^4 - 84944*T^3 - 749117*T^2 - 33372*T + 2898841)^2
$97$
\( T^{16} + 3 T^{15} + \cdots + 11\!\cdots\!16 \)
T^16 + 3*T^15 - 27*T^14 + 823*T^13 + 92904*T^12 - 700457*T^11 + 11656342*T^10 + 133481947*T^9 + 2024752498*T^8 - 36315947943*T^7 + 661310292237*T^6 - 3420418777536*T^5 + 38545545769705*T^4 - 36791839053396*T^3 + 1325619333416864*T^2 + 1358872115871744*T + 11155817610178816
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