[N,k,chi] = [546,2,Mod(121,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.121");
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 \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
\(n\)
\(157\)
\(365\)
\(379\)
\(\chi(n)\)
\(-\beta_{3}\)
\(1\)
\(1 - \beta_{3}\)
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
\( T_{5}^{16} - 20 T_{5}^{14} + 335 T_{5}^{12} - 390 T_{5}^{11} - 1112 T_{5}^{10} + 1212 T_{5}^{9} + 3608 T_{5}^{8} + 1110 T_{5}^{7} - 2168 T_{5}^{6} - 1146 T_{5}^{5} + 1375 T_{5}^{4} + 1680 T_{5}^{3} + 708 T_{5}^{2} + 126 T_{5} + 9 \)
T5^16 - 20*T5^14 + 335*T5^12 - 390*T5^11 - 1112*T5^10 + 1212*T5^9 + 3608*T5^8 + 1110*T5^7 - 2168*T5^6 - 1146*T5^5 + 1375*T5^4 + 1680*T5^3 + 708*T5^2 + 126*T5 + 9
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{4} \)
(T^4 - T^2 + 1)^4
$3$
\( (T + 1)^{16} \)
(T + 1)^16
$5$
\( T^{16} - 20 T^{14} + 335 T^{12} - 390 T^{11} + \cdots + 9 \)
T^16 - 20*T^14 + 335*T^12 - 390*T^11 - 1112*T^10 + 1212*T^9 + 3608*T^8 + 1110*T^7 - 2168*T^6 - 1146*T^5 + 1375*T^4 + 1680*T^3 + 708*T^2 + 126*T + 9
$7$
\( T^{16} - 8 T^{15} + 33 T^{14} + \cdots + 5764801 \)
T^16 - 8*T^15 + 33*T^14 - 116*T^13 + 322*T^12 - 570*T^11 + 419*T^10 + 1346*T^9 - 6261*T^8 + 9422*T^7 + 20531*T^6 - 195510*T^5 + 773122*T^4 - 1949612*T^3 + 3882417*T^2 - 6588344*T + 5764801
$11$
\( T^{16} + 68 T^{14} + 1716 T^{12} + \cdots + 1089 \)
T^16 + 68*T^14 + 1716*T^12 + 20446*T^10 + 123150*T^8 + 364580*T^6 + 463621*T^4 + 203262*T^2 + 1089
$13$
\( T^{16} + 10 T^{15} + \cdots + 815730721 \)
T^16 + 10*T^15 + 23*T^14 - 78*T^13 - 506*T^12 - 1726*T^11 - 4623*T^10 + 17920*T^9 + 163851*T^8 + 232960*T^7 - 781287*T^6 - 3792022*T^5 - 14451866*T^4 - 28960854*T^3 + 111016607*T^2 + 627485170*T + 815730721
$17$
\( T^{16} + 54 T^{14} - 4 T^{13} + 2160 T^{12} + \cdots + 729 \)
T^16 + 54*T^14 - 4*T^13 + 2160*T^12 - 162*T^11 + 37050*T^10 - 5454*T^9 + 469665*T^8 + 27968*T^7 + 1436184*T^6 + 1816560*T^5 + 3527497*T^4 + 2139210*T^3 + 1234953*T^2 + 30618*T + 729
$19$
\( T^{16} + 176 T^{14} + \cdots + 50936769 \)
T^16 + 176*T^14 + 12186*T^12 + 422464*T^10 + 7706091*T^8 + 70036928*T^6 + 258776362*T^4 + 225306000*T^2 + 50936769
$23$
\( T^{16} + 16 T^{15} + 250 T^{14} + \cdots + 55935441 \)
T^16 + 16*T^15 + 250*T^14 + 2068*T^13 + 20626*T^12 + 144994*T^11 + 1084222*T^10 + 5065840*T^9 + 20568661*T^8 + 47568240*T^7 + 106883922*T^6 + 130044672*T^5 + 291981087*T^4 + 251246232*T^3 + 451056357*T^2 - 135429732*T + 55935441
$29$
\( T^{16} + 4 T^{15} + 146 T^{14} + \cdots + 845588241 \)
T^16 + 4*T^15 + 146*T^14 + 784*T^13 + 15874*T^12 + 78974*T^11 + 819338*T^10 + 3634502*T^9 + 28746529*T^8 + 104710404*T^7 + 470194938*T^6 + 777506310*T^5 + 2033942643*T^4 + 752910714*T^3 + 6682606677*T^2 + 2338126074*T + 845588241
$31$
\( T^{16} - 12 T^{15} + \cdots + 5963391729 \)
T^16 - 12*T^15 - 70*T^14 + 1416*T^13 + 5094*T^12 - 160170*T^11 + 581374*T^10 + 3419496*T^9 - 22174851*T^8 - 73152960*T^7 + 934633838*T^6 - 2628610344*T^5 + 945192091*T^4 + 5668545912*T^3 - 120481959*T^2 - 8371590984*T + 5963391729
$37$
\( T^{16} - 30 T^{15} + \cdots + 18014203089 \)
T^16 - 30*T^15 + 234*T^14 + 1980*T^13 - 30870*T^12 - 212706*T^11 + 6135750*T^10 - 38129130*T^9 - 15456987*T^8 + 1033630146*T^7 - 113745870*T^6 - 50453705088*T^5 + 319891238307*T^4 - 890018930592*T^3 + 1076992316235*T^2 - 233434501344*T + 18014203089
$41$
\( T^{16} + 18 T^{15} + \cdots + 24389381241 \)
T^16 + 18*T^15 + 30*T^14 - 1404*T^13 - 4692*T^12 + 80166*T^11 + 428258*T^10 - 2078556*T^9 - 14175063*T^8 + 39397266*T^7 + 353842224*T^6 - 144591792*T^5 - 3962896775*T^4 - 754806966*T^3 + 32623530423*T^2 + 49136706414*T + 24389381241
$43$
\( T^{16} + 32 T^{15} + \cdots + 3870808348969 \)
T^16 + 32*T^15 + 730*T^14 + 10212*T^13 + 118268*T^12 + 1026662*T^11 + 8320602*T^10 + 55620622*T^9 + 366695941*T^8 + 1941903776*T^7 + 10024795764*T^6 + 40453591852*T^5 + 170241737333*T^4 + 536390504046*T^3 + 1666204105117*T^2 + 2804065975006*T + 3870808348969
$47$
\( T^{16} - 66 T^{15} + \cdots + 24307197434289 \)
T^16 - 66*T^15 + 1925*T^14 - 31218*T^13 + 282285*T^12 - 996288*T^11 - 5223668*T^10 + 50329134*T^9 + 211792059*T^8 - 4225637976*T^7 + 15365240918*T^6 + 37702562436*T^5 - 319301019236*T^4 - 542824551192*T^3 + 8716448907600*T^2 - 24417462095334*T + 24307197434289
$53$
\( T^{16} - 2 T^{15} + \cdots + 689748521121 \)
T^16 - 2*T^15 + 229*T^14 - 1954*T^13 + 42019*T^12 - 321650*T^11 + 3974186*T^10 - 29397920*T^9 + 261948253*T^8 - 1444002202*T^7 + 7545464746*T^6 - 22871615102*T^5 + 68495393026*T^4 - 101343009648*T^3 + 320399410950*T^2 - 389237251392*T + 689748521121
$59$
\( T^{16} + 36 T^{15} + \cdots + 63426911409 \)
T^16 + 36*T^15 + 366*T^14 - 2376*T^13 - 51547*T^12 + 534240*T^11 + 19046766*T^10 + 198279660*T^9 + 1002175036*T^8 + 1990641204*T^7 - 2823362790*T^6 - 14616496440*T^5 + 17268058845*T^4 + 104302583496*T^3 + 50453741322*T^2 - 137058159564*T + 63426911409
$61$
\( (T^{8} + 4 T^{7} - 156 T^{6} - 1200 T^{5} + \cdots - 2816)^{2} \)
(T^8 + 4*T^7 - 156*T^6 - 1200*T^5 - 224*T^4 + 14592*T^3 + 24960*T^2 + 5888*T - 2816)^2
$67$
\( T^{16} + 698 T^{14} + \cdots + 59619415692321 \)
T^16 + 698*T^14 + 199671*T^12 + 30027608*T^10 + 2526460285*T^8 + 116584132584*T^6 + 2679933282399*T^4 + 25032524425614*T^2 + 59619415692321
$71$
\( T^{16} + \cdots + 610341027216561 \)
T^16 + 30*T^15 - 72*T^14 - 11160*T^13 - 18693*T^12 + 2858598*T^11 + 15662484*T^10 - 400741830*T^9 - 2548554516*T^8 + 41313331044*T^7 + 292923808596*T^6 - 2522510071326*T^5 - 15147190490049*T^4 + 97567384514976*T^3 + 602891034017400*T^2 - 1096953326616492*T + 610341027216561
$73$
\( T^{16} + 18 T^{15} + \cdots + 42\!\cdots\!81 \)
T^16 + 18*T^15 - 283*T^14 - 7038*T^13 + 66096*T^12 + 2349576*T^11 + 4915447*T^10 - 290298816*T^9 - 1675889709*T^8 + 28880778672*T^7 + 401805924479*T^6 + 1211693176248*T^5 - 7407240542996*T^4 - 42970769322450*T^3 + 181056304529817*T^2 + 1891523913852150*T + 4284129113721081
$79$
\( T^{16} + 24 T^{15} + \cdots + 1257624369 \)
T^16 + 24*T^15 + 516*T^14 + 5060*T^13 + 55203*T^12 + 306798*T^11 + 3324984*T^10 + 13373676*T^9 + 102182940*T^8 + 28485086*T^7 + 932604096*T^6 + 376447506*T^5 + 5283824647*T^4 - 2763353436*T^3 + 7260138876*T^2 + 2519078742*T + 1257624369
$83$
\( T^{16} + 844 T^{14} + \cdots + 40\!\cdots\!89 \)
T^16 + 844*T^14 + 296316*T^12 + 56648614*T^10 + 6458734966*T^8 + 450095163060*T^6 + 18726082612965*T^4 + 424910533723542*T^2 + 4016014566553089
$89$
\( T^{16} + 42 T^{15} + \cdots + 25608042872721 \)
T^16 + 42*T^15 + 511*T^14 - 3234*T^13 - 98835*T^12 + 571644*T^11 + 31180348*T^10 + 310157970*T^9 + 705231385*T^8 - 7217360688*T^7 - 30777258534*T^6 + 241139942730*T^5 + 1969110015336*T^4 + 3236458963500*T^3 - 5403656958030*T^2 - 11588658331950*T + 25608042872721
$97$
\( T^{16} + 6 T^{15} + \cdots + 11594767041 \)
T^16 + 6*T^15 - 273*T^14 - 1710*T^13 + 56787*T^12 + 373218*T^11 - 4739482*T^10 - 35766252*T^9 + 294651795*T^8 + 2751107274*T^7 + 2623979814*T^6 - 27315611616*T^5 - 14621940542*T^4 + 193273582080*T^3 + 383888415408*T^2 + 112518955692*T + 11594767041
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