[N,k,chi] = [136,2,Mod(9,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.9");
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 \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
\(n\)
\(69\)
\(103\)
\(105\)
\(\chi(n)\)
\(1\)
\(1\)
\(\beta_{7}\)
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
\( T_{3}^{12} + 4 T_{3}^{10} - 8 T_{3}^{9} + 8 T_{3}^{8} + 40 T_{3}^{7} - 224 T_{3}^{6} + 96 T_{3}^{5} + 3652 T_{3}^{4} + 464 T_{3}^{3} + 304 T_{3}^{2} + 192 T_{3} + 32 \)
T3^12 + 4*T3^10 - 8*T3^9 + 8*T3^8 + 40*T3^7 - 224*T3^6 + 96*T3^5 + 3652*T3^4 + 464*T3^3 + 304*T3^2 + 192*T3 + 32
acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( T^{12} + 4 T^{10} - 8 T^{9} + 8 T^{8} + \cdots + 32 \)
T^12 + 4*T^10 - 8*T^9 + 8*T^8 + 40*T^7 - 224*T^6 + 96*T^5 + 3652*T^4 + 464*T^3 + 304*T^2 + 192*T + 32
$5$
\( T^{12} - 4 T^{11} + 16 T^{10} - 56 T^{9} + \cdots + 128 \)
T^12 - 4*T^11 + 16*T^10 - 56*T^9 + 160*T^8 - 328*T^7 + 288*T^6 + 2624*T^5 + 4228*T^4 + 3232*T^3 + 1120*T^2 - 896*T + 128
$7$
\( T^{12} - 22 T^{10} - 12 T^{9} + \cdots + 147968 \)
T^12 - 22*T^10 - 12*T^9 + 242*T^8 + 632*T^7 + 3352*T^6 + 9024*T^5 + 25488*T^4 + 49408*T^3 + 62848*T^2 + 95744*T + 147968
$11$
\( T^{12} + 12 T^{11} + 48 T^{10} + \cdots + 16928 \)
T^12 + 12*T^11 + 48*T^10 - 32*T^9 - 960*T^8 - 2592*T^7 + 10088*T^6 + 68416*T^5 + 107044*T^4 + 6640*T^3 + 118608*T^2 + 29440*T + 16928
$13$
\( T^{12} + 92 T^{10} + 3060 T^{8} + \cdots + 262144 \)
T^12 + 92*T^10 + 3060*T^8 + 44448*T^6 + 265280*T^4 + 483328*T^2 + 262144
$17$
\( T^{12} - 4 T^{11} - 40 T^{10} + \cdots + 24137569 \)
T^12 - 4*T^11 - 40*T^10 + 180*T^9 + 559*T^8 - 1936*T^7 - 5136*T^6 - 32912*T^5 + 161551*T^4 + 884340*T^3 - 3340840*T^2 - 5679428*T + 24137569
$19$
\( T^{12} + 4 T^{11} + 8 T^{10} + 8 T^{9} + \cdots + 1024 \)
T^12 + 4*T^11 + 8*T^10 + 8*T^9 + 676*T^8 + 2912*T^7 + 6272*T^6 + 128*T^5 + 59776*T^4 + 250368*T^3 + 524288*T^2 + 32768*T + 1024
$23$
\( T^{12} + 8 T^{11} - 6 T^{10} + \cdots + 43655168 \)
T^12 + 8*T^11 - 6*T^10 + 68*T^9 + 1970*T^8 - 5416*T^7 - 24200*T^6 + 227040*T^5 - 277488*T^4 - 2947072*T^3 + 21729792*T^2 - 29601792*T + 43655168
$29$
\( T^{12} - 8 T^{11} + 20 T^{10} + \cdots + 118210688 \)
T^12 - 8*T^11 + 20*T^10 - 8*T^9 + 8*T^8 + 96*T^7 + 17480*T^6 + 70160*T^5 + 571140*T^4 - 2766336*T^3 + 2452512*T^2 - 25462656*T + 118210688
$31$
\( T^{12} + 32 T^{11} + \cdots + 103219712 \)
T^12 + 32*T^11 + 386*T^10 + 1868*T^9 + 1154*T^8 - 15752*T^7 - 39864*T^6 - 39200*T^5 + 1676816*T^4 + 1519616*T^3 + 18451328*T^2 + 5747200*T + 103219712
$37$
\( T^{12} - 4 T^{11} + 56 T^{10} - 328 T^{9} + \cdots + 512 \)
T^12 - 4*T^11 + 56*T^10 - 328*T^9 + 2048*T^8 - 8616*T^7 + 24832*T^6 - 72992*T^5 + 200260*T^4 - 267872*T^3 + 147520*T^2 - 15360*T + 512
$41$
\( T^{12} - 16 T^{11} + \cdots + 591542408 \)
T^12 - 16*T^11 + 174*T^10 - 404*T^9 - 6558*T^8 + 61328*T^7 + 25104*T^6 - 3730464*T^5 + 28852132*T^4 - 49805920*T^3 + 150106584*T^2 - 497916496*T + 591542408
$43$
\( T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 148254976 \)
T^12 - 8*T^11 + 32*T^10 + 336*T^9 + 2264*T^8 - 6560*T^7 + 36480*T^6 + 433984*T^5 + 2328464*T^4 + 1186816*T^3 + 739328*T^2 + 14806016*T + 148254976
$47$
\( T^{12} + 360 T^{10} + \cdots + 440664064 \)
T^12 + 360*T^10 + 46032*T^8 + 2438912*T^6 + 46576640*T^4 + 288784384*T^2 + 440664064
$53$
\( T^{12} + 8 T^{11} + 32 T^{10} + \cdots + 25080064 \)
T^12 + 8*T^11 + 32*T^10 - 512*T^9 + 1880*T^8 + 896*T^7 + 78080*T^6 - 1028224*T^5 + 5908880*T^4 - 19102336*T^3 + 38158848*T^2 - 43749888*T + 25080064
$59$
\( T^{12} - 16 T^{11} + 128 T^{10} + \cdots + 9339136 \)
T^12 - 16*T^11 + 128*T^10 - 592*T^9 + 15192*T^8 - 216288*T^7 + 1691264*T^6 - 8008640*T^5 + 24599952*T^4 - 47992960*T^3 + 56775168*T^2 - 32564736*T + 9339136
$61$
\( T^{12} - 44 T^{11} + \cdots + 118949888 \)
T^12 - 44*T^11 + 824*T^10 - 8632*T^9 + 61056*T^8 - 338280*T^7 + 1608928*T^6 - 7793088*T^5 + 32975108*T^4 + 9096640*T^3 + 511760768*T^2 + 455810048*T + 118949888
$67$
\( (T^{6} + 20 T^{5} + 58 T^{4} - 880 T^{3} + \cdots + 28928)^{2} \)
(T^6 + 20*T^5 + 58*T^4 - 880*T^3 - 4784*T^2 + 1664*T + 28928)^2
$71$
\( T^{12} - 32 T^{11} + 578 T^{10} + \cdots + 10913792 \)
T^12 - 32*T^11 + 578*T^10 - 6868*T^9 + 57090*T^8 - 336872*T^7 + 1384040*T^6 - 3716800*T^5 + 7623056*T^4 - 14803584*T^3 + 24433920*T^2 - 24817664*T + 10913792
$73$
\( T^{12} - 8 T^{11} + 158 T^{10} + \cdots + 545424392 \)
T^12 - 8*T^11 + 158*T^10 - 460*T^9 + 7074*T^8 - 77360*T^7 - 429104*T^6 + 5009888*T^5 + 43060708*T^4 + 86347968*T^3 + 196591256*T^2 - 757133872*T + 545424392
$79$
\( T^{12} + 8 T^{11} + \cdots + 1130596352 \)
T^12 + 8*T^11 - 118*T^10 - 2524*T^9 - 4654*T^8 + 385432*T^7 + 5894168*T^6 + 1103008*T^5 + 32598160*T^4 - 293761152*T^3 + 374777600*T^2 + 101951488*T + 1130596352
$83$
\( T^{12} - 40 T^{11} + 800 T^{10} + \cdots + 256 \)
T^12 - 40*T^11 + 800*T^10 - 8192*T^9 + 51672*T^8 - 208640*T^7 + 562432*T^6 - 999296*T^5 + 1150352*T^4 - 774528*T^3 + 270848*T^2 - 11776*T + 256
$89$
\( T^{12} + 552 T^{10} + \cdots + 12558340096 \)
T^12 + 552*T^10 + 107672*T^8 + 9151008*T^6 + 325147152*T^4 + 3825710592*T^2 + 12558340096
$97$
\( T^{12} + 16 T^{11} + 334 T^{10} + \cdots + 30451208 \)
T^12 + 16*T^11 + 334*T^10 + 20*T^9 - 13022*T^8 - 77296*T^7 + 235472*T^6 + 5368608*T^5 + 27214564*T^4 + 57668896*T^3 + 42001240*T^2 - 905264*T + 30451208
show more
show less