[N,k,chi] = [456,2,Mod(65,456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(456, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("456.65");
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 \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).
\(n\)
\(97\)
\(229\)
\(305\)
\(343\)
\(\chi(n)\)
\(-\beta_{3}\)
\(1\)
\(-1\)
\(1\)
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
\( T_{5}^{16} - 3 T_{5}^{15} - 21 T_{5}^{14} + 72 T_{5}^{13} + 326 T_{5}^{12} - 1188 T_{5}^{11} - 2224 T_{5}^{10} + 10632 T_{5}^{9} + 9624 T_{5}^{8} - 72960 T_{5}^{7} + 25760 T_{5}^{6} + 242688 T_{5}^{5} - 252000 T_{5}^{4} + \cdots + 430336 \)
T5^16 - 3*T5^15 - 21*T5^14 + 72*T5^13 + 326*T5^12 - 1188*T5^11 - 2224*T5^10 + 10632*T5^9 + 9624*T5^8 - 72960*T5^7 + 25760*T5^6 + 242688*T5^5 - 252000*T5^4 - 544320*T5^3 + 1443392*T5^2 - 1275264*T5 + 430336
acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} \)
T^16
$3$
\( T^{16} - T^{15} + 3 T^{14} - 4 T^{13} + \cdots + 6561 \)
T^16 - T^15 + 3*T^14 - 4*T^13 + 9*T^12 - 13*T^11 - 10*T^10 - 15*T^9 - 14*T^8 - 45*T^7 - 90*T^6 - 351*T^5 + 729*T^4 - 972*T^3 + 2187*T^2 - 2187*T + 6561
$5$
\( T^{16} - 3 T^{15} - 21 T^{14} + \cdots + 430336 \)
T^16 - 3*T^15 - 21*T^14 + 72*T^13 + 326*T^12 - 1188*T^11 - 2224*T^10 + 10632*T^9 + 9624*T^8 - 72960*T^7 + 25760*T^6 + 242688*T^5 - 252000*T^4 - 544320*T^3 + 1443392*T^2 - 1275264*T + 430336
$7$
\( (T^{8} - 30 T^{6} + 22 T^{5} + 223 T^{4} + \cdots - 32)^{2} \)
(T^8 - 30*T^6 + 22*T^5 + 223*T^4 - 234*T^3 - 278*T^2 + 332*T - 32)^2
$11$
\( T^{16} + 121 T^{14} + 5883 T^{12} + \cdots + 2637376 \)
T^16 + 121*T^14 + 5883*T^12 + 146207*T^10 + 1945988*T^8 + 13143656*T^6 + 37229952*T^4 + 22890544*T^2 + 2637376
$13$
\( T^{16} + 3 T^{15} - 61 T^{14} - 192 T^{13} + \cdots + 64 \)
T^16 + 3*T^15 - 61*T^14 - 192*T^13 + 3117*T^12 + 9105*T^11 - 45991*T^10 - 114492*T^9 + 680559*T^8 - 213441*T^7 - 474235*T^6 + 164334*T^5 + 351032*T^4 - 152712*T^3 + 26016*T^2 - 2016*T + 64
$17$
\( T^{16} + 3 T^{15} + \cdots + 2945449984 \)
T^16 + 3*T^15 - 69*T^14 - 216*T^13 + 3260*T^12 + 13344*T^11 - 73808*T^10 - 402432*T^9 + 1128960*T^8 + 9556992*T^7 + 2434048*T^6 - 99373056*T^5 - 150208512*T^4 + 748683264*T^3 + 3228762112*T^2 + 4960026624*T + 2945449984
$19$
\( T^{16} - 11 T^{15} + \cdots + 16983563041 \)
T^16 - 11*T^15 + 105*T^14 - 726*T^13 + 4390*T^12 - 23056*T^11 + 109983*T^10 - 500817*T^9 + 2164434*T^8 - 9515523*T^7 + 39703863*T^6 - 158141104*T^5 + 572109190*T^4 - 1797647874*T^3 + 4939817505*T^2 - 9832589129*T + 16983563041
$23$
\( T^{16} + 3 T^{15} + \cdots + 1328456704 \)
T^16 + 3*T^15 - 67*T^14 - 210*T^13 + 3158*T^12 + 10680*T^11 - 71536*T^10 - 251592*T^9 + 1232920*T^8 + 3811056*T^7 - 13037888*T^6 - 35166336*T^5 + 106375712*T^4 + 188404608*T^3 - 419170112*T^2 - 515228928*T + 1328456704
$29$
\( T^{16} - 5 T^{15} + 153 T^{14} + \cdots + 554696704 \)
T^16 - 5*T^15 + 153*T^14 - 636*T^13 + 15842*T^12 - 65592*T^11 + 808964*T^10 - 3108008*T^9 + 29191056*T^8 - 102023680*T^7 + 383674880*T^6 - 537397248*T^5 + 885616640*T^4 - 595329024*T^3 + 1067581440*T^2 - 627048448*T + 554696704
$31$
\( T^{16} + 300 T^{14} + \cdots + 5554422784 \)
T^16 + 300*T^14 + 36674*T^12 + 2351020*T^10 + 84484833*T^8 + 1677358288*T^6 + 16672154004*T^4 + 61498640944*T^2 + 5554422784
$37$
\( T^{16} + 80 T^{14} + 2386 T^{12} + \cdots + 262144 \)
T^16 + 80*T^14 + 2386*T^12 + 32764*T^10 + 213681*T^8 + 717644*T^6 + 1250404*T^4 + 1018624*T^2 + 262144
$41$
\( T^{16} - 6 T^{15} + 188 T^{14} + \cdots + 27541504 \)
T^16 - 6*T^15 + 188*T^14 + 60*T^13 + 19621*T^12 + 13968*T^11 + 1103348*T^10 + 4562790*T^9 + 31383385*T^8 + 53716980*T^7 + 143594624*T^6 + 31802688*T^5 + 319632256*T^4 + 121285632*T^3 + 212092928*T^2 - 58441728*T + 27541504
$43$
\( T^{16} - 13 T^{15} + \cdots + 334805733376 \)
T^16 - 13*T^15 + 307*T^14 - 2706*T^13 + 46497*T^12 - 356587*T^11 + 4243425*T^10 - 21327546*T^9 + 192325795*T^8 - 631863037*T^7 + 6058267489*T^6 - 11909811440*T^5 + 112720485480*T^4 - 27149448576*T^3 + 1365036800576*T^2 - 665968450048*T + 334805733376
$47$
\( T^{16} - 27 T^{15} + \cdots + 18653553688576 \)
T^16 - 27*T^15 + 131*T^14 + 3024*T^13 - 27246*T^12 - 285936*T^11 + 4287548*T^10 + 2019096*T^9 - 245084688*T^8 + 389611968*T^7 + 11147836928*T^6 - 54578422272*T^5 - 82711192576*T^4 + 864711475200*T^3 + 776729444352*T^2 - 11153323622400*T + 18653553688576
$53$
\( T^{16} + 7 T^{15} + 139 T^{14} + \cdots + 50176 \)
T^16 + 7*T^15 + 139*T^14 + 546*T^13 + 11016*T^12 + 42676*T^11 + 422076*T^10 + 526296*T^9 + 5963680*T^8 + 16046464*T^7 + 35681872*T^6 + 41881184*T^5 + 37527552*T^4 + 17388672*T^3 + 5722112*T^2 - 455168*T + 50176
$59$
\( T^{16} + 10 T^{15} + \cdots + 933720229264 \)
T^16 + 10*T^15 + 334*T^14 + 1600*T^13 + 57071*T^12 + 233596*T^11 + 5897038*T^10 + 15445562*T^9 + 391587469*T^8 + 983796684*T^7 + 16733105140*T^6 + 29436754944*T^5 + 448451791852*T^4 + 773262094192*T^3 + 6026592306688*T^2 - 2290030871472*T + 933720229264
$61$
\( T^{16} + T^{15} + \cdots + 215225477776 \)
T^16 + T^15 + 287*T^14 + 282*T^13 + 63135*T^12 + 31631*T^11 + 5131093*T^10 - 12157766*T^9 + 301064797*T^8 - 359496063*T^7 + 3496995873*T^6 - 1946108732*T^5 + 28469199738*T^4 - 9351307320*T^3 + 99379753636*T^2 + 49406978152*T + 215225477776
$67$
\( T^{16} + 24 T^{15} + \cdots + 5605723904881 \)
T^16 + 24*T^15 + 50*T^14 - 3408*T^13 - 13668*T^12 + 385056*T^11 + 3059332*T^10 - 12145140*T^9 - 166047523*T^8 + 246239568*T^7 + 7508390500*T^6 + 20798447580*T^5 - 56876612628*T^4 - 283878827016*T^3 + 465507857186*T^2 + 3845494100508*T + 5605723904881
$71$
\( T^{16} - 27 T^{15} + \cdots + 59895709696 \)
T^16 - 27*T^15 + 631*T^14 - 6814*T^13 + 76552*T^12 - 419880*T^11 + 4415776*T^10 - 16819296*T^9 + 183219904*T^8 - 97254656*T^7 + 3256228864*T^6 - 1869303808*T^5 + 33798250496*T^4 + 18901434368*T^3 + 113279500288*T^2 - 67413999616*T + 59895709696
$73$
\( T^{16} - 2 T^{15} + \cdots + 24812542725961 \)
T^16 - 2*T^15 + 234*T^14 - 576*T^13 + 35588*T^12 - 87738*T^11 + 3261260*T^10 - 7582058*T^9 + 217615629*T^8 - 438026146*T^7 + 9396294764*T^6 - 13270008162*T^5 + 280886922092*T^4 - 202367268528*T^3 + 4026371584194*T^2 + 5409374697926*T + 24812542725961
$79$
\( T^{16} + 21 T^{15} + \cdots + 3046449086464 \)
T^16 + 21*T^15 - 91*T^14 - 4998*T^13 + 11797*T^12 + 877083*T^11 + 3029431*T^10 - 50750478*T^9 - 210402513*T^8 + 2160318693*T^7 + 11719725227*T^6 - 37226119008*T^5 - 212721773336*T^4 + 381880492800*T^3 + 3481411316416*T^2 + 5465570611200*T + 3046449086464
$83$
\( T^{16} + 585 T^{14} + \cdots + 8981961048064 \)
T^16 + 585*T^14 + 134491*T^12 + 15513207*T^10 + 961447668*T^8 + 32415125952*T^6 + 567557450752*T^4 + 4414635147264*T^2 + 8981961048064
$89$
\( T^{16} - 25 T^{15} + 627 T^{14} + \cdots + 6885376 \)
T^16 - 25*T^15 + 627*T^14 - 4030*T^13 + 45204*T^12 + 274044*T^11 + 3783708*T^10 + 12917992*T^9 + 58972272*T^8 + 120120288*T^7 + 458330832*T^6 + 821744736*T^5 + 1874654912*T^4 + 1044334336*T^3 + 449920512*T^2 + 62724096*T + 6885376
$97$
\( T^{16} + 60 T^{15} + \cdots + 14736876810496 \)
T^16 + 60*T^15 + 1498*T^14 + 17880*T^13 + 61659*T^12 - 704856*T^11 - 926566*T^10 + 190391220*T^9 + 2833108337*T^8 + 19708747680*T^7 + 69723025688*T^6 + 69882543552*T^5 - 294809494224*T^4 - 450461124480*T^3 + 3573632212096*T^2 + 13411762379520*T + 14736876810496
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