[N,k,chi] = [816,2,Mod(49,816)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(816, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("816.49");
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 \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).
\(n\)
\(241\)
\(511\)
\(545\)
\(613\)
\(\chi(n)\)
\(-\beta_{8}\)
\(1\)
\(1\)
\(1\)
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
\( T_{5}^{16} + 4 T_{5}^{14} + 32 T_{5}^{13} + 8 T_{5}^{12} - 56 T_{5}^{11} + 336 T_{5}^{10} - 72 T_{5}^{9} + 69 T_{5}^{8} + 2920 T_{5}^{7} + 14908 T_{5}^{6} + 34536 T_{5}^{5} + 46856 T_{5}^{4} + 41056 T_{5}^{3} + 23672 T_{5}^{2} + \cdots + 1156 \)
T5^16 + 4*T5^14 + 32*T5^13 + 8*T5^12 - 56*T5^11 + 336*T5^10 - 72*T5^9 + 69*T5^8 + 2920*T5^7 + 14908*T5^6 + 34536*T5^5 + 46856*T5^4 + 41056*T5^3 + 23672*T5^2 + 7888*T5 + 1156
acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} \)
T^16
$3$
\( (T^{8} + 1)^{2} \)
(T^8 + 1)^2
$5$
\( T^{16} + 4 T^{14} + 32 T^{13} + \cdots + 1156 \)
T^16 + 4*T^14 + 32*T^13 + 8*T^12 - 56*T^11 + 336*T^10 - 72*T^9 + 69*T^8 + 2920*T^7 + 14908*T^6 + 34536*T^5 + 46856*T^4 + 41056*T^3 + 23672*T^2 + 7888*T + 1156
$7$
\( T^{16} - 12 T^{14} - 40 T^{13} + 72 T^{12} + \cdots + 64 \)
T^16 - 12*T^14 - 40*T^13 + 72*T^12 + 848*T^11 + 1880*T^10 - 144*T^9 - 3948*T^8 - 3264*T^7 + 4608*T^6 + 15616*T^5 + 17536*T^4 + 19456*T^3 + 86656*T^2 - 1792*T + 64
$11$
\( T^{16} - 8 T^{15} + 64 T^{14} + \cdots + 73984 \)
T^16 - 8*T^15 + 64*T^14 - 312*T^13 + 1472*T^12 - 5560*T^11 + 14896*T^10 - 20680*T^9 - 13567*T^8 + 132160*T^7 - 29360*T^6 - 271744*T^5 + 734336*T^4 - 589056*T^3 - 91904*T^2 + 226304*T + 73984
$13$
\( T^{16} + 100 T^{14} + 3470 T^{12} + \cdots + 150544 \)
T^16 + 100*T^14 + 3470*T^12 + 51356*T^10 + 322209*T^8 + 956728*T^6 + 1360232*T^4 + 836704*T^2 + 150544
$17$
\( T^{16} + 52 T^{14} + \cdots + 6975757441 \)
T^16 + 52*T^14 - 80*T^13 + 1398*T^12 - 3328*T^11 + 34788*T^10 - 63664*T^9 + 714594*T^8 - 1082288*T^7 + 10053732*T^6 - 16350464*T^5 + 116762358*T^4 - 113588560*T^3 + 1255153588*T^2 + 6975757441
$19$
\( T^{16} - 8 T^{15} + 32 T^{14} + 88 T^{13} + \cdots + 64 \)
T^16 - 8*T^15 + 32*T^14 + 88*T^13 + 274*T^12 - 808*T^11 + 1568*T^10 + 2360*T^9 + 6785*T^8 - 16800*T^7 + 21760*T^6 + 4832*T^5 + 10896*T^4 - 23808*T^3 + 25088*T^2 - 1792*T + 64
$23$
\( T^{16} - 8 T^{15} + 56 T^{14} + \cdots + 4624 \)
T^16 - 8*T^15 + 56*T^14 - 136*T^13 + 96*T^12 - 312*T^11 + 744*T^10 + 2760*T^9 + 5841*T^8 - 132800*T^7 + 599552*T^6 - 1456800*T^5 + 2075648*T^4 - 1629888*T^3 + 594304*T^2 - 50048*T + 4624
$29$
\( T^{16} - 32 T^{15} + \cdots + 277155904 \)
T^16 - 32*T^15 + 516*T^14 - 5944*T^13 + 57096*T^12 - 466224*T^11 + 3209416*T^10 - 17954032*T^9 + 79163156*T^8 - 271528256*T^7 + 720217216*T^6 - 1461850112*T^5 + 2233053312*T^4 - 2509555712*T^3 + 2001953664*T^2 - 1044695296*T + 277155904
$31$
\( T^{16} + 8 T^{15} + 136 T^{14} + \cdots + 75203584 \)
T^16 + 8*T^15 + 136*T^14 + 1344*T^13 + 12320*T^12 + 65280*T^11 + 102080*T^10 - 1439104*T^9 - 4187568*T^8 - 14674560*T^7 + 299725440*T^6 - 220733952*T^5 + 683303424*T^4 - 1685881856*T^3 + 1556046848*T^2 - 563888128*T + 75203584
$37$
\( T^{16} - 8 T^{15} + \cdots + 8599223824 \)
T^16 - 8*T^15 + 48*T^14 - 400*T^13 + 2304*T^12 - 6032*T^11 + 13744*T^10 + 100160*T^9 + 243144*T^8 - 759456*T^7 - 112576*T^6 - 35504192*T^5 + 242301056*T^4 - 291706176*T^3 + 1013300416*T^2 + 7412625152*T + 8599223824
$41$
\( T^{16} - 40 T^{15} + 796 T^{14} + \cdots + 6728836 \)
T^16 - 40*T^15 + 796*T^14 - 10352*T^13 + 97288*T^12 - 693272*T^11 + 3926112*T^10 - 18541816*T^9 + 75941797*T^8 - 272342416*T^7 + 813821092*T^6 - 1782130936*T^5 + 3774742216*T^4 - 7266994416*T^3 + 7298827320*T^2 - 404684752*T + 6728836
$43$
\( T^{16} - 24 T^{15} + \cdots + 4773151744 \)
T^16 - 24*T^15 + 288*T^14 - 2600*T^13 + 33154*T^12 - 495944*T^11 + 5734304*T^10 - 46840552*T^9 + 273397857*T^8 - 1145506928*T^7 + 3404129408*T^6 - 6845948032*T^5 + 8434616384*T^4 - 4615941632*T^3 + 90316800*T^2 - 928542720*T + 4773151744
$47$
\( T^{16} + 320 T^{14} + \cdots + 33926692864 \)
T^16 + 320*T^14 + 39600*T^12 + 2430848*T^10 + 78405440*T^8 + 1295664128*T^6 + 10049284096*T^4 + 33415151616*T^2 + 33926692864
$53$
\( T^{16} + 24 T^{15} + \cdots + 143598555136 \)
T^16 + 24*T^15 + 288*T^14 + 1168*T^13 + 7672*T^12 + 159136*T^11 + 2291840*T^10 + 8051648*T^9 + 13889552*T^8 + 154473472*T^7 + 4190633984*T^6 + 9943587840*T^5 + 6255189504*T^4 - 104875491328*T^3 + 309237645312*T^2 - 298013687808*T + 143598555136
$59$
\( T^{16} - 16 T^{15} + \cdots + 29302041354496 \)
T^16 - 16*T^15 + 128*T^14 + 80*T^13 + 23196*T^12 - 353504*T^11 + 2690176*T^10 + 715552*T^9 + 140288516*T^8 - 1923928448*T^7 + 12964385792*T^6 - 8258330752*T^5 + 233189495616*T^4 - 2728695135232*T^3 + 15399918028800*T^2 - 30041605647360*T + 29302041354496
$61$
\( T^{16} + \cdots + 307387995210256 \)
T^16 + 72*T^14 + 1024*T^13 + 2592*T^12 - 42400*T^11 + 546496*T^10 + 36814336*T^9 + 758021576*T^8 + 9728990848*T^7 + 110819820832*T^6 + 1067522693632*T^5 + 8063181639680*T^4 + 44834910822528*T^3 + 166621597900288*T^2 + 346522673526272*T + 307387995210256
$67$
\( (T^{8} - 316 T^{6} - 304 T^{5} + \cdots + 2236384)^{2} \)
(T^8 - 316*T^6 - 304*T^5 + 30268*T^4 + 62560*T^3 - 943776*T^2 - 2796288*T + 2236384)^2
$71$
\( T^{16} - 16 T^{15} + \cdots + 4226040064 \)
T^16 - 16*T^15 + 256*T^14 - 544*T^13 - 7680*T^12 - 44096*T^11 + 381952*T^10 + 6582144*T^9 + 45016096*T^8 - 146997504*T^7 + 1715838976*T^6 - 3044008448*T^5 + 7750688768*T^4 - 21619622912*T^3 + 29688266752*T^2 - 18098227200*T + 4226040064
$73$
\( T^{16} - 24 T^{15} + \cdots + 130269021184 \)
T^16 - 24*T^15 + 204*T^14 + 648*T^13 - 29304*T^12 + 153472*T^11 + 2057512*T^10 - 29806992*T^9 + 139255940*T^8 - 219413696*T^7 - 352653376*T^6 + 4301922048*T^5 + 1301926400*T^4 - 12482152448*T^3 + 93848671232*T^2 - 96601657344*T + 130269021184
$79$
\( T^{16} - 96 T^{15} + \cdots + 25305004646464 \)
T^16 - 96*T^15 + 4436*T^14 - 130344*T^13 + 2703560*T^12 - 41381680*T^11 + 475248088*T^10 - 4096327248*T^9 + 26021726228*T^8 - 114777377344*T^7 + 292889510784*T^6 - 67709121280*T^5 - 1978040057728*T^4 + 3032010739712*T^3 + 18849119341696*T^2 + 24085593504000*T + 25305004646464
$83$
\( T^{16} - 160 T^{13} + \cdots + 797549019136 \)
T^16 - 160*T^13 + 28272*T^12 - 23552*T^11 + 12800*T^10 - 1928960*T^9 + 201764672*T^8 - 399310848*T^7 + 224100352*T^6 + 14367764480*T^5 + 151126026240*T^4 + 150818947072*T^3 + 47817162752*T^2 - 276175781888*T + 797549019136
$89$
\( T^{16} + 1040 T^{14} + \cdots + 47876546426944 \)
T^16 + 1040*T^14 + 415148*T^12 + 80091216*T^10 + 7698035700*T^8 + 337001054496*T^6 + 4962618377632*T^4 + 27801322739584*T^2 + 47876546426944
$97$
\( T^{16} + \cdots + 343239946813696 \)
T^16 - 56*T^15 + 1548*T^14 - 30584*T^13 + 514952*T^12 - 7249936*T^11 + 79544536*T^10 - 482284720*T^9 - 1194378748*T^8 + 36979900256*T^7 - 71415011680*T^6 - 1812248538624*T^5 + 30098729985152*T^4 - 145437369324288*T^3 + 532528882314496*T^2 - 717986016227328*T + 343239946813696
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