[N,k,chi] = [161,4,Mod(1,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.1");
S:= CuspForms(chi, 4);
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
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
\( p \)
Sign
\(7\)
\(1\)
\(23\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 60T_{2}^{7} - 22T_{2}^{6} + 1179T_{2}^{5} + 694T_{2}^{4} - 7936T_{2}^{3} - 4352T_{2}^{2} + 11008T_{2} + 3072 \)
T2^9 - 60*T2^7 - 22*T2^6 + 1179*T2^5 + 694*T2^4 - 7936*T2^3 - 4352*T2^2 + 11008*T2 + 3072
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\).
$p$
$F_p(T)$
$2$
\( T^{9} - 60 T^{7} - 22 T^{6} + \cdots + 3072 \)
T^9 - 60*T^7 - 22*T^6 + 1179*T^5 + 694*T^4 - 7936*T^3 - 4352*T^2 + 11008*T + 3072
$3$
\( T^{9} - 9 T^{8} - 142 T^{7} + \cdots - 561568 \)
T^9 - 9*T^8 - 142*T^7 + 1148*T^6 + 6892*T^5 - 46670*T^4 - 116376*T^3 + 596668*T^2 + 64040*T - 561568
$5$
\( T^{9} + 4 T^{8} + \cdots + 1135409856 \)
T^9 + 4*T^8 - 702*T^7 - 2106*T^6 + 159546*T^5 + 459352*T^4 - 14284180*T^3 - 42045264*T^2 + 414497008*T + 1135409856
$7$
\( (T + 7)^{9} \)
(T + 7)^9
$11$
\( T^{9} + 8 T^{8} + \cdots + 43666792951552 \)
T^9 + 8*T^8 - 6728*T^7 - 90916*T^6 + 14085260*T^5 + 251866032*T^4 - 10068956160*T^3 - 195854425920*T^2 + 2237741040192*T + 43666792951552
$13$
\( T^{9} - 25 T^{8} + \cdots - 6636351117312 \)
T^9 - 25*T^8 - 14280*T^7 + 329192*T^6 + 62052560*T^5 - 1309720624*T^4 - 76302442304*T^3 + 1233039945856*T^2 - 1949534477312*T - 6636351117312
$17$
\( T^{9} + 28 T^{8} + \cdots + 80\!\cdots\!88 \)
T^9 + 28*T^8 - 26230*T^7 - 895142*T^6 + 209333982*T^5 + 8881038332*T^4 - 499651068396*T^3 - 24499808469192*T^2 + 106997589728992*T + 8029303646834688
$19$
\( T^{9} - 254 T^{8} + \cdots + 95\!\cdots\!20 \)
T^9 - 254*T^8 + 9044*T^7 + 2223352*T^6 - 160386528*T^5 - 4847896576*T^4 + 568304670464*T^3 - 2025368325632*T^2 - 566800526245888*T + 9546131941662720
$23$
\( (T + 23)^{9} \)
(T + 23)^9
$29$
\( T^{9} - 25 T^{8} + \cdots + 87\!\cdots\!60 \)
T^9 - 25*T^8 - 122364*T^7 + 6510612*T^6 + 4522373176*T^5 - 245596589836*T^4 - 58387581724864*T^3 + 1066203877850672*T^2 + 318439858280782976*T + 8724737103597980160
$31$
\( T^{9} - 1171 T^{8} + \cdots - 42\!\cdots\!44 \)
T^9 - 1171*T^8 + 510780*T^7 - 88645504*T^6 - 1153413160*T^5 + 2327873598450*T^4 - 209638246064456*T^3 - 8806716318921652*T^2 + 1686514993112134440*T - 42711191784233233344
$37$
\( T^{9} - 70 T^{8} + \cdots + 78\!\cdots\!00 \)
T^9 - 70*T^8 - 211956*T^7 + 11467416*T^6 + 10183509808*T^5 - 368954025184*T^4 - 143636465240896*T^3 + 2158251344624000*T^2 + 600512492680358400*T + 7866982070659584000
$41$
\( T^{9} - 221 T^{8} + \cdots + 51\!\cdots\!48 \)
T^9 - 221*T^8 - 258732*T^7 + 28359480*T^6 + 19544146352*T^5 - 314031661888*T^4 - 325080591713856*T^3 - 4501546730060672*T^2 + 1552415764951730176*T + 51760382205475746048
$43$
\( T^{9} + 42 T^{8} + \cdots + 13\!\cdots\!28 \)
T^9 + 42*T^8 - 235256*T^7 - 4931568*T^6 + 18313337488*T^5 + 98042896352*T^4 - 549243639943040*T^3 + 235000420718592*T^2 + 5234628504222736384*T + 13183241313782857728
$47$
\( T^{9} - 159 T^{8} + \cdots + 15\!\cdots\!76 \)
T^9 - 159*T^8 - 299612*T^7 + 36030476*T^6 + 17577676936*T^5 - 2090579782382*T^4 - 50625868409232*T^3 + 6877331225543740*T^2 + 66032292840099208*T + 152181368220430176
$53$
\( T^{9} + 774 T^{8} + \cdots + 27\!\cdots\!12 \)
T^9 + 774*T^8 - 707932*T^7 - 648737816*T^6 + 91291303248*T^5 + 153443219504544*T^4 + 11251942444053952*T^3 - 10701820366619954816*T^2 - 1759578488595015484928*T + 2768699210207179456512
$59$
\( T^{9} - 1080 T^{8} + \cdots + 12\!\cdots\!40 \)
T^9 - 1080*T^8 - 363146*T^7 + 376515254*T^6 + 119914233634*T^5 - 20061092051724*T^4 - 8698441195097428*T^3 - 46873210108527896*T^2 + 177226320928199041536*T + 12972580946814264935040
$61$
\( T^{9} - 686 T^{8} + \cdots + 12\!\cdots\!56 \)
T^9 - 686*T^8 - 873394*T^7 + 408173962*T^6 + 262062175094*T^5 - 44822469370460*T^4 - 27626438816569220*T^3 - 623493760919841976*T^2 + 427884178780425616304*T + 12429251080209907847456
$67$
\( T^{9} + 370 T^{8} + \cdots - 11\!\cdots\!56 \)
T^9 + 370*T^8 - 1612156*T^7 - 839360076*T^6 + 719703249620*T^5 + 524158653676904*T^4 - 23280064721534704*T^3 - 80993581471382202272*T^2 - 18524347040567926265856*T - 1149984502372420216979456
$71$
\( T^{9} - 1035 T^{8} + \cdots - 36\!\cdots\!72 \)
T^9 - 1035*T^8 - 1897564*T^7 + 1927641464*T^6 + 1300505291520*T^5 - 1277424899810176*T^4 - 386252800030816256*T^3 + 359212731143339365376*T^2 + 41820488781964465410048*T - 36447935716418082709635072
$73$
\( T^{9} - 1979 T^{8} + \cdots + 17\!\cdots\!12 \)
T^9 - 1979*T^8 + 11520*T^7 + 1917302696*T^6 - 691060773968*T^5 - 541303082402624*T^4 + 271864596454457088*T^3 + 31965874609671460224*T^2 - 24746415902615478601728*T + 1794756947543074060422912
$79$
\( T^{9} - 2336 T^{8} + \cdots - 92\!\cdots\!20 \)
T^9 - 2336*T^8 + 215828*T^7 + 3152033860*T^6 - 2272873143684*T^5 - 483809970981984*T^4 + 1075775596591715376*T^3 - 411754664031703103360*T^2 + 53795135486898393970688*T - 920149935305286657310720
$83$
\( T^{9} - 130 T^{8} + \cdots + 20\!\cdots\!56 \)
T^9 - 130*T^8 - 1241276*T^7 + 34680264*T^6 + 505193301184*T^5 + 20026566748416*T^4 - 77220705260801280*T^3 - 5260218225917762304*T^2 + 3219014387429336919040*T + 20514665993589830393856
$89$
\( T^{9} + 1328 T^{8} + \cdots + 79\!\cdots\!60 \)
T^9 + 1328*T^8 - 2461522*T^7 - 3203026154*T^6 + 1900226342198*T^5 + 2338155090225276*T^4 - 474482966778991100*T^3 - 469061727718533474456*T^2 + 42089265023583614379488*T + 79695931763760801045760
$97$
\( T^{9} + 104 T^{8} + \cdots - 44\!\cdots\!92 \)
T^9 + 104*T^8 - 3209450*T^7 + 739930158*T^6 + 2268727411234*T^5 - 480303366635008*T^4 - 447047070948431316*T^3 + 15342712850214666144*T^2 + 8313689430787324904752*T - 447903321912536555572992
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