[N,k,chi] = [241,2,Mod(1,241)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(241, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("241.1");
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
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
\(241\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 3 T_{2}^{11} - 14 T_{2}^{10} + 44 T_{2}^{9} + 65 T_{2}^{8} - 219 T_{2}^{7} - 123 T_{2}^{6} + 444 T_{2}^{5} + 105 T_{2}^{4} - 328 T_{2}^{3} - 45 T_{2}^{2} + 18 T_{2} - 1 \)
T2^12 - 3*T2^11 - 14*T2^10 + 44*T2^9 + 65*T2^8 - 219*T2^7 - 123*T2^6 + 444*T2^5 + 105*T2^4 - 328*T2^3 - 45*T2^2 + 18*T2 - 1
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(241))\).
$p$
$F_p(T)$
$2$
\( T^{12} - 3 T^{11} - 14 T^{10} + 44 T^{9} + \cdots - 1 \)
T^12 - 3*T^11 - 14*T^10 + 44*T^9 + 65*T^8 - 219*T^7 - 123*T^6 + 444*T^5 + 105*T^4 - 328*T^3 - 45*T^2 + 18*T - 1
$3$
\( T^{12} - T^{11} - 25 T^{10} + 25 T^{9} + \cdots + 64 \)
T^12 - T^11 - 25*T^10 + 25*T^9 + 224*T^8 - 210*T^7 - 888*T^6 + 725*T^5 + 1540*T^4 - 960*T^3 - 992*T^2 + 400*T + 64
$5$
\( T^{12} - 6 T^{11} - 14 T^{10} + 134 T^{9} + \cdots + 62 \)
T^12 - 6*T^11 - 14*T^10 + 134*T^9 - 68*T^8 - 797*T^7 + 1301*T^6 + 497*T^5 - 2193*T^4 + 1071*T^3 + 339*T^2 - 347*T + 62
$7$
\( T^{12} - 3 T^{11} - 33 T^{10} + 96 T^{9} + \cdots + 4 \)
T^12 - 3*T^11 - 33*T^10 + 96*T^9 + 245*T^8 - 854*T^7 + 263*T^6 + 855*T^5 - 588*T^4 - 131*T^3 + 200*T^2 - 53*T + 4
$11$
\( T^{12} - 22 T^{11} + 177 T^{10} + \cdots + 128 \)
T^12 - 22*T^11 + 177*T^10 - 553*T^9 - 215*T^8 + 5545*T^7 - 12739*T^6 + 9811*T^5 + 3100*T^4 - 9672*T^3 + 5900*T^2 - 1460*T + 128
$13$
\( T^{12} + 5 T^{11} - 62 T^{10} + \cdots - 52672 \)
T^12 + 5*T^11 - 62*T^10 - 296*T^9 + 1425*T^8 + 6470*T^7 - 15049*T^6 - 64645*T^5 + 69802*T^4 + 288472*T^3 - 90512*T^2 - 441248*T - 52672
$17$
\( T^{12} + 4 T^{11} - 97 T^{10} + \cdots + 154144 \)
T^12 + 4*T^11 - 97*T^10 - 370*T^9 + 2997*T^8 + 9972*T^7 - 35221*T^6 - 87027*T^5 + 159474*T^4 + 295792*T^3 - 261264*T^2 - 302576*T + 154144
$19$
\( T^{12} + 6 T^{11} - 86 T^{10} + \cdots - 3556280 \)
T^12 + 6*T^11 - 86*T^10 - 524*T^9 + 2538*T^8 + 16891*T^7 - 28947*T^6 - 247081*T^5 + 58969*T^4 + 1614089*T^3 + 903711*T^2 - 3617301*T - 3556280
$23$
\( T^{12} - 32 T^{11} + \cdots - 116949436 \)
T^12 - 32*T^11 + 304*T^10 + 627*T^9 - 28306*T^8 + 138011*T^7 + 372702*T^6 - 5191210*T^5 + 11455889*T^4 + 31479187*T^3 - 182523158*T^2 + 276824423*T - 116949436
$29$
\( T^{12} - 6 T^{11} - 213 T^{10} + \cdots + 58109390 \)
T^12 - 6*T^11 - 213*T^10 + 1375*T^9 + 15216*T^8 - 116722*T^7 - 355685*T^6 + 4236578*T^5 - 3169769*T^4 - 50865568*T^3 + 164527164*T^2 - 179077009*T + 58109390
$31$
\( T^{12} - 8 T^{11} - 262 T^{10} + \cdots - 318193616 \)
T^12 - 8*T^11 - 262*T^10 + 2167*T^9 + 22930*T^8 - 208450*T^7 - 688338*T^6 + 8192365*T^5 + 841016*T^4 - 108211396*T^3 + 77270368*T^2 + 468437780*T - 318193616
$37$
\( T^{12} + 8 T^{11} - 159 T^{10} + \cdots + 50796928 \)
T^12 + 8*T^11 - 159*T^10 - 928*T^9 + 10466*T^8 + 36249*T^7 - 323567*T^6 - 619307*T^5 + 4698614*T^4 + 5067040*T^3 - 29177888*T^2 - 18033984*T + 50796928
$41$
\( T^{12} + T^{11} - 262 T^{10} + \cdots - 63338 \)
T^12 + T^11 - 262*T^10 - 708*T^9 + 21111*T^8 + 92737*T^7 - 471938*T^6 - 2920817*T^5 - 976432*T^4 + 9341574*T^3 - 930334*T^2 - 4034251*T - 63338
$43$
\( T^{12} + 2 T^{11} - 237 T^{10} + \cdots + 12503272 \)
T^12 + 2*T^11 - 237*T^10 - 26*T^9 + 18808*T^8 - 18272*T^7 - 569920*T^6 + 657869*T^5 + 6500883*T^4 - 4519982*T^3 - 25360675*T^2 - 1488169*T + 12503272
$47$
\( T^{12} - 34 T^{11} + 332 T^{10} + \cdots + 53297792 \)
T^12 - 34*T^11 + 332*T^10 + 851*T^9 - 34508*T^8 + 179952*T^7 + 233524*T^6 - 4772881*T^5 + 11628792*T^4 + 11362328*T^3 - 69239488*T^2 + 37689968*T + 53297792
$53$
\( T^{12} - 5 T^{11} - 195 T^{10} + \cdots - 3014 \)
T^12 - 5*T^11 - 195*T^10 + 1019*T^9 + 12170*T^8 - 65448*T^7 - 270515*T^6 + 1538756*T^5 + 1527793*T^4 - 11787807*T^3 + 6874728*T^2 + 298853*T - 3014
$59$
\( T^{12} - 26 T^{11} + 22 T^{10} + \cdots - 25476160 \)
T^12 - 26*T^11 + 22*T^10 + 5338*T^9 - 57444*T^8 + 67258*T^7 + 2412075*T^6 - 17302787*T^5 + 47289076*T^4 - 36571744*T^3 - 59605584*T^2 + 96501648*T - 25476160
$61$
\( T^{12} + 26 T^{11} + 20 T^{10} + \cdots + 10893274 \)
T^12 + 26*T^11 + 20*T^10 - 3955*T^9 - 22505*T^8 + 117122*T^7 + 801476*T^6 - 1560634*T^5 - 8091392*T^4 + 10617016*T^3 + 16419914*T^2 - 29191311*T + 10893274
$67$
\( T^{12} - 6 T^{11} + \cdots + 4538509504 \)
T^12 - 6*T^11 - 429*T^10 + 2947*T^9 + 62307*T^8 - 474596*T^7 - 3555172*T^6 + 29459511*T^5 + 75258180*T^4 - 669419464*T^3 - 678403520*T^2 + 4714883120*T + 4538509504
$71$
\( T^{12} - 94 T^{11} + \cdots - 12017198348 \)
T^12 - 94*T^11 + 3737*T^10 - 79770*T^9 + 918997*T^8 - 3647013*T^7 - 46063490*T^6 + 801669175*T^5 - 5606812300*T^4 + 21246049133*T^3 - 42976926619*T^2 + 39886445545*T - 12017198348
$73$
\( T^{12} + 22 T^{11} - 208 T^{10} + \cdots + 2219968 \)
T^12 + 22*T^11 - 208*T^10 - 7860*T^9 - 28097*T^8 + 533877*T^7 + 3607447*T^6 - 7168229*T^5 - 83908922*T^4 - 76860088*T^3 + 64034288*T^2 + 25741920*T + 2219968
$79$
\( T^{12} - 9 T^{11} + \cdots - 1277319040 \)
T^12 - 9*T^11 - 581*T^10 + 5783*T^9 + 109307*T^8 - 1163461*T^7 - 7904508*T^6 + 90986869*T^5 + 163831840*T^4 - 2331826216*T^3 + 1017318496*T^2 + 3418562576*T - 1277319040
$83$
\( T^{12} + 8 T^{11} + \cdots + 98860915136 \)
T^12 + 8*T^11 - 548*T^10 - 4386*T^9 + 100342*T^8 + 669374*T^7 - 9197429*T^6 - 42544271*T^5 + 452061900*T^4 + 1151271176*T^3 - 10895951696*T^2 - 10813065520*T + 98860915136
$89$
\( T^{12} + 3 T^{11} + \cdots - 1500609440 \)
T^12 + 3*T^11 - 479*T^10 - 1663*T^9 + 79002*T^8 + 305681*T^7 - 5233031*T^6 - 21479633*T^5 + 124536598*T^4 + 490376448*T^3 - 829552176*T^2 - 3427797584*T - 1500609440
$97$
\( T^{12} + 29 T^{11} + \cdots + 107861318 \)
T^12 + 29*T^11 + 85*T^10 - 5577*T^9 - 70766*T^8 - 118070*T^7 + 3390903*T^6 + 27963948*T^5 + 85302025*T^4 + 64432913*T^3 - 162786822*T^2 - 194920563*T + 107861318
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