[N,k,chi] = [1320,2,Mod(361,1320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1320, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1320.361");
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 \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).
\(n\)
\(661\)
\(881\)
\(991\)
\(1057\)
\(1201\)
\(\chi(n)\)
\(1\)
\(1\)
\(1\)
\(1\)
\(\beta_{7}\)
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
\( T_{7}^{12} + 16 T_{7}^{10} + 60 T_{7}^{9} + 76 T_{7}^{8} - 190 T_{7}^{7} + 421 T_{7}^{6} + 1370 T_{7}^{5} + 2551 T_{7}^{4} + 2360 T_{7}^{3} + 1265 T_{7}^{2} + 150 T_{7} + 25 \)
T7^12 + 16*T7^10 + 60*T7^9 + 76*T7^8 - 190*T7^7 + 421*T7^6 + 1370*T7^5 + 2551*T7^4 + 2360*T7^3 + 1265*T7^2 + 150*T7 + 25
acting on \(S_{2}^{\mathrm{new}}(1320, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \)
(T^4 - T^3 + T^2 - T + 1)^3
$5$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{3} \)
(T^4 + T^3 + T^2 + T + 1)^3
$7$
\( T^{12} + 16 T^{10} + 60 T^{9} + 76 T^{8} + \cdots + 25 \)
T^12 + 16*T^10 + 60*T^9 + 76*T^8 - 190*T^7 + 421*T^6 + 1370*T^5 + 2551*T^4 + 2360*T^3 + 1265*T^2 + 150*T + 25
$11$
\( T^{12} + 4 T^{11} + 32 T^{10} + \cdots + 1771561 \)
T^12 + 4*T^11 + 32*T^10 + 143*T^9 + 658*T^8 + 2588*T^7 + 8483*T^6 + 28468*T^5 + 79618*T^4 + 190333*T^3 + 468512*T^2 + 644204*T + 1771561
$13$
\( T^{12} - 3 T^{11} + 33 T^{10} + \cdots + 83521 \)
T^12 - 3*T^11 + 33*T^10 - 154*T^9 + 628*T^8 - 1151*T^7 + 1457*T^6 + 931*T^5 + 17358*T^4 - 56236*T^3 + 80053*T^2 + 4913*T + 83521
$17$
\( T^{12} - 12 T^{11} + 125 T^{10} + \cdots + 20736 \)
T^12 - 12*T^11 + 125*T^10 - 820*T^9 + 4501*T^8 - 20900*T^7 + 77341*T^6 - 190940*T^5 + 307321*T^4 - 314460*T^3 + 226080*T^2 - 96768*T + 20736
$19$
\( T^{12} + 4 T^{11} + 7 T^{10} + \cdots + 249001 \)
T^12 + 4*T^11 + 7*T^10 + 88*T^9 + 3203*T^8 - 4282*T^7 + 88513*T^6 + 562778*T^5 + 2120518*T^4 + 4107098*T^3 + 5406472*T^2 + 871254*T + 249001
$23$
\( (T^{6} + 6 T^{5} - 41 T^{4} - 275 T^{3} + \cdots + 3275)^{2} \)
(T^6 + 6*T^5 - 41*T^4 - 275*T^3 + 175*T^2 + 2700*T + 3275)^2
$29$
\( T^{12} - 16 T^{11} + 203 T^{10} + \cdots + 250000 \)
T^12 - 16*T^11 + 203*T^10 - 1575*T^9 + 9486*T^8 - 47925*T^7 + 208157*T^6 - 601627*T^5 + 1027201*T^4 - 833400*T^3 + 1486000*T^2 - 925000*T + 250000
$31$
\( T^{12} - 3 T^{11} - T^{10} + \cdots + 156250000 \)
T^12 - 3*T^11 - T^10 + 173*T^9 + 10226*T^8 - 37955*T^7 + 891215*T^6 - 419725*T^5 + 21120775*T^4 - 28736250*T^3 + 59125000*T^2 - 106250000*T + 156250000
$37$
\( T^{12} + 19 T^{11} + 213 T^{10} + \cdots + 198025 \)
T^12 + 19*T^11 + 213*T^10 + 1370*T^9 + 4876*T^8 + 1795*T^7 - 9503*T^6 + 126038*T^5 + 818716*T^4 + 964670*T^3 + 1357160*T^2 + 418300*T + 198025
$41$
\( T^{12} - 22 T^{11} + 278 T^{10} + \cdots + 45495025 \)
T^12 - 22*T^11 + 278*T^10 - 2544*T^9 + 29070*T^8 - 177414*T^7 + 682733*T^6 - 1121302*T^5 + 56278561*T^4 - 284325660*T^3 + 3129504565*T^2 + 229802150*T + 45495025
$43$
\( (T^{6} - 26 T^{5} + 238 T^{4} - 857 T^{3} + \cdots + 20)^{2} \)
(T^6 - 26*T^5 + 238*T^4 - 857*T^3 + 629*T^2 + 1650*T + 20)^2
$47$
\( T^{12} - 25 T^{11} + 359 T^{10} + \cdots + 10208025 \)
T^12 - 25*T^11 + 359*T^10 - 3910*T^9 + 43586*T^8 - 312905*T^7 + 1457519*T^6 - 3442295*T^5 + 5750986*T^4 + 18609600*T^3 + 28588365*T^2 + 20080575*T + 10208025
$53$
\( T^{12} + T^{11} + 122 T^{10} + \cdots + 10773402025 \)
T^12 + T^11 + 122*T^10 - 242*T^9 + 13815*T^8 + 144452*T^7 + 2021827*T^6 + 16254839*T^5 + 146286711*T^4 + 798652645*T^3 + 4013421085*T^2 + 9649821150*T + 10773402025
$59$
\( T^{12} + 9 T^{11} + 45 T^{10} + \cdots + 128881 \)
T^12 + 9*T^11 + 45*T^10 - 410*T^9 + 12456*T^8 + 276285*T^7 + 4039669*T^6 + 16866330*T^5 + 39845766*T^4 + 52005580*T^3 + 46603110*T^2 - 1397946*T + 128881
$61$
\( T^{12} - 21 T^{11} + 138 T^{10} + \cdots + 32400 \)
T^12 - 21*T^11 + 138*T^10 + 685*T^9 + 1821*T^8 - 104225*T^7 + 736332*T^6 - 685707*T^5 + 26020021*T^4 - 27195180*T^3 + 90953640*T^2 + 1053000*T + 32400
$67$
\( (T^{6} + 9 T^{5} + 2 T^{4} - 88 T^{3} + \cdots + 44)^{2} \)
(T^6 + 9*T^5 + 2*T^4 - 88*T^3 - 37*T^2 + 94*T + 44)^2
$71$
\( T^{12} - 17 T^{11} + 299 T^{10} + \cdots + 24010000 \)
T^12 - 17*T^11 + 299*T^10 - 4303*T^9 + 59016*T^8 - 279895*T^7 + 1527825*T^6 + 3472625*T^5 + 43254075*T^4 - 655590250*T^3 + 3606890000*T^2 + 178360000*T + 24010000
$73$
\( T^{12} - 37 T^{11} + \cdots + 356605456 \)
T^12 - 37*T^11 + 860*T^10 - 12965*T^9 + 140911*T^8 - 1089785*T^7 + 6634576*T^6 - 29728605*T^5 + 116298551*T^4 - 374962330*T^3 + 874607500*T^2 - 846796328*T + 356605456
$79$
\( T^{12} + 18 T^{11} + \cdots + 365758848400 \)
T^12 + 18*T^11 + 287*T^10 + 2950*T^9 + 40246*T^8 + 295300*T^7 + 3625408*T^6 + 23598624*T^5 + 455302921*T^4 + 4406732810*T^3 + 33488588940*T^2 - 25697102200*T + 365758848400
$83$
\( T^{12} - 19 T^{11} + \cdots + 237285894400 \)
T^12 - 19*T^11 + 493*T^10 - 4355*T^9 + 66396*T^8 - 213325*T^7 + 6929337*T^6 + 64033767*T^5 + 573528541*T^4 + 4915335980*T^3 + 40044510560*T^2 + 145434547200*T + 237285894400
$89$
\( (T^{6} - T^{5} - 174 T^{4} + 259 T^{3} + \cdots - 1375)^{2} \)
(T^6 - T^5 - 174*T^4 + 259*T^3 + 5206*T^2 + 5875*T - 1375)^2
$97$
\( T^{12} - T^{11} - 58 T^{10} + \cdots + 6990736 \)
T^12 - T^11 - 58*T^10 + 503*T^9 + 22493*T^8 + 224633*T^7 + 1528798*T^6 + 6204323*T^5 + 17760483*T^4 + 31173138*T^3 + 30551612*T^2 - 3104056*T + 6990736
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