[N,k,chi] = [1336,2,Mod(1,1336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1336.1");
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
\(2\)
\(1\)
\(167\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 5 T_{3}^{11} - 15 T_{3}^{10} + 100 T_{3}^{9} + 36 T_{3}^{8} - 641 T_{3}^{7} + 129 T_{3}^{6} + 1804 T_{3}^{5} - 433 T_{3}^{4} - 2399 T_{3}^{3} + 148 T_{3}^{2} + 1224 T_{3} + 224 \)
T3^12 - 5*T3^11 - 15*T3^10 + 100*T3^9 + 36*T3^8 - 641*T3^7 + 129*T3^6 + 1804*T3^5 - 433*T3^4 - 2399*T3^3 + 148*T3^2 + 1224*T3 + 224
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( T^{12} - 5 T^{11} - 15 T^{10} + 100 T^{9} + \cdots + 224 \)
T^12 - 5*T^11 - 15*T^10 + 100*T^9 + 36*T^8 - 641*T^7 + 129*T^6 + 1804*T^5 - 433*T^4 - 2399*T^3 + 148*T^2 + 1224*T + 224
$5$
\( T^{12} + 2 T^{11} - 37 T^{10} - 62 T^{9} + \cdots - 128 \)
T^12 + 2*T^11 - 37*T^10 - 62*T^9 + 503*T^8 + 658*T^7 - 2991*T^6 - 2638*T^5 + 7020*T^4 + 2696*T^3 - 3840*T^2 - 1696*T - 128
$7$
\( T^{12} - 10 T^{11} - 2 T^{10} + \cdots + 10696 \)
T^12 - 10*T^11 - 2*T^10 + 294*T^9 - 615*T^8 - 2515*T^7 + 8184*T^6 + 6047*T^5 - 36610*T^4 + 11219*T^3 + 51154*T^2 - 45795*T + 10696
$11$
\( T^{12} - 14 T^{11} + 28 T^{10} + \cdots + 8192 \)
T^12 - 14*T^11 + 28*T^10 + 460*T^9 - 2684*T^8 + 1147*T^7 + 29358*T^6 - 95707*T^5 + 112864*T^4 - 21016*T^3 - 48256*T^2 + 15872*T + 8192
$13$
\( T^{12} + 9 T^{11} - 32 T^{10} + \cdots + 14336 \)
T^12 + 9*T^11 - 32*T^10 - 503*T^9 - 859*T^8 + 4608*T^7 + 16156*T^6 + 3376*T^5 - 42112*T^4 - 42208*T^3 + 19968*T^2 + 42240*T + 14336
$17$
\( T^{12} - 4 T^{11} - 107 T^{10} + \cdots - 152320 \)
T^12 - 4*T^11 - 107*T^10 + 484*T^9 + 3457*T^8 - 17768*T^7 - 30376*T^6 + 204152*T^5 - 4912*T^4 - 718688*T^3 + 667456*T^2 + 65792*T - 152320
$19$
\( T^{12} - 9 T^{11} - 101 T^{10} + \cdots + 256 \)
T^12 - 9*T^11 - 101*T^10 + 882*T^9 + 3877*T^8 - 27602*T^7 - 83321*T^6 + 302125*T^5 + 913360*T^4 - 339000*T^3 - 1444272*T^2 + 263568*T + 256
$23$
\( T^{12} - 15 T^{11} - 8 T^{10} + \cdots - 3584 \)
T^12 - 15*T^11 - 8*T^10 + 897*T^9 - 2475*T^8 - 8500*T^7 + 22816*T^6 + 38864*T^5 - 43440*T^4 - 66400*T^3 + 10112*T^2 + 18560*T - 3584
$29$
\( T^{12} - 17 T^{11} - 40 T^{10} + \cdots + 86030192 \)
T^12 - 17*T^11 - 40*T^10 + 2339*T^9 - 11122*T^8 - 63700*T^7 + 699610*T^6 - 1478717*T^5 - 4525851*T^4 + 22060503*T^3 - 10471766*T^2 - 66811048*T + 86030192
$31$
\( T^{12} - 11 T^{11} - 104 T^{10} + \cdots - 10048 \)
T^12 - 11*T^11 - 104*T^10 + 1505*T^9 + 236*T^8 - 46766*T^7 + 62684*T^6 + 526471*T^5 - 754599*T^4 - 2459803*T^3 + 2307886*T^2 + 3887668*T - 10048
$37$
\( T^{12} + 29 T^{11} + 187 T^{10} + \cdots + 32082176 \)
T^12 + 29*T^11 + 187*T^10 - 1858*T^9 - 25223*T^8 - 20659*T^7 + 767515*T^6 + 2565136*T^5 - 4925704*T^4 - 26380840*T^3 + 2846944*T^2 + 69495200*T + 32082176
$41$
\( T^{12} - 14 T^{11} + \cdots - 401387264 \)
T^12 - 14*T^11 - 162*T^10 + 2760*T^9 + 5209*T^8 - 173906*T^7 + 126608*T^6 + 4580528*T^5 - 8166384*T^4 - 49913344*T^3 + 113933120*T^2 + 160080768*T - 401387264
$43$
\( T^{12} - 15 T^{11} - 258 T^{10} + \cdots + 226048 \)
T^12 - 15*T^11 - 258*T^10 + 5191*T^9 + 5713*T^8 - 527448*T^7 + 2344292*T^6 + 10938760*T^5 - 112802896*T^4 + 309703584*T^3 - 305662720*T^2 + 96022016*T + 226048
$47$
\( T^{12} - 17 T^{11} - 148 T^{10} + \cdots - 23909920 \)
T^12 - 17*T^11 - 148*T^10 + 3567*T^9 - 1082*T^8 - 198498*T^7 + 508200*T^6 + 2998773*T^5 - 8528665*T^4 - 17209117*T^3 + 38361686*T^2 + 37880532*T - 23909920
$53$
\( T^{12} + 7 T^{11} + \cdots + 109447774208 \)
T^12 + 7*T^11 - 546*T^10 - 3593*T^9 + 116851*T^8 + 711706*T^7 - 12427128*T^6 - 68736136*T^5 + 675012496*T^4 + 3259971360*T^3 - 16573697600*T^2 - 61063433856*T + 109447774208
$59$
\( T^{12} - 32 T^{11} + \cdots - 25602668800 \)
T^12 - 32*T^11 + 2*T^10 + 9214*T^9 - 67827*T^8 - 765862*T^7 + 9087088*T^6 + 13278448*T^5 - 384829696*T^4 + 356254496*T^3 + 5884546368*T^2 - 7874267776*T - 25602668800
$61$
\( T^{12} + 8 T^{11} - 139 T^{10} + \cdots + 50608 \)
T^12 + 8*T^11 - 139*T^10 - 1194*T^9 + 4014*T^8 + 42707*T^7 - 182*T^6 - 423432*T^5 - 604336*T^4 + 310735*T^3 + 826598*T^2 + 383480*T + 50608
$67$
\( T^{12} - 39 T^{11} + \cdots + 4208021440 \)
T^12 - 39*T^11 + 223*T^10 + 8790*T^9 - 117857*T^8 - 339725*T^7 + 12291001*T^6 - 37566494*T^5 - 339741500*T^4 + 2296669344*T^3 - 3591797376*T^2 - 1716830272*T + 4208021440
$71$
\( T^{12} - 35 T^{11} + \cdots - 7322525696 \)
T^12 - 35*T^11 + 113*T^10 + 8508*T^9 - 92868*T^8 - 297496*T^7 + 7608800*T^6 - 12528768*T^5 - 191171584*T^4 + 553608320*T^3 + 1628939776*T^2 - 3922143232*T - 7322525696
$73$
\( T^{12} + 12 T^{11} + \cdots - 21136762880 \)
T^12 + 12*T^11 - 507*T^10 - 5698*T^9 + 94639*T^8 + 994734*T^7 - 8009820*T^6 - 80600136*T^5 + 300242048*T^4 + 3038733728*T^3 - 3524321216*T^2 - 42816505856*T - 21136762880
$79$
\( T^{12} - 36 T^{11} + \cdots - 744310784 \)
T^12 - 36*T^11 + 208*T^10 + 5258*T^9 - 54517*T^8 - 213878*T^7 + 3263212*T^6 + 2071736*T^5 - 66303808*T^4 - 16996640*T^3 + 483149312*T^2 + 188994816*T - 744310784
$83$
\( T^{12} - 25 T^{11} + \cdots - 6437720512 \)
T^12 - 25*T^11 - 213*T^10 + 8050*T^9 + 9753*T^8 - 949333*T^7 + 593979*T^6 + 47677612*T^5 - 64425744*T^4 - 910305256*T^3 + 1648604560*T^2 + 3736088320*T - 6437720512
$89$
\( T^{12} - 23 T^{11} + \cdots - 947344672 \)
T^12 - 23*T^11 - 243*T^10 + 6764*T^9 + 23917*T^8 - 703854*T^7 - 1352151*T^6 + 31904735*T^5 + 49077526*T^4 - 584623364*T^3 - 831182584*T^2 + 2284871568*T - 947344672
$97$
\( T^{12} + 2 T^{11} + \cdots + 5304470056 \)
T^12 + 2*T^11 - 565*T^10 - 1000*T^9 + 113556*T^8 + 136087*T^7 - 9773680*T^6 - 1294596*T^5 + 342924422*T^4 - 370715511*T^3 - 2817018836*T^2 + 2767434936*T + 5304470056
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