[N,k,chi] = [2013,2,Mod(1,2013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, 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("2013.1");
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
\(3\)
\(-1\)
\(11\)
\(-1\)
\(61\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} - 4 T_{2}^{12} - 11 T_{2}^{11} + 57 T_{2}^{10} + 28 T_{2}^{9} - 290 T_{2}^{8} + 51 T_{2}^{7} + 644 T_{2}^{6} - 259 T_{2}^{5} - 640 T_{2}^{4} + 274 T_{2}^{3} + 256 T_{2}^{2} - 74 T_{2} - 35 \)
T2^13 - 4*T2^12 - 11*T2^11 + 57*T2^10 + 28*T2^9 - 290*T2^8 + 51*T2^7 + 644*T2^6 - 259*T2^5 - 640*T2^4 + 274*T2^3 + 256*T2^2 - 74*T2 - 35
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).
$p$
$F_p(T)$
$2$
\( T^{13} - 4 T^{12} - 11 T^{11} + 57 T^{10} + \cdots - 35 \)
T^13 - 4*T^12 - 11*T^11 + 57*T^10 + 28*T^9 - 290*T^8 + 51*T^7 + 644*T^6 - 259*T^5 - 640*T^4 + 274*T^3 + 256*T^2 - 74*T - 35
$3$
\( (T - 1)^{13} \)
(T - 1)^13
$5$
\( T^{13} - 7 T^{12} - 9 T^{11} + 162 T^{10} + \cdots + 44 \)
T^13 - 7*T^12 - 9*T^11 + 162*T^10 - 200*T^9 - 1053*T^8 + 2484*T^7 + 1485*T^6 - 7767*T^5 + 3167*T^4 + 5167*T^3 - 2159*T^2 - 1440*T + 44
$7$
\( T^{13} - 7 T^{12} - 21 T^{11} + \cdots - 17156 \)
T^13 - 7*T^12 - 21*T^11 + 219*T^10 + 83*T^9 - 2642*T^8 + 910*T^7 + 15225*T^6 - 8136*T^5 - 41588*T^4 + 18966*T^3 + 45455*T^2 - 10924*T - 17156
$11$
\( (T - 1)^{13} \)
(T - 1)^13
$13$
\( T^{13} - 9 T^{12} - 27 T^{11} + \cdots - 18548 \)
T^13 - 9*T^12 - 27*T^11 + 441*T^10 - 431*T^9 - 6413*T^8 + 16456*T^7 + 22113*T^6 - 114402*T^5 + 73876*T^4 + 149071*T^3 - 241211*T^2 + 118656*T - 18548
$17$
\( T^{13} - 19 T^{12} + 65 T^{11} + \cdots + 244288 \)
T^13 - 19*T^12 + 65*T^11 + 846*T^10 - 6776*T^9 + 2530*T^8 + 117576*T^7 - 352441*T^6 - 43429*T^5 + 1314467*T^4 - 876576*T^3 - 1167623*T^2 + 553456*T + 244288
$19$
\( T^{13} - 14 T^{12} + T^{11} + 768 T^{10} + \cdots - 3150 \)
T^13 - 14*T^12 + T^11 + 768*T^10 - 2666*T^9 - 10650*T^8 + 66401*T^7 - 16912*T^6 - 436785*T^5 + 786592*T^4 - 210990*T^3 - 271921*T^2 + 61140*T - 3150
$23$
\( T^{13} - 5 T^{12} - 84 T^{11} + \cdots - 6016 \)
T^13 - 5*T^12 - 84*T^11 + 280*T^10 + 2259*T^9 - 4159*T^8 - 25296*T^7 + 19647*T^6 + 105004*T^5 - 50024*T^4 - 133641*T^3 + 98991*T^2 - 3104*T - 6016
$29$
\( T^{13} - 10 T^{12} - 106 T^{11} + \cdots - 242600 \)
T^13 - 10*T^12 - 106*T^11 + 1160*T^10 + 2536*T^9 - 32203*T^8 - 46873*T^7 + 329739*T^6 + 583218*T^5 - 914834*T^4 - 1934680*T^3 + 416501*T^2 + 1260520*T - 242600
$31$
\( T^{13} + T^{12} - 115 T^{11} + \cdots - 8931814 \)
T^13 + T^12 - 115*T^11 - 244*T^10 + 4882*T^9 + 15068*T^8 - 89809*T^7 - 374872*T^6 + 589080*T^5 + 4054817*T^4 + 1505129*T^3 - 15173429*T^2 - 23348214*T - 8931814
$37$
\( T^{13} + 8 T^{12} - 206 T^{11} + \cdots - 15250532 \)
T^13 + 8*T^12 - 206*T^11 - 1851*T^10 + 11675*T^9 + 127471*T^8 - 132851*T^7 - 2949916*T^6 - 1819493*T^5 + 24534353*T^4 + 33843369*T^3 - 50652281*T^2 - 91103632*T - 15250532
$41$
\( T^{13} - 21 T^{12} + 7 T^{11} + \cdots - 5681822 \)
T^13 - 21*T^12 + 7*T^11 + 2696*T^10 - 17450*T^9 - 56137*T^8 + 891382*T^7 - 2310515*T^6 - 3811943*T^5 + 23000503*T^4 - 17353643*T^3 - 31686665*T^2 + 39806862*T - 5681822
$43$
\( T^{13} - 11 T^{12} + \cdots + 347470436 \)
T^13 - 11*T^12 - 216*T^11 + 2942*T^10 + 12007*T^9 - 278232*T^8 + 309044*T^7 + 10313887*T^6 - 42373555*T^5 - 79155633*T^4 + 821283450*T^3 - 1702751331*T^2 + 932846452*T + 347470436
$47$
\( T^{13} - 22 T^{12} + 31 T^{11} + \cdots + 926644 \)
T^13 - 22*T^12 + 31*T^11 + 2312*T^10 - 13766*T^9 - 55003*T^8 + 587878*T^7 - 297060*T^6 - 7118511*T^5 + 14494308*T^4 + 7624198*T^3 - 24291353*T^2 + 5010412*T + 926644
$53$
\( T^{13} - 16 T^{12} + 14 T^{11} + \cdots - 483498 \)
T^13 - 16*T^12 + 14*T^11 + 869*T^10 - 3139*T^9 - 14596*T^8 + 78560*T^7 + 60712*T^6 - 672465*T^5 + 259425*T^4 + 1768633*T^3 - 527905*T^2 - 1840338*T - 483498
$59$
\( T^{13} - 19 T^{12} - 179 T^{11} + \cdots - 22000 \)
T^13 - 19*T^12 - 179*T^11 + 5065*T^10 - 5024*T^9 - 348312*T^8 + 1446691*T^7 + 2750303*T^6 - 8803660*T^5 - 20283191*T^4 - 13447421*T^3 - 3779335*T^2 - 475400*T - 22000
$61$
\( (T - 1)^{13} \)
(T - 1)^13
$67$
\( T^{13} - 12 T^{12} + \cdots - 1402879138 \)
T^13 - 12*T^12 - 395*T^11 + 4151*T^10 + 57725*T^9 - 488263*T^8 - 3891589*T^7 + 25131988*T^6 + 110091623*T^5 - 630401147*T^4 - 821397010*T^3 + 6701931509*T^2 - 6801644312*T - 1402879138
$71$
\( T^{13} - 5 T^{12} - 256 T^{11} + \cdots - 5920072 \)
T^13 - 5*T^12 - 256*T^11 + 1470*T^10 + 17164*T^9 - 101485*T^8 - 389819*T^7 + 2643443*T^6 + 2068116*T^5 - 26174959*T^4 + 16776210*T^3 + 66266179*T^2 - 70636912*T - 5920072
$73$
\( T^{13} - 18 T^{12} + \cdots - 157498316 \)
T^13 - 18*T^12 - 227*T^11 + 4604*T^10 + 16575*T^9 - 408104*T^8 - 347087*T^7 + 15988263*T^6 - 8329802*T^5 - 274577652*T^4 + 369932341*T^3 + 1561639569*T^2 - 2682475280*T - 157498316
$79$
\( T^{13} + T^{12} - 435 T^{11} + \cdots - 6611280300 \)
T^13 + T^12 - 435*T^11 + 734*T^10 + 66586*T^9 - 246958*T^8 - 4164261*T^7 + 22541426*T^6 + 82986731*T^5 - 652033354*T^4 + 188991557*T^3 + 3689403043*T^2 - 1712950440*T - 6611280300
$83$
\( T^{13} - 48 T^{12} + \cdots + 129082984 \)
T^13 - 48*T^12 + 867*T^11 - 6266*T^10 - 6627*T^9 + 373824*T^8 - 1763459*T^7 - 2191890*T^6 + 39196913*T^5 - 83280586*T^4 - 106523923*T^3 + 556398179*T^2 - 568547600*T + 129082984
$89$
\( T^{13} - 15 T^{12} + \cdots - 227388350 \)
T^13 - 15*T^12 - 371*T^11 + 4455*T^10 + 56355*T^9 - 419796*T^8 - 3822884*T^7 + 13529793*T^6 + 86277786*T^5 - 192309328*T^4 - 415516164*T^3 + 750964883*T^2 + 292922300*T - 227388350
$97$
\( T^{13} + 17 T^{12} - 323 T^{11} + \cdots + 17107184 \)
T^13 + 17*T^12 - 323*T^11 - 5647*T^10 + 35458*T^9 + 665406*T^8 - 1505393*T^7 - 33401059*T^6 + 22902176*T^5 + 692957134*T^4 - 249873104*T^3 - 5011252697*T^2 + 3355675112*T + 17107184
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