/*
  This code can be loaded, or copied and paste using cpaste, into Sage.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
*/

P.<x> = PolynomialRing(QQ)
g = P([-2, 4, 3, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([19, 19, -w^3 + w^2 + 5*w - 1])

primes_array = [
[2, 2, w],\
[7, 7, -2*w^4 + w^3 + 12*w^2 + w - 5],\
[8, 2, w^4 - 7*w^2 - 3*w + 5],\
[19, 19, -2*w^4 + w^3 + 12*w^2 - 5],\
[19, 19, -w^3 + w^2 + 5*w - 1],\
[29, 29, 2*w^4 - 2*w^3 - 11*w^2 + 4*w + 3],\
[31, 31, -2*w^4 + w^3 + 12*w^2 + 2*w - 5],\
[37, 37, -2*w^4 + 13*w^2 + 5*w - 7],\
[53, 53, 3*w^4 - 2*w^3 - 18*w^2 + 3*w + 9],\
[59, 59, 2*w^4 - 13*w^2 - 6*w + 5],\
[61, 61, -3*w^4 + 2*w^3 + 18*w^2 - 2*w - 11],\
[61, 61, w^2 - w - 3],\
[61, 61, 6*w^4 - 2*w^3 - 38*w^2 - 6*w + 23],\
[67, 67, -w^4 + 7*w^2 + 4*w - 5],\
[67, 67, -w^4 + 6*w^2 + 2*w - 1],\
[71, 71, -w^2 + 3],\
[73, 73, 3*w^4 - w^3 - 19*w^2 - 3*w + 9],\
[79, 79, 3*w^4 - w^3 - 19*w^2 - 4*w + 11],\
[83, 83, -w^4 - w^3 + 7*w^2 + 8*w - 3],\
[97, 97, -2*w^4 + w^3 + 12*w^2 + 2*w - 3],\
[97, 97, w^4 - w^3 - 5*w^2 + 2*w + 3],\
[103, 103, -4*w^4 + w^3 + 25*w^2 + 7*w - 13],\
[103, 103, 7*w^4 - 3*w^3 - 43*w^2 - 4*w + 23],\
[107, 107, 4*w^4 - 2*w^3 - 24*w^2 - w + 9],\
[113, 113, 5*w^4 - 3*w^3 - 30*w^2 + w + 15],\
[137, 137, -3*w^4 + 2*w^3 + 17*w^2 - 7],\
[139, 139, w^2 - 5],\
[149, 149, 4*w^4 - w^3 - 26*w^2 - 5*w + 17],\
[151, 151, -4*w^4 + 2*w^3 + 24*w^2 + w - 13],\
[157, 157, -4*w^4 + w^3 + 26*w^2 + 5*w - 15],\
[157, 157, 5*w^4 - 2*w^3 - 31*w^2 - 4*w + 15],\
[157, 157, -3*w^4 + 2*w^3 + 19*w^2 - 2*w - 15],\
[169, 13, -w^4 + 2*w^3 + 5*w^2 - 8*w - 3],\
[173, 173, w^4 - w^3 - 5*w^2 + w - 1],\
[179, 179, -2*w^4 + 2*w^3 + 12*w^2 - 5*w - 7],\
[191, 191, -4*w^4 + 2*w^3 + 24*w^2 + 2*w - 11],\
[193, 193, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 5],\
[197, 197, w^4 - 6*w^2 - w + 1],\
[197, 197, -w^4 + 8*w^2 + 2*w - 11],\
[199, 199, -3*w^4 + w^3 + 20*w^2 + 3*w - 17],\
[211, 211, 4*w^4 - 2*w^3 - 25*w^2 + 13],\
[223, 223, w^4 - 6*w^2 - 3*w - 1],\
[223, 223, 2*w^4 - 13*w^2 - 4*w + 9],\
[223, 223, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 3],\
[227, 227, 2*w^4 - 12*w^2 - 7*w + 5],\
[227, 227, -w^4 + 8*w^2 + w - 7],\
[229, 229, -2*w^4 - w^3 + 14*w^2 + 11*w - 11],\
[229, 229, -3*w^4 + 19*w^2 + 8*w - 11],\
[233, 233, -w^4 + 8*w^2 + 2*w - 7],\
[233, 233, 3*w^4 - w^3 - 20*w^2 - w + 11],\
[239, 239, -w^4 + 6*w^2 + 5*w - 3],\
[243, 3, -3],\
[251, 251, -w^4 + w^3 + 5*w^2 - 3*w - 3],\
[263, 263, 3*w^4 - 2*w^3 - 18*w^2 + 2*w + 7],\
[269, 269, -3*w^4 + 20*w^2 + 8*w - 11],\
[271, 271, 4*w^4 - 2*w^3 - 25*w^2 - w + 13],\
[283, 283, -w^2 + 3*w + 1],\
[283, 283, 6*w^4 - 4*w^3 - 36*w^2 + 3*w + 19],\
[283, 283, -6*w^4 + 3*w^3 + 36*w^2 + 2*w - 17],\
[289, 17, -8*w^4 + 2*w^3 + 49*w^2 + 12*w - 21],\
[293, 293, -3*w^4 + 2*w^3 + 17*w^2 - 2*w - 7],\
[293, 293, 2*w^4 - 2*w^3 - 12*w^2 + 4*w + 7],\
[307, 307, -2*w^4 + 13*w^2 + 7*w - 9],\
[311, 311, -w^4 + 5*w^2 + 4*w + 3],\
[311, 311, 3*w^4 - w^3 - 18*w^2 - 2*w + 7],\
[317, 317, -w^4 + w^3 + 6*w^2 - 4*w - 3],\
[337, 337, 4*w^4 - w^3 - 25*w^2 - 7*w + 15],\
[347, 347, -2*w^4 + w^3 + 13*w^2 + w - 11],\
[347, 347, -3*w^4 + w^3 + 20*w^2 + 2*w - 17],\
[349, 349, -w^3 + w^2 + 3*w + 3],\
[359, 359, 3*w^4 - w^3 - 18*w^2 - 3*w + 11],\
[367, 367, -w^4 - w^3 + 8*w^2 + 6*w - 7],\
[367, 367, 4*w^4 - 3*w^3 - 24*w^2 + 3*w + 11],\
[373, 373, -w^3 + 6*w + 1],\
[379, 379, 2*w^4 - 14*w^2 - 5*w + 9],\
[397, 397, -2*w^4 - w^3 + 15*w^2 + 11*w - 13],\
[397, 397, w^4 - 6*w^2 - 2*w - 1],\
[397, 397, -w^4 - w^3 + 8*w^2 + 8*w - 5],\
[401, 401, -2*w^4 + 14*w^2 + 7*w - 13],\
[409, 409, -4*w^4 + 2*w^3 + 24*w^2 - 11],\
[421, 421, 3*w^4 - 2*w^3 - 18*w^2 + 9],\
[433, 433, -2*w^4 + 2*w^3 + 12*w^2 - 5*w - 5],\
[433, 433, 6*w^4 - w^3 - 38*w^2 - 12*w + 21],\
[439, 439, -5*w^4 + 2*w^3 + 30*w^2 + 3*w - 15],\
[443, 443, 4*w^4 - 2*w^3 - 26*w^2 - w + 15],\
[443, 443, -w^4 + w^3 + 5*w^2 - w - 5],\
[443, 443, -w^4 + w^3 + 6*w^2 - 4*w - 5],\
[443, 443, -w^4 + 6*w^2 + 5*w - 1],\
[443, 443, w^4 + w^3 - 7*w^2 - 7*w + 1],\
[461, 461, 2*w^4 - 14*w^2 - 6*w + 9],\
[461, 461, -w^4 + w^3 + 4*w^2 - 2*w + 5],\
[463, 463, w^4 - 7*w^2 - 2*w + 1],\
[479, 479, -6*w^4 + 3*w^3 + 37*w^2 + w - 19],\
[499, 499, -4*w^4 + 25*w^2 + 11*w - 13],\
[499, 499, 5*w^4 - 3*w^3 - 31*w^2 + 17],\
[503, 503, -4*w^4 + 3*w^3 + 23*w^2 - 3*w - 7],\
[503, 503, -3*w^4 + w^3 + 18*w^2 + w - 9],\
[503, 503, 2*w^4 - w^3 - 14*w^2 - w + 11],\
[509, 509, 8*w^4 - 2*w^3 - 52*w^2 - 11*w + 35],\
[521, 521, -2*w^4 + 2*w^3 + 13*w^2 - 5*w - 11],\
[521, 521, 10*w^4 - 4*w^3 - 63*w^2 - 6*w + 39],\
[529, 23, -w^4 - w^3 + 7*w^2 + 7*w - 3],\
[547, 547, 9*w^4 - 5*w^3 - 55*w^2 + w + 29],\
[547, 547, -w^4 + 2*w^3 + 6*w^2 - 9*w - 7],\
[557, 557, -5*w^4 + w^3 + 32*w^2 + 6*w - 17],\
[563, 563, 4*w^4 - w^3 - 26*w^2 - 4*w + 19],\
[587, 587, -2*w^3 + 3*w^2 + 10*w - 5],\
[587, 587, 7*w^4 - w^3 - 45*w^2 - 15*w + 25],\
[593, 593, -9*w^4 + 3*w^3 + 57*w^2 + 10*w - 33],\
[599, 599, w^4 + w^3 - 6*w^2 - 7*w + 3],\
[601, 601, w^4 - 2*w^3 - 4*w^2 + 9*w - 1],\
[601, 601, 3*w^4 - 3*w^3 - 16*w^2 + 5*w + 3],\
[607, 607, 6*w^4 - 2*w^3 - 38*w^2 - 5*w + 23],\
[607, 607, 4*w^4 - 2*w^3 - 24*w^2 - 3*w + 11],\
[613, 613, -w^4 + w^3 + 7*w^2 - 3*w - 5],\
[641, 641, 2*w^4 - 2*w^3 - 11*w^2 + 4*w + 7],\
[641, 641, -w^3 + 6*w - 1],\
[647, 647, 7*w^4 - 3*w^3 - 45*w^2 - 2*w + 31],\
[659, 659, -3*w^4 + w^3 + 19*w^2 + w - 9],\
[661, 661, -2*w^4 + 15*w^2 + 5*w - 13],\
[683, 683, -w^4 + 8*w^2 + 2*w - 9],\
[683, 683, -4*w^4 + 3*w^3 + 24*w^2 - 4*w - 17],\
[683, 683, 6*w^4 - 3*w^3 - 36*w^2 - w + 15],\
[691, 691, -3*w^4 + w^3 + 18*w^2 + 4*w - 11],\
[709, 709, w^4 - w^3 - 4*w^2 + 4*w - 3],\
[733, 733, -5*w^4 + 2*w^3 + 30*w^2 + 2*w - 15],\
[743, 743, -9*w^4 + 5*w^3 + 55*w^2 - 31],\
[751, 751, -4*w^4 + 2*w^3 + 23*w^2 + 2*w - 9],\
[751, 751, 5*w^4 - w^3 - 32*w^2 - 8*w + 19],\
[761, 761, w^4 + w^3 - 7*w^2 - 6*w + 3],\
[761, 761, -6*w^4 + w^3 + 38*w^2 + 11*w - 19],\
[787, 787, -w^3 + w^2 + 3*w - 5],\
[787, 787, -6*w^4 + 3*w^3 + 35*w^2 + 3*w - 13],\
[809, 809, -7*w^4 + 5*w^3 + 42*w^2 - 6*w - 23],\
[821, 821, 3*w^4 - 2*w^3 - 18*w^2 + 2*w + 5],\
[827, 827, 7*w^4 - 3*w^3 - 43*w^2 - 3*w + 23],\
[829, 829, 4*w^4 - w^3 - 26*w^2 - 7*w + 19],\
[829, 829, 5*w^4 - 2*w^3 - 30*w^2 - 4*w + 17],\
[841, 29, w^4 - w^3 - 5*w^2 - 1],\
[841, 29, 6*w^4 - 2*w^3 - 38*w^2 - 6*w + 19],\
[853, 853, 3*w^4 - 20*w^2 - 7*w + 13],\
[857, 857, 2*w^3 - 3*w^2 - 8*w + 3],\
[859, 859, -5*w^4 + 2*w^3 + 33*w^2 + 2*w - 27],\
[863, 863, -3*w^4 + 2*w^3 + 17*w^2 - 3],\
[881, 881, -5*w^4 + w^3 + 32*w^2 + 10*w - 21],\
[883, 883, w^2 - 2*w - 5],\
[887, 887, 3*w^4 + w^3 - 21*w^2 - 12*w + 13],\
[911, 911, -4*w^4 + w^3 + 26*w^2 + 7*w - 17],\
[911, 911, 6*w^4 - 2*w^3 - 37*w^2 - 5*w + 19],\
[919, 919, 14*w^4 - 3*w^3 - 86*w^2 - 26*w + 37],\
[919, 919, -7*w^4 + 4*w^3 + 42*w^2 - 21],\
[937, 937, 3*w^4 - 2*w^3 - 19*w^2 + 4*w + 13],\
[947, 947, 6*w^4 - 3*w^3 - 36*w^2 - 2*w + 19],\
[953, 953, w^4 - 2*w^3 - 3*w^2 + 6*w - 5],\
[953, 953, -9*w^4 + 3*w^3 + 56*w^2 + 10*w - 31],\
[977, 977, -5*w^4 + w^3 + 33*w^2 + 7*w - 23],\
[977, 977, -5*w^4 + 2*w^3 + 31*w^2 + 2*w - 13],\
[997, 997, 6*w^4 - w^3 - 38*w^2 - 13*w + 21],\
[997, 997, 4*w^4 - 2*w^3 - 24*w^2 - 4*w + 13]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^16 - 4*x^15 - 22*x^14 + 101*x^13 + 177*x^12 - 1035*x^11 - 551*x^10 + 5544*x^9 - 306*x^8 - 16618*x^7 + 6443*x^6 + 27546*x^5 - 16204*x^4 - 23056*x^3 + 16288*x^2 + 7488*x - 5760
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 27/32*e^15 - 35/16*e^14 - 347/16*e^13 + 1755/32*e^12 + 7281/32*e^11 - 17807/32*e^10 - 40195/32*e^9 + 47053/16*e^8 + 62531/16*e^7 - 138553/16*e^6 - 216795/32*e^5 + 56105/4*e^4 + 47913/8*e^3 - 45503/4*e^2 - 2089*e + 3521, 213/16*e^15 - 531/16*e^14 - 2743/8*e^13 + 13243/16*e^12 + 28829/8*e^11 - 66757/8*e^10 - 159309/8*e^9 + 700333/16*e^8 + 495361/8*e^7 - 511307/4*e^6 - 1711779/16*e^5 + 3285149/16*e^4 + 751449/8*e^3 - 660989/4*e^2 - 32378*e + 50847, 17*e^15 - 339/8*e^14 - 438*e^13 + 4229/4*e^12 + 36841/8*e^11 - 85311/8*e^10 - 203667/8*e^9 + 447705/8*e^8 + 158395/2*e^7 - 654041/4*e^6 - 547649/4*e^5 + 2101991/8*e^4 + 481143/4*e^3 - 423075/2*e^2 - 41496*e + 65108, 1, -63/32*e^15 + 19/4*e^14 + 815/16*e^13 - 3787/32*e^12 - 17211/32*e^11 + 38133/32*e^10 + 95505/32*e^9 - 6242*e^8 - 148899/16*e^7 + 291251/16*e^6 + 514667/32*e^5 - 467221/16*e^4 - 56295/4*e^3 + 23468*e^2 + 4819*e - 7209, -29/32*e^15 + 37/16*e^14 + 373/16*e^13 - 1853/32*e^12 - 7831/32*e^11 + 18777/32*e^10 + 43237/32*e^9 - 49555/16*e^8 - 67229/16*e^7 + 145775/16*e^6 + 232797/32*e^5 - 58987/4*e^4 - 51363/8*e^3 + 47789/4*e^2 + 2235*e - 3685, 31/4*e^15 - 77/4*e^14 - 799/4*e^13 + 1921/4*e^12 + 2101*e^11 - 19375/4*e^10 - 46479/4*e^9 + 50839/2*e^8 + 144653/4*e^7 - 74277*e^6 - 250179/4*e^5 + 477577/4*e^4 + 219875/4*e^3 - 192373/2*e^2 - 18962*e + 29636, 149/16*e^15 - 93/4*e^14 - 1919/8*e^13 + 9281/16*e^12 + 40341/16*e^11 - 93607/16*e^10 - 222939/16*e^9 + 30700*e^8 + 346615/8*e^7 - 717477/8*e^6 - 1197673/16*e^5 + 1152617/8*e^4 + 262797/4*e^3 - 231875/2*e^2 - 22632*e + 35658, -75/16*e^15 + 93/8*e^14 + 967/8*e^13 - 4643/16*e^12 - 20349/16*e^11 + 46859/16*e^10 + 112543/16*e^9 - 123039/8*e^8 - 175023/8*e^7 + 359749/8*e^6 + 604419/16*e^5 - 289239/4*e^4 - 132401/4*e^3 + 116477/2*e^2 + 11376*e - 17922, -123/16*e^15 + 153/8*e^14 + 1585/8*e^13 - 7635/16*e^12 - 33341/16*e^11 + 77015/16*e^10 + 184395/16*e^9 - 202105/8*e^8 - 286957/8*e^7 + 590597/8*e^6 + 992715/16*e^5 - 474647/4*e^4 - 109075/2*e^3 + 95576*e^2 + 18816*e - 29440, -91/32*e^15 + 57/8*e^14 + 1171/16*e^13 - 5695/32*e^12 - 24587/32*e^11 + 57525/32*e^10 + 135665/32*e^9 - 75605/8*e^8 - 210543/16*e^7 + 442691/16*e^6 + 726175/32*e^5 - 712823/16*e^4 - 19888*e^3 + 71857/2*e^2 + 6847*e - 11065, -17*e^15 + 169/4*e^14 + 438*e^13 - 2107/2*e^12 - 18419/4*e^11 + 42473/4*e^10 + 101801/4*e^9 - 222703/4*e^8 - 79132*e^7 + 325037/2*e^6 + 273347/2*e^5 - 1043673/4*e^4 - 239819/2*e^3 + 209896*e^2 + 41298*e - 64558, -103/4*e^15 + 257/4*e^14 + 663*e^13 - 6409/4*e^12 - 13931/2*e^11 + 16152*e^10 + 38475*e^9 - 338849/4*e^8 - 119583*e^7 + 247347*e^6 + 826133/4*e^5 - 1588831/4*e^4 - 181276*e^3 + 319580*e^2 + 62478*e - 98284, 1065/32*e^15 - 1325/16*e^14 - 13721/16*e^13 + 66105/32*e^12 + 288547/32*e^11 - 666621/32*e^10 - 1595225/32*e^9 + 1748771/16*e^8 + 2481169/16*e^7 - 5108203/16*e^6 - 8577209/32*e^5 + 1025809/2*e^4 + 1883231/8*e^3 - 1651333/4*e^2 - 81161*e + 127037, 229/32*e^15 - 283/16*e^14 - 2949/16*e^13 + 14093/32*e^12 + 61987/32*e^11 - 141821/32*e^10 - 342521/32*e^9 + 371213/16*e^8 + 532437/16*e^7 - 1081979/16*e^6 - 1839245/32*e^5 + 867617/8*e^4 + 403477/8*e^3 - 348843/4*e^2 - 17369*e + 26835, -63/8*e^15 + 39/2*e^14 + 203*e^13 - 3891/8*e^12 - 17083/8*e^11 + 39235/8*e^10 + 94479/8*e^9 - 102929/4*e^8 - 73501/2*e^7 + 300719/4*e^6 + 508367/8*e^5 - 483353/4*e^4 - 223353/4*e^3 + 194701/2*e^2 + 19262*e - 29992, 77/16*e^15 - 95/8*e^14 - 993/8*e^13 + 4741/16*e^12 + 20899/16*e^11 - 47829/16*e^10 - 115585/16*e^9 + 125541/8*e^8 + 179721/8*e^7 - 366971/8*e^6 - 620437/16*e^5 + 295011/4*e^4 + 135883/4*e^3 - 118793/2*e^2 - 11682*e + 18278, 785/32*e^15 - 245/4*e^14 - 10109/16*e^13 + 48901/32*e^12 + 212485/32*e^11 - 493235/32*e^10 - 1174167/32*e^9 + 323561/4*e^8 + 1825585/16*e^7 - 3781549/16*e^6 - 6309877/32*e^5 + 6076819/16*e^4 + 173204*e^3 - 611565/2*e^2 - 59747*e + 94113, -1375/32*e^15 + 1717/16*e^14 + 17699/16*e^13 - 85639/32*e^12 - 371841/32*e^11 + 863351/32*e^10 + 2053675/32*e^9 - 2264127/16*e^8 - 3191259/16*e^7 + 6611225/16*e^6 + 11023543/32*e^5 - 5308489/8*e^4 - 2419237/8*e^3 + 2135519/4*e^2 + 104265*e - 164215, -315/16*e^15 + 395/8*e^14 + 4053/8*e^13 - 19707/16*e^12 - 85113/16*e^11 + 198739/16*e^10 + 469879/16*e^9 - 521387/8*e^8 - 729889/8*e^7 + 1522993/8*e^6 + 2520579/16*e^5 - 305794*e^4 - 276541/2*e^3 + 246028*e^2 + 47676*e - 75646, -203/16*e^15 + 63/2*e^14 + 2615/8*e^13 - 12567/16*e^12 - 54983/16*e^11 + 126673/16*e^10 + 303901/16*e^9 - 83043/2*e^8 - 472507/8*e^7 + 970015/8*e^6 + 1632439/16*e^5 - 1558305/8*e^4 - 89522*e^3 + 156823*e^2 + 30826*e - 48274, 541/32*e^15 - 335/8*e^14 - 6969/16*e^13 + 33401/32*e^12 + 146525/32*e^11 - 336539/32*e^10 - 809823/32*e^9 + 441041/8*e^8 + 1259061/16*e^7 - 2574533/16*e^6 - 4350089/32*e^5 + 4133785/16*e^4 + 477237/4*e^3 - 207923*e^2 - 41099*e + 64001, -429/16*e^15 + 267/4*e^14 + 5525/8*e^13 - 26633/16*e^12 - 116141/16*e^11 + 268483/16*e^10 + 641799/16*e^9 - 352043/4*e^8 - 997769/8*e^7 + 2056069/8*e^6 + 3447577/16*e^5 - 3302569/8*e^4 - 189164*e^3 + 332279*e^2 + 65210*e - 102264, 1/16*e^15 - 1/4*e^14 - 13/8*e^13 + 109/16*e^12 + 281/16*e^11 - 1215/16*e^10 - 1651/16*e^9 + 1775/4*e^8 + 2873/8*e^7 - 11545/8*e^6 - 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180325*e^3 + 310265*e^2 + 61942*e - 95452, 2933/32*e^15 - 3663/16*e^14 - 37757/16*e^13 + 182733/32*e^12 + 793323/32*e^11 - 1842597/32*e^10 - 4382049/32*e^9 + 4833521/16*e^8 + 6810509/16*e^7 - 14118563/16*e^6 - 23531565/32*e^5 + 11341083/8*e^4 + 5166405/8*e^3 - 4564471/4*e^2 - 222795*e + 351131, -2015/32*e^15 + 629/4*e^14 + 25947/16*e^13 - 125547/32*e^12 - 545339/32*e^11 + 1266333/32*e^10 + 3013049/32*e^9 - 830727/4*e^8 - 4683671/16*e^7 + 9709219/16*e^6 + 16183643/32*e^5 - 15602965/16*e^4 - 888155/2*e^3 + 1570313/2*e^2 + 153143*e - 241645, 2453/32*e^15 - 1525/8*e^14 - 31597/16*e^13 + 152129/32*e^12 + 664301/32*e^11 - 1533683/32*e^10 - 3671351/32*e^9 + 2011069/8*e^8 + 5707825/16*e^7 - 11745093/16*e^6 - 19720193/32*e^5 + 18863413/16*e^4 + 1081681/2*e^3 - 1897425/2*e^2 - 186325*e + 291867, 59/16*e^15 - 77/8*e^14 - 757/8*e^13 + 3859/16*e^12 + 15861/16*e^11 - 39143/16*e^10 - 87483/16*e^9 + 103425/8*e^8 + 136121/8*e^7 - 304589/8*e^6 - 472907/16*e^5 + 246705/4*e^4 + 26261*e^3 - 50006*e^2 - 9236*e + 15462, 1623/32*e^15 - 2025/16*e^14 - 20903/16*e^13 + 101039/32*e^12 + 439425/32*e^11 - 1019007/32*e^10 - 2428499/32*e^9 + 2673391/16*e^8 + 3776055/16*e^7 - 7809097/16*e^6 - 13050543/32*e^5 + 6272099/8*e^4 + 2865015/8*e^3 - 2523525/4*e^2 - 123473*e + 194055, 539/16*e^15 - 335/4*e^14 - 6947/8*e^13 + 33439/16*e^12 + 146147/16*e^11 - 337357/16*e^10 - 808201/16*e^9 + 442731/4*e^8 + 1257207/8*e^7 - 2588019/8*e^6 - 4345631/16*e^5 + 4160579/8*e^4 + 238471*e^3 - 418919*e^2 - 82218*e + 129018, -211/4*e^15 + 1047/8*e^14 + 5435/4*e^13 - 3262*e^12 - 114247/8*e^11 + 262911/8*e^10 + 631267/8*e^9 - 1378003/8*e^8 - 981151/4*e^7 + 2010559/4*e^6 + 847135/2*e^5 - 6454539/8*e^4 - 371551*e^3 + 649065*e^2 + 127916*e - 199702, -719/16*e^15 + 897/8*e^14 + 9255/8*e^13 - 44727/16*e^12 - 194441/16*e^11 + 450759/16*e^10 + 1073883/16*e^9 - 1181703/8*e^8 - 1668599/8*e^7 + 3449449/8*e^6 + 5762487/16*e^5 - 2769059/4*e^4 - 1264063/4*e^3 + 1113763/2*e^2 + 108884*e - 171258, 397/16*e^15 - 495/8*e^14 - 5113/8*e^13 + 24693/16*e^12 + 107475/16*e^11 - 248949/16*e^10 - 593809/16*e^9 + 652773/8*e^8 + 922833/8*e^7 - 1905267/8*e^6 - 3186629/16*e^5 + 1528595/4*e^4 + 698647/4*e^3 - 614185/2*e^2 - 60122*e + 94326, -1421/16*e^15 + 1769/8*e^14 + 18301/8*e^13 - 88237/16*e^12 - 384703/16*e^11 + 889585/16*e^10 + 2125821/16*e^9 - 2333039/8*e^8 - 3304725/8*e^7 + 6812839/8*e^6 + 11417885/16*e^5 - 1367697*e^4 - 2505607/4*e^3 + 2200973/2*e^2 + 215882*e - 338514, 103*e^15 - 1025/4*e^14 - 2654*e^13 + 12785/2*e^12 + 111623/4*e^11 - 257873/4*e^10 - 617081/4*e^9 + 1353123/4*e^8 + 479863*e^7 - 1976571/2*e^6 - 1658685/2*e^5 + 6352245/4*e^4 + 1456575/2*e^3 - 1278567*e^2 - 251082*e + 393524, -221/4*e^15 + 138*e^14 + 5691/4*e^13 - 13769/4*e^12 - 59799/4*e^11 + 34711*e^10 + 82589*e^9 - 728451/4*e^8 - 1026857/4*e^7 + 1063917/2*e^6 + 1773435/4*e^5 - 854637*e^4 - 1556265/4*e^3 + 1375827/2*e^2 + 134068*e - 211648, 1313/32*e^15 - 821/8*e^14 - 16905/16*e^13 + 81949/32*e^12 + 355241/32*e^11 - 826743/32*e^10 - 1962347/32*e^9 + 1084917/8*e^8 + 3049565/16*e^7 - 6340929/16*e^6 - 10532829/32*e^5 + 10189609/16*e^4 + 577657/2*e^3 - 1025061/2*e^2 - 99531*e + 157595, -1139/32*e^15 + 1411/16*e^14 + 14683/16*e^13 - 70387/32*e^12 - 308969/32*e^11 + 709703/32*e^10 + 1709147/32*e^9 - 1861525/16*e^8 - 2659651/16*e^7 + 5436945/16*e^6 + 9196691/32*e^5 - 2183555/4*e^4 - 2019269/8*e^3 + 1757567/4*e^2 + 87011*e - 135243, -237/8*e^15 + 293/4*e^14 + 3057/4*e^13 - 14621/8*e^12 - 64367/8*e^11 + 147481/8*e^10 + 356277/8*e^9 - 387031/4*e^8 - 554705/4*e^7 + 1131139/4*e^6 + 1918893/8*e^5 - 454695*e^4 - 421461/2*e^3 + 366455*e^2 + 72670*e - 112948, -1515/32*e^15 + 949/8*e^14 + 19503/16*e^13 - 94735/32*e^12 - 409787/32*e^11 + 955853/32*e^10 + 2263529/32*e^9 - 1254539/8*e^8 - 3517811/16*e^7 + 7333763/16*e^6 + 12152959/32*e^5 - 11788375/16*e^4 - 1333663/4*e^3 + 593237*e^2 + 114991*e - 182563, -103/16*e^15 + 131/8*e^14 + 1331/8*e^13 - 6583/16*e^12 - 28093/16*e^11 + 66939/16*e^10 + 156031/16*e^9 - 177225/8*e^8 - 244187/8*e^7 + 522637/8*e^6 + 851415/16*e^5 - 105903*e^4 - 189131/4*e^3 + 171813/2*e^2 + 16540*e - 26592, 179/32*e^15 - 235/16*e^14 - 2287/16*e^13 + 11731/32*e^12 + 47657/32*e^11 - 118367/32*e^10 - 260979/32*e^9 + 310617/16*e^8 + 402191/16*e^7 - 906881/16*e^6 - 1378819/32*e^5 + 363427/4*e^4 + 300575/8*e^3 - 291077/4*e^2 - 12895*e + 22203, 33/16*e^15 - 5*e^14 - 425/8*e^13 + 1989/16*e^12 + 8941/16*e^11 - 20003/16*e^10 - 49511/16*e^9 + 13101/2*e^8 + 77277/8*e^7 - 153253/8*e^6 - 268757/16*e^5 + 247487/8*e^4 + 29783/2*e^3 - 25176*e^2 - 5198*e + 7886, -1103/32*e^15 + 1369/16*e^14 + 14215/16*e^13 - 68295/32*e^12 - 299033/32*e^11 + 688631/32*e^10 + 1653707/32*e^9 - 1806231/16*e^8 - 2572759/16*e^7 + 5274929/16*e^6 + 8894855/32*e^5 - 4236031/8*e^4 - 1952731/8*e^3 + 1704205/4*e^2 + 84091*e - 131047, -589/32*e^15 + 723/16*e^14 + 7605/16*e^13 - 36053/32*e^12 - 160315/32*e^11 + 363349/32*e^10 + 888465/32*e^9 - 952573/16*e^8 - 1384869/16*e^7 + 2781075/16*e^6 + 4794069/32*e^5 - 2233729/8*e^4 - 1052857/8*e^3 + 899359/4*e^2 + 45329*e - 69261, -1671/16*e^15 + 2079/8*e^14 + 21527/8*e^13 - 103719/16*e^12 - 452661/16*e^11 + 1045883/16*e^10 + 2502191/16*e^9 - 2743537/8*e^8 - 3891127/8*e^7 + 8013373/8*e^6 + 13448135/16*e^5 - 3218239/2*e^4 - 2951881/4*e^3 + 2590323/2*e^2 + 254350*e - 398556, -589/32*e^15 + 743/16*e^14 + 7581/16*e^13 - 37125/32*e^12 - 159251/32*e^11 + 375021/32*e^10 + 879433/32*e^9 - 985585/16*e^8 - 1366557/16*e^7 + 2883739/16*e^6 + 4721957/32*e^5 - 2319139/8*e^4 - 1037169/8*e^3 + 933675/4*e^2 + 44767*e - 71787, 183/8*e^15 - 227/4*e^14 - 589*e^13 + 11311/8*e^12 + 49505/8*e^11 - 113901/8*e^10 - 273433/8*e^9 + 74584*e^8 + 212425/2*e^7 - 870057/4*e^6 - 1467035/8*e^5 + 697853/2*e^4 + 643491/4*e^3 - 561013/2*e^2 - 55392*e + 86240, 1317/16*e^15 - 819/4*e^14 - 16967/8*e^13 + 81713/16*e^12 + 356797/16*e^11 - 823935/16*e^10 - 1972483/16*e^9 + 540313/2*e^8 + 3067839/8*e^7 - 6312589/8*e^6 - 10604745/16*e^5 + 10141109/8*e^4 + 2328209/4*e^3 - 2040773/2*e^2 - 200632*e + 314042, 361/16*e^15 - 453/8*e^14 - 4647/8*e^13 + 22617/16*e^12 + 97643/16*e^11 - 228297/16*e^10 - 539445/16*e^9 + 599637/8*e^8 + 838755/8*e^7 - 1754131/8*e^6 - 2900497/16*e^5 + 705655/2*e^4 + 318897/2*e^3 - 284461*e^2 - 55154*e + 87662, -165/16*e^15 + 103/4*e^14 + 2125/8*e^13 - 10289/16*e^12 - 44661/16*e^11 + 103899/16*e^10 + 246687/16*e^9 - 136499/4*e^8 - 383217/8*e^7 + 798829/8*e^6 + 1322593/16*e^5 - 1285545/8*e^4 - 72444*e^3 + 129530*e^2 + 24912*e - 39892, -215/2*e^15 + 2143/8*e^14 + 11077/4*e^13 - 26731/4*e^12 - 232877/8*e^11 + 539177/8*e^10 + 1287061/8*e^9 - 2829171/8*e^8 - 2001281/4*e^7 + 4132389/4*e^6 + 3458353/4*e^5 - 13278127/8*e^4 - 759228*e^3 + 1335922*e^2 + 261762*e - 411076, -297/4*e^15 + 1485/8*e^14 + 7647/4*e^13 - 4631*e^12 - 160677/8*e^11 + 373657/8*e^10 + 887501/8*e^9 - 1960773/8*e^8 - 1379155/4*e^7 + 2864097/4*e^6 + 1190903/2*e^5 - 9202497/8*e^4 - 1045117/2*e^3 + 925670*e^2 + 180068*e - 284700, 583/16*e^15 - 90*e^14 - 7519/8*e^13 + 35923/16*e^12 + 158283/16*e^11 - 362261/16*e^10 - 875809/16*e^9 + 237569/2*e^8 + 1362851/8*e^7 - 2775323/8*e^6 - 4710323/16*e^5 + 4457209/8*e^4 + 516467/2*e^3 - 448190*e^2 - 88814*e + 137810, 3991/32*e^15 - 4981/16*e^14 - 51383/16*e^13 + 248447/32*e^12 + 1079785/32*e^11 - 2504759/32*e^10 - 5965355/32*e^9 + 6568979/16*e^8 + 9272551/16*e^7 - 19182497/16*e^6 - 32039391/32*e^5 + 15404169/8*e^4 + 7032843/8*e^3 - 6197841/4*e^2 - 303073*e + 476679, 775/16*e^15 - 120*e^14 - 9987/8*e^13 + 47875/16*e^12 + 210059/16*e^11 - 482557/16*e^10 - 1161369/16*e^9 + 158160*e^8 + 1806063/8*e^7 - 3694219/8*e^6 - 6240323/16*e^5 + 5932945/8*e^4 + 342241*e^3 - 596849*e^2 - 117874*e + 183684, -559/32*e^15 + 43*e^14 + 7219/16*e^13 - 34347/32*e^12 - 152179/32*e^11 + 346581/32*e^10 + 843185/32*e^9 - 113715/2*e^8 - 1313695/16*e^7 + 2658731/16*e^6 + 4545051/32*e^5 - 4273681/16*e^4 - 249413/2*e^3 + 430237/2*e^2 + 42947*e - 66253, -63/4*e^15 + 159/4*e^14 + 404*e^13 - 3957/4*e^12 - 4227*e^11 + 19897/2*e^10 + 46499/2*e^9 - 208123/4*e^8 - 71969*e^7 + 151429*e^6 + 495473/4*e^5 - 969093/4*e^4 - 108438*e^3 + 194156*e^2 + 37312*e - 59496, -2309/32*e^15 + 1439/8*e^14 + 29737/16*e^13 - 143585/32*e^12 - 625093/32*e^11 + 1447971/32*e^10 + 3454247/32*e^9 - 1899317/8*e^8 - 5370197/16*e^7 + 11096413/16*e^6 + 18556689/32*e^5 - 17827601/16*e^4 - 2036591/4*e^3 + 896874*e^2 + 175559*e - 275979, -4053/32*e^15 + 5045/16*e^14 + 52213/16*e^13 - 251701/32*e^12 - 1097935/32*e^11 + 2538289/32*e^10 + 6069533/32*e^9 - 6659083/16*e^8 - 9440093/16*e^7 + 19452615/16*e^6 + 32634389/32*e^5 - 7813465/4*e^4 - 7165975/8*e^3 + 6289725/4*e^2 + 308893*e - 483939, -1331/32*e^15 + 823/8*e^14 + 17159/16*e^13 - 82087/32*e^12 - 361083/32*e^11 + 827405/32*e^10 + 1997385/32*e^9 - 1084737/8*e^8 - 3107707/16*e^7 + 6334019/16*e^6 + 10741559/32*e^5 - 10171843/16*e^4 - 1178225/4*e^3 + 511562*e^2 + 101391*e - 157381, -2143/32*e^15 + 2669/16*e^14 + 27599/16*e^13 - 133095/32*e^12 - 580161/32*e^11 + 1341375/32*e^10 + 3206067/32*e^9 - 3516395/16*e^8 - 4984527/16*e^7 + 10263481/16*e^6 + 17223911/32*e^5 - 8238249/8*e^4 - 3780187/8*e^3 + 3313813/4*e^2 + 162823*e - 254899, -21/2*e^15 + 26*e^14 + 271*e^13 - 1299/2*e^12 - 5709/2*e^11 + 13121/2*e^10 + 31617/2*e^9 - 34480*e^8 - 49252*e^7 + 100863*e^6 + 170437/2*e^5 - 162127*e^4 - 74846*e^3 + 130340*e^2 + 25776*e - 39994, -1595/16*e^15 + 997/4*e^14 + 20531/8*e^13 - 99471/16*e^12 - 431339/16*e^11 + 1002949/16*e^10 + 2382257/16*e^9 - 1315273/4*e^8 - 3701759/8*e^7 + 7681667/8*e^6 + 12786351/16*e^5 - 12335743/8*e^4 - 701458*e^3 + 1240467*e^2 + 241798*e - 381474, -227/8*e^15 + 283/4*e^14 + 2923/4*e^13 - 14115/8*e^12 - 61433/8*e^11 + 142295/8*e^10 + 339411/8*e^9 - 373169/4*e^8 - 527539/4*e^7 + 1089729/4*e^6 + 1822259/8*e^5 - 437585*e^4 - 399765/2*e^3 + 352171*e^2 + 68852*e - 108366, -1003/32*e^15 + 625/8*e^14 + 12915/16*e^13 - 62351/32*e^12 - 271419/32*e^11 + 628613/32*e^10 + 1499393/32*e^9 - 824277/8*e^8 - 2330079/16*e^7 + 4813475/16*e^6 + 8046895/32*e^5 - 7728535/16*e^4 - 220617*e^3 + 776941/2*e^2 + 76021*e - 119397, -533/32*e^15 + 669/16*e^14 + 6857/16*e^13 - 33365/32*e^12 - 143983/32*e^11 + 336297/32*e^10 + 794869/32*e^9 - 881615/16*e^8 - 1234857/16*e^7 + 2572711/16*e^6 + 4265413/32*e^5 - 1031833/4*e^4 - 936033/8*e^3 + 828855/4*e^2 + 40299*e - 63597, -2811/32*e^15 + 219*e^14 + 36195/16*e^13 - 174791/32*e^12 - 760671/32*e^11 + 1762369/32*e^10 + 4202381/32*e^9 - 1155637/4*e^8 - 6531591/16*e^7 + 13500359/16*e^6 + 22564615/32*e^5 - 21685197/16*e^4 - 2476015/4*e^3 + 1090758*e^2 + 213381*e - 335613, -1007/8*e^15 + 1253/4*e^14 + 3242*e^13 - 62487/8*e^12 - 272573/8*e^11 + 629841/8*e^10 + 1506045/8*e^9 - 825721/2*e^8 - 1170471/2*e^7 + 4821377/4*e^6 + 8086563/8*e^5 - 1935471*e^4 - 3548155/4*e^3 + 3114405/2*e^2 + 305568*e - 479008, -1825/16*e^15 + 2275/8*e^14 + 23501/8*e^13 - 113481/16*e^12 - 493951/16*e^11 + 1144153/16*e^10 + 2729253/16*e^9 - 3000897/8*e^8 - 4242613/8*e^7 + 8764015/8*e^6 + 14658761/16*e^5 - 7038715/4*e^4 - 3217179/4*e^3 + 2832445/2*e^2 + 277236*e - 435772]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([19, 19, -w^3 + w^2 + 5*w - 1])] = -1

# EXAMPLE:
# pp = ZF.ideal(2).factor()[0][0]
# hecke_eigenvalues[pp]