/*
  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([-1, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

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

primes_array = [
[3, 3, w + 1],\
[7, 7, w - 1],\
[8, 2, 2],\
[9, 3, -w^2 + 2*w + 4],\
[11, 11, -w^2 + 2*w + 2],\
[13, 13, -w^2 + w + 4],\
[19, 19, w + 3],\
[19, 19, -w^2 + 2*w + 5],\
[19, 19, -w^2 + 3*w + 2],\
[23, 23, w^2 - w - 3],\
[23, 23, -w^2 + 2],\
[23, 23, -w + 4],\
[31, 31, w^2 - 5],\
[43, 43, w^2 - 3*w - 3],\
[49, 7, w^2 - 6],\
[53, 53, 2*w - 5],\
[61, 61, 2*w^2 - 2*w - 9],\
[71, 71, 2*w - 3],\
[73, 73, 2*w^2 - 5*w - 5],\
[83, 83, w^2 - w - 9],\
[97, 97, 2*w^2 - 3*w - 7],\
[103, 103, w^2 - 3*w - 6],\
[109, 109, w^2 - 4*w - 3],\
[121, 11, 2*w^2 - w - 7],\
[125, 5, -5],\
[127, 127, w^2 + w - 5],\
[131, 131, w^2 - 11],\
[137, 137, 3*w^2 - 7*w - 8],\
[137, 137, 2*w^2 - 5*w - 6],\
[137, 137, -w^2 + w - 2],\
[139, 139, 3*w^2 - 2*w - 16],\
[139, 139, 3*w - 2],\
[139, 139, -w^2 + 3*w - 3],\
[151, 151, w^2 + w - 11],\
[157, 157, 2*w^2 - 3*w - 3],\
[163, 163, w^2 - w - 10],\
[169, 13, 2*w^2 - 3*w - 4],\
[173, 173, -3*w^2 + 6*w + 8],\
[191, 191, -4*w^2 + 7*w + 15],\
[193, 193, 2*w^2 - 3],\
[197, 197, w^2 - 3*w - 11],\
[199, 199, 3*w^2 - 5*w - 16],\
[211, 211, w^2 - 4*w - 9],\
[223, 223, 2*w^2 - w - 4],\
[227, 227, 3*w - 4],\
[229, 229, w^2 + w - 8],\
[241, 241, w^2 - 4*w - 6],\
[251, 251, w - 7],\
[257, 257, w^2 - 4*w - 7],\
[263, 263, 3*w^2 - 5*w - 10],\
[263, 263, 2*w^2 - 5*w - 15],\
[263, 263, -w^2 + 6*w - 1],\
[269, 269, -w^2 - w + 14],\
[277, 277, 2*w^2 - 5*w - 8],\
[277, 277, w^2 + 3*w - 3],\
[277, 277, 3*w^2 - 4*w - 12],\
[281, 281, -2*w - 7],\
[293, 293, 3*w^2 - 2*w - 12],\
[307, 307, 2*w^2 + w - 8],\
[313, 313, w^2 + 4*w - 2],\
[317, 317, 3*w^2 - 6*w - 5],\
[331, 331, 4*w^2 - 4*w - 19],\
[337, 337, -5*w^2 + 8*w + 21],\
[347, 347, 3*w^2 - 6*w - 13],\
[347, 347, 3*w^2 - 5*w - 22],\
[347, 347, 3*w^2 - 5*w - 9],\
[349, 349, w^2 - 5*w - 11],\
[353, 353, -w^2 + 6*w - 7],\
[359, 359, -4*w^2 + 8*w + 11],\
[367, 367, 3*w^2 - 4*w - 11],\
[373, 373, w^2 + 2*w - 6],\
[379, 379, 5*w^2 - 9*w - 18],\
[409, 409, -w^2 - 4],\
[439, 439, 3*w^2 - 6*w - 14],\
[443, 443, -w^2 - 2*w + 13],\
[443, 443, 2*w^2 - 15],\
[443, 443, 3*w^2 - 13],\
[449, 449, w^2 + 2*w - 7],\
[449, 449, 2*w^2 - 6*w - 7],\
[449, 449, 3*w^2 - 3*w - 11],\
[461, 461, -w^2 - 2*w - 5],\
[467, 467, 3*w^2 - 5*w - 7],\
[479, 479, -w^2 + 6*w - 4],\
[491, 491, 3*w^2 - 5*w - 6],\
[499, 499, w^2 - 5*w - 8],\
[509, 509, 3*w^2 - 6*w - 19],\
[521, 521, 3*w^2 - 2*w - 10],\
[523, 523, 4*w - 9],\
[541, 541, 3*w^2 - 3*w - 10],\
[547, 547, 2*w^2 + w - 20],\
[547, 547, w^2 - 6*w - 5],\
[547, 547, 6*w^2 - 11*w - 24],\
[557, 557, -5*w^2 + 11*w + 11],\
[563, 563, 3*w^2 - 6*w - 16],\
[569, 569, 4*w^2 - 5*w - 17],\
[587, 587, 5*w^2 - 8*w - 20],\
[593, 593, 5*w - 3],\
[593, 593, 3*w^2 - w - 17],\
[593, 593, w - 9],\
[613, 613, 2*w^2 - w - 19],\
[613, 613, 3*w^2 - 4*w - 5],\
[613, 613, 3*w^2 - w - 8],\
[617, 617, 3*w^2 - 2*w - 9],\
[617, 617, -w^2 + w - 5],\
[617, 617, 4*w^2 - 6*w - 15],\
[619, 619, -w^2 + 3*w - 6],\
[631, 631, 4*w^2 - 7*w - 21],\
[641, 641, -w^2 - 3*w + 20],\
[643, 643, 3*w^2 - 14],\
[643, 643, w^2 - w - 13],\
[643, 643, 2*w^2 - 9*w + 2],\
[647, 647, 2*w^2 - 6*w - 9],\
[653, 653, 3*w^2 - 5],\
[659, 659, 6*w^2 - 11*w - 21],\
[661, 661, w^2 - 6*w - 6],\
[661, 661, w^2 + 3*w - 6],\
[661, 661, 5*w^2 - 7*w - 22],\
[673, 673, w^2 - 6*w - 12],\
[677, 677, 4*w^2 - 7*w - 22],\
[677, 677, 3*w^2 - 7*w - 12],\
[677, 677, -2*w^2 - 3],\
[683, 683, -2*w - 9],\
[683, 683, 4*w^2 - 7*w - 12],\
[683, 683, 3*w^2 - 3*w - 8],\
[691, 691, 2*w^2 + w - 11],\
[701, 701, 3*w^2 - 3*w - 5],\
[709, 709, -w^2 - 2*w - 6],\
[727, 727, 3*w^2 - w - 6],\
[727, 727, 3*w^2 - w - 27],\
[727, 727, 3*w^2 - w - 5],\
[733, 733, -2*w^2 + 9*w - 5],\
[739, 739, 3*w^2 - 4*w - 23],\
[743, 743, 2*w^2 - 7*w - 7],\
[743, 743, 3*w^2 - w - 20],\
[743, 743, 3*w^2 - 2*w - 7],\
[757, 757, -w - 9],\
[757, 757, 4*w^2 - 2*w - 29],\
[757, 757, 2*w^2 - 3*w - 18],\
[761, 761, 3*w^2 - 2*w - 6],\
[761, 761, 3*w^2 - 2*w - 25],\
[773, 773, -5*w - 12],\
[787, 787, -3*w - 10],\
[809, 809, 4*w^2 - w - 19],\
[821, 821, 5*w^2 - 10*w - 19],\
[823, 823, -4*w - 11],\
[827, 827, -7*w - 5],\
[827, 827, 2*w^2 - 6*w - 13],\
[827, 827, -w^2 - 2*w + 18],\
[839, 839, 6*w^2 - 9*w - 26],\
[839, 839, 5*w^2 - 12*w - 13],\
[839, 839, w - 10],\
[857, 857, -w^2 + 4*w - 8],\
[859, 859, 3*w^2 - 8*w - 10],\
[859, 859, 4*w^2 - 9*w - 14],\
[859, 859, 3*w^2 - 26],\
[863, 863, 2*w^2 + 3*w - 7],\
[877, 877, -5*w^2 + 11*w + 9],\
[877, 877, -5*w^2 + 10*w + 13],\
[877, 877, -w^2 + w - 6],\
[881, 881, w^2 - 3*w - 14],\
[883, 883, -w^2 + 3*w - 7],\
[887, 887, -3*w^2 + 6*w - 2],\
[907, 907, w^2 + 3*w - 8],\
[911, 911, 4*w^2 - 6*w - 29],\
[937, 937, -6*w^2 + 11*w + 25],\
[953, 953, 6*w^2 - 6*w - 29],\
[961, 31, -w^2 + 7*w - 5],\
[967, 967, 2*w^2 + w - 14],\
[971, 971, 5*w^2 - 3*w - 27],\
[971, 971, 4*w - 15],\
[971, 971, w^2 - 2*w - 14],\
[977, 977, 3*w^2 - 5*w - 24],\
[983, 983, w^2 - 4*w - 15],\
[991, 991, 6*w^2 - 10*w - 23],\
[991, 991, 5*w^2 - 6*w - 22],\
[991, 991, 3*w^2 + w - 13]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 + 3*x^5 - 18*x^4 - 20*x^3 + 113*x^2 - 87*x + 9
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-17/375*e^5 - 24/125*e^4 + 13/25*e^3 + 5/3*e^2 - 796/375*e - 298/125, e, -89/375*e^5 - 133/125*e^4 + 71/25*e^3 + 26/3*e^2 - 6307/375*e + 359/125, -14/125*e^5 - 74/125*e^4 + 13/25*e^3 + 4*e^2 - 207/125*e - 398/125, 4/75*e^5 + 13/25*e^4 + 4/5*e^3 - 11/3*e^2 - 448/75*e + 101/25, 37/125*e^5 + 142/125*e^4 - 104/25*e^3 - 9*e^2 + 2931/125*e - 966/125, -103/375*e^5 - 116/125*e^4 + 117/25*e^3 + 25/3*e^2 - 10139/375*e + 1018/125, 3/25*e^5 + 23/25*e^4 + 4/5*e^3 - 5*e^2 - 111/25*e + 21/25, -1, -9/25*e^5 - 44/25*e^4 + 13/5*e^3 + 11*e^2 - 342/25*e + 112/25, 16/125*e^5 + 31/125*e^4 - 72/25*e^3 - 3*e^2 + 1683/125*e - 438/125, 208/375*e^5 + 301/125*e^4 - 162/25*e^3 - 55/3*e^2 + 14129/375*e - 898/125, -32/125*e^5 - 187/125*e^4 + 19/25*e^3 + 10*e^2 - 241/125*e - 499/125, -157/375*e^5 - 229/125*e^4 + 123/25*e^3 + 40/3*e^2 - 10991/375*e + 1417/125, 436/375*e^5 + 667/125*e^4 - 304/25*e^3 - 121/3*e^2 + 25268/375*e - 1291/125, -7/25*e^5 - 37/25*e^4 + 9/5*e^3 + 9*e^2 - 291/25*e + 226/25, 16/75*e^5 + 27/25*e^4 - 4/5*e^3 - 14/3*e^2 + 383/75*e - 271/25, 116/375*e^5 + 127/125*e^4 - 149/25*e^3 - 32/3*e^2 + 13108/375*e - 1496/125, 97/125*e^5 + 477/125*e^4 - 149/25*e^3 - 25*e^2 + 3961/125*e - 921/125, 77/125*e^5 + 282/125*e^4 - 234/25*e^3 - 18*e^2 + 6451/125*e - 2561/125, 352/375*e^5 + 519/125*e^4 - 278/25*e^3 - 100/3*e^2 + 23276/375*e - 1837/125, -23/25*e^5 - 118/25*e^4 + 36/5*e^3 + 36*e^2 - 949/25*e - 186/25, -49/125*e^5 - 134/125*e^4 + 208/25*e^3 + 10*e^2 - 6787/125*e + 2357/125, -61/125*e^5 - 251/125*e^4 + 187/25*e^3 + 18*e^2 - 5893/125*e + 1373/125, 433/375*e^5 + 751/125*e^4 - 237/25*e^3 - 142/3*e^2 + 19304/375*e + 1502/125, 51/125*e^5 + 341/125*e^4 + 8/25*e^3 - 17*e^2 - 112/125*e + 57/125, 31/25*e^5 + 146/25*e^4 - 62/5*e^3 - 43*e^2 + 1778/25*e - 583/25, -6/125*e^5 - 121/125*e^4 - 73/25*e^3 + 5*e^2 + 2072/125*e + 508/125, -46/75*e^5 - 87/25*e^4 + 14/5*e^3 + 71/3*e^2 - 773/75*e - 74/25, -79/125*e^5 - 364/125*e^4 + 118/25*e^3 + 16*e^2 - 3052/125*e + 2147/125, 79/75*e^5 + 138/25*e^4 - 36/5*e^3 - 119/3*e^2 + 2627/75*e + 251/25, -74/75*e^5 - 128/25*e^4 + 36/5*e^3 + 109/3*e^2 - 2962/75*e - 106/25, -46/125*e^5 - 261/125*e^4 + 57/25*e^3 + 15*e^2 - 2073/125*e - 147/125, 67/125*e^5 + 372/125*e^4 - 39/25*e^3 - 17*e^2 + 821/125*e - 756/125, 248/125*e^5 + 1168/125*e^4 - 466/25*e^3 - 68*e^2 + 12524/125*e - 1414/125, -377/375*e^5 - 569/125*e^4 + 253/25*e^3 + 101/3*e^2 - 20476/375*e - 13/125, -418/375*e^5 - 671/125*e^4 + 277/25*e^3 + 124/3*e^2 - 24734/375*e - 92/125, -11/125*e^5 - 76/125*e^4 - 13/25*e^3 + 2*e^2 + 257/125*e - 402/125, -76/125*e^5 - 366/125*e^4 + 142/25*e^3 + 20*e^2 - 4463/125*e + 1643/125, -129/125*e^5 - 539/125*e^4 + 318/25*e^3 + 33*e^2 - 8327/125*e + 2172/125, -72/125*e^5 - 327/125*e^4 + 99/25*e^3 + 15*e^2 - 1886/125*e + 346/125, -19/25*e^5 - 79/25*e^4 + 48/5*e^3 + 23*e^2 - 1472/25*e + 417/25, 4/25*e^5 + 14/25*e^4 - 18/5*e^3 - 8*e^2 + 677/25*e + 178/25, 256/125*e^5 + 1121/125*e^4 - 577/25*e^3 - 67*e^2 + 16428/125*e - 3633/125, -152/125*e^5 - 732/125*e^4 + 284/25*e^3 + 45*e^2 - 7176/125*e - 1089/125, -59/125*e^5 - 294/125*e^4 + 128/25*e^3 + 21*e^2 - 3667/125*e - 1338/125, -34/25*e^5 - 169/25*e^4 + 53/5*e^3 + 49*e^2 - 1317/25*e - 38/25, -616/375*e^5 - 877/125*e^4 + 449/25*e^3 + 148/3*e^2 - 37358/375*e + 3496/125, -74/125*e^5 - 409/125*e^4 + 83/25*e^3 + 27*e^2 - 487/125*e - 3693/125, 72/125*e^5 + 327/125*e^4 - 149/25*e^3 - 20*e^2 + 3886/125*e - 1596/125, -394/375*e^5 - 593/125*e^4 + 241/25*e^3 + 85/3*e^2 - 20897/375*e + 3814/125, -71/75*e^5 - 112/25*e^4 + 49/5*e^3 + 106/3*e^2 - 4123/75*e + 26/25, 347/375*e^5 + 659/125*e^4 - 58/25*e^3 - 92/3*e^2 + 1336/375*e - 557/125, -241/125*e^5 - 1006/125*e^4 + 597/25*e^3 + 60*e^2 - 17233/125*e + 4613/125, 131/125*e^5 + 621/125*e^4 - 202/25*e^3 - 31*e^2 + 5178/125*e - 1758/125, 116/125*e^5 + 631/125*e^4 - 147/25*e^3 - 36*e^2 + 4233/125*e + 387/125, -38/375*e^5 - 61/125*e^4 + 32/25*e^3 + 5/3*e^2 - 5794/375*e + 2378/125, 17/25*e^5 + 72/25*e^4 - 54/5*e^3 - 28*e^2 + 1721/25*e - 31/25, -29/25*e^5 - 139/25*e^4 + 58/5*e^3 + 39*e^2 - 1927/25*e + 672/25, -28/15*e^5 - 41/5*e^4 + 21*e^3 + 181/3*e^2 - 1799/15*e + 193/5, -27/125*e^5 - 232/125*e^4 - 66/25*e^3 + 10*e^2 + 2449/125*e - 1839/125, 154/75*e^5 + 238/25*e^4 - 101/5*e^3 - 215/3*e^2 + 7877/75*e - 124/25, 118/375*e^5 + 71/125*e^4 - 177/25*e^3 - 16/3*e^2 + 15584/375*e - 1858/125, 2/5*e^5 + 12/5*e^4 - 2*e^3 - 16*e^2 + 96/5*e - 66/5, 94/375*e^5 + 118/125*e^4 - 116/25*e^3 - 25/3*e^2 + 10997/375*e - 2764/125, -173/125*e^5 - 968/125*e^4 + 216/25*e^3 + 60*e^2 - 5174/125*e - 3186/125, -84/125*e^5 - 319/125*e^4 + 278/25*e^3 + 24*e^2 - 7617/125*e + 2112/125, 14/375*e^5 - 142/125*e^4 - 221/25*e^3 + 7/3*e^2 + 22582/375*e - 2909/125, -4/125*e^5 + 86/125*e^4 + 93/25*e^3 - 5*e^2 - 2577/125*e + 672/125, -254/125*e^5 - 1039/125*e^4 + 643/25*e^3 + 62*e^2 - 18702/125*e + 4672/125, -136/125*e^5 - 576/125*e^4 + 362/25*e^3 + 43*e^2 - 8993/125*e - 1277/125, 368/125*e^5 + 1588/125*e^4 - 831/25*e^3 - 94*e^2 + 22834/125*e - 5699/125, -82/375*e^5 - 204/125*e^4 - 27/25*e^3 + 43/3*e^2 + 7984/375*e - 3158/125, -84/125*e^5 - 444/125*e^4 + 103/25*e^3 + 27*e^2 - 1117/125*e - 3888/125, 19/375*e^5 + 93/125*e^4 + 34/25*e^3 - 31/3*e^2 - 8353/375*e + 5186/125, -38/125*e^5 - 183/125*e^4 + 71/25*e^3 + 17*e^2 + 206/125*e - 3741/125, -364/125*e^5 - 1674/125*e^4 + 738/25*e^3 + 98*e^2 - 20382/125*e + 3902/125, -94/375*e^5 + 7/125*e^4 + 216/25*e^3 + 13/3*e^2 - 16997/375*e - 611/125, 793/375*e^5 + 1421/125*e^4 - 327/25*e^3 - 235/3*e^2 + 27359/375*e - 283/125, -4/75*e^5 + 12/25*e^4 + 26/5*e^3 + 14/3*e^2 - 1802/75*e - 701/25, -131/125*e^5 - 496/125*e^4 + 427/25*e^3 + 33*e^2 - 14678/125*e + 4383/125, 238/125*e^5 + 1133/125*e^4 - 446/25*e^3 - 65*e^2 + 12644/125*e - 3234/125, -124/75*e^5 - 228/25*e^4 + 51/5*e^3 + 203/3*e^2 - 3962/75*e - 281/25, 64/125*e^5 + 249/125*e^4 - 188/25*e^3 - 16*e^2 + 4482/125*e - 4252/125, -4/75*e^5 - 13/25*e^4 - 4/5*e^3 + 5/3*e^2 - 2/75*e + 124/25, 3/125*e^5 + 123/125*e^4 + 74/25*e^3 - 8*e^2 - 1786/125*e + 2871/125, -149/75*e^5 - 203/25*e^4 + 131/5*e^3 + 202/3*e^2 - 10687/75*e + 419/25, -128/125*e^5 - 623/125*e^4 + 176/25*e^3 + 28*e^2 - 4964/125*e + 3754/125, -3*e^5 - 13*e^4 + 36*e^3 + 103*e^2 - 194*e + 28, -282/125*e^5 - 1187/125*e^4 + 694/25*e^3 + 76*e^2 - 17866/125*e + 3251/125, -214/125*e^5 - 1149/125*e^4 + 288/25*e^3 + 69*e^2 - 7307/125*e - 2048/125, -29/125*e^5 - 189/125*e^4 - 57/25*e^3 + 4*e^2 + 2723/125*e + 1872/125, 197/125*e^5 + 952/125*e^4 - 349/25*e^3 - 57*e^2 + 7761/125*e + 779/125, -84/125*e^5 - 319/125*e^4 + 228/25*e^3 + 20*e^2 - 5242/125*e + 1987/125, 67/375*e^5 + 124/125*e^4 + 37/25*e^3 + 5/3*e^2 - 3679/375*e - 3127/125, 77/125*e^5 + 407/125*e^4 - 134/25*e^3 - 26*e^2 + 3201/125*e - 2061/125, 31/125*e^5 + 271/125*e^4 + 23/25*e^3 - 18*e^2 + 378/125*e + 4542/125, 62/125*e^5 + 42/125*e^4 - 429/25*e^3 - 9*e^2 + 14631/125*e - 6916/125, -26/125*e^5 + 59/125*e^4 + 217/25*e^3 - 4*e^2 - 7563/125*e + 4618/125, 6/25*e^5 + 21/25*e^4 - 12/5*e^3 - 5*e^2 + 178/25*e - 83/25, 391/125*e^5 + 1531/125*e^4 - 1047/25*e^3 - 95*e^2 + 28558/125*e - 8563/125, -23/25*e^5 - 143/25*e^4 + 6/5*e^3 + 39*e^2 + 151/25*e - 711/25, -21/25*e^5 - 111/25*e^4 + 17/5*e^3 + 31*e^2 + 52/25*e - 522/25, 56/125*e^5 + 171/125*e^4 - 202/25*e^3 - 10*e^2 + 6453/125*e - 3783/125, -89/75*e^5 - 133/25*e^4 + 56/5*e^3 + 115/3*e^2 - 4057/75*e - 316/25, e^5 + 4*e^4 - 12*e^3 - 34*e^2 + 51*e + 15, -136/125*e^5 - 951/125*e^4 - 38/25*e^3 + 50*e^2 + 1257/125*e - 3777/125, 194/125*e^5 + 1079/125*e^4 - 173/25*e^3 - 59*e^2 + 3172/125*e + 1158/125, 46/25*e^5 + 186/25*e^4 - 122/5*e^3 - 57*e^2 + 3398/25*e - 1053/25, 138/125*e^5 + 408/125*e^4 - 446/25*e^3 - 20*e^2 + 12969/125*e - 5559/125, 8/25*e^5 + 53/25*e^4 + 14/5*e^3 - 9*e^2 - 721/25*e + 556/25, -14/125*e^5 + 176/125*e^4 + 313/25*e^3 - 3*e^2 - 10082/125*e + 5352/125, -352/375*e^5 - 519/125*e^4 + 278/25*e^3 + 100/3*e^2 - 26276/375*e + 1462/125, 47/25*e^5 + 202/25*e^4 - 119/5*e^3 - 61*e^2 + 3711/25*e - 1196/25, -179/75*e^5 - 238/25*e^4 + 161/5*e^3 + 229/3*e^2 - 12877/75*e + 899/25, 314/125*e^5 + 1499/125*e^4 - 638/25*e^3 - 88*e^2 + 19357/125*e - 7377/125, 37/25*e^5 + 142/25*e^4 - 109/5*e^3 - 47*e^2 + 3031/25*e - 1191/25, 233/375*e^5 + 351/125*e^4 - 162/25*e^3 - 44/3*e^2 + 18454/375*e - 5048/125, 232/375*e^5 + 379/125*e^4 - 98/25*e^3 - 64/3*e^2 + 1091/375*e + 383/125, 716/375*e^5 + 1077/125*e^4 - 549/25*e^3 - 203/3*e^2 + 45283/375*e - 4346/125, -778/375*e^5 - 1216/125*e^4 + 467/25*e^3 + 214/3*e^2 - 33539/375*e - 1682/125, 77/125*e^5 + 532/125*e^4 - 9/25*e^3 - 34*e^2 - 424/125*e + 3564/125, 31/125*e^5 + 21/125*e^4 - 227/25*e^3 - 14*e^2 + 5253/125*e + 2167/125, 352/375*e^5 + 394/125*e^4 - 353/25*e^3 - 61/3*e^2 + 32276/375*e - 5962/125, -47/125*e^5 - 302/125*e^4 - 76/25*e^3 + 12*e^2 + 4814/125*e - 3104/125, 2/3*e^5 + 3*e^4 - 10*e^3 - 76/3*e^2 + 202/3*e - 56, -4/5*e^5 - 14/5*e^4 + 15*e^3 + 29*e^2 - 477/5*e + 72/5, -34/375*e^5 + 77/125*e^4 + 101/25*e^3 - 35/3*e^2 - 12842/375*e + 7279/125, -13/75*e^5 + 14/25*e^4 + 47/5*e^3 + 17/3*e^2 - 3419/75*e + 178/25, 82/125*e^5 + 487/125*e^4 + 6/25*e^3 - 22*e^2 - 1234/125*e + 1099/125, -231/125*e^5 - 971/125*e^4 + 602/25*e^3 + 67*e^2 - 15853/125*e + 3183/125, -234/125*e^5 - 969/125*e^4 + 628/25*e^3 + 66*e^2 - 17067/125*e + 1562/125, -219/125*e^5 - 1229/125*e^4 + 173/25*e^3 + 62*e^2 - 4747/125*e + 1542/125, 79/25*e^5 + 389/25*e^4 - 113/5*e^3 - 98*e^2 + 2877/25*e - 1297/25, 1193/375*e^5 + 1721/125*e^4 - 902/25*e^3 - 299/3*e^2 + 77809/375*e - 7433/125, 47/125*e^5 + 302/125*e^4 - 24/25*e^3 - 13*e^2 + 2186/125*e - 4521/125, 413/375*e^5 + 436/125*e^4 - 557/25*e^3 - 101/3*e^2 + 54169/375*e - 6503/125, 162/125*e^5 + 892/125*e^4 - 129/25*e^3 - 46*e^2 + 2181/125*e + 2284/125, 513/125*e^5 + 2283/125*e^4 - 1096/25*e^3 - 134*e^2 + 29844/125*e - 4559/125, -479/375*e^5 - 963/125*e^4 + 56/25*e^3 + 146/3*e^2 - 3127/375*e - 676/125, -106/25*e^5 - 471/25*e^4 + 237/5*e^3 + 141*e^2 - 6678/25*e + 1683/25, 349/125*e^5 + 1434/125*e^4 - 933/25*e^3 - 94*e^2 + 25687/125*e - 4382/125, -1006/375*e^5 - 1582/125*e^4 + 609/25*e^3 + 277/3*e^2 - 46178/375*e - 2039/125, 526/125*e^5 + 2566/125*e^4 - 992/25*e^3 - 156*e^2 + 28063/125*e - 1618/125, 319/125*e^5 + 1454/125*e^4 - 698/25*e^3 - 90*e^2 + 18547/125*e - 2842/125, 54/25*e^5 + 264/25*e^4 - 93/5*e^3 - 84*e^2 + 1977/25*e + 828/25, 361/125*e^5 + 1676/125*e^4 - 662/25*e^3 - 94*e^2 + 16918/125*e - 3648/125, -49/25*e^5 - 259/25*e^4 + 68/5*e^3 + 78*e^2 - 1612/25*e - 743/25, -1247/375*e^5 - 1959/125*e^4 + 808/25*e^3 + 347/3*e^2 - 67786/375*e + 2582/125, 203/375*e^5 + 441/125*e^4 + 8/25*e^3 - 53/3*e^2 + 3064/375*e - 2993/125, -148/75*e^5 - 181/25*e^4 + 152/5*e^3 + 185/3*e^2 - 12224/75*e + 1063/25, 71/125*e^5 + 161/125*e^4 - 307/25*e^3 - 10*e^2 + 10898/125*e - 4803/125, -1018/375*e^5 - 1371/125*e^4 + 927/25*e^3 + 277/3*e^2 - 76034/375*e + 4633/125, 249/125*e^5 + 1459/125*e^4 - 183/25*e^3 - 82*e^2 + 3387/125*e + 5668/125, -593/375*e^5 - 896/125*e^4 + 477/25*e^3 + 176/3*e^2 - 40759/375*e + 3458/125, 113/375*e^5 + 86/125*e^4 - 182/25*e^3 - 29/3*e^2 + 14644/375*e - 203/125, -258/125*e^5 - 1453/125*e^4 + 311/25*e^3 + 88*e^2 - 8404/125*e - 2781/125, 3/25*e^5 - 2/25*e^4 - 21/5*e^3 + 2*e^2 + 664/25*e - 804/25, -236/125*e^5 - 801/125*e^4 + 762/25*e^3 + 52*e^2 - 21168/125*e + 7023/125, 12/125*e^5 + 242/125*e^4 + 146/25*e^3 - 16*e^2 - 5894/125*e + 5484/125, 1231/375*e^5 + 1907/125*e^4 - 809/25*e^3 - 340/3*e^2 + 67478/375*e - 1561/125, 911/375*e^5 + 1367/125*e^4 - 679/25*e^3 - 251/3*e^2 + 59443/375*e - 2891/125, -49/125*e^5 - 259/125*e^4 + 83/25*e^3 + 15*e^2 - 2912/125*e + 607/125, -49/25*e^5 - 184/25*e^4 + 148/5*e^3 + 57*e^2 - 4612/25*e + 1707/25, -262/125*e^5 - 1367/125*e^4 + 379/25*e^3 + 83*e^2 - 9356/125*e - 3609/125, 52/125*e^5 + 382/125*e^4 - 34/25*e^3 - 32*e^2 + 501/125*e + 6264/125, -37/25*e^5 - 167/25*e^4 + 74/5*e^3 + 52*e^2 - 1681/25*e - 359/25, -564/125*e^5 - 2624/125*e^4 + 1163/25*e^3 + 161*e^2 - 32232/125*e + 2377/125, 267/125*e^5 + 1447/125*e^4 - 264/25*e^3 - 74*e^2 + 5671/125*e - 356/125, 99/25*e^5 + 459/25*e^4 - 203/5*e^3 - 140*e^2 + 5737/25*e - 682/25, -133/125*e^5 - 828/125*e^4 + 86/25*e^3 + 49*e^2 - 2779/125*e - 3906/125, -571/125*e^5 - 2661/125*e^4 + 1107/25*e^3 + 156*e^2 - 30148/125*e + 2553/125, 143/125*e^5 + 613/125*e^4 - 281/25*e^3 - 36*e^2 + 5409/125*e + 976/125, 376/125*e^5 + 1791/125*e^4 - 692/25*e^3 - 99*e^2 + 19988/125*e - 5043/125, -12/5*e^5 - 62/5*e^4 + 17*e^3 + 88*e^2 - 451/5*e + 21/5, 287/125*e^5 + 1017/125*e^4 - 904/25*e^3 - 64*e^2 + 27806/125*e - 10341/125]
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^2 + 3*w + 2])] = 1

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