/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![2, 12, -11, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {8, 2, w^3 - 2*w^2 - 11*w + 12}>;

primesArray := [
[2, 2, 4/9*w^3 - 2/3*w^2 - 47/9*w + 20/9],
[2, 2, 5/9*w^3 - 4/3*w^2 - 52/9*w + 79/9],
[9, 3, 1/3*w^3 - 11/3*w - 1/3],
[9, 3, w^3 - w^2 - 12*w - 1],
[17, 17, -1/9*w^3 - 1/3*w^2 + 14/9*w + 49/9],
[17, 17, -1/9*w^3 + 2/3*w^2 + 5/9*w - 59/9],
[25, 5, -7/9*w^3 + 2/3*w^2 + 89/9*w + 19/9],
[25, 5, -5/9*w^3 + 1/3*w^2 + 52/9*w + 11/9],
[47, 47, -1/9*w^3 + 2/3*w^2 + 5/9*w - 41/9],
[47, 47, 1/9*w^3 + 1/3*w^2 - 14/9*w - 13/9],
[47, 47, -1/9*w^3 + 2/3*w^2 + 5/9*w - 23/9],
[47, 47, 1/9*w^3 + 1/3*w^2 - 14/9*w - 31/9],
[49, 7, -4/9*w^3 + 2/3*w^2 + 38/9*w - 29/9],
[49, 7, 4/9*w^3 - 2/3*w^2 - 38/9*w + 11/9],
[89, 89, 20/9*w^3 - 16/3*w^2 - 208/9*w + 325/9],
[89, 89, 2/3*w^3 - w^2 - 25/3*w + 7/3],
[89, 89, -2/3*w^3 + w^2 + 25/3*w - 19/3],
[89, 89, -2*w^3 + 3*w^2 + 23*w - 11],
[103, 103, 8/9*w^3 - 4/3*w^2 - 94/9*w + 67/9],
[103, 103, -38/9*w^3 + 28/3*w^2 + 406/9*w - 541/9],
[103, 103, 20/9*w^3 - 7/3*w^2 - 235/9*w + 19/9],
[103, 103, -8/9*w^3 + 4/3*w^2 + 94/9*w - 31/9],
[121, 11, -7/3*w^3 + 2*w^2 + 83/3*w + 7/3],
[121, 11, -31/9*w^3 + 23/3*w^2 + 326/9*w - 443/9],
[127, 127, -8/9*w^3 + 7/3*w^2 + 67/9*w - 103/9],
[127, 127, -2/9*w^3 + 1/3*w^2 + 37/9*w - 55/9],
[127, 127, -8/9*w^3 + 7/3*w^2 + 67/9*w - 85/9],
[127, 127, -2/3*w^3 + w^2 + 25/3*w - 25/3],
[137, 137, 7/9*w^3 - 2/3*w^2 - 89/9*w + 17/9],
[137, 137, 1/9*w^3 - 5/3*w^2 + 4/9*w + 167/9],
[137, 137, -29/9*w^3 + 19/3*w^2 + 316/9*w - 325/9],
[137, 137, -7/9*w^3 + 5/3*w^2 + 80/9*w - 71/9],
[151, 151, 2/9*w^3 + 2/3*w^2 - 28/9*w - 89/9],
[151, 151, 8/9*w^3 - 7/3*w^2 - 85/9*w + 157/9],
[151, 151, -4/3*w^3 + 2*w^2 + 44/3*w - 29/3],
[151, 151, -20/9*w^3 + 13/3*w^2 + 217/9*w - 235/9],
[169, 13, -4/9*w^3 + 2/3*w^2 + 56/9*w - 11/9],
[169, 13, -4/9*w^3 + 2/3*w^2 + 56/9*w - 47/9],
[191, 191, 7/9*w^3 - 2/3*w^2 - 71/9*w - 1/9],
[191, 191, 7/9*w^3 - 5/3*w^2 - 62/9*w + 53/9],
[191, 191, -7/9*w^3 + 2/3*w^2 + 71/9*w - 17/9],
[191, 191, 7/9*w^3 - 5/3*w^2 - 62/9*w + 71/9],
[223, 223, -59/9*w^3 + 43/3*w^2 + 628/9*w - 835/9],
[223, 223, 7/9*w^3 - 5/3*w^2 - 80/9*w + 125/9],
[223, 223, -7/9*w^3 + 2/3*w^2 + 89/9*w + 37/9],
[223, 223, -41/9*w^3 + 31/3*w^2 + 430/9*w - 601/9],
[239, 239, 1/9*w^3 + 1/3*w^2 - 14/9*w - 67/9],
[239, 239, 1/9*w^3 - 2/3*w^2 - 5/9*w - 13/9],
[239, 239, -1/9*w^3 - 1/3*w^2 + 14/9*w - 23/9],
[239, 239, 1/9*w^3 - 2/3*w^2 - 5/9*w + 77/9],
[257, 257, -13/9*w^3 + 11/3*w^2 + 110/9*w - 137/9],
[257, 257, -13/9*w^3 + 2/3*w^2 + 137/9*w + 25/9],
[257, 257, -13/9*w^3 + 11/3*w^2 + 110/9*w - 155/9],
[257, 257, -5/9*w^3 + 4/3*w^2 + 61/9*w - 97/9],
[263, 263, 22/9*w^3 - 17/3*w^2 - 227/9*w + 353/9],
[263, 263, -10/9*w^3 + 5/3*w^2 + 113/9*w - 41/9],
[263, 263, 14/9*w^3 - 10/3*w^2 - 142/9*w + 205/9],
[263, 263, -20/9*w^3 + 10/3*w^2 + 226/9*w - 109/9],
[271, 271, -17/9*w^3 + 13/3*w^2 + 184/9*w - 283/9],
[271, 271, 23/9*w^3 - 7/3*w^2 - 268/9*w - 11/9],
[271, 271, -49/9*w^3 + 35/3*w^2 + 524/9*w - 677/9],
[271, 271, -17/9*w^3 + 7/3*w^2 + 184/9*w - 85/9],
[281, 281, -5/9*w^3 + 4/3*w^2 + 43/9*w - 79/9],
[281, 281, 5/9*w^3 - 4/3*w^2 - 43/9*w + 25/9],
[281, 281, 1/3*w^3 - 17/3*w - 7/3],
[281, 281, 5/9*w^3 - 4/3*w^2 - 61/9*w + 43/9],
[353, 353, -13/9*w^3 + 5/3*w^2 + 146/9*w - 47/9],
[353, 353, 1/3*w^3 + w^2 - 14/3*w - 43/3],
[353, 353, -29/9*w^3 + 19/3*w^2 + 316/9*w - 343/9],
[353, 353, 17/9*w^3 - 13/3*w^2 - 184/9*w + 265/9],
[359, 359, -5/9*w^3 + 4/3*w^2 + 43/9*w - 115/9],
[359, 359, -23/9*w^3 + 13/3*w^2 + 250/9*w - 205/9],
[359, 359, 41/9*w^3 - 28/3*w^2 - 439/9*w + 511/9],
[359, 359, -7/3*w^3 + 5*w^2 + 74/3*w - 101/3],
[361, 19, -8/9*w^3 + 4/3*w^2 + 112/9*w - 121/9],
[361, 19, -4/9*w^3 + 2/3*w^2 + 56/9*w - 83/9],
[383, 383, 32/9*w^3 - 25/3*w^2 - 331/9*w + 493/9],
[383, 383, 14/9*w^3 - 4/3*w^2 - 178/9*w - 47/9],
[383, 383, -40/9*w^3 + 29/3*w^2 + 425/9*w - 533/9],
[383, 383, -38/9*w^3 + 25/3*w^2 + 415/9*w - 451/9],
[409, 409, -5/9*w^3 + 4/3*w^2 + 43/9*w - 7/9],
[409, 409, -w^3 + 2*w^2 + 11*w - 9],
[409, 409, -5/9*w^3 + 4/3*w^2 + 43/9*w - 97/9],
[409, 409, -5/9*w^3 + 1/3*w^2 + 52/9*w - 43/9],
[433, 433, -8/9*w^3 + 7/3*w^2 + 85/9*w - 175/9],
[433, 433, 40/9*w^3 - 29/3*w^2 - 425/9*w + 569/9],
[433, 433, 22/9*w^3 - 8/3*w^2 - 254/9*w + 29/9],
[433, 433, -16/9*w^3 + 8/3*w^2 + 170/9*w - 125/9],
[457, 457, -28/9*w^3 + 17/3*w^2 + 311/9*w - 275/9],
[457, 457, 2/9*w^3 + 2/3*w^2 - 28/9*w - 71/9],
[457, 457, 2/9*w^3 - 4/3*w^2 - 10/9*w + 91/9],
[457, 457, -8/9*w^3 + 10/3*w^2 + 76/9*w - 247/9],
[463, 463, 8/9*w^3 - 4/3*w^2 - 94/9*w + 85/9],
[463, 463, -2*w + 5],
[463, 463, 2*w + 3],
[463, 463, 8/9*w^3 - 4/3*w^2 - 94/9*w + 13/9],
[529, 23, 2/9*w^3 - 1/3*w^2 - 19/9*w - 35/9],
[529, 23, -2/9*w^3 + 1/3*w^2 + 19/9*w - 55/9],
[569, 569, -14/9*w^3 + 7/3*w^2 + 151/9*w - 97/9],
[569, 569, -20/9*w^3 + 16/3*w^2 + 208/9*w - 343/9],
[569, 569, -22/9*w^3 + 11/3*w^2 + 245/9*w - 137/9],
[569, 569, -10/9*w^3 + 8/3*w^2 + 104/9*w - 185/9],
[577, 577, 41/9*w^3 - 31/3*w^2 - 430/9*w + 619/9],
[577, 577, -11/9*w^3 + 7/3*w^2 + 118/9*w - 91/9],
[577, 577, -11/9*w^3 + 4/3*w^2 + 127/9*w - 37/9],
[577, 577, 29/9*w^3 - 13/3*w^2 - 334/9*w + 109/9],
[593, 593, 4/9*w^3 + 1/3*w^2 - 47/9*w - 25/9],
[593, 593, -4/9*w^3 + 5/3*w^2 + 29/9*w - 83/9],
[593, 593, 4/9*w^3 + 1/3*w^2 - 47/9*w - 43/9],
[593, 593, -4/9*w^3 + 5/3*w^2 + 29/9*w - 65/9],
[599, 599, 2/3*w^3 - 22/3*w - 17/3],
[599, 599, 2/3*w^3 - 2*w^2 - 16/3*w + 19/3],
[599, 599, -2/3*w^3 + 22/3*w - 1/3],
[599, 599, -2/3*w^3 + 2*w^2 + 16/3*w - 37/3],
[631, 631, -3*w^3 + 6*w^2 + 33*w - 35],
[631, 631, 17/9*w^3 - 10/3*w^2 - 193/9*w + 157/9],
[631, 631, -7/9*w^3 + 8/3*w^2 + 71/9*w - 179/9],
[631, 631, 1/9*w^3 + 4/3*w^2 - 23/9*w - 139/9],
[647, 647, -29/9*w^3 + 22/3*w^2 + 307/9*w - 451/9],
[647, 647, -19/9*w^3 + 14/3*w^2 + 203/9*w - 293/9],
[647, 647, -5/3*w^3 + 2*w^2 + 55/3*w - 19/3],
[647, 647, -23/9*w^3 + 10/3*w^2 + 259/9*w - 97/9],
[727, 727, -5/3*w^3 + 3*w^2 + 58/3*w - 55/3],
[727, 727, -67/9*w^3 + 47/3*w^2 + 722/9*w - 875/9],
[727, 727, 7/3*w^3 - w^2 - 86/3*w - 43/3],
[727, 727, -1/9*w^3 + 5/3*w^2 + 14/9*w - 113/9],
[761, 761, -13/3*w^3 + 9*w^2 + 140/3*w - 161/3],
[761, 761, -13/9*w^3 + 11/3*w^2 + 128/9*w - 227/9],
[761, 761, -5/3*w^3 + 3*w^2 + 58/3*w - 37/3],
[761, 761, 7/9*w^3 + 1/3*w^2 - 98/9*w - 127/9],
[769, 769, -7/9*w^3 + 2/3*w^2 + 89/9*w - 35/9],
[769, 769, 1/9*w^3 + 1/3*w^2 - 32/9*w - 31/9],
[769, 769, -1/9*w^3 + 2/3*w^2 + 23/9*w - 59/9],
[769, 769, -7/9*w^3 + 5/3*w^2 + 80/9*w - 53/9],
[841, 29, 5/3*w^3 - 2*w^2 - 61/3*w - 5/3],
[841, 29, -5/3*w^3 + w^2 + 58/3*w + 5/3],
[863, 863, 2/3*w^3 - 28/3*w + 1/3],
[863, 863, 2/9*w^3 + 2/3*w^2 - 46/9*w - 53/9],
[863, 863, 2/9*w^3 - 4/3*w^2 - 28/9*w + 91/9],
[863, 863, -2/3*w^3 + 2*w^2 + 22/3*w - 25/3],
[919, 919, -5/3*w^3 + 4*w^2 + 43/3*w - 49/3],
[919, 919, 5/9*w^3 + 2/3*w^2 - 43/9*w - 29/9],
[919, 919, 5/9*w^3 - 7/3*w^2 - 16/9*w + 61/9],
[919, 919, -13/9*w^3 + 11/3*w^2 + 128/9*w - 191/9],
[937, 937, -5/9*w^3 + 1/3*w^2 + 70/9*w - 61/9],
[937, 937, -1/3*w^3 + w^2 + 14/3*w - 35/3],
[937, 937, 1/3*w^3 - 17/3*w - 19/3],
[937, 937, -5/9*w^3 + 4/3*w^2 + 61/9*w - 7/9],
[953, 953, -2/9*w^3 + 4/3*w^2 + 10/9*w - 73/9],
[953, 953, -2/9*w^3 + 4/3*w^2 + 10/9*w - 55/9],
[953, 953, 2/9*w^3 + 2/3*w^2 - 28/9*w - 35/9],
[953, 953, 2/9*w^3 + 2/3*w^2 - 28/9*w - 53/9],
[961, 31, -8/9*w^3 + 4/3*w^2 + 76/9*w - 31/9],
[961, 31, -8/9*w^3 + 4/3*w^2 + 76/9*w - 49/9],
[967, 967, -25/9*w^3 + 8/3*w^2 + 305/9*w + 19/9],
[967, 967, -7/9*w^3 - 1/3*w^2 + 80/9*w + 19/9],
[967, 967, 7/9*w^3 - 5/3*w^2 - 98/9*w - 19/9],
[967, 967, -35/9*w^3 + 10/3*w^2 + 409/9*w + 41/9],
[977, 977, -8/9*w^3 + 4/3*w^2 + 94/9*w - 121/9],
[977, 977, 14/9*w^3 - 1/3*w^2 - 151/9*w - 47/9],
[977, 977, 14/9*w^3 - 13/3*w^2 - 115/9*w + 187/9],
[977, 977, -14/9*w^3 + 13/3*w^2 + 115/9*w - 169/9],
[1033, 1033, -2/9*w^3 + 1/3*w^2 + 55/9*w - 55/9],
[1033, 1033, 14/9*w^3 - 7/3*w^2 - 169/9*w + 61/9],
[1033, 1033, -14/9*w^3 + 7/3*w^2 + 169/9*w - 115/9],
[1033, 1033, -2/9*w^3 + 1/3*w^2 + 55/9*w - 1/9],
[1039, 1039, -4/9*w^3 + 5/3*w^2 + 29/9*w - 137/9],
[1039, 1039, 4/3*w^3 - 4*w^2 - 38/3*w + 95/3],
[1039, 1039, -28/9*w^3 + 17/3*w^2 + 311/9*w - 257/9],
[1039, 1039, -4/9*w^3 - 1/3*w^2 + 47/9*w + 97/9],
[1063, 1063, -w^3 + 3*w^2 + 8*w - 15],
[1063, 1063, w^3 - 3*w^2 - 8*w + 13],
[1063, 1063, w^3 - 11*w - 3],
[1063, 1063, 19/9*w^3 - 14/3*w^2 - 185/9*w + 221/9],
[1087, 1087, -4/9*w^3 - 1/3*w^2 + 65/9*w + 61/9],
[1087, 1087, -4/9*w^3 + 5/3*w^2 + 47/9*w - 47/9],
[1087, 1087, 4/9*w^3 + 1/3*w^2 - 65/9*w + 11/9],
[1087, 1087, -4/9*w^3 + 5/3*w^2 + 47/9*w - 119/9],
[1097, 1097, -26/9*w^3 + 16/3*w^2 + 292/9*w - 247/9],
[1097, 1097, -16/9*w^3 + 8/3*w^2 + 188/9*w - 107/9],
[1097, 1097, 14/9*w^3 - 13/3*w^2 - 151/9*w + 259/9],
[1097, 1097, 10/9*w^3 - 11/3*w^2 - 95/9*w + 257/9],
[1103, 1103, 8/9*w^3 - 1/3*w^2 - 85/9*w + 13/9],
[1103, 1103, 8/9*w^3 - 1/3*w^2 - 85/9*w - 41/9],
[1103, 1103, 8/9*w^3 - 7/3*w^2 - 67/9*w + 121/9],
[1103, 1103, 8/9*w^3 - 7/3*w^2 - 67/9*w + 67/9],
[1223, 1223, 10/9*w^3 - 8/3*w^2 - 104/9*w + 113/9],
[1223, 1223, -10/9*w^3 + 5/3*w^2 + 113/9*w - 113/9],
[1223, 1223, 10/9*w^3 - 5/3*w^2 - 113/9*w + 5/9],
[1223, 1223, -10/9*w^3 + 2/3*w^2 + 122/9*w - 5/9],
[1249, 1249, -23/9*w^3 + 13/3*w^2 + 268/9*w - 151/9],
[1249, 1249, 5/3*w^3 - w^2 - 64/3*w - 29/3],
[1249, 1249, -49/9*w^3 + 35/3*w^2 + 524/9*w - 641/9],
[1249, 1249, 23/3*w^3 - 16*w^2 - 247/3*w + 301/3],
[1279, 1279, 22/3*w^3 - 16*w^2 - 236/3*w + 311/3],
[1279, 1279, 10/9*w^3 + 1/3*w^2 - 113/9*w - 103/9],
[1279, 1279, -4*w^3 + 8*w^2 + 44*w - 51],
[1279, 1279, 34/9*w^3 - 11/3*w^2 - 395/9*w + 17/9],
[1327, 1327, -28/9*w^3 + 17/3*w^2 + 311/9*w - 293/9],
[1327, 1327, 4*w^3 - 9*w^2 - 43*w + 59],
[1327, 1327, -22/9*w^3 + 8/3*w^2 + 254/9*w - 47/9],
[1327, 1327, -4/9*w^3 + 8/3*w^2 + 38/9*w - 209/9],
[1361, 1361, -14/3*w^3 + 11*w^2 + 145/3*w - 217/3],
[1361, 1361, 14/9*w^3 - 4/3*w^2 - 178/9*w - 65/9],
[1361, 1361, 14/9*w^3 - 10/3*w^2 - 160/9*w + 241/9],
[1361, 1361, -2/9*w^3 - 2/3*w^2 + 64/9*w - 73/9],
[1369, 37, 13/9*w^3 - 2/3*w^2 - 173/9*w - 115/9],
[1369, 37, -43/9*w^3 + 32/3*w^2 + 449/9*w - 611/9],
[1409, 1409, -71/9*w^3 + 52/3*w^2 + 751/9*w - 1003/9],
[1409, 1409, -61/9*w^3 + 44/3*w^2 + 647/9*w - 809/9],
[1409, 1409, 7/3*w^3 - 2*w^2 - 89/3*w - 25/3],
[1409, 1409, -37/9*w^3 + 14/3*w^2 + 437/9*w - 41/9],
[1447, 1447, -4/3*w^3 + 2*w^2 + 50/3*w - 47/3],
[1447, 1447, 2/3*w^3 + 2*w^2 - 34/3*w - 101/3],
[1447, 1447, -70/9*w^3 + 47/3*w^2 + 755/9*w - 827/9],
[1447, 1447, -20/9*w^3 + 16/3*w^2 + 190/9*w - 271/9],
[1471, 1471, -5/3*w^3 + 3*w^2 + 52/3*w - 43/3],
[1471, 1471, 7/9*w^3 - 5/3*w^2 - 62/9*w - 19/9],
[1471, 1471, -61/9*w^3 + 41/3*w^2 + 656/9*w - 737/9],
[1471, 1471, -7/9*w^3 + 5/3*w^2 + 62/9*w - 143/9],
[1481, 1481, 10/9*w^3 - 5/3*w^2 - 131/9*w + 23/9],
[1481, 1481, 46/9*w^3 - 23/3*w^2 - 527/9*w + 257/9],
[1481, 1481, -50/9*w^3 + 40/3*w^2 + 520/9*w - 817/9],
[1481, 1481, -10/9*w^3 + 5/3*w^2 + 131/9*w - 113/9],
[1487, 1487, 11/9*w^3 - 4/3*w^2 - 145/9*w + 1/9],
[1487, 1487, -11/9*w^3 + 1/3*w^2 + 136/9*w + 35/9],
[1487, 1487, -11/9*w^3 + 10/3*w^2 + 109/9*w - 163/9],
[1487, 1487, -7/9*w^3 + 5/3*w^2 + 116/9*w + 37/9],
[1511, 1511, -5/3*w^3 + 3*w^2 + 58/3*w - 49/3],
[1511, 1511, 1/9*w^3 + 7/3*w^2 - 32/9*w - 247/9],
[1511, 1511, 1/3*w^3 - 2*w^2 - 11/3*w + 23/3],
[1511, 1511, -5/9*w^3 + 7/3*w^2 + 52/9*w - 151/9],
[1543, 1543, -1/9*w^3 + 2/3*w^2 - 13/9*w + 49/9],
[1543, 1543, w^3 - w^2 - 12*w - 5],
[1543, 1543, -w^3 + 2*w^2 + 11*w - 17],
[1543, 1543, -1/9*w^3 - 1/3*w^2 - 4/9*w - 41/9],
[1583, 1583, 1/9*w^3 + 1/3*w^2 - 14/9*w - 85/9],
[1583, 1583, 1/9*w^3 - 2/3*w^2 - 5/9*w - 31/9],
[1583, 1583, -1/9*w^3 - 1/3*w^2 + 14/9*w - 41/9],
[1583, 1583, -1/9*w^3 + 2/3*w^2 + 5/9*w - 95/9],
[1607, 1607, -20/9*w^3 + 13/3*w^2 + 217/9*w - 199/9],
[1607, 1607, -32/9*w^3 + 25/3*w^2 + 331/9*w - 511/9],
[1607, 1607, -28/9*w^3 + 14/3*w^2 + 320/9*w - 149/9],
[1607, 1607, -2/3*w^3 + 2*w^2 + 16/3*w - 55/3],
[1657, 1657, 1/9*w^3 + 1/3*w^2 - 32/9*w - 49/9],
[1657, 1657, -7/9*w^3 + 5/3*w^2 + 80/9*w - 35/9],
[1657, 1657, -7/9*w^3 + 2/3*w^2 + 89/9*w - 53/9],
[1657, 1657, -1/9*w^3 + 2/3*w^2 + 23/9*w - 77/9],
[1681, 41, 10/9*w^3 - 5/3*w^2 - 95/9*w + 23/9],
[1681, 41, -10/9*w^3 + 5/3*w^2 + 95/9*w - 77/9],
[1721, 1721, 7/9*w^3 - 8/3*w^2 - 53/9*w + 71/9],
[1721, 1721, -7/9*w^3 + 8/3*w^2 + 53/9*w - 161/9],
[1721, 1721, 7/9*w^3 + 1/3*w^2 - 80/9*w - 91/9],
[1721, 1721, -7/9*w^3 - 1/3*w^2 + 80/9*w + 1/9],
[1753, 1753, -4/3*w^3 + 4*w^2 + 38/3*w - 101/3],
[1753, 1753, 2/9*w^3 - 4/3*w^2 - 10/9*w + 145/9],
[1753, 1753, 2/9*w^3 + 2/3*w^2 - 28/9*w - 125/9],
[1753, 1753, -32/9*w^3 + 19/3*w^2 + 349/9*w - 295/9],
[1759, 1759, -19/9*w^3 + 14/3*w^2 + 221/9*w - 221/9],
[1759, 1759, 37/9*w^3 - 14/3*w^2 - 437/9*w + 59/9],
[1759, 1759, 67/9*w^3 - 50/3*w^2 - 713/9*w + 965/9],
[1759, 1759, 1/3*w^3 - 2*w^2 - 5/3*w + 47/3],
[1777, 1777, -13/9*w^3 + 8/3*w^2 + 155/9*w - 83/9],
[1777, 1777, 7/9*w^3 + 4/3*w^2 - 107/9*w - 235/9],
[1777, 1777, 1/3*w^3 - w^2 - 20/3*w + 29/3],
[1777, 1777, 13/9*w^3 - 5/3*w^2 - 164/9*w + 83/9],
[1783, 1783, -1/9*w^3 + 5/3*w^2 - 4/9*w - 185/9],
[1783, 1783, -13/9*w^3 + 11/3*w^2 + 128/9*w - 263/9],
[1783, 1783, -23/9*w^3 + 13/3*w^2 + 250/9*w - 187/9],
[1783, 1783, -11/3*w^3 + 7*w^2 + 118/3*w - 121/3],
[1801, 1801, 14/9*w^3 - 4/3*w^2 - 142/9*w - 11/9],
[1801, 1801, -14/9*w^3 + 7/3*w^2 + 187/9*w - 187/9],
[1801, 1801, 14/9*w^3 - 10/3*w^2 - 124/9*w + 97/9],
[1801, 1801, -14/9*w^3 + 10/3*w^2 + 124/9*w - 151/9],
[1823, 1823, 7/9*w^3 - 2/3*w^2 - 89/9*w + 107/9],
[1823, 1823, -1/9*w^3 + 2/3*w^2 + 23/9*w - 131/9],
[1823, 1823, 1/9*w^3 + 1/3*w^2 - 32/9*w - 103/9],
[1823, 1823, -7/9*w^3 + 5/3*w^2 + 80/9*w + 19/9],
[1849, 43, 8/9*w^3 - 4/3*w^2 - 112/9*w + 13/9],
[1849, 43, -4/3*w^3 + 2*w^2 + 56/3*w + 13/3],
[1871, 1871, -58/9*w^3 + 38/3*w^2 + 632/9*w - 677/9],
[1871, 1871, -16/9*w^3 + 11/3*w^2 + 161/9*w - 143/9],
[1871, 1871, 16/9*w^3 - 5/3*w^2 - 179/9*w + 35/9],
[1871, 1871, 4/9*w^3 + 7/3*w^2 - 65/9*w - 277/9],
[1879, 1879, 14/9*w^3 - 1/3*w^2 - 133/9*w - 29/9],
[1879, 1879, -22/9*w^3 + 5/3*w^2 + 245/9*w + 7/9],
[1879, 1879, -22/9*w^3 + 17/3*w^2 + 209/9*w - 245/9],
[1879, 1879, -14/9*w^3 + 13/3*w^2 + 97/9*w - 151/9],
[1889, 1889, 2/9*w^3 - 1/3*w^2 - 37/9*w + 91/9],
[1889, 1889, 2/3*w^3 - w^2 - 25/3*w - 11/3],
[1889, 1889, 2/3*w^3 - w^2 - 25/3*w + 37/3],
[1889, 1889, 2/9*w^3 - 1/3*w^2 - 37/9*w - 53/9],
[1913, 1913, 11/9*w^3 - 7/3*w^2 - 118/9*w + 73/9],
[1913, 1913, 1/3*w^3 - 5/3*w - 13/3],
[1913, 1913, -1/3*w^3 + w^2 + 2/3*w - 17/3],
[1913, 1913, -11/9*w^3 + 4/3*w^2 + 127/9*w - 55/9],
[1951, 1951, -4/3*w^3 + 3*w^2 + 41/3*w - 41/3],
[1951, 1951, -4/9*w^3 - 1/3*w^2 + 29/9*w + 43/9],
[1951, 1951, 4/9*w^3 - 5/3*w^2 - 11/9*w + 65/9],
[1951, 1951, -4/3*w^3 + w^2 + 47/3*w - 5/3],
[1993, 1993, 38/9*w^3 - 16/3*w^2 - 424/9*w + 163/9],
[1993, 1993, -58/9*w^3 + 41/3*w^2 + 623/9*w - 803/9],
[1993, 1993, 44/9*w^3 - 31/3*w^2 - 481/9*w + 589/9],
[1993, 1993, -46/9*w^3 + 35/3*w^2 + 491/9*w - 725/9]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - 6*x + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 0, e, e - 6, e - 4, e - 2, e, e - 6, -2*e + 8, -2*e + 8, 2*e - 4, 2*e - 4, -4*e + 10, 4*e - 14, 10, -6*e + 22, 10, 6*e - 14, -4*e + 14, -4*e + 10, -4*e + 20, -4*e + 4, -3*e - 6, -3*e + 24, -4*e + 6, -4*e + 18, -4*e + 18, -4*e + 6, -3*e + 6, 9*e - 24, -3*e + 12, 9*e - 30, -4*e + 8, -4*e + 26, -4*e + 16, -4*e - 2, 10, -2, -6*e + 8, 6*e - 16, 6*e - 28, -6*e + 20, -8*e + 24, 8*e - 24, -8*e + 24, 8*e - 24, -2*e - 8, 4*e - 8, 2*e - 20, -4*e + 16, -7*e + 12, 5*e, -7*e + 30, 5*e - 30, 4*e - 28, 4*e + 4, -8*e + 14, -8*e + 34, 2*e - 16, -4*e + 32, -2*e - 4, 4*e + 8, 5*e - 10, 5*e - 20, 5*e - 4, 5*e - 26, e + 30, e - 36, -11*e + 42, -11*e + 24, 8*e - 8, -8*e + 40, 8*e - 32, -8*e + 16, -18, 18, 16, 4, -16, -4, e - 22, -11*e + 44, e + 16, -11*e + 22, -14, -14, -12*e + 34, 12*e - 38, -4*e + 34, 8*e - 38, 4*e + 10, -8*e + 10, -4*e + 10, -4*e + 14, 8*e - 2, 8*e - 46, -4*e - 6, 4*e - 30, -4*e + 18, 14*e - 30, 4*e - 6, -14*e + 54, -7*e - 2, -7*e + 44, 5*e - 50, 5*e + 20, 12*e - 22, -22, -12*e + 50, -22, 4*e - 46, 4*e - 40, 4*e + 22, 4*e + 16, 4*e + 24, -4*e + 48, 10*e - 36, -10*e + 24, 40, 40, -6*e - 8, 6*e - 44, 2*e + 16, -2*e + 28, 2*e + 4, -2*e + 16, 13*e - 22, 13*e - 56, -11*e + 20, -11*e + 46, e + 2, -11*e + 50, e - 8, -11*e + 16, -3*e + 12, -3*e + 6, -16*e + 46, -4*e - 14, -16*e + 50, -4*e + 38, 14*e - 48, -14*e + 36, 14*e - 36, -14*e + 48, -7*e + 24, -7*e + 18, -7*e - 18, -7*e + 60, 14*e - 62, -14*e + 22, 8*e - 62, -8*e - 14, -8*e + 6, 8*e - 42, 6*e - 20, -6*e + 16, 12*e - 32, -12*e + 40, -6*e + 30, 6*e - 6, 6, 6, -4*e + 34, 8*e - 50, 4*e + 10, -8*e - 2, 12*e - 26, 22, 12*e - 46, -22, 6*e + 20, -6*e + 56, 24*e - 64, -24*e + 80, 24*e - 72, 24*e - 72, 12, -12, 10*e - 30, -20*e + 66, -10*e + 30, 20*e - 54, -12, 12, 12, -12, -4*e + 56, -4*e - 32, -4*e + 38, -4*e - 14, 9*e - 10, 9*e - 44, 21*e - 46, 21*e - 80, -4*e + 62, 8*e - 34, -4*e - 38, 8*e - 14, 16*e - 64, 16*e - 32, -8*e + 32, -8*e + 16, -18*e + 58, 18*e - 50, 22, 22, -15*e + 24, -15*e + 66, 5*e + 24, 5*e, 5*e - 54, 5*e - 30, -12*e + 14, -12*e + 44, -12*e + 58, -12*e + 28, -12*e + 24, -18*e + 48, 12*e - 48, 18*e - 60, 4*e + 34, -4*e + 58, -2*e - 14, 2*e - 26, 4*e + 16, -4*e + 40, 10*e - 80, -10*e - 20, 12*e - 16, -6*e - 4, -12*e + 56, 6*e - 40, -24*e + 64, 12*e - 56, 24*e - 80, -12*e + 16, -8*e + 8, -2*e - 40, 8*e - 40, 2*e - 52, -4*e - 8, -4*e + 32, -4*e + 46, -4*e - 22, -19*e + 58, -19*e + 56, -7*e + 28, -7*e + 14, -8*e - 6, 8*e - 54, 5*e, 5*e - 30, 17*e - 54, 17*e - 48, 2*e + 54, -2*e + 66, 20*e - 54, -20*e + 66, 8*e - 24, -8*e + 24, -4*e + 72, 4*e + 48, 13*e - 4, 13*e - 74, 25*e - 100, 25*e - 50, 8*e - 32, -8*e + 16, -10*e + 40, 10*e - 20, -8*e + 34, 8*e - 14, -8*e + 58, 8*e + 10, -2*e - 8, 2*e - 20, 4*e - 8, -4*e + 16, 10, -50, -4*e + 42, -4*e - 18, 20*e - 42, 20*e - 78, -12*e + 20, -12*e + 52, 74, -74, 34*e - 102, 10*e - 18, -34*e + 102, -10*e + 42, -11*e + 74, -11*e + 32, -11*e - 8, -11*e + 34, 32*e - 98, -4*e + 46, 32*e - 94, -4*e - 22, -16*e + 66, 2*e - 42, 16*e - 30, -2*e - 30];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, 4/9*w^3 - 2/3*w^2 - 47/9*w + 20/9}>] := -1;
ALEigenvalues[ideal<ZF | {2, 2, 5/9*w^3 - 4/3*w^2 - 52/9*w + 79/9}>] := -1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;