/*
  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![-4, -9, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {2, 2, w + 2}>;

primesArray := [
[2, 2, w + 2],
[3, 3, w + 1],
[4, 2, -w^2 + 3*w + 3],
[9, 3, -w^2 + 2*w + 7],
[13, 13, w + 3],
[13, 13, -w + 3],
[13, 13, -w + 1],
[17, 17, -w^2 - 2*w + 1],
[23, 23, w^2 - 2*w - 5],
[31, 31, 2*w + 3],
[31, 31, -2*w^2 + 3*w + 15],
[31, 31, 3*w + 7],
[37, 37, 3*w^2 - 4*w - 27],
[41, 41, -2*w^2 + 7*w + 1],
[59, 59, w^2 - 3],
[61, 61, 4*w^2 - 12*w - 11],
[71, 71, w^2 - 2*w - 11],
[97, 97, 3*w + 5],
[101, 101, 2*w^2 - 6*w - 7],
[103, 103, 2*w^2 - 3*w - 19],
[107, 107, -3*w^2 + 10*w + 3],
[109, 109, -w - 5],
[109, 109, 5*w^2 - 8*w - 41],
[109, 109, 2*w^2 - 2*w - 21],
[113, 113, -4*w - 3],
[125, 5, -5],
[127, 127, 2*w^2 - 2*w - 17],
[131, 131, 2*w - 3],
[137, 137, 2*w^2 - 5*w - 5],
[137, 137, w^2 - 4*w - 13],
[137, 137, 2*w - 5],
[139, 139, 2*w^2 - 3*w - 13],
[151, 151, 2*w^2 - 9],
[163, 163, 2*w^2 - 5*w - 11],
[173, 173, -2*w^2 + 8*w - 1],
[179, 179, w^2 - 15],
[181, 181, -w^2 + 11],
[191, 191, 3*w - 1],
[193, 193, -6*w^2 + 16*w + 23],
[197, 197, 2*w^2 - 8*w + 3],
[199, 199, 4*w^2 - 5*w - 35],
[211, 211, 2*w - 9],
[223, 223, 3*w^2 - 8*w - 13],
[227, 227, 3*w^2 + 2*w - 11],
[227, 227, w^2 - 4*w - 15],
[227, 227, w - 7],
[229, 229, -4*w^2 + 15*w - 1],
[239, 239, -4*w - 5],
[257, 257, 2*w^2 + w - 9],
[263, 263, -w^2 - 6*w - 7],
[269, 269, w^2 - 4*w - 7],
[277, 277, 4*w^2 - 7*w - 33],
[281, 281, 6*w^2 - 17*w - 19],
[281, 281, 3*w^2 - 8*w - 9],
[281, 281, -5*w - 1],
[289, 17, -2*w^2 + 9*w - 5],
[293, 293, -3*w^2 + 8*w + 7],
[293, 293, 2*w^2 + 5*w - 1],
[293, 293, 6*w^2 - 10*w - 47],
[307, 307, 2*w^2 - 2*w - 3],
[313, 313, 2*w^2 - 4*w - 7],
[317, 317, 3*w^2 - 6*w - 17],
[317, 317, 2*w^2 - 7*w - 7],
[317, 317, 2*w^2 + 2*w - 7],
[331, 331, 2*w^2 + 7*w + 7],
[337, 337, w^2 - 4*w - 9],
[337, 337, 3*w^2 - 4*w - 29],
[337, 337, w^2 - 6*w - 5],
[343, 7, -7],
[347, 347, 2*w^2 + 4*w - 3],
[349, 349, 2*w^2 - w - 25],
[359, 359, -2*w^2 - w + 5],
[379, 379, 4*w^2 - 9*w - 23],
[379, 379, -5*w^2 + 18*w + 3],
[379, 379, 2*w^2 - 3*w - 21],
[383, 383, -w^2 + 6*w - 7],
[397, 397, 2*w^2 - 7],
[401, 401, 9*w^2 - 28*w - 21],
[419, 419, 3*w^2 - 6*w - 25],
[419, 419, 4*w^2 - 7*w - 27],
[419, 419, 4*w^2 - 6*w - 29],
[433, 433, 8*w^2 - 13*w - 61],
[433, 433, -6*w^2 + 23*w - 3],
[433, 433, -w^2 - 3],
[443, 443, -2*w^2 + 5*w + 15],
[449, 449, -7*w^2 + 12*w + 51],
[457, 457, 2*w^2 - 2*w - 9],
[457, 457, -w^2 - 6*w - 1],
[457, 457, 6*w^2 - 11*w - 43],
[461, 461, -5*w^2 + 6*w + 45],
[463, 463, 2*w^2 - 5*w - 17],
[463, 463, 3*w - 5],
[463, 463, 2*w^2 - 3*w - 7],
[467, 467, -3*w^2 + 10*w + 9],
[479, 479, 3*w - 7],
[487, 487, 7*w^2 - 20*w - 23],
[487, 487, -4*w - 11],
[487, 487, 5*w + 7],
[509, 509, 2*w^2 - 2*w - 5],
[523, 523, -8*w^2 + 28*w + 7],
[529, 23, 2*w^2 - w - 7],
[541, 541, 3*w^2 + 2*w - 9],
[547, 547, -6*w - 1],
[557, 557, 2*w^2 - 13],
[563, 563, w - 9],
[569, 569, -8*w^2 + 27*w + 9],
[571, 571, 5*w^2 - 14*w - 15],
[577, 577, w^2 + 4*w + 7],
[587, 587, 4*w^2 - 4*w - 27],
[593, 593, 2*w^2 + 5*w + 5],
[599, 599, -2*w - 9],
[599, 599, -w^2 + 6*w - 1],
[599, 599, 2*w^2 - w - 19],
[607, 607, 4*w - 13],
[617, 617, -5*w - 13],
[617, 617, -w^2 - 2*w - 5],
[617, 617, 4*w^2 - 16*w + 5],
[619, 619, 3*w^2 - 8*w - 15],
[643, 643, -6*w - 11],
[647, 647, -w^2 + 17],
[647, 647, 2*w^2 - 9*w - 7],
[647, 647, 2*w^2 - 6*w - 21],
[659, 659, 4*w^2 - 8*w - 23],
[661, 661, 2*w - 11],
[673, 673, 3*w^2 - 13],
[677, 677, 5*w^2 - 16*w - 13],
[677, 677, -3*w^2 - 4*w + 3],
[677, 677, 5*w^2 - 6*w - 43],
[691, 691, 6*w^2 - 12*w - 37],
[701, 701, -3*w^2 + 12*w - 1],
[709, 709, -6*w^2 + 21*w + 7],
[719, 719, -w^2 + 6*w - 3],
[733, 733, -w - 9],
[733, 733, 2*w^2 + w - 11],
[733, 733, -2*w^2 + 9*w - 3],
[739, 739, 6*w^2 - 17*w - 21],
[743, 743, -2*w^2 - 6*w + 1],
[751, 751, 4*w^2 - 8*w - 29],
[751, 751, -w^2 - 8*w - 11],
[751, 751, 4*w^2 - 21],
[757, 757, 2*w^2 - 9*w - 1],
[761, 761, -4*w^2 + 10*w + 15],
[809, 809, w^2 + 2*w - 19],
[811, 811, -7*w - 17],
[823, 823, 3*w^2 - 17],
[827, 827, -9*w - 19],
[857, 857, 4*w^2 + 14*w + 9],
[857, 857, 4*w^2 - 16*w + 3],
[857, 857, 4*w^2 - 6*w - 27],
[859, 859, -2*w^2 + 5*w - 1],
[863, 863, 12*w^2 - 37*w - 29],
[877, 877, 3*w^2 - 6*w - 7],
[877, 877, 3*w^2 - 6*w - 31],
[877, 877, 3*w^2 - 6*w - 13],
[907, 907, 8*w^2 - 14*w - 57],
[907, 907, 4*w^2 - 7*w - 25],
[907, 907, 8*w^2 - 24*w - 19],
[911, 911, -3*w - 11],
[919, 919, -16*w^2 + 53*w + 23],
[929, 929, 2*w^2 - 6*w - 19],
[937, 937, w^2 - 6*w - 15],
[937, 937, 3*w^2 - 4*w - 17],
[937, 937, -3*w^2 + 12*w + 1],
[941, 941, -2*w^2 + 6*w + 15],
[947, 947, -2*w^2 - 3*w + 7],
[953, 953, 2*w^2 - 25],
[971, 971, 3*w^2 - 10*w - 11],
[971, 971, 3*w^2 - 12*w + 5],
[971, 971, -4*w^2 + 10*w + 9],
[977, 977, 2*w^2 - 8*w - 9],
[983, 983, -5*w^2 + 8*w + 35],
[991, 991, 4*w^2 + 5*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 2*x^5 - 16*x^4 + 26*x^3 + 60*x^2 - 66*x + 12;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -2/9*e^5 + 1/3*e^4 + 29/9*e^3 - 11/3*e^2 - 32/3*e + 19/3, 1/6*e^5 - 8/3*e^3 + 10*e - 3, 1/18*e^5 - 1/3*e^4 - 14/9*e^3 + 11/3*e^2 + 32/3*e - 13/3, -1/18*e^5 + 1/3*e^4 + 5/9*e^3 - 11/3*e^2 - 5/3*e + 19/3, 1/18*e^5 - 1/3*e^4 - 14/9*e^3 + 11/3*e^2 + 32/3*e - 13/3, 1/6*e^5 - 8/3*e^3 + e^2 + 9*e - 7, 1/3*e^5 - 16/3*e^3 + e^2 + 18*e - 10, 2/9*e^5 - 1/3*e^4 - 38/9*e^3 + 14/3*e^2 + 59/3*e - 34/3, -2/3*e^5 + e^4 + 32/3*e^3 - 12*e^2 - 40*e + 20, -2/9*e^5 + 1/3*e^4 + 38/9*e^3 - 14/3*e^2 - 53/3*e + 40/3, 2/9*e^5 - 1/3*e^4 - 29/9*e^3 + 8/3*e^2 + 29/3*e + 8/3, 1/2*e^5 - 8*e^3 + e^2 + 30*e - 13, -1/3*e^5 + e^4 + 16/3*e^3 - 10*e^2 - 24*e + 6, -11/18*e^5 + 2/3*e^4 + 82/9*e^3 - 19/3*e^2 - 94/3*e + 23/3, -1/3*e^5 + e^4 + 19/3*e^3 - 12*e^2 - 31*e + 14, 5/18*e^5 - 2/3*e^4 - 52/9*e^3 + 25/3*e^2 + 88/3*e - 41/3, 1/2*e^5 - e^4 - 8*e^3 + 11*e^2 + 30*e - 19, -4/9*e^5 + 2/3*e^4 + 76/9*e^3 - 25/3*e^2 - 118/3*e + 38/3, -1/3*e^5 + 10/3*e^3 + 2*e^2 - e - 8, 5/18*e^5 - 2/3*e^4 - 52/9*e^3 + 28/3*e^2 + 88/3*e - 53/3, 2/9*e^5 - 4/3*e^4 - 38/9*e^3 + 44/3*e^2 + 68/3*e - 34/3, -5/18*e^5 + 2/3*e^4 + 43/9*e^3 - 19/3*e^2 - 61/3*e + 5/3, -1/2*e^5 + 8*e^3 - e^2 - 30*e + 7, -1/3*e^5 + 16/3*e^3 - 2*e^2 - 18*e + 14, 1/9*e^5 - 2/3*e^4 - 19/9*e^3 + 28/3*e^2 + 31/3*e - 44/3, -1/3*e^5 + 16/3*e^3 - e^2 - 18*e + 4, 1/2*e^5 - 10*e^3 + e^2 + 44*e - 7, -1/6*e^5 + 5/3*e^3 + e^2 + e - 7, -1/6*e^5 + 8/3*e^3 - e^2 - 12*e + 13, 1/9*e^5 - 2/3*e^4 - 10/9*e^3 + 22/3*e^2 + 10/3*e - 20/3, 11/9*e^5 - 4/3*e^4 - 191/9*e^3 + 50/3*e^2 + 269/3*e - 130/3, -1/9*e^5 - 1/3*e^4 + 10/9*e^3 + 14/3*e^2 - 4/3*e - 34/3, 1/6*e^5 - 8/3*e^3 + e^2 + 6*e - 1, 2*e^3 - 3*e^2 - 20*e + 18, 1/6*e^5 - 8/3*e^3 - e^2 + 12*e + 15, 1/3*e^5 - 2*e^4 - 16/3*e^3 + 23*e^2 + 24*e - 30, 1/3*e^5 - 10/3*e^3 - 4*e^2 + 22, -5/6*e^5 + e^4 + 40/3*e^3 - 11*e^2 - 51*e + 25, e^4 - e^3 - 12*e^2 + 3*e + 26, e^5 - 2*e^4 - 16*e^3 + 22*e^2 + 62*e - 38, -8/9*e^5 + 4/3*e^4 + 134/9*e^3 - 50/3*e^2 - 170/3*e + 82/3, 1/3*e^5 - 10/3*e^3 - e^2 + 4*e - 8, -e^5 + 14*e^3 + 2*e^2 - 40*e - 2, -e^4 - 2*e^3 + 14*e^2 + 23*e - 28, 1/18*e^5 + 2/3*e^4 - 14/9*e^3 - 31/3*e^2 + 32/3*e + 53/3, -2/3*e^5 + 29/3*e^3 - 35*e + 12, 1/2*e^5 - 6*e^3 + e^2 + 10*e - 7, -e^5 + 3*e^4 + 20*e^3 - 36*e^2 - 100*e + 54, -e^4 + 12*e^2 + 6*e - 14, 7/18*e^5 - 4/3*e^4 - 71/9*e^3 + 53/3*e^2 + 125/3*e - 115/3, 3/2*e^5 - 3*e^4 - 27*e^3 + 37*e^2 + 117*e - 61, -1/2*e^5 + 8*e^3 - e^2 - 30*e + 13, 2/3*e^5 - e^4 - 32/3*e^3 + 10*e^2 + 42*e - 6, 1/2*e^5 - 3*e^3 - 3*e^2 - 17*e + 5, -1/2*e^5 + 10*e^3 - 3*e^2 - 50*e + 9, 5/6*e^5 - 28/3*e^3 - 3*e^2 + 14*e + 3, 2/3*e^5 - e^4 - 26/3*e^3 + 8*e^2 + 22*e - 10, 10/9*e^5 - 5/3*e^4 - 136/9*e^3 + 52/3*e^2 + 130/3*e - 92/3, -16/9*e^5 + 8/3*e^4 + 250/9*e^3 - 88/3*e^2 - 310/3*e + 158/3, -1/6*e^5 + 8/3*e^3 + 3*e^2 - 15*e - 21, -7/6*e^5 + e^4 + 56/3*e^3 - 13*e^2 - 69*e + 33, -1/3*e^5 + 2*e^4 + 16/3*e^3 - 22*e^2 - 24*e + 38, 2*e, 8/9*e^5 + 2/3*e^4 - 134/9*e^3 - 16/3*e^2 + 176/3*e - 34/3, -25/18*e^5 + 4/3*e^4 + 197/9*e^3 - 53/3*e^2 - 221/3*e + 139/3, 7/6*e^5 - 2*e^4 - 62/3*e^3 + 23*e^2 + 88*e - 39, e^5 - e^4 - 14*e^3 + 10*e^2 + 40*e - 10, -e^5 + e^4 + 16*e^3 - 12*e^2 - 54*e + 14, -5/9*e^5 + 4/3*e^4 + 59/9*e^3 - 38/3*e^2 - 53/3*e + 52/3, 1/3*e^5 - 2*e^4 - 16/3*e^3 + 24*e^2 + 24*e - 40, e^5 - 2*e^4 - 18*e^3 + 26*e^2 + 82*e - 48, 5/9*e^5 - 4/3*e^4 - 122/9*e^3 + 59/3*e^2 + 224/3*e - 130/3, -8/9*e^5 + 4/3*e^4 + 116/9*e^3 - 44/3*e^2 - 110/3*e + 112/3, 1/3*e^5 - 10/3*e^3 - 2*e - 6, -13/18*e^5 + 4/3*e^4 + 110/9*e^3 - 47/3*e^2 - 152/3*e + 79/3, -1/6*e^5 + 2/3*e^3 - e^2 + 5*e + 13, 2*e^5 - 4*e^4 - 32*e^3 + 44*e^2 + 120*e - 64, -e^5 + 2*e^4 + 16*e^3 - 22*e^2 - 66*e + 38, e^4 - 10*e^2, 20/9*e^5 - 4/3*e^4 - 317/9*e^3 + 50/3*e^2 + 389/3*e - 148/3, -29/18*e^5 + 2/3*e^4 + 208/9*e^3 - 16/3*e^2 - 214/3*e + 11/3, -1/6*e^5 + e^4 + 11/3*e^3 - 11*e^2 - 19*e + 15, -1/3*e^5 + e^4 + 19/3*e^3 - 8*e^2 - 31*e - 2, -2/3*e^5 + e^4 + 44/3*e^3 - 14*e^2 - 76*e + 34, -13/18*e^5 - 2/3*e^4 + 74/9*e^3 + 19/3*e^2 - 32/3*e + 55/3, 2/3*e^5 - e^4 - 35/3*e^3 + 12*e^2 + 55*e - 12, -13/18*e^5 + 1/3*e^4 + 110/9*e^3 - 23/3*e^2 - 152/3*e + 157/3, 1/6*e^5 - 14/3*e^3 + e^2 + 32*e - 13, 5/9*e^5 + 2/3*e^4 - 68/9*e^3 - 34/3*e^2 + 62/3*e + 56/3, -4/9*e^5 + 2/3*e^4 + 76/9*e^3 - 10/3*e^2 - 136/3*e - 22/3, 2*e^5 - 3*e^4 - 34*e^3 + 34*e^2 + 140*e - 68, 1/3*e^5 - 2*e^4 - 22/3*e^3 + 22*e^2 + 44*e - 14, 5/3*e^5 - 3*e^4 - 80/3*e^3 + 32*e^2 + 102*e - 50, 10/9*e^5 - 2/3*e^4 - 172/9*e^3 + 25/3*e^2 + 250/3*e - 86/3, -2/9*e^5 - 5/3*e^4 + 38/9*e^3 + 46/3*e^2 - 35/3*e - 32/3, 4/9*e^5 - 2/3*e^4 - 58/9*e^3 + 22/3*e^2 + 82/3*e - 74/3, -2/3*e^5 + 2*e^4 + 44/3*e^3 - 25*e^2 - 82*e + 38, -10/9*e^5 + 2/3*e^4 + 154/9*e^3 - 28/3*e^2 - 148/3*e + 68/3, -41/18*e^5 + 11/3*e^4 + 322/9*e^3 - 121/3*e^2 - 400/3*e + 197/3, 16/9*e^5 - 2/3*e^4 - 268/9*e^3 + 34/3*e^2 + 349/3*e - 140/3, -10/9*e^5 + 8/3*e^4 + 172/9*e^3 - 88/3*e^2 - 262/3*e + 122/3, 11/6*e^5 - 4*e^4 - 94/3*e^3 + 46*e^2 + 134*e - 75, 2*e^3 - 2*e^2 - 8*e + 14, -1/3*e^5 - e^4 + 16/3*e^3 + 10*e^2 - 9*e, -8/9*e^5 + 4/3*e^4 + 134/9*e^3 - 44/3*e^2 - 170/3*e + 58/3, 5/6*e^5 - e^4 - 28/3*e^3 + 13*e^2 + 14*e - 37, 2/3*e^5 - 26/3*e^3 + 3*e^2 + 16*e - 30, -5/6*e^5 + 2*e^4 + 40/3*e^3 - 19*e^2 - 60*e + 5, -4/3*e^5 + 4*e^4 + 70/3*e^3 - 44*e^2 - 110*e + 52, -2/3*e^5 + 38/3*e^3 - 6*e^2 - 56*e + 36, -5/3*e^5 + 2*e^4 + 68/3*e^3 - 24*e^2 - 62*e + 58, 1/3*e^5 - 28/3*e^3 + 2*e^2 + 60*e - 2, -1/2*e^5 + 2*e^4 + 10*e^3 - 25*e^2 - 56*e + 35, e^5 - 2*e^4 - 20*e^3 + 26*e^2 + 94*e - 54, -3/2*e^5 + 2*e^4 + 26*e^3 - 25*e^2 - 104*e + 65, -16/9*e^5 + 8/3*e^4 + 232/9*e^3 - 88/3*e^2 - 232/3*e + 176/3, -11/9*e^5 + 7/3*e^4 + 182/9*e^3 - 68/3*e^2 - 251/3*e + 52/3, -2/3*e^5 + 2*e^4 + 20/3*e^3 - 23*e^2 - 8*e + 36, e^5 - 3*e^4 - 15*e^3 + 34*e^2 + 53*e - 52, -4/3*e^5 + 2*e^4 + 76/3*e^3 - 26*e^2 - 118*e + 66, -4/3*e^5 + 2*e^4 + 58/3*e^3 - 24*e^2 - 64*e + 52, 11/18*e^5 - 2/3*e^4 - 100/9*e^3 + 43/3*e^2 + 142/3*e - 149/3, 5/6*e^5 - 2*e^4 - 46/3*e^3 + 24*e^2 + 76*e - 49, 1/3*e^5 - 19/3*e^3 + 2*e^2 + 25*e + 16, -1/2*e^5 - e^4 + 8*e^3 + 9*e^2 - 24*e - 5, -3/2*e^5 + 3*e^4 + 26*e^3 - 33*e^2 - 116*e + 45, -13/9*e^5 + 8/3*e^4 + 202/9*e^3 - 82/3*e^2 - 256/3*e + 170/3, -3/2*e^5 + 2*e^4 + 24*e^3 - 23*e^2 - 90*e + 27, 1/6*e^5 + 2*e^4 + 4/3*e^3 - 27*e^2 - 32*e + 43, -1/3*e^5 + e^4 + 10/3*e^3 - 6*e^2 - 4*e + 2, -1/2*e^5 + 8*e^3 - 3*e^2 - 28*e + 3, 19/9*e^5 - 8/3*e^4 - 298/9*e^3 + 94/3*e^2 + 352/3*e - 194/3, 1/18*e^5 + 2/3*e^4 + 4/9*e^3 - 31/3*e^2 - 46/3*e + 35/3, -e^5 + e^4 + 18*e^3 - 12*e^2 - 72*e + 26, e^5 - 2*e^4 - 14*e^3 + 22*e^2 + 43*e - 44, -5/9*e^5 + 4/3*e^4 + 50/9*e^3 - 50/3*e^2 - 23/3*e + 64/3, e^5 - 14*e^3 - 4*e^2 + 48*e + 10, -28/9*e^5 + 14/3*e^4 + 442/9*e^3 - 157/3*e^2 - 526/3*e + 284/3, -26/9*e^5 + 10/3*e^4 + 422/9*e^3 - 116/3*e^2 - 548/3*e + 286/3, -1/6*e^5 - e^4 - 4/3*e^3 + 9*e^2 + 25*e + 31, -1/2*e^5 + e^4 + 6*e^3 - 11*e^2 - 16*e + 31, 10/9*e^5 - 8/3*e^4 - 172/9*e^3 + 100/3*e^2 + 232/3*e - 164/3, 2/3*e^5 + 2*e^4 - 32/3*e^3 - 24*e^2 + 29*e + 40, -5/3*e^5 + e^4 + 62/3*e^3 - 6*e^2 - 48*e + 2, 4/3*e^5 - 2*e^4 - 70/3*e^3 + 25*e^2 + 98*e - 50, -1/2*e^5 + e^4 + 10*e^3 - 11*e^2 - 56*e - 17, 5/6*e^5 - 3*e^4 - 52/3*e^3 + 37*e^2 + 88*e - 61, 5/3*e^5 - 2*e^4 - 65/3*e^3 + 20*e^2 + 57*e - 36, -7/3*e^5 + 4*e^4 + 118/3*e^3 - 48*e^2 - 158*e + 86, 1/18*e^5 + 5/3*e^4 + 4/9*e^3 - 49/3*e^2 - 55/3*e + 23/3, 35/18*e^5 - 5/3*e^4 - 256/9*e^3 + 55/3*e^2 + 292/3*e - 131/3, 2*e^4 + 4*e^3 - 28*e^2 - 38*e + 62, 5/3*e^5 - 2*e^4 - 80/3*e^3 + 22*e^2 + 102*e - 42, -14/9*e^5 + 4/3*e^4 + 194/9*e^3 - 56/3*e^2 - 158/3*e + 196/3, 4/9*e^5 - 5/3*e^4 - 58/9*e^3 + 52/3*e^2 + 100/3*e - 56/3, 5/3*e^5 - 3*e^4 - 68/3*e^3 + 32*e^2 + 68*e - 62, 1/9*e^5 - 2/3*e^4 + 8/9*e^3 + 4/3*e^2 - 50/3*e + 34/3, e^4 + 4*e^3 - 16*e^2 - 43*e + 36, -7/18*e^5 + 7/3*e^4 + 44/9*e^3 - 71/3*e^2 - 56/3*e + 19/3, -e^5 + 2*e^4 + 20*e^3 - 26*e^2 - 104*e + 30, -47/18*e^5 + 8/3*e^4 + 388/9*e^3 - 97/3*e^2 - 496/3*e + 185/3, 11/6*e^5 - 3*e^4 - 94/3*e^3 + 37*e^2 + 122*e - 79, 4/3*e^5 - e^4 - 70/3*e^3 + 12*e^2 + 104*e - 32, -5/3*e^5 + 74/3*e^3 - 6*e^2 - 70*e + 54, -1/3*e^5 - e^4 + 10/3*e^3 + 8*e^2 - e + 44, -e^5 + 18*e^3 - 74*e + 6, e^5 - 2*e^4 - 14*e^3 + 26*e^2 + 46*e - 54, 7/6*e^5 - 50/3*e^3 + 7*e^2 + 46*e - 67, 7/3*e^5 - e^4 - 118/3*e^3 + 12*e^2 + 158*e - 50, -e^5 + 14*e^3 - 5*e^2 - 40*e + 64];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, w + 2}>] := -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;