/*
  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 | {3, 3, w + 1}>;

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 - 11*x^4 + 22*x^3 + 17*x^2 - 36*x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/4*e^5 + 1/4*e^4 + 3*e^3 - 5/2*e^2 - 23/4*e + 9/4, -1, e, -1/2*e^5 + 11/2*e^3 + 1/2*e^2 - 10*e - 7/2, -1/2*e^5 + 1/2*e^4 + 11/2*e^3 - 11/2*e^2 - 9*e + 5, -1/2*e^3 - 1/2*e^2 + 9/2*e + 1/2, e^5 - 1/2*e^4 - 23/2*e^3 + 9/2*e^2 + 41/2*e - 2, 1/2*e^5 - 1/2*e^4 - 11/2*e^3 + 11/2*e^2 + 8*e - 8, -1/2*e^5 + 1/2*e^4 + 6*e^3 - 6*e^2 - 27/2*e + 19/2, 1/2*e^4 - 4*e^2 - e + 1/2, 3/2*e^5 - e^4 - 17*e^3 + 9*e^2 + 61/2*e - 7, e^2 - 5, 1/2*e^5 - 1/2*e^4 - 5*e^3 + 5*e^2 + 9/2*e - 13/2, -1/2*e^4 + 1/2*e^3 + 11/2*e^2 - 3/2*e - 12, -1/2*e^5 - 1/2*e^4 + 7*e^3 + 6*e^2 - 41/2*e - 15/2, -2*e^5 + 1/2*e^4 + 47/2*e^3 - 11/2*e^2 - 93/2*e + 2, -e^5 + 1/2*e^4 + 11*e^3 - 3*e^2 - 17*e - 21/2, -3/2*e^5 + 35/2*e^3 - 1/2*e^2 - 36*e + 1/2, 1/2*e^5 + 1/2*e^4 - 13/2*e^3 - 15/2*e^2 + 16*e + 17, 5/2*e^5 - 3/2*e^4 - 28*e^3 + 14*e^2 + 95/2*e - 21/2, -2*e^5 + 2*e^4 + 23*e^3 - 18*e^2 - 45*e + 16, 2*e^5 - 3/2*e^4 - 47/2*e^3 + 31/2*e^2 + 95/2*e - 16, 1/2*e^5 - e^4 - 5*e^3 + 10*e^2 + 7/2*e - 6, e^5 - 21/2*e^3 - 7/2*e^2 + 25/2*e + 33/2, 3/2*e^5 + 1/2*e^4 - 37/2*e^3 - 13/2*e^2 + 37*e + 14, -3*e^5 + 2*e^4 + 34*e^3 - 18*e^2 - 61*e + 16, 7/2*e^5 - 3/2*e^4 - 41*e^3 + 14*e^2 + 155/2*e - 25/2, -e^5 + 12*e^3 - 25*e - 2, 5/2*e^5 - 57/2*e^3 - 1/2*e^2 + 54*e + 9/2, -2*e^5 + 3/2*e^4 + 43/2*e^3 - 29/2*e^2 - 59/2*e + 15, 1/2*e^5 - 9/2*e^3 - 5/2*e^2 + 21/2, -2*e^5 + e^4 + 23*e^3 - 9*e^2 - 39*e + 10, -3/2*e^5 + 1/2*e^4 + 18*e^3 - 2*e^2 - 69/2*e - 25/2, 1/2*e^5 - 2*e^4 - 6*e^3 + 18*e^2 + 25/2*e - 11, 7/2*e^5 - e^4 - 83/2*e^3 + 13/2*e^2 + 86*e - 11/2, -2*e^5 + 2*e^4 + 21*e^3 - 19*e^2 - 29*e + 19, -3/2*e^5 + 1/2*e^4 + 33/2*e^3 - 7/2*e^2 - 29*e - 3, 1/2*e^5 - 3/2*e^4 - 5*e^3 + 15*e^2 + 13/2*e - 47/2, -1/2*e^5 + 6*e^3 - 2*e^2 - 21/2*e + 1, 1/2*e^5 - 1/2*e^4 - 11/2*e^3 + 13/2*e^2 + 13*e - 10, 1/2*e^4 + e^3 - 7*e^2 - 6*e + 23/2, 1/2*e^5 + 1/2*e^4 - 6*e^3 - 6*e^2 + 35/2*e + 19/2, 9/2*e^5 - 3/2*e^4 - 50*e^3 + 12*e^2 + 167/2*e + 23/2, -e^5 + e^4 + 12*e^3 - 7*e^2 - 23*e - 6, 3*e^2 - 27, -1/2*e^4 + 4*e^2 + 3*e - 13/2, 1/2*e^4 + 1/2*e^3 - 11/2*e^2 - 17/2*e + 1, -2*e^5 + e^4 + 23*e^3 - 11*e^2 - 41*e + 14, -3*e^5 + e^4 + 65/2*e^3 - 13/2*e^2 - 105/2*e - 23/2, -5/2*e^5 + 1/2*e^4 + 29*e^3 - 2*e^2 - 105/2*e - 1/2, e^3 - 2*e^2 - 7*e + 14, e^5 - e^4 - 21/2*e^3 + 21/2*e^2 + 13/2*e - 5/2, -9/2*e^5 + 5/2*e^4 + 101/2*e^3 - 49/2*e^2 - 84*e + 8, -3/2*e^5 + 35/2*e^3 + 3/2*e^2 - 30*e - 31/2, -3/2*e^5 + 3/2*e^4 + 17*e^3 - 15*e^2 - 63/2*e + 31/2, 7/2*e^5 - 81/2*e^3 - 1/2*e^2 + 79*e + 5/2, -4*e^5 + 95/2*e^3 + 3/2*e^2 - 191/2*e - 35/2, -3*e^5 + 3/2*e^4 + 69/2*e^3 - 27/2*e^2 - 135/2*e, 1/2*e^5 + 3/2*e^4 - 7*e^3 - 19*e^2 + 45/2*e + 55/2, e^5 + e^4 - 14*e^3 - 11*e^2 + 41*e + 22, -3/2*e^5 + 2*e^4 + 18*e^3 - 18*e^2 - 67/2*e + 3, 4*e^5 - 3/2*e^4 - 91/2*e^3 + 29/2*e^2 + 171/2*e - 17, 3/2*e^5 - 1/2*e^4 - 41/2*e^3 + 9/2*e^2 + 55*e - 4, -e^5 + 11*e^3 + 2*e^2 - 24*e - 22, -2*e^5 + e^4 + 23*e^3 - 7*e^2 - 45*e - 2, 3/2*e^5 + e^4 - 19*e^3 - 14*e^2 + 85/2*e + 30, -3*e^5 - 1/2*e^4 + 71/2*e^3 + 19/2*e^2 - 153/2*e - 21, 3*e^5 - 5/2*e^4 - 69/2*e^3 + 47/2*e^2 + 131/2*e - 31, 2*e^5 - 3*e^4 - 21*e^3 + 28*e^2 + 25*e - 23, -3/2*e^5 + 3*e^4 + 19*e^3 - 31*e^2 - 85/2*e + 33, 2*e^5 - e^4 - 22*e^3 + 10*e^2 + 28*e - 11, -3/2*e^5 + e^4 + 15*e^3 - 8*e^2 - 33/2*e - 2, 1/2*e^5 - 3/2*e^4 - 6*e^3 + 14*e^2 + 39/2*e - 37/2, 1/2*e^5 - 5*e^3 + 2*e^2 + 19/2*e - 11, -3/2*e^5 - 3/2*e^4 + 20*e^3 + 14*e^2 - 101/2*e - 25/2, 9/2*e^5 - 5/2*e^4 - 52*e^3 + 22*e^2 + 191/2*e - 39/2, 9/2*e^5 - 5/2*e^4 - 103/2*e^3 + 49/2*e^2 + 91*e - 30, -3/2*e^4 - 1/2*e^3 + 35/2*e^2 + 13/2*e - 22, -1/2*e^5 + 3/2*e^4 + 4*e^3 - 14*e^2 + 9/2*e + 17/2, -4*e^5 + e^4 + 46*e^3 - 4*e^2 - 80*e - 23, -3/2*e^5 + 2*e^4 + 16*e^3 - 21*e^2 - 35/2*e + 34, 5/2*e^5 - 2*e^4 - 29*e^3 + 20*e^2 + 103/2*e - 1, -1/2*e^5 + 7/2*e^3 + 9/2*e^2 + 12*e - 31/2, -4*e^5 + 3*e^4 + 91/2*e^3 - 59/2*e^2 - 183/2*e + 65/2, -e^3 - 4*e^2 + 7*e + 10, -4*e^5 + 49*e^3 - 109*e + 2, 7/2*e^5 - 2*e^4 - 81/2*e^3 + 33/2*e^2 + 79*e - 25/2, -5/2*e^5 + 1/2*e^4 + 31*e^3 - e^2 - 137/2*e - 27/2, -4*e^5 + 2*e^4 + 95/2*e^3 - 35/2*e^2 - 195/2*e + 35/2, -e^5 - 1/2*e^4 + 25/2*e^3 + 17/2*e^2 - 59/2*e - 14, -3*e^5 + 1/2*e^4 + 36*e^3 - e^2 - 76*e - 41/2, -3/2*e^5 - e^4 + 18*e^3 + 11*e^2 - 83/2*e - 13, -7/2*e^5 + 5/2*e^4 + 41*e^3 - 24*e^2 - 155/2*e + 51/2, -e^5 + 12*e^3 + 3*e^2 - 31*e - 11, -1/2*e^5 - 1/2*e^4 + 6*e^3 + 5*e^2 - 15/2*e + 11/2, 5*e^5 - 58*e^3 - 2*e^2 + 113*e + 22, 1/2*e^5 - 3/2*e^4 - 8*e^3 + 15*e^2 + 51/2*e - 47/2, -7/2*e^5 + 1/2*e^4 + 42*e^3 - 3*e^2 - 181/2*e + 13/2, -3*e^5 + 4*e^4 + 32*e^3 - 38*e^2 - 51*e + 30, -1/2*e^5 + 2*e^4 + 3*e^3 - 22*e^2 + 33/2*e + 25, -5/2*e^5 + 5/2*e^4 + 57/2*e^3 - 45/2*e^2 - 52*e + 22, -e^5 + e^4 + 8*e^3 - 8*e^2 + 9*e + 13, 9/2*e^5 - 5/2*e^4 - 50*e^3 + 18*e^2 + 167/2*e + 5/2, -1/2*e^5 - e^4 + 13/2*e^3 + 25/2*e^2 - 14*e - 87/2, -5/2*e^5 - 5/2*e^4 + 29*e^3 + 23*e^2 - 113/2*e - 53/2, 5/2*e^5 + 1/2*e^4 - 29*e^3 - 9*e^2 + 117/2*e - 3/2, -3/2*e^5 - 1/2*e^4 + 15*e^3 + 11*e^2 - 27/2*e - 77/2, 5/2*e^5 + 1/2*e^4 - 65/2*e^3 - 13/2*e^2 + 86*e + 10, -3*e^5 + 38*e^3 + 2*e^2 - 83*e - 2, 5/2*e^5 - 5/2*e^4 - 59/2*e^3 + 55/2*e^2 + 58*e - 12, 7/2*e^5 - 7/2*e^4 - 39*e^3 + 34*e^2 + 127/2*e - 53/2, 3/2*e^5 - 16*e^3 - 5*e^2 + 55/2*e + 36, 7/2*e^5 - 3/2*e^4 - 44*e^3 + 13*e^2 + 209/2*e - 39/2, -3/2*e^4 + 2*e^3 + 15*e^2 - 27*e - 33/2, 7/2*e^5 - e^4 - 81/2*e^3 + 11/2*e^2 + 81*e + 31/2, -3/2*e^5 - 1/2*e^4 + 20*e^3 + 5*e^2 - 113/2*e - 1/2, 6*e^5 - 7/2*e^4 - 137/2*e^3 + 71/2*e^2 + 237/2*e - 24, e^4 - 3*e^3 - 7*e^2 + 25*e - 4, 3*e^5 - 2*e^4 - 35*e^3 + 12*e^2 + 68*e + 22, 13/2*e^5 + 1/2*e^4 - 77*e^3 - 7*e^2 + 313/2*e + 65/2, -e^5 - e^4 + 15*e^3 + 12*e^2 - 56*e - 17, 5/2*e^5 + 2*e^4 - 29*e^3 - 22*e^2 + 119/2*e + 23, 3/2*e^5 + 1/2*e^4 - 17*e^3 - 6*e^2 + 55/2*e - 13/2, -8*e^5 + 2*e^4 + 183/2*e^3 - 33/2*e^2 - 343/2*e - 11/2, 13/2*e^5 - 2*e^4 - 149/2*e^3 + 39/2*e^2 + 138*e - 7/2, 1/2*e^5 - 3*e^3 - 2*e^2 - 25/2*e + 11, e^5 - 3/2*e^4 - 21/2*e^3 + 17/2*e^2 + 21/2*e + 12, e^5 - 23/2*e^3 - 1/2*e^2 + 69/2*e + 25/2, 1/2*e^5 - 3/2*e^4 - 6*e^3 + 8*e^2 + 31/2*e + 39/2, 17/2*e^5 - 3*e^4 - 199/2*e^3 + 59/2*e^2 + 201*e - 73/2, 7/2*e^5 + 1/2*e^4 - 85/2*e^3 - 15/2*e^2 + 94*e + 24, -11/2*e^5 + 7/2*e^4 + 62*e^3 - 34*e^2 - 233/2*e + 61/2, 3/2*e^5 - 2*e^4 - 27/2*e^3 + 29/2*e^2 + 6*e + 19/2, e^5 - e^4 - 9*e^3 + 4*e^2 + 3, -5/2*e^5 + 53/2*e^3 + 7/2*e^2 - 43*e - 73/2, -4*e^5 + 3/2*e^4 + 43*e^3 - 14*e^2 - 62*e - 1/2, 5/2*e^5 - 1/2*e^4 - 29*e^3 + 7*e^2 + 125/2*e - 37/2, -5/2*e^5 + 5/2*e^4 + 29*e^3 - 22*e^2 - 125/2*e + 31/2, 4*e^5 - 49*e^3 - 7*e^2 + 107*e + 25, -5/2*e^5 + 3*e^4 + 27*e^3 - 24*e^2 - 91/2*e - 14, -7*e^5 + 3*e^4 + 79*e^3 - 34*e^2 - 140*e + 49, -6*e^5 + e^4 + 135/2*e^3 - 3/2*e^2 - 247/2*e - 75/2, 3/2*e^5 - 1/2*e^4 - 43/2*e^3 + 19/2*e^2 + 62*e - 27, -5/2*e^5 + 9/2*e^4 + 29*e^3 - 47*e^2 - 89/2*e + 113/2, -2*e^5 + 21*e^3 - e^2 - 19*e - 7, -5/2*e^5 + 9/2*e^4 + 27*e^3 - 41*e^2 - 97/2*e + 49/2, -13/2*e^5 + 7/2*e^4 + 78*e^3 - 33*e^2 - 315/2*e + 67/2, 3*e^5 - 2*e^4 - 73/2*e^3 + 41/2*e^2 + 147/2*e - 77/2, -2*e^5 - e^4 + 43/2*e^3 + 17/2*e^2 - 59/2*e - 35/2, e^3 - e^2 - 11*e + 7, -e^5 + 3*e^4 + 7*e^3 - 31*e^2 + 6*e + 40, 6*e^5 - 139/2*e^3 - 13/2*e^2 + 275/2*e + 73/2, -2*e^5 + 39/2*e^3 - 5/2*e^2 - 27/2*e + 5/2, -13/2*e^5 + 4*e^4 + 73*e^3 - 34*e^2 - 243/2*e + 3, -2*e^5 + e^4 + 19*e^3 - 11*e^2 - 17*e + 18, -3*e^5 + 2*e^4 + 31*e^3 - 23*e^2 - 40*e + 41, 11*e^5 - 7*e^4 - 124*e^3 + 69*e^2 + 209*e - 54, 3/2*e^5 - 2*e^4 - 19*e^3 + 23*e^2 + 85/2*e - 38, -5/2*e^5 + 3/2*e^4 + 27*e^3 - 11*e^2 - 77/2*e - 65/2, 3/2*e^5 + 1/2*e^4 - 19*e^3 - 9*e^2 + 115/2*e + 37/2, e^5 - e^4 - 23/2*e^3 + 37/2*e^2 + 37/2*e - 75/2, -5*e^5 + 2*e^4 + 60*e^3 - 18*e^2 - 135*e + 18, -3/2*e^5 - e^4 + 31/2*e^3 + 21/2*e^2 - 17*e - 53/2, 9/2*e^5 - 3/2*e^4 - 101/2*e^3 + 29/2*e^2 + 92*e - 3, 6*e^5 - 3*e^4 - 71*e^3 + 33*e^2 + 147*e - 56, e^5 + e^4 - 12*e^3 - 14*e^2 + 25*e + 57, 11/2*e^5 - 3*e^4 - 66*e^3 + 30*e^2 + 287/2*e - 58, 11/2*e^5 - 3*e^4 - 61*e^3 + 26*e^2 + 205/2*e - 18, -5*e^5 + 5/2*e^4 + 55*e^3 - 26*e^2 - 77*e + 5/2, 15/2*e^5 - 5*e^4 - 169/2*e^3 + 87/2*e^2 + 151*e - 73/2, 2*e^5 - 3/2*e^4 - 24*e^3 + 16*e^2 + 55*e - 23/2, 2*e^5 - 3*e^4 - 21*e^3 + 30*e^2 + 23*e - 15];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w + 1}>] := 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;