/*
  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![-3, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w^2 - w + 4}>;

primesArray := [
[3, 3, w],
[3, 3, w + 1],
[3, 3, w + 2],
[8, 2, 2],
[11, 11, -w^2 + 5],
[13, 13, w^2 - w - 7],
[17, 17, -w^2 + w + 4],
[19, 19, -w^2 - w + 4],
[29, 29, 2*w^2 - 2*w - 11],
[31, 31, w^2 - 2*w - 4],
[37, 37, -2*w^2 + 3*w + 7],
[41, 41, w^2 - 2],
[59, 59, 2*w^2 - 13],
[61, 61, w^2 + w - 10],
[67, 67, 2*w^2 - w - 11],
[73, 73, -w^2 - 1],
[83, 83, w^2 - w - 10],
[89, 89, -w^2 + 4*w - 2],
[97, 97, w^2 - 2*w - 7],
[97, 97, -w^2 - 4*w - 5],
[97, 97, 2*w^2 + w - 8],
[101, 101, -4*w^2 + 2*w + 25],
[103, 103, w^2 + w - 7],
[107, 107, -3*w - 4],
[109, 109, 5*w^2 - 3*w - 31],
[121, 11, 2*w^2 - 3*w - 4],
[125, 5, -5],
[137, 137, -5*w^2 + 3*w + 34],
[139, 139, w^2 + 2*w - 4],
[151, 151, 3*w^2 - 23],
[151, 151, w^2 - 3*w - 5],
[151, 151, 3*w^2 - 3*w - 17],
[157, 157, 4*w^2 - 9*w - 8],
[157, 157, w^2 + 3*w - 2],
[157, 157, 2*w^2 - w - 2],
[167, 167, 2*w^2 + w - 11],
[167, 167, -w^2 + 11],
[167, 167, -3*w^2 + 19],
[169, 13, w^2 - 4*w - 4],
[173, 173, 2*w^2 - 3*w - 10],
[179, 179, -3*w + 7],
[191, 191, -w^2 + 5*w - 5],
[193, 193, -w^2 + w - 2],
[199, 199, 3*w - 2],
[211, 211, w^2 - 3*w - 11],
[223, 223, 3*w^2 - 20],
[223, 223, w^2 - 5*w - 4],
[223, 223, 2*w^2 - w - 8],
[229, 229, 4*w^2 - 3*w - 26],
[229, 229, 2*w^2 + w - 20],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, -6*w^2 + 3*w + 38],
[241, 241, -w^2 + 4*w - 5],
[257, 257, -2*w^2 + 5*w - 1],
[263, 263, -2*w^2 - w + 17],
[269, 269, -w^2 - w + 13],
[269, 269, 3*w - 4],
[269, 269, -4*w^2 + w + 31],
[271, 271, 2*w^2 - 7],
[271, 271, 3*w - 5],
[271, 271, w^2 - 3*w - 8],
[289, 17, 2*w^2 + w - 5],
[307, 307, 2*w^2 - w - 5],
[311, 311, 3*w^2 - 3*w - 19],
[311, 311, -7*w^2 + 13*w + 19],
[311, 311, w^2 - 2*w - 13],
[313, 313, w^2 + 2*w - 7],
[317, 317, -5*w^2 + w + 35],
[331, 331, -w^2 + 3*w - 4],
[337, 337, 3*w^2 - 3*w - 13],
[343, 7, -7],
[347, 347, 2*w^2 - 3*w - 13],
[349, 349, 3*w^2 - 3*w - 20],
[359, 359, -w^2 + 5*w - 2],
[361, 19, -2*w^2 + 7*w - 1],
[389, 389, -2*w^2 + w - 1],
[397, 397, w^2 - 6*w + 10],
[401, 401, 6*w^2 - 3*w - 44],
[419, 419, -3*w^2 + 6*w + 10],
[419, 419, 4*w^2 - 2*w - 31],
[419, 419, -4*w^2 + 4*w + 19],
[421, 421, 6*w^2 - 12*w - 13],
[431, 431, 4*w^2 - 7*w - 10],
[433, 433, w^2 - 4*w - 7],
[443, 443, 2*w^2 + 4*w - 5],
[443, 443, 5*w^2 - 11*w - 11],
[443, 443, 3*w^2 - 14],
[449, 449, -2*w^2 + 2*w - 1],
[457, 457, w^2 - 6*w - 5],
[457, 457, w^2 + 3*w - 5],
[457, 457, 6*w^2 - 3*w - 37],
[463, 463, w^2 + 5*w - 1],
[467, 467, w^2 - w - 13],
[479, 479, 5*w^2 - 9*w - 16],
[487, 487, -6*w - 5],
[499, 499, 3*w^2 - 3*w - 4],
[499, 499, 3*w^2 + 3*w - 1],
[499, 499, w^2 - 5*w + 8],
[503, 503, 4*w^2 - 5*w - 16],
[509, 509, 2*w^2 - 5*w - 8],
[521, 521, 4*w^2 - w - 22],
[523, 523, w^2 - 5*w - 16],
[547, 547, 5*w^2 - 2*w - 38],
[547, 547, 8*w^2 - 3*w - 58],
[547, 547, 4*w^2 - 29],
[557, 557, 7*w^2 - 4*w - 43],
[557, 557, 3*w - 11],
[557, 557, 2*w^2 - 4*w - 11],
[569, 569, 5*w^2 - 4*w - 32],
[569, 569, 3*w^2 - 3*w - 11],
[569, 569, w^2 + 2*w - 16],
[571, 571, w^2 - 4*w - 10],
[593, 593, 2*w^2 - 6*w - 7],
[601, 601, -w^2 - 3*w + 14],
[607, 607, 3*w^2 + 3*w - 7],
[619, 619, 3*w^2 - 6*w - 11],
[631, 631, -w^2 - w - 5],
[641, 641, 3*w^2 - 3*w - 5],
[647, 647, -w^2 + 6*w - 1],
[653, 653, -5*w^2 + 4*w + 35],
[661, 661, 3*w^2 - 3*w - 10],
[673, 673, w^2 + 4*w - 4],
[677, 677, 5*w^2 - 6*w - 25],
[677, 677, w^2 - 14],
[677, 677, 4*w^2 - 3*w - 20],
[691, 691, 5*w^2 - 2*w - 29],
[701, 701, 5*w^2 - 10*w - 14],
[719, 719, -3*w - 11],
[727, 727, -2*w^2 + 9*w - 8],
[733, 733, -w^2 - 4*w - 8],
[739, 739, w^2 + 3*w - 8],
[743, 743, 2*w^2 - 4*w - 17],
[757, 757, -w^2 + w - 5],
[769, 769, 3*w^2 - 3*w - 7],
[769, 769, -w^2 - 3*w - 7],
[769, 769, 5*w^2 - 3*w - 28],
[773, 773, 7*w^2 - 6*w - 38],
[787, 787, 2*w^2 - w - 20],
[797, 797, -5*w^2 + 14*w - 1],
[821, 821, -w^2 + 6*w - 4],
[823, 823, 7*w^2 - 6*w - 41],
[839, 839, 4*w^2 - 6*w - 11],
[841, 29, 5*w^2 - 4*w - 26],
[857, 857, 3*w^2 - 11],
[863, 863, 2*w^2 + 2*w - 17],
[881, 881, 5*w^2 - 5*w - 29],
[883, 883, -4*w^2 + 8*w + 13],
[907, 907, 4*w^2 - 9*w - 11],
[911, 911, 2*w^2 + 3*w - 10],
[947, 947, 2*w^2 + 5*w - 5],
[953, 953, 2*w^2 - 5*w - 11],
[961, 31, w^2 - 5*w - 13],
[967, 967, 2*w^2 - 3*w - 22],
[967, 967, 5*w^2 - 9*w - 10],
[967, 967, 3*w^2 - 10],
[977, 977, 5*w^2 - 37],
[983, 983, 7*w^2 - 5*w - 40],
[997, 997, -7*w^2 + 5*w + 49]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 24*x^8 + 202*x^6 - 704*x^4 + 864*x^2 - 128;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/32*e^8 - 5/8*e^6 + 61/16*e^4 - 27/4*e^2, e, 1/64*e^9 - 5/16*e^7 + 61/32*e^5 - 27/8*e^3 - e, -1/8*e^8 + 9/4*e^6 - 47/4*e^4 + 33/2*e^2 + 2, -1/64*e^9 + 5/16*e^7 - 69/32*e^5 + 51/8*e^3 - 13/2*e, 1/8*e^7 - 9/4*e^5 + 45/4*e^3 - 27/2*e, 1/16*e^9 - 9/8*e^7 + 45/8*e^5 - 21/4*e^3 - 9*e, 1, -3/32*e^9 + 15/8*e^7 - 195/16*e^5 + 121/4*e^3 - 51/2*e, 1/16*e^7 - 5/4*e^5 + 61/8*e^3 - 23/2*e, -3/32*e^9 + 2*e^7 - 223/16*e^5 + 71/2*e^3 - 23*e, 5/64*e^9 - 23/16*e^7 + 249/32*e^5 - 85/8*e^3 - 19/2*e, 1/4*e^6 - 4*e^4 + 33/2*e^2 - 6, -2*e^2 + 6, 1/4*e^7 - 9/2*e^5 + 45/2*e^3 - 28*e, -5/16*e^8 + 23/4*e^6 - 249/8*e^4 + 101/2*e^2 - 18, -5/16*e^8 + 6*e^6 - 281/8*e^4 + 66*e^2 - 14, -3/16*e^8 + 7/2*e^6 - 155/8*e^4 + 32*e^2 - 7, 1/8*e^7 - 3/2*e^5 + 5/4*e^3 + 14*e, -3/16*e^8 + 13/4*e^6 - 127/8*e^4 + 45/2*e^2 - 16, 1/64*e^9 - 13/16*e^7 + 333/32*e^5 - 347/8*e^3 + 45*e, 1/64*e^9 - 5/16*e^7 + 85/32*e^5 - 107/8*e^3 + 49/2*e, 5/32*e^8 - 3*e^6 + 289/16*e^4 - 75/2*e^2 + 20, 1/8*e^9 - 41/16*e^7 + 17*e^5 - 325/8*e^3 + 41/2*e, 1/32*e^9 - 3/4*e^7 + 93/16*e^5 - 16*e^3 + 12*e, -1/16*e^9 + 9/8*e^7 - 45/8*e^5 + 17/4*e^3 + 16*e, 1/4*e^8 - 35/8*e^6 + 45/2*e^4 - 151/4*e^2 + 26, 3/16*e^9 - 15/4*e^7 + 195/8*e^5 - 121/2*e^3 + 51*e, 9/32*e^8 - 21/4*e^6 + 469/16*e^4 - 49*e^2 + 12, 3/8*e^8 - 57/8*e^6 + 167/4*e^4 - 317/4*e^2 + 26, -3/32*e^9 + 15/8*e^7 - 191/16*e^5 + 117/4*e^3 - 32*e, -1/16*e^9 + 9/8*e^7 - 47/8*e^5 + 37/4*e^3 - 7/2*e, -9/64*e^9 + 39/16*e^7 - 373/32*e^5 + 109/8*e^3 - 3*e, -1/16*e^8 + 5/4*e^6 - 69/8*e^4 + 51/2*e^2 - 12, -3/32*e^9 + 7/4*e^7 - 151/16*e^5 + 11*e^3 + 18*e, e^4 - 10*e^2 + 6, -1/16*e^8 + 5/4*e^6 - 61/8*e^4 + 27/2*e^2, -9/16*e^8 + 43/4*e^6 - 501/8*e^4 + 231/2*e^2 - 32, 1/4*e^7 - 5*e^5 + 57/2*e^3 - 42*e, 11/32*e^9 - 7*e^7 + 739/16*e^5 - 225/2*e^3 + 159/2*e, -3/32*e^8 + 2*e^6 - 231/16*e^4 + 79/2*e^2 - 14, 1/4*e^8 - 19/4*e^6 + 53/2*e^4 - 83/2*e^2 + 4, -1/2*e^8 + 37/4*e^6 - 50*e^4 + 151/2*e^2, -7/32*e^8 + 31/8*e^6 - 315/16*e^4 + 101/4*e^2 + 22, 3/32*e^9 - 37/16*e^7 + 315/16*e^5 - 525/8*e^3 + 113/2*e, 1/16*e^8 - 5/4*e^6 + 61/8*e^4 - 27/2*e^2 - 4, -1/16*e^9 + 3/2*e^7 - 97/8*e^5 + 35*e^3 - 20*e, -1/32*e^9 + 1/4*e^7 + 43/16*e^5 - 25*e^3 + 44*e, 1/16*e^9 - 3/8*e^7 - 55/8*e^5 + 201/4*e^3 - 56*e, -1/4*e^8 + 19/4*e^6 - 53/2*e^4 + 89/2*e^2 - 26, 3/32*e^9 - 19/8*e^7 + 335/16*e^5 - 297/4*e^3 + 80*e, -13/64*e^9 + 59/16*e^7 - 601/32*e^5 + 153/8*e^3 + 31*e, 15/32*e^8 - 17/2*e^6 + 707/16*e^4 - 117/2*e^2 - 10, 1/16*e^8 - 5/8*e^6 - 11/8*e^4 + 67/4*e^2 - 6, 5/16*e^8 - 47/8*e^6 + 265/8*e^4 - 239/4*e^2 + 20, 1/4*e^6 - 9/2*e^4 + 49/2*e^2 - 27, 3/4*e^8 - 55/4*e^6 + 149/2*e^4 - 243/2*e^2 + 34, 1/4*e^8 - 5*e^6 + 31*e^4 - 62*e^2 + 29, -1/16*e^9 + 29/16*e^7 - 139/8*e^5 + 481/8*e^3 - 109/2*e, -1/8*e^8 + 2*e^6 - 33/4*e^4 + 3*e^2 + 26, -1/8*e^9 + 5/2*e^7 - 59/4*e^5 + 21*e^3 + 19*e, 9/32*e^9 - 43/8*e^7 + 509/16*e^5 - 261/4*e^3 + 38*e, -1/8*e^9 + 19/8*e^7 - 59/4*e^5 + 159/4*e^3 - 57*e, -1/16*e^9 + 13/8*e^7 - 113/8*e^5 + 177/4*e^3 - 36*e, 1/4*e^9 - 19/4*e^7 + 28*e^5 - 115/2*e^3 + 37*e, 1/4*e^8 - 5*e^6 + 63/2*e^4 - 66*e^2 + 18, 5/16*e^8 - 21/4*e^6 + 185/8*e^4 - 35/2*e^2 - 12, 1/4*e^8 - 5*e^6 + 31*e^4 - 63*e^2 + 27, 5/8*e^8 - 12*e^6 + 281/4*e^4 - 128*e^2 + 24, -1/4*e^9 + 19/4*e^7 - 55/2*e^5 + 103/2*e^3 - 22*e, -3/16*e^8 + 15/4*e^6 - 195/8*e^4 + 115/2*e^2 - 15, 7/32*e^9 - 19/4*e^7 + 555/16*e^5 - 99*e^3 + 89*e, -1/4*e^9 + 19/4*e^7 - 28*e^5 + 119/2*e^3 - 52*e, 1/8*e^8 - 15/8*e^6 + 21/4*e^4 + 53/4*e^2 - 14, -29/32*e^8 + 67/4*e^6 - 1465/16*e^4 + 149*e^2 - 30, 11/16*e^8 - 51/4*e^6 + 559/8*e^4 - 219/2*e^2 + 2, -11/16*e^8 + 13*e^6 - 599/8*e^4 + 141*e^2 - 56, 7/32*e^8 - 5*e^6 + 587/16*e^4 - 175/2*e^2 + 28, -3/64*e^9 + 9/16*e^7 + 41/32*e^5 - 221/8*e^3 + 63*e, -3/4*e^6 + 12*e^4 - 99/2*e^2 + 24, -1/32*e^9 + 3/2*e^7 - 301/16*e^5 + 155/2*e^3 - 74*e, 3/16*e^9 - 13/4*e^7 + 123/8*e^5 - 29/2*e^3 - 6*e, -1/32*e^9 + 1/2*e^7 - 21/16*e^5 - 17/2*e^3 + 19*e, -1/16*e^9 + 23/16*e^7 - 99/8*e^5 + 387/8*e^3 - 141/2*e, 1/16*e^8 - 5/4*e^6 + 53/8*e^4 - 9/2*e^2 - 2, 3/8*e^9 - 13/2*e^7 + 125/4*e^5 - 35*e^3 - e, -1/64*e^9 + 9/16*e^7 - 189/32*e^5 + 167/8*e^3 - 24*e, 1/4*e^8 - 17/4*e^6 + 21*e^4 - 69/2*e^2 + 27, 13/32*e^9 - 17/2*e^7 + 929/16*e^5 - 291/2*e^3 + 99*e, 13/32*e^8 - 53/8*e^6 + 425/16*e^4 - 27/4*e^2 - 24, -1/32*e^9 + 3/4*e^7 - 85/16*e^5 + 8*e^3 + 20*e, 3/16*e^8 - 5/2*e^6 + 23/8*e^4 + 39*e^2 - 40, -5/16*e^8 + 13/2*e^6 - 345/8*e^4 + 97*e^2 - 38, -1/8*e^9 + 21/8*e^7 - 79/4*e^5 + 269/4*e^3 - 88*e, 11/64*e^9 - 63/16*e^7 + 983/32*e^5 - 729/8*e^3 + 141/2*e, 23/64*e^9 - 113/16*e^7 + 1403/32*e^5 - 779/8*e^3 + 70*e, -1/8*e^9 + 9/4*e^7 - 45/4*e^5 + 25/2*e^3 + 10*e, -3/16*e^8 + 4*e^6 - 207/8*e^4 + 44*e^2 + 14, -19/64*e^9 + 93/16*e^7 - 1191/32*e^5 + 783/8*e^3 - 114*e, -3/8*e^9 + 59/8*e^7 - 185/4*e^5 + 423/4*e^3 - 70*e, 7/16*e^9 - 71/8*e^7 + 463/8*e^5 - 543/4*e^3 + 78*e, 9/64*e^9 - 59/16*e^7 + 1053/32*e^5 - 881/8*e^3 + 199/2*e, 3/4*e^8 - 55/4*e^6 + 149/2*e^4 - 239/2*e^2 + 38, 3/16*e^8 - 17/4*e^6 + 243/8*e^4 - 135/2*e^2 + 17, -7/32*e^8 + 35/8*e^6 - 443/16*e^4 + 217/4*e^2 - 6, -17/64*e^9 + 85/16*e^7 - 1085/32*e^5 + 627/8*e^3 - 59*e, -13/16*e^8 + 31/2*e^6 - 713/8*e^4 + 158*e^2 - 42, -1/4*e^9 + 11/2*e^7 - 40*e^5 + 107*e^3 - 78*e, -3/16*e^9 + 31/8*e^7 - 207/8*e^5 + 247/4*e^3 - 44*e, 3/64*e^9 - 31/16*e^7 + 775/32*e^5 - 873/8*e^3 + 129*e, -9/16*e^8 + 10*e^6 - 401/8*e^4 + 65*e^2 - 17, -3/32*e^9 + e^7 + 65/16*e^5 - 119/2*e^3 + 112*e, -11/32*e^9 + 51/8*e^7 - 567/16*e^5 + 253/4*e^3 - 34*e, 3/8*e^8 - 27/4*e^6 + 147/4*e^4 - 141/2*e^2 + 36, 3/32*e^9 - 5/4*e^7 + 7/16*e^5 + 34*e^3 - 65*e, 7/32*e^9 - 61/16*e^7 + 287/16*e^5 - 93/8*e^3 - 79/2*e, -9/16*e^8 + 87/8*e^6 - 509/8*e^4 + 455/4*e^2 - 38, -3/32*e^9 + 2*e^7 - 231/16*e^5 + 79/2*e^3 - 28*e, -3/16*e^8 + 4*e^6 - 207/8*e^4 + 49*e^2 - 6, -1/8*e^9 + 2*e^7 - 6*e^5 - 24*e^3 + 185/2*e, -3/32*e^9 + 17/8*e^7 - 255/16*e^5 + 195/4*e^3 - 68*e, -3/16*e^8 + 5*e^6 - 343/8*e^4 + 121*e^2 - 62, -7/64*e^9 + 25/16*e^7 - 59/32*e^5 - 301/8*e^3 + 92*e, 23/32*e^8 - 101/8*e^6 + 1003/16*e^4 - 343/4*e^2 + 48, -9/32*e^9 + 6*e^7 - 661/16*e^5 + 193/2*e^3 - 39*e, -11/32*e^9 + 61/8*e^7 - 903/16*e^5 + 639/4*e^3 - 137*e, 19/64*e^9 - 83/16*e^7 + 783/32*e^5 - 149/8*e^3 - 77/2*e, 7/8*e^8 - 33/2*e^6 + 383/4*e^4 - 184*e^2 + 62, -1/16*e^8 + 1/2*e^6 + 11/8*e^4 - 6*e^2 - 10, -1/16*e^8 + 1/4*e^6 + 55/8*e^4 - 73/2*e^2 + 23, 7/8*e^8 - 65/4*e^6 + 367/4*e^4 - 333/2*e^2 + 44, -3/16*e^9 + 27/8*e^7 - 151/8*e^5 + 183/4*e^3 - 77*e, -3/16*e^8 + 15/4*e^6 - 179/8*e^4 + 73/2*e^2 + 25, 5/32*e^9 - 19/8*e^7 + 113/16*e^5 + 79/4*e^3 - 73*e, -3/16*e^8 + 25/8*e^6 - 127/8*e^4 + 113/4*e^2 - 2, 3/16*e^9 - 29/8*e^7 + 175/8*e^5 - 197/4*e^3 + 58*e, -7/16*e^9 + 19/2*e^7 - 547/8*e^5 + 186*e^3 - 144*e, -1/4*e^8 + 31/8*e^6 - 25/2*e^4 - 53/4*e^2 + 26, -1/16*e^8 + 2*e^6 - 157/8*e^4 + 63*e^2 - 30, 3/4*e^6 - 10*e^4 + 55/2*e^2 + 2, 7/32*e^9 - 69/16*e^7 + 439/16*e^5 - 533/8*e^3 + 91/2*e, -31/64*e^9 + 155/16*e^7 - 1979/32*e^5 + 1165/8*e^3 - 229/2*e, 5/32*e^9 - 2*e^7 - 7/16*e^5 + 121/2*e^3 - 114*e, 15/16*e^7 - 67/4*e^5 + 675/8*e^3 - 209/2*e, -7/16*e^8 + 35/4*e^6 - 435/8*e^4 + 225/2*e^2 - 36, 1/16*e^9 - 31/16*e^7 + 151/8*e^5 - 523/8*e^3 + 131/2*e, -5/64*e^9 + 27/16*e^7 - 393/32*e^5 + 281/8*e^3 - 57/2*e, -1/16*e^9 + 13/8*e^7 - 125/8*e^5 + 261/4*e^3 - 92*e, 3/4*e^8 - 107/8*e^6 + 135/2*e^4 - 343/4*e^2 + 8, 3/16*e^8 - 15/4*e^6 + 175/8*e^4 - 67/2*e^2 - 10, -5/32*e^9 + 21/8*e^7 - 185/16*e^5 + 15/4*e^3 + 36*e, -1/2*e^9 + 79/8*e^7 - 61*e^5 + 503/4*e^3 - 45*e, 1/32*e^8 + 9/8*e^6 - 339/16*e^4 + 295/4*e^2 - 14, 3/8*e^9 - 109/16*e^7 + 36*e^5 - 441/8*e^3 + 31/2*e, 5/16*e^9 - 25/4*e^7 + 327/8*e^5 - 199/2*e^3 + 119/2*e, -17/16*e^8 + 77/4*e^6 - 821/8*e^4 + 321/2*e^2 - 44, -11/32*e^9 + 59/8*e^7 - 839/16*e^5 + 549/4*e^3 - 89*e, 11/32*e^9 - 47/8*e^7 + 423/16*e^5 - 65/4*e^3 - 31*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^2 - w + 4}>] := -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;