/*
  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, -3, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^3 + w^2 + 3*w - 1}>;

primesArray := [
[3, 3, w],
[5, 5, w - 1],
[7, 7, w^3 - w^2 - 4*w + 1],
[11, 11, -w^2 + w + 2],
[13, 13, -w^3 + w^2 + 3*w - 1],
[16, 2, 2],
[17, 17, -w^3 + 5*w + 2],
[29, 29, -w^2 + w + 1],
[37, 37, -w^3 + 2*w^2 + 4*w - 4],
[41, 41, -w^3 + w^2 + 5*w + 1],
[41, 41, w^2 - 5],
[47, 47, w^3 - w^2 - 5*w - 2],
[47, 47, -w^3 + 2*w^2 + 4*w - 1],
[53, 53, w^3 - 4*w - 4],
[53, 53, 2*w^3 - 2*w^2 - 9*w + 1],
[61, 61, -w^3 + w^2 + 6*w - 2],
[71, 71, w^3 - w^2 - 6*w - 2],
[103, 103, 2*w^3 - 2*w^2 - 8*w + 1],
[103, 103, -3*w^3 + 3*w^2 + 13*w - 5],
[109, 109, 2*w^3 - w^2 - 8*w - 4],
[109, 109, 2*w^3 - w^2 - 9*w - 1],
[113, 113, -2*w^3 + w^2 + 11*w - 1],
[113, 113, 2*w^3 - w^2 - 10*w - 4],
[125, 5, 2*w^3 - 3*w^2 - 10*w + 4],
[131, 131, -2*w^3 + w^2 + 9*w - 1],
[139, 139, 2*w^3 - 4*w^2 - 7*w + 5],
[139, 139, 3*w^3 - w^2 - 16*w - 5],
[149, 149, w^2 + w - 4],
[149, 149, 3*w^3 - 3*w^2 - 13*w + 1],
[151, 151, w^2 - 2*w - 8],
[157, 157, w^3 - w^2 - 7*w - 2],
[157, 157, -2*w^2 + w + 7],
[167, 167, 2*w^3 - 2*w^2 - 7*w + 5],
[173, 173, -w^3 + 6*w - 2],
[173, 173, -w^3 + 2*w^2 + 5*w - 7],
[181, 181, -3*w^3 + 2*w^2 + 14*w + 1],
[191, 191, -w^3 + 2*w^2 + 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 5],
[193, 193, -w^3 + 3*w^2 + 4*w - 4],
[193, 193, 2*w^3 + 3*w^2 - 13*w - 20],
[197, 197, w^3 - 2*w - 2],
[199, 199, 2*w^3 - w^2 - 11*w - 2],
[211, 211, -2*w^3 + 9*w + 5],
[223, 223, -w^3 + w^2 + 4*w - 5],
[223, 223, w^2 - 3*w - 2],
[229, 229, 2*w^3 - 2*w^2 - 7*w + 1],
[241, 241, 3*w^2 - 2*w - 13],
[241, 241, -w^3 + w^2 + 5*w + 4],
[257, 257, -3*w^3 + w^2 + 15*w + 8],
[263, 263, -3*w^3 + 5*w^2 + 9*w - 7],
[263, 263, 2*w^3 - w^2 - 9*w + 2],
[263, 263, -w^3 + 4*w^2 + 2*w - 16],
[263, 263, w^3 - w^2 - 2*w - 2],
[271, 271, -3*w - 1],
[281, 281, 2*w^3 - 3*w^2 - 8*w + 2],
[281, 281, 3*w^3 - 4*w^2 - 12*w + 4],
[283, 283, w^3 - 5*w - 7],
[293, 293, -2*w^3 + w^2 + 10*w - 1],
[307, 307, w^3 - 3*w^2 - 3*w + 8],
[307, 307, 3*w^3 - 15*w - 10],
[311, 311, 2*w^3 - 3*w^2 - 7*w + 7],
[313, 313, -w^3 + 7*w + 2],
[337, 337, 3*w^3 - 17*w - 10],
[337, 337, 2*w^3 - 2*w^2 - 9*w - 5],
[343, 7, -2*w^3 + w^2 + 8*w + 2],
[347, 347, -3*w^3 + 17*w + 14],
[347, 347, -3*w^3 - w^2 + 17*w + 13],
[349, 349, 2*w^3 - w^2 - 8*w - 1],
[349, 349, -w^3 + 3*w^2 + 4*w - 8],
[353, 353, -w^3 - w^2 + 7*w + 4],
[359, 359, -2*w^3 + 9*w + 10],
[367, 367, 3*w^3 - 4*w^2 - 9*w + 4],
[367, 367, -w^3 + w^2 + 4*w + 4],
[367, 367, -w^3 + 3*w^2 + 3*w - 7],
[373, 373, -w^2 + w - 2],
[373, 373, -2*w^3 + 2*w^2 + 11*w - 2],
[379, 379, -2*w^3 + 4*w^2 + 7*w - 8],
[379, 379, -w^2 - 2],
[389, 389, -w^3 - 2*w^2 + 5*w + 11],
[397, 397, -2*w^3 + 4*w^2 + 7*w - 7],
[401, 401, w^3 - 3*w^2 - 2*w + 10],
[401, 401, -2*w^3 + 11*w + 4],
[401, 401, 3*w^3 - 2*w^2 - 14*w - 4],
[401, 401, -3*w^3 + 4*w^2 + 15*w - 5],
[421, 421, -w^3 + w^2 + 7*w - 1],
[421, 421, -2*w^3 + 2*w^2 + 9*w - 7],
[433, 433, w^2 - w - 8],
[439, 439, -w^3 + 3*w^2 + 4*w - 7],
[443, 443, -2*w^3 + w^2 + 10*w - 2],
[449, 449, w^2 + 2*w - 4],
[449, 449, 2*w^3 - 2*w^2 - 10*w - 1],
[461, 461, 2*w^3 - 2*w^2 - 6*w + 5],
[461, 461, -2*w^3 + 4*w^2 + 7*w - 13],
[463, 463, -3*w^3 + 3*w^2 + 12*w - 4],
[463, 463, w - 5],
[467, 467, -3*w^3 + w^2 + 16*w + 8],
[467, 467, 2*w^2 - 3*w - 10],
[479, 479, 2*w^2 - 4*w - 1],
[479, 479, -w^3 + 2*w^2 + 2*w - 7],
[487, 487, -3*w^3 + 5*w^2 + 12*w - 10],
[503, 503, w^3 + 2*w^2 - 8*w - 8],
[509, 509, -3*w^3 + 5*w^2 + 11*w - 5],
[521, 521, 2*w^2 - 3*w - 11],
[523, 523, -2*w^3 - w^2 + 13*w + 11],
[523, 523, -2*w^3 + 4*w^2 + 8*w - 7],
[541, 541, 3*w^3 - 2*w^2 - 13*w + 1],
[563, 563, w^3 - 3*w - 5],
[563, 563, -w^3 + 3*w^2 + 2*w - 11],
[569, 569, -2*w^3 + 4*w^2 + 9*w - 4],
[569, 569, 3*w^2 - w - 14],
[571, 571, -w^3 + 8*w - 1],
[577, 577, 3*w^3 - 2*w^2 - 16*w + 1],
[587, 587, -4*w^3 + 5*w^2 + 18*w - 7],
[587, 587, 2*w^3 + w^2 - 10*w - 7],
[599, 599, 2*w^3 + w^2 - 11*w - 8],
[607, 607, w^3 - 3*w + 4],
[613, 613, 3*w^3 - w^2 - 14*w - 5],
[613, 613, 3*w^2 - w - 11],
[619, 619, 2*w^2 - 5*w - 8],
[631, 631, -2*w^3 - 2*w^2 + 12*w + 19],
[647, 647, 2*w^3 - 2*w^2 - 11*w - 2],
[653, 653, 3*w^3 - 5*w^2 - 8*w + 7],
[677, 677, -2*w^3 + 4*w^2 + 5*w - 8],
[683, 683, 2*w^3 + w^2 - 12*w - 8],
[683, 683, -w^3 + 3*w^2 - 8],
[683, 683, -2*w^3 + 3*w^2 + 11*w - 8],
[683, 683, -2*w^3 + 3*w^2 + 10*w - 2],
[691, 691, w^2 - 4*w - 1],
[709, 709, 3*w^3 - 2*w^2 - 13*w - 2],
[709, 709, w^3 - 2*w^2 - 3*w - 2],
[727, 727, -w^3 + 4*w^2 + 2*w - 7],
[727, 727, -w^3 + 8*w + 5],
[733, 733, 3*w^2 - 2*w - 10],
[733, 733, -w^3 + 5*w + 8],
[757, 757, 3*w^3 - 14*w - 8],
[761, 761, -w^3 + 3*w^2 + 5*w - 10],
[773, 773, -2*w^3 + 3*w^2 + 9*w - 1],
[787, 787, -w^3 + 4*w - 4],
[797, 797, w^2 + 2*w - 5],
[797, 797, w^3 + w^2 - 5*w - 1],
[809, 809, w^3 + 3*w^2 - 9*w - 17],
[811, 811, -4*w^2 + 3*w + 17],
[811, 811, w^3 - 2*w^2 - 8*w + 2],
[823, 823, -4*w^3 + 4*w^2 + 17*w - 4],
[857, 857, 3*w^2 - w - 16],
[881, 881, -2*w^3 + 9*w + 1],
[883, 883, w^3 - w^2 - 4*w - 5],
[887, 887, 3*w^2 + w - 10],
[911, 911, w^3 - 3*w - 7],
[919, 919, -w^3 + w^2 + 3*w - 7],
[919, 919, 3*w^3 - 2*w^2 - 12*w + 2],
[929, 929, 3*w^3 - w^2 - 16*w - 2],
[929, 929, w^3 + w^2 - 6*w - 2],
[941, 941, -3*w^3 + 5*w^2 + 14*w - 14],
[941, 941, -w^3 + 2*w^2 + 6*w - 8],
[947, 947, 3*w^3 - 3*w^2 - 13*w - 5],
[947, 947, 3*w^3 - 4*w^2 - 12*w + 2],
[953, 953, -3*w^3 + 5*w^2 + 10*w - 10],
[961, 31, -w^3 + 4*w^2 + 3*w - 13],
[961, 31, -4*w^3 + 3*w^2 + 17*w - 4],
[967, 967, -w^3 - 4*w^2 + 9*w + 19],
[967, 967, w^3 + 5*w^2 - 9*w - 26],
[971, 971, -3*w^3 + 6*w^2 + 13*w - 7],
[971, 971, w^3 + 2*w^2 - 7*w - 7],
[977, 977, 2*w^3 - 3*w^2 - 4*w - 2],
[983, 983, 3*w^3 - 5*w^2 - 7*w + 7],
[991, 991, 2*w^2 - w - 13],
[991, 991, w^3 - 8*w - 4]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 15*x^4 - 2*x^3 + 45*x^2 + 38*x + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^5 + 15/2*e^3 - 45/2*e - 10, -1/2*e^5 + 15/2*e^3 - 45/2*e - 10, -3/2*e^5 + e^4 + 45/2*e^3 - 10*e^2 - 133/2*e - 26, -1, 2*e^5 - e^4 - 29*e^3 + 11*e^2 + 80*e + 33, -3/2*e^5 + e^4 + 43/2*e^3 - 12*e^2 - 113/2*e - 18, -e^2 - 2*e + 8, -1/2*e^5 + e^4 + 15/2*e^3 - 10*e^2 - 39/2*e - 4, -3/2*e^5 + e^4 + 41/2*e^3 - 12*e^2 - 95/2*e - 14, 3/2*e^5 - e^4 - 41/2*e^3 + 12*e^2 + 95/2*e + 14, 1/2*e^5 - 15/2*e^3 + 49/2*e + 18, 1/2*e^5 - e^4 - 15/2*e^3 + 12*e^2 + 37/2*e - 6, 2*e^3 + 2*e^2 - 19*e - 12, -e^3 + 7*e - 4, -e^5 + 14*e^3 - 38*e - 22, e^5 - 15*e^3 + 45*e + 20, e^5 - 16*e^3 - 2*e^2 + 54*e + 32, 2*e^5 - e^4 - 28*e^3 + 12*e^2 + 70*e + 24, -4*e^5 + 2*e^4 + 56*e^3 - 24*e^2 - 144*e - 50, 2*e^3 + 2*e^2 - 16*e - 14, -5*e^5 + 2*e^4 + 72*e^3 - 24*e^2 - 196*e - 80, -2*e - 8, -e^5 + 15*e^3 - 47*e - 28, -3/2*e^5 + e^4 + 45/2*e^3 - 12*e^2 - 127/2*e - 14, 2*e^5 - 2*e^4 - 30*e^3 + 20*e^2 + 84*e + 36, e^5 - 17*e^3 - 3*e^2 + 67*e + 40, -9/2*e^5 + 3*e^4 + 131/2*e^3 - 32*e^2 - 361/2*e - 66, -3*e^5 + 2*e^4 + 43*e^3 - 24*e^2 - 115*e - 44, e^5 - 13*e^3 + 25*e + 28, 7/2*e^5 - 3*e^4 - 105/2*e^3 + 32*e^2 + 307/2*e + 52, 11/2*e^5 - 3*e^4 - 165/2*e^3 + 32*e^2 + 487/2*e + 96, 2*e^5 - 30*e^3 + 94*e + 56, -e^5 + 13*e^3 - 29*e - 20, 11/2*e^5 - 3*e^4 - 161/2*e^3 + 32*e^2 + 451/2*e + 86, 9/2*e^5 - 3*e^4 - 127/2*e^3 + 36*e^2 + 313/2*e + 44, -1/2*e^5 + e^4 + 19/2*e^3 - 7*e^2 - 77/2*e - 26, -e^5 + 13*e^3 - 4*e^2 - 25*e - 4, 3/2*e^5 - e^4 - 37/2*e^3 + 14*e^2 + 63/2*e, -1/2*e^5 + e^4 + 11/2*e^3 - 12*e^2 - 1/2*e + 12, -6*e^5 + 4*e^4 + 88*e^3 - 43*e^2 - 244*e - 96, 3*e^5 - 43*e^3 + 123*e + 60, -9/2*e^5 + e^4 + 129/2*e^3 - 14*e^2 - 351/2*e - 66, 1/2*e^5 + e^4 - 19/2*e^3 - 14*e^2 + 81/2*e + 42, -3/2*e^5 + e^4 + 45/2*e^3 - 8*e^2 - 127/2*e - 30, 2*e^3 + 4*e^2 - 20*e - 22, 4*e^5 - 2*e^4 - 58*e^3 + 22*e^2 + 154*e + 74, 6*e^5 - 5*e^4 - 88*e^3 + 56*e^2 + 240*e + 78, 3*e^5 - e^4 - 45*e^3 + 8*e^2 + 137*e + 68, -9/2*e^5 + e^4 + 135/2*e^3 - 8*e^2 - 409/2*e - 98, 4*e^5 - 2*e^4 - 56*e^3 + 25*e^2 + 144*e + 48, 2*e^5 - 2*e^4 - 30*e^3 + 22*e^2 + 92*e + 16, 3/2*e^5 - e^4 - 45/2*e^3 + 12*e^2 + 141/2*e + 10, -6*e^5 + 3*e^4 + 88*e^3 - 36*e^2 - 250*e - 88, -2*e - 8, -e^5 + 2*e^4 + 17*e^3 - 18*e^2 - 55*e - 28, -9/2*e^5 + e^4 + 135/2*e^3 - 8*e^2 - 415/2*e - 98, -1/2*e^5 + e^4 + 15/2*e^3 - 14*e^2 - 49/2*e + 14, -7*e^5 + 4*e^4 + 105*e^3 - 39*e^2 - 313*e - 128, -e^5 + 2*e^4 + 12*e^3 - 28*e^2 - 8*e + 28, -3*e^5 + 45*e^3 + 2*e^2 - 131*e - 76, -13/2*e^5 + 5*e^4 + 195/2*e^3 - 50*e^2 - 573/2*e - 116, -9*e^5 + 4*e^4 + 132*e^3 - 44*e^2 - 380*e - 158, -5/2*e^5 + 3*e^4 + 71/2*e^3 - 32*e^2 - 169/2*e - 20, 5*e^5 - 2*e^4 - 75*e^3 + 20*e^2 + 225*e + 100, 2*e^5 - 2*e^4 - 26*e^3 + 24*e^2 + 50*e + 8, -4*e^5 + 2*e^4 + 56*e^3 - 23*e^2 - 140*e - 64, 4*e^5 - 2*e^4 - 62*e^3 + 18*e^2 + 192*e + 98, -e^5 + 15*e^3 + e^2 - 49*e - 22, -1/2*e^5 + e^4 + 15/2*e^3 - 14*e^2 - 35/2*e + 10, -2*e^5 + 2*e^4 + 28*e^3 - 24*e^2 - 68*e - 16, 7*e^5 - 4*e^4 - 101*e^3 + 44*e^2 + 275*e + 116, 1/2*e^5 - e^4 - 15/2*e^3 + 12*e^2 + 47/2*e - 2, e^5 - 14*e^3 + 2*e^2 + 36*e + 8, -2*e^2 + 4*e + 26, -7*e^5 + 4*e^4 + 107*e^3 - 40*e^2 - 331*e - 130, -6*e^5 + 4*e^4 + 88*e^3 - 44*e^2 - 242*e - 84, 3/2*e^5 - e^4 - 47/2*e^3 + 8*e^2 + 145/2*e + 46, e^5 - 17*e^3 - 4*e^2 + 67*e + 44, 2*e^5 - 2*e^4 - 28*e^3 + 22*e^2 + 72*e + 14, 11/2*e^5 - 3*e^4 - 165/2*e^3 + 30*e^2 + 507/2*e + 102, 6*e^5 - 2*e^4 - 88*e^3 + 24*e^2 + 250*e + 104, 2*e^5 - 32*e^3 - 3*e^2 + 116*e + 64, -e^5 + 2*e^4 + 15*e^3 - 24*e^2 - 42*e + 24, -5*e^3 - 6*e^2 + 47*e + 34, -e^5 - e^4 + 13*e^3 + 10*e^2 - 35*e - 18, 2*e^5 - 30*e^3 - 6*e^2 + 90*e + 70, 11/2*e^5 - e^4 - 161/2*e^3 + 10*e^2 + 463/2*e + 110, 5/2*e^5 - e^4 - 75/2*e^3 + 11*e^2 + 217/2*e + 50, -2*e^5 + 30*e^3 + 6*e^2 - 90*e - 72, -e^5 + 13*e^3 + 2*e^2 - 25*e - 36, -7*e^5 + 4*e^4 + 103*e^3 - 44*e^2 - 289*e - 100, 5*e^5 - 2*e^4 - 75*e^3 + 18*e^2 + 228*e + 112, -19/2*e^5 + 5*e^4 + 277/2*e^3 - 52*e^2 - 779/2*e - 166, 3*e^5 - 2*e^4 - 47*e^3 + 20*e^2 + 146*e + 64, -3*e^5 + 2*e^4 + 45*e^3 - 20*e^2 - 129*e - 36, 7*e^5 - 4*e^4 - 105*e^3 + 40*e^2 + 315*e + 140, -3*e^5 + 45*e^3 + 2*e^2 - 142*e - 88, 5/2*e^5 + e^4 - 71/2*e^3 - 12*e^2 + 205/2*e + 74, e^5 - 9*e^3 + 6*e^2 - 9*e - 20, 2*e^5 - 2*e^4 - 30*e^3 + 23*e^2 + 82*e + 24, 5*e^5 - 3*e^4 - 75*e^3 + 32*e^2 + 219*e + 76, -3/2*e^5 + e^4 + 45/2*e^3 - 10*e^2 - 123/2*e - 38, 6*e^5 - 2*e^4 - 90*e^3 + 22*e^2 + 274*e + 100, 4*e^5 - 4*e^4 - 60*e^3 + 44*e^2 + 178*e + 44, 6*e^5 - 4*e^4 - 90*e^3 + 42*e^2 + 261*e + 82, 2*e^5 - 30*e^3 + 90*e + 40, -8*e^5 + 4*e^4 + 118*e^3 - 45*e^2 - 340*e - 128, 13/2*e^5 - 3*e^4 - 183/2*e^3 + 30*e^2 + 469/2*e + 122, 5*e^5 - 3*e^4 - 75*e^3 + 30*e^2 + 215*e + 92, 9*e^5 - 6*e^4 - 135*e^3 + 64*e^2 + 397*e + 152, -3*e^5 + 45*e^3 - 135*e - 62, -15/2*e^5 + 3*e^4 + 213/2*e^3 - 38*e^2 - 567/2*e - 94, 9*e^5 - 4*e^4 - 129*e^3 + 44*e^2 + 349*e + 148, 8*e^5 - 3*e^4 - 114*e^3 + 34*e^2 + 310*e + 120, -3*e^5 + 2*e^4 + 43*e^3 - 22*e^2 - 117*e - 52, 15/2*e^5 - 3*e^4 - 217/2*e^3 + 36*e^2 + 595/2*e + 112, 1/2*e^5 - e^4 - 11/2*e^3 + 13*e^2 + 13/2*e, -3*e^5 + 2*e^4 + 43*e^3 - 24*e^2 - 111*e - 32, 9/2*e^5 - 3*e^4 - 133/2*e^3 + 32*e^2 + 371/2*e + 46, 3*e^5 - 4*e^4 - 46*e^3 + 40*e^2 + 126*e + 56, 25/2*e^5 - 9*e^4 - 369/2*e^3 + 98*e^2 + 1035/2*e + 198, -2*e^5 + 4*e^4 + 32*e^3 - 42*e^2 - 102*e + 8, 11/2*e^5 - e^4 - 153/2*e^3 + 18*e^2 + 391/2*e + 62, 17/2*e^5 - 3*e^4 - 247/2*e^3 + 36*e^2 + 685/2*e + 138, 23/2*e^5 - 7*e^4 - 341/2*e^3 + 72*e^2 + 987/2*e + 198, -6*e^5 + 2*e^4 + 88*e^3 - 22*e^2 - 244*e - 112, -e^5 + 13*e^3 - 2*e^2 - 23*e, 5/2*e^5 + e^4 - 75/2*e^3 - 10*e^2 + 233/2*e + 72, -6*e^5 + 4*e^4 + 90*e^3 - 42*e^2 - 266*e - 98, -4*e^5 + 2*e^4 + 62*e^3 - 21*e^2 - 206*e - 64, -5*e^5 + 4*e^4 + 73*e^3 - 44*e^2 - 189*e - 68, 11*e^5 - 6*e^4 - 161*e^3 + 66*e^2 + 449*e + 166, 2*e^4 + 4*e^3 - 16*e^2 - 34*e - 6, 5*e^5 - 2*e^4 - 75*e^3 + 18*e^2 + 222*e + 110, 3*e^5 - 41*e^3 - 2*e^2 + 101*e + 84, 13/2*e^5 - 2*e^4 - 203/2*e^3 + 16*e^2 + 665/2*e + 162, 4*e^5 - 2*e^4 - 56*e^3 + 24*e^2 + 142*e + 60, 29/2*e^5 - 9*e^4 - 427/2*e^3 + 98*e^2 + 1205/2*e + 250, -e^5 + 2*e^4 + 16*e^3 - 20*e^2 - 40*e - 32, 8*e^5 - 6*e^4 - 123*e^3 + 62*e^2 + 373*e + 148, 17/2*e^5 - 4*e^4 - 259/2*e^3 + 36*e^2 + 793/2*e + 182, 4*e^5 - 4*e^4 - 60*e^3 + 44*e^2 + 168*e + 36, 19/2*e^5 - 5*e^4 - 281/2*e^3 + 56*e^2 + 815/2*e + 158, 6*e^5 - 4*e^4 - 92*e^3 + 41*e^2 + 272*e + 112, 2*e^5 - 4*e^4 - 30*e^3 + 44*e^2 + 78*e - 8, 5/2*e^5 - e^4 - 75/2*e^3 + 4*e^2 + 237/2*e + 78, -3*e^5 + 3*e^4 + 41*e^3 - 40*e^2 - 87*e + 4, -13/2*e^5 + 5*e^4 + 187/2*e^3 - 58*e^2 - 473/2*e - 66, -12*e^5 + 6*e^4 + 178*e^3 - 64*e^2 - 516*e - 216, 9/2*e^5 - 2*e^4 - 123/2*e^3 + 28*e^2 + 297/2*e + 26, 5*e^5 - 6*e^4 - 73*e^3 + 67*e^2 + 191*e + 44, 5*e^5 - 4*e^4 - 75*e^3 + 42*e^2 + 213*e + 84, 2*e^2 - e - 4, -2*e^4 + 2*e^3 + 26*e^2 - 34*e - 40, -1/2*e^5 + e^4 + 15/2*e^3 - 8*e^2 - 57/2*e - 34, 7*e^5 - 2*e^4 - 103*e^3 + 18*e^2 + 291*e + 140, -3*e^5 - 2*e^4 + 45*e^3 + 25*e^2 - 141*e - 100, -1/2*e^5 + 3*e^4 + 19/2*e^3 - 34*e^2 - 61/2*e + 36, -14*e^5 + 8*e^4 + 199*e^3 - 94*e^2 - 521*e - 182, -2*e^4 - 2*e^3 + 24*e^2 + 6*e - 32, 12*e^5 - 8*e^4 - 174*e^3 + 92*e^2 + 476*e + 168, 3*e^5 + 2*e^4 - 44*e^3 - 22*e^2 + 142*e + 88, -14*e^5 + 9*e^4 + 206*e^3 - 96*e^2 - 586*e - 232, e^5 - 13*e^3 + 25*e + 4, -4*e^5 + 6*e^4 + 58*e^3 - 66*e^2 - 140*e - 16, -3/2*e^5 - e^4 + 53/2*e^3 + 18*e^2 - 235/2*e - 102, -17/2*e^5 + 5*e^4 + 239/2*e^3 - 60*e^2 - 613/2*e - 98];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^3 + w^2 + 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;