/*
  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![5, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {5, 5, w}>;

primesArray := [
[5, 5, w],
[5, 5, -w^2 + w + 2],
[7, 7, -w^2 + 2],
[13, 13, -w^2 + 3],
[13, 13, w^2 - w - 4],
[16, 2, 2],
[19, 19, -w^2 + w + 1],
[23, 23, -w^3 + 4*w + 2],
[25, 5, -w^3 + w^2 + 3*w - 1],
[29, 29, w^3 - w^2 - 4*w + 1],
[29, 29, -w + 3],
[31, 31, -w^3 + w^2 + 5*w - 2],
[37, 37, w^3 - 4*w - 1],
[37, 37, w^3 - 3*w + 1],
[53, 53, -w^3 + 2*w^2 + 4*w - 6],
[59, 59, w^2 + w - 4],
[61, 61, -2*w^2 + w + 8],
[73, 73, -w^3 + 5*w - 1],
[79, 79, 2*w^2 + w - 6],
[79, 79, 2*w^3 - w^2 - 9*w + 3],
[81, 3, -3],
[83, 83, -w^3 + w^2 + 3*w - 4],
[97, 97, -w^3 + 6*w + 2],
[97, 97, -2*w^3 + w^2 + 10*w - 1],
[97, 97, 3*w^3 - 4*w^2 - 15*w + 13],
[97, 97, w^3 - w^2 - 6*w + 2],
[103, 103, -2*w^3 + 3*w^2 + 11*w - 9],
[109, 109, w^2 + 3*w - 3],
[137, 137, -w^2 + 7],
[149, 149, w^3 - 2*w^2 - 5*w + 4],
[149, 149, w^3 + 2*w^2 - 6*w - 7],
[151, 151, w^3 + w^2 - 5*w - 3],
[163, 163, w^3 - 3*w - 4],
[163, 163, -w^3 + 6*w - 3],
[167, 167, w^3 - 2*w^2 - 6*w + 4],
[169, 13, w^2 + 2*w - 4],
[191, 191, w^3 - w^2 - 5*w - 2],
[191, 191, 2*w^3 - 9*w - 2],
[197, 197, w^3 - 4*w + 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 8],
[223, 223, -w^3 + 3*w^2 + 4*w - 6],
[229, 229, w^3 - w^2 - 5*w - 1],
[251, 251, 2*w^3 - 2*w^2 - 7*w + 3],
[257, 257, 3*w^2 - 14],
[263, 263, w^2 - 3*w - 1],
[263, 263, w^2 - 2*w - 7],
[269, 269, -w^3 + w^2 + 2*w - 4],
[269, 269, 2*w^3 - 11*w - 1],
[271, 271, -w^3 + w^2 + 4*w - 7],
[277, 277, w^3 - w^2 - 3*w + 8],
[311, 311, w^3 + 2*w^2 - 7*w - 11],
[311, 311, -3*w^2 - 2*w + 8],
[317, 317, -2*w^3 + 2*w^2 + 7*w - 4],
[331, 331, w^2 - w - 8],
[331, 331, 3*w^3 - 2*w^2 - 12*w + 9],
[337, 337, -2*w^3 + 2*w^2 + 11*w - 8],
[343, 7, w^3 - 3*w^2 - 4*w + 11],
[347, 347, -2*w^3 + 2*w^2 + 9*w - 4],
[349, 349, -w^3 + 2*w^2 + 2*w - 6],
[353, 353, 2*w^2 - w - 12],
[353, 353, 2*w^3 - w^2 - 9*w - 1],
[367, 367, -2*w^3 + w^2 + 10*w - 6],
[367, 367, 2*w^3 - 11*w - 3],
[379, 379, -3*w - 1],
[379, 379, w^3 - 2*w - 3],
[383, 383, -w^3 + 2*w^2 + 4*w - 11],
[383, 383, -w^3 - w^2 + 7*w + 4],
[389, 389, 2*w^3 - w^2 - 11*w + 4],
[401, 401, w^3 + 2*w^2 - 7*w - 6],
[401, 401, 2*w^3 - 11*w - 6],
[419, 419, -2*w^3 + w^2 + 8*w - 1],
[421, 421, -w^3 + 3*w^2 + 3*w - 12],
[421, 421, -w^3 - 2*w^2 + 2*w + 6],
[433, 433, w^3 - 2*w^2 - 6*w + 11],
[433, 433, w^2 - 8],
[439, 439, -w^3 + 3*w^2 + 3*w - 7],
[443, 443, w^3 - w^2 - 2*w - 4],
[443, 443, 2*w^3 + w^2 - 7*w - 2],
[443, 443, 2*w^3 - w^2 - 8*w + 2],
[443, 443, -w^2 - 2],
[467, 467, 3*w^3 - 5*w^2 - 15*w + 19],
[467, 467, 2*w^3 - 3*w^2 - 8*w + 13],
[467, 467, -w^3 + 2*w^2 + 3*w - 9],
[467, 467, -w^3 + 3*w^2 + 3*w - 8],
[479, 479, -w^3 - 3*w^2 + 5*w + 13],
[479, 479, -w^3 + w^2 + w - 4],
[487, 487, 3*w^3 - w^2 - 13*w - 2],
[491, 491, w^2 + w - 9],
[499, 499, -3*w^3 + 3*w^2 + 13*w - 7],
[503, 503, 2*w^2 + w - 9],
[509, 509, -2*w^2 - 3*w + 6],
[509, 509, -w^3 + w^2 - 4],
[521, 521, -3*w^2 + w + 12],
[523, 523, 2*w^3 - 8*w - 3],
[529, 23, 2*w^3 - w^2 - 6*w - 1],
[541, 541, -w^3 + w^2 + 7*w - 4],
[547, 547, 2*w^3 - 3*w^2 - 9*w + 7],
[557, 557, 3*w^2 - 2*w - 9],
[557, 557, w^3 - 6*w - 7],
[563, 563, -2*w^3 + w^2 + 6*w - 6],
[577, 577, -w^3 + 4*w^2 + 2*w - 13],
[587, 587, w^2 - 2*w + 3],
[587, 587, w^2 + 2*w - 6],
[593, 593, -2*w^3 + 2*w^2 + 11*w - 3],
[599, 599, w^3 + 3*w^2 - 6*w - 12],
[601, 601, -2*w^3 + w^2 + 9*w + 2],
[607, 607, -w^3 + 3*w^2 + 4*w - 9],
[617, 617, -2*w^3 + w^2 + 7*w - 3],
[619, 619, -2*w^3 + 4*w^2 + 9*w - 11],
[631, 631, 3*w^2 - w - 7],
[631, 631, -w^3 - w^2 + 6*w + 1],
[641, 641, -w^3 + 2*w^2 + 5*w - 1],
[641, 641, -w^3 + 2*w^2 + 2*w - 7],
[643, 643, -3*w^3 + w^2 + 14*w - 4],
[647, 647, -3*w^3 + 5*w^2 + 14*w - 18],
[647, 647, -w^3 + 7*w + 2],
[647, 647, -2*w^3 + 7*w - 1],
[647, 647, 2*w^3 - w^2 - 10*w - 1],
[653, 653, -2*w^3 - w^2 + 11*w + 4],
[659, 659, -w^3 - w^2 + 5*w - 1],
[673, 673, -w^3 + w^2 + 7*w - 2],
[673, 673, -w^3 + w^2 + 2*w - 8],
[683, 683, 2*w^3 - 11*w - 8],
[683, 683, 2*w^3 - 7*w - 3],
[701, 701, 2*w^3 - 4*w^2 - 10*w + 17],
[727, 727, -w^3 + w^2 + 4*w - 8],
[727, 727, 3*w^2 - 11],
[739, 739, 2*w^3 + 3*w^2 - 8*w - 13],
[743, 743, 2*w^2 + w - 12],
[743, 743, 3*w^2 - 2*w - 14],
[751, 751, w^3 + w^2 - 3*w - 7],
[757, 757, -2*w^3 + 10*w - 1],
[761, 761, -2*w^3 + w^2 + 7*w - 1],
[761, 761, 3*w^2 + w - 7],
[773, 773, -2*w^3 + 2*w^2 + 8*w - 1],
[797, 797, -2*w^3 + w^2 + 9*w + 3],
[797, 797, 3*w^3 - 14*w - 8],
[809, 809, 5*w^3 - 5*w^2 - 24*w + 16],
[821, 821, -4*w^3 + 3*w^2 + 18*w - 9],
[821, 821, w^3 - w - 3],
[823, 823, 3*w^2 + w - 8],
[829, 829, -2*w^3 + 3*w^2 + 10*w - 7],
[829, 829, 3*w^2 - 2*w - 16],
[841, 29, 2*w^2 + w - 11],
[853, 853, 3*w^3 - 2*w^2 - 14*w + 3],
[853, 853, 3*w^3 - 5*w^2 - 14*w + 17],
[857, 857, 3*w^2 - w - 8],
[857, 857, 3*w^3 - w^2 - 13*w + 3],
[859, 859, w^3 - 3*w^2 - 6*w + 12],
[877, 877, w^3 + 2*w^2 - 5*w - 4],
[877, 877, w^2 - w - 9],
[881, 881, w^3 + 2*w^2 - 4*w - 12],
[881, 881, w^2 - 4*w - 4],
[883, 883, -3*w^3 + 2*w^2 + 11*w - 8],
[887, 887, -2*w^3 + w^2 + 6*w - 1],
[911, 911, 3*w^2 - w - 17],
[911, 911, 2*w^3 - 2*w^2 - 12*w + 3],
[919, 919, 5*w^3 - 6*w^2 - 22*w + 24],
[919, 919, -2*w^3 + w^2 + 6*w - 7],
[929, 929, 2*w^3 - 7*w - 1],
[937, 937, -w^3 - w^2 + 8*w - 2],
[947, 947, 3*w^3 + w^2 - 15*w - 6],
[947, 947, -2*w^3 + 9*w + 9],
[953, 953, w^3 - 2*w^2 - 6*w + 1],
[977, 977, w^3 - w^2 - 3*w + 9],
[977, 977, w^3 + w^2 - 3*w - 8],
[983, 983, w^3 - 5*w - 7],
[991, 991, -w^3 + 2*w^2 + 2*w - 8],
[991, 991, w^2 + 4*w - 1],
[997, 997, 2*w^3 + w^2 - 12*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 13*x^3 + 14*x^2 + 15*x - 18;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, 5/3*e^4 + 2*e^3 - 56/3*e^2 + 4/3*e + 21, -4/3*e^4 - 2*e^3 + 43/3*e^2 + 4/3*e - 18, 1/3*e^4 - 13/3*e^2 + 11/3*e + 5, -1/3*e^4 - e^3 + 7/3*e^2 + 16/3*e - 3, 4/3*e^4 + 2*e^3 - 40/3*e^2 - 4/3*e + 14, -3*e^4 - 4*e^3 + 34*e^2 + 4*e - 42, 5/3*e^4 + 2*e^3 - 59/3*e^2 - 2/3*e + 27, 3*e^4 + 4*e^3 - 33*e^2 - 2*e + 39, 2*e^4 + 2*e^3 - 24*e^2 + 2*e + 30, -13/3*e^4 - 6*e^3 + 148/3*e^2 + 22/3*e - 66, 4/3*e^4 + 2*e^3 - 40/3*e^2 + 2/3*e + 8, 10/3*e^4 + 4*e^3 - 112/3*e^2 + 2/3*e + 44, -e^4 - 2*e^3 + 11*e^2 + 8*e - 15, 2*e + 6, 4/3*e^4 + 2*e^3 - 46/3*e^2 - 22/3*e + 26, -8/3*e^4 - 4*e^3 + 86/3*e^2 + 14/3*e - 34, 20/3*e^4 + 9*e^3 - 224/3*e^2 - 23/3*e + 96, -8/3*e^4 - 4*e^3 + 83/3*e^2 + 8/3*e - 28, -5*e^4 - 6*e^3 + 59*e^2 - 77, e^3 + 4*e^2 - 3*e - 18, 4/3*e^4 + 2*e^3 - 40/3*e^2 + 8/3*e + 14, 4/3*e^4 + 2*e^3 - 46/3*e^2 - 13/3*e + 20, -e^4 - 2*e^3 + 9*e^2 + 6*e - 1, -2*e^4 - 2*e^3 + 24*e^2 - 28, -5/3*e^4 - 2*e^3 + 59/3*e^2 + 8/3*e - 31, -5/3*e^4 - 3*e^3 + 53/3*e^2 + 17/3*e - 31, -2*e^4 - 3*e^3 + 22*e^2 + 3*e - 36, -6*e^4 - 6*e^3 + 70*e^2 - 10*e - 84, -e^4 - 3*e^3 + 9*e^2 + 15*e - 9, -8*e^4 - 10*e^3 + 92*e^2 + 6*e - 118, 10/3*e^4 + 6*e^3 - 103/3*e^2 - 40/3*e + 50, 8/3*e^4 + 3*e^3 - 92/3*e^2 + 1/3*e + 30, -4*e^4 - 6*e^3 + 42*e^2 + 2*e - 42, -7/3*e^4 - 4*e^3 + 73/3*e^2 + 34/3*e - 21, -7*e^4 - 8*e^3 + 84*e^2 - 108, -4*e^4 - 6*e^3 + 41*e^2 + 8*e - 45, 4*e^4 + 4*e^3 - 47*e^2 + 6*e + 69, -25/3*e^4 - 10*e^3 + 289/3*e^2 - 8/3*e - 111, -11/3*e^4 - 5*e^3 + 119/3*e^2 + 5/3*e - 37, -14/3*e^4 - 6*e^3 + 152/3*e^2 - 4/3*e - 58, -4*e^4 - 4*e^3 + 48*e^2 - 2*e - 66, 2*e^4 + 4*e^3 - 19*e^2 - 12*e + 30, 4*e^4 + 4*e^3 - 46*e^2 + 12*e + 48, e^4 - 14*e^2 + 4*e + 6, -6*e^4 - 10*e^3 + 62*e^2 + 22*e - 78, -7*e^4 - 8*e^3 + 83*e^2 - 105, 29/3*e^4 + 14*e^3 - 323/3*e^2 - 47/3*e + 141, 32/3*e^4 + 14*e^3 - 362/3*e^2 - 29/3*e + 156, 2*e^4 + 6*e^3 - 14*e^2 - 27*e + 18, -2*e^4 - 2*e^3 + 24*e^2 + 2*e - 30, -3*e^4 - 6*e^3 + 29*e^2 + 20*e - 51, -e^4 - 2*e^3 + 11*e^2 + 9*e - 19, -14/3*e^4 - 4*e^3 + 158/3*e^2 - 64/3*e - 58, -2*e^4 - e^3 + 30*e^2 - 9*e - 58, -25/3*e^4 - 10*e^3 + 283/3*e^2 - 14/3*e - 111, -8*e^4 - 11*e^3 + 88*e^2 + 3*e - 114, -10/3*e^4 - 6*e^3 + 88/3*e^2 + 34/3*e - 18, -8*e^4 - 12*e^3 + 86*e^2 + 18*e - 120, -3*e^4 - 4*e^3 + 35*e^2 - e - 45, -19/3*e^4 - 8*e^3 + 208/3*e^2 - 8/3*e - 57, -19/3*e^4 - 10*e^3 + 211/3*e^2 + 70/3*e - 87, -25/3*e^4 - 12*e^3 + 283/3*e^2 + 52/3*e - 123, 14/3*e^4 + 8*e^3 - 152/3*e^2 - 71/3*e + 66, -5*e^4 - 6*e^3 + 57*e^2 - 4*e - 81, -4*e^4 - 2*e^3 + 53*e^2 - 16*e - 72, 7*e^4 + 8*e^3 - 80*e^2 + 4*e + 84, 3*e^4 - 40*e^2 + 28*e + 42, e^3 + 4*e^2 - 11*e - 12, 6*e^4 + 8*e^3 - 70*e^2 - 7*e + 90, 14/3*e^4 + 10*e^3 - 131/3*e^2 - 92/3*e + 60, -7/3*e^4 - 4*e^3 + 61/3*e^2 + 22/3*e + 3, -2/3*e^4 + 20/3*e^2 - 34/3*e + 8, -28/3*e^4 - 14*e^3 + 292/3*e^2 + 46/3*e - 114, 16/3*e^4 + 8*e^3 - 181/3*e^2 - 34/3*e + 89, 9*e^4 + 14*e^3 - 100*e^2 - 26*e + 132, 8*e^4 + 12*e^3 - 88*e^2 - 16*e + 102, 3*e^4 + 6*e^3 - 28*e^2 - 16*e + 21, 2*e^4 + 2*e^3 - 20*e^2 + 6*e + 12, -4*e^3 - 7*e^2 + 32*e + 18, 13*e^4 + 18*e^3 - 145*e^2 - 12*e + 171, 6*e^4 + 8*e^3 - 66*e^2 - 5*e + 90, 6*e^4 + 10*e^3 - 62*e^2 - 22*e + 84, 5*e^4 + 8*e^3 - 49*e^2 - 12*e + 39, 2*e^3 + 8*e^2 - 4*e - 42, -2/3*e^4 - 2*e^3 + 14/3*e^2 + 20/3*e - 16, 6*e^4 + 8*e^3 - 68*e^2 - 2*e + 84, -32/3*e^4 - 14*e^3 + 359/3*e^2 + 44/3*e - 148, -7*e^4 - 8*e^3 + 79*e^2 - 2*e - 75, 2*e^3 + 2*e^2 - 12*e + 6, -2*e^4 - 5*e^3 + 18*e^2 + 21*e - 24, -8*e^4 - 10*e^3 + 92*e^2 - 4*e - 126, -11*e^4 - 16*e^3 + 119*e^2 + 18*e - 133, 35/3*e^4 + 16*e^3 - 395/3*e^2 - 32/3*e + 177, -19/3*e^4 - 10*e^3 + 211/3*e^2 + 46/3*e - 111, 8/3*e^4 - 110/3*e^2 + 94/3*e + 36, -6*e^4 - 9*e^3 + 66*e^2 + 7*e - 84, 2*e^4 - 30*e^2 + 6*e + 54, 7*e^4 + 8*e^3 - 79*e^2 - 3*e + 81, 11/3*e^4 + 8*e^3 - 83/3*e^2 - 56/3*e + 3, 14*e^4 + 20*e^3 - 152*e^2 - 22*e + 186, 7*e^4 + 10*e^3 - 75*e^2 - 8*e + 93, 12*e^4 + 16*e^3 - 133*e^2 - 2*e + 159, 20*e^4 + 28*e^3 - 220*e^2 - 22*e + 270, 37/3*e^4 + 14*e^3 - 430/3*e^2 + 26/3*e + 176, -23/3*e^4 - 9*e^3 + 275/3*e^2 + 17/3*e - 121, 2*e^4 - 28*e^2 + 12*e + 48, 17/3*e^4 + 8*e^3 - 173/3*e^2 - 5/3*e + 45, -28/3*e^4 - 8*e^3 + 340/3*e^2 - 68/3*e - 144, -3*e^4 - 2*e^3 + 39*e^2 - 12*e - 61, 7*e^4 + 11*e^3 - 79*e^2 - 17*e + 117, -e^4 + 12*e^2 - 12*e + 9, 25/3*e^4 + 12*e^3 - 268/3*e^2 - 4/3*e + 107, e^4 + 2*e^3 - 10*e^2 - 22*e, -2*e^4 - 8*e^3 + 10*e^2 + 36*e, -6*e^4 - 9*e^3 + 66*e^2 + 9*e - 78, -22*e^4 - 29*e^3 + 242*e^2 - e - 276, 2*e^4 - 25*e^2 + 30*e + 39, -14*e^4 - 18*e^3 + 155*e^2 - 4*e - 168, 1/3*e^4 - 3*e^3 - 25/3*e^2 + 59/3*e - 7, -62/3*e^4 - 26*e^3 + 707/3*e^2 + 20/3*e - 292, 14*e^4 + 14*e^3 - 164*e^2 + 26*e + 192, -6*e^4 - 10*e^3 + 64*e^2 + 16*e - 72, -6*e^4 - 6*e^3 + 72*e^2 - 7*e - 114, 14/3*e^4 + 5*e^3 - 170/3*e^2 + 19/3*e + 72, -20/3*e^4 - 8*e^3 + 212/3*e^2 - 28/3*e - 70, 5/3*e^4 + e^3 - 65/3*e^2 + 25/3*e + 57, 10*e^4 + 14*e^3 - 116*e^2 - 27*e + 150, -e^3 - 8*e^2 - 5*e + 30, 20/3*e^4 + 10*e^3 - 224/3*e^2 - 44/3*e + 108, 7/3*e^4 + 6*e^3 - 67/3*e^2 - 76/3*e + 53, -9*e^4 - 12*e^3 + 105*e^2 + 4*e - 153, 8*e^4 + 16*e^3 - 80*e^2 - 48*e + 102, -15*e^4 - 22*e^3 + 169*e^2 + 37*e - 225, e^4 - 4*e^3 - 22*e^2 + 40*e + 30, -16*e^4 - 20*e^3 + 188*e^2 + 22*e - 258, -8*e^4 - 14*e^3 + 78*e^2 + 25*e - 72, 4*e^4 + 4*e^3 - 42*e^2 + 26*e + 30, 2*e^4 - 26*e^2 + 25*e + 6, 55/3*e^4 + 20*e^3 - 652/3*e^2 + 56/3*e + 248, -10/3*e^4 + 136/3*e^2 - 98/3*e - 42, 6*e^4 + 4*e^3 - 82*e^2 + 18*e + 128, 34/3*e^4 + 16*e^3 - 367/3*e^2 - 40/3*e + 134, 3*e^4 + 6*e^3 - 22*e^2 - 6*e - 28, -53/3*e^4 - 24*e^3 + 608/3*e^2 + 92/3*e - 280, -3*e^4 - 8*e^3 + 23*e^2 + 24*e - 27, -e^4 + 2*e^3 + 18*e^2 - 24*e - 33, 71/3*e^4 + 32*e^3 - 797/3*e^2 - 62/3*e + 327, 16/3*e^4 + 2*e^3 - 202/3*e^2 + 128/3*e + 80, 56/3*e^4 + 24*e^3 - 635/3*e^2 - 32/3*e + 264, -22*e^4 - 29*e^3 + 246*e^2 + 7*e - 282, -8*e^4 - 8*e^3 + 91*e^2 - 20*e - 102, 37/3*e^4 + 14*e^3 - 445/3*e^2 - 22/3*e + 209, 11*e^4 + 18*e^3 - 119*e^2 - 32*e + 153, 20*e^4 + 22*e^3 - 232*e^2 + 24*e + 264, 2*e^4 + 2*e^3 - 24*e^2 + 3*e + 30, -10/3*e^4 - 4*e^3 + 130/3*e^2 + 31/3*e - 78, -47/3*e^4 - 18*e^3 + 545/3*e^2 - 28/3*e - 217, 7*e^4 + 6*e^3 - 85*e^2 + 16*e + 75, -53/3*e^4 - 22*e^3 + 611/3*e^2 + 5/3*e - 253, -8*e^4 - 10*e^3 + 88*e^2 - 2*e - 90, 7*e^4 + 8*e^3 - 90*e^2 - 8*e + 150, 6*e^4 + 8*e^3 - 64*e^2 + 10*e + 48, 10*e^4 + 10*e^3 - 115*e^2 + 24*e + 135, -2*e^4 - 3*e^3 + 18*e^2 + e + 12, 19*e^4 + 26*e^3 - 209*e^2 - 10*e + 261, -1/3*e^4 - 4*e^3 - 29/3*e^2 + 52/3*e + 33, e^4 + 2*e^3 - 11*e^2 - 18*e + 11, -20/3*e^4 - 14*e^3 + 206/3*e^2 + 128/3*e - 100];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, w}>] := 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;