/*
  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![12, -9, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {12, 6, w^2 + w - 5}>;

primesArray := [
[2, 2, w - 2],
[3, 3, -w + 3],
[3, 3, w - 1],
[4, 2, w^2 + w - 7],
[19, 19, w + 1],
[23, 23, w^2 - 2*w - 1],
[31, 31, -2*w^2 + 19],
[31, 31, -w^2 + 5],
[31, 31, -3*w + 5],
[41, 41, w^2 + 2*w - 7],
[43, 43, w^2 - 11],
[47, 47, 3*w - 7],
[53, 53, -3*w^2 - 6*w + 11],
[59, 59, 2*w - 1],
[67, 67, 2*w^2 + w - 19],
[67, 67, 3*w^2 + 2*w - 25],
[67, 67, w - 5],
[73, 73, -4*w^2 - 3*w + 29],
[73, 73, 2*w^2 - w - 11],
[73, 73, w^2 + 2*w - 11],
[79, 79, 2*w^2 + 2*w - 11],
[83, 83, 2*w^2 - 13],
[89, 89, -2*w + 7],
[97, 97, -2*w^2 + 4*w - 1],
[101, 101, -2*w - 5],
[103, 103, 2*w^2 + w - 13],
[107, 107, -2*w^2 - 2*w + 17],
[109, 109, -4*w^2 - 2*w + 31],
[113, 113, w^2 + 2*w - 1],
[125, 5, -5],
[127, 127, 2*w^2 + 3*w - 13],
[137, 137, -2*w^2 + 7*w - 7],
[139, 139, 2*w^2 + w - 7],
[149, 149, -6*w^2 - 9*w + 31],
[157, 157, w^2 - 4*w + 1],
[163, 163, 2*w^2 + 5*w - 5],
[173, 173, 5*w^2 + 4*w - 37],
[179, 179, 2*w^2 + w - 11],
[179, 179, 3*w^2 - 2*w - 17],
[179, 179, -4*w^2 - w + 37],
[191, 191, -6*w^2 - 9*w + 29],
[193, 193, -2*w^2 - 4*w + 5],
[223, 223, -2*w^2 + 2*w + 7],
[227, 227, 6*w^2 + 4*w - 47],
[233, 233, 2*w^2 + 4*w - 13],
[233, 233, -4*w^2 - 4*w + 25],
[233, 233, 4*w + 13],
[239, 239, -4*w^2 - 8*w + 13],
[239, 239, 2*w^2 + 2*w - 5],
[239, 239, -w^2 + 8*w - 11],
[241, 241, 3*w - 1],
[251, 251, w^2 + 4*w - 19],
[263, 263, -2*w^2 + 2*w + 25],
[263, 263, -5*w^2 - 4*w + 35],
[263, 263, 3*w^2 - 25],
[269, 269, 6*w^2 + 3*w - 47],
[269, 269, -w^2 - 1],
[269, 269, 2*w^2 - 7],
[277, 277, 2*w^2 - w - 5],
[277, 277, -5*w^2 - 6*w + 31],
[277, 277, 2*w^2 - 2*w - 11],
[281, 281, 4*w^2 + 5*w - 25],
[307, 307, 4*w^2 + 2*w - 37],
[317, 317, -w - 7],
[317, 317, 3*w^2 + 2*w - 19],
[317, 317, -2*w^2 + 2*w + 5],
[331, 331, 2*w^2 + 3*w - 19],
[337, 337, w^2 - 4*w + 7],
[343, 7, -7],
[347, 347, 4*w^2 + 2*w - 29],
[349, 349, 3*w^2 - 19],
[361, 19, w^2 - 2*w - 7],
[367, 367, 4*w^2 + 5*w - 19],
[367, 367, -6*w + 11],
[373, 373, -4*w^2 + w + 25],
[379, 379, -5*w^2 - 10*w + 19],
[383, 383, 4*w^2 + 6*w - 23],
[401, 401, -3*w + 11],
[401, 401, -3*w - 1],
[401, 401, -3*w - 7],
[409, 409, -w^2 - 4*w + 13],
[419, 419, 2*w^2 - w - 17],
[421, 421, -3*w^2 + 31],
[431, 431, w^2 + 4*w - 1],
[449, 449, -w^2 + 4*w + 1],
[457, 457, -w^2 + 2*w + 13],
[461, 461, 4*w^2 - 4*w - 19],
[463, 463, 2*w^2 + 4*w - 23],
[467, 467, 2*w^2 - 5*w + 5],
[479, 479, 3*w + 11],
[491, 491, 8*w^2 + 9*w - 49],
[499, 499, -6*w^2 - 6*w + 41],
[499, 499, 2*w^2 + 2*w - 23],
[499, 499, 2*w^2 + 5*w - 29],
[523, 523, -4*w^2 + w + 41],
[529, 23, 6*w^2 + 7*w - 35],
[547, 547, -2*w^2 + 6*w + 1],
[557, 557, 3*w^2 + 2*w - 13],
[563, 563, -w^2 + 4*w + 23],
[569, 569, 10*w^2 + 8*w - 73],
[587, 587, 2*w^2 - 2*w - 13],
[593, 593, -6*w^2 - 8*w + 31],
[601, 601, 4*w^2 + 5*w - 41],
[601, 601, 10*w^2 + 16*w - 49],
[601, 601, w^2 - 2*w + 5],
[613, 613, -8*w^2 - 7*w + 55],
[617, 617, -8*w^2 - 12*w + 37],
[617, 617, 2*w^2 + 3*w - 25],
[617, 617, 3*w^2 - 17],
[641, 641, 4*w^2 - 35],
[643, 643, 4*w^2 + 4*w - 37],
[659, 659, -3*w^2 - 4*w + 23],
[661, 661, -w^2 - 4*w - 5],
[673, 673, 4*w^2 + 4*w - 31],
[683, 683, 2*w^2 + 4*w - 19],
[683, 683, 2*w^2 + w - 23],
[683, 683, 3*w^2 + 6*w - 19],
[701, 701, -6*w^2 - 12*w + 23],
[701, 701, -2*w^2 + 7*w - 1],
[701, 701, 3*w^2 + 2*w - 31],
[709, 709, 5*w^2 + 4*w - 41],
[719, 719, -6*w^2 - 2*w + 55],
[719, 719, w^2 + 2*w - 17],
[719, 719, 3*w^2 + 4*w - 29],
[727, 727, -9*w^2 - 12*w + 49],
[733, 733, 13*w^2 + 20*w - 65],
[739, 739, 11*w^2 + 6*w - 91],
[743, 743, -2*w^2 - 8*w + 35],
[751, 751, 3*w^2 - 4*w - 13],
[769, 769, 2*w^2 - 1],
[797, 797, 2*w^2 - 8*w + 11],
[811, 811, 4*w^2 - w - 29],
[821, 821, 2*w^2 + 5*w - 17],
[827, 827, -3*w^2 - 6*w + 7],
[827, 827, -11*w^2 - 8*w + 83],
[827, 827, -5*w^2 - 6*w + 25],
[829, 829, -4*w^2 + 41],
[829, 829, 4*w^2 + 8*w - 23],
[829, 829, 8*w - 19],
[839, 839, 2*w^2 + 6*w - 7],
[853, 853, 10*w^2 + 6*w - 77],
[859, 859, -5*w^2 - 2*w + 47],
[863, 863, 12*w^2 + 19*w - 59],
[877, 877, 8*w^2 + 6*w - 61],
[907, 907, -4*w - 1],
[907, 907, 9*w^2 + 4*w - 71],
[907, 907, 2*w^2 + 7*w - 17],
[919, 919, -10*w^2 - 5*w + 79],
[929, 929, 2*w^2 + 7*w - 11],
[937, 937, -4*w^2 + 6*w + 11],
[941, 941, 3*w^2 - 11],
[947, 947, 6*w - 5],
[967, 967, -2*w^2 + 5*w + 5],
[971, 971, 5*w^2 - 2*w - 29],
[991, 991, 4*w^2 + 2*w - 25],
[997, 997, 3*w^2 - 2*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 2*x^6 - 11*x^5 + 20*x^4 + 32*x^3 - 45*x^2 - 32*x + 21;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^6 - 1/2*e^5 - 5*e^4 + 4*e^3 + 10*e^2 - 7/2*e - 3/2, 1, -1, 1/2*e^6 - 1/2*e^5 - 5*e^4 + 3*e^3 + 9*e^2 + 5/2*e + 5/2, -1/2*e^6 + 1/2*e^5 + 4*e^4 - 3*e^3 - 3*e^2 - 3/2*e - 9/2, -e^5 + e^4 + 10*e^3 - 8*e^2 - 17*e + 7, -e^5 + 2*e^4 + 9*e^3 - 17*e^2 - 12*e + 19, 1/2*e^6 + 3/2*e^5 - 7*e^4 - 15*e^3 + 25*e^2 + 61/2*e - 23/2, -e^5 + 2*e^4 + 9*e^3 - 17*e^2 - 16*e + 21, -e^5 + 2*e^4 + 9*e^3 - 15*e^2 - 12*e + 13, 1/2*e^6 + 3/2*e^5 - 8*e^4 - 15*e^3 + 35*e^2 + 55/2*e - 51/2, e^6 - e^5 - 10*e^4 + 8*e^3 + 22*e^2 - 7*e - 15, e^6 - 3*e^5 - 8*e^4 + 26*e^3 + 4*e^2 - 33*e + 9, -1/2*e^6 + 5/2*e^5 + e^4 - 23*e^3 + 21*e^2 + 75/2*e - 53/2, e^5 - e^4 - 8*e^3 + 8*e^2 + 7*e - 11, -2*e^6 + e^5 + 21*e^4 - 4*e^3 - 50*e^2 - 13*e + 19, 3/2*e^6 - 1/2*e^5 - 17*e^4 + 2*e^3 + 48*e^2 + 7/2*e - 55/2, -e^6 - e^5 + 12*e^4 + 10*e^3 - 34*e^2 - 19*e + 7, -2*e^6 + 2*e^5 + 20*e^4 - 16*e^3 - 42*e^2 + 14*e + 14, -3/2*e^6 + 3/2*e^5 + 15*e^4 - 11*e^3 - 29*e^2 + 13/2*e + 5/2, -e^6 - e^5 + 12*e^4 + 10*e^3 - 36*e^2 - 19*e + 15, -1/2*e^6 - 1/2*e^5 + 5*e^4 + 7*e^3 - 7*e^2 - 45/2*e - 15/2, -3/2*e^6 + 1/2*e^5 + 16*e^4 - e^3 - 41*e^2 - 29/2*e + 47/2, 1/2*e^6 - 3/2*e^5 - 5*e^4 + 13*e^3 + 11*e^2 - 35/2*e - 21/2, -e^6 - 2*e^5 + 14*e^4 + 21*e^3 - 51*e^2 - 45*e + 32, -e^6 - e^5 + 12*e^4 + 12*e^3 - 36*e^2 - 29*e + 15, -3/2*e^6 + 5/2*e^5 + 14*e^4 - 21*e^3 - 23*e^2 + 55/2*e + 7/2, 1/2*e^6 + 5/2*e^5 - 9*e^4 - 25*e^3 + 41*e^2 + 97/2*e - 57/2, e^6 - 10*e^4 - 3*e^3 + 19*e^2 + 17*e + 6, 3*e^5 - 4*e^4 - 29*e^3 + 31*e^2 + 52*e - 29, -e^6 + 5*e^5 + 6*e^4 - 46*e^3 + 10*e^2 + 71*e - 15, e^6 - 11*e^4 + 26*e^2 + 2*e + 2, -e^6 + 3*e^5 + 8*e^4 - 28*e^3 - 8*e^2 + 49*e + 3, 3/2*e^6 - 5/2*e^5 - 13*e^4 + 22*e^3 + 14*e^2 - 65/2*e + 25/2, -5/2*e^6 + 5/2*e^5 + 25*e^4 - 21*e^3 - 53*e^2 + 43/2*e + 47/2, 2*e^6 - 5*e^5 - 15*e^4 + 44*e^3 - 63*e + 27, -2*e^5 + 2*e^4 + 18*e^3 - 16*e^2 - 32*e + 18, -3*e^6 + 6*e^5 + 28*e^4 - 51*e^3 - 47*e^2 + 65*e + 6, 3/2*e^6 + 5/2*e^5 - 19*e^4 - 28*e^3 + 60*e^2 + 131/2*e - 45/2, -3/2*e^6 + 5/2*e^5 + 13*e^4 - 19*e^3 - 15*e^2 + 29/2*e - 21/2, e^6 - e^5 - 10*e^4 + 6*e^3 + 16*e^2 + 7*e + 11, -3*e^6 + 2*e^5 + 30*e^4 - 11*e^3 - 61*e^2 - 11*e + 14, -1/2*e^6 - 3/2*e^5 + 6*e^4 + 15*e^3 - 15*e^2 - 67/2*e + 3/2, -9/2*e^6 + 11/2*e^5 + 43*e^4 - 43*e^3 - 75*e^2 + 75/2*e - 3/2, 5*e^5 - 6*e^4 - 47*e^3 + 45*e^2 + 76*e - 39, -e^6 - 5*e^5 + 18*e^4 + 48*e^3 - 84*e^2 - 83*e + 69, e^6 - e^5 - 10*e^4 + 10*e^3 + 18*e^2 - 19*e + 9, -e^6 + e^5 + 10*e^4 - 8*e^3 - 22*e^2 + 5*e + 15, -e^6 + 14*e^4 + e^3 - 51*e^2 - e + 30, e^6 - e^5 - 10*e^4 + 8*e^3 + 22*e^2 - 11*e + 5, 3/2*e^6 + 1/2*e^5 - 17*e^4 - 10*e^3 + 46*e^2 + 67/2*e - 21/2, -1/2*e^6 + 1/2*e^5 + 4*e^4 - 3*e^3 - e^2 - 3/2*e - 45/2, 2*e^5 - 2*e^4 - 20*e^3 + 14*e^2 + 42*e - 12, 1/2*e^6 + 1/2*e^5 - 7*e^4 - 7*e^3 + 29*e^2 + 41/2*e - 57/2, 3/2*e^6 + 3/2*e^5 - 19*e^4 - 17*e^3 + 61*e^2 + 83/2*e - 87/2, 3*e^6 - 2*e^5 - 31*e^4 + 14*e^3 + 70*e^2 - 8*e - 24, 1/2*e^6 - 3/2*e^5 - 5*e^4 + 13*e^3 + 9*e^2 - 43/2*e + 15/2, -1/2*e^6 + 7/2*e^5 - 33*e^3 + 33*e^2 + 117/2*e - 79/2, -2*e^6 + 4*e^5 + 18*e^4 - 34*e^3 - 26*e^2 + 42*e - 8, 2*e^5 - 2*e^4 - 18*e^3 + 12*e^2 + 30*e + 2, -7/2*e^6 + 5/2*e^5 + 35*e^4 - 17*e^3 - 73*e^2 + 9/2*e + 27/2, e^6 - 4*e^5 - 4*e^4 + 33*e^3 - 29*e^2 - 35*e + 50, e^6 - 2*e^5 - 6*e^4 + 15*e^3 - 9*e^2 - 11*e + 30, e^6 - 3*e^5 - 6*e^4 + 24*e^3 - 12*e^2 - 25*e + 39, -5/2*e^6 - 5/2*e^5 + 31*e^4 + 27*e^3 - 103*e^2 - 113/2*e + 129/2, 1/2*e^6 - 5/2*e^5 - 3*e^4 + 25*e^3 - e^2 - 103/2*e - 7/2, -7/2*e^6 + 9/2*e^5 + 35*e^4 - 38*e^3 - 76*e^2 + 85/2*e + 67/2, -5/2*e^6 + 5/2*e^5 + 25*e^4 - 21*e^3 - 49*e^2 + 55/2*e + 19/2, -e^6 - 2*e^5 + 13*e^4 + 24*e^3 - 44*e^2 - 60*e + 18, 2*e^6 + 5*e^5 - 29*e^4 - 52*e^3 + 110*e^2 + 107*e - 61, 9/2*e^6 - 15/2*e^5 - 41*e^4 + 62*e^3 + 64*e^2 - 139/2*e - 13/2, 3/2*e^6 - 3/2*e^5 - 15*e^4 + 11*e^3 + 33*e^2 - 5/2*e - 5/2, -1/2*e^6 + 1/2*e^5 + 3*e^4 - 3*e^3 + e^2 - 9/2*e + 7/2, -5/2*e^6 - 1/2*e^5 + 26*e^4 + 11*e^3 - 57*e^2 - 107/2*e + 37/2, 7/2*e^6 - 7/2*e^5 - 33*e^4 + 25*e^3 + 57*e^2 - 33/2*e + 7/2, -1/2*e^6 - 7/2*e^5 + 12*e^4 + 35*e^3 - 65*e^2 - 143/2*e + 123/2, e^6 - 11*e^4 + 30*e^2 - 6, e^6 - 3*e^5 - 6*e^4 + 24*e^3 - 10*e^2 - 21*e + 33, -5/2*e^6 + 11/2*e^5 + 23*e^4 - 47*e^3 - 35*e^2 + 127/2*e - 3/2, -e^6 + 3*e^5 + 6*e^4 - 26*e^3 + 14*e^2 + 29*e - 47, -1/2*e^6 + 1/2*e^5 + 3*e^4 - 2*e^3 + 8*e^2 - 33/2*e - 57/2, -2*e^6 - 2*e^5 + 26*e^4 + 20*e^3 - 84*e^2 - 34*e + 38, e^6 + e^5 - 10*e^4 - 14*e^3 + 26*e^2 + 43*e - 15, -2*e^6 - e^5 + 24*e^4 + 13*e^3 - 65*e^2 - 34*e + 3, -5/2*e^6 + 9/2*e^5 + 23*e^4 - 35*e^3 - 33*e^2 + 55/2*e - 13/2, -4*e^6 + 2*e^5 + 44*e^4 - 8*e^3 - 116*e^2 - 24*e + 60, 2*e^6 - 2*e^5 - 22*e^4 + 18*e^3 + 62*e^2 - 32*e - 34, 6*e^5 - 8*e^4 - 56*e^3 + 68*e^2 + 90*e - 72, 4*e^6 - 6*e^5 - 36*e^4 + 50*e^3 + 52*e^2 - 58*e - 6, -e^6 - e^5 + 16*e^4 + 10*e^3 - 70*e^2 - 23*e + 57, 6*e^5 - 10*e^4 - 58*e^3 + 76*e^2 + 104*e - 58, -2*e^6 + 2*e^5 + 16*e^4 - 14*e^3 - 8*e^2 - 14, -5/2*e^6 + 9/2*e^5 + 25*e^4 - 39*e^3 - 49*e^2 + 99/2*e - 1/2, -3/2*e^6 - 9/2*e^5 + 25*e^4 + 41*e^3 - 111*e^2 - 135/2*e + 181/2, 2*e^6 - 8*e^5 - 16*e^4 + 72*e^3 + 10*e^2 - 104*e + 10, 2*e^6 - 24*e^4 + 74*e^2 - 32, -3*e^6 + 6*e^5 + 28*e^4 - 47*e^3 - 51*e^2 + 45*e + 12, e^6 + e^5 - 16*e^4 - 8*e^3 + 66*e^2 + 5*e - 45, -e^6 + 13*e^4 + 2*e^3 - 50*e^2 - 20*e + 42, -11/2*e^6 + 7/2*e^5 + 57*e^4 - 22*e^3 - 132*e^2 - 27/2*e + 105/2, 4*e^6 - e^5 - 45*e^4 + 2*e^3 + 118*e^2 + 19*e - 27, 3*e^6 - 3*e^5 - 30*e^4 + 26*e^3 + 64*e^2 - 37*e - 13, -4*e^6 + 9*e^5 + 33*e^4 - 78*e^3 - 32*e^2 + 105*e - 23, -3*e^6 + 5*e^5 + 28*e^4 - 42*e^3 - 50*e^2 + 59*e + 13, -9*e^5 + 11*e^4 + 84*e^3 - 90*e^2 - 139*e + 89, 1/2*e^6 + 5/2*e^5 - 9*e^4 - 27*e^3 + 47*e^2 + 129/2*e - 105/2, -7*e^5 + 7*e^4 + 64*e^3 - 52*e^2 - 107*e + 33, 1/2*e^6 + 5/2*e^5 - 9*e^4 - 19*e^3 + 45*e^2 + 37/2*e - 105/2, -5/2*e^6 - 9/2*e^5 + 33*e^4 + 47*e^3 - 119*e^2 - 201/2*e + 177/2, 7/2*e^6 + 5/2*e^5 - 41*e^4 - 31*e^3 + 117*e^2 + 171/2*e - 73/2, -e^6 + 7*e^4 + 6*e^3 - 36*e - 12, e^6 + e^5 - 10*e^4 - 10*e^3 + 24*e^2 + 17*e - 17, 1/2*e^6 + 11/2*e^5 - 15*e^4 - 53*e^3 + 89*e^2 + 189/2*e - 151/2, 6*e^6 - 6*e^5 - 58*e^4 + 44*e^3 + 114*e^2 - 26*e - 30, e^6 + 4*e^5 - 15*e^4 - 42*e^3 + 56*e^2 + 104*e - 24, -3/2*e^6 + 11/2*e^5 + 11*e^4 - 50*e^3 - 4*e^2 + 149/2*e - 15/2, 9/2*e^6 - 15/2*e^5 - 37*e^4 + 61*e^3 + 31*e^2 - 135/2*e + 75/2, -5/2*e^6 - 1/2*e^5 + 31*e^4 + 5*e^3 - 101*e^2 - 29/2*e + 105/2, -6*e^6 + 3*e^5 + 65*e^4 - 18*e^3 - 166*e^2 - 13*e + 81, -e^6 + 3*e^5 + 10*e^4 - 26*e^3 - 20*e^2 + 41*e - 1, -3*e^6 + 6*e^5 + 27*e^4 - 50*e^3 - 36*e^2 + 62*e - 30, -e^6 - 8*e^5 + 22*e^4 + 77*e^3 - 109*e^2 - 137*e + 84, 6*e^6 - 4*e^5 - 62*e^4 + 26*e^3 + 136*e^2 + 8*e - 30, 7/2*e^6 - 7/2*e^5 - 35*e^4 + 25*e^3 + 71*e^2 - 9/2*e - 17/2, 7*e^6 - 7*e^5 - 66*e^4 + 50*e^3 + 114*e^2 - 19*e - 1, 6*e^6 - 9*e^5 - 59*e^4 + 74*e^3 + 116*e^2 - 87*e - 29, -1/2*e^6 + 5/2*e^5 + 2*e^4 - 25*e^3 + 9*e^2 + 101/2*e + 3/2, -4*e^6 + 6*e^5 + 40*e^4 - 52*e^3 - 84*e^2 + 74*e + 28, 2*e^6 + 3*e^5 - 23*e^4 - 36*e^3 + 68*e^2 + 89*e - 41, -7*e^6 + 6*e^5 + 70*e^4 - 43*e^3 - 143*e^2 + 21*e + 18, -e^6 + 2*e^5 + 7*e^4 - 14*e^3 + 4*e^2 + 4*e - 22, 5*e^6 + 3*e^5 - 58*e^4 - 40*e^3 + 162*e^2 + 113*e - 51, -2*e^6 + 6*e^5 + 16*e^4 - 48*e^3 - 4*e^2 + 40*e - 36, 2*e^6 - 8*e^5 - 14*e^4 + 74*e^3 - 2*e^2 - 126*e + 6, 4*e^5 - 4*e^4 - 40*e^3 + 32*e^2 + 74*e - 30, 2*e^6 - 2*e^5 - 18*e^4 + 14*e^3 + 26*e^2 - 10*e + 34, 2*e^6 + 2*e^5 - 24*e^4 - 18*e^3 + 76*e^2 + 26*e - 46, -2*e^6 + 7*e^5 + 20*e^4 - 63*e^3 - 45*e^2 + 98*e + 19, -3/2*e^6 + 11/2*e^5 + 10*e^4 - 49*e^3 + 13*e^2 + 139/2*e - 111/2, -5*e^6 - e^5 + 56*e^4 + 20*e^3 - 148*e^2 - 77*e + 49, -e^6 + 2*e^5 + 6*e^4 - 13*e^3 + 17*e^2 - 5*e - 50, 2*e^5 - 20*e^3 + 2*e^2 + 36*e + 12, 2*e^6 - 8*e^5 - 8*e^4 + 72*e^3 - 60*e^2 - 106*e + 106, -4*e^6 - 3*e^5 + 48*e^4 + 35*e^3 - 149*e^2 - 84*e + 73, 4*e^5 - 4*e^4 - 40*e^3 + 26*e^2 + 68*e + 22, e^6 - 6*e^5 - 6*e^4 + 53*e^3 - 11*e^2 - 67*e + 32, 11/2*e^6 - 11/2*e^5 - 55*e^4 + 45*e^3 + 117*e^2 - 113/2*e - 53/2, e^6 + 5*e^5 - 18*e^4 - 52*e^3 + 82*e^2 + 107*e - 51, -6*e^6 + 9*e^5 + 55*e^4 - 72*e^3 - 84*e^2 + 67*e - 11, e^6 + 6*e^5 - 23*e^4 - 56*e^3 + 124*e^2 + 92*e - 102, 7/2*e^6 + 3/2*e^5 - 41*e^4 - 17*e^3 + 123*e^2 + 79/2*e - 147/2, -4*e^6 + e^5 + 42*e^4 + 3*e^3 - 99*e^2 - 60*e + 29, -1/2*e^6 + 13/2*e^5 - 3*e^4 - 60*e^3 + 52*e^2 + 167/2*e - 117/2, e^6 - 9*e^5 + 2*e^4 + 86*e^3 - 74*e^2 - 155*e + 77, 2*e^6 - 6*e^5 - 16*e^4 + 52*e^3 + 10*e^2 - 68*e + 4];
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;
ALEigenvalues[ideal<ZF | {4, 2, w^2 + w - 7}>] := 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;