/*
  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![-34, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, 3}>;

primesArray := [
[2, 2, w - 6],
[3, 3, w + 1],
[3, 3, w + 2],
[5, 5, w + 2],
[5, 5, w + 3],
[11, 11, w + 1],
[11, 11, w + 10],
[17, 17, -3*w + 17],
[29, 29, w + 11],
[29, 29, w + 18],
[37, 37, w + 16],
[37, 37, w + 21],
[47, 47, -w - 9],
[47, 47, w - 9],
[49, 7, -7],
[61, 61, w + 20],
[61, 61, w + 41],
[89, 89, 2*w - 15],
[89, 89, -2*w - 15],
[103, 103, -14*w + 81],
[103, 103, 4*w - 21],
[107, 107, w + 26],
[107, 107, w + 81],
[109, 109, w + 19],
[109, 109, w + 90],
[127, 127, 2*w - 3],
[127, 127, -2*w - 3],
[131, 131, w + 54],
[131, 131, w + 77],
[137, 137, -3*w - 13],
[137, 137, 3*w - 13],
[139, 139, w + 27],
[139, 139, w + 112],
[151, 151, 8*w - 45],
[151, 151, -10*w + 57],
[163, 163, w + 69],
[163, 163, w + 94],
[169, 13, -13],
[173, 173, w + 42],
[173, 173, w + 131],
[181, 181, w + 45],
[181, 181, w + 136],
[191, 191, -w - 15],
[191, 191, w - 15],
[197, 197, w + 25],
[197, 197, w + 172],
[211, 211, w + 33],
[211, 211, w + 178],
[223, 223, -3*w - 23],
[223, 223, 3*w - 23],
[227, 227, w + 48],
[227, 227, w + 179],
[239, 239, -5*w + 33],
[239, 239, -23*w + 135],
[257, 257, -3*w - 7],
[257, 257, 3*w - 7],
[263, 263, -24*w + 139],
[263, 263, 6*w - 31],
[269, 269, w + 29],
[269, 269, w + 240],
[271, 271, -9*w + 55],
[271, 271, -15*w + 89],
[277, 277, w + 119],
[277, 277, w + 158],
[281, 281, 3*w - 5],
[281, 281, -3*w - 5],
[283, 283, w + 113],
[283, 283, w + 170],
[317, 317, w + 44],
[317, 317, w + 273],
[347, 347, w + 46],
[347, 347, w + 301],
[353, 353, 9*w - 49],
[353, 353, -21*w + 121],
[359, 359, -25*w + 147],
[359, 359, -7*w + 45],
[361, 19, -19],
[379, 379, w + 105],
[379, 379, w + 274],
[383, 383, 6*w - 29],
[383, 383, -36*w + 209],
[397, 397, w + 35],
[397, 397, w + 362],
[409, 409, -5*w - 21],
[409, 409, 5*w - 21],
[419, 419, w + 156],
[419, 419, w + 263],
[433, 433, -12*w + 73],
[433, 433, -18*w + 107],
[457, 457, 6*w + 41],
[457, 457, -6*w + 41],
[463, 463, -4*w - 9],
[463, 463, 4*w - 9],
[499, 499, w + 212],
[499, 499, w + 287],
[529, 23, -23],
[541, 541, w + 121],
[541, 541, w + 420],
[547, 547, w + 157],
[547, 547, w + 390],
[569, 569, -28*w + 165],
[569, 569, -10*w + 63],
[571, 571, w + 264],
[571, 571, w + 307],
[577, 577, -47*w + 273],
[577, 577, 7*w - 33],
[593, 593, 2*w - 27],
[593, 593, -2*w - 27],
[599, 599, -6*w - 25],
[599, 599, 6*w - 25],
[619, 619, w + 167],
[619, 619, w + 452],
[631, 631, -15*w + 91],
[631, 631, -21*w + 125],
[643, 643, w + 57],
[643, 643, w + 586],
[647, 647, -11*w + 69],
[647, 647, -29*w + 171],
[653, 653, w + 120],
[653, 653, w + 533],
[677, 677, w + 64],
[677, 677, w + 613],
[683, 683, w + 248],
[683, 683, w + 435],
[691, 691, w + 322],
[691, 691, w + 369],
[709, 709, w + 265],
[709, 709, w + 444],
[727, 727, -9*w + 59],
[727, 727, -39*w + 229],
[761, 761, -27*w + 155],
[761, 761, 15*w - 83],
[769, 769, 5*w - 9],
[769, 769, -5*w - 9],
[787, 787, w + 63],
[787, 787, w + 724],
[811, 811, w + 70],
[811, 811, w + 741],
[821, 821, w + 149],
[821, 821, w + 672],
[827, 827, w + 193],
[827, 827, w + 634],
[853, 853, w + 253],
[853, 853, w + 600],
[863, 863, -6*w - 19],
[863, 863, 6*w - 19],
[877, 877, w + 339],
[877, 877, w + 538],
[907, 907, w + 74],
[907, 907, w + 833],
[919, 919, -3*w - 35],
[919, 919, 3*w - 35],
[937, 937, 7*w - 27],
[937, 937, -7*w - 27],
[941, 941, w + 144],
[941, 941, w + 797],
[947, 947, w + 292],
[947, 947, w + 655],
[953, 953, 2*w - 33],
[953, 953, -2*w - 33],
[961, 31, -31],
[967, 967, 9*w - 61],
[967, 967, -9*w - 61],
[977, 977, -4*w - 39],
[977, 977, 4*w - 39],
[997, 997, w + 55],
[997, 997, w + 942]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - 19*x^7 + 120*x^5 - 274*x^3 + 4*x^2 + 152*x - 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1, 1, 1/8*e^8 + 1/4*e^7 - 11/8*e^6 - 11/4*e^5 + 3*e^4 + 7*e^3 + 11/4*e^2 - 2*e - 3, 1/8*e^8 + 1/4*e^7 - 11/8*e^6 - 11/4*e^5 + 3*e^4 + 7*e^3 + 11/4*e^2 - 2*e - 3, -1/4*e^8 - 1/2*e^7 + 13/4*e^6 + 11/2*e^5 - 25/2*e^4 - 14*e^3 + 33/2*e^2 + 4*e - 6, -1/4*e^8 - 1/2*e^7 + 13/4*e^6 + 11/2*e^5 - 25/2*e^4 - 14*e^3 + 33/2*e^2 + 4*e - 6, 1/2*e^7 + e^6 - 11/2*e^5 - 11*e^4 + 14*e^3 + 28*e^2 - 5*e - 8, -1/8*e^8 - 1/4*e^7 + 15/8*e^6 + 11/4*e^5 - 17/2*e^4 - 7*e^3 + 41/4*e^2 + 2*e + 3, -1/8*e^8 - 1/4*e^7 + 15/8*e^6 + 11/4*e^5 - 17/2*e^4 - 7*e^3 + 41/4*e^2 + 2*e + 3, -1/8*e^8 - 1/4*e^7 + 15/8*e^6 + 11/4*e^5 - 17/2*e^4 - 7*e^3 + 49/4*e^2 + 2*e - 7, -1/8*e^8 - 1/4*e^7 + 15/8*e^6 + 11/4*e^5 - 17/2*e^4 - 7*e^3 + 49/4*e^2 + 2*e - 7, -1/2*e^8 - e^7 + 13/2*e^6 + 12*e^5 - 25*e^4 - 39*e^3 + 33*e^2 + 34*e - 12, -1/2*e^8 - e^7 + 13/2*e^6 + 12*e^5 - 25*e^4 - 39*e^3 + 33*e^2 + 34*e - 12, 3/4*e^8 + 2*e^7 - 37/4*e^6 - 24*e^5 + 31*e^4 + 76*e^3 - 55/2*e^2 - 57*e + 14, -1/8*e^8 - 1/4*e^7 + 15/8*e^6 + 15/4*e^5 - 17/2*e^4 - 18*e^3 + 49/4*e^2 + 28*e - 7, -1/8*e^8 - 1/4*e^7 + 15/8*e^6 + 15/4*e^5 - 17/2*e^4 - 18*e^3 + 49/4*e^2 + 28*e - 7, -1/2*e^7 + 13/2*e^5 - 23*e^3 - 2*e^2 + 17*e + 8, -1/2*e^7 + 13/2*e^5 - 23*e^3 - 2*e^2 + 17*e + 8, -1/2*e^6 - e^5 + 11/2*e^4 + 9*e^3 - 13*e^2 - 14*e - 2, -1/2*e^6 - e^5 + 11/2*e^4 + 9*e^3 - 13*e^2 - 14*e - 2, -1/2*e^6 + 9/2*e^4 - 2*e^3 - 6*e^2 + 12*e - 4, -1/2*e^6 + 9/2*e^4 - 2*e^3 - 6*e^2 + 12*e - 4, 5/8*e^8 + 9/4*e^7 - 51/8*e^6 - 107/4*e^5 + 23/2*e^4 + 83*e^3 + 51/4*e^2 - 56*e - 9, 5/8*e^8 + 9/4*e^7 - 51/8*e^6 - 107/4*e^5 + 23/2*e^4 + 83*e^3 + 51/4*e^2 - 56*e - 9, 1/2*e^8 + 2*e^7 - 5*e^6 - 23*e^5 + 17/2*e^4 + 65*e^3 + 10*e^2 - 28*e - 6, 1/2*e^8 + 2*e^7 - 5*e^6 - 23*e^5 + 17/2*e^4 + 65*e^3 + 10*e^2 - 28*e - 6, -1/2*e^6 + 13/2*e^4 - 2*e^3 - 24*e^2 + 12*e + 20, -1/2*e^6 + 13/2*e^4 - 2*e^3 - 24*e^2 + 12*e + 20, e^6 + e^5 - 11*e^4 - 9*e^3 + 30*e^2 + 16*e - 18, e^6 + e^5 - 11*e^4 - 9*e^3 + 30*e^2 + 16*e - 18, 1/4*e^8 + 1/2*e^7 - 15/4*e^6 - 13/2*e^5 + 19*e^4 + 25*e^3 - 73/2*e^2 - 28*e + 10, 1/4*e^8 + 1/2*e^7 - 15/4*e^6 - 13/2*e^5 + 19*e^4 + 25*e^3 - 73/2*e^2 - 28*e + 10, -1/2*e^8 - 2*e^7 + 11/2*e^6 + 24*e^5 - 14*e^4 - 74*e^3 + 5*e^2 + 46*e - 6, -1/2*e^8 - 2*e^7 + 11/2*e^6 + 24*e^5 - 14*e^4 - 74*e^3 + 5*e^2 + 46*e - 6, -1/4*e^8 - 1/2*e^7 + 11/4*e^6 + 11/2*e^5 - 6*e^4 - 16*e^3 - 7/2*e^2 + 16*e - 2, -1/4*e^8 - 1/2*e^7 + 11/4*e^6 + 11/2*e^5 - 6*e^4 - 16*e^3 - 7/2*e^2 + 16*e - 2, -1/2*e^6 + e^5 + 9/2*e^4 - 11*e^3 - 6*e^2 + 28*e + 10, 3/8*e^8 + 3/4*e^7 - 41/8*e^6 - 33/4*e^5 + 22*e^4 + 23*e^3 - 135/4*e^2 - 22*e + 11, 3/8*e^8 + 3/4*e^7 - 41/8*e^6 - 33/4*e^5 + 22*e^4 + 23*e^3 - 135/4*e^2 - 22*e + 11, -3/8*e^8 - 7/4*e^7 + 21/8*e^6 + 81/4*e^5 + 11/2*e^4 - 58*e^3 - 133/4*e^2 + 30*e - 1, -3/8*e^8 - 7/4*e^7 + 21/8*e^6 + 81/4*e^5 + 11/2*e^4 - 58*e^3 - 133/4*e^2 + 30*e - 1, 1/2*e^8 + 2*e^7 - 11/2*e^6 - 25*e^5 + 14*e^4 + 85*e^3 - 7*e^2 - 68*e + 12, 1/2*e^8 + 2*e^7 - 11/2*e^6 - 25*e^5 + 14*e^4 + 85*e^3 - 7*e^2 - 68*e + 12, 5/8*e^8 + 5/4*e^7 - 71/8*e^6 - 63/4*e^5 + 39*e^4 + 53*e^3 - 225/4*e^2 - 38*e + 13, 5/8*e^8 + 5/4*e^7 - 71/8*e^6 - 63/4*e^5 + 39*e^4 + 53*e^3 - 225/4*e^2 - 38*e + 13, 1/4*e^8 + 1/2*e^7 - 11/4*e^6 - 11/2*e^5 + 6*e^4 + 16*e^3 + 15/2*e^2 - 20*e - 14, 1/4*e^8 + 1/2*e^7 - 11/4*e^6 - 11/2*e^5 + 6*e^4 + 16*e^3 + 15/2*e^2 - 20*e - 14, 1/2*e^8 + e^7 - 6*e^6 - 10*e^5 + 39/2*e^4 + 19*e^3 - 20*e^2 + 2*e + 10, 1/2*e^8 + e^7 - 6*e^6 - 10*e^5 + 39/2*e^4 + 19*e^3 - 20*e^2 + 2*e + 10, 5/4*e^8 + 5/2*e^7 - 65/4*e^6 - 59/2*e^5 + 121/2*e^4 + 92*e^3 - 137/2*e^2 - 68*e + 22, 5/4*e^8 + 5/2*e^7 - 65/4*e^6 - 59/2*e^5 + 121/2*e^4 + 92*e^3 - 137/2*e^2 - 68*e + 22, 1/2*e^8 + e^7 - 15/2*e^6 - 12*e^5 + 36*e^4 + 39*e^3 - 59*e^2 - 38*e + 16, 1/2*e^8 + e^7 - 15/2*e^6 - 12*e^5 + 36*e^4 + 39*e^3 - 59*e^2 - 38*e + 16, e^8 + 2*e^7 - 13*e^6 - 23*e^5 + 48*e^4 + 65*e^3 - 48*e^2 - 28*e - 2, e^8 + 2*e^7 - 13*e^6 - 23*e^5 + 48*e^4 + 65*e^3 - 48*e^2 - 28*e - 2, e^7 + e^6 - 13*e^5 - 11*e^4 + 46*e^3 + 30*e^2 - 34*e - 12, e^7 + e^6 - 13*e^5 - 11*e^4 + 46*e^3 + 30*e^2 - 34*e - 12, -9/8*e^8 - 9/4*e^7 + 115/8*e^6 + 107/4*e^5 - 53*e^4 - 83*e^3 + 245/4*e^2 + 54*e - 13, -9/8*e^8 - 9/4*e^7 + 115/8*e^6 + 107/4*e^5 - 53*e^4 - 83*e^3 + 245/4*e^2 + 54*e - 13, -1/2*e^8 - e^7 + 7*e^6 + 12*e^5 - 61/2*e^4 - 37*e^3 + 46*e^2 + 18*e - 14, -1/2*e^8 - e^7 + 7*e^6 + 12*e^5 - 61/2*e^4 - 37*e^3 + 46*e^2 + 18*e - 14, 5/8*e^8 + 5/4*e^7 - 75/8*e^6 - 67/4*e^5 + 89/2*e^4 + 62*e^3 - 293/4*e^2 - 52*e + 23, 5/8*e^8 + 5/4*e^7 - 75/8*e^6 - 67/4*e^5 + 89/2*e^4 + 62*e^3 - 293/4*e^2 - 52*e + 23, 1/2*e^8 + 3/2*e^7 - 11/2*e^6 - 35/2*e^5 + 12*e^4 + 51*e^3 + 9*e^2 - 25*e - 4, 1/2*e^8 + 3/2*e^7 - 11/2*e^6 - 35/2*e^5 + 12*e^4 + 51*e^3 + 9*e^2 - 25*e - 4, -1/4*e^8 - 1/2*e^7 + 15/4*e^6 + 15/2*e^5 - 19*e^4 - 34*e^3 + 81/2*e^2 + 40*e - 26, -1/4*e^8 - 1/2*e^7 + 15/4*e^6 + 15/2*e^5 - 19*e^4 - 34*e^3 + 81/2*e^2 + 40*e - 26, 3/8*e^8 + 11/4*e^7 - 13/8*e^6 - 129/4*e^5 - 33/2*e^4 + 95*e^3 + 253/4*e^2 - 46*e - 13, 3/8*e^8 + 11/4*e^7 - 13/8*e^6 - 129/4*e^5 - 33/2*e^4 + 95*e^3 + 253/4*e^2 - 46*e - 13, 3/4*e^8 + 7/2*e^7 - 29/4*e^6 - 85/2*e^5 + 11*e^4 + 136*e^3 + 33/2*e^2 - 100*e - 14, 3/4*e^8 + 7/2*e^7 - 29/4*e^6 - 85/2*e^5 + 11*e^4 + 136*e^3 + 33/2*e^2 - 100*e - 14, 1/2*e^8 + 1/2*e^7 - 13/2*e^6 - 11/2*e^5 + 23*e^4 + 14*e^3 - 21*e^2 - 3*e + 12, 1/2*e^8 + 1/2*e^7 - 13/2*e^6 - 11/2*e^5 + 23*e^4 + 14*e^3 - 21*e^2 - 3*e + 12, -1/2*e^8 - e^7 + 13/2*e^6 + 11*e^5 - 27*e^4 - 28*e^3 + 51*e^2 + 4*e - 32, -1/2*e^8 - e^7 + 13/2*e^6 + 11*e^5 - 27*e^4 - 28*e^3 + 51*e^2 + 4*e - 32, -1/2*e^8 - e^7 + 8*e^6 + 12*e^5 - 85/2*e^4 - 39*e^3 + 83*e^2 + 36*e - 14, 1/4*e^8 + 1/2*e^7 - 3/4*e^6 - 9/2*e^5 - 16*e^4 + 3*e^3 + 119/2*e^2 + 24*e - 14, 1/4*e^8 + 1/2*e^7 - 3/4*e^6 - 9/2*e^5 - 16*e^4 + 3*e^3 + 119/2*e^2 + 24*e - 14, -e^8 - 3*e^7 + 11*e^6 + 34*e^5 - 30*e^4 - 91*e^3 + 24*e^2 + 24*e - 16, -e^8 - 3*e^7 + 11*e^6 + 34*e^5 - 30*e^4 - 91*e^3 + 24*e^2 + 24*e - 16, -9/8*e^8 - 9/4*e^7 + 111/8*e^6 + 103/4*e^5 - 95/2*e^4 - 70*e^3 + 193/4*e^2 + 12*e - 15, -9/8*e^8 - 9/4*e^7 + 111/8*e^6 + 103/4*e^5 - 95/2*e^4 - 70*e^3 + 193/4*e^2 + 12*e - 15, -1/4*e^8 - e^7 + 13/4*e^6 + 12*e^5 - 29/2*e^4 - 41*e^3 + 65/2*e^2 + 45*e - 22, -1/4*e^8 - e^7 + 13/4*e^6 + 12*e^5 - 29/2*e^4 - 41*e^3 + 65/2*e^2 + 45*e - 22, -3/4*e^8 - 7/2*e^7 + 29/4*e^6 + 85/2*e^5 - 9*e^4 - 136*e^3 - 53/2*e^2 + 92*e - 2, -3/4*e^8 - 7/2*e^7 + 29/4*e^6 + 85/2*e^5 - 9*e^4 - 136*e^3 - 53/2*e^2 + 92*e - 2, 1/4*e^8 - 13/4*e^6 + e^5 + 25/2*e^4 - 9*e^3 - 37/2*e^2 + 9*e + 14, 1/4*e^8 - 13/4*e^6 + e^5 + 25/2*e^4 - 9*e^3 - 37/2*e^2 + 9*e + 14, -7/4*e^8 - 3*e^7 + 91/4*e^6 + 34*e^5 - 171/2*e^4 - 93*e^3 + 199/2*e^2 + 35*e - 26, -7/4*e^8 - 3*e^7 + 91/4*e^6 + 34*e^5 - 171/2*e^4 - 93*e^3 + 199/2*e^2 + 35*e - 26, -e^8 - 2*e^7 + 13*e^6 + 24*e^5 - 50*e^4 - 78*e^3 + 68*e^2 + 64*e - 30, -e^8 - 2*e^7 + 13*e^6 + 24*e^5 - 50*e^4 - 78*e^3 + 68*e^2 + 64*e - 30, -5/4*e^8 - 5/2*e^7 + 71/4*e^6 + 61/2*e^5 - 78*e^4 - 99*e^3 + 229/2*e^2 + 80*e - 26, -5/4*e^8 - 5/2*e^7 + 71/4*e^6 + 61/2*e^5 - 78*e^4 - 99*e^3 + 229/2*e^2 + 80*e - 26, -1/4*e^8 + 21/4*e^6 + e^5 - 73/2*e^4 - 9*e^3 + 177/2*e^2 + 19*e - 14, 5/8*e^8 + 5/4*e^7 - 59/8*e^6 - 55/4*e^5 + 45/2*e^4 + 37*e^3 - 85/4*e^2 - 30*e + 23, 5/8*e^8 + 5/4*e^7 - 59/8*e^6 - 55/4*e^5 + 45/2*e^4 + 37*e^3 - 85/4*e^2 - 30*e + 23, 3/2*e^8 + 3*e^7 - 39/2*e^6 - 36*e^5 + 71*e^4 + 115*e^3 - 63*e^2 - 84*e - 8, 3/2*e^8 + 3*e^7 - 39/2*e^6 - 36*e^5 + 71*e^4 + 115*e^3 - 63*e^2 - 84*e - 8, e^8 + 2*e^7 - 14*e^6 - 23*e^5 + 61*e^4 + 65*e^3 - 92*e^2 - 32*e + 26, e^8 + 2*e^7 - 14*e^6 - 23*e^5 + 61*e^4 + 65*e^3 - 92*e^2 - 32*e + 26, 1/2*e^8 + e^7 - 15/2*e^6 - 12*e^5 + 34*e^4 + 37*e^3 - 41*e^2 - 24*e - 8, 1/2*e^8 + e^7 - 15/2*e^6 - 12*e^5 + 34*e^4 + 37*e^3 - 41*e^2 - 24*e - 8, 1/2*e^8 + e^7 - 6*e^6 - 12*e^5 + 41/2*e^4 + 39*e^3 - 27*e^2 - 28*e + 14, 1/2*e^8 + e^7 - 6*e^6 - 12*e^5 + 41/2*e^4 + 39*e^3 - 27*e^2 - 28*e + 14, 2*e^6 + 3*e^5 - 24*e^4 - 31*e^3 + 74*e^2 + 64*e - 38, 2*e^6 + 3*e^5 - 24*e^4 - 31*e^3 + 74*e^2 + 64*e - 38, -e^8 - 3*e^7 + 11*e^6 + 38*e^5 - 26*e^4 - 131*e^3 - 8*e^2 + 108*e + 20, -e^8 - 3*e^7 + 11*e^6 + 38*e^5 - 26*e^4 - 131*e^3 - 8*e^2 + 108*e + 20, -3/2*e^8 - 5*e^7 + 33/2*e^6 + 59*e^5 - 40*e^4 - 176*e^3 - e^2 + 88*e + 8, -3/2*e^8 - 5*e^7 + 33/2*e^6 + 59*e^5 - 40*e^4 - 176*e^3 - e^2 + 88*e + 8, -2*e^8 - 5*e^7 + 47/2*e^6 + 56*e^5 - 141/2*e^4 - 149*e^3 + 45*e^2 + 52*e - 6, -2*e^8 - 5*e^7 + 47/2*e^6 + 56*e^5 - 141/2*e^4 - 149*e^3 + 45*e^2 + 52*e - 6, 3/4*e^8 + 3/2*e^7 - 45/4*e^6 - 39/2*e^5 + 53*e^4 + 71*e^3 - 171/2*e^2 - 60*e + 30, 3/4*e^8 + 3/2*e^7 - 45/4*e^6 - 39/2*e^5 + 53*e^4 + 71*e^3 - 171/2*e^2 - 60*e + 30, -3/2*e^8 - 2*e^7 + 45/2*e^6 + 23*e^5 - 104*e^4 - 67*e^3 + 153*e^2 + 40*e - 48, -3/2*e^8 - 2*e^7 + 45/2*e^6 + 23*e^5 - 104*e^4 - 67*e^3 + 153*e^2 + 40*e - 48, 1/8*e^8 + 1/4*e^7 + 5/8*e^6 - 3/4*e^5 - 19*e^4 - 11*e^3 + 251/4*e^2 + 34*e - 47, 1/8*e^8 + 1/4*e^7 + 5/8*e^6 - 3/4*e^5 - 19*e^4 - 11*e^3 + 251/4*e^2 + 34*e - 47, 3/8*e^8 + 3/4*e^7 - 49/8*e^6 - 41/4*e^5 + 33*e^4 + 37*e^3 - 271/4*e^2 - 26*e + 43, 3/8*e^8 + 3/4*e^7 - 49/8*e^6 - 41/4*e^5 + 33*e^4 + 37*e^3 - 271/4*e^2 - 26*e + 43, -1/4*e^8 - 1/2*e^7 + 9/4*e^6 + 11/2*e^5 - 7/2*e^4 - 14*e^3 + 17/2*e^2 + 12*e - 22, -1/4*e^8 - 1/2*e^7 + 9/4*e^6 + 11/2*e^5 - 7/2*e^4 - 14*e^3 + 17/2*e^2 + 12*e - 22, -1/2*e^8 + e^7 + 19/2*e^6 - 13*e^5 - 56*e^4 + 46*e^3 + 101*e^2 - 44*e - 16, -1/2*e^8 + e^7 + 19/2*e^6 - 13*e^5 - 56*e^4 + 46*e^3 + 101*e^2 - 44*e - 16, -11/8*e^8 - 15/4*e^7 + 133/8*e^6 + 177/4*e^5 - 111/2*e^4 - 136*e^3 + 219/4*e^2 + 102*e - 9, -11/8*e^8 - 15/4*e^7 + 133/8*e^6 + 177/4*e^5 - 111/2*e^4 - 136*e^3 + 219/4*e^2 + 102*e - 9, e^8 + e^7 - 15*e^6 - 11*e^5 + 74*e^4 + 32*e^3 - 134*e^2 - 30*e + 26, e^8 + e^7 - 15*e^6 - 11*e^5 + 74*e^4 + 32*e^3 - 134*e^2 - 30*e + 26, -2*e^8 - 4*e^7 + 24*e^6 + 46*e^5 - 74*e^4 - 134*e^3 + 40*e^2 + 84*e + 14, -2*e^8 - 4*e^7 + 24*e^6 + 46*e^5 - 74*e^4 - 134*e^3 + 40*e^2 + 84*e + 14, -1/2*e^8 - e^7 + 4*e^6 + 10*e^5 + 11/2*e^4 - 21*e^3 - 57*e^2 + 12*e + 18, -1/2*e^8 - e^7 + 4*e^6 + 10*e^5 + 11/2*e^4 - 21*e^3 - 57*e^2 + 12*e + 18, -e^8 - 2*e^7 + 11*e^6 + 25*e^5 - 24*e^4 - 87*e^3 - 22*e^2 + 84*e + 28, -e^8 - 2*e^7 + 11*e^6 + 25*e^5 - 24*e^4 - 87*e^3 - 22*e^2 + 84*e + 28, -3/4*e^8 - 7/2*e^7 + 17/4*e^6 + 83/2*e^5 + 22*e^4 - 127*e^3 - 197/2*e^2 + 76*e + 26, -3/4*e^8 - 7/2*e^7 + 17/4*e^6 + 83/2*e^5 + 22*e^4 - 127*e^3 - 197/2*e^2 + 76*e + 26, -5/8*e^8 - 13/4*e^7 + 47/8*e^6 + 159/4*e^5 - 4*e^4 - 129*e^3 - 167/4*e^2 + 94*e + 11, -5/8*e^8 - 13/4*e^7 + 47/8*e^6 + 159/4*e^5 - 4*e^4 - 129*e^3 - 167/4*e^2 + 94*e + 11, -2*e^7 - 9/2*e^6 + 22*e^5 + 101/2*e^4 - 50*e^3 - 136*e^2 - 32*e + 60, -2*e^7 - 9/2*e^6 + 22*e^5 + 101/2*e^4 - 50*e^3 - 136*e^2 - 32*e + 60, 9/8*e^8 + 9/4*e^7 - 111/8*e^6 - 99/4*e^5 + 91/2*e^4 + 61*e^3 - 113/4*e^2 - 14*e - 33, 9/8*e^8 + 9/4*e^7 - 111/8*e^6 - 99/4*e^5 + 91/2*e^4 + 61*e^3 - 113/4*e^2 - 14*e - 33, -2*e^8 - 6*e^7 + 23*e^6 + 67*e^5 - 65*e^4 - 179*e^3 + 24*e^2 + 78*e + 28, -2*e^8 - 6*e^7 + 23*e^6 + 67*e^5 - 65*e^4 - 179*e^3 + 24*e^2 + 78*e + 28, 17/8*e^8 + 21/4*e^7 - 199/8*e^6 - 235/4*e^5 + 147/2*e^4 + 160*e^3 - 153/4*e^2 - 82*e - 25, 17/8*e^8 + 21/4*e^7 - 199/8*e^6 - 235/4*e^5 + 147/2*e^4 + 160*e^3 - 153/4*e^2 - 82*e - 25, 3/2*e^8 + 3*e^7 - 41/2*e^6 - 34*e^5 + 84*e^4 + 93*e^3 - 111*e^2 - 28*e + 32, 3/2*e^8 + 3*e^7 - 41/2*e^6 - 34*e^5 + 84*e^4 + 93*e^3 - 111*e^2 - 28*e + 32, -1/2*e^8 - 3*e^7 + 3*e^6 + 36*e^5 + 19/2*e^4 - 115*e^3 - 38*e^2 + 98*e + 2, -1/2*e^8 - 3*e^7 + 3*e^6 + 36*e^5 + 19/2*e^4 - 115*e^3 - 38*e^2 + 98*e + 2, 3/4*e^8 + 2*e^7 - 37/4*e^6 - 22*e^5 + 35*e^4 + 58*e^3 - 119/2*e^2 - 25*e + 34, 3/4*e^8 + 2*e^7 - 37/4*e^6 - 22*e^5 + 35*e^4 + 58*e^3 - 119/2*e^2 - 25*e + 34, -9/8*e^8 - 9/4*e^7 + 119/8*e^6 + 115/4*e^5 - 109/2*e^4 - 107*e^3 + 177/4*e^2 + 114*e - 1, -9/8*e^8 - 9/4*e^7 + 119/8*e^6 + 115/4*e^5 - 109/2*e^4 - 107*e^3 + 177/4*e^2 + 114*e - 1, 2*e^7 + 2*e^6 - 24*e^5 - 22*e^4 + 78*e^3 + 56*e^2 - 76*e - 20, 2*e^7 + 2*e^6 - 24*e^5 - 22*e^4 + 78*e^3 + 56*e^2 - 76*e - 20, 3/2*e^8 + 7/2*e^7 - 35/2*e^6 - 85/2*e^5 + 47*e^4 + 142*e^3 + 5*e^2 - 129*e - 20, 3/2*e^8 + 7/2*e^7 - 35/2*e^6 - 85/2*e^5 + 47*e^4 + 142*e^3 + 5*e^2 - 129*e - 20, -5/4*e^8 - 4*e^7 + 51/4*e^6 + 46*e^5 - 23*e^4 - 132*e^3 - 51/2*e^2 + 63*e + 54, 1/2*e^8 + e^7 - 9*e^6 - 14*e^5 + 105/2*e^4 + 59*e^3 - 106*e^2 - 70*e + 46, 1/2*e^8 + e^7 - 9*e^6 - 14*e^5 + 105/2*e^4 + 59*e^3 - 106*e^2 - 70*e + 46, e^8 + 4*e^7 - 10*e^6 - 46*e^5 + 15*e^4 + 130*e^3 + 38*e^2 - 56*e - 26, e^8 + 4*e^7 - 10*e^6 - 46*e^5 + 15*e^4 + 130*e^3 + 38*e^2 - 56*e - 26, 1/8*e^8 - 3/4*e^7 - 23/8*e^6 + 33/4*e^5 + 43/2*e^4 - 19*e^3 - 217/4*e^2 + 4*e + 7, 1/8*e^8 - 3/4*e^7 - 23/8*e^6 + 33/4*e^5 + 43/2*e^4 - 19*e^3 - 217/4*e^2 + 4*e + 7];
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 | {3, 3, w + 2}>] := -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;