/*
  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![-98, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {6,6,5*w - 52}>;

primesArray := [
[2, 2, -17*w - 160],
[2, 2, -17*w + 177],
[3, 3, -842*w + 8767],
[7, 7, -2*w + 21],
[7, 7, 2*w + 19],
[13, 13, -12*w - 113],
[13, 13, 12*w - 125],
[17, 17, 182*w - 1895],
[17, 17, 182*w + 1713],
[23, 23, -512*w - 4819],
[23, 23, 512*w - 5331],
[25, 5, -5],
[29, 29, 22*w - 229],
[29, 29, 22*w + 207],
[43, 43, 114*w + 1073],
[43, 43, 114*w - 1187],
[47, 47, 8*w - 83],
[47, 47, -8*w - 75],
[61, 61, -1172*w - 11031],
[61, 61, 1172*w - 12203],
[71, 71, 216*w + 2033],
[71, 71, 216*w - 2249],
[83, 83, 8932*w - 93001],
[83, 83, -2196*w + 22865],
[109, 109, -4*w - 39],
[109, 109, 4*w - 43],
[121, 11, -11],
[131, 131, 5564*w - 57933],
[137, 137, 2*w - 17],
[137, 137, -2*w - 15],
[149, 149, 18*w - 187],
[149, 149, 18*w + 169],
[151, 151, -1502*w - 14137],
[151, 151, -1502*w + 15639],
[173, 173, -6*w - 55],
[173, 173, 6*w - 61],
[193, 193, 808*w - 8413],
[193, 193, -808*w - 7605],
[197, 197, 2*w - 15],
[197, 197, -2*w - 13],
[211, 211, -70*w + 729],
[211, 211, -70*w - 659],
[227, 227, 284*w + 2673],
[227, 227, 284*w - 2957],
[251, 251, 124*w - 1291],
[251, 251, 124*w + 1167],
[257, 257, 42*w + 395],
[257, 257, 42*w - 437],
[271, 271, 410*w + 3859],
[271, 271, 410*w - 4269],
[277, 277, 4*w - 45],
[277, 277, -4*w - 41],
[281, 281, -3550*w - 33413],
[281, 281, 3550*w - 36963],
[283, 283, 2*w - 27],
[283, 283, -2*w - 25],
[293, 293, 2*w - 11],
[293, 293, -2*w - 9],
[307, 307, -6*w - 59],
[307, 307, 6*w - 65],
[337, 337, 376*w - 3915],
[337, 337, 376*w + 3539],
[347, 347, -4*w - 33],
[347, 347, -4*w + 37],
[359, 359, -8*w - 73],
[359, 359, -8*w + 81],
[361, 19, -19],
[367, 367, -774*w + 8059],
[367, 367, -774*w - 7285],
[379, 379, -30*w + 313],
[379, 379, -30*w - 283],
[389, 389, 2*w - 3],
[389, 389, -2*w - 1],
[397, 397, -172*w + 1791],
[397, 397, -172*w - 1619],
[401, 401, 6*w - 59],
[401, 401, 6*w + 53],
[409, 409, -40*w - 377],
[409, 409, 40*w - 417],
[419, 419, 100*w - 1041],
[419, 419, -100*w - 941],
[421, 421, 308*w + 2899],
[421, 421, 308*w - 3207],
[439, 439, 274*w - 2853],
[439, 439, 274*w + 2579],
[443, 443, -52*w - 489],
[443, 443, 52*w - 541],
[449, 449, 15338*w - 159701],
[449, 449, -6918*w + 72031],
[457, 457, 17352*w - 180671],
[457, 457, 6224*w - 64805],
[461, 461, 614*w - 6393],
[461, 461, 614*w + 5779],
[479, 479, 192*w - 1999],
[479, 479, 192*w + 1807],
[487, 487, -94*w + 979],
[487, 487, -94*w - 885],
[491, 491, -20*w - 187],
[491, 491, 20*w - 207],
[503, 503, -16*w + 165],
[503, 503, 16*w + 149],
[509, 509, 2890*w - 30091],
[509, 509, -2890*w - 27201],
[521, 521, 11970*w - 124633],
[521, 521, -10286*w + 107099],
[577, 577, -1104*w - 10391],
[577, 577, 1104*w - 11495],
[593, 593, 134*w + 1261],
[593, 593, 134*w - 1395],
[601, 601, -8*w - 79],
[601, 601, 8*w - 87],
[613, 613, -2492*w - 23455],
[613, 613, 2492*w - 25947],
[617, 617, 226*w - 2353],
[617, 617, 226*w + 2127],
[631, 631, 2*w - 33],
[631, 631, -2*w - 31],
[677, 677, -62*w - 583],
[677, 677, 62*w - 645],
[691, 691, 1798*w - 18721],
[691, 691, -1798*w - 16923],
[739, 739, 74*w + 697],
[739, 739, 74*w - 771],
[743, 743, 48*w - 499],
[743, 743, 48*w + 451],
[757, 757, -44*w + 459],
[757, 757, -44*w - 415],
[761, 761, 14*w + 129],
[761, 761, 14*w - 143],
[769, 769, 16*w + 153],
[769, 769, 16*w - 169],
[773, 773, 4574*w - 47625],
[773, 773, -39938*w + 415839],
[787, 787, 3846*w + 36199],
[787, 787, 3846*w - 40045],
[809, 809, 30*w + 281],
[809, 809, -30*w + 311],
[811, 811, 1070*w - 11141],
[811, 811, -1070*w - 10071],
[829, 829, -4*w - 47],
[829, 829, 4*w - 51],
[857, 857, -18*w + 185],
[857, 857, 18*w + 167],
[877, 877, -196*w - 1845],
[877, 877, -196*w + 2041],
[881, 881, -26*w - 243],
[881, 881, 26*w - 269],
[907, 907, -14*w - 135],
[907, 907, -14*w + 149],
[937, 937, 8*w - 89],
[937, 937, -8*w - 81],
[941, 941, 294*w - 3061],
[941, 941, 294*w + 2767],
[947, 947, 4*w - 27],
[947, 947, -4*w - 23],
[961, 31, -31],
[971, 971, -1604*w + 16701],
[971, 971, 1604*w + 15097],
[983, 983, 144*w - 1499],
[983, 983, 144*w + 1355],
[991, 991, -10*w - 99],
[991, 991, 10*w - 109],
[997, 997, 604*w - 6289],
[997, 997, 604*w + 5685]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 12*x^5 + x^4 + 40*x^3 - 6*x^2 - 28*x + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -1, -1/2*e^5 + 4*e^3 - 1/2*e^2 - 6*e + 1, -1/4*e^6 + 3*e^4 + 3/4*e^3 - 10*e^2 - 9/2*e + 6, 1/4*e^6 - 2*e^4 + 5/4*e^3 + 3*e^2 - 11/2*e - 1, 1/4*e^6 - 3*e^4 - 3/4*e^3 + 10*e^2 + 9/2*e - 7, 1/4*e^6 - 4*e^4 - 3/4*e^3 + 16*e^2 + 7/2*e - 9, 1/4*e^6 + 1/2*e^5 - 3*e^4 - 19/4*e^3 + 19/2*e^2 + 19/2*e - 6, -1/4*e^6 + 3*e^4 - 1/4*e^3 - 11*e^2 + 1/2*e + 9, 1/2*e^5 - 4*e^3 + 1/2*e^2 + 8*e - 1, -1/2*e^6 - 1/2*e^5 + 5*e^4 + 7/2*e^3 - 27/2*e^2 - 5*e + 9, 1/2*e^5 - 4*e^3 + 1/2*e^2 + 4*e - 4, -3/4*e^6 - 1/2*e^5 + 9*e^4 + 17/4*e^3 - 57/2*e^2 - 15/2*e + 12, 1/2*e^6 + 1/2*e^5 - 4*e^4 - 7/2*e^3 + 17/2*e^2 + 5*e - 13, -1/2*e^6 + 5*e^4 - 3/2*e^3 - 14*e^2 + 6*e + 9, 1/4*e^6 - 3*e^4 - 3/4*e^3 + 10*e^2 + 5/2*e - 10, -1/4*e^6 + 2*e^4 - 1/4*e^3 - 2*e^2 - 3/2*e - 5, -1/2*e^6 - 1/2*e^5 + 6*e^4 + 9/2*e^3 - 41/2*e^2 - 7*e + 17, 1/4*e^6 - 1/2*e^5 - 4*e^4 + 13/4*e^3 + 33/2*e^2 - 3/2*e - 12, -5/4*e^6 + 14*e^4 - 1/4*e^3 - 42*e^2 + 1/2*e + 20, e^6 + 1/2*e^5 - 11*e^4 - 4*e^3 + 63/2*e^2 + 8*e - 12, 3/2*e^6 - 16*e^4 + 1/2*e^3 + 45*e^2 - 17, 3/4*e^6 + e^5 - 9*e^4 - 33/4*e^3 + 29*e^2 + 27/2*e - 16, e^6 - 10*e^4 - e^3 + 23*e^2 + 5*e - 1, 1/2*e^6 - 6*e^4 + 1/2*e^3 + 18*e^2 - e - 9, 3/2*e^6 + e^5 - 16*e^4 - 13/2*e^3 + 47*e^2 + 3*e - 25, -1/2*e^6 - 1/2*e^5 + 5*e^4 + 3/2*e^3 - 25/2*e^2 + 4*e + 3, e^6 + 3/2*e^5 - 10*e^4 - 14*e^3 + 55/2*e^2 + 30*e - 11, 1/2*e^6 + 3/2*e^5 - 4*e^4 - 27/2*e^3 + 17/2*e^2 + 24*e - 8, -e^6 - e^5 + 11*e^4 + 11*e^3 - 34*e^2 - 32*e + 17, 3/2*e^6 + 3/2*e^5 - 16*e^4 - 27/2*e^3 + 93/2*e^2 + 32*e - 35, -3/2*e^6 - e^5 + 18*e^4 + 21/2*e^3 - 56*e^2 - 26*e + 28, 1/2*e^6 + 1/2*e^5 - 4*e^4 - 7/2*e^3 + 11/2*e^2 + 8*e + 3, e^6 + e^5 - 11*e^4 - 11*e^3 + 30*e^2 + 32*e - 15, -1/2*e^6 - 1/2*e^5 + 2*e^4 + 1/2*e^3 + 17/2*e^2 + 8*e - 9, -5/4*e^6 - 3/2*e^5 + 14*e^4 + 55/4*e^3 - 85/2*e^2 - 49/2*e + 20, 1/2*e^6 + 1/2*e^5 - 7*e^4 - 5/2*e^3 + 53/2*e^2 - 6*e - 15, 1/2*e^6 + 2*e^5 - 2*e^4 - 31/2*e^3 - 4*e^2 + 21*e + 6, 3/4*e^6 - e^5 - 8*e^4 + 31/4*e^3 + 19*e^2 - 15/2*e + 6, -5/4*e^6 - 1/2*e^5 + 14*e^4 + 15/4*e^3 - 73/2*e^2 - 11/2*e + 5, 5/4*e^6 + 3/2*e^5 - 13*e^4 - 43/4*e^3 + 73/2*e^2 + 31/2*e - 14, 1/2*e^6 - 6*e^4 + 1/2*e^3 + 19*e^2 - 2*e - 6, -3/4*e^6 + 1/2*e^5 + 8*e^4 - 27/4*e^3 - 49/2*e^2 + 33/2*e + 14, -3/4*e^6 + e^5 + 8*e^4 - 27/4*e^3 - 21*e^2 + 11/2*e + 7, 3/4*e^6 + 3/2*e^5 - 6*e^4 - 53/4*e^3 + 15/2*e^2 + 57/2*e + 6, 1/2*e^6 + 1/2*e^5 - 7*e^4 - 7/2*e^3 + 57/2*e^2 + 6*e - 27, 1/2*e^6 + 1/2*e^5 - 5*e^4 - 5/2*e^3 + 31/2*e^2 + e - 24, 1/4*e^6 + 1/2*e^5 - 2*e^4 - 19/4*e^3 + 1/2*e^2 + 15/2*e + 7, -2*e^6 - 3/2*e^5 + 18*e^4 + 8*e^3 - 71/2*e^2 - 4*e + 1, -2*e^6 + 3/2*e^5 + 26*e^4 - 11*e^3 - 167/2*e^2 + 7*e + 33, -7/4*e^6 + 1/2*e^5 + 21*e^4 + 1/4*e^3 - 131/2*e^2 - 35/2*e + 23, 1/2*e^6 + 3/2*e^5 - 5*e^4 - 19/2*e^3 + 33/2*e^2 + 5*e - 11, -2*e^6 + 24*e^4 + 5*e^3 - 78*e^2 - 32*e + 49, -2*e^6 - e^5 + 21*e^4 + 11*e^3 - 59*e^2 - 27*e + 37, -13/4*e^6 - 5/2*e^5 + 32*e^4 + 75/4*e^3 - 163/2*e^2 - 61/2*e + 36, -e^4 - 2*e^3 + 5*e^2 + 18*e - 9, -e^6 - 2*e^5 + 12*e^4 + 16*e^3 - 35*e^2 - 19*e + 6, 3/4*e^6 - 1/2*e^5 - 9*e^4 + 27/4*e^3 + 51/2*e^2 - 53/2*e - 8, -3/4*e^6 - 1/2*e^5 + 10*e^4 + 29/4*e^3 - 67/2*e^2 - 31/2*e + 8, -5/4*e^6 - 2*e^5 + 10*e^4 + 55/4*e^3 - 17*e^2 - 33/2*e + 5, -3/2*e^6 - 1/2*e^5 + 17*e^4 + 11/2*e^3 - 103/2*e^2 - 15*e + 19, 1/2*e^6 + 1/2*e^5 - 5*e^4 - 9/2*e^3 + 33/2*e^2 + 4*e - 26, 1/4*e^6 + 1/2*e^5 - 3*e^4 - 19/4*e^3 + 13/2*e^2 + 13/2*e + 3, e^6 - 1/2*e^5 - 11*e^4 + 2*e^3 + 59/2*e^2 + 7*e - 14, 1/2*e^6 - 1/2*e^5 - 7*e^4 + 15/2*e^3 + 41/2*e^2 - 25*e + 4, -2*e^6 - 7/2*e^5 + 21*e^4 + 25*e^3 - 113/2*e^2 - 30*e + 9, 3/2*e^6 - 19*e^4 + 3/2*e^3 + 64*e^2 - 12*e - 32, -3/4*e^6 - 1/2*e^5 + 12*e^4 + 17/4*e^3 - 95/2*e^2 - 7/2*e + 21, 7/4*e^6 + 5/2*e^5 - 18*e^4 - 77/4*e^3 + 109/2*e^2 + 45/2*e - 35, -3/2*e^6 + 3/2*e^5 + 13*e^4 - 35/2*e^3 - 53/2*e^2 + 44*e + 15, 1/4*e^6 - 1/2*e^5 - 6*e^4 + 37/4*e^3 + 63/2*e^2 - 69/2*e - 20, -5/2*e^6 - 5/2*e^5 + 27*e^4 + 41/2*e^3 - 153/2*e^2 - 42*e + 35, 1/2*e^6 - 3/2*e^5 - 3*e^4 + 25/2*e^3 - 3/2*e^2 - 16*e + 19, -1/4*e^6 + 1/2*e^5 + 3*e^4 + 7/4*e^3 - 15/2*e^2 - 53/2*e + 8, e^6 + 5/2*e^5 - 8*e^4 - 18*e^3 + 17/2*e^2 + 24*e + 7, -3/4*e^6 - 3/2*e^5 + 6*e^4 + 37/4*e^3 - 25/2*e^2 - 7/2*e + 19, -7/4*e^6 - 1/2*e^5 + 16*e^4 + 5/4*e^3 - 71/2*e^2 + 17/2*e + 5, -1/2*e^6 + 3/2*e^5 + 8*e^4 - 27/2*e^3 - 63/2*e^2 + 24*e + 14, -e^6 + 1/2*e^5 + 14*e^4 - 3*e^3 - 95/2*e^2 - 4*e + 5, 1/4*e^6 - e^5 - e^4 + 41/4*e^3 - 3*e^2 - 51/2*e - 13, 1/2*e^6 - 8*e^4 - 3/2*e^3 + 34*e^2 + 9*e - 27, 1/2*e^5 - 3*e^3 + 15/2*e^2 + e - 15, 5/2*e^6 + e^5 - 29*e^4 - 23/2*e^3 + 88*e^2 + 37*e - 48, -1/4*e^6 + 1/2*e^5 + 5*e^4 - 1/4*e^3 - 29/2*e^2 - 27/2*e - 18, 5/4*e^6 - 3/2*e^5 - 15*e^4 + 37/4*e^3 + 93/2*e^2 - 7/2*e - 40, -5/4*e^6 - e^5 + 12*e^4 + 39/4*e^3 - 25*e^2 - 63/2*e + 6, 3/4*e^6 - 13*e^4 - 9/4*e^3 + 55*e^2 + 17/2*e - 31, -2*e^5 - 3*e^4 + 14*e^3 + 20*e^2 - 13*e - 19, -e^5 - 2*e^4 + 8*e^3 + 19*e^2 - 6*e - 32, -1/2*e^6 + 7*e^4 + 17/2*e^3 - 24*e^2 - 44*e + 17, 1/2*e^6 + 1/2*e^5 - 9*e^4 - 13/2*e^3 + 81/2*e^2 + 24*e - 28, -3/4*e^6 + e^5 + 4*e^4 - 27/4*e^3 + 9*e^2 + 11/2*e - 25, -e^6 - e^5 + 12*e^4 + 10*e^3 - 39*e^2 - 22*e + 17, -5/2*e^6 - e^5 + 24*e^4 + 15/2*e^3 - 61*e^2 - 15*e + 46, 11/2*e^6 + 3/2*e^5 - 58*e^4 - 19/2*e^3 + 315/2*e^2 + 19*e - 62, -5/2*e^5 + 22*e^3 + 3/2*e^2 - 46*e - 11, -11/4*e^6 + 3/2*e^5 + 30*e^4 - 51/4*e^3 - 167/2*e^2 + 37/2*e + 30, e^6 + 1/2*e^5 - 14*e^4 - 7*e^3 + 113/2*e^2 + 14*e - 47, -1/2*e^6 - 2*e^5 - e^4 + 23/2*e^3 + 29*e^2 - e - 44, 15/4*e^6 - 1/2*e^5 - 42*e^4 + 23/4*e^3 + 239/2*e^2 - 11/2*e - 50, 7/2*e^6 + 1/2*e^5 - 40*e^4 - 13/2*e^3 + 231/2*e^2 + 26*e - 57, 7/4*e^6 + 1/2*e^5 - 18*e^4 - 1/4*e^3 + 87/2*e^2 - 29/2*e - 2, e^6 + 3/2*e^5 - 8*e^4 - 10*e^3 + 25/2*e^2 + 17*e - 1, 7/2*e^6 + 1/2*e^5 - 38*e^4 - 5/2*e^3 + 207/2*e^2 - 50, -3*e^6 - 2*e^5 + 34*e^4 + 17*e^3 - 103*e^2 - 41*e + 45, 7/2*e^6 + 1/2*e^5 - 37*e^4 - 1/2*e^3 + 191/2*e^2 - 7*e - 28, 3/2*e^6 - 17*e^4 - 1/2*e^3 + 49*e^2 + 7*e - 33, 9/4*e^6 + e^5 - 24*e^4 - 35/4*e^3 + 65*e^2 + 51/2*e - 25, 7/4*e^6 - 20*e^4 - 1/4*e^3 + 53*e^2 - 5/2*e - 10, -1/2*e^6 + 5/2*e^5 + 8*e^4 - 35/2*e^3 - 57/2*e^2 + 16*e + 3, -5/4*e^6 - 1/2*e^5 + 17*e^4 + 27/4*e^3 - 127/2*e^2 - 47/2*e + 47, -e^6 + 1/2*e^5 + 9*e^4 - 9*e^3 - 35/2*e^2 + 25*e + 4, 1/4*e^6 - 1/2*e^5 - e^4 + 21/4*e^3 - 13/2*e^2 - 23/2*e + 39, 2*e^6 - 3/2*e^5 - 21*e^4 + 14*e^3 + 115/2*e^2 - 26*e - 29, -11/4*e^6 - 5*e^5 + 25*e^4 + 173/4*e^3 - 59*e^2 - 163/2*e + 27, -1/2*e^6 + 3/2*e^5 + 7*e^4 - 27/2*e^3 - 57/2*e^2 + 36*e + 28, e^6 - 12*e^4 - e^3 + 33*e^2 + 11*e - 2, -2*e^6 + e^5 + 20*e^4 - 9*e^3 - 46*e^2 + 13*e + 13, 1/2*e^6 + 2*e^5 - 3*e^4 - 27/2*e^3 - 5*e^2 + 7*e + 22, -1/2*e^6 - e^5 + 6*e^4 + 19/2*e^3 - 27*e^2 - 21*e + 28, -7/2*e^5 - 3*e^4 + 24*e^3 + 27/2*e^2 - 22*e + 1, -5/2*e^6 - 4*e^5 + 23*e^4 + 61/2*e^3 - 50*e^2 - 54*e + 15, -e^6 + 3/2*e^5 + 9*e^4 - 14*e^3 - 23/2*e^2 + 24*e - 38, -2*e^5 - 2*e^4 + 13*e^3 + 16*e^2 - 33, e^6 - 5/2*e^5 - 16*e^4 + 21*e^3 + 117/2*e^2 - 35*e - 32, 2*e^6 + 3/2*e^5 - 23*e^4 - 14*e^3 + 145/2*e^2 + 46*e - 50, -3/4*e^6 - 9/2*e^5 + 8*e^4 + 161/4*e^3 - 55/2*e^2 - 155/2*e + 20, 1/4*e^6 + 3/2*e^5 + 2*e^4 - 43/4*e^3 - 55/2*e^2 + 41/2*e + 42, -2*e^6 - 3*e^5 + 17*e^4 + 27*e^3 - 30*e^2 - 54*e + 10, 7/4*e^6 + 5/2*e^5 - 23*e^4 - 101/4*e^3 + 159/2*e^2 + 117/2*e - 32, 2*e^6 - 3/2*e^5 - 22*e^4 + 9*e^3 + 119/2*e^2 - 11*e - 23, -3*e^6 - 2*e^5 + 31*e^4 + 18*e^3 - 79*e^2 - 38*e + 46, e^6 + 7/2*e^5 - 12*e^4 - 35*e^3 + 87/2*e^2 + 86*e - 35, 3/4*e^6 - 8*e^4 + 15/4*e^3 + 28*e^2 - 63/2*e - 20, -5/2*e^6 - 11/2*e^5 + 23*e^4 + 85/2*e^3 - 117/2*e^2 - 61*e + 36, 11/4*e^6 + 13/2*e^5 - 28*e^4 - 205/4*e^3 + 169/2*e^2 + 145/2*e - 64, 3/4*e^6 - 13*e^4 + 11/4*e^3 + 59*e^2 - 23/2*e - 51, -1/4*e^6 - 5/2*e^5 + 3*e^4 + 103/4*e^3 - 21/2*e^2 - 101/2*e + 20, -11/4*e^6 + 5/2*e^5 + 35*e^4 - 63/4*e^3 - 221/2*e^2 - 5/2*e + 36, -2*e^6 - 1/2*e^5 + 21*e^4 + 5*e^3 - 107/2*e^2 - 2*e + 1, 3/4*e^6 + 3*e^5 - 5*e^4 - 117/4*e^3 + 137/2*e - 10, -9/4*e^6 + 22*e^4 + 3/4*e^3 - 58*e^2 - 19/2*e + 37, e^6 + e^5 - 13*e^4 - 9*e^3 + 52*e^2 + 20*e - 58, -1/2*e^6 - 1/2*e^5 + 13*e^4 + 23/2*e^3 - 141/2*e^2 - 36*e + 53, -7/4*e^6 - 1/2*e^5 + 21*e^4 + 13/4*e^3 - 139/2*e^2 - 25/2*e + 21, 1/2*e^6 - 5/2*e^5 - 9*e^4 + 25/2*e^3 + 79/2*e^2 + 20*e - 35, 7/4*e^6 + 3*e^5 - 25*e^4 - 97/4*e^3 + 97*e^2 + 79/2*e - 59, 2*e^6 + 2*e^5 - 21*e^4 - 15*e^3 + 49*e^2 + 28*e + 13, -5/2*e^6 - 1/2*e^5 + 28*e^4 + 11/2*e^3 - 179/2*e^2 - 18*e + 60, -1/4*e^6 - 4*e^5 + 4*e^4 + 127/4*e^3 - 22*e^2 - 107/2*e + 30, -3/2*e^6 - 5/2*e^5 + 13*e^4 + 39/2*e^3 - 55/2*e^2 - 23*e + 15, 11/4*e^6 + 2*e^5 - 25*e^4 - 41/4*e^3 + 43*e^2 - 3/2*e + 19, -e^6 - 1/2*e^5 + 16*e^4 + 6*e^3 - 133/2*e^2 - 24*e + 68, 3/2*e^6 + 1/2*e^5 - 14*e^4 - 25/2*e^3 + 67/2*e^2 + 58*e - 23, -2*e^6 - 5/2*e^5 + 21*e^4 + 21*e^3 - 117/2*e^2 - 47*e + 15, -e^5 - 2*e^4 + 6*e^3 + 7*e^2 + 2*e, 5/2*e^6 + 1/2*e^5 - 29*e^4 - 23/2*e^3 + 173/2*e^2 + 50*e - 35, -21/4*e^6 - e^5 + 54*e^4 - 5/4*e^3 - 141*e^2 + 41/2*e + 63, -9/4*e^6 - 1/2*e^5 + 22*e^4 + 15/4*e^3 - 85/2*e^2 + 5/2*e - 24, 9/4*e^6 + 3*e^5 - 18*e^4 - 63/4*e^3 + 27*e^2 - 21/2*e + 21, 1/4*e^6 + 3*e^4 - 7/4*e^3 - 28*e^2 + 25/2*e - 9, -5/4*e^6 - 1/2*e^5 + 15*e^4 + 31/4*e^3 - 91/2*e^2 - 51/2*e + 14, 3/4*e^6 + e^5 - 4*e^4 - 25/4*e^3 - 17*e^2 + 17/2*e + 43];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,17*w + 160}>] := -1;
ALEigenvalues[ideal<ZF | {3,3,842*w + 7925}>] := 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;