/*
  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![-13, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {36, 6, 6}>;

primesArray := [
[4, 2, 2],
[7, 7, w + 2],
[7, 7, w - 3],
[9, 3, 3],
[11, 11, w + 1],
[11, 11, w - 2],
[13, 13, w],
[13, 13, w - 1],
[17, 17, -w - 5],
[17, 17, w - 6],
[25, 5, -5],
[29, 29, -w - 6],
[29, 29, w - 7],
[37, 37, 2*w - 5],
[37, 37, -2*w - 3],
[43, 43, -w - 7],
[43, 43, w - 8],
[47, 47, 3*w + 7],
[47, 47, 3*w - 10],
[53, 53, 2*w - 1],
[59, 59, -w - 8],
[59, 59, w - 9],
[89, 89, -3*w - 4],
[89, 89, 3*w - 7],
[97, 97, -w - 10],
[97, 97, w - 11],
[107, 107, 3*w - 5],
[107, 107, -3*w - 2],
[113, 113, -3*w - 1],
[113, 113, 3*w - 4],
[131, 131, 4*w - 11],
[131, 131, -4*w - 7],
[149, 149, 4*w - 21],
[149, 149, -4*w - 17],
[163, 163, -4*w - 5],
[163, 163, 4*w - 9],
[197, 197, -w - 14],
[197, 197, w - 15],
[199, 199, 5*w - 14],
[199, 199, -5*w - 9],
[211, 211, 4*w - 3],
[211, 211, 4*w - 1],
[223, 223, -3*w - 17],
[223, 223, 3*w - 20],
[227, 227, -w - 15],
[227, 227, w - 16],
[229, 229, -4*w - 19],
[229, 229, 4*w - 23],
[241, 241, 5*w - 12],
[241, 241, -5*w - 7],
[269, 269, -5*w - 22],
[269, 269, 5*w - 27],
[271, 271, 2*w - 19],
[271, 271, -2*w - 17],
[281, 281, 6*w - 17],
[281, 281, -6*w - 11],
[293, 293, -w - 17],
[293, 293, w - 18],
[307, 307, -7*w - 15],
[307, 307, 7*w - 22],
[311, 311, 5*w - 7],
[311, 311, -5*w - 2],
[317, 317, 4*w - 25],
[317, 317, -4*w - 21],
[331, 331, 5*w - 2],
[331, 331, 5*w - 3],
[347, 347, 2*w - 21],
[347, 347, -2*w - 19],
[361, 19, -19],
[367, 367, -w - 19],
[367, 367, w - 20],
[409, 409, -7*w - 12],
[409, 409, 7*w - 19],
[431, 431, 2*w - 23],
[431, 431, -2*w - 21],
[433, 433, 3*w - 25],
[433, 433, -3*w - 22],
[439, 439, -7*w - 11],
[439, 439, 7*w - 18],
[449, 449, -w - 21],
[449, 449, w - 22],
[461, 461, -6*w - 1],
[461, 461, 6*w - 7],
[467, 467, 7*w - 17],
[467, 467, -7*w - 10],
[487, 487, 8*w - 23],
[487, 487, -8*w - 15],
[521, 521, -9*w - 19],
[521, 521, 9*w - 28],
[523, 523, 2*w - 25],
[523, 523, -2*w - 23],
[529, 23, -23],
[541, 541, 10*w - 33],
[541, 541, 10*w + 23],
[547, 547, -6*w - 29],
[547, 547, 6*w - 35],
[577, 577, -7*w - 5],
[577, 577, 7*w - 12],
[587, 587, -w - 24],
[587, 587, w - 25],
[593, 593, -7*w - 4],
[593, 593, 7*w - 11],
[599, 599, 5*w - 33],
[599, 599, -5*w - 28],
[607, 607, 7*w - 10],
[607, 607, -7*w - 3],
[619, 619, 7*w - 9],
[619, 619, -7*w - 2],
[643, 643, 7*w - 6],
[643, 643, 7*w - 1],
[647, 647, 7*w - 2],
[647, 647, 7*w - 5],
[653, 653, -9*w - 16],
[653, 653, 9*w - 25],
[661, 661, -5*w - 29],
[661, 661, 5*w - 34],
[673, 673, 8*w - 43],
[673, 673, -8*w - 35],
[683, 683, 7*w - 40],
[683, 683, -7*w - 33],
[727, 727, 8*w - 15],
[727, 727, -8*w - 7],
[733, 733, -9*w - 38],
[733, 733, 9*w - 47],
[743, 743, -w - 27],
[743, 743, w - 28],
[751, 751, 3*w - 31],
[751, 751, -3*w - 28],
[757, 757, -7*w - 34],
[757, 757, 7*w - 41],
[811, 811, -3*w - 29],
[811, 811, 3*w - 32],
[823, 823, -8*w - 1],
[823, 823, 8*w - 9],
[839, 839, 8*w - 7],
[839, 839, 8*w - 1],
[857, 857, -w - 29],
[857, 857, w - 30],
[859, 859, -5*w - 32],
[859, 859, 5*w - 37],
[863, 863, -9*w - 10],
[863, 863, 9*w - 19],
[877, 877, -4*w - 31],
[877, 877, 4*w - 35],
[907, 907, 13*w - 43],
[907, 907, 13*w + 30],
[911, 911, 7*w - 43],
[911, 911, -7*w - 36],
[929, 929, 5*w - 38],
[929, 929, -5*w - 33],
[937, 937, 3*w - 34],
[937, 937, -3*w - 31],
[941, 941, -9*w - 7],
[941, 941, 9*w - 16],
[947, 947, -12*w - 25],
[947, 947, 12*w - 37],
[953, 953, -11*w - 20],
[953, 953, 11*w - 31],
[961, 31, -31],
[967, 967, -6*w - 35],
[967, 967, 6*w - 41],
[971, 971, 2*w - 33],
[971, 971, -2*w - 31],
[983, 983, -9*w - 5],
[983, 983, 9*w - 14],
[991, 991, -7*w - 37],
[991, 991, 7*w - 44],
[997, 997, 9*w - 50],
[997, 997, -9*w - 41],
[1013, 1013, 4*w - 37],
[1013, 1013, -4*w - 33],
[1031, 1031, 9*w - 11],
[1031, 1031, -9*w - 2],
[1049, 1049, 13*w - 41],
[1049, 1049, -13*w - 28],
[1051, 1051, 11*w - 29],
[1051, 1051, -11*w - 18],
[1061, 1061, 9*w - 8],
[1061, 1061, 9*w - 1],
[1069, 1069, -10*w - 11],
[1069, 1069, 10*w - 21],
[1097, 1097, -11*w - 17],
[1097, 1097, 11*w - 28],
[1103, 1103, 2*w - 35],
[1103, 1103, -2*w - 33],
[1109, 1109, -w - 33],
[1109, 1109, w - 34],
[1117, 1117, -13*w - 27],
[1117, 1117, 13*w - 40],
[1123, 1123, 6*w - 43],
[1123, 1123, -6*w - 37],
[1129, 1129, 10*w - 19],
[1129, 1129, -10*w - 9],
[1151, 1151, 5*w - 41],
[1151, 1151, -5*w - 36],
[1153, 1153, -14*w - 31],
[1153, 1153, 14*w - 45],
[1181, 1181, -10*w - 7],
[1181, 1181, 10*w - 17],
[1213, 1213, -3*w - 35],
[1213, 1213, 3*w - 38],
[1223, 1223, -11*w - 14],
[1223, 1223, 11*w - 25],
[1229, 1229, -5*w - 37],
[1229, 1229, 5*w - 42],
[1259, 1259, 15*w + 34],
[1259, 1259, 15*w - 49],
[1279, 1279, -9*w - 44],
[1279, 1279, 9*w - 53],
[1283, 1283, 12*w - 31],
[1283, 1283, -12*w - 19],
[1289, 1289, -10*w - 1],
[1289, 1289, 10*w - 11],
[1297, 1297, -11*w - 12],
[1297, 1297, 11*w - 23],
[1301, 1301, -14*w - 29],
[1301, 1301, 14*w - 43],
[1319, 1319, -w - 36],
[1319, 1319, w - 37],
[1321, 1321, 10*w - 3],
[1321, 1321, 10*w - 7],
[1361, 1361, 8*w - 51],
[1361, 1361, -8*w - 43],
[1367, 1367, -11*w - 49],
[1367, 1367, 11*w - 60],
[1427, 1427, -13*w - 22],
[1427, 1427, 13*w - 35],
[1447, 1447, 11*w - 18],
[1447, 1447, -11*w - 7],
[1459, 1459, -6*w - 41],
[1459, 1459, 6*w - 47],
[1471, 1471, -11*w - 6],
[1471, 1471, 11*w - 17],
[1483, 1483, -13*w - 21],
[1483, 1483, 13*w - 34],
[1493, 1493, 11*w - 16],
[1493, 1493, -11*w - 5],
[1499, 1499, -15*w - 31],
[1499, 1499, 15*w - 46],
[1531, 1531, -11*w - 3],
[1531, 1531, 11*w - 14],
[1543, 1543, -13*w - 55],
[1543, 1543, 13*w - 68],
[1553, 1553, -8*w - 45],
[1553, 1553, 8*w - 53],
[1579, 1579, -9*w - 47],
[1579, 1579, 9*w - 56],
[1583, 1583, 11*w - 10],
[1583, 1583, 11*w - 1],
[1597, 1597, 11*w - 8],
[1597, 1597, 11*w - 3],
[1601, 1601, 11*w - 7],
[1601, 1601, 11*w - 4],
[1607, 1607, -7*w - 44],
[1607, 1607, 7*w - 51],
[1619, 1619, 12*w - 23],
[1619, 1619, -12*w - 11],
[1627, 1627, -w - 40],
[1627, 1627, w - 41],
[1637, 1637, 4*w - 45],
[1637, 1637, -4*w - 41],
[1667, 1667, -13*w - 56],
[1667, 1667, 13*w - 69],
[1681, 41, -41],
[1697, 1697, 14*w - 37],
[1697, 1697, -14*w - 23],
[1709, 1709, -w - 41],
[1709, 1709, w - 42],
[1721, 1721, -15*w - 28],
[1721, 1721, 15*w - 43],
[1733, 1733, 13*w - 29],
[1733, 1733, -13*w - 16],
[1753, 1753, -8*w - 47],
[1753, 1753, 8*w - 55],
[1759, 1759, -17*w - 37],
[1759, 1759, 17*w - 54],
[1777, 1777, 13*w - 28],
[1777, 1777, -13*w - 15],
[1787, 1787, -12*w - 5],
[1787, 1787, 12*w - 17],
[1789, 1789, -9*w - 49],
[1789, 1789, 9*w - 58],
[1801, 1801, 7*w - 53],
[1801, 1801, -7*w - 46],
[1811, 1811, 10*w - 61],
[1811, 1811, -10*w - 51],
[1831, 1831, -5*w - 44],
[1831, 1831, 5*w - 49],
[1861, 1861, 3*w - 46],
[1861, 1861, -3*w - 43],
[1871, 1871, 16*w - 47],
[1871, 1871, -16*w - 31],
[1879, 1879, -w - 43],
[1879, 1879, w - 44],
[1901, 1901, 7*w - 54],
[1901, 1901, -7*w - 47],
[1907, 1907, 12*w - 7],
[1907, 1907, 12*w - 5],
[1933, 1933, -13*w - 11],
[1933, 1933, 13*w - 24],
[1951, 1951, -3*w - 44],
[1951, 1951, 3*w - 47],
[1997, 1997, 4*w - 49],
[1997, 1997, -4*w - 45],
[1999, 1999, -13*w - 9],
[1999, 1999, 13*w - 22],
[2003, 2003, -7*w - 48],
[2003, 2003, 7*w - 55],
[2027, 2027, 19*w + 43],
[2027, 2027, 19*w - 62],
[2029, 2029, -13*w - 8],
[2029, 2029, 13*w - 21],
[2039, 2039, -10*w - 53],
[2039, 2039, 10*w - 63],
[2063, 2063, 2*w - 47],
[2063, 2063, -2*w - 45],
[2083, 2083, 13*w - 19],
[2083, 2083, -13*w - 6],
[2111, 2111, 15*w - 37],
[2111, 2111, -15*w - 22],
[2113, 2113, -14*w - 15],
[2113, 2113, 14*w - 29],
[2129, 2129, 13*w - 17],
[2129, 2129, -13*w - 4],
[2131, 2131, 15*w - 79],
[2131, 2131, -15*w - 64],
[2137, 2137, 3*w - 49],
[2137, 2137, -3*w - 46],
[2179, 2179, 11*w - 67],
[2179, 2179, -11*w - 56],
[2213, 2213, 7*w - 57],
[2213, 2213, -7*w - 50],
[2237, 2237, 13*w - 5],
[2237, 2237, 13*w - 8],
[2239, 2239, 13*w - 6],
[2239, 2239, 13*w - 7],
[2243, 2243, -w - 47],
[2243, 2243, w - 48],
[2251, 2251, 2*w - 49],
[2251, 2251, -2*w - 47],
[2269, 2269, 17*w - 48],
[2269, 2269, -17*w - 31],
[2273, 2273, 14*w - 25],
[2273, 2273, -14*w - 11],
[2333, 2333, 19*w - 59],
[2333, 2333, -19*w - 40],
[2339, 2339, -w - 48],
[2339, 2339, w - 49],
[2341, 2341, 14*w - 23],
[2341, 2341, -14*w - 9],
[2347, 2347, -17*w - 30],
[2347, 2347, 17*w - 47],
[2357, 2357, 18*w - 53],
[2357, 2357, -18*w - 35],
[2381, 2381, 15*w - 32],
[2381, 2381, -15*w - 17],
[2389, 2389, 4*w - 53],
[2389, 2389, -4*w - 49],
[2423, 2423, 17*w - 46],
[2423, 2423, -17*w - 29],
[2437, 2437, -w - 49],
[2437, 2437, w - 50],
[2447, 2447, 2*w - 51],
[2447, 2447, -2*w - 49],
[2467, 2467, 9*w - 64],
[2467, 2467, -9*w - 55],
[2531, 2531, 5*w - 56],
[2531, 2531, -5*w - 51],
[2543, 2543, 7*w - 60],
[2543, 2543, -7*w - 53],
[2551, 2551, 16*w - 37],
[2551, 2551, -16*w - 21],
[2557, 2557, 11*w - 70],
[2557, 2557, -11*w - 59],
[2591, 2591, 13*w - 76],
[2591, 2591, -13*w - 63],
[2593, 2593, 14*w - 9],
[2593, 2593, 14*w - 5],
[2621, 2621, 19*w - 56],
[2621, 2621, -19*w - 37],
[2633, 2633, -8*w - 55],
[2633, 2633, 8*w - 63],
[2657, 2657, 7*w - 61],
[2657, 2657, -7*w - 54],
[2659, 2659, -6*w - 53],
[2659, 2659, 6*w - 59],
[2663, 2663, 16*w - 35],
[2663, 2663, -16*w - 19],
[2687, 2687, -11*w - 60],
[2687, 2687, 11*w - 71],
[2693, 2693, -18*w - 31],
[2693, 2693, 18*w - 49],
[2699, 2699, -20*w - 41],
[2699, 2699, 20*w - 61],
[2707, 2707, 17*w - 42],
[2707, 2707, -17*w - 25],
[2713, 2713, -19*w - 36],
[2713, 2713, 19*w - 55],
[2719, 2719, 15*w - 83],
[2719, 2719, -15*w - 68],
[2731, 2731, -13*w - 64],
[2731, 2731, 13*w - 77],
[2741, 2741, -15*w - 8],
[2741, 2741, 15*w - 23],
[2749, 2749, 5*w - 58],
[2749, 2749, -5*w - 53],
[2767, 2767, -16*w - 17],
[2767, 2767, 16*w - 33],
[2803, 2803, -19*w - 35],
[2803, 2803, 19*w - 54],
[2819, 2819, 11*w - 72],
[2819, 2819, -11*w - 61],
[2833, 2833, -9*w - 58],
[2833, 2833, 9*w - 67],
[2837, 2837, -17*w - 23],
[2837, 2837, 17*w - 40],
[2851, 2851, -3*w - 53],
[2851, 2851, 3*w - 56],
[2861, 2861, -5*w - 54],
[2861, 2861, 5*w - 59],
[2879, 2879, -14*w - 67],
[2879, 2879, 14*w - 81],
[2887, 2887, 6*w - 61],
[2887, 2887, -6*w - 55],
[2909, 2909, -15*w - 1],
[2909, 2909, 15*w - 16],
[2939, 2939, 15*w - 14],
[2939, 2939, 15*w - 1],
[2953, 2953, -11*w - 62],
[2953, 2953, 11*w - 73],
[2957, 2957, -w - 54],
[2957, 2957, w - 55],
[2969, 2969, 15*w - 11],
[2969, 2969, 15*w - 4],
[3011, 3011, 7*w - 64],
[3011, 3011, -7*w - 57],
[3037, 3037, -4*w - 55],
[3037, 3037, 4*w - 59],
[3049, 3049, 22*w - 69],
[3049, 3049, -22*w - 47],
[3061, 3061, -19*w - 32],
[3061, 3061, 19*w - 51],
[3067, 3067, -w - 55],
[3067, 3067, w - 56],
[3083, 3083, 2*w - 57],
[3083, 3083, -2*w - 55],
[3089, 3089, -11*w - 63],
[3089, 3089, 11*w - 74],
[3121, 3121, 8*w - 67],
[3121, 3121, -8*w - 59],
[3137, 3137, 18*w - 43],
[3137, 3137, -18*w - 25],
[3163, 3163, -13*w - 67],
[3163, 3163, 13*w - 80],
[3167, 3167, -16*w - 7],
[3167, 3167, 16*w - 23],
[3169, 3169, -16*w - 73],
[3169, 3169, 16*w - 89],
[3181, 3181, -15*w - 71],
[3181, 3181, 15*w - 86],
[3187, 3187, -3*w - 56],
[3187, 3187, 3*w - 59],
[3191, 3191, 21*w - 62],
[3191, 3191, -21*w - 41],
[3209, 3209, -5*w - 57],
[3209, 3209, 5*w - 62],
[3217, 3217, 9*w - 70],
[3217, 3217, -9*w - 61],
[3229, 3229, 17*w - 33],
[3229, 3229, -17*w - 16],
[3257, 3257, -7*w - 59],
[3257, 3257, 7*w - 66],
[3271, 3271, 16*w - 19],
[3271, 3271, -16*w - 3],
[3299, 3299, -10*w - 63],
[3299, 3299, 10*w - 73],
[3301, 3301, -19*w - 29],
[3301, 3301, 19*w - 48],
[3323, 3323, 17*w - 31],
[3323, 3323, -17*w - 14],
[3329, 3329, 5*w - 63],
[3329, 3329, -5*w - 58],
[3343, 3343, 16*w - 15],
[3343, 3343, 16*w - 1],
[3391, 3391, 16*w - 9],
[3391, 3391, 16*w - 7],
[3407, 3407, 19*w - 100],
[3407, 3407, -19*w - 81],
[3449, 3449, 17*w - 28],
[3449, 3449, -17*w - 11],
[3461, 3461, -13*w - 69],
[3461, 3461, 13*w - 82],
[3469, 3469, 23*w - 71],
[3469, 3469, -23*w - 48],
[3491, 3491, -21*w - 38],
[3491, 3491, 21*w - 59],
[3499, 3499, 15*w - 88],
[3499, 3499, -15*w - 73],
[3511, 3511, 7*w - 68],
[3511, 3511, -7*w - 61],
[3527, 3527, -w - 59],
[3527, 3527, w - 60],
[3541, 3541, -3*w - 59],
[3541, 3541, 3*w - 62],
[3547, 3547, 2*w - 61],
[3547, 3547, -2*w - 59],
[3557, 3557, 17*w - 25],
[3557, 3557, -17*w - 8],
[3593, 3593, 19*w - 44],
[3593, 3593, -19*w - 25],
[3613, 3613, -13*w - 70],
[3613, 3613, 13*w - 83],
[3617, 3617, 18*w - 35],
[3617, 3617, -18*w - 17],
[3673, 3673, 17*w - 21],
[3673, 3673, -17*w - 4],
[3697, 3697, -17*w - 3],
[3697, 3697, 17*w - 20],
[3701, 3701, 5*w - 66],
[3701, 3701, -5*w - 61],
[3709, 3709, -22*w - 41],
[3709, 3709, 22*w - 63],
[3719, 3719, 17*w - 19],
[3719, 3719, -17*w - 2],
[3721, 61, -61],
[3727, 3727, -19*w - 23],
[3727, 3727, 19*w - 42],
[3739, 3739, -17*w - 1],
[3739, 3739, 17*w - 18],
[3767, 3767, -13*w - 71],
[3767, 3767, 13*w - 84],
[3769, 3769, -w - 61],
[3769, 3769, w - 62],
[3779, 3779, -20*w - 29],
[3779, 3779, 20*w - 49],
[3803, 3803, 14*w - 87],
[3803, 3803, -14*w - 73],
[3823, 3823, 17*w - 11],
[3823, 3823, 17*w - 6],
[3833, 3833, 16*w - 93],
[3833, 3833, -16*w - 77],
[3853, 3853, 19*w - 40],
[3853, 3853, -19*w - 21],
[3863, 3863, 21*w - 55],
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[8527, 8527, -32*w - 55],
[8527, 8527, 32*w - 87],
[8537, 8537, 27*w - 47],
[8537, 8537, -27*w - 20],
[8539, 8539, -28*w - 29],
[8539, 8539, 28*w - 57],
[8543, 8543, -w - 92],
[8543, 8543, w - 93],
[8573, 8573, 31*w - 80],
[8573, 8573, -31*w - 49],
[8597, 8597, 39*w + 88],
[8597, 8597, 39*w - 127],
[8599, 8599, -5*w - 92],
[8599, 8599, 5*w - 97],
[8623, 8623, -3*w - 92],
[8623, 8623, 3*w - 95],
[8629, 8629, -9*w - 94],
[8629, 8629, 9*w - 103],
[8663, 8663, -7*w - 93],
[8663, 8663, 7*w - 100],
[8677, 8677, -21*w - 110],
[8677, 8677, 21*w - 131],
[8681, 8681, 13*w - 111],
[8681, 8681, -13*w - 98],
[8693, 8693, -29*w - 35],
[8693, 8693, 29*w - 64],
[8699, 8699, -11*w - 96],
[8699, 8699, 11*w - 107],
[8707, 8707, 28*w - 55],
[8707, 8707, -28*w - 27],
[8741, 8741, -23*w - 114],
[8741, 8741, 23*w - 137],
[8761, 8761, -26*w - 1],
[8761, 8761, 26*w - 27],
[8783, 8783, 2*w - 95],
[8783, 8783, -2*w - 93],
[8807, 8807, 32*w - 85],
[8807, 8807, -32*w - 53],
[8861, 8861, 20*w - 129],
[8861, 8861, -20*w - 109],
[8867, 8867, -28*w - 25],
[8867, 8867, 28*w - 53],
[8887, 8887, 29*w - 62],
[8887, 8887, -29*w - 33],
[8893, 8893, 26*w - 5],
[8893, 8893, 26*w - 21],
[8929, 8929, 16*w - 119],
[8929, 8929, -16*w - 103],
[8933, 8933, 28*w - 153],
[8933, 8933, -28*w - 125],
[8941, 8941, 26*w - 17],
[8941, 8941, 26*w - 9],
[8951, 8951, 31*w - 77],
[8951, 8951, -31*w - 46],
[8963, 8963, -22*w - 113],
[8963, 8963, 22*w - 135],
[8999, 8999, -26*w - 121],
[8999, 8999, 26*w - 147],
[9001, 9001, 3*w - 97],
[9001, 9001, -3*w - 94],
[9011, 9011, 25*w - 144],
[9011, 9011, -25*w - 119],
[9059, 9059, -27*w - 11],
[9059, 9059, 27*w - 38],
[9067, 9067, 14*w - 115],
[9067, 9067, -14*w - 101],
[9091, 9091, 35*w - 102],
[9091, 9091, -35*w - 67],
[9103, 9103, 13*w - 113],
[9103, 9103, -13*w - 100],
[9109, 9109, -11*w - 98],
[9109, 9109, 11*w - 109],
[9127, 9127, -6*w - 95],
[9127, 9127, 6*w - 101],
[9133, 9133, 38*w - 119],
[9133, 9133, -38*w - 81],
[9173, 9173, 33*w - 89],
[9173, 9173, -33*w - 56],
[9209, 9209, 37*w - 113],
[9209, 9209, -37*w - 76],
[9221, 9221, -30*w - 37],
[9221, 9221, 30*w - 67],
[9239, 9239, -27*w - 7],
[9239, 9239, 27*w - 34],
[9281, 9281, -17*w - 106],
[9281, 9281, 17*w - 123],
[9311, 9311, -31*w - 43],
[9311, 9311, 31*w - 74],
[9319, 9319, 31*w - 164],
[9319, 9319, -31*w - 133],
[9337, 9337, 29*w - 57],
[9337, 9337, -29*w - 28],
[9341, 9341, -20*w - 111],
[9341, 9341, 20*w - 131],
[9343, 9343, 32*w - 81],
[9343, 9343, -32*w - 49],
[9371, 9371, 5*w - 101],
[9371, 9371, -5*w - 96],
[9377, 9377, -16*w - 105],
[9377, 9377, 16*w - 121],
[9391, 9391, 41*w + 93],
[9391, 9391, 41*w - 134],
[9397, 9397, -37*w - 75],
[9397, 9397, 37*w - 112],
[9419, 9419, 27*w - 29],
[9419, 9419, -27*w - 2],
[9421, 9421, -29*w - 27],
[9421, 9421, 29*w - 56],
[9433, 9433, -9*w - 98],
[9433, 9433, 9*w - 107],
[9463, 9463, 22*w - 137],
[9463, 9463, -22*w - 115],
[9491, 9491, -10*w - 99],
[9491, 9491, 10*w - 109],
[9497, 9497, -19*w - 110],
[9497, 9497, 19*w - 129],
[9511, 9511, 17*w - 124],
[9511, 9511, -17*w - 107],
[9533, 9533, 13*w - 115],
[9533, 9533, -13*w - 102],
[9539, 9539, 26*w - 149],
[9539, 9539, -26*w - 123],
[9547, 9547, 28*w - 43],
[9547, 9547, -28*w - 15],
[9551, 9551, 2*w - 99],
[9551, 9551, -2*w - 97],
[9587, 9587, 27*w - 22],
[9587, 9587, 27*w - 5],
[9629, 9629, 27*w - 8],
[9629, 9629, 27*w - 19],
[9631, 9631, 18*w - 127],
[9631, 9631, -18*w - 109],
[9661, 9661, 29*w - 53],
[9661, 9661, -29*w - 24],
[9689, 9689, -w - 98],
[9689, 9689, w - 99],
[9739, 9739, -11*w - 101],
[9739, 9739, 11*w - 112],
[9743, 9743, 17*w - 125],
[9743, 9743, -17*w - 108],
[9767, 9767, 37*w - 110],
[9767, 9767, -37*w - 73],
[9769, 9769, 5*w - 103],
[9769, 9769, -5*w - 98],
[9781, 9781, -3*w - 98],
[9781, 9781, 3*w - 101],
[9811, 9811, -29*w - 22],
[9811, 9811, 29*w - 51],
[9829, 9829, -20*w - 113],
[9829, 9829, 20*w - 133],
[9833, 9833, 16*w - 123],
[9833, 9833, -16*w - 107],
[9851, 9851, -39*w - 82],
[9851, 9851, 39*w - 121],
[9857, 9857, 7*w - 106],
[9857, 9857, -7*w - 99],
[9859, 9859, 28*w - 37],
[9859, 9859, -28*w - 9],
[9871, 9871, -31*w - 38],
[9871, 9871, 31*w - 69],
[9883, 9883, 29*w - 50],
[9883, 9883, -29*w - 21],
[9887, 9887, -w - 99],
[9887, 9887, w - 100],
[9901, 9901, 15*w - 121],
[9901, 9901, -15*w - 106],
[9907, 9907, -21*w - 115],
[9907, 9907, 21*w - 136],
[9949, 9949, -37*w - 72],
[9949, 9949, 37*w - 109],
[9973, 9973, -12*w - 103],
[9973, 9973, 12*w - 115]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [-1, 1, 1, -1, 5, 5, 2, 2, 0, 0, -6, 5, 5, -10, -10, 6, 6, 6, 6, 14, -12, -12, -14, -14, 5, 5, -12, -12, -8, -8, -7, -7, 22, 22, 2, 2, 5, 5, 13, 13, 10, 10, 21, 21, -3, -3, -24, -24, -1, -1, -14, -14, 31, 31, -16, -16, -21, -21, 0, 0, 32, 32, -33, -33, 28, 28, 0, 0, 37, 8, 8, 1, 1, 18, 18, 21, 21, -24, -24, 34, 34, 1, 1, -3, -3, -8, -8, -24, -24, -16, -16, 37, -8, -8, 22, 22, -11, -11, 25, 25, -30, -30, -28, -28, -28, -28, -10, -10, -24, -24, 14, 14, 6, 6, -28, -28, -41, -41, 13, 13, -37, -37, -22, -22, 34, 34, -43, -43, 34, 34, -38, -38, 23, 23, -4, -4, -42, -42, 44, 44, 2, 2, 22, 22, 2, 2, -6, -6, -10, -10, -7, -7, 45, 45, 11, 11, -54, -54, 26, 29, 29, 11, 11, -38, -38, -1, -1, 60, 60, 25, 25, 20, 20, 8, 8, -36, -36, 29, 29, 0, 0, 12, 12, -42, -42, -3, -3, -10, -10, 34, 34, -29, -29, -42, -42, -5, -5, -13, -13, 8, 8, 48, 48, 46, 46, -21, -21, 8, 8, 59, 59, 34, 34, 50, 50, 57, 57, 10, 10, -38, -38, -62, -62, 52, 52, -15, -15, 31, 31, 42, 42, -21, -21, -44, -44, 19, 19, 3, 3, -10, -10, -16, -16, 44, 44, 0, 0, 8, 8, 12, 12, -34, -34, -18, -18, 68, 68, 28, 28, 35, 35, 9, 9, 73, 48, 48, 6, 6, -44, -44, -63, -63, 63, 63, -55, -55, -23, -23, -68, -68, 0, 0, -42, -42, -36, -36, 33, 33, -44, -44, -6, -6, 77, 77, 3, 3, 15, 15, 54, 54, 35, 35, 78, 78, 24, 24, -59, -59, -57, -57, 80, 80, -64, -64, -50, -50, -44, -44, -28, -28, -66, -66, 0, 0, -34, -34, 61, 61, 30, 30, -51, -51, -51, -51, 53, 53, 23, 23, -18, -18, -26, -26, 6, 6, 69, 69, 36, 36, 24, 24, -18, -18, -55, -55, -31, -31, -2, -2, -26, -26, 52, 52, -28, -28, 10, 10, -45, -45, 74, 74, 25, 25, -4, -4, -64, -64, 85, 85, -30, -30, -68, -68, 2, 2, -8, -8, -18, -18, 58, 58, 10, 10, 84, 84, 28, 28, -23, -23, 44, 44, -30, -30, -13, -13, 66, 66, -31, -31, -20, -20, 45, 45, -35, -35, -39, -39, -22, -22, 14, 14, -66, -66, -32, -32, -13, -13, -77, -77, 42, 42, 14, 14, -84, -84, 69, 69, 22, 22, 59, 59, 20, 20, 20, 20, -4, -4, 0, 0, -74, -74, -54, -54, -18, -18, 70, 70, -11, -11, 78, 78, 74, 74, 22, 22, 4, 4, 49, 49, -96, -96, -18, -18, 33, 33, -21, -21, -44, -44, -60, -60, -12, -12, 24, 24, 96, 96, -10, -10, -52, -52, 90, 90, 34, 34, 12, 12, -22, -22, -35, -35, -98, -98, -108, -108, -32, -32, -45, -45, -18, -18, 32, 32, 18, 18, -106, -106, -43, -43, 39, 39, 20, 20, 14, 14, -47, -73, -73, 92, 92, -42, -42, -103, -103, 79, 79, -100, -100, 0, 0, 34, 34, 56, 56, -52, -52, 116, 116, 76, 76, -107, -107, 78, 78, -56, -56, -8, -8, 56, 56, -83, -83, -59, -59, -16, -16, 76, 76, -93, -93, -63, -63, -24, -24, -86, -86, 53, 53, -9, -9, 39, 39, -90, -90, -28, -28, -38, -38, 83, 83, 117, 117, 78, 78, 50, 50, -2, -2, -56, -56, 4, 4, 118, 118, -20, -20, -99, -99, 71, 71, -27, -27, -66, -66, -102, -102, -35, -3, -3, 4, 4, 57, 57, -6, -6, -68, -68, 79, 79, 24, 24, -18, -18, -49, -49, -81, -81, 18, 18, -42, -42, 104, 104, -62, -62, 27, 27, 69, 69, -59, -59, -63, -63, -8, -8, -58, -58, 124, 124, -20, -20, -19, -19, 60, 60, 36, 36, -92, -92, 58, 58, 28, 28, -13, -13, -81, -81, 62, 62, 55, 55, -44, -44, 6, 6, 93, 105, 105, 88, 88, -82, -82, 66, 66, -94, -94, -84, -84, 82, 82, 6, 6, -3, -3, -20, -20, -78, -78, -58, -58, -23, -23, 31, 31, 130, -96, -96, -6, -6, 60, 60, 68, 68, -83, -83, -58, -58, -38, -38, -74, -74, 91, 91, -34, -34, -66, -66, 83, 83, 130, 130, 139, 139, 30, 30, -77, -77, -7, -7, 129, 129, 42, 42, 140, 140, -12, -12, 111, 111, 15, 15, -93, -93, 46, 46, 126, 126, -78, -78, -3, -3, 46, 46, -35, -35, -111, -111, 59, 59, 98, 98, 130, 130, -12, -12, 129, 129, 86, 86, -116, -116, 34, 34, 92, 92, -105, -105, -60, -60, 0, 0, -33, -33, -61, -61, -54, -54, 46, 46, -123, -123, 70, 70, 32, 32, 125, 125, -50, -50, -98, 92, 92, 126, 126, -91, -91, -117, -117, 78, 78, -72, -72, -35, -35, -111, -111, 104, 104, 26, 26, -36, -36, 91, 91, -71, -71, -16, -16, 97, 97, -8, -8, -42, -42, 12, 12, -54, -54, -119, -119, -31, -31, -31, -31, 98, 98, 84, 84, -154, -154, -99, -99, 82, 82, -126, -126, 120, 120, -79, -79, 46, 46, 26, 26, -29, -29, 62, 62, 110, 110, -11, -11, -87, -87, -78, -78, 25, 25, 82, 82, 22, 12, 12, -128, -128, 89, 89, 44, 44, 78, 78, 65, 65, 20, 20, 74, 74, 126, 126, 106, 106, 25, 25, 83, 83, -34, -34, -49, -49, 26, 26, -30, -30, 59, 59, -106, -106, -80, -80, 130, 130, -28, -28, 118, 118, 91, 91, -15, -15, 11, 11, 160, 160, 14, 14, 110, 110, 32, 32, 106, 106, -122, -122, -49, -49, -171, -171, 15, 15, 86, 86, -82, -82, 54, 54, 52, 52, 94, 94, -154, -154, -27, -27, 108, 108, 30, 30, -10, -10, 173, 173, 15, 15, -168, -168, 54, 54, 117, 117, 10, 10, 11, 11, -171, -171, -23, -23, 45, 45, -154, -154, -128, -128, -50, -50, -40, -40, -150, -150, 119, 119, 60, 60, -100, -100, 117, 117, -120, -120, 3, 3, 72, 72, 10, 10, 51, 51, 124, 124, 0, 0, 31, 31, -91, -91, 80, 80, 160, 160, 29, 29, -146, -146, 76, 76, 130, 130, 88, 88, 84, 84, 24, 24, -45, -45, -90, -90, -88, -88, 155, 155, -82, -82, 176, 176, -97, -97, 24, 24, 140, 140, 154, 154, 127, 127, 181, 181, 88, 88, -37, -37, -42, -42, 136, 136, -40, -40, 24, 24, -101, -101, -123, -123, -72, -72, 102, 102, 105, 105, -180, -180, -120, -120, -21, -21, -15, -15, -1, -1, -60, -60, -29, -29, 3, 3, -76, -76, 18, 18, 164, 164, -40, -40, 17, 17, 132, 132, -128, -128, -48, -48, 142, 142, 140, 140, 84, 84, -35, -35, -78, -78, -54, -54, 6, 6, -141, -141, -30, -30, -16, -16, 178, 178, -64, -64, 7, 7, 66, 66, -92, -92, -179, -179, 56, 56, 1, 1, 50, 50, -65, -65, 8, 8, 5, 5, -141, -141, 17, 17, -4, -4, 135, 135, -2, -2, 116, 116, 182, 182, 64, 64, -172, -172, 69, 69, 0, 0, 136, 136, -96, -96, -90, -90, 172, 172, 172, 172, -101, -101, -18, -18, -88, -88, -56, -56, -180, -180, 51, 51, 20, 20, 44, 44, 109, 109, 32, 32, -184, -184, 56, 56, 130, 130, 124, 124, -30, -30];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, 2}>] := 1;
ALEigenvalues[ideal<ZF | {9, 3, 3}>] := 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;