/*
  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![-10, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {576, 24, 24}>;

primesArray := [
[2, 2, w - 4],
[2, 2, -w - 3],
[5, 5, -2*w - 5],
[5, 5, -2*w + 7],
[9, 3, 3],
[23, 23, 2*w - 9],
[23, 23, -2*w - 7],
[31, 31, -6*w - 17],
[31, 31, 6*w - 23],
[37, 37, 2*w - 3],
[37, 37, -2*w - 1],
[41, 41, 2*w - 1],
[43, 43, -4*w - 9],
[43, 43, 4*w - 13],
[49, 7, -7],
[59, 59, 2*w - 11],
[59, 59, -2*w - 9],
[61, 61, -4*w + 17],
[61, 61, 4*w + 13],
[73, 73, 8*w + 23],
[73, 73, -8*w + 31],
[83, 83, 4*w - 11],
[83, 83, -4*w - 7],
[103, 103, 2*w - 13],
[103, 103, -2*w - 11],
[107, 107, 14*w + 39],
[107, 107, -14*w + 53],
[113, 113, -6*w - 13],
[113, 113, 6*w - 19],
[121, 11, -11],
[127, 127, -8*w - 19],
[127, 127, -8*w + 27],
[131, 131, 10*w + 29],
[131, 131, -10*w + 39],
[139, 139, 4*w - 7],
[139, 139, -4*w - 3],
[163, 163, 4*w - 3],
[163, 163, 4*w - 1],
[169, 13, -13],
[173, 173, 6*w - 17],
[173, 173, -6*w - 11],
[197, 197, 4*w - 21],
[197, 197, -4*w - 17],
[223, 223, -16*w + 57],
[223, 223, 16*w + 41],
[241, 241, 10*w - 33],
[241, 241, -10*w - 23],
[251, 251, -12*w - 29],
[251, 251, -12*w + 41],
[269, 269, -6*w - 7],
[269, 269, 6*w - 13],
[271, 271, -10*w + 41],
[271, 271, 10*w + 31],
[277, 277, -4*w - 19],
[277, 277, 4*w - 23],
[283, 283, 2*w - 19],
[283, 283, -2*w - 17],
[289, 17, -17],
[307, 307, 6*w + 23],
[307, 307, 6*w - 29],
[337, 337, 24*w + 67],
[337, 337, -24*w + 91],
[349, 349, -10*w - 21],
[349, 349, 10*w - 31],
[353, 353, -6*w - 1],
[353, 353, 6*w - 7],
[359, 359, 2*w - 21],
[359, 359, -2*w - 19],
[361, 19, -19],
[367, 367, 8*w - 21],
[367, 367, -8*w - 13],
[373, 373, -12*w + 49],
[373, 373, 12*w + 37],
[379, 379, -20*w + 71],
[379, 379, 20*w + 51],
[389, 389, 20*w + 57],
[389, 389, -20*w + 77],
[401, 401, 16*w + 47],
[401, 401, -16*w + 63],
[409, 409, -14*w - 33],
[409, 409, -14*w + 47],
[419, 419, 10*w + 33],
[419, 419, -10*w + 43],
[431, 431, -8*w - 11],
[431, 431, 8*w - 19],
[433, 433, 8*w + 29],
[433, 433, -8*w + 37],
[443, 443, 2*w - 23],
[443, 443, -2*w - 21],
[449, 449, 10*w - 29],
[449, 449, -10*w - 19],
[461, 461, -4*w - 23],
[461, 461, 4*w - 27],
[467, 467, -26*w + 99],
[467, 467, 28*w - 101],
[487, 487, -8*w - 9],
[487, 487, 8*w - 17],
[491, 491, -14*w + 57],
[491, 491, 14*w + 43],
[523, 523, 18*w + 53],
[523, 523, -18*w + 71],
[541, 541, -10*w - 17],
[541, 541, 10*w - 27],
[569, 569, -8*w - 31],
[569, 569, 8*w - 39],
[599, 599, 16*w - 53],
[599, 599, -16*w - 37],
[607, 607, -8*w - 3],
[607, 607, 8*w - 11],
[613, 613, -28*w + 107],
[613, 613, 34*w - 123],
[617, 617, -18*w + 61],
[617, 617, -18*w - 43],
[619, 619, -20*w - 49],
[619, 619, -20*w + 69],
[631, 631, -8*w - 1],
[631, 631, 8*w - 9],
[647, 647, 8*w - 7],
[647, 647, 8*w - 1],
[661, 661, 20*w + 59],
[661, 661, -20*w + 79],
[677, 677, -4*w - 27],
[677, 677, 4*w - 31],
[701, 701, 10*w - 23],
[701, 701, -10*w - 13],
[733, 733, -12*w + 53],
[733, 733, 12*w + 41],
[739, 739, 10*w - 47],
[739, 739, 10*w + 37],
[743, 743, 2*w - 29],
[743, 743, -2*w - 27],
[761, 761, -30*w + 107],
[761, 761, 30*w + 77],
[769, 769, -10*w - 11],
[769, 769, 10*w - 21],
[787, 787, 6*w - 37],
[787, 787, -6*w - 31],
[797, 797, 4*w - 33],
[797, 797, -4*w - 29],
[811, 811, -28*w + 99],
[811, 811, 28*w + 71],
[821, 821, 18*w - 59],
[821, 821, -18*w - 41],
[829, 829, 10*w - 19],
[829, 829, -10*w - 9],
[841, 29, -29],
[853, 853, 14*w - 41],
[853, 853, 14*w + 27],
[857, 857, 16*w + 51],
[857, 857, -16*w + 67],
[859, 859, 2*w - 31],
[859, 859, -2*w - 29],
[863, 863, -24*w - 59],
[863, 863, -24*w + 83],
[877, 877, -36*w + 137],
[877, 877, 38*w - 137],
[881, 881, -10*w - 7],
[881, 881, 10*w - 17],
[907, 907, -14*w + 61],
[907, 907, 14*w + 47],
[911, 911, 10*w + 39],
[911, 911, 10*w - 49],
[941, 941, -20*w + 81],
[941, 941, 20*w + 61],
[947, 947, -12*w - 17],
[947, 947, 12*w - 29],
[953, 953, 32*w + 91],
[953, 953, -32*w + 123],
[983, 983, 2*w - 33],
[983, 983, -2*w - 31],
[1009, 1009, 10*w - 9],
[1009, 1009, 10*w - 1],
[1021, 1021, 10*w - 3],
[1021, 1021, 10*w - 7],
[1033, 1033, 34*w + 87],
[1033, 1033, -34*w + 121],
[1061, 1061, 4*w - 37],
[1061, 1061, -4*w - 33],
[1087, 1087, -32*w + 113],
[1087, 1087, 32*w + 81],
[1091, 1091, 10*w - 51],
[1091, 1091, -10*w - 41],
[1097, 1097, 16*w + 53],
[1097, 1097, -16*w + 69],
[1103, 1103, 16*w - 47],
[1103, 1103, 16*w + 31],
[1109, 1109, 14*w - 37],
[1109, 1109, -14*w - 23],
[1117, 1117, 22*w - 73],
[1117, 1117, -22*w - 51],
[1123, 1123, -22*w + 89],
[1123, 1123, 22*w + 67],
[1153, 1153, -26*w + 89],
[1153, 1153, -26*w - 63],
[1171, 1171, 26*w + 77],
[1171, 1171, -26*w + 103],
[1181, 1181, 44*w + 123],
[1181, 1181, -44*w + 167],
[1187, 1187, 12*w - 23],
[1187, 1187, -12*w - 11],
[1193, 1193, -8*w - 39],
[1193, 1193, 8*w - 47],
[1229, 1229, -20*w + 83],
[1229, 1229, 20*w + 63],
[1231, 1231, 6*w - 43],
[1231, 1231, -6*w - 37],
[1279, 1279, -10*w - 43],
[1279, 1279, 10*w - 53],
[1289, 1289, 50*w + 131],
[1289, 1289, 50*w - 181],
[1291, 1291, -20*w - 43],
[1291, 1291, 20*w - 63],
[1303, 1303, 18*w + 59],
[1303, 1303, -18*w + 77],
[1307, 1307, 12*w - 19],
[1307, 1307, -12*w - 7],
[1321, 1321, -24*w + 97],
[1321, 1321, 24*w + 73],
[1361, 1361, -22*w - 49],
[1361, 1361, 22*w - 71],
[1373, 1373, 28*w + 83],
[1373, 1373, -28*w + 111],
[1399, 1399, -16*w - 27],
[1399, 1399, 16*w - 43],
[1427, 1427, -12*w - 1],
[1427, 1427, 12*w - 13],
[1433, 1433, -14*w - 17],
[1433, 1433, 14*w - 31],
[1439, 1439, 22*w + 69],
[1439, 1439, -22*w + 91],
[1451, 1451, 12*w - 11],
[1451, 1451, 12*w - 1],
[1453, 1453, -26*w - 61],
[1453, 1453, -26*w + 87],
[1471, 1471, -32*w - 79],
[1471, 1471, -32*w + 111],
[1481, 1481, -30*w - 73],
[1481, 1481, -30*w + 103],
[1499, 1499, -20*w - 41],
[1499, 1499, 20*w - 61],
[1549, 1549, 12*w - 61],
[1549, 1549, -12*w - 49],
[1553, 1553, 8*w - 51],
[1553, 1553, -8*w - 43],
[1559, 1559, 2*w - 41],
[1559, 1559, -2*w - 39],
[1567, 1567, -6*w - 41],
[1567, 1567, 6*w - 47],
[1579, 1579, 18*w + 61],
[1579, 1579, -18*w + 79],
[1583, 1583, -38*w + 147],
[1583, 1583, 38*w + 109],
[1597, 1597, 22*w - 69],
[1597, 1597, -22*w - 47],
[1601, 1601, 16*w + 57],
[1601, 1601, -16*w + 73],
[1607, 1607, -34*w + 133],
[1607, 1607, 34*w + 99],
[1609, 1609, 14*w - 27],
[1609, 1609, -14*w - 13],
[1619, 1619, -44*w + 157],
[1619, 1619, 44*w + 113],
[1663, 1663, -16*w - 23],
[1663, 1663, 16*w - 39],
[1697, 1697, 42*w + 107],
[1697, 1697, -42*w + 149],
[1699, 1699, 20*w - 59],
[1699, 1699, 20*w + 39],
[1721, 1721, -18*w - 31],
[1721, 1721, 18*w - 49],
[1723, 1723, 2*w - 43],
[1723, 1723, -2*w - 41],
[1747, 1747, 6*w - 49],
[1747, 1747, -6*w - 43],
[1753, 1753, 14*w - 23],
[1753, 1753, -14*w - 9],
[1759, 1759, -40*w + 141],
[1759, 1759, 40*w + 101],
[1783, 1783, 16*w - 37],
[1783, 1783, -16*w - 21],
[1847, 1847, -32*w - 77],
[1847, 1847, -32*w + 109],
[1861, 1861, -4*w - 43],
[1861, 1861, 4*w - 47],
[1877, 1877, -18*w - 29],
[1877, 1877, 18*w - 47],
[1907, 1907, -26*w + 107],
[1907, 1907, 26*w + 81],
[1931, 1931, -50*w + 191],
[1931, 1931, 60*w - 217],
[1973, 1973, 14*w - 13],
[1973, 1973, 14*w - 1],
[1993, 1993, 14*w - 11],
[1993, 1993, 14*w - 3],
[1999, 1999, -16*w - 17],
[1999, 1999, 16*w - 33],
[2011, 2011, -30*w + 121],
[2011, 2011, 30*w + 91],
[2017, 2017, -24*w + 101],
[2017, 2017, 24*w + 77],
[2027, 2027, -46*w + 177],
[2027, 2027, 46*w + 131],
[2029, 2029, 26*w - 83],
[2029, 2029, -26*w - 57],
[2081, 2081, -56*w + 213],
[2081, 2081, 58*w - 209],
[2083, 2083, 42*w + 121],
[2083, 2083, -42*w + 163],
[2087, 2087, 14*w - 71],
[2087, 2087, -14*w - 57],
[2089, 2089, -46*w + 163],
[2089, 2089, 46*w + 117],
[2099, 2099, 38*w + 111],
[2099, 2099, -38*w + 149],
[2111, 2111, 10*w - 61],
[2111, 2111, -10*w - 51],
[2131, 2131, -6*w - 47],
[2131, 2131, 6*w - 53],
[2137, 2137, 16*w - 77],
[2137, 2137, 16*w + 61],
[2141, 2141, -20*w + 89],
[2141, 2141, 20*w + 69],
[2153, 2153, 8*w - 57],
[2153, 2153, -8*w - 49],
[2209, 47, -47],
[2213, 2213, 66*w + 173],
[2213, 2213, 66*w - 239],
[2237, 2237, -4*w - 47],
[2237, 2237, 4*w - 51],
[2239, 2239, -40*w - 99],
[2239, 2239, -40*w + 139],
[2251, 2251, 20*w - 53],
[2251, 2251, -20*w - 33],
[2273, 2273, -32*w + 129],
[2273, 2273, 32*w + 97],
[2287, 2287, 26*w + 83],
[2287, 2287, -26*w + 109],
[2297, 2297, 18*w - 41],
[2297, 2297, -18*w - 23],
[2333, 2333, 54*w + 139],
[2333, 2333, -54*w + 193],
[2339, 2339, -10*w - 53],
[2339, 2339, 10*w - 63],
[2341, 2341, 36*w + 107],
[2341, 2341, -36*w + 143],
[2347, 2347, 14*w - 73],
[2347, 2347, -14*w - 59],
[2357, 2357, 44*w + 127],
[2357, 2357, -44*w + 171],
[2377, 2377, -24*w + 103],
[2377, 2377, 24*w + 79],
[2383, 2383, -58*w + 221],
[2383, 2383, 64*w - 231],
[2399, 2399, -16*w - 7],
[2399, 2399, 16*w - 23],
[2411, 2411, 28*w - 89],
[2411, 2411, -28*w - 61],
[2417, 2417, 16*w + 63],
[2417, 2417, 16*w - 79],
[2423, 2423, 24*w - 71],
[2423, 2423, 24*w + 47],
[2437, 2437, 4*w - 53],
[2437, 2437, -4*w - 49],
[2459, 2459, 2*w - 51],
[2459, 2459, -2*w - 49],
[2503, 2503, 16*w - 19],
[2503, 2503, -16*w - 3],
[2521, 2521, 50*w + 127],
[2521, 2521, -50*w + 177],
[2543, 2543, -16*w - 1],
[2543, 2543, 16*w - 17],
[2551, 2551, -34*w + 137],
[2551, 2551, 34*w + 103],
[2579, 2579, -20*w - 29],
[2579, 2579, 20*w - 49],
[2591, 2591, -48*w + 169],
[2591, 2591, 48*w + 121],
[2593, 2593, -8*w - 53],
[2593, 2593, 8*w - 61],
[2633, 2633, 34*w - 113],
[2633, 2633, -34*w - 79],
[2647, 2647, -46*w + 179],
[2647, 2647, 46*w + 133],
[2657, 2657, -22*w - 37],
[2657, 2657, 22*w - 59],
[2663, 2663, 2*w - 53],
[2663, 2663, -2*w - 51],
[2683, 2683, -44*w - 109],
[2683, 2683, -44*w + 153],
[2707, 2707, 28*w - 87],
[2707, 2707, -28*w - 59],
[2711, 2711, -40*w + 137],
[2711, 2711, -40*w - 97],
[2729, 2729, -32*w + 131],
[2729, 2729, 32*w + 99],
[2731, 2731, -20*w - 27],
[2731, 2731, 20*w - 47],
[2749, 2749, -12*w - 59],
[2749, 2749, 12*w - 71],
[2767, 2767, -6*w - 53],
[2767, 2767, 6*w - 59],
[2789, 2789, 20*w + 73],
[2789, 2789, -20*w + 93],
[2797, 2797, -58*w + 207],
[2797, 2797, 58*w + 149],
[2809, 53, -53],
[2819, 2819, 10*w - 67],
[2819, 2819, -10*w - 57],
[2833, 2833, 26*w - 77],
[2833, 2833, 26*w + 51],
[2837, 2837, -18*w - 13],
[2837, 2837, 18*w - 31],
[2861, 2861, 4*w - 57],
[2861, 2861, -4*w - 53],
[2879, 2879, 24*w - 67],
[2879, 2879, -24*w - 43],
[2903, 2903, -56*w + 199],
[2903, 2903, 56*w + 143],
[2909, 2909, 52*w + 149],
[2909, 2909, -52*w + 201],
[2927, 2927, 32*w - 103],
[2927, 2927, -32*w - 71],
[2953, 2953, -48*w + 187],
[2953, 2953, 48*w + 139],
[2957, 2957, 44*w + 129],
[2957, 2957, -44*w + 173],
[3001, 3001, -16*w - 67],
[3001, 3001, 16*w - 83],
[3011, 3011, -20*w - 23],
[3011, 3011, 20*w - 43],
[3067, 3067, -52*w + 183],
[3067, 3067, 52*w + 131],
[3079, 3079, 18*w + 71],
[3079, 3079, 18*w - 89],
[3083, 3083, 36*w - 119],
[3083, 3083, -36*w - 83],
[3121, 3121, -24*w + 107],
[3121, 3121, 24*w + 83],
[3137, 3137, -38*w - 89],
[3137, 3137, -38*w + 127],
[3167, 3167, -48*w - 119],
[3167, 3167, -48*w + 167],
[3203, 3203, -44*w - 107],
[3203, 3203, -44*w + 151],
[3221, 3221, -18*w - 1],
[3221, 3221, 18*w - 19],
[3229, 3229, 74*w - 267],
[3229, 3229, -68*w + 259],
[3257, 3257, 18*w - 17],
[3257, 3257, 18*w - 1],
[3259, 3259, 20*w - 39],
[3259, 3259, -20*w - 19],
[3271, 3271, -32*w - 69],
[3271, 3271, 32*w - 101],
[3301, 3301, -62*w + 221],
[3301, 3301, 62*w + 159],
[3313, 3313, 8*w - 67],
[3313, 3313, -8*w - 59],
[3319, 3319, -30*w + 127],
[3319, 3319, 30*w + 97],
[3323, 3323, 2*w - 59],
[3323, 3323, -2*w - 57],
[3329, 3329, -26*w - 47],
[3329, 3329, 26*w - 73],
[3331, 3331, 10*w - 71],
[3331, 3331, -10*w - 61],
[3361, 3361, 22*w - 51],
[3361, 3361, -22*w - 29],
[3371, 3371, 20*w - 37],
[3371, 3371, -20*w - 17],
[3407, 3407, 74*w + 207],
[3407, 3407, -74*w + 281],
[3413, 3413, 28*w + 93],
[3413, 3413, -28*w + 121],
[3449, 3449, -30*w - 61],
[3449, 3449, 30*w - 91],
[3467, 3467, -14*w - 67],
[3467, 3467, 14*w - 81],
[3469, 3469, 20*w + 77],
[3469, 3469, 20*w - 97],
[3517, 3517, -22*w - 27],
[3517, 3517, 22*w - 49],
[3527, 3527, -34*w + 141],
[3527, 3527, 34*w + 107],
[3547, 3547, -28*w - 53],
[3547, 3547, 28*w - 81],
[3557, 3557, -4*w - 59],
[3557, 3557, 4*w - 63],
[3559, 3559, 2*w - 61],
[3559, 3559, -2*w - 59],
[3571, 3571, 20*w - 33],
[3571, 3571, -20*w - 13],
[3583, 3583, -56*w + 197],
[3583, 3583, 56*w + 141],
[3607, 3607, -32*w - 67],
[3607, 3607, 32*w - 99],
[3613, 3613, 52*w + 151],
[3613, 3613, -52*w + 203],
[3617, 3617, 16*w - 87],
[3617, 3617, -16*w - 71],
[3631, 3631, 40*w - 133],
[3631, 3631, -40*w - 93],
[3659, 3659, 20*w - 31],
[3659, 3659, -20*w - 11],
[3691, 3691, -52*w - 129],
[3691, 3691, -52*w + 181],
[3727, 3727, 6*w - 67],
[3727, 3727, -6*w - 61],
[3733, 3733, -46*w - 111],
[3733, 3733, -46*w + 157],
[3739, 3739, -20*w - 9],
[3739, 3739, 20*w - 29],
[3767, 3767, -14*w - 69],
[3767, 3767, 14*w - 83],
[3793, 3793, 26*w - 69],
[3793, 3793, -26*w - 43],
[3797, 3797, -76*w + 289],
[3797, 3797, 76*w + 213],
[3803, 3803, 2*w - 63],
[3803, 3803, -2*w - 61],
[3821, 3821, -20*w - 79],
[3821, 3821, 20*w - 99],
[3823, 3823, -42*w + 169],
[3823, 3823, 42*w + 127],
[3833, 3833, -8*w - 63],
[3833, 3833, 8*w - 71],
[3853, 3853, -12*w - 67],
[3853, 3853, 12*w - 79],
[3863, 3863, 62*w + 177],
[3863, 3863, -62*w + 239],
[3877, 3877, 36*w + 113],
[3877, 3877, -36*w + 149],
[3911, 3911, 46*w + 137],
[3911, 3911, -46*w + 183],
[3931, 3931, -20*w - 3],
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[8387, 8387, -74*w + 291],
[8389, 8389, 130*w + 341],
[8389, 8389, 130*w - 471],
[8423, 8423, 2*w - 93],
[8423, 8423, -2*w - 91],
[8447, 8447, -26*w - 111],
[8447, 8447, 26*w - 137],
[8467, 8467, 6*w - 97],
[8467, 8467, -6*w - 91],
[8527, 8527, 54*w + 169],
[8527, 8527, -54*w + 223],
[8537, 8537, -78*w - 193],
[8537, 8537, -78*w + 271],
[8573, 8573, -54*w - 119],
[8573, 8573, 54*w - 173],
[8609, 8609, -70*w - 169],
[8609, 8609, -70*w + 239],
[8641, 8641, -64*w + 257],
[8641, 8641, 64*w + 193],
[8647, 8647, 32*w - 59],
[8647, 8647, -32*w - 27],
[8669, 8669, 20*w - 123],
[8669, 8669, -20*w - 103],
[8693, 8693, 46*w - 137],
[8693, 8693, -46*w - 91],
[8713, 8713, -34*w - 39],
[8713, 8713, 34*w - 73],
[8731, 8731, -30*w - 119],
[8731, 8731, 30*w - 149],
[8737, 8737, -24*w - 109],
[8737, 8737, 24*w - 133],
[8741, 8741, -30*w - 7],
[8741, 8741, 30*w - 37],
[8753, 8753, -8*w - 93],
[8753, 8753, 8*w - 101],
[8779, 8779, 18*w - 119],
[8779, 8779, -18*w - 101],
[8783, 8783, 58*w + 179],
[8783, 8783, -58*w + 237],
[8807, 8807, -14*w - 97],
[8807, 8807, 14*w - 111],
[8819, 8819, 44*w - 127],
[8819, 8819, -44*w - 83],
[8831, 8831, -40*w - 67],
[8831, 8831, 40*w - 107],
[8861, 8861, 4*w - 97],
[8861, 8861, -4*w - 93],
[8887, 8887, 34*w - 161],
[8887, 8887, 34*w + 127],
[8893, 8893, 132*w + 367],
[8893, 8893, -132*w + 499],
[8929, 8929, 80*w + 233],
[8929, 8929, -80*w + 313],
[8933, 8933, 34*w - 71],
[8933, 8933, -34*w - 37],
[8963, 8963, 22*w - 129],
[8963, 8963, -22*w - 107],
[8969, 8969, -30*w - 1],
[8969, 8969, 30*w - 31],
[8971, 8971, 50*w + 161],
[8971, 8971, -50*w + 211],
[8999, 8999, -62*w + 251],
[8999, 8999, 62*w + 189],
[9011, 9011, 60*w - 197],
[9011, 9011, -60*w - 137],
[9029, 9029, 30*w - 29],
[9029, 9029, 30*w - 1],
[9041, 9041, 90*w + 227],
[9041, 9041, -90*w + 317],
[9043, 9043, -122*w + 463],
[9043, 9043, 122*w + 341],
[9059, 9059, 36*w - 83],
[9059, 9059, -36*w - 47],
[9103, 9103, -104*w + 371],
[9103, 9103, 104*w + 267],
[9127, 9127, 32*w - 53],
[9127, 9127, -32*w - 21],
[9133, 9133, 12*w - 109],
[9133, 9133, -12*w - 97],
[9151, 9151, -88*w + 309],
[9151, 9151, 88*w + 221],
[9161, 9161, 30*w - 23],
[9161, 9161, 30*w - 7],
[9209, 9209, 30*w - 19],
[9209, 9209, 30*w - 11],
[9221, 9221, 30*w - 17],
[9221, 9221, 30*w - 13],
[9227, 9227, 14*w - 113],
[9227, 9227, -14*w - 99],
[9241, 9241, -50*w - 103],
[9241, 9241, 50*w - 153],
[9257, 9257, -16*w - 101],
[9257, 9257, 16*w - 117],
[9311, 9311, 70*w + 209],
[9311, 9311, -70*w + 279],
[9323, 9323, -84*w - 209],
[9323, 9323, -84*w + 293],
[9343, 9343, 56*w - 179],
[9343, 9343, -56*w - 123],
[9349, 9349, 34*w - 67],
[9349, 9349, -34*w - 33],
[9371, 9371, -68*w + 229],
[9371, 9371, -68*w - 161],
[9391, 9391, 90*w + 259],
[9391, 9391, -90*w + 349],
[9397, 9397, -44*w + 193],
[9397, 9397, 44*w + 149],
[9409, 97, -97],
[9421, 9421, -70*w - 167],
[9421, 9421, -70*w + 237],
[9431, 9431, -80*w - 197],
[9431, 9431, -80*w + 277],
[9439, 9439, -22*w - 109],
[9439, 9439, 22*w - 131],
[9461, 9461, -78*w - 191],
[9461, 9461, -78*w + 269],
[9463, 9463, -78*w + 307],
[9463, 9463, 78*w + 229],
[9467, 9467, 34*w + 129],
[9467, 9467, 34*w - 163],
[9473, 9473, -74*w + 253],
[9473, 9473, -74*w - 179],
[9479, 9479, -82*w + 321],
[9479, 9479, 82*w + 239],
[9491, 9491, -52*w - 109],
[9491, 9491, 52*w - 161],
[9511, 9511, -40*w - 63],
[9511, 9511, 40*w - 103],
[9521, 9521, 40*w + 141],
[9521, 9521, -40*w + 181],
[9533, 9533, -52*w + 219],
[9533, 9533, 52*w + 167],
[9551, 9551, 134*w + 373],
[9551, 9551, -134*w + 507],
[9619, 9619, -58*w + 239],
[9619, 9619, 58*w + 181],
[9631, 9631, 6*w - 103],
[9631, 9631, -6*w - 97],
[9643, 9643, 44*w - 123],
[9643, 9643, -44*w - 79],
[9677, 9677, 124*w + 347],
[9677, 9677, -124*w + 471],
[9697, 9697, 16*w - 119],
[9697, 9697, -16*w - 103],
[9719, 9719, 50*w + 163],
[9719, 9719, -50*w + 213],
[9721, 9721, -24*w - 113],
[9721, 9721, 24*w - 137],
[9733, 9733, 34*w - 63],
[9733, 9733, -34*w - 29],
[9749, 9749, -50*w - 101],
[9749, 9749, 50*w - 151],
[9767, 9767, -32*w - 11],
[9767, 9767, 32*w - 43],
[9781, 9781, 110*w + 283],
[9781, 9781, -110*w + 393],
[9791, 9791, -10*w - 99],
[9791, 9791, 10*w - 109],
[9803, 9803, 36*w - 77],
[9803, 9803, -36*w - 41],
[9817, 9817, -62*w - 141],
[9817, 9817, 62*w - 203],
[9839, 9839, 40*w - 101],
[9839, 9839, -40*w - 61],
[9871, 9871, 32*w - 41],
[9871, 9871, -32*w - 9],
[9883, 9883, -62*w + 253],
[9883, 9883, 62*w + 191],
[9901, 9901, -38*w - 51],
[9901, 9901, 38*w - 89],
[9923, 9923, 22*w - 133],
[9923, 9923, -22*w - 111],
[9931, 9931, 140*w - 507],
[9931, 9931, 140*w + 367],
[9967, 9967, 32*w - 39],
[9967, 9967, -32*w - 7],
[9973, 9973, 12*w - 113],
[9973, 9973, -12*w - 101]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [0, 0, -2, -2, 1, -8, -8, 8, 8, 6, 6, -6, 4, 4, -14, 4, 4, -2, -2, 10, 10, -4, -4, 16, 16, -12, -12, 18, 18, -6, -8, -8, -4, -4, -12, -12, 12, 12, -22, 6, 6, -18, -18, -8, -8, 18, 18, 20, 20, -10, -10, 8, 8, -26, -26, -28, -28, -30, 12, 12, 18, 18, 30, 30, 2, 2, -24, -24, -22, -8, -8, -10, -10, 20, 20, -2, -2, -30, -30, -6, -6, 12, 12, 32, 32, -14, -14, 20, 20, -14, -14, -26, -26, -36, -36, -32, -32, -12, -12, 4, 4, -18, -18, 10, 10, 24, 24, -40, -40, 38, 38, 42, 42, -44, -44, 16, 16, 8, 8, -10, -10, -2, -2, 6, 6, 14, 14, -4, -4, -8, -8, -22, -22, 2, 2, 28, 28, 22, 22, 4, 4, 30, 30, -50, -50, -22, -10, -10, 42, 42, -12, -12, -32, -32, -18, -18, 50, 50, 4, 4, 16, 16, 6, 6, 12, 12, -54, -54, -24, -24, -46, -46, 30, 30, 10, 10, 30, 30, 8, 8, -52, -52, -54, -54, 48, 48, -18, -18, -50, -50, -36, -36, -62, -62, -20, -20, 22, 22, -36, -36, -22, -22, 38, 38, 8, 8, 56, 56, -22, -22, 4, 4, -32, -32, -28, -28, -22, -22, -62, -62, 54, 54, 0, 0, 60, 60, -6, -6, 48, 48, 52, 52, -66, -66, -40, -40, 10, 10, 20, 20, 46, 46, 18, 18, -40, -40, -56, -56, 52, 52, -16, -16, 78, 78, -30, -30, 8, 8, 10, 10, 12, 12, -8, -8, -14, -14, -4, -4, 26, 26, -76, -76, -52, -52, -38, -38, 40, 40, -64, -64, -8, -8, -58, -58, -18, -18, 12, 12, -12, -12, -66, -66, 74, 74, -40, -40, 84, 84, 2, 2, 36, 36, -34, -34, -46, -46, -20, -20, -8, -8, -22, -22, -84, -84, 48, 48, -20, -20, -38, -38, -58, -58, 58, 58, -94, 30, 30, -42, -42, 24, 24, -12, -12, 66, 66, 72, 72, -54, -54, -26, -26, -20, -20, -42, -42, 52, 52, -34, -34, -22, -22, -24, -24, 64, 64, -76, -76, 34, 34, -24, -24, 38, 38, 20, 20, -32, -32, -38, -38, 80, 80, 16, 16, 60, 60, -96, -96, 34, 34, -70, -70, 48, 48, -30, -30, 24, 24, 4, 4, 60, 60, 72, 72, -6, -6, -28, -28, 62, 62, 40, 40, 46, 46, -2, -2, -102, 12, 12, -46, -46, 46, 46, 22, 22, 0, 0, -40, -40, -10, -10, -16, -16, 10, 10, -10, -10, -38, -38, 60, 60, 20, 20, -80, -80, 36, 36, -14, -14, -30, -30, 16, 16, 28, 28, -18, -18, 14, 14, 90, 90, 20, 20, -32, -32, 70, 70, -14, -14, 64, 64, 4, 4, 50, 50, -4, -4, -30, -30, -12, -12, -64, -64, 14, 14, -102, -102, -60, -60, -98, -98, -98, -98, -88, -88, 36, 36, -50, -50, 0, 0, 60, 60, -56, -56, -64, -64, 30, 30, 114, 114, -24, -24, -60, -60, -92, -92, -24, -24, 70, 70, 100, 100, -56, -56, -14, -14, 94, 94, 36, 36, 22, 22, 56, 56, -70, -70, 94, 94, 24, 24, 102, 102, 120, 120, -60, -60, 88, 88, 54, 54, 44, 44, 52, 52, -46, -46, -100, -100, 90, 90, -80, -80, -60, -60, -4, -4, -66, -66, 116, 116, 54, 54, 104, 104, 18, 18, 36, 36, 112, 112, 34, 34, -52, -52, -52, -52, -14, -14, -30, -30, -54, -54, -30, -30, 24, 24, -90, -90, 16, 16, 124, 124, -118, 16, 16, 92, 92, 70, 70, 18, 18, -112, -112, 88, 88, -40, -40, 118, 118, 38, 38, 44, 44, -38, -38, -28, -28, -78, -78, 104, 104, 44, 44, 38, 38, 64, 64, 12, 12, -74, -74, 10, 10, 96, 96, 66, 66, -50, -50, -78, -78, 78, 78, 40, 40, -42, -42, 74, 74, -56, -56, -32, -32, 128, 128, -82, -82, -86, -86, 2, 2, -124, -124, -116, -116, -24, -24, -48, -48, -78, -44, -44, -20, -20, 76, 76, 104, 104, 12, 12, -82, -82, 14, 14, -38, -38, -28, -28, 10, 10, 16, 16, 98, 98, -78, -78, -90, -90, 24, 24, -34, -34, 84, 84, 8, 8, 118, 118, -102, -102, -82, -82, -20, -20, 74, 74, -80, -80, 104, 104, 16, 16, -16, -16, 148, 148, 110, 110, -10, -10, -54, -54, -28, -28, -84, -84, -6, -6, -10, -10, -82, -82, -10, -10, -90, -90, -100, -100, 24, 24, -88, -88, -102, -102, 14, 14, 14, 14, 76, 76, -84, -84, -114, -114, 68, 68, -24, -24, -6, -6, -96, -96, -24, -24, 98, 98, -10, -10, -84, -84, 64, 64, 52, 52, 102, 102, 86, 86, -44, -44, -112, -112, 76, 76, -6, -6, -22, -22, 100, 100, -18, -18, -14, -14, -32, -32, -10, -10, 128, 128, -116, -116, -94, 130, 130, 24, 24, -138, -138, 120, 120, 70, 70, 92, 92, -62, -62, 18, 18, 72, 72, -154, -154, 30, 30, 70, 70, -124, -124, 54, 54, -70, -70, -70, -70, -62, -62, -56, -56, -86, -86, -34, -34, 88, 88, -124, -124, 14, 14, 50, 50, 80, 80, 124, 124, 38, 38, 8, 8, -96, -96, 30, 30, 42, 42, -140, -140, 14, 14, -156, -156, 46, 46, -22, -22, 96, 96, 124, 124, 0, 0, -132, -132, 54, 54, -110, -110, -76, -76, -88, -88, -86, -86, 126, 126, 36, 36, -4, -4, 12, 12, -110, -110, 16, 16, 30, 30, -38, -38, 48, 48, 10, 10, 106, 106, 32, 32, -156, -156, -138, -138, 112, 112, -2, -2, -158, -158, -140, -140, 154, 154, -36, -36, -34, -34, -68, -68, 122, 122, 18, 18, 16, 16, -124, -124, 28, 28, 140, 140, 94, 94, -102, -102, -48, -48, 30, 30, 92, 92, 22, 22, 18, 18, 70, 70, -32, -32, -116, -116, -120, -120, -74, -74, -82, -82, 10, 10, -90, -90, -24, -24, -114, -114, -144, -144, 14, 14, 66, 66, 164, 164, 34, 34, 158, 158, -142, 62, 62, 150, 150, -44, -44, -70, -70, 132, 132, -92, -92, 30, 30, 146, 146, 88, 88, -90, -90, -114, -114, 36, 36, 98, 98, 160, 160, 44, 44, 168, 168, -46, -46, -2, -2, -72, -72, -86, -86, 6, 6, -52, -52, -14, -14, 8, 8, 60, 60, -60, -60, -110, -110, 140, 140, 70, 70, -24, -24, 48, 48, 28, 28, -56, -56, 26, 26, -138, -138, -30, -30, 66, 66, -112, -112, -122, -122, -66, -66, 74, 74, 148, 148, 98, 98, -18, -18, -126, -126, -60, -60, -48, -48, 136, 136, 60, 60, 80, 80, -122, -122, 0, 0, 126, 126, -30, -30, 78, 78, 12, 12, 170, 170, -28, -28, 24, 24, -36, -36, 126, 126, -126, -126, 28, 28, -132, -132, 88, 88, 0, 0, 94, 94, 8, 8, -134, -134, 122, 122, -18, -18, 20, 20, -102, -102, -6, -6, 80, 80, -44, -44, 104, 104, -154, -154, 132, 132, -184, -184, 150, 150, -190, 94, 94, 8, 8, -8, -8, -18, -18, -16, -16, -44, -44, -94, -94, 56, 56, -20, -20, -32, -32, 82, 82, -138, -138, -32, -32, -116, -116, -152, -152, -140, -140, 54, 54, 34, 34, -120, -120, 122, 122, 102, 102, -18, -18, -88, -88, -10, -10, 0, 0, 100, 100, -6, -6, -16, -16, 72, 72, 148, 148, -130, -130, 156, 156, 148, 148, -8, -8, 6, 6];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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