/*
  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![-15, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {61, 61, 2*w - 1}>;

primesArray := [
[3, 3, -w - 3],
[3, 3, -w + 4],
[4, 2, 2],
[5, 5, w - 5],
[5, 5, -w - 4],
[13, 13, -w - 1],
[13, 13, w - 2],
[19, 19, 3*w - 14],
[19, 19, -3*w - 11],
[41, 41, -w - 7],
[41, 41, w - 8],
[47, 47, 3*w - 11],
[47, 47, -3*w - 8],
[49, 7, -7],
[61, 61, 2*w - 1],
[73, 73, -3*w + 16],
[73, 73, 3*w + 13],
[83, 83, 2*w - 13],
[83, 83, -2*w - 11],
[97, 97, 7*w + 22],
[97, 97, 7*w - 29],
[103, 103, -3*w - 14],
[103, 103, 3*w - 17],
[107, 107, -3*w - 4],
[107, 107, 3*w - 7],
[109, 109, 5*w - 19],
[109, 109, 5*w + 14],
[113, 113, 9*w - 38],
[113, 113, 9*w + 29],
[121, 11, -11],
[127, 127, -6*w - 23],
[127, 127, 6*w - 29],
[131, 131, -3*w - 1],
[131, 131, 3*w - 4],
[137, 137, 3*w - 2],
[137, 137, 3*w - 1],
[149, 149, 6*w - 23],
[149, 149, 6*w + 17],
[163, 163, 4*w - 11],
[163, 163, -4*w - 7],
[167, 167, -w - 13],
[167, 167, w - 14],
[179, 179, 9*w + 28],
[179, 179, 9*w - 37],
[197, 197, -4*w - 19],
[197, 197, 4*w - 23],
[199, 199, -5*w - 11],
[199, 199, 5*w - 16],
[229, 229, 10*w + 31],
[229, 229, 10*w - 41],
[239, 239, 10*w - 47],
[239, 239, 10*w + 37],
[241, 241, 7*w + 19],
[241, 241, 7*w - 26],
[257, 257, -w - 16],
[257, 257, w - 17],
[263, 263, 2*w - 19],
[263, 263, -2*w - 17],
[269, 269, -5*w + 28],
[269, 269, 5*w + 23],
[271, 271, -5*w - 8],
[271, 271, 5*w - 13],
[283, 283, 3*w - 22],
[283, 283, -3*w - 19],
[289, 17, -17],
[293, 293, -6*w - 13],
[293, 293, 6*w - 19],
[317, 317, 11*w + 41],
[317, 317, 11*w - 52],
[347, 347, 12*w + 37],
[347, 347, 12*w - 49],
[353, 353, 6*w - 17],
[353, 353, -6*w - 11],
[367, 367, 7*w - 23],
[367, 367, -7*w - 16],
[379, 379, 5*w - 4],
[379, 379, 5*w - 1],
[431, 431, 5*w - 31],
[431, 431, -5*w - 26],
[439, 439, 11*w - 43],
[439, 439, 11*w + 32],
[443, 443, -7*w - 31],
[443, 443, -7*w + 38],
[449, 449, -6*w - 7],
[449, 449, 6*w - 13],
[461, 461, 15*w - 62],
[461, 461, 15*w + 47],
[463, 463, -3*w - 23],
[463, 463, 3*w - 26],
[479, 479, 9*w + 23],
[479, 479, 9*w - 32],
[487, 487, -9*w + 46],
[487, 487, -9*w - 37],
[491, 491, -w - 22],
[491, 491, w - 23],
[503, 503, 13*w - 62],
[503, 503, -13*w - 49],
[529, 23, -23],
[563, 563, 17*w + 62],
[563, 563, 17*w - 79],
[569, 569, 15*w + 46],
[569, 569, 15*w - 61],
[571, 571, -9*w - 38],
[571, 571, -9*w + 47],
[601, 601, 15*w + 56],
[601, 601, 15*w - 71],
[607, 607, 6*w - 37],
[607, 607, -6*w - 31],
[613, 613, -12*w - 47],
[613, 613, 12*w - 59],
[619, 619, -3*w - 26],
[619, 619, 3*w - 29],
[659, 659, -7*w - 34],
[659, 659, 7*w - 41],
[683, 683, -9*w - 19],
[683, 683, 9*w - 28],
[691, 691, -7*w - 4],
[691, 691, 7*w - 11],
[719, 719, 21*w - 88],
[719, 719, 21*w + 67],
[727, 727, -7*w - 1],
[727, 727, 7*w - 8],
[733, 733, 3*w - 31],
[733, 733, -3*w - 28],
[751, 751, -8*w - 11],
[751, 751, 8*w - 19],
[757, 757, 21*w + 76],
[757, 757, 21*w - 97],
[773, 773, 9*w - 26],
[773, 773, -9*w - 17],
[797, 797, -w - 28],
[797, 797, w - 29],
[809, 809, -5*w - 32],
[809, 809, 5*w - 37],
[827, 827, 12*w - 43],
[827, 827, 12*w + 31],
[829, 829, -12*w + 61],
[829, 829, -12*w - 49],
[839, 839, 2*w - 31],
[839, 839, -2*w - 29],
[841, 29, -29],
[853, 853, -11*w - 26],
[853, 853, 11*w - 37],
[857, 857, 16*w - 77],
[857, 857, -16*w - 61],
[859, 859, 15*w - 73],
[859, 859, -15*w - 58],
[863, 863, 24*w + 77],
[863, 863, 24*w - 101],
[881, 881, 15*w + 43],
[881, 881, 15*w - 58],
[911, 911, -11*w + 58],
[911, 911, -11*w - 47],
[919, 919, 3*w - 34],
[919, 919, -3*w - 31],
[929, 929, 9*w - 22],
[929, 929, -9*w - 13],
[937, 937, 13*w + 34],
[937, 937, 13*w - 47],
[961, 31, -31],
[967, 967, 8*w - 7],
[967, 967, 8*w - 1],
[971, 971, -12*w - 29],
[971, 971, 12*w - 41],
[977, 977, -w - 31],
[977, 977, w - 32],
[991, 991, -15*w - 59],
[991, 991, 15*w - 74],
[1021, 1021, 9*w - 52],
[1021, 1021, -9*w - 43],
[1033, 1033, -11*w - 23],
[1033, 1033, 11*w - 34],
[1049, 1049, 8*w - 49],
[1049, 1049, -8*w - 41],
[1051, 1051, 6*w - 43],
[1051, 1051, -6*w - 37],
[1093, 1093, 26*w - 109],
[1093, 1093, 26*w + 83],
[1097, 1097, 18*w - 71],
[1097, 1097, 18*w + 53],
[1103, 1103, -9*w - 7],
[1103, 1103, 9*w - 16],
[1117, 1117, -9*w - 44],
[1117, 1117, 9*w - 53],
[1123, 1123, 3*w - 37],
[1123, 1123, -3*w - 34],
[1163, 1163, 9*w - 13],
[1163, 1163, -9*w - 4],
[1171, 1171, 13*w - 44],
[1171, 1171, -13*w - 31],
[1181, 1181, 25*w - 116],
[1181, 1181, 25*w + 91],
[1193, 1193, 9*w - 11],
[1193, 1193, -9*w - 2],
[1201, 1201, 10*w - 23],
[1201, 1201, -10*w - 13],
[1217, 1217, 27*w + 86],
[1217, 1217, 27*w - 113],
[1223, 1223, 9*w - 8],
[1223, 1223, 9*w - 1],
[1229, 1229, 9*w - 2],
[1229, 1229, 9*w - 7],
[1259, 1259, 5*w - 43],
[1259, 1259, -5*w - 38],
[1277, 1277, 4*w - 41],
[1277, 1277, -4*w - 37],
[1297, 1297, 22*w + 67],
[1297, 1297, 22*w - 89],
[1301, 1301, 19*w - 92],
[1301, 1301, -19*w - 73],
[1303, 1303, 16*w + 43],
[1303, 1303, 16*w - 59],
[1327, 1327, 17*w + 47],
[1327, 1327, 17*w - 64],
[1361, 1361, 15*w + 38],
[1361, 1361, 15*w - 53],
[1367, 1367, -14*w + 73],
[1367, 1367, -14*w - 59],
[1369, 37, -37],
[1381, 1381, -10*w - 7],
[1381, 1381, 10*w - 17],
[1399, 1399, -15*w - 62],
[1399, 1399, -15*w + 77],
[1423, 1423, -3*w - 38],
[1423, 1423, 3*w - 41],
[1439, 1439, 23*w + 86],
[1439, 1439, 23*w - 109],
[1451, 1451, 15*w - 52],
[1451, 1451, -15*w - 37],
[1459, 1459, -18*w - 71],
[1459, 1459, 18*w - 89],
[1483, 1483, 29*w + 92],
[1483, 1483, 29*w - 121],
[1489, 1489, -10*w - 1],
[1489, 1489, 10*w - 11],
[1511, 1511, 5*w - 46],
[1511, 1511, -5*w - 41],
[1559, 1559, 24*w + 73],
[1559, 1559, 24*w - 97],
[1567, 1567, 6*w - 49],
[1567, 1567, -6*w - 43],
[1571, 1571, 12*w - 31],
[1571, 1571, -12*w - 19],
[1583, 1583, 13*w + 58],
[1583, 1583, -13*w + 71],
[1601, 1601, 33*w - 139],
[1601, 1601, 33*w + 106],
[1613, 1613, 21*w + 61],
[1613, 1613, 21*w - 82],
[1627, 1627, -9*w - 49],
[1627, 1627, 9*w - 58],
[1663, 1663, 11*w - 19],
[1663, 1663, -11*w - 8],
[1667, 1667, -12*w - 17],
[1667, 1667, 12*w - 29],
[1669, 1669, -3*w - 41],
[1669, 1669, 3*w - 44],
[1693, 1693, -14*w - 29],
[1693, 1693, 14*w - 43],
[1699, 1699, 25*w + 76],
[1699, 1699, 25*w - 101],
[1709, 1709, -15*w - 34],
[1709, 1709, 15*w - 49],
[1721, 1721, -16*w - 67],
[1721, 1721, -16*w + 83],
[1723, 1723, 19*w - 71],
[1723, 1723, 19*w + 52],
[1733, 1733, 28*w - 131],
[1733, 1733, 28*w + 103],
[1747, 1747, 23*w - 91],
[1747, 1747, 23*w + 68],
[1753, 1753, 34*w + 109],
[1753, 1753, 34*w - 143],
[1783, 1783, -18*w + 91],
[1783, 1783, -18*w - 73],
[1789, 1789, 11*w - 13],
[1789, 1789, -11*w - 2],
[1811, 1811, 25*w - 119],
[1811, 1811, -25*w - 94],
[1831, 1831, 21*w - 103],
[1831, 1831, -21*w - 82],
[1849, 43, -43],
[1871, 1871, 15*w - 47],
[1871, 1871, -15*w - 32],
[1877, 1877, -w - 43],
[1877, 1877, w - 44],
[1879, 1879, 16*w - 53],
[1879, 1879, -16*w - 37],
[1907, 1907, 12*w - 23],
[1907, 1907, -12*w - 11],
[1913, 1913, -17*w + 88],
[1913, 1913, -17*w - 71],
[1933, 1933, -3*w - 44],
[1933, 1933, 3*w - 47],
[1949, 1949, -15*w - 31],
[1949, 1949, 15*w - 46],
[1951, 1951, -6*w - 47],
[1951, 1951, 6*w - 53],
[1979, 1979, 13*w - 74],
[1979, 1979, -13*w - 61],
[1993, 1993, 27*w + 101],
[1993, 1993, 27*w - 128],
[1997, 1997, 27*w + 82],
[1997, 1997, 27*w - 109],
[1999, 1999, 31*w + 97],
[1999, 1999, 31*w - 128],
[2017, 2017, -21*w - 83],
[2017, 2017, 21*w - 104],
[2027, 2027, 12*w - 19],
[2027, 2027, -12*w - 7],
[2029, 2029, -12*w - 59],
[2029, 2029, 12*w - 71],
[2069, 2069, -5*w - 47],
[2069, 2069, 5*w - 52],
[2083, 2083, 19*w + 49],
[2083, 2083, 19*w - 68],
[2087, 2087, 33*w - 137],
[2087, 2087, 33*w + 104],
[2089, 2089, 14*w - 37],
[2089, 2089, -14*w - 23],
[2099, 2099, 15*w - 44],
[2099, 2099, -15*w - 29],
[2113, 2113, 29*w - 118],
[2113, 2113, 29*w + 89],
[2131, 2131, 20*w - 73],
[2131, 2131, 20*w + 53],
[2221, 2221, 25*w + 73],
[2221, 2221, 25*w - 98],
[2237, 2237, 18*w - 61],
[2237, 2237, -18*w - 43],
[2243, 2243, 2*w - 49],
[2243, 2243, -2*w - 47],
[2269, 2269, -15*w + 83],
[2269, 2269, 15*w + 68],
[2273, 2273, -8*w - 53],
[2273, 2273, 8*w - 61],
[2293, 2293, 12*w - 73],
[2293, 2293, -12*w - 61],
[2309, 2309, 15*w - 41],
[2309, 2309, -15*w - 26],
[2333, 2333, -7*w - 52],
[2333, 2333, 7*w - 59],
[2357, 2357, 4*w - 53],
[2357, 2357, -4*w - 49],
[2383, 2383, 16*w - 47],
[2383, 2383, -16*w - 31],
[2393, 2393, -13*w - 64],
[2393, 2393, 13*w - 77],
[2399, 2399, -17*w + 91],
[2399, 2399, -17*w - 74],
[2437, 2437, 36*w - 167],
[2437, 2437, 36*w + 131],
[2441, 2441, 18*w - 59],
[2441, 2441, -18*w - 41],
[2459, 2459, 25*w - 122],
[2459, 2459, -25*w - 97],
[2467, 2467, 13*w - 17],
[2467, 2467, -13*w - 4],
[2521, 2521, -13*w - 1],
[2521, 2521, 13*w - 14],
[2543, 2543, 24*w - 91],
[2543, 2543, 24*w + 67],
[2549, 2549, -20*w + 103],
[2549, 2549, -20*w - 83],
[2557, 2557, 13*w - 2],
[2557, 2557, 13*w - 11],
[2609, 2609, 32*w - 151],
[2609, 2609, 32*w + 119],
[2657, 2657, 16*w + 73],
[2657, 2657, -16*w + 89],
[2659, 2659, -30*w - 113],
[2659, 2659, 30*w - 143],
[2671, 2671, -9*w - 58],
[2671, 2671, 9*w - 67],
[2683, 2683, 41*w - 172],
[2683, 2683, 41*w + 131],
[2687, 2687, -26*w - 101],
[2687, 2687, 26*w - 127],
[2689, 2689, -27*w - 104],
[2689, 2689, 27*w - 131],
[2693, 2693, 21*w + 53],
[2693, 2693, 21*w - 74],
[2699, 2699, 5*w - 58],
[2699, 2699, -5*w - 53],
[2711, 2711, -11*w - 62],
[2711, 2711, 11*w - 73],
[2729, 2729, -15*w - 19],
[2729, 2729, 15*w - 34],
[2731, 2731, -15*w - 71],
[2731, 2731, 15*w - 86],
[2741, 2741, -w - 52],
[2741, 2741, w - 53],
[2749, 2749, -19*w - 43],
[2749, 2749, 19*w - 62],
[2767, 2767, -17*w - 32],
[2767, 2767, 17*w - 49],
[2791, 2791, 40*w - 167],
[2791, 2791, 40*w + 127],
[2797, 2797, 9*w - 68],
[2797, 2797, -9*w - 59],
[2801, 2801, 33*w - 134],
[2801, 2801, 33*w + 101],
[2803, 2803, -18*w - 79],
[2803, 2803, -18*w + 97],
[2809, 53, -53],
[2819, 2819, 21*w - 73],
[2819, 2819, -21*w - 52],
[2833, 2833, -3*w - 53],
[2833, 2833, 3*w - 56],
[2851, 2851, -20*w - 47],
[2851, 2851, 20*w - 67],
[2879, 2879, -15*w - 16],
[2879, 2879, 15*w - 31],
[2887, 2887, 39*w + 142],
[2887, 2887, 39*w - 181],
[2903, 2903, -17*w + 94],
[2903, 2903, 17*w + 77],
[2909, 2909, 30*w - 119],
[2909, 2909, 30*w + 89],
[2927, 2927, 27*w - 104],
[2927, 2927, 27*w + 77],
[2953, 2953, 14*w - 13],
[2953, 2953, 14*w - 1],
[2969, 2969, 15*w - 29],
[2969, 2969, -15*w - 14],
[3001, 3001, 17*w - 46],
[3001, 3001, -17*w - 29],
[3011, 3011, -15*w - 13],
[3011, 3011, 15*w - 28],
[3023, 3023, -22*w - 91],
[3023, 3023, -22*w + 113],
[3037, 3037, 36*w - 169],
[3037, 3037, 36*w + 133],
[3041, 3041, 5*w - 61],
[3041, 3041, -5*w - 56],
[3049, 3049, 15*w - 88],
[3049, 3049, -15*w - 73],
[3089, 3089, 15*w - 26],
[3089, 3089, -15*w - 11],
[3163, 3163, 27*w - 133],
[3163, 3163, -27*w - 106],
[3167, 3167, 31*w - 149],
[3167, 3167, -31*w - 118],
[3169, 3169, -3*w - 56],
[3169, 3169, 3*w - 59],
[3181, 3181, 35*w - 142],
[3181, 3181, 35*w + 107],
[3187, 3187, -9*w - 62],
[3187, 3187, 9*w - 71],
[3191, 3191, -15*w - 8],
[3191, 3191, 15*w - 23],
[3217, 3217, -17*w - 26],
[3217, 3217, 17*w - 43],
[3221, 3221, 15*w - 22],
[3221, 3221, -15*w - 7],
[3229, 3229, 34*w + 103],
[3229, 3229, 34*w - 137],
[3253, 3253, 26*w + 71],
[3253, 3253, 26*w - 97],
[3299, 3299, -15*w - 4],
[3299, 3299, 15*w - 19],
[3307, 3307, 28*w + 79],
[3307, 3307, 28*w - 107],
[3313, 3313, 31*w + 91],
[3313, 3313, 31*w - 122],
[3319, 3319, 29*w + 83],
[3319, 3319, 29*w - 112],
[3343, 3343, 19*w - 56],
[3343, 3343, -19*w - 37],
[3359, 3359, -15*w - 1],
[3359, 3359, 15*w - 16],
[3371, 3371, 35*w + 131],
[3371, 3371, 35*w - 166],
[3389, 3389, 15*w - 14],
[3389, 3389, 15*w - 1],
[3391, 3391, 43*w - 179],
[3391, 3391, 43*w + 136],
[3407, 3407, -w - 58],
[3407, 3407, w - 59],
[3413, 3413, 7*w - 68],
[3413, 3413, -7*w - 61],
[3457, 3457, 9*w - 73],
[3457, 3457, -9*w - 64],
[3461, 3461, 27*w - 101],
[3461, 3461, 27*w + 74],
[3463, 3463, -16*w - 13],
[3463, 3463, 16*w - 29],
[3481, 59, -59],
[3491, 3491, 45*w + 143],
[3491, 3491, 45*w - 188],
[3499, 3499, -20*w - 41],
[3499, 3499, 20*w - 61],
[3511, 3511, -23*w - 56],
[3511, 3511, 23*w - 79],
[3529, 3529, 45*w + 163],
[3529, 3529, 45*w - 208],
[3533, 3533, 21*w - 67],
[3533, 3533, -21*w - 46],
[3539, 3539, 2*w - 61],
[3539, 3539, -2*w - 59],
[3541, 3541, 15*w - 91],
[3541, 3541, -15*w - 76],
[3547, 3547, 6*w - 67],
[3547, 3547, -6*w - 61],
[3557, 3557, -11*w - 68],
[3557, 3557, 11*w - 79],
[3583, 3583, 41*w - 169],
[3583, 3583, 41*w + 128],
[3613, 3613, 19*w - 53],
[3613, 3613, -19*w - 34],
[3659, 3659, 10*w - 77],
[3659, 3659, -10*w - 67],
[3673, 3673, 49*w + 157],
[3673, 3673, 49*w - 206],
[3701, 3701, -20*w + 109],
[3701, 3701, 20*w + 89],
[3709, 3709, -15*w - 77],
[3709, 3709, 15*w - 92],
[3733, 3733, -12*w - 71],
[3733, 3733, 12*w - 83],
[3767, 3767, -w - 61],
[3767, 3767, w - 62],
[3769, 3769, 3*w - 64],
[3769, 3769, -3*w - 61],
[3779, 3779, -5*w - 62],
[3779, 3779, 5*w - 67],
[3797, 3797, 42*w + 131],
[3797, 3797, 42*w - 173],
[3821, 3821, 45*w + 142],
[3821, 3821, 45*w - 187],
[3823, 3823, -16*w - 1],
[3823, 3823, 16*w - 17],
[3847, 3847, 37*w + 112],
[3847, 3847, 37*w - 149],
[3863, 3863, -21*w - 43],
[3863, 3863, 21*w - 64],
[3877, 3877, -9*w - 67],
[3877, 3877, 9*w - 76],
[3889, 3889, -27*w + 136],
[3889, 3889, -27*w - 109],
[3907, 3907, -23*w - 53],
[3907, 3907, 23*w - 76],
[3917, 3917, 18*w - 41],
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[8431, 8431, 54*w - 257],
[8443, 8443, -42*w - 167],
[8443, 8443, 42*w - 209],
[8467, 8467, -9*w - 94],
[8467, 8467, 9*w - 103],
[8501, 8501, -44*w - 173],
[8501, 8501, 44*w - 217],
[8513, 8513, 51*w + 151],
[8513, 8513, 51*w - 202],
[8521, 8521, 45*w - 221],
[8521, 8521, -45*w - 176],
[8527, 8527, 21*w - 134],
[8527, 8527, -21*w - 113],
[8537, 8537, -19*w - 109],
[8537, 8537, 19*w - 128],
[8539, 8539, -25*w - 19],
[8539, 8539, 25*w - 44],
[8543, 8543, -47*w - 182],
[8543, 8543, 47*w - 229],
[8581, 8581, 35*w - 118],
[8581, 8581, -35*w - 83],
[8597, 8597, 33*w - 106],
[8597, 8597, -33*w - 73],
[8623, 8623, 27*w - 154],
[8623, 8623, -27*w - 127],
[8647, 8647, 66*w + 241],
[8647, 8647, 66*w - 307],
[8663, 8663, 24*w - 23],
[8663, 8663, 24*w - 1],
[8677, 8677, 67*w - 278],
[8677, 8677, 67*w + 211],
[8681, 8681, 71*w - 328],
[8681, 8681, 71*w + 257],
[8689, 8689, 55*w + 166],
[8689, 8689, 55*w - 221],
[8707, 8707, 28*w - 71],
[8707, 8707, -28*w - 43],
[8719, 8719, -25*w - 16],
[8719, 8719, 25*w - 41],
[8737, 8737, 43*w - 161],
[8737, 8737, 43*w + 118],
[8779, 8779, 15*w - 118],
[8779, 8779, -15*w - 103],
[8783, 8783, 24*w - 13],
[8783, 8783, 24*w - 11],
[8803, 8803, 31*w - 92],
[8803, 8803, -31*w - 61],
[8831, 8831, 72*w - 301],
[8831, 8831, 72*w + 229],
[8849, 8849, 69*w - 287],
[8849, 8849, 69*w + 218],
[8861, 8861, -27*w - 34],
[8861, 8861, 27*w - 61],
[8867, 8867, -11*w - 98],
[8867, 8867, 11*w - 109],
[8887, 8887, 37*w - 128],
[8887, 8887, -37*w - 91],
[8893, 8893, 49*w - 191],
[8893, 8893, 49*w + 142],
[8933, 8933, 17*w - 124],
[8933, 8933, -17*w - 107],
[8951, 8951, 33*w - 104],
[8951, 8951, -33*w - 71],
[8963, 8963, 23*w - 142],
[8963, 8963, -23*w - 119],
[8971, 8971, 65*w - 268],
[8971, 8971, 65*w + 203],
[9001, 9001, -15*w - 104],
[9001, 9001, 15*w - 119],
[9013, 9013, 26*w - 49],
[9013, 9013, -26*w - 23],
[9029, 9029, -20*w - 113],
[9029, 9029, 20*w - 133],
[9041, 9041, -35*w + 184],
[9041, 9041, -35*w - 149],
[9043, 9043, 29*w - 76],
[9043, 9043, -29*w - 47],
[9067, 9067, 9*w - 106],
[9067, 9067, -9*w - 97],
[9103, 9103, 64*w - 263],
[9103, 9103, 64*w + 199],
[9109, 9109, 60*w - 283],
[9109, 9109, 60*w + 223],
[9137, 9137, -27*w - 31],
[9137, 9137, 27*w - 58],
[9151, 9151, 25*w - 32],
[9151, 9151, -25*w - 7],
[9199, 9199, -42*w + 211],
[9199, 9199, -42*w - 169],
[9227, 9227, -29*w - 134],
[9227, 9227, 29*w - 163],
[9257, 9257, 19*w - 131],
[9257, 9257, -19*w - 112],
[9277, 9277, 21*w - 137],
[9277, 9277, -21*w - 116],
[9281, 9281, 65*w + 239],
[9281, 9281, 65*w - 304],
[9311, 9311, 27*w - 56],
[9311, 9311, -27*w - 29],
[9319, 9319, 45*w - 223],
[9319, 9319, -45*w - 178],
[9337, 9337, 62*w + 191],
[9337, 9337, 62*w - 253],
[9349, 9349, -25*w - 1],
[9349, 9349, 25*w - 26],
[9391, 9391, 75*w - 346],
[9391, 9391, 75*w + 271],
[9397, 9397, 51*w - 247],
[9397, 9397, -51*w - 196],
[9403, 9403, -29*w - 44],
[9403, 9403, 29*w - 73],
[9413, 9413, -13*w - 103],
[9413, 9413, 13*w - 116],
[9419, 9419, 45*w + 124],
[9419, 9419, 45*w - 169],
[9421, 9421, 25*w - 2],
[9421, 9421, 25*w - 23],
[9433, 9433, 63*w + 233],
[9433, 9433, 63*w - 296],
[9439, 9439, 61*w - 248],
[9439, 9439, 61*w + 187],
[9467, 9467, 33*w - 101],
[9467, 9467, -33*w - 68],
[9491, 9491, -w - 97],
[9491, 9491, w - 98],
[9497, 9497, -11*w - 101],
[9497, 9497, 11*w - 112],
[9511, 9511, 25*w - 8],
[9511, 9511, 25*w - 17],
[9521, 9521, -41*w + 208],
[9521, 9521, -41*w - 167],
[9613, 9613, 59*w - 238],
[9613, 9613, 59*w + 179],
[9619, 9619, 79*w + 253],
[9619, 9619, 79*w - 332],
[9623, 9623, 39*w - 136],
[9623, 9623, -39*w - 97],
[9629, 9629, 30*w - 79],
[9629, 9629, -30*w - 49],
[9643, 9643, 44*w - 163],
[9643, 9643, 44*w + 119],
[9677, 9677, 63*w - 257],
[9677, 9677, 63*w + 194],
[9679, 9679, -15*w - 107],
[9679, 9679, 15*w - 122],
[9719, 9719, 5*w - 103],
[9719, 9719, -5*w - 98],
[9721, 9721, -45*w - 179],
[9721, 9721, 45*w - 224],
[9733, 9733, 26*w - 37],
[9733, 9733, -26*w - 11],
[9769, 9769, 46*w - 173],
[9769, 9769, 46*w + 127],
[9787, 9787, -21*w - 118],
[9787, 9787, 21*w - 139],
[9817, 9817, 47*w + 131],
[9817, 9817, 47*w - 178],
[9833, 9833, -53*w - 203],
[9833, 9833, 53*w - 256],
[9857, 9857, 27*w - 49],
[9857, 9857, -27*w - 22],
[9883, 9883, 49*w + 139],
[9883, 9883, 49*w - 188],
[9887, 9887, 72*w - 299],
[9887, 9887, 72*w + 227],
[9901, 9901, 50*w + 143],
[9901, 9901, 50*w - 193],
[9907, 9907, 52*w - 203],
[9907, 9907, 52*w + 151],
[9923, 9923, -37*w - 157],
[9923, 9923, -37*w + 194],
[9929, 9929, -25*w - 127],
[9929, 9929, 25*w - 152],
[9931, 9931, 28*w - 59],
[9931, 9931, -28*w - 31]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-2, -2, -3, -3, -3, 1, 1, -4, -4, 5, 5, 4, 4, -13, -1, -11, -11, 4, 4, -14, -14, 4, 4, -2, -2, -17, -17, 1, 1, 3, 6, 6, -16, -16, 9, 9, 19, 19, 18, 18, -12, -12, -18, -18, -3, -3, 6, 6, 9, 9, 2, 2, -3, -3, -22, -22, -16, -16, -18, -18, 14, 14, 6, 6, -18, 18, 18, -30, -30, 8, 8, 19, 19, 14, 14, -14, -14, 6, 6, 0, 0, 12, 12, -35, -35, -18, -18, -4, -4, 14, 14, -12, -12, -28, -28, 10, 10, 35, -10, -10, -15, -15, 38, 38, -43, -43, -26, -26, -6, -6, -28, -28, -34, -34, 26, 26, 16, 16, -46, -46, 8, 8, 49, 49, -10, -10, 27, 27, -46, -46, -23, -23, -15, -15, 8, 8, 3, 3, -30, -30, -22, -43, -43, 37, 37, 6, 6, 16, 16, -29, -29, 2, 2, -56, -56, -3, -3, -21, -21, -62, 14, 14, -6, -6, -18, -18, -48, -48, 19, 19, -54, -54, 55, 55, 8, 8, 59, 59, 18, 18, -14, -14, 29, 29, 52, 52, -2, -2, -64, -64, 21, 21, -5, -5, -2, -2, 30, 30, -48, -48, -19, -19, -42, -42, -39, -39, -7, -7, 27, 27, -10, -10, -26, -26, -33, -33, 12, 12, -10, 65, 65, 68, 68, -54, -54, 10, 10, 72, 72, 10, 10, 68, 68, 51, 51, -34, -34, -34, -34, -2, -2, 14, 14, 36, 36, 62, 62, -30, -30, -30, -30, -62, -62, -6, -6, -54, -54, -29, -29, -74, -74, -33, -33, 42, 42, 40, 40, -61, -61, -66, -66, -19, -19, 8, 8, 3, 3, 60, 60, -22, -22, -22, 28, 28, -3, -3, -44, -44, -66, -66, 51, 51, 29, 29, -37, -37, 34, 34, -32, -32, -42, -42, -10, -10, 28, 28, -63, -63, -84, -84, -5, -5, -63, -63, 20, 20, 48, 48, 65, 65, 76, 76, 49, 49, 48, 48, -10, -10, 13, 13, 8, 8, 30, 30, -5, -5, -1, -1, 30, 30, 21, 21, -41, -41, 74, 74, 3, 3, -90, -90, -9, -9, -5, -5, -84, -84, 50, 50, 71, 71, -24, -24, -42, -42, 78, 78, 9, 9, 86, 86, 88, 88, -26, -26, 24, 24, 84, 84, 13, 13, -99, -99, -28, -28, -78, -78, 35, 35, -74, -74, 63, 63, 5, 5, 28, 28, 20, 20, 23, 23, -57, -57, 10, 10, -70, 76, 76, -37, -37, -16, -16, -54, -54, 82, 82, -50, -50, 45, 45, -62, -62, 61, 61, 30, 30, -98, -98, -90, -90, 0, 0, 10, 10, 59, 59, 58, 58, -34, -34, -70, -70, -52, -52, -41, -41, -77, -77, 76, 76, -60, -60, -3, -3, 73, 73, 63, 63, 34, 34, 4, 4, 14, 14, -26, -26, -82, -82, -4, -4, 50, 50, -66, -66, -78, -78, 4, 4, -20, -20, -29, -29, 43, 43, 46, 46, 70, 70, -37, -18, -18, 0, 0, 26, 26, 65, 65, -91, -91, 14, 14, -109, -109, -56, -56, 67, 67, -26, -26, 61, 61, 54, 54, 43, 43, 27, 27, 21, 21, 23, 23, 2, 2, 31, 31, 54, 54, -69, -69, -97, -97, 82, 82, 104, 104, 42, 42, 62, 62, -82, -82, 56, 56, 111, 111, -44, -44, 78, 78, -70, -70, -30, -30, -116, -116, 66, 66, -12, -12, -34, -34, -97, -97, -102, -102, 28, 28, 90, 90, 0, 0, 2, 2, -94, -94, -103, -103, -30, -30, 33, 33, -30, -30, -14, -14, 4, 4, 50, 50, 79, 79, -18, -18, 23, 23, -42, -42, 63, 63, 13, 13, 22, 22, -27, -27, -12, -12, 27, 27, -18, -18, 98, 98, -85, 19, 19, 89, 89, -14, -14, 116, 116, -105, -105, -48, -48, -118, -118, -69, -69, 27, 27, -94, -94, -55, -55, 80, 80, -26, -26, 66, 66, -100, -100, 72, 72, 16, 16, -20, -20, -115, -115, -136, -136, -89, -89, 42, 42, 104, 104, -45, -45, 122, 122, -22, -22, -34, -34, 78, 78, 102, 102, -116, -116, 87, 87, -78, -122, -122, -104, -104, -41, -41, 78, 78, 42, 42, -46, -46, -11, -11, -6, -6, 72, 72, 102, 102, -69, -69, 91, 91, -19, -19, -30, -30, 36, 36, -100, -100, -135, -135, 12, 12, -101, -101, 62, 62, -95, -95, -69, -69, 63, 63, -50, -50, -86, -86, -130, -130, -73, -73, -52, -52, -48, -48, 92, 92, 126, 126, -30, -30, 0, 0, 91, 91, -31, -31, 130, 130, -99, -99, -61, -61, -70, -70, 50, 50, -54, -54, 121, 121, 50, 50, -72, -72, 32, 32, -24, -24, 86, 86, -60, -60, 43, 43, -51, -51, 94, 94, 34, 34, 63, 63, -90, -90, 92, 92, -119, -119, 39, 39, 112, 112, -40, -40, -109, -109, -19, -19, -66, -66, -90, -90, -90, -90, 30, 30, -15, -15, 146, 146, -105, -105, 66, 66, -149, 42, 42, 150, 150, -82, -82, -32, -32, -90, -90, 148, 148, -105, -105, 59, 59, -46, -46, -78, -78, -56, -56, 154, 154, -50, -50, 64, 64, -80, -80, -1, -1, 21, 21, 64, 64, -2, -2, 88, 88, 71, 71, 42, 42, -13, -13, 75, 75, 115, 115, 154, 154, 3, 3, -10, -10, -126, -126, -67, -67, 84, 84, -29, -29, -146, -146, 78, 78, -2, -2, -34, -34, -81, -81, 97, 97, -110, -110, -72, -72, 6, 6, -66, -66, 46, 46, -78, -78, -22, -22, -78, -78, -82, -82, -152, -152, 66, 66, 109, 109, 90, 90, 16, 16, -139, -139, -94, -94, 96, 96, -85, -85, 74, 74, -112, -112, 148, 148, -24, -24, 33, 33, 79, 79, 37, 37, 139, 139, 138, 138, 11, 11, 102, 102, -94, -94, 6, 6, 84, 84, -39, -39, 30, 30, 66, 66, 39, 39, -94, -94, -106, -106, 115, 115, -80, -80, -27, -27, -43, -43, -26, -26, 48, 48, -161, -161, -28, -28, 108, 108, 70, 70, 22, 22, -163, -163, 95, 95, 122, 122, -162, -162, 56, 56, -112, -112, 80, 80, -47, -47, 141, 141, 9, 9, 48, 48, 43, 43, 98, 98, 66, 66, -162, -110, -110, -133, -133, -118, -118, -176, -176, 34, 34, -13, -13, -79, -79, 31, 31, 35, 35, 106, 106, -170, -170, -152, -152, -124, -124, -68, -68, -23, -23, -108, -108, 61, 61, -2, -2, 162, 162, -122, -122, -17, -17, -88, -88, 5, 5, 93, 93, -55, -55, -16, -16, 20, 20, -182, -182, -114, -114, -22, -22, 142, 142, 117, 117, 101, 101, 54, 54, 98, 98, -6, -6, 36, 36, 26, 26, 73, 73, -160, -160, -16, -16, 148, 148, -58, -58, 23, 23, 79, 79, 58, 58, 134, 134, 103, 103, 36, 36, 68, 68, 26, 26, -38, -38, -99, -99, -157, -157, -182, -182, 26, 26, -66, -66, 19, 19, 40, 40, 76, 76, -76, -76, -173, -173, -54, -54, 69, 69, 73, 73, 82, 82, 46, 46, 190, 190, -181, -181, -105, -105, -164, -164, -48, -48, -168, -168, -38, -38, -170, -170, -5, -5, -8, -8, 0, 0, 111, 111, -26, -26, 62, 62, -49, -49, -106, -106, 39, 39, 122, 122, -94, -94, -157, -157, -74, -74, 116, 116, 18, 18, -90, -90, 146, 146, 45, 45, -179, -179, -10, -10, 8, 8, 15, 15, -166, -166, 61, 61, 106, 106, -50, -50, 143, 143, 111, 111, 185, 185, 88, 88, 137, 137, 6, 6, -73, -73, -166, -166, -22, -22, -187, -187, 48, 48, -180, -180, 87, 87, 32, 32];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {61, 61, 2*w - 1}>] := 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;