/*
  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![-19, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {44, 22, 2*w + 10}>;

primesArray := [
[4, 2, 2],
[7, 7, -w - 3],
[9, 3, 3],
[11, 11, w + 5],
[13, 13, w + 2],
[13, 13, w - 3],
[17, 17, w + 1],
[17, 17, -w + 2],
[19, 19, w],
[19, 19, w - 1],
[23, 23, w + 6],
[23, 23, -w + 7],
[25, 5, -5],
[37, 37, -w - 7],
[37, 37, w - 8],
[41, 41, 2*w - 7],
[41, 41, -2*w - 5],
[53, 53, -w - 8],
[53, 53, w - 9],
[61, 61, 2*w - 5],
[61, 61, -2*w - 3],
[67, 67, 2*w - 13],
[67, 67, -2*w - 11],
[71, 71, -w - 9],
[71, 71, w - 10],
[73, 73, 2*w - 3],
[73, 73, -2*w - 1],
[83, 83, 3*w - 11],
[83, 83, -3*w - 8],
[101, 101, -3*w - 7],
[101, 101, 3*w - 10],
[113, 113, -w - 11],
[113, 113, w - 12],
[131, 131, 3*w - 8],
[131, 131, -3*w - 5],
[137, 137, -w - 12],
[137, 137, w - 13],
[139, 139, 4*w - 15],
[139, 139, -4*w - 11],
[163, 163, -w - 13],
[163, 163, w - 14],
[167, 167, -3*w - 1],
[167, 167, 3*w - 4],
[173, 173, 3*w - 2],
[173, 173, 3*w - 1],
[179, 179, 2*w - 17],
[179, 179, -2*w - 15],
[191, 191, -w - 14],
[191, 191, w - 15],
[227, 227, 4*w - 11],
[227, 227, -4*w - 7],
[241, 241, -5*w - 13],
[241, 241, 5*w - 18],
[271, 271, 5*w - 17],
[271, 271, -5*w - 12],
[283, 283, 4*w - 7],
[283, 283, -4*w - 3],
[293, 293, 6*w - 23],
[293, 293, -6*w - 17],
[307, 307, 4*w - 3],
[307, 307, 4*w - 1],
[317, 317, -4*w - 23],
[317, 317, 4*w - 27],
[331, 331, 5*w - 31],
[331, 331, -5*w - 26],
[349, 349, 5*w - 14],
[349, 349, -5*w - 9],
[379, 379, 3*w - 25],
[379, 379, -3*w - 22],
[389, 389, -5*w - 27],
[389, 389, 5*w - 32],
[401, 401, -w - 20],
[401, 401, w - 21],
[409, 409, -5*w - 6],
[409, 409, 5*w - 11],
[421, 421, 4*w - 29],
[421, 421, -4*w - 25],
[439, 439, 5*w - 9],
[439, 439, -5*w - 4],
[443, 443, -w - 21],
[443, 443, w - 22],
[449, 449, 5*w - 33],
[449, 449, -5*w - 28],
[461, 461, 5*w - 7],
[461, 461, -5*w - 2],
[463, 463, -7*w - 34],
[463, 463, 7*w - 41],
[479, 479, 5*w - 4],
[479, 479, 5*w - 1],
[487, 487, -w - 22],
[487, 487, w - 23],
[499, 499, 2*w - 25],
[499, 499, -2*w - 23],
[503, 503, 9*w + 28],
[503, 503, 8*w + 23],
[523, 523, 7*w - 24],
[523, 523, -7*w - 17],
[563, 563, 7*w - 23],
[563, 563, -7*w - 16],
[593, 593, -6*w - 7],
[593, 593, 6*w - 13],
[599, 599, 2*w - 27],
[599, 599, -2*w - 25],
[601, 601, -7*w - 15],
[601, 601, 7*w - 22],
[607, 607, 8*w - 29],
[607, 607, -8*w - 21],
[617, 617, -8*w - 39],
[617, 617, 8*w - 47],
[631, 631, -w - 25],
[631, 631, w - 26],
[641, 641, 5*w - 36],
[641, 641, -5*w - 31],
[653, 653, 4*w - 33],
[653, 653, -4*w - 29],
[677, 677, -6*w - 1],
[677, 677, 6*w - 7],
[683, 683, -w - 26],
[683, 683, w - 27],
[709, 709, -5*w - 32],
[709, 709, 5*w - 37],
[733, 733, -7*w - 11],
[733, 733, 7*w - 18],
[751, 751, -6*w - 35],
[751, 751, 6*w - 41],
[757, 757, -3*w - 29],
[757, 757, 3*w - 32],
[761, 761, 7*w - 17],
[761, 761, -7*w - 10],
[769, 769, 11*w + 34],
[769, 769, 10*w + 29],
[787, 787, 7*w - 16],
[787, 787, -7*w - 9],
[811, 811, -7*w - 8],
[811, 811, 7*w - 15],
[823, 823, 2*w - 31],
[823, 823, -2*w - 29],
[841, 29, -29],
[853, 853, 7*w - 13],
[853, 853, -7*w - 6],
[857, 857, 9*w - 31],
[857, 857, -9*w - 22],
[863, 863, 7*w - 46],
[863, 863, -7*w - 39],
[883, 883, 3*w - 34],
[883, 883, -3*w - 31],
[887, 887, -7*w - 4],
[887, 887, 7*w - 11],
[907, 907, 6*w - 43],
[907, 907, -6*w - 37],
[911, 911, -w - 30],
[911, 911, w - 31],
[937, 937, 7*w - 6],
[937, 937, 7*w - 1],
[941, 941, 7*w - 2],
[941, 941, 7*w - 5],
[947, 947, 2*w - 33],
[947, 947, -2*w - 31],
[961, 31, -31],
[977, 977, 8*w - 51],
[977, 977, -8*w - 43],
[991, 991, -10*w - 49],
[991, 991, 10*w - 59],
[997, 997, -11*w - 31],
[997, 997, 11*w - 42],
[1061, 1061, -4*w - 35],
[1061, 1061, 4*w - 39],
[1063, 1063, -8*w - 9],
[1063, 1063, 8*w - 17],
[1069, 1069, 11*w - 41],
[1069, 1069, -11*w - 30],
[1087, 1087, 3*w - 37],
[1087, 1087, -3*w - 34],
[1091, 1091, 13*w + 40],
[1091, 1091, 12*w + 35],
[1093, 1093, -9*w - 47],
[1093, 1093, 9*w - 56],
[1097, 1097, 9*w - 26],
[1097, 1097, -9*w - 17],
[1103, 1103, -w - 33],
[1103, 1103, w - 34],
[1151, 1151, 8*w - 13],
[1151, 1151, -8*w - 5],
[1171, 1171, -w - 34],
[1171, 1171, w - 35],
[1213, 1213, 4*w - 41],
[1213, 1213, -4*w - 37],
[1217, 1217, -9*w - 14],
[1217, 1217, 9*w - 23],
[1223, 1223, 8*w - 7],
[1223, 1223, 8*w - 1],
[1231, 1231, 8*w - 5],
[1231, 1231, 8*w - 3],
[1249, 1249, -10*w - 21],
[1249, 1249, 10*w - 31],
[1303, 1303, -9*w - 49],
[1303, 1303, 9*w - 58],
[1319, 1319, 9*w - 20],
[1319, 1319, -9*w - 11],
[1361, 1361, -13*w - 37],
[1361, 1361, 13*w - 50],
[1367, 1367, 2*w - 39],
[1367, 1367, -2*w - 37],
[1373, 1373, -4*w - 39],
[1373, 1373, 4*w - 43],
[1399, 1399, -11*w - 25],
[1399, 1399, 11*w - 36],
[1409, 1409, -7*w - 45],
[1409, 1409, 7*w - 52],
[1423, 1423, 6*w - 49],
[1423, 1423, -6*w - 43],
[1427, 1427, -9*w - 7],
[1427, 1427, 9*w - 16],
[1439, 1439, -10*w - 53],
[1439, 1439, 10*w - 63],
[1447, 1447, -13*w - 36],
[1447, 1447, 13*w - 49],
[1453, 1453, -12*w - 59],
[1453, 1453, 12*w - 71],
[1459, 1459, 11*w - 35],
[1459, 1459, -11*w - 24],
[1487, 1487, 9*w - 13],
[1487, 1487, -9*w - 4],
[1499, 1499, -5*w - 42],
[1499, 1499, 5*w - 47],
[1523, 1523, 2*w - 41],
[1523, 1523, -2*w - 39],
[1531, 1531, -13*w - 35],
[1531, 1531, 13*w - 48],
[1549, 1549, 3*w - 43],
[1549, 1549, -3*w - 40],
[1553, 1553, 9*w - 2],
[1553, 1553, 9*w - 7],
[1559, 1559, 9*w - 4],
[1559, 1559, 9*w - 5],
[1601, 1601, 10*w - 23],
[1601, 1601, -10*w - 13],
[1607, 1607, 7*w - 54],
[1607, 1607, -7*w - 47],
[1613, 1613, -13*w - 34],
[1613, 1613, 13*w - 47],
[1621, 1621, -w - 40],
[1621, 1621, w - 41],
[1627, 1627, 11*w - 32],
[1627, 1627, -11*w - 21],
[1657, 1657, 14*w - 53],
[1657, 1657, -14*w - 39],
[1669, 1669, -10*w - 11],
[1669, 1669, 10*w - 21],
[1693, 1693, -13*w - 33],
[1693, 1693, 13*w - 46],
[1709, 1709, -7*w - 48],
[1709, 1709, 7*w - 55],
[1747, 1747, -9*w - 53],
[1747, 1747, 9*w - 62],
[1777, 1777, 11*w - 29],
[1777, 1777, -11*w - 18],
[1787, 1787, -w - 42],
[1787, 1787, w - 43],
[1811, 1811, -12*w - 25],
[1811, 1811, 12*w - 37],
[1823, 1823, -11*w - 17],
[1823, 1823, 11*w - 28],
[1831, 1831, 11*w - 70],
[1831, 1831, -11*w - 59],
[1847, 1847, 13*w - 44],
[1847, 1847, -13*w - 31],
[1849, 43, -43],
[1861, 1861, 10*w - 13],
[1861, 1861, -10*w - 3],
[1867, 1867, 11*w - 27],
[1867, 1867, -11*w - 16],
[1871, 1871, 5*w - 51],
[1871, 1871, -5*w - 46],
[1873, 1873, -w - 43],
[1873, 1873, w - 44],
[1889, 1889, -10*w - 1],
[1889, 1889, 10*w - 11],
[1901, 1901, 4*w - 49],
[1901, 1901, -4*w - 45],
[1931, 1931, 12*w - 35],
[1931, 1931, -12*w - 23],
[1949, 1949, -11*w - 14],
[1949, 1949, 11*w - 25],
[1979, 1979, 15*w - 56],
[1979, 1979, -15*w - 41],
[1987, 1987, 11*w - 24],
[1987, 1987, -11*w - 13],
[1993, 1993, 13*w - 42],
[1993, 1993, -13*w - 29],
[2003, 2003, -14*w - 69],
[2003, 2003, 14*w - 83],
[2011, 2011, 6*w - 55],
[2011, 2011, -6*w - 49],
[2017, 2017, -8*w - 53],
[2017, 2017, 8*w - 61],
[2027, 2027, -7*w - 51],
[2027, 2027, 7*w - 58],
[2039, 2039, 2*w - 47],
[2039, 2039, -2*w - 45],
[2063, 2063, 13*w - 41],
[2063, 2063, -13*w - 28],
[2069, 2069, 5*w - 53],
[2069, 2069, -5*w - 48],
[2083, 2083, 3*w - 49],
[2083, 2083, -3*w - 46],
[2089, 2089, 11*w - 21],
[2089, 2089, -11*w - 10],
[2131, 2131, -13*w - 27],
[2131, 2131, 13*w - 40],
[2137, 2137, -7*w - 52],
[2137, 2137, 7*w - 59],
[2141, 2141, 18*w + 55],
[2141, 2141, 17*w + 50],
[2143, 2143, -w - 46],
[2143, 2143, w - 47],
[2179, 2179, -3*w - 47],
[2179, 2179, 3*w - 50],
[2209, 47, -47],
[2237, 2237, -w - 47],
[2237, 2237, w - 48],
[2239, 2239, -11*w - 4],
[2239, 2239, 11*w - 15],
[2243, 2243, -12*w - 17],
[2243, 2243, 12*w - 29],
[2269, 2269, 12*w - 77],
[2269, 2269, -12*w - 65],
[2273, 2273, 11*w - 13],
[2273, 2273, -11*w - 2],
[2287, 2287, -11*w - 1],
[2287, 2287, 11*w - 12],
[2293, 2293, 4*w - 53],
[2293, 2293, -4*w - 49],
[2297, 2297, -13*w - 68],
[2297, 2297, 13*w - 81],
[2309, 2309, 11*w - 10],
[2309, 2309, 11*w - 1],
[2311, 2311, -15*w - 74],
[2311, 2311, 15*w - 89],
[2333, 2333, -w - 48],
[2333, 2333, w - 49],
[2347, 2347, -9*w - 58],
[2347, 2347, 9*w - 67],
[2351, 2351, 15*w - 52],
[2351, 2351, -15*w - 37],
[2371, 2371, 17*w - 65],
[2371, 2371, -17*w - 48],
[2377, 2377, 3*w - 52],
[2377, 2377, -3*w - 49],
[2381, 2381, 5*w - 56],
[2381, 2381, -5*w - 51],
[2383, 2383, -13*w - 23],
[2383, 2383, 13*w - 36],
[2393, 2393, 19*w + 58],
[2393, 2393, 18*w + 53],
[2411, 2411, -12*w - 13],
[2411, 2411, 12*w - 25],
[2423, 2423, 2*w - 51],
[2423, 2423, -2*w - 49],
[2441, 2441, -13*w - 22],
[2441, 2441, 13*w - 35],
[2447, 2447, -13*w - 69],
[2447, 2447, 13*w - 82],
[2473, 2473, 9*w - 68],
[2473, 2473, -9*w - 59],
[2477, 2477, -14*w - 29],
[2477, 2477, 14*w - 43],
[2531, 2531, -w - 50],
[2531, 2531, w - 51],
[2551, 2551, -13*w - 20],
[2551, 2551, 13*w - 33],
[2557, 2557, -12*w - 67],
[2557, 2557, 12*w - 79],
[2593, 2593, 17*w - 63],
[2593, 2593, -17*w - 46],
[2609, 2609, -15*w - 34],
[2609, 2609, 15*w - 49],
[2617, 2617, 14*w - 41],
[2617, 2617, -14*w - 27],
[2633, 2633, -w - 51],
[2633, 2633, w - 52],
[2659, 2659, 20*w + 61],
[2659, 2659, 19*w + 56],
[2671, 2671, 6*w - 61],
[2671, 2671, -6*w - 55],
[2689, 2689, 3*w - 55],
[2689, 2689, -3*w - 52],
[2699, 2699, -10*w - 63],
[2699, 2699, 10*w - 73],
[2711, 2711, -5*w - 54],
[2711, 2711, 5*w - 59],
[2719, 2719, -16*w - 39],
[2719, 2719, 16*w - 55],
[2731, 2731, 9*w - 70],
[2731, 2731, -9*w - 61],
[2749, 2749, 14*w - 39],
[2749, 2749, -14*w - 25],
[2753, 2753, -13*w - 71],
[2753, 2753, 13*w - 84],
[2789, 2789, 19*w + 55],
[2789, 2789, 21*w + 65],
[2791, 2791, 13*w - 28],
[2791, 2791, -13*w - 15],
[2797, 2797, -3*w - 53],
[2797, 2797, 3*w - 56],
[2833, 2833, -13*w - 14],
[2833, 2833, 13*w - 27],
[2843, 2843, -w - 53],
[2843, 2843, w - 54],
[2903, 2903, 16*w - 53],
[2903, 2903, -16*w - 37],
[2909, 2909, 13*w - 85],
[2909, 2909, -13*w - 72],
[2917, 2917, -19*w - 54],
[2917, 2917, 19*w - 73],
[2927, 2927, 11*w - 78],
[2927, 2927, -11*w - 67],
[2939, 2939, 21*w + 64],
[2939, 2939, 20*w + 59],
[2963, 2963, -7*w - 59],
[2963, 2963, 7*w - 66],
[2999, 2999, 15*w - 44],
[2999, 2999, -15*w - 29],
[3019, 3019, 3*w - 58],
[3019, 3019, -3*w - 55],
[3061, 3061, -w - 55],
[3061, 3061, w - 56],
[3067, 3067, 13*w - 86],
[3067, 3067, -13*w - 73],
[3079, 3079, 16*w - 51],
[3079, 3079, -16*w - 35],
[3089, 3089, 7*w - 67],
[3089, 3089, -7*w - 60],
[3121, 3121, 13*w - 18],
[3121, 3121, -13*w - 5],
[3163, 3163, 13*w - 16],
[3163, 3163, -13*w - 3],
[3167, 3167, 19*w - 71],
[3167, 3167, -19*w - 52],
[3181, 3181, 13*w - 15],
[3181, 3181, -13*w - 2],
[3209, 3209, 15*w - 41],
[3209, 3209, -15*w - 26],
[3217, 3217, 7*w - 68],
[3217, 3217, -7*w - 61],
[3221, 3221, 11*w - 80],
[3221, 3221, -11*w - 69],
[3251, 3251, 13*w - 5],
[3251, 3251, 13*w - 8],
[3253, 3253, 13*w - 6],
[3253, 3253, 13*w - 7],
[3257, 3257, -8*w - 63],
[3257, 3257, 8*w - 71],
[3259, 3259, 10*w - 77],
[3259, 3259, -10*w - 67],
[3271, 3271, -9*w - 65],
[3271, 3271, 9*w - 74],
[3301, 3301, -5*w - 59],
[3301, 3301, 5*w - 64],
[3307, 3307, -17*w - 39],
[3307, 3307, 17*w - 56],
[3347, 3347, -7*w - 62],
[3347, 3347, 7*w - 69],
[3371, 3371, 11*w - 81],
[3371, 3371, -11*w - 70],
[3373, 3373, 14*w - 27],
[3373, 3373, -14*w - 13],
[3389, 3389, 13*w - 88],
[3389, 3389, -13*w - 75],
[3407, 3407, 16*w - 47],
[3407, 3407, -16*w - 31],
[3413, 3413, -4*w - 59],
[3413, 3413, 4*w - 63],
[3449, 3449, 14*w - 25],
[3449, 3449, -14*w - 11],
[3461, 3461, 15*w - 37],
[3461, 3461, -15*w - 22],
[3469, 3469, 12*w - 85],
[3469, 3469, -12*w - 73],
[3481, 59, -59],
[3517, 3517, 14*w - 23],
[3517, 3517, -14*w - 9],
[3527, 3527, -19*w - 49],
[3527, 3527, 19*w - 68],
[3529, 3529, 8*w - 73],
[3529, 3529, -8*w - 65],
[3533, 3533, 18*w - 61],
[3533, 3533, -18*w - 43],
[3541, 3541, 23*w + 70],
[3541, 3541, 22*w + 65],
[3557, 3557, 19*w - 112],
[3557, 3557, -19*w - 93],
[3559, 3559, -16*w - 29],
[3559, 3559, 16*w - 45],
[3583, 3583, 17*w - 53],
[3583, 3583, -17*w - 36],
[3613, 3613, 7*w - 71],
[3613, 3613, -7*w - 64],
[3623, 3623, -14*w - 79],
[3623, 3623, 14*w - 93],
[3643, 3643, -19*w - 48],
[3643, 3643, 19*w - 67],
[3659, 3659, -21*w - 59],
[3659, 3659, 21*w - 80],
[3671, 3671, -17*w - 35],
[3671, 3671, 17*w - 52],
[3673, 3673, -14*w - 3],
[3673, 3673, 14*w - 17],
[3677, 3677, 11*w - 83],
[3677, 3677, -11*w - 72],
[3697, 3697, 9*w - 77],
[3697, 3697, -9*w - 68],
[3709, 3709, -14*w - 1],
[3709, 3709, 14*w - 15],
[3719, 3719, -13*w - 77],
[3719, 3719, 13*w - 90],
[3733, 3733, 3*w - 64],
[3733, 3733, -3*w - 61],
[3767, 3767, 2*w - 63],
[3767, 3767, -2*w - 61],
[3769, 3769, 14*w - 9],
[3769, 3769, 14*w - 5],
[3779, 3779, -15*w - 16],
[3779, 3779, 15*w - 31],
[3797, 3797, -21*w - 58],
[3797, 3797, 21*w - 79],
[3833, 3833, -11*w - 73],
[3833, 3833, 11*w - 84],
[3851, 3851, 10*w - 81],
[3851, 3851, -10*w - 71],
[3863, 3863, 24*w + 73],
[3863, 3863, 23*w + 68],
[3911, 3911, -15*w - 13],
[3911, 3911, 15*w - 28],
[3917, 3917, -4*w - 63],
[3917, 3917, 4*w - 67],
[3923, 3923, -17*w - 32],
[3923, 3923, 17*w - 49],
[3931, 3931, -6*w - 65],
[3931, 3931, 6*w - 71],
[3943, 3943, -17*w - 89],
[3943, 3943, 17*w - 106],
[3967, 3967, -16*w - 23],
[3967, 3967, 16*w - 39],
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[8599, 8599, 31*w - 115],
[8599, 8599, -31*w - 84],
[8609, 8609, -25*w - 46],
[8609, 8609, 25*w - 71],
[8623, 8623, 23*w - 51],
[8623, 8623, -23*w - 28],
[8641, 8641, -22*w - 15],
[8641, 8641, 22*w - 37],
[8647, 8647, -18*w - 113],
[8647, 8647, 18*w - 131],
[8677, 8677, 12*w - 113],
[8677, 8677, -12*w - 101],
[8707, 8707, 29*w - 101],
[8707, 8707, -29*w - 72],
[8737, 8737, -13*w - 103],
[8737, 8737, 13*w - 116],
[8741, 8741, 22*w - 35],
[8741, 8741, -22*w - 13],
[8753, 8753, -33*w - 94],
[8753, 8753, 33*w - 127],
[8761, 8761, 16*w - 125],
[8761, 8761, -16*w - 109],
[8779, 8779, -15*w - 107],
[8779, 8779, 15*w - 122],
[8803, 8803, 21*w - 142],
[8803, 8803, -21*w - 121],
[8819, 8819, 36*w + 109],
[8819, 8819, 35*w + 104],
[8831, 8831, -5*w - 94],
[8831, 8831, 5*w - 99],
[8839, 8839, -25*w - 44],
[8839, 8839, 25*w - 69],
[8849, 8849, -25*w - 132],
[8849, 8849, 25*w - 157],
[8861, 8861, 30*w - 107],
[8861, 8861, -30*w - 77],
[8923, 8923, -23*w - 24],
[8923, 8923, 23*w - 47],
[8933, 8933, -29*w - 144],
[8933, 8933, 29*w - 173],
[8941, 8941, -20*w - 119],
[8941, 8941, 20*w - 139],
[8951, 8951, -25*w - 43],
[8951, 8951, 25*w - 68],
[8969, 8969, 8*w - 105],
[8969, 8969, -8*w - 97],
[8999, 8999, 23*w - 150],
[8999, 8999, -23*w - 127],
[9013, 9013, -11*w - 101],
[9013, 9013, 11*w - 112],
[9049, 9049, 29*w - 99],
[9049, 9049, -29*w - 70],
[9067, 9067, 21*w - 143],
[9067, 9067, -21*w - 122],
[9103, 9103, -32*w - 87],
[9103, 9103, 32*w - 119],
[9109, 9109, -12*w - 103],
[9109, 9109, 12*w - 115],
[9127, 9127, -23*w - 21],
[9127, 9127, 23*w - 44],
[9157, 9157, -7*w - 97],
[9157, 9157, 7*w - 104],
[9173, 9173, -22*w - 1],
[9173, 9173, 22*w - 23],
[9187, 9187, 31*w - 112],
[9187, 9187, -31*w - 81],
[9199, 9199, -17*w - 113],
[9199, 9199, 17*w - 130],
[9203, 9203, 27*w - 83],
[9203, 9203, -27*w - 56],
[9221, 9221, 5*w - 101],
[9221, 9221, -5*w - 96],
[9227, 9227, 11*w - 113],
[9227, 9227, -11*w - 102],
[9239, 9239, -24*w - 31],
[9239, 9239, 24*w - 55],
[9241, 9241, 15*w - 124],
[9241, 9241, -15*w - 109],
[9257, 9257, 34*w - 131],
[9257, 9257, -34*w - 97],
[9277, 9277, -23*w - 128],
[9277, 9277, 23*w - 151],
[9281, 9281, 22*w - 17],
[9281, 9281, 22*w - 5],
[9293, 9293, -w - 96],
[9293, 9293, w - 97],
[9311, 9311, 10*w - 111],
[9311, 9311, -10*w - 101],
[9323, 9323, 37*w + 112],
[9323, 9323, 36*w + 107],
[9341, 9341, -27*w - 55],
[9341, 9341, 27*w - 82],
[9371, 9371, 23*w - 40],
[9371, 9371, -23*w - 17],
[9377, 9377, 8*w - 107],
[9377, 9377, -8*w - 99],
[9403, 9403, -13*w - 106],
[9403, 9403, 13*w - 119],
[9409, 97, -97],
[9413, 9413, -26*w - 47],
[9413, 9413, 26*w - 73],
[9419, 9419, -5*w - 97],
[9419, 9419, 5*w - 102],
[9431, 9431, -25*w - 134],
[9431, 9431, 25*w - 159],
[9461, 9461, 20*w - 141],
[9461, 9461, -20*w - 121],
[9467, 9467, 28*w - 89],
[9467, 9467, -28*w - 61],
[9511, 9511, 32*w - 117],
[9511, 9511, -32*w - 85],
[9533, 9533, -23*w - 14],
[9533, 9533, 23*w - 37],
[9539, 9539, 35*w - 136],
[9539, 9539, -35*w - 101],
[9547, 9547, -29*w - 67],
[9547, 9547, 29*w - 96],
[9601, 9601, 21*w - 145],
[9601, 9601, -21*w - 124],
[9619, 9619, 5*w - 103],
[9619, 9619, -5*w - 98],
[9629, 9629, -13*w - 107],
[9629, 9629, 13*w - 120],
[9631, 9631, -23*w - 12],
[9631, 9631, 23*w - 35],
[9649, 9649, -26*w - 45],
[9649, 9649, 26*w - 71],
[9661, 9661, -11*w - 104],
[9661, 9661, 11*w - 115],
[9677, 9677, 23*w - 34],
[9677, 9677, -23*w - 11],
[9679, 9679, 25*w - 61],
[9679, 9679, -25*w - 36],
[9689, 9689, 17*w - 132],
[9689, 9689, -17*w - 115],
[9719, 9719, -24*w - 25],
[9719, 9719, 24*w - 49],
[9721, 9721, -23*w - 10],
[9721, 9721, 23*w - 33],
[9739, 9739, -10*w - 103],
[9739, 9739, 10*w - 113],
[9743, 9743, -27*w - 52],
[9743, 9743, 27*w - 79],
[9769, 9769, -7*w - 100],
[9769, 9769, 7*w - 107],
[9803, 9803, -23*w - 8],
[9803, 9803, 23*w - 31],
[9833, 9833, 33*w - 122],
[9833, 9833, -33*w - 89],
[9839, 9839, -23*w - 130],
[9839, 9839, 23*w - 153],
[9857, 9857, -13*w - 108],
[9857, 9857, 13*w - 121],
[9871, 9871, -21*w - 125],
[9871, 9871, 21*w - 146],
[9923, 9923, 2*w - 101],
[9923, 9923, -2*w - 99],
[9929, 9929, -30*w - 71],
[9929, 9929, 30*w - 101],
[9949, 9949, -15*w - 112],
[9949, 9949, 15*w - 127],
[9973, 9973, -23*w - 3],
[9973, 9973, 23*w - 26]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [1, 0, 2, 1, -2, 2, 0, 0, -2, 2, 0, 0, 6, 2, 2, 8, -8, 2, 2, 10, -10, 12, 12, -4, -4, -12, 12, -18, 18, -6, 6, -10, -10, -2, 2, 22, 22, -14, 14, -12, -12, -12, 12, 6, -6, 12, 12, 12, 12, -6, 6, -8, 8, -16, 16, -2, 2, -26, 26, -14, 14, 30, 30, 12, 12, 14, -14, -28, -28, -10, -10, -18, -18, -16, 16, 34, 34, -16, 16, 12, 12, 30, 30, 6, -6, -4, -4, -36, 36, -8, -8, -20, -20, -8, 8, -34, 34, 22, -22, -20, 20, -28, -28, 4, -4, -32, 32, 18, 18, -28, -28, 18, 18, -18, -18, 42, -42, 4, 4, 14, 14, 6, -6, 0, 0, 30, 30, 8, -8, -40, 40, -14, 14, -50, 50, -20, -20, -22, 42, -42, -48, 48, -52, -52, -52, -52, 20, -20, -4, -4, 32, 32, -20, 20, 42, -42, 12, 12, 46, 2, 2, 52, 52, -62, 62, -38, -38, -40, 40, 10, -10, -56, -56, 42, -42, 54, 54, 52, -52, -40, -40, -4, 4, -20, -20, -30, -30, -4, 4, 16, -16, 28, -28, -12, 12, -40, -40, 16, -16, -24, 24, 12, 12, -34, -34, -48, 48, 46, 46, -8, -8, -18, 18, 8, 8, -40, 40, 26, 26, -46, 46, -32, 32, -4, -4, -4, -4, -14, 14, 34, 34, 44, -44, 44, -44, 12, -12, -4, -4, -74, 74, -62, -62, -66, 66, 56, -56, -66, 66, 14, -14, -78, -78, -52, -52, -52, 52, -52, -52, 26, -26, 24, -24, -52, -52, 4, -4, 58, -62, 62, -30, 30, 48, 48, -34, -34, -12, 12, 50, 50, 26, -26, 74, -74, 74, -74, -42, 42, 16, -16, 36, 36, 52, 52, -2, -2, -28, -28, -72, -72, -24, 24, -42, -42, 76, 76, 76, -76, 18, -18, 42, 42, -66, 66, -68, -68, -36, -36, -50, 58, 58, 44, -44, 2, -2, -78, -78, 4, -4, -52, 52, -58, -58, 42, 42, 30, -30, 12, 12, 30, 30, -4, -4, 8, -8, 62, -62, 38, 38, -26, -26, 60, -60, 44, -44, -46, 46, -24, -24, -12, 12, -32, -32, -14, -14, -22, 22, -4, -4, -32, 32, -38, -38, -52, 52, -40, 40, -56, 56, -54, -54, 34, -34, 100, 100, 26, 26, -52, -52, 48, 48, 20, -20, -4, -4, 18, -18, 34, 34, 30, -30, 12, -12, -10, -10, -12, 12, -12, -12, -56, 56, 54, 54, 14, -14, -68, -68, 66, -66, -36, -36, -52, 52, 28, 28, 62, 62, -68, -68, 108, -108, 10, 10, 12, -12, 38, -38, -92, 92, 62, -62, -88, 88, 58, 58, 6, 6, 30, -30, -50, 50, -82, -82, -60, -60, 112, 112, 22, 22, 42, -42, -12, -12, -60, -60, -86, 86, -14, -14, 104, -104, -30, -30, 60, -60, 38, -38, 58, 58, 18, 98, -98, -112, 112, 6, 6, -46, 46, 114, -114, 38, 38, -4, 4, 40, -40, -34, -34, -36, -36, -26, 26, -110, 110, 16, -16, 76, -76, 22, 22, -66, -66, 86, -86, -24, -24, 86, 86, -60, -60, 72, -72, -98, 98, -70, 70, -22, -22, 44, 44, 16, -16, 32, -32, 30, 30, -6, 6, -60, -60, 24, 24, -44, 44, 6, -6, 34, -34, 82, 82, 52, 52, 110, -110, -4, -4, -2, -2, -34, 34, 94, -94, -52, -52, 30, -30, -40, -40, -68, 68, -12, -12, -106, 106, 54, 54, 68, -68, 88, -88, -98, 98, 40, 40, -68, 68, -108, 108, 116, 116, 50, 50, -42, -42, 34, -34, -104, 104, -92, -92, -90, 90, 12, -12, 6, 6, 10, -10, 106, -106, 8, 8, 12, 12, -106, 106, 8, -8, -36, 36, 110, -110, -132, -132, -42, -42, -42, 42, 88, -88, -132, -132, -2, -2, -110, -110, 124, 124, 60, -60, 52, -52, 106, 106, 36, -36, -84, 84, 36, 36, 98, -98, -90, -90, -40, 40, 36, 36, -30, 30, -28, 28, 106, 106, 4, -4, -66, -66, 32, 32, 116, 116, -32, 32, 48, 48, -18, -18, 18, -18, -94, -94, 86, -86, 108, -108, 126, -126, 22, -22, -20, -20, -52, -52, 30, 30, -86, 86, 98, 98, 38, 38, -22, -22, 132, -132, 84, 84, -26, 26, 46, -46, -102, 102, 68, 68, -52, 52, 42, 42, 140, -140, 76, 76, -108, -108, 62, -62, 84, 84, 52, 52, 90, -90, 48, -48, -56, 56, -68, -68, 52, 52, 30, -30, -126, 126, 10, 10, -74, -74, -38, -38, -74, 74, -68, 68, -4, 4, 102, -102, -108, -108, 60, 60, 2, -2, -20, -20, -126, 126, 54, -54, -54, -54, -60, -60, 62, -62, 60, 60, -42, 42, 48, -48, 74, -74, 132, 132, -44, -44, 98, 98, -124, -124, 108, -108, -46, 46, -38, -38, 112, -112, -92, 92, 72, 72, -104, 104, -50, 50, 58, 58, 70, -70, -158, -112, 112, 46, -46, 50, -50, 110, 110, 124, 124, 150, 150, -82, -82, 56, 56, -126, 126, -140, -140, 38, 38, 140, 140, 134, 134, -68, 68, -36, -36, 6, 6, -36, 36, 0, 0, -140, 140, -70, -70, -48, 48, 68, -68, 142, 142, 84, 84, 66, 66, -80, -80, -158, 158, 52, -52, -20, -20, 12, 12, -124, 124, 26, 26, -54, -54, 76, -76, 62, 62, -26, 26, 24, 24, 90, 90, -62, 62, 110, -110, 76, 76, -2, 2, 36, 36, -100, 100, -26, -26, 6, 6, 102, -102, 4, 4, 2, -2, 96, -96, 102, 102, 30, 30, 140, 140, 10, 10, -114, 114, -20, -20, -54, 54, 78, -78, 112, 112, 74, 74, 18, 18, 24, -24, 132, 132, 8, 8, -8, 8, -14, -14, -78, 78, -138, -138, -96, 96, 4, 4, 106, -106, -162, -162, 64, -64, -36, -36, 8, -8, 70, 70, -48, -48, 36, -36, 32, 32, 126, 126, 76, 76, -30, 30, -98, 98, -86, 86, 92, 92, 50, -50, 82, 82, -126, -126, -76, 76, 150, -150, 90, 90, 30, -30, 118, -118, 64, -64, -18, -18, 4, 4, -164, -164, 178, 36, -36, -144, 144, -20, 20, -10, -10, -82, -82, 90, -90, 132, -132, 10, 10, -170, -170, -74, 74, 20, -20, 84, 84, -98, 98, 86, 86, -76, -76, 144, -144, 126, -126, -94, 94, 54, 54, -6, -6, 120, -120, 146, 146, 62, 62, 8, -8, -52, -52, 2, -2, -146, -146, 84, -84, -14, 14, 162, 162, 180, -180, -124, -124, 48, -48, -124, 124, -4, 4, -160, 160, 108, 108, 58, 58, 70, -70, -114, -114, 134, -134, 144, -144, 86, 86, -140, -140, 124, 124, 106, -106, 8, 8, -32, 32, 14, 14, -170, 170, 86, -86, 114, 114, -50, -50, 84, -84, -10, -10, -8, -8, -154, -154, 56, -56, 132, 132, -132, 132, -70, -70, -68, 68, 158, 158, 102, -102, 158, -158, 184, 184, -114, 114, 94, 94, -84, -84, 24, -24, -10, -10, 96, -96, -38, -38, 28, -28, -42, -42, -28, -28, -66, 66, 142, -142, -174, 174, 94, 94, -68, -68, 50, -102, 102, -124, -124, 168, 168, -18, -18, -62, 62, -52, 52, -10, 10, -122, 122, -134, 134, -10, -10, -28, -28, 138, 138, 40, -40, 164, -164, -138, -138, 138, -138, -88, 88, 122, 122, -80, 80, -60, 60, -84, -84, -108, 108, 18, 18, 10, -10, -64, 64, 120, 120, 18, 18, -100, -100, 20, 20, 40, -40, -78, -78, -134, 134];
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 | {11, 11, w + 5}>] := -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;