/*
  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![-7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {900, 30, 30}>;

primesArray := [
[4, 2, 2],
[5, 5, w + 1],
[5, 5, w - 2],
[7, 7, w],
[7, 7, -w + 1],
[9, 3, 3],
[13, 13, w + 4],
[13, 13, w - 5],
[23, 23, -w - 5],
[23, 23, w - 6],
[29, 29, 2*w - 1],
[53, 53, 3*w - 5],
[53, 53, -3*w - 2],
[59, 59, -3*w - 1],
[59, 59, 3*w - 4],
[67, 67, 3*w - 13],
[67, 67, -3*w - 10],
[71, 71, 2*w - 11],
[71, 71, -2*w - 9],
[83, 83, -w - 9],
[83, 83, w - 10],
[103, 103, -w - 10],
[103, 103, w - 11],
[107, 107, -4*w - 1],
[107, 107, 4*w - 5],
[109, 109, 4*w - 17],
[109, 109, -4*w - 13],
[121, 11, -11],
[139, 139, 5*w - 9],
[139, 139, -5*w - 4],
[149, 149, -w - 12],
[149, 149, w - 13],
[151, 151, -5*w - 3],
[151, 151, 5*w - 8],
[167, 167, 2*w - 15],
[167, 167, -2*w - 13],
[173, 173, 7*w - 17],
[173, 173, 4*w - 19],
[179, 179, 5*w - 4],
[179, 179, 5*w - 1],
[181, 181, 5*w - 2],
[181, 181, 5*w - 3],
[197, 197, 6*w - 11],
[197, 197, -6*w - 5],
[199, 199, -5*w - 17],
[199, 199, 5*w - 22],
[223, 223, 6*w - 25],
[223, 223, -6*w - 19],
[227, 227, 2*w - 17],
[227, 227, -2*w - 15],
[233, 233, -w - 15],
[233, 233, w - 16],
[239, 239, 5*w - 23],
[239, 239, 8*w - 19],
[241, 241, 3*w - 19],
[241, 241, -3*w - 16],
[257, 257, 6*w - 5],
[257, 257, 6*w - 1],
[277, 277, -3*w - 17],
[277, 277, 3*w - 20],
[281, 281, 9*w - 22],
[281, 281, -9*w - 13],
[283, 283, -7*w - 5],
[283, 283, 7*w - 12],
[289, 17, -17],
[313, 313, 7*w - 10],
[313, 313, -7*w - 3],
[347, 347, -7*w - 23],
[347, 347, 7*w - 30],
[349, 349, 7*w - 6],
[349, 349, 7*w - 1],
[353, 353, 7*w - 2],
[353, 353, 7*w - 5],
[361, 19, -19],
[373, 373, -w - 19],
[373, 373, w - 20],
[383, 383, 8*w - 13],
[383, 383, -8*w - 5],
[397, 397, -3*w - 20],
[397, 397, 3*w - 23],
[401, 401, -7*w - 24],
[401, 401, 7*w - 31],
[419, 419, -5*w - 22],
[419, 419, 5*w - 27],
[431, 431, 9*w - 17],
[431, 431, -9*w - 8],
[439, 439, -8*w - 1],
[439, 439, 8*w - 9],
[457, 457, 7*w - 32],
[457, 457, 11*w - 26],
[463, 463, 8*w - 5],
[463, 463, 8*w - 3],
[487, 487, 3*w - 25],
[487, 487, -3*w - 22],
[499, 499, -w - 22],
[499, 499, w - 23],
[509, 509, -4*w - 23],
[509, 509, 4*w - 27],
[521, 521, -5*w - 24],
[521, 521, 5*w - 29],
[523, 523, 13*w + 20],
[523, 523, 13*w - 33],
[547, 547, 2*w - 25],
[547, 547, -2*w - 23],
[557, 557, -9*w - 1],
[557, 557, 9*w - 10],
[571, 571, 10*w - 41],
[571, 571, -10*w - 31],
[587, 587, 9*w - 4],
[587, 587, 9*w - 5],
[593, 593, -w - 24],
[593, 593, w - 25],
[613, 613, 4*w - 29],
[613, 613, -4*w - 25],
[631, 631, 5*w - 31],
[631, 631, -5*w - 26],
[643, 643, -w - 25],
[643, 643, w - 26],
[647, 647, 2*w - 27],
[647, 647, -2*w - 25],
[661, 661, 10*w - 13],
[661, 661, -10*w - 3],
[673, 673, -9*w - 31],
[673, 673, 9*w - 40],
[683, 683, 11*w - 45],
[683, 683, -11*w - 34],
[691, 691, -3*w - 26],
[691, 691, 3*w - 29],
[701, 701, 15*w + 23],
[701, 701, 15*w - 38],
[709, 709, 10*w - 9],
[709, 709, 10*w - 1],
[719, 719, -10*w - 33],
[719, 719, 10*w - 43],
[761, 761, 15*w - 37],
[761, 761, -15*w - 22],
[787, 787, -11*w - 4],
[787, 787, 11*w - 15],
[811, 811, -5*w - 29],
[811, 811, 5*w - 34],
[821, 821, 11*w - 13],
[821, 821, -11*w - 2],
[857, 857, 11*w - 10],
[857, 857, 11*w - 1],
[863, 863, -w - 29],
[863, 863, w - 30],
[877, 877, 11*w - 5],
[877, 877, 11*w - 6],
[883, 883, 13*w - 25],
[883, 883, -13*w - 12],
[919, 919, -13*w - 11],
[919, 919, 13*w - 24],
[929, 929, 11*w - 48],
[929, 929, -11*w - 37],
[937, 937, 13*w - 53],
[937, 937, -13*w - 40],
[941, 941, 5*w - 36],
[941, 941, -5*w - 31],
[953, 953, -13*w - 10],
[953, 953, 13*w - 23],
[961, 31, -31],
[977, 977, 7*w - 40],
[977, 977, -7*w - 33],
[991, 991, 3*w - 34],
[991, 991, -3*w - 31],
[1009, 1009, -5*w - 32],
[1009, 1009, 5*w - 37],
[1019, 1019, 12*w - 11],
[1019, 1019, 12*w - 1],
[1021, 1021, 14*w - 27],
[1021, 1021, -14*w - 13],
[1031, 1031, -13*w - 41],
[1031, 1031, 13*w - 54],
[1039, 1039, 10*w - 47],
[1039, 1039, 17*w - 41],
[1049, 1049, -w - 32],
[1049, 1049, w - 33],
[1051, 1051, -7*w - 34],
[1051, 1051, 7*w - 41],
[1069, 1069, 13*w - 19],
[1069, 1069, -13*w - 6],
[1093, 1093, 13*w - 18],
[1093, 1093, -13*w - 5],
[1097, 1097, 14*w - 25],
[1097, 1097, -14*w - 11],
[1103, 1103, 11*w - 50],
[1103, 1103, 17*w - 40],
[1109, 1109, 4*w - 37],
[1109, 1109, -4*w - 33],
[1151, 1151, -5*w - 34],
[1151, 1151, 5*w - 39],
[1153, 1153, 13*w - 15],
[1153, 1153, -13*w - 2],
[1193, 1193, 18*w - 43],
[1193, 1193, 11*w - 51],
[1213, 1213, 13*w - 3],
[1213, 1213, 13*w - 10],
[1217, 1217, -8*w - 37],
[1217, 1217, 8*w - 45],
[1223, 1223, 13*w - 5],
[1223, 1223, 13*w - 8],
[1231, 1231, 15*w - 61],
[1231, 1231, -15*w - 46],
[1277, 1277, -14*w - 5],
[1277, 1277, 14*w - 19],
[1283, 1283, 14*w - 59],
[1283, 1283, -14*w - 45],
[1289, 1289, 15*w - 26],
[1289, 1289, -15*w - 11],
[1301, 1301, 5*w - 41],
[1301, 1301, -5*w - 36],
[1321, 1321, -14*w - 3],
[1321, 1321, 14*w - 17],
[1327, 1327, 16*w - 31],
[1327, 1327, -16*w - 15],
[1367, 1367, -7*w - 38],
[1367, 1367, 7*w - 45],
[1369, 37, -37],
[1399, 1399, -w - 37],
[1399, 1399, w - 38],
[1427, 1427, -13*w - 45],
[1427, 1427, 13*w - 58],
[1451, 1451, 7*w - 46],
[1451, 1451, -7*w - 39],
[1459, 1459, 5*w - 43],
[1459, 1459, -5*w - 38],
[1483, 1483, 9*w - 50],
[1483, 1483, -9*w - 41],
[1499, 1499, -15*w - 4],
[1499, 1499, 15*w - 19],
[1531, 1531, 13*w - 59],
[1531, 1531, 20*w - 47],
[1543, 1543, -17*w - 15],
[1543, 1543, 17*w - 32],
[1553, 1553, -w - 39],
[1553, 1553, w - 40],
[1559, 1559, -15*w - 1],
[1559, 1559, 15*w - 16],
[1567, 1567, 16*w - 25],
[1567, 1567, -16*w - 9],
[1571, 1571, 2*w - 41],
[1571, 1571, -2*w - 39],
[1579, 1579, -10*w - 43],
[1579, 1579, 10*w - 53],
[1601, 1601, 15*w - 2],
[1601, 1601, 15*w - 13],
[1619, 1619, 15*w - 11],
[1619, 1619, 15*w - 4],
[1637, 1637, 13*w - 60],
[1637, 1637, 21*w - 50],
[1657, 1657, 3*w - 43],
[1657, 1657, -3*w - 40],
[1669, 1669, 9*w - 52],
[1669, 1669, -9*w - 43],
[1681, 41, -41],
[1733, 1733, 4*w - 45],
[1733, 1733, -4*w - 41],
[1741, 1741, -3*w - 41],
[1741, 1741, 3*w - 44],
[1747, 1747, 18*w - 73],
[1747, 1747, -18*w - 55],
[1753, 1753, 17*w - 27],
[1753, 1753, -17*w - 10],
[1789, 1789, -17*w - 9],
[1789, 1789, 17*w - 26],
[1811, 1811, 17*w - 71],
[1811, 1811, -17*w - 54],
[1823, 1823, 17*w - 25],
[1823, 1823, -17*w - 8],
[1831, 1831, 16*w - 3],
[1831, 1831, 16*w - 13],
[1847, 1847, 16*w - 5],
[1847, 1847, 16*w - 11],
[1849, 43, -43],
[1861, 1861, 19*w - 37],
[1861, 1861, -19*w - 18],
[1879, 1879, 11*w - 58],
[1879, 1879, -11*w - 47],
[1889, 1889, 5*w - 48],
[1889, 1889, -5*w - 43],
[1901, 1901, -7*w - 44],
[1901, 1901, 7*w - 51],
[1907, 1907, 2*w - 45],
[1907, 1907, -2*w - 43],
[1913, 1913, -17*w - 5],
[1913, 1913, 17*w - 22],
[1949, 1949, -18*w - 11],
[1949, 1949, 18*w - 29],
[1973, 1973, -w - 44],
[1973, 1973, w - 45],
[1979, 1979, 10*w - 57],
[1979, 1979, -10*w - 47],
[1997, 1997, -7*w - 45],
[1997, 1997, 7*w - 52],
[2017, 2017, 19*w - 34],
[2017, 2017, -19*w - 15],
[2029, 2029, 15*w - 68],
[2029, 2029, 23*w - 54],
[2039, 2039, 17*w - 16],
[2039, 2039, 17*w - 1],
[2053, 2053, 17*w - 2],
[2053, 2053, 17*w - 15],
[2063, 2063, -w - 45],
[2063, 2063, w - 46],
[2081, 2081, 25*w - 62],
[2081, 2081, -25*w - 37],
[2083, 2083, 17*w - 12],
[2083, 2083, 17*w - 5],
[2087, 2087, 2*w - 47],
[2087, 2087, -2*w - 45],
[2089, 2089, 17*w - 11],
[2089, 2089, 17*w - 6],
[2111, 2111, -19*w - 13],
[2111, 2111, 19*w - 32],
[2113, 2113, 16*w - 71],
[2113, 2113, -16*w - 55],
[2137, 2137, -8*w - 47],
[2137, 2137, 8*w - 55],
[2141, 2141, 20*w - 81],
[2141, 2141, -20*w - 61],
[2153, 2153, 18*w - 23],
[2153, 2153, -18*w - 5],
[2179, 2179, 25*w - 61],
[2179, 2179, -25*w - 36],
[2203, 2203, 11*w - 61],
[2203, 2203, -11*w - 50],
[2209, 47, -47],
[2213, 2213, -19*w - 60],
[2213, 2213, 19*w - 79],
[2237, 2237, -19*w - 10],
[2237, 2237, 19*w - 29],
[2239, 2239, -6*w - 47],
[2239, 2239, 6*w - 53],
[2267, 2267, 21*w - 41],
[2267, 2267, -21*w - 20],
[2269, 2269, -5*w - 47],
[2269, 2269, 5*w - 52],
[2287, 2287, -3*w - 47],
[2287, 2287, 3*w - 50],
[2297, 2297, -7*w - 48],
[2297, 2297, 7*w - 55],
[2311, 2311, 19*w - 27],
[2311, 2311, -19*w - 8],
[2333, 2333, 18*w - 5],
[2333, 2333, 18*w - 13],
[2371, 2371, 20*w - 33],
[2371, 2371, -20*w - 13],
[2377, 2377, 19*w - 25],
[2377, 2377, -19*w - 6],
[2383, 2383, 9*w - 59],
[2383, 2383, -9*w - 50],
[2411, 2411, 10*w - 61],
[2411, 2411, -10*w - 51],
[2423, 2423, 27*w - 67],
[2423, 2423, -27*w - 40],
[2437, 2437, 12*w - 65],
[2437, 2437, -12*w - 53],
[2441, 2441, -21*w - 17],
[2441, 2441, 21*w - 38],
[2459, 2459, 20*w - 31],
[2459, 2459, -20*w - 11],
[2503, 2503, -21*w - 65],
[2503, 2503, 21*w - 86],
[2539, 2539, -20*w - 9],
[2539, 2539, 20*w - 29],
[2543, 2543, -w - 50],
[2543, 2543, w - 51],
[2551, 2551, -18*w - 61],
[2551, 2551, 18*w - 79],
[2557, 2557, 29*w - 74],
[2557, 2557, 29*w + 45],
[2609, 2609, -22*w - 19],
[2609, 2609, 22*w - 41],
[2617, 2617, 19*w - 9],
[2617, 2617, 19*w - 10],
[2633, 2633, 16*w - 75],
[2633, 2633, 27*w - 65],
[2659, 2659, -28*w - 41],
[2659, 2659, 28*w - 69],
[2663, 2663, -11*w - 54],
[2663, 2663, 11*w - 65],
[2677, 2677, -12*w - 55],
[2677, 2677, 12*w - 67],
[2693, 2693, -4*w - 51],
[2693, 2693, 4*w - 55],
[2713, 2713, 23*w - 45],
[2713, 2713, -23*w - 22],
[2719, 2719, -9*w - 53],
[2719, 2719, 9*w - 62],
[2731, 2731, -20*w - 3],
[2731, 2731, 20*w - 23],
[2749, 2749, -w - 52],
[2749, 2749, w - 53],
[2777, 2777, 21*w - 31],
[2777, 2777, -21*w - 10],
[2789, 2789, -5*w - 52],
[2789, 2789, 5*w - 57],
[2791, 2791, 29*w - 72],
[2791, 2791, -29*w - 43],
[2797, 2797, 3*w - 55],
[2797, 2797, -3*w - 52],
[2819, 2819, 20*w - 19],
[2819, 2819, 20*w - 1],
[2833, 2833, -22*w - 15],
[2833, 2833, 22*w - 37],
[2837, 2837, 7*w - 60],
[2837, 2837, -7*w - 53],
[2843, 2843, 23*w - 43],
[2843, 2843, -23*w - 20],
[2851, 2851, 20*w - 17],
[2851, 2851, 20*w - 3],
[2887, 2887, 2*w - 55],
[2887, 2887, -2*w - 53],
[2909, 2909, 4*w - 57],
[2909, 2909, -4*w - 53],
[2953, 2953, 9*w - 64],
[2953, 2953, -9*w - 55],
[2957, 2957, 21*w - 26],
[2957, 2957, -21*w - 5],
[2963, 2963, -w - 54],
[2963, 2963, w - 55],
[2971, 2971, -23*w - 71],
[2971, 2971, 23*w - 94],
[3011, 3011, -5*w - 54],
[3011, 3011, 5*w - 59],
[3023, 3023, 23*w - 40],
[3023, 3023, -23*w - 17],
[3041, 3041, 21*w - 23],
[3041, 3041, -21*w - 2],
[3049, 3049, -25*w - 26],
[3049, 3049, 25*w - 51],
[3061, 3061, 31*w + 47],
[3061, 3061, 31*w - 78],
[3067, 3067, -7*w - 55],
[3067, 3067, 7*w - 62],
[3079, 3079, 23*w - 39],
[3079, 3079, -23*w - 16],
[3083, 3083, -19*w - 66],
[3083, 3083, 19*w - 85],
[3109, 3109, -22*w - 9],
[3109, 3109, 22*w - 31],
[3119, 3119, 10*w - 67],
[3119, 3119, -10*w - 57],
[3137, 3137, 23*w - 95],
[3137, 3137, -23*w - 72],
[3167, 3167, 21*w - 5],
[3167, 3167, 21*w - 16],
[3181, 3181, -12*w - 59],
[3181, 3181, 12*w - 71],
[3191, 3191, 21*w - 13],
[3191, 3191, 21*w - 8],
[3203, 3203, 14*w - 75],
[3203, 3203, -14*w - 61],
[3253, 3253, -22*w - 5],
[3253, 3253, 22*w - 27],
[3257, 3257, 8*w - 65],
[3257, 3257, -8*w - 57],
[3271, 3271, -25*w - 23],
[3271, 3271, 25*w - 48],
[3299, 3299, -w - 57],
[3299, 3299, w - 58],
[3301, 3301, 25*w - 101],
[3301, 3301, -25*w - 76],
[3307, 3307, 31*w - 76],
[3307, 3307, -31*w - 45],
[3313, 3313, 22*w - 25],
[3313, 3313, -22*w - 3],
[3319, 3319, -9*w - 58],
[3319, 3319, 9*w - 67],
[3329, 3329, 23*w - 34],
[3329, 3329, -23*w - 11],
[3331, 3331, 13*w - 74],
[3331, 3331, -13*w - 61],
[3359, 3359, -5*w - 57],
[3359, 3359, 5*w - 62],
[3371, 3371, -10*w - 59],
[3371, 3371, 10*w - 69],
[3373, 3373, -23*w - 10],
[3373, 3373, 23*w - 33],
[3389, 3389, 19*w - 87],
[3389, 3389, 30*w - 71],
[3413, 3413, -11*w - 60],
[3413, 3413, 11*w - 71],
[3457, 3457, 26*w - 51],
[3457, 3457, -26*w - 25],
[3467, 3467, 13*w - 75],
[3467, 3467, -13*w - 62],
[3529, 3529, -23*w - 6],
[3529, 3529, 23*w - 29],
[3533, 3533, -w - 59],
[3533, 3533, w - 60],
[3539, 3539, -25*w - 19],
[3539, 3539, 25*w - 44],
[3547, 3547, -22*w - 73],
[3547, 3547, 22*w - 95],
[3571, 3571, 2*w - 61],
[3571, 3571, -2*w - 59],
[3583, 3583, -6*w - 59],
[3583, 3583, 6*w - 65],
[3631, 3631, 10*w - 71],
[3631, 3631, -10*w - 61],
[3659, 3659, -25*w - 78],
[3659, 3659, 25*w - 103],
[3677, 3677, 7*w - 67],
[3677, 3677, -7*w - 60],
[3719, 3719, -25*w - 16],
[3719, 3719, 25*w - 41],
[3721, 61, -61],
[3761, 3761, -8*w - 61],
[3761, 3761, 8*w - 69],
[3769, 3769, 35*w - 89],
[3769, 3769, 35*w + 54],
[3779, 3779, 23*w - 19],
[3779, 3779, 23*w - 4],
[3793, 3793, 23*w - 5],
[3793, 3793, 23*w - 18],
[3803, 3803, -27*w - 25],
[3803, 3803, 27*w - 52],
[3821, 3821, 23*w - 99],
[3821, 3821, -23*w - 76],
[3823, 3823, 23*w - 15],
[3823, 3823, 23*w - 8],
[3833, 3833, 23*w - 13],
[3833, 3833, 23*w - 10],
[3851, 3851, 5*w - 66],
[3851, 3851, -5*w - 61],
[3853, 3853, 4*w - 65],
[3853, 3853, -4*w - 61],
[3863, 3863, 33*w - 80],
[3863, 3863, 19*w - 90],
[3877, 3877, -26*w - 19],
[3877, 3877, 26*w - 45],
[3881, 3881, -25*w - 13],
[3881, 3881, 25*w - 38],
[3911, 3911, 35*w + 53],
[3911, 3911, 35*w - 88],
[3919, 3919, -15*w - 67],
[3919, 3919, 15*w - 82],
[3931, 3931, 25*w - 37],
[3931, 3931, -25*w - 12],
[3943, 3943, 21*w - 95],
[3943, 3943, 32*w - 75],
[3967, 3967, -3*w - 62],
[3967, 3967, 3*w - 65],
[3989, 3989, 33*w - 79],
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[8501, 8501, -35*w - 2],
[8513, 8513, 19*w - 115],
[8513, 8513, -19*w - 96],
[8521, 8521, 50*w - 123],
[8521, 8521, -50*w - 73],
[8527, 8527, 41*w - 81],
[8527, 8527, -41*w - 40],
[8539, 8539, -35*w - 1],
[8539, 8539, 35*w - 36],
[8597, 8597, 23*w - 123],
[8597, 8597, -23*w - 100],
[8609, 8609, 35*w - 34],
[8609, 8609, 35*w - 1],
[8629, 8629, 11*w - 103],
[8629, 8629, -11*w - 92],
[8641, 8641, 35*w - 2],
[8641, 8641, 35*w - 33],
[8647, 8647, -41*w - 39],
[8647, 8647, 41*w - 80],
[8677, 8677, -3*w - 92],
[8677, 8677, 3*w - 95],
[8693, 8693, -42*w - 43],
[8693, 8693, 42*w - 85],
[8699, 8699, 35*w - 31],
[8699, 8699, 35*w - 4],
[8707, 8707, 37*w - 155],
[8707, 8707, -37*w - 118],
[8713, 8713, 39*w - 160],
[8713, 8713, -39*w - 121],
[8753, 8753, -16*w - 95],
[8753, 8753, 16*w - 111],
[8783, 8783, 31*w - 141],
[8783, 8783, 48*w - 113],
[8803, 8803, -37*w - 15],
[8803, 8803, 37*w - 52],
[8807, 8807, 2*w - 95],
[8807, 8807, -2*w - 93],
[8821, 8821, -23*w - 101],
[8821, 8821, 23*w - 124],
[8839, 8839, 35*w - 24],
[8839, 8839, 35*w - 11],
[8849, 8849, -40*w - 123],
[8849, 8849, 40*w - 163],
[8861, 8861, 35*w - 13],
[8861, 8861, 35*w - 22],
[8867, 8867, 36*w - 41],
[8867, 8867, -36*w - 5],
[8887, 8887, -18*w - 97],
[8887, 8887, 18*w - 115],
[8923, 8923, -w - 94],
[8923, 8923, w - 95],
[8933, 8933, 37*w - 50],
[8933, 8933, -37*w - 13],
[8941, 8941, 35*w - 151],
[8941, 8941, -35*w - 116],
[8999, 8999, 40*w - 71],
[8999, 8999, -40*w - 31],
[9013, 9013, 21*w - 121],
[9013, 9013, -21*w - 100],
[9041, 9041, 54*w - 137],
[9041, 9041, 54*w + 83],
[9043, 9043, -34*w - 115],
[9043, 9043, 34*w - 149],
[9049, 9049, 24*w - 127],
[9049, 9049, -24*w - 103],
[9151, 9151, -7*w - 94],
[9151, 9151, 7*w - 101],
[9157, 9157, -12*w - 95],
[9157, 9157, 12*w - 107],
[9173, 9173, -38*w - 17],
[9173, 9173, 38*w - 55],
[9187, 9187, 2*w - 97],
[9187, 9187, -2*w - 95],
[9199, 9199, 40*w - 69],
[9199, 9199, -40*w - 29],
[9209, 9209, 35*w - 152],
[9209, 9209, -35*w - 117],
[9221, 9221, -39*w - 23],
[9221, 9221, 39*w - 62],
[9227, 9227, 36*w - 5],
[9227, 9227, 36*w - 31],
[9257, 9257, -37*w - 120],
[9257, 9257, 37*w - 157],
[9281, 9281, -13*w - 96],
[9281, 9281, 13*w - 109],
[9293, 9293, -4*w - 95],
[9293, 9293, 4*w - 99],
[9337, 9337, -8*w - 95],
[9337, 9337, 8*w - 103],
[9343, 9343, -6*w - 95],
[9343, 9343, 6*w - 101],
[9371, 9371, 36*w - 13],
[9371, 9371, 36*w - 23],
[9391, 9391, -40*w - 27],
[9391, 9391, 40*w - 67],
[9397, 9397, -49*w + 114],
[9397, 9397, 33*w - 148],
[9403, 9403, 52*w - 127],
[9403, 9403, -52*w - 75],
[9409, 97, -97],
[9419, 9419, 37*w - 41],
[9419, 9419, -37*w - 4],
[9421, 9421, 20*w - 121],
[9421, 9421, -20*w - 101],
[9431, 9431, -41*w - 32],
[9431, 9431, 41*w - 73],
[9461, 9461, -17*w - 99],
[9461, 9461, 17*w - 116],
[9463, 9463, 37*w - 40],
[9463, 9463, -37*w - 3],
[9467, 9467, -39*w - 20],
[9467, 9467, 39*w - 59],
[9479, 9479, -35*w - 118],
[9479, 9479, 35*w - 153],
[9511, 9511, -25*w - 106],
[9511, 9511, 25*w - 131],
[9521, 9521, 5*w - 101],
[9521, 9521, -5*w - 96],
[9547, 9547, -44*w - 45],
[9547, 9547, 44*w - 89],
[9619, 9619, 37*w - 36],
[9619, 9619, 37*w - 1],
[9623, 9623, -43*w - 40],
[9623, 9623, 43*w - 83],
[9629, 9629, 11*w - 108],
[9629, 9629, -11*w - 97],
[9661, 9661, 33*w - 149],
[9661, 9661, 50*w - 117],
[9677, 9677, 17*w - 117],
[9677, 9677, -17*w - 100],
[9679, 9679, 10*w - 107],
[9679, 9679, -10*w - 97],
[9719, 9719, -5*w - 97],
[9719, 9719, 5*w - 102],
[9721, 9721, 55*w + 83],
[9721, 9721, 55*w - 138],
[9739, 9739, 6*w - 103],
[9739, 9739, -6*w - 97],
[9743, 9743, 37*w - 32],
[9743, 9743, 37*w - 5],
[9749, 9749, -25*w - 107],
[9749, 9749, 25*w - 132],
[9767, 9767, 39*w - 55],
[9767, 9767, -39*w - 16],
[9769, 9769, 37*w - 6],
[9769, 9769, 37*w - 31],
[9803, 9803, 31*w - 145],
[9803, 9803, 52*w - 125],
[9811, 9811, 44*w - 87],
[9811, 9811, -44*w - 43],
[9851, 9851, 45*w - 92],
[9851, 9851, -45*w - 47],
[9859, 9859, 53*w - 129],
[9859, 9859, -53*w - 76],
[9883, 9883, 37*w - 12],
[9883, 9883, 37*w - 25],
[9923, 9923, 37*w - 17],
[9923, 9923, 37*w - 20],
[9931, 9931, 41*w - 68],
[9931, 9931, -41*w - 27],
[9941, 9941, 55*w - 137],
[9941, 9941, -55*w - 82],
[9967, 9967, 56*w - 141],
[9967, 9967, 56*w + 85]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [1, -1, -1, -4, -4, 1, 2, 2, 0, 0, -6, -6, -6, 0, 0, -4, -4, 0, 0, 12, 12, -4, -4, -12, -12, -10, -10, -22, -4, -4, -6, -6, 8, 8, 0, 0, 18, 18, 24, 24, 14, 14, -6, -6, 8, 8, 20, 20, -12, -12, -18, -18, 24, 24, 2, 2, -18, -18, 2, 2, 18, 18, -28, -28, 2, 2, 2, 12, 12, -10, -10, 6, 6, -22, 26, 26, 0, 0, -22, -22, -6, -6, 0, 0, 0, 0, 8, 8, 26, 26, -4, -4, -28, -28, -4, -4, -6, -6, 18, 18, 20, 20, -28, -28, 18, 18, 20, 20, -12, -12, 30, 30, 2, 2, 32, 32, -4, -4, 24, 24, 14, 14, 26, 26, -12, -12, 44, 44, -6, -6, 38, 38, 0, 0, 18, 18, -4, -4, -4, -4, 18, 18, -18, -18, 24, 24, 2, 2, -4, -4, -16, -16, -6, -6, 26, 26, 18, 18, 6, 6, 2, -42, -42, -16, -16, 2, 2, 48, 48, 14, 14, 24, 24, 56, 56, 18, 18, -52, -52, 14, 14, 50, 50, 6, 6, 0, 0, -6, -6, -24, -24, -22, -22, 6, 6, -46, -46, 30, 30, -24, -24, 8, 8, -30, -30, -12, -12, -30, -30, 42, 42, 26, 26, 20, 20, 24, 24, -70, -40, -40, 12, 12, 48, 48, 20, 20, -28, -28, 48, 48, 20, 20, 68, 68, 6, 6, -48, -48, 68, 68, 24, 24, -28, -28, 66, 66, -48, -48, 42, 42, 2, 2, -58, -58, -46, -6, -6, -34, -34, 20, 20, -46, -46, 38, 38, 0, 0, 72, 72, -40, -40, -72, -72, -70, -10, -10, 8, 8, 66, 66, -30, -30, 60, 60, -66, -66, 18, 18, 42, 42, 0, 0, -78, -78, -46, -46, 14, 14, 48, 48, 74, 74, 72, 72, -54, -54, -28, -28, -72, -72, 26, 26, 24, 24, -70, -70, 2, 2, -54, -54, -66, -66, 20, 20, 44, 44, -94, -6, -6, 66, 66, -88, -88, 36, 36, -34, -34, 20, 20, 78, 78, -40, -40, 18, 18, 20, 20, -70, -70, -28, -28, -48, -48, -48, -48, -70, -70, 42, 42, 0, 0, -52, -52, 44, 44, -48, -48, -40, -40, -22, -22, 42, 42, 26, 26, 6, 6, 68, 68, -48, -48, -94, -94, -6, -6, -22, -22, 32, 32, 20, 20, 14, 14, 54, 54, -30, -30, -16, -16, 74, 74, -48, -48, 2, 2, -54, -54, -60, -60, 44, 44, 92, 92, -78, -78, 2, 2, -78, -78, -84, -84, 20, 20, 24, 24, 72, 72, -78, -78, 26, 26, -34, -34, -28, -28, -16, -16, -36, -36, 86, 86, 72, 72, 102, 102, -72, -72, 14, 14, 96, 96, -12, -12, -22, -22, -90, -90, -16, -16, -24, -24, 38, 38, -100, -100, 26, 26, 80, 80, -30, -30, -4, -4, 96, 96, 72, 72, 26, 26, -78, -78, 42, 42, 2, 2, 12, 12, 74, 74, 18, 18, 0, 0, -28, -28, -76, -76, 20, 20, -112, -112, 24, 24, 66, 66, 72, 72, -22, 90, 90, -70, -70, 48, 48, -94, -94, 84, 84, -6, -6, -28, -28, 30, 30, 96, 96, 74, 74, 24, 24, 2, 2, -30, -30, -24, -24, 32, 32, -100, -100, -76, -76, -28, -28, 18, 18, 66, 66, 68, 68, 48, 48, -28, -28, -100, -100, 6, 6, 98, 98, 32, 32, -48, -48, 74, 74, 74, 74, -70, -70, -72, -72, -78, -78, 18, 18, -124, -124, 0, 0, 108, 108, -22, -22, 68, 68, 54, 54, 42, 42, 74, 74, -4, -4, -6, -6, -6, -6, -102, -102, 74, 74, 6, 6, 90, 90, 128, 128, 108, 108, 60, 60, 14, 14, 96, 96, 80, 80, -16, -16, -6, -6, -52, -52, -114, -114, -48, -48, 48, 48, -54, -54, 68, 68, -78, -78, -48, -48, -34, -34, 50, 50, -46, -46, 0, 0, -126, -126, 48, 48, 78, 78, 48, 48, -100, -100, 26, 26, 44, 44, -30, -30, 20, 20, 72, 72, 96, 96, -124, -124, 42, 42, -48, -48, 74, 74, 78, 78, 44, 44, 0, 0, 122, 122, 44, 44, 50, 50, -66, -66, -24, -24, 48, 48, -4, -4, -142, -6, -6, 12, 12, 102, 102, -24, -24, 44, 44, 54, 54, -100, -100, -100, -100, -6, -6, 138, 138, -28, -28, -72, -72, 44, 44, 98, 98, -6, -6, 86, 86, 48, 48, 0, 0, 24, 24, -28, -28, 116, 116, -70, -70, 66, 66, 42, 42, -94, -94, -6, -6, -76, -76, 14, 14, -40, -40, 42, 42, 24, 24, -54, -54, 90, 90, -52, -52, -142, -142, 12, 12, -70, -70, -144, -144, 116, 116, 0, 0, 114, 114, -36, -36, 20, 20, -70, -70, 92, 92, 42, 42, 44, 44, -18, -18, 24, 24, -126, -126, -54, -54, -112, -112, 92, 92, -106, -106, -94, 126, 126, -96, -96, 18, 18, 8, 8, 98, 98, 72, 72, -96, -96, -78, -78, -108, -108, -30, -30, -100, -100, 74, 74, -28, -28, 42, 42, 44, 44, -66, -66, -96, -96, 90, 90, 50, 50, -4, -4, -70, -70, 108, 108, -46, -46, 120, 120, 20, 20, 44, 44, -94, -94, 62, 62, 152, 152, 44, 44, -120, -120, -94, -94, -66, -66, -102, -102, 68, 68, -24, -24, -34, -34, -72, -72, -22, -22, -18, -18, -54, -54, 68, 68, 24, 24, 108, 108, 0, 0, -142, -142, 92, 92, -144, -144, -102, -102, 120, 120, 116, 116, -36, -36, -130, -130, -102, -102, 90, 90, -72, -72, 74, 74, -16, -16, 132, 132, 54, 54, -70, -70, 20, 20, 12, 12, 108, 108, -10, -10, -118, -118, -168, -168, -46, -46, 20, 20, 2, 2, -42, -42, 6, 6, -124, -124, -22, -22, -78, -78, 144, 144, -94, -94, -124, -124, 162, 162, -6, -6, -108, -108, 134, 134, -94, -94, -6, -6, -40, -40, -100, -100, 72, 72, 134, 134, -156, -156, -126, -126, 110, 110, 50, 50, 0, 0, 164, 164, 72, 72, 104, 104, -162, -162, -48, -48, 138, 138, 114, 114, -136, -136, 156, 156, 114, 114, 146, 146, 146, 54, 54, -40, -40, -54, -54, -52, -52, -22, -22, -48, -48, -118, -118, -6, -6, 24, 24, -96, -96, 96, 96, -76, -76, 128, 128, 168, 168, -78, -78, 180, 180, -34, -34, 140, 140, 146, 146, 2, 2, 74, 74, -46, -46, 26, 26, -84, -84, 140, 140, -48, -48, 116, 116, -34, -34, 68, 68, -30, -30, -90, -90, -118, -118, 68, 68, -28, -28, 90, 90, -126, -126, 134, 134, -94, -94, 20, 20, 146, 146, 138, 138, -72, -72, -148, -148, -46, -46, -114, -114, 120, 120, 92, 92, 72, 72, 86, 86, 32, 32, 162, 162, 90, 90, 12, 12, -28, -28, -4, -4, 42, 42, -130, -130, 72, 72, 98, 98, 66, 66, 140, 140, -70, -70, 8, 8, -166, -166, -54, -54, -4, -4, -112, -112, -150, -150, 18, 18, 108, 108, -42, -42, 90, 90, -174, -174, 122, 122, -124, -124, 96, 96, -88, -88, 50, 50, -28, -28, -190, 120, 120, 62, 62, 72, 72, -6, -6, -4, -4, 84, 84, -96, -96, 80, 80, 114, 114, 20, 20, -100, -100, 96, 96, 42, 42, 134, 134, -126, -126, -112, -112, -144, -144, 26, 26, 44, 44, 168, 168, 42, 42, 48, 48, 74, 74, 108, 108, 140, 140, 0, 0, 140, 140, -4, -4, 60, 60, -76, -76, -6, -6, 44, 44];
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 | {5, 5, w + 1}>] := 1;
ALEigenvalues[ideal<ZF | {5, 5, w - 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;