/*
  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![-23, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {200, 20, 10*w + 50}>;

primesArray := [
[2, 2, -w - 5],
[7, 7, -w + 4],
[7, 7, w + 4],
[9, 3, 3],
[11, 11, -2*w + 9],
[11, 11, -2*w - 9],
[13, 13, w + 6],
[13, 13, -w + 6],
[19, 19, -w - 2],
[19, 19, w - 2],
[23, 23, -w],
[25, 5, -5],
[29, 29, 7*w + 34],
[29, 29, 2*w + 11],
[41, 41, -w - 8],
[41, 41, w - 8],
[43, 43, 2*w - 7],
[43, 43, -2*w - 7],
[67, 67, 2*w - 5],
[67, 67, -2*w - 5],
[73, 73, 9*w + 44],
[73, 73, 4*w + 21],
[79, 79, -4*w - 17],
[79, 79, 11*w + 52],
[83, 83, 2*w - 3],
[83, 83, -2*w - 3],
[101, 101, 10*w + 49],
[101, 101, 5*w + 26],
[103, 103, -7*w - 32],
[103, 103, 8*w + 37],
[107, 107, -3*w - 10],
[107, 107, 3*w - 10],
[173, 173, -w - 14],
[173, 173, w - 14],
[191, 191, -3*w - 4],
[191, 191, 3*w - 4],
[193, 193, -3*w - 20],
[193, 193, 3*w - 20],
[197, 197, 2*w - 17],
[197, 197, -2*w - 17],
[199, 199, -4*w - 13],
[199, 199, 4*w - 13],
[227, 227, 18*w + 85],
[227, 227, -7*w - 30],
[233, 233, -w - 16],
[233, 233, w - 16],
[251, 251, -5*w - 18],
[251, 251, 5*w - 18],
[257, 257, 4*w - 25],
[257, 257, -4*w - 25],
[263, 263, 16*w + 75],
[263, 263, -9*w - 40],
[269, 269, 2*w - 19],
[269, 269, -2*w - 19],
[277, 277, 3*w - 22],
[277, 277, -3*w - 22],
[283, 283, -11*w - 50],
[283, 283, 14*w + 65],
[289, 17, -17],
[317, 317, 7*w + 38],
[317, 317, 22*w + 107],
[349, 349, 2*w - 21],
[349, 349, -2*w - 21],
[353, 353, 16*w + 79],
[353, 353, 11*w + 56],
[359, 359, 4*w - 3],
[359, 359, -4*w - 3],
[367, 367, -4*w - 1],
[367, 367, 4*w - 1],
[379, 379, 5*w - 14],
[379, 379, -5*w - 14],
[383, 383, 8*w + 33],
[383, 383, -8*w + 33],
[397, 397, -6*w - 35],
[397, 397, 6*w - 35],
[409, 409, 17*w + 84],
[409, 409, 12*w + 61],
[419, 419, 26*w + 123],
[419, 419, -9*w - 38],
[431, 431, 5*w - 12],
[431, 431, -5*w - 12],
[449, 449, -5*w - 32],
[449, 449, 5*w - 32],
[461, 461, -w - 22],
[461, 461, w - 22],
[467, 467, -6*w - 19],
[467, 467, 6*w - 19],
[479, 479, -11*w - 48],
[479, 479, 24*w + 113],
[503, 503, -12*w - 53],
[503, 503, 23*w + 108],
[509, 509, 10*w + 53],
[509, 509, 25*w + 122],
[523, 523, 22*w + 103],
[523, 523, -13*w - 58],
[541, 541, 41*w + 198],
[541, 541, 6*w + 37],
[563, 563, 18*w + 83],
[563, 563, -17*w - 78],
[571, 571, 5*w - 2],
[571, 571, -5*w - 2],
[577, 577, 3*w - 28],
[577, 577, -3*w - 28],
[593, 593, -4*w - 31],
[593, 593, 4*w - 31],
[601, 601, 20*w + 99],
[601, 601, 15*w + 76],
[619, 619, 10*w + 41],
[619, 619, -10*w + 41],
[631, 631, 8*w - 29],
[631, 631, -8*w - 29],
[643, 643, -7*w - 22],
[643, 643, 7*w - 22],
[653, 653, -w - 26],
[653, 653, w - 26],
[659, 659, -6*w - 13],
[659, 659, 6*w - 13],
[673, 673, 21*w + 104],
[673, 673, 16*w + 81],
[727, 727, 7*w - 20],
[727, 727, -7*w - 20],
[743, 743, -8*w - 27],
[743, 743, 8*w - 27],
[751, 751, 32*w + 151],
[751, 751, -13*w - 56],
[761, 761, -w - 28],
[761, 761, w - 28],
[787, 787, -14*w - 61],
[787, 787, 31*w + 146],
[809, 809, -7*w - 44],
[809, 809, 7*w - 44],
[821, 821, 29*w + 142],
[821, 821, 14*w + 73],
[827, 827, -6*w - 1],
[827, 827, 6*w - 1],
[829, 829, 23*w + 114],
[829, 829, 18*w + 91],
[839, 839, -9*w - 32],
[839, 839, 9*w - 32],
[853, 853, -6*w - 41],
[853, 853, 6*w - 41],
[857, 857, -4*w - 35],
[857, 857, 4*w - 35],
[877, 877, -w - 30],
[877, 877, w - 30],
[907, 907, -19*w - 86],
[907, 907, 26*w + 121],
[911, 911, 12*w + 49],
[911, 911, -12*w + 49],
[919, 919, 25*w + 116],
[919, 919, -20*w - 91],
[929, 929, 53*w + 256],
[929, 929, 8*w + 49],
[961, 31, -31],
[971, 971, 42*w + 199],
[971, 971, -13*w - 54],
[983, 983, -7*w - 12],
[983, 983, 7*w - 12],
[997, 997, 2*w - 33],
[997, 997, -2*w - 33],
[1013, 1013, 13*w + 70],
[1013, 1013, 38*w + 185],
[1019, 1019, 11*w + 42],
[1019, 1019, -11*w + 42],
[1021, 1021, 6*w - 43],
[1021, 1021, -6*w - 43],
[1031, 1031, 8*w - 21],
[1031, 1031, -8*w - 21],
[1061, 1061, -11*w - 62],
[1061, 1061, 11*w - 62],
[1063, 1063, -7*w - 8],
[1063, 1063, 7*w - 8],
[1091, 1091, 7*w - 6],
[1091, 1091, -7*w - 6],
[1093, 1093, 26*w + 129],
[1093, 1093, 21*w + 106],
[1097, 1097, 32*w + 157],
[1097, 1097, 17*w + 88],
[1103, 1103, -53*w - 252],
[1103, 1103, 12*w + 47],
[1117, 1117, 39*w + 190],
[1117, 1117, 14*w + 75],
[1123, 1123, 7*w - 2],
[1123, 1123, -7*w - 2],
[1129, 1129, 8*w - 51],
[1129, 1129, -8*w - 51],
[1153, 1153, -4*w - 39],
[1153, 1153, 4*w - 39],
[1171, 1171, -17*w - 74],
[1171, 1171, 38*w + 179],
[1181, 1181, 10*w + 59],
[1181, 1181, 55*w + 266],
[1187, 1187, 9*w - 26],
[1187, 1187, -9*w - 26],
[1237, 1237, -3*w - 38],
[1237, 1237, 3*w - 38],
[1259, 1259, 14*w + 57],
[1259, 1259, -14*w + 57],
[1277, 1277, 2*w - 37],
[1277, 1277, -2*w - 37],
[1279, 1279, -20*w - 89],
[1279, 1279, 35*w + 164],
[1289, 1289, 28*w + 139],
[1289, 1289, 23*w + 116],
[1297, 1297, 48*w + 233],
[1297, 1297, 13*w + 72],
[1301, 1301, 19*w + 98],
[1301, 1301, 34*w + 167],
[1303, 1303, 8*w - 13],
[1303, 1303, -8*w - 13],
[1307, 1307, 34*w + 159],
[1307, 1307, -21*w - 94],
[1361, 1361, -5*w - 44],
[1361, 1361, 5*w - 44],
[1367, 1367, 31*w + 144],
[1367, 1367, -24*w - 109],
[1369, 37, -37],
[1373, 1373, 7*w - 50],
[1373, 1373, -7*w - 50],
[1381, 1381, -6*w - 47],
[1381, 1381, 6*w - 47],
[1399, 1399, 49*w + 232],
[1399, 1399, -16*w - 67],
[1409, 1409, 35*w + 172],
[1409, 1409, 20*w + 103],
[1423, 1423, 8*w - 7],
[1423, 1423, -8*w - 7],
[1429, 1429, 2*w - 39],
[1429, 1429, -2*w - 39],
[1447, 1447, 8*w - 5],
[1447, 1447, -8*w - 5],
[1453, 1453, 42*w + 205],
[1453, 1453, 17*w + 90],
[1459, 1459, 10*w - 29],
[1459, 1459, -10*w - 29],
[1471, 1471, -8*w - 1],
[1471, 1471, 8*w - 1],
[1481, 1481, -4*w - 43],
[1481, 1481, 4*w - 43],
[1483, 1483, -14*w + 55],
[1483, 1483, 14*w + 55],
[1487, 1487, -11*w - 36],
[1487, 1487, 11*w - 36],
[1523, 1523, 47*w + 222],
[1523, 1523, -18*w - 77],
[1549, 1549, 15*w + 82],
[1549, 1549, 50*w + 243],
[1553, 1553, -8*w - 55],
[1553, 1553, 8*w - 55],
[1571, 1571, -10*w - 27],
[1571, 1571, 10*w - 27],
[1579, 1579, 46*w + 217],
[1579, 1579, -19*w - 82],
[1583, 1583, 13*w + 48],
[1583, 1583, -13*w + 48],
[1607, 1607, -9*w - 16],
[1607, 1607, 9*w - 16],
[1613, 1613, 31*w + 154],
[1613, 1613, 26*w + 131],
[1627, 1627, -11*w - 34],
[1627, 1627, 11*w - 34],
[1637, 1637, 22*w + 113],
[1637, 1637, 37*w + 182],
[1657, 1657, 4*w - 45],
[1657, 1657, -4*w - 45],
[1663, 1663, 16*w + 65],
[1663, 1663, -16*w + 65],
[1667, 1667, -9*w - 14],
[1667, 1667, 9*w - 14],
[1669, 1669, -10*w - 63],
[1669, 1669, 10*w - 63],
[1697, 1697, 44*w + 215],
[1697, 1697, 19*w + 100],
[1699, 1699, -14*w + 53],
[1699, 1699, 14*w + 53],
[1723, 1723, -22*w - 97],
[1723, 1723, 43*w + 202],
[1733, 1733, -14*w - 79],
[1733, 1733, 14*w - 79],
[1741, 1741, -w - 42],
[1741, 1741, w - 42],
[1747, 1747, -17*w + 70],
[1747, 1747, 17*w + 70],
[1759, 1759, 11*w - 32],
[1759, 1759, -11*w - 32],
[1777, 1777, 8*w - 57],
[1777, 1777, -8*w - 57],
[1789, 1789, 7*w - 54],
[1789, 1789, -7*w - 54],
[1811, 1811, 15*w + 58],
[1811, 1811, -70*w - 333],
[1831, 1831, 40*w + 187],
[1831, 1831, -25*w - 112],
[1847, 1847, 9*w - 4],
[1847, 1847, -9*w - 4],
[1889, 1889, 13*w - 76],
[1889, 1889, -13*w - 76],
[1907, 1907, 14*w - 51],
[1907, 1907, -14*w - 51],
[1913, 1913, -w - 44],
[1913, 1913, w - 44],
[1931, 1931, 35*w + 162],
[1931, 1931, -30*w - 137],
[1933, 1933, 2*w - 45],
[1933, 1933, -2*w - 45],
[1951, 1951, 13*w - 44],
[1951, 1951, -13*w - 44],
[1973, 1973, 34*w + 169],
[1973, 1973, 29*w + 146],
[1999, 1999, -11*w - 28],
[1999, 1999, 11*w - 28],
[2011, 2011, -10*w - 17],
[2011, 2011, 10*w - 17],
[2017, 2017, 77*w + 372],
[2017, 2017, 12*w + 73],
[2039, 2039, -15*w + 56],
[2039, 2039, 15*w + 56],
[2053, 2053, -14*w - 81],
[2053, 2053, 14*w - 81],
[2087, 2087, 12*w - 35],
[2087, 2087, -12*w - 35],
[2129, 2129, -5*w - 52],
[2129, 2129, 5*w - 52],
[2131, 2131, 10*w - 13],
[2131, 2131, -10*w - 13],
[2141, 2141, 41*w + 202],
[2141, 2141, 26*w + 133],
[2179, 2179, -10*w - 11],
[2179, 2179, 10*w - 11],
[2207, 2207, 11*w - 24],
[2207, 2207, -11*w - 24],
[2209, 47, -47],
[2221, 2221, -15*w - 86],
[2221, 2221, 15*w - 86],
[2237, 2237, -7*w - 58],
[2237, 2237, 7*w - 58],
[2251, 2251, -10*w - 7],
[2251, 2251, 10*w - 7],
[2281, 2281, -w - 48],
[2281, 2281, w - 48],
[2287, 2287, -13*w - 40],
[2287, 2287, 13*w - 40],
[2293, 2293, -3*w - 50],
[2293, 2293, 3*w - 50],
[2309, 2309, 2*w - 49],
[2309, 2309, -2*w - 49],
[2311, 2311, -20*w - 83],
[2311, 2311, 65*w + 308],
[2341, 2341, -5*w - 54],
[2341, 2341, 5*w - 54],
[2351, 2351, 12*w - 31],
[2351, 2351, -12*w - 31],
[2377, 2377, 49*w + 240],
[2377, 2377, 24*w + 125],
[2381, 2381, 14*w + 83],
[2381, 2381, 79*w + 382],
[2383, 2383, -11*w - 20],
[2383, 2383, 11*w - 20],
[2393, 2393, 16*w - 91],
[2393, 2393, -16*w - 91],
[2399, 2399, 64*w + 303],
[2399, 2399, -21*w - 88],
[2411, 2411, -18*w + 71],
[2411, 2411, 18*w + 71],
[2417, 2417, 28*w + 143],
[2417, 2417, 43*w + 212],
[2441, 2441, 4*w - 53],
[2441, 2441, -4*w - 53],
[2459, 2459, 11*w - 18],
[2459, 2459, -11*w - 18],
[2473, 2473, 7*w - 60],
[2473, 2473, -7*w - 60],
[2477, 2477, -w - 50],
[2477, 2477, w - 50],
[2503, 2503, -31*w - 140],
[2503, 2503, 44*w + 205],
[2551, 2551, 17*w + 64],
[2551, 2551, -17*w + 64],
[2557, 2557, 22*w + 117],
[2557, 2557, 57*w + 278],
[2591, 2591, -13*w - 36],
[2591, 2591, 13*w - 36],
[2617, 2617, 12*w - 77],
[2617, 2617, -12*w - 77],
[2657, 2657, -4*w - 55],
[2657, 2657, 4*w - 55],
[2659, 2659, -14*w - 43],
[2659, 2659, 14*w - 43],
[2677, 2677, 26*w + 135],
[2677, 2677, 51*w + 250],
[2683, 2683, 11*w - 10],
[2683, 2683, -11*w - 10],
[2687, 2687, -12*w - 25],
[2687, 2687, 12*w - 25],
[2693, 2693, -11*w - 74],
[2693, 2693, 11*w - 74],
[2711, 2711, 60*w + 283],
[2711, 2711, -25*w - 108],
[2719, 2719, 11*w - 8],
[2719, 2719, -11*w - 8],
[2731, 2731, -13*w - 34],
[2731, 2731, 13*w - 34],
[2741, 2741, 10*w - 71],
[2741, 2741, -10*w - 71],
[2749, 2749, 18*w - 101],
[2749, 2749, -18*w - 101],
[2753, 2753, 8*w - 65],
[2753, 2753, -8*w - 65],
[2767, 2767, -11*w - 4],
[2767, 2767, 11*w - 4],
[2789, 2789, 5*w - 58],
[2789, 2789, -5*w - 58],
[2801, 2801, 40*w + 199],
[2801, 2801, 35*w + 176],
[2803, 2803, 17*w - 62],
[2803, 2803, -17*w - 62],
[2809, 53, -53],
[2833, 2833, 52*w + 255],
[2833, 2833, 27*w + 140],
[2837, 2837, -13*w - 82],
[2837, 2837, 13*w - 82],
[2843, 2843, 58*w + 273],
[2843, 2843, -27*w - 118],
[2851, 2851, 73*w + 346],
[2851, 2851, -22*w - 91],
[2861, 2861, 31*w + 158],
[2861, 2861, 46*w + 227],
[2903, 2903, 57*w + 268],
[2903, 2903, -28*w - 123],
[2953, 2953, 41*w + 204],
[2953, 2953, 36*w + 181],
[2957, 2957, 17*w - 98],
[2957, 2957, -17*w - 98],
[2963, 2963, 18*w + 67],
[2963, 2963, -18*w + 67],
[2969, 2969, 7*w - 64],
[2969, 2969, -7*w - 64],
[3011, 3011, -30*w - 133],
[3011, 3011, 55*w + 258],
[3023, 3023, -12*w - 17],
[3023, 3023, 12*w - 17],
[3037, 3037, 9*w - 70],
[3037, 3037, -9*w - 70],
[3049, 3049, 25*w + 132],
[3049, 3049, 60*w + 293],
[3061, 3061, 14*w - 87],
[3061, 3061, -14*w - 87],
[3079, 3079, 16*w - 53],
[3079, 3079, -16*w - 53],
[3109, 3109, 42*w + 209],
[3109, 3109, 37*w + 186],
[3119, 3119, -19*w + 72],
[3119, 3119, 19*w + 72],
[3121, 3121, 20*w + 111],
[3121, 3121, 75*w + 364],
[3137, 3137, 16*w + 95],
[3137, 3137, 91*w + 440],
[3169, 3169, -13*w - 84],
[3169, 3169, 13*w - 84],
[3191, 3191, 12*w - 11],
[3191, 3191, -12*w - 11],
[3209, 3209, 23*w + 124],
[3209, 3209, 68*w + 331],
[3221, 3221, 26*w + 137],
[3221, 3221, 61*w + 298],
[3229, 3229, -7*w - 66],
[3229, 3229, 7*w - 66],
[3271, 3271, -95*w - 452],
[3271, 3271, 20*w + 77],
[3299, 3299, -39*w - 178],
[3299, 3299, 46*w + 213],
[3301, 3301, 11*w - 78],
[3301, 3301, -11*w - 78],
[3313, 3313, 76*w + 369],
[3313, 3313, 21*w + 116],
[3319, 3319, -41*w - 188],
[3319, 3319, 44*w + 203],
[3323, 3323, 43*w + 198],
[3323, 3323, -42*w - 193],
[3331, 3331, -23*w + 94],
[3331, 3331, 23*w + 94],
[3361, 3361, 19*w - 108],
[3361, 3361, -19*w - 108],
[3389, 3389, 2*w - 59],
[3389, 3389, -2*w - 59],
[3391, 3391, 67*w + 316],
[3391, 3391, -28*w - 121],
[3413, 3413, 14*w - 89],
[3413, 3413, -14*w - 89],
[3433, 3433, 44*w + 219],
[3433, 3433, 39*w + 196],
[3467, 3467, -29*w - 126],
[3467, 3467, 66*w + 311],
[3481, 59, -59],
[3511, 3511, -17*w - 56],
[3511, 3511, 17*w - 56],
[3539, 3539, -30*w - 131],
[3539, 3539, 65*w + 306],
[3547, 3547, -14*w - 31],
[3547, 3547, 14*w - 31],
[3559, 3559, 80*w + 379],
[3559, 3559, -25*w - 104],
[3581, 3581, 25*w + 134],
[3581, 3581, 70*w + 341],
[3607, 3607, -31*w - 136],
[3607, 3607, 64*w + 301],
[3613, 3613, -9*w - 74],
[3613, 3613, 9*w - 74],
[3617, 3617, 11*w - 80],
[3617, 3617, -11*w - 80],
[3631, 3631, 13*w - 16],
[3631, 3631, -13*w - 16],
[3637, 3637, -3*w - 62],
[3637, 3637, 3*w - 62],
[3671, 3671, -32*w - 141],
[3671, 3671, 63*w + 296],
[3673, 3673, 57*w + 280],
[3673, 3673, 32*w + 165],
[3691, 3691, -13*w - 14],
[3691, 3691, 13*w - 14],
[3709, 3709, 23*w + 126],
[3709, 3709, 78*w + 379],
[3721, 61, -61],
[3761, 3761, 29*w + 152],
[3761, 3761, 64*w + 313],
[3779, 3779, 14*w - 27],
[3779, 3779, -14*w - 27],
[3797, 3797, -19*w - 110],
[3797, 3797, 19*w - 110],
[3821, 3821, -w - 62],
[3821, 3821, w - 62],
[3823, 3823, -13*w - 8],
[3823, 3823, 13*w - 8],
[3851, 3851, 13*w - 6],
[3851, 3851, -13*w - 6],
[3853, 3853, 33*w + 170],
[3853, 3853, 58*w + 285],
[3863, 3863, -16*w - 45],
[3863, 3863, 16*w - 45],
[3877, 3877, 2*w - 63],
[3877, 3877, -2*w - 63],
[3889, 3889, 3*w - 64],
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[8599, 8599, 25*w - 76],
[8623, 8623, -28*w - 97],
[8623, 8623, 28*w - 97],
[8629, 8629, 3*w - 94],
[8629, 8629, -3*w - 94],
[8641, 8641, 5*w - 96],
[8641, 8641, -5*w - 96],
[8647, 8647, -31*w + 116],
[8647, 8647, 31*w + 116],
[8663, 8663, 33*w + 128],
[8663, 8663, -152*w - 723],
[8677, 8677, 18*w - 127],
[8677, 8677, -18*w - 127],
[8689, 8689, 35*w + 192],
[8689, 8689, 120*w + 583],
[8699, 8699, -21*w - 38],
[8699, 8699, 21*w - 38],
[8731, 8731, -22*w - 49],
[8731, 8731, 22*w - 49],
[8741, 8741, 74*w + 367],
[8741, 8741, 59*w + 298],
[8747, 8747, -51*w - 226],
[8747, 8747, 94*w + 441],
[8753, 8753, -16*w - 121],
[8753, 8753, 16*w - 121],
[8783, 8783, 36*w + 145],
[8783, 8783, -36*w + 145],
[8803, 8803, -23*w - 58],
[8803, 8803, 23*w - 58],
[8807, 8807, -44*w - 189],
[8807, 8807, 111*w + 524],
[8819, 8819, 30*w - 109],
[8819, 8819, -30*w - 109],
[8821, 8821, 86*w + 423],
[8821, 8821, 51*w + 262],
[8831, 8831, 93*w + 436],
[8831, 8831, -52*w - 231],
[8839, 8839, 20*w - 19],
[8839, 8839, -20*w - 19],
[8861, 8861, 106*w + 517],
[8861, 8861, 41*w + 218],
[8923, 8923, 22*w - 47],
[8923, 8923, -22*w - 47],
[8933, 8933, 2*w - 95],
[8933, 8933, -2*w - 95],
[9001, 9001, 121*w + 588],
[9001, 9001, 36*w + 197],
[9007, 9007, 28*w - 95],
[9007, 9007, -28*w - 95],
[9029, 9029, 5*w - 98],
[9029, 9029, -5*w - 98],
[9041, 9041, 4*w - 97],
[9041, 9041, -4*w - 97],
[9059, 9059, 90*w + 421],
[9059, 9059, -55*w - 246],
[9067, 9067, 109*w + 514],
[9067, 9067, -46*w - 199],
[9109, 9109, 13*w - 114],
[9109, 9109, -13*w - 114],
[9127, 9127, -56*w - 251],
[9127, 9127, 89*w + 416],
[9133, 9133, 129*w + 626],
[9133, 9133, 34*w + 189],
[9137, 9137, -8*w - 103],
[9137, 9137, 8*w - 103],
[9151, 9151, -20*w - 7],
[9151, 9151, 20*w - 7],
[9157, 9157, 107*w + 522],
[9157, 9157, 42*w + 223],
[9181, 9181, 14*w - 117],
[9181, 9181, -14*w - 117],
[9187, 9187, 38*w + 155],
[9187, 9187, -38*w + 155],
[9199, 9199, -20*w - 1],
[9199, 9199, 20*w - 1],
[9209, 9209, -32*w - 181],
[9209, 9209, 32*w - 181],
[9241, 9241, 16*w - 123],
[9241, 9241, -16*w - 123],
[9277, 9277, 7*w - 102],
[9277, 9277, -7*w - 102],
[9281, 9281, 61*w + 308],
[9281, 9281, 76*w + 377],
[9283, 9283, 22*w - 43],
[9283, 9283, -22*w - 43],
[9293, 9293, 94*w + 461],
[9293, 9293, 49*w + 254],
[9311, 9311, -48*w - 209],
[9311, 9311, 107*w + 504],
[9341, 9341, 25*w + 154],
[9341, 9341, 170*w + 821],
[9343, 9343, -29*w - 100],
[9343, 9343, 29*w - 100],
[9371, 9371, -27*w - 86],
[9371, 9371, 27*w - 86],
[9377, 9377, 53*w + 272],
[9377, 9377, 88*w + 433],
[9391, 9391, 32*w + 119],
[9391, 9391, -32*w + 119],
[9397, 9397, -3*w - 98],
[9397, 9397, 3*w - 98],
[9403, 9403, -43*w - 182],
[9403, 9403, 122*w + 577],
[9409, 97, -97],
[9413, 9413, 101*w + 494],
[9413, 9413, 46*w + 241],
[9433, 9433, -4*w - 99],
[9433, 9433, 4*w - 99],
[9461, 9461, 35*w + 194],
[9461, 9461, 130*w + 631],
[9463, 9463, -23*w - 52],
[9463, 9463, 23*w - 52],
[9467, 9467, 21*w - 26],
[9467, 9467, -21*w - 26],
[9491, 9491, -63*w - 286],
[9491, 9491, 82*w + 381],
[9539, 9539, 105*w + 494],
[9539, 9539, -50*w - 219],
[9587, 9587, -66*w - 301],
[9587, 9587, 79*w + 366],
[9619, 9619, 26*w - 77],
[9619, 9619, -26*w - 77],
[9631, 9631, -68*w - 311],
[9631, 9631, 77*w + 356],
[9649, 9649, 72*w + 359],
[9649, 9649, 67*w + 336],
[9661, 9661, 54*w + 277],
[9661, 9661, 89*w + 438],
[9679, 9679, -160*w - 761],
[9679, 9679, 35*w + 136],
[9689, 9689, -7*w - 104],
[9689, 9689, 7*w - 104],
[9733, 9733, 171*w + 826],
[9733, 9733, 26*w + 159],
[9739, 9739, -29*w - 98],
[9739, 9739, 29*w - 98],
[9743, 9743, 21*w - 20],
[9743, 9743, -21*w - 20],
[9767, 9767, -24*w - 59],
[9767, 9767, 24*w - 59],
[9781, 9781, 6*w - 103],
[9781, 9781, -6*w - 103],
[9803, 9803, 38*w + 153],
[9803, 9803, -38*w + 153],
[9829, 9829, -5*w - 102],
[9829, 9829, 5*w - 102],
[9833, 9833, 4*w - 101],
[9833, 9833, -4*w - 101],
[9851, 9851, 102*w + 479],
[9851, 9851, -53*w - 234],
[9857, 9857, 28*w + 167],
[9857, 9857, 163*w + 788],
[9859, 9859, 119*w + 562],
[9859, 9859, -46*w - 197],
[9887, 9887, -21*w - 16],
[9887, 9887, 21*w - 16],
[9907, 9907, 22*w - 35],
[9907, 9907, -22*w - 35],
[9923, 9923, 26*w - 75],
[9923, 9923, -26*w - 75],
[9929, 9929, 73*w + 364],
[9929, 9929, 68*w + 341],
[9949, 9949, 90*w + 443],
[9949, 9949, 55*w + 282]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [0, -4, -4, -6, 4, 4, -2, -2, 4, 4, 4, 1, -2, -2, -6, -6, -8, -8, 8, 8, -6, -6, 0, 0, -16, -16, 6, 6, 4, 4, 0, 0, 14, 14, 8, 8, -14, -14, 22, 22, 8, 8, -24, -24, -6, -6, -12, -12, -30, -30, -12, -12, 14, 14, -10, -10, 8, 8, -30, -18, -18, 30, 30, 2, 2, -24, -24, 20, 20, -20, -20, -36, -36, -2, -2, 10, 10, 36, 36, -40, -40, 18, 18, -18, -18, 8, 8, 16, 16, -12, -12, -2, -2, -8, -8, -2, -2, 16, 16, 4, 4, 2, 2, 18, 18, 10, 10, -4, -4, 16, 16, -48, -48, -34, -34, -12, -12, 18, 18, 44, 44, 36, 36, -24, -24, -6, -6, 8, 8, 42, 42, 22, 22, 48, 48, -2, -2, -40, -40, 38, 38, -6, -6, 46, 46, 48, 48, 40, 40, 8, 8, -14, -14, 2, 20, 20, 36, 36, -42, -42, 38, 38, 44, 44, -2, -2, -16, -16, 38, 38, -44, -44, 12, 12, -10, -10, 58, 58, -4, -4, 62, 62, -16, -16, 26, 26, 2, 2, -52, -52, -34, -34, -8, -8, -58, -58, 12, 12, 62, 62, 64, 64, 42, 42, 34, 34, -10, -10, -28, -28, 0, 0, -46, -46, -36, -36, -38, 46, 46, 38, 38, -56, -56, 34, 34, -36, -36, 22, 22, -4, -4, -2, -2, 36, 36, 8, 8, 58, 58, 72, 72, 36, 36, 16, 16, -50, -50, 50, 50, -4, -4, -20, -20, -52, -52, 12, 12, 14, 14, 48, 48, 6, 6, -6, -6, -20, -20, 8, 8, 6, 6, 50, 50, -44, -44, 24, 24, -26, -26, 14, 14, -56, -56, 32, 32, 50, 50, -34, -34, -36, -36, -48, -48, 44, 44, -78, -78, -24, -24, -54, -54, 36, 36, -50, -50, 56, 56, 54, 54, 32, 32, -12, -12, 82, 82, 24, 24, 38, 38, -52, -52, 34, 34, 44, 44, 30, 30, -44, -44, -76, -76, -78, 14, 14, -2, -2, 4, 4, -38, -38, -12, -12, -10, -10, -26, -26, -64, -64, -26, -26, -24, -24, 42, 42, -50, -50, -4, -4, 10, 10, 48, 48, 52, 52, 34, 34, 10, 10, -36, -36, -38, -38, -66, -66, 20, 20, -48, -48, -82, -82, -40, -40, 26, 26, -30, -30, 84, 84, -42, -42, -72, -72, 52, 52, -90, -90, 80, 80, -16, -16, 68, 68, 54, 54, 30, 30, 18, 18, 52, 52, -90, -90, -78, -78, 16, 16, -70, -14, -14, 54, 54, 8, 8, -20, -20, 78, 78, 84, 84, -54, -54, -18, -18, 0, 0, -54, -54, -84, -84, -52, -52, -18, -18, 10, 10, -74, -74, -40, -40, 6, 6, 64, 64, 50, 50, -62, -62, 2, 2, -96, -96, -70, -70, 54, 54, 30, 30, -16, -16, 36, 36, -26, -26, 50, 50, -8, -8, 56, 56, -68, -68, 82, 82, -66, -66, 8, 8, -42, -42, 74, 74, 48, 48, -102, 16, 16, -60, -60, -80, -80, -88, -88, 30, 30, -20, -20, -50, -50, 18, 18, 56, 56, 22, 22, 96, 96, 10, 10, -12, -12, -2, -2, -118, 34, 34, 68, 68, -10, -10, -114, -114, 60, 60, -12, -12, 46, 46, 36, 36, 70, 70, -62, -62, 104, 104, -44, -44, -76, -76, 32, 32, -76, -76, -68, -68, 20, 20, -110, -110, 106, 106, -54, -54, 36, 36, 52, 52, -8, -8, 68, 68, -14, -14, 86, 86, -100, -100, 106, 106, -48, -48, -54, -54, -20, -20, 64, 64, 34, 34, 64, 64, -26, -26, -46, -46, -88, -88, -126, -126, 116, 116, -98, -98, -90, -90, 96, 96, -82, -82, 42, 42, 68, 68, -38, -38, -70, -70, 128, 128, 110, 110, -96, -96, -10, -10, 24, 24, -104, -104, 102, 102, -56, -56, -32, -32, 106, 106, -44, -44, -12, -12, 114, 114, -8, -8, -20, -20, -116, -116, 34, 34, 94, 94, 40, 40, 124, 124, 90, 90, 16, 16, 62, 62, 62, 62, 94, 94, 10, 10, 8, 8, 124, 124, 78, 78, 108, 108, 90, 90, 80, 80, -62, -62, 50, 50, 60, 60, -142, 132, 132, -124, -124, 46, 46, 82, 82, -92, -92, 108, 108, -24, -24, -46, -46, -106, -106, 26, 26, -136, -136, -88, -88, -16, -16, -48, -48, 24, 24, 134, 134, -118, -118, -20, -20, 14, 14, 66, 66, 64, 64, 24, 24, 6, 6, 56, 56, -2, -2, 56, 56, -80, -80, 98, 98, -20, -20, -60, -60, 136, 136, 82, 82, 52, 52, -102, -102, -74, -74, 90, 90, -66, -66, 56, 56, -42, -42, -108, -108, 84, 84, -34, -34, 80, 80, 46, 46, -56, -56, 106, 106, 10, 10, -36, -36, 4, 4, 62, 62, 40, 40, -18, -18, 40, 40, 132, 132, 102, 102, 26, 26, 0, 0, -60, -60, 86, 86, 50, 50, -38, -38, 48, 48, 0, 0, -114, -114, 76, 76, 34, 34, -44, -44, -34, -34, 120, 120, -4, -4, 96, 96, 42, 42, -142, -142, -56, -56, 26, 26, 100, 100, 102, 102, 150, 150, -34, -34, 54, 54, -16, -16, 66, 66, -36, -36, -58, -58, 18, 18, 52, 52, 58, 58, 120, 120, -64, -64, 150, 150, 56, 56, 110, 110, -114, -114, -62, -62, -100, -100, 46, 46, -4, -4, -74, -74, 108, 108, 122, 122, -76, -76, -128, -128, -62, -62, 58, 58, -160, -160, -60, -60, 0, 0, -120, -120, 38, 38, -116, -116, -14, -14, -156, -156, 40, 40, -22, -22, 0, 0, -2, -2, 44, 44, -90, -90, 124, 124, -120, -120, 10, 10, -40, -40, -44, -44, -8, -8, -10, -10, 48, 48, 18, 18, -82, -82, -36, -36, 54, 54, 96, 96, -118, -118, 124, 124, 10, 10, 52, 52, 100, 100, -58, -58, 42, 42, -38, -38, 130, 130, 24, 24, 102, 102, -84, -84, 6, 6, -104, -104, -126, -126, 92, 92, -124, -124, 148, 148, -154, -154, -28, -28, 126, 126, -70, -70, -66, -66, -74, -74, -24, -24, -66, -66, 64, 64, -142, -68, -68, -110, -110, -88, -88, 42, 42, 84, 84, 2, 2, -10, -10, 18, 18, 12, 12, -54, -54, -136, -136, -168, -168, 4, 4, 64, 64, -18, -18, 142, 142, 146, 146, -76, -76, -180, -180, 22, 22, 10, 10, -142, -142, -104, -104, -8, -8, -108, -108, 114, 114, -140, -140, -22, -22, 44, 44, 128, 128, 102, 102, -42, -42, -40, -40, 76, 76, 54, 54, 130, 130, -20, -20, -60, -60, 38, 38, 82, 82, 76, 76, -92, -92, -58, -58, -128, -128, -126, -126, -36, -36, 96, 96, 60, 60, -124, -124, -10, -10, 168, 168, 56, 56, -2, -2, -120, -120, -106, -106, 106, 106, -12, -12, -58, -58, 98, 98, 36, 36, -160, -160, 22, 22, 92, 92, -66, -66, 18, 18, -24, -24, 182, 182, 30, 30, 56, 56, 160, 160, -150, -150, -118, -118, -130, -130, -78, -78, -96, -96, 14, 14, 168, 168, 126, 126, -100, -100, 164, 164, 162, 162, -8, -8, 86, 86, -8, -8, 2, 6, 6, -38, -38, 150, 150, -156, -156, -96, -96, -132, -132, -12, -12, 168, 168, -60, -60, 56, 56, 130, 130, -130, -130, 0, 0, 90, 90, 182, 182, -20, -20, -4, -4, 44, 44, -106, -106, -56, -56, 70, 70, 106, 106, 116, 116, 114, 114, -156, -156, 84, 84, -136, -136, 96, 96, 74, 74, 62, 62];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, -w - 5}>] := -1;
ALEigenvalues[ideal<ZF | {25, 5, -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;