/*
  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![-16, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {196, 196, 3*w + 17}>;

primesArray := [
[2, 2, w],
[2, 2, w + 1],
[5, 5, w + 2],
[7, 7, w + 1],
[7, 7, w + 5],
[9, 3, 3],
[13, 13, w + 6],
[29, 29, -2*w + 7],
[29, 29, 2*w + 5],
[37, 37, w + 9],
[37, 37, w + 27],
[47, 47, w + 10],
[47, 47, w + 36],
[61, 61, 2*w - 3],
[61, 61, -2*w - 1],
[67, 67, w + 23],
[67, 67, w + 43],
[73, 73, w + 24],
[73, 73, w + 48],
[79, 79, 2*w - 13],
[79, 79, -2*w - 11],
[83, 83, w + 13],
[83, 83, w + 69],
[97, 97, w + 14],
[97, 97, w + 82],
[101, 101, 4*w - 21],
[101, 101, 6*w - 25],
[121, 11, -11],
[131, 131, 2*w - 15],
[131, 131, -2*w - 13],
[137, 137, w + 52],
[137, 137, w + 84],
[139, 139, -4*w - 9],
[139, 139, 4*w - 13],
[163, 163, w + 18],
[163, 163, w + 144],
[167, 167, w + 68],
[167, 167, w + 98],
[179, 179, 4*w - 11],
[179, 179, -4*w - 7],
[181, 181, 4*w + 19],
[181, 181, 10*w + 33],
[191, 191, 2*w - 17],
[191, 191, -2*w - 15],
[193, 193, w + 39],
[193, 193, w + 153],
[197, 197, w + 79],
[197, 197, w + 117],
[199, 199, 6*w - 31],
[199, 199, 8*w - 33],
[211, 211, -4*w - 5],
[211, 211, 4*w - 9],
[223, 223, w + 21],
[223, 223, w + 201],
[227, 227, w + 67],
[227, 227, w + 159],
[251, 251, -4*w - 1],
[251, 251, 4*w - 5],
[269, 269, 4*w - 25],
[269, 269, -4*w - 21],
[289, 17, -17],
[293, 293, w + 123],
[293, 293, w + 169],
[307, 307, w + 133],
[307, 307, w + 173],
[311, 311, 8*w - 31],
[311, 311, 8*w + 23],
[317, 317, w + 25],
[317, 317, w + 291],
[353, 353, w + 99],
[353, 353, w + 253],
[361, 19, -19],
[383, 383, w + 55],
[383, 383, w + 327],
[389, 389, 6*w - 17],
[389, 389, -6*w - 11],
[397, 397, w + 56],
[397, 397, w + 340],
[419, 419, 2*w - 23],
[419, 419, -2*w - 21],
[439, 439, 6*w + 29],
[439, 439, 16*w + 53],
[457, 457, w + 30],
[457, 457, w + 426],
[463, 463, w + 192],
[463, 463, w + 270],
[487, 487, w + 218],
[487, 487, w + 268],
[491, 491, 10*w - 51],
[491, 491, 12*w - 49],
[521, 521, 6*w - 11],
[521, 521, -6*w - 5],
[529, 23, -23],
[557, 557, w + 197],
[557, 557, w + 359],
[569, 569, -6*w - 1],
[569, 569, 6*w - 7],
[571, 571, 6*w - 37],
[571, 571, -6*w - 31],
[577, 577, w + 107],
[577, 577, w + 469],
[587, 587, w + 34],
[587, 587, w + 552],
[593, 593, w + 185],
[593, 593, w + 407],
[599, 599, -8*w - 17],
[599, 599, 8*w - 25],
[601, 601, -16*w + 77],
[601, 601, 10*w - 37],
[613, 613, w + 273],
[613, 613, w + 339],
[617, 617, w + 131],
[617, 617, w + 485],
[641, 641, 18*w - 77],
[641, 641, 18*w + 59],
[643, 643, w + 143],
[643, 643, w + 499],
[659, 659, 12*w - 47],
[659, 659, 12*w + 35],
[683, 683, w + 188],
[683, 683, w + 494],
[701, 701, 4*w - 33],
[701, 701, -4*w - 29],
[719, 719, 2*w - 29],
[719, 719, -2*w - 27],
[733, 733, w + 38],
[733, 733, w + 694],
[743, 743, w + 288],
[743, 743, w + 454],
[751, 751, 8*w - 21],
[751, 751, -8*w - 13],
[773, 773, w + 349],
[773, 773, w + 423],
[787, 787, w + 148],
[787, 787, w + 638],
[809, 809, -8*w - 39],
[809, 809, 8*w - 47],
[827, 827, w + 361],
[827, 827, w + 465],
[829, 829, -4*w - 31],
[829, 829, 4*w - 35],
[853, 853, w + 41],
[853, 853, w + 811],
[859, 859, -6*w - 35],
[859, 859, 6*w - 41],
[863, 863, w + 155],
[863, 863, w + 707],
[877, 877, w + 132],
[877, 877, w + 744],
[881, 881, 14*w - 55],
[881, 881, 14*w + 41],
[911, 911, 16*w + 49],
[911, 911, 16*w - 65],
[919, 919, 8*w - 15],
[919, 919, -8*w - 7],
[947, 947, w + 257],
[947, 947, w + 689],
[961, 31, -31],
[967, 967, w + 448],
[967, 967, w + 518],
[971, 971, 12*w - 43],
[971, 971, -12*w - 31],
[977, 977, w + 88],
[977, 977, w + 888],
[983, 983, w + 357],
[983, 983, w + 625],
[991, 991, -8*w - 3],
[991, 991, 8*w - 11],
[1031, 1031, 8*w - 7],
[1031, 1031, 8*w - 1],
[1033, 1033, w + 287],
[1033, 1033, w + 745],
[1039, 1039, 8*w - 5],
[1039, 1039, 8*w - 3],
[1049, 1049, 10*w - 29],
[1049, 1049, -10*w - 19],
[1069, 1069, 14*w - 53],
[1069, 1069, 14*w + 39],
[1087, 1087, w + 147],
[1087, 1087, w + 939],
[1091, 1091, 2*w - 35],
[1091, 1091, -2*w - 33],
[1097, 1097, w + 504],
[1097, 1097, w + 592],
[1103, 1103, w + 351],
[1103, 1103, w + 751],
[1109, 1109, -4*w - 35],
[1109, 1109, 4*w - 39],
[1123, 1123, w + 459],
[1123, 1123, w + 663],
[1163, 1163, w + 180],
[1163, 1163, w + 982],
[1171, 1171, 14*w - 73],
[1171, 1171, 20*w - 83],
[1217, 1217, w + 49],
[1217, 1217, w + 1167],
[1231, 1231, 2*w - 37],
[1231, 1231, -2*w - 35],
[1237, 1237, w + 474],
[1237, 1237, w + 762],
[1249, 1249, 14*w - 51],
[1249, 1249, 14*w + 37],
[1291, 1291, -10*w - 49],
[1291, 1291, 10*w - 59],
[1301, 1301, 10*w - 23],
[1301, 1301, -10*w - 13],
[1307, 1307, w + 302],
[1307, 1307, w + 1004],
[1361, 1361, -8*w - 45],
[1361, 1361, 8*w - 53],
[1367, 1367, w + 525],
[1367, 1367, w + 841],
[1373, 1373, w + 422],
[1373, 1373, w + 950],
[1381, 1381, 26*w - 111],
[1381, 1381, 26*w + 85],
[1423, 1423, w + 53],
[1423, 1423, w + 1369],
[1429, 1429, 10*w - 19],
[1429, 1429, -10*w - 9],
[1439, 1439, 24*w + 77],
[1439, 1439, 24*w - 101],
[1459, 1459, 20*w + 61],
[1459, 1459, 20*w - 81],
[1481, 1481, -10*w - 7],
[1481, 1481, 10*w - 17],
[1487, 1487, w + 172],
[1487, 1487, w + 1314],
[1493, 1493, w + 204],
[1493, 1493, w + 1288],
[1499, 1499, 12*w - 35],
[1499, 1499, -12*w - 23],
[1511, 1511, 10*w - 61],
[1511, 1511, -10*w - 51],
[1523, 1523, w + 526],
[1523, 1523, w + 996],
[1531, 1531, 6*w - 49],
[1531, 1531, -6*w - 43],
[1553, 1553, w + 352],
[1553, 1553, w + 1200],
[1559, 1559, -26*w + 125],
[1559, 1559, 16*w - 59],
[1567, 1567, w + 301],
[1567, 1567, w + 1265],
[1597, 1597, w + 211],
[1597, 1597, w + 1385],
[1607, 1607, w + 758],
[1607, 1607, w + 848],
[1609, 1609, 10*w - 9],
[1609, 1609, 10*w - 1],
[1621, 1621, 10*w - 3],
[1621, 1621, 10*w - 7],
[1627, 1627, w + 337],
[1627, 1627, w + 1289],
[1657, 1657, w + 774],
[1657, 1657, w + 882],
[1681, 41, -41],
[1697, 1697, w + 585],
[1697, 1697, w + 1111],
[1699, 1699, 2*w - 43],
[1699, 1699, -2*w - 41],
[1723, 1723, w + 458],
[1723, 1723, w + 1264],
[1741, 1741, -14*w - 31],
[1741, 1741, 14*w - 45],
[1747, 1747, w + 508],
[1747, 1747, w + 1238],
[1753, 1753, w + 414],
[1753, 1753, w + 1338],
[1759, 1759, 16*w - 57],
[1759, 1759, -16*w - 41],
[1783, 1783, w + 119],
[1783, 1783, w + 1663],
[1787, 1787, w + 418],
[1787, 1787, w + 1368],
[1811, 1811, -12*w - 17],
[1811, 1811, 12*w - 29],
[1849, 43, -43],
[1867, 1867, w + 855],
[1867, 1867, w + 1011],
[1871, 1871, 2*w - 45],
[1871, 1871, -2*w - 43],
[1877, 1877, w + 362],
[1877, 1877, w + 1514],
[1889, 1889, -14*w - 29],
[1889, 1889, 14*w - 43],
[1901, 1901, -28*w + 135],
[1901, 1901, 18*w - 67],
[1913, 1913, w + 898],
[1913, 1913, w + 1014],
[1949, 1949, 4*w - 49],
[1949, 1949, -4*w - 45],
[1951, 1951, -16*w - 39],
[1951, 1951, 16*w - 55],
[1979, 1979, -12*w - 13],
[1979, 1979, 12*w - 25],
[1987, 1987, w + 943],
[1987, 1987, w + 1043],
[1997, 1997, w + 236],
[1997, 1997, w + 1760],
[1999, 1999, 32*w - 137],
[1999, 1999, 32*w + 105],
[2011, 2011, 20*w - 77],
[2011, 2011, 20*w + 57],
[2017, 2017, w + 854],
[2017, 2017, w + 1162],
[2029, 2029, 14*w - 41],
[2029, 2029, -14*w - 27],
[2081, 2081, 30*w + 97],
[2081, 2081, 30*w - 127],
[2087, 2087, w + 904],
[2087, 2087, w + 1182],
[2089, 2089, 22*w - 87],
[2089, 2089, 22*w + 65],
[2113, 2113, w + 331],
[2113, 2113, w + 1781],
[2129, 2129, -18*w - 47],
[2129, 2129, 18*w - 65],
[2131, 2131, 28*w - 117],
[2131, 2131, 18*w - 95],
[2137, 2137, w + 65],
[2137, 2137, w + 2071],
[2141, 2141, -4*w - 47],
[2141, 2141, 4*w - 51],
[2143, 2143, w + 843],
[2143, 2143, w + 1299],
[2153, 2153, w + 564],
[2153, 2153, w + 1588],
[2161, 2161, 14*w - 39],
[2161, 2161, -14*w - 25],
[2203, 2203, w + 66],
[2203, 2203, w + 2136],
[2239, 2239, 2*w - 49],
[2239, 2239, -2*w - 47],
[2243, 2243, w + 792],
[2243, 2243, w + 1450],
[2267, 2267, w + 598],
[2267, 2267, w + 1668],
[2273, 2273, w + 780],
[2273, 2273, w + 1492],
[2293, 2293, w + 135],
[2293, 2293, w + 2157],
[2311, 2311, 16*w - 51],
[2311, 2311, -16*w - 35],
[2333, 2333, w + 686],
[2333, 2333, w + 1646],
[2339, 2339, 12*w - 7],
[2339, 2339, 12*w - 5],
[2341, 2341, 4*w - 53],
[2341, 2341, -4*w - 49],
[2347, 2347, w + 966],
[2347, 2347, w + 1380],
[2377, 2377, w + 668],
[2377, 2377, w + 1708],
[2389, 2389, 22*w + 63],
[2389, 2389, 22*w - 85],
[2423, 2423, w + 260],
[2423, 2423, w + 2162],
[2437, 2437, w + 899],
[2437, 2437, w + 1537],
[2441, 2441, -8*w - 55],
[2441, 2441, 8*w - 63],
[2477, 2477, w + 70],
[2477, 2477, w + 2406],
[2503, 2503, w + 447],
[2503, 2503, w + 2055],
[2521, 2521, 26*w + 79],
[2521, 2521, 26*w - 105],
[2531, 2531, 20*w - 73],
[2531, 2531, 20*w + 53],
[2539, 2539, 28*w - 115],
[2539, 2539, 22*w - 113],
[2543, 2543, w + 1180],
[2543, 2543, w + 1362],
[2549, 2549, -4*w - 51],
[2549, 2549, 4*w - 55],
[2551, 2551, -6*w - 53],
[2551, 2551, 6*w - 59],
[2591, 2591, -14*w - 69],
[2591, 2591, 14*w - 83],
[2593, 2593, w + 619],
[2593, 2593, w + 1973],
[2609, 2609, -14*w - 17],
[2609, 2609, 14*w - 31],
[2633, 2633, w + 1248],
[2633, 2633, w + 1384],
[2647, 2647, w + 544],
[2647, 2647, w + 2102],
[2657, 2657, w + 1117],
[2657, 2657, w + 1539],
[2663, 2663, w + 1174],
[2663, 2663, w + 1488],
[2683, 2683, w + 394],
[2683, 2683, w + 2288],
[2693, 2693, w + 73],
[2693, 2693, w + 2619],
[2729, 2729, 38*w + 125],
[2729, 2729, 38*w - 163],
[2731, 2731, 10*w - 71],
[2731, 2731, -10*w - 61],
[2767, 2767, w + 74],
[2767, 2767, w + 2692],
[2777, 2777, w + 900],
[2777, 2777, w + 1876],
[2791, 2791, -16*w - 29],
[2791, 2791, 16*w - 45],
[2797, 2797, w + 442],
[2797, 2797, w + 2354],
[2803, 2803, w + 665],
[2803, 2803, w + 2137],
[2809, 53, -53],
[2851, 2851, 2*w - 55],
[2851, 2851, -2*w - 53],
[2861, 2861, 14*w - 25],
[2861, 2861, -14*w - 11],
[2897, 2897, w + 304],
[2897, 2897, w + 2592],
[2909, 2909, 34*w + 109],
[2909, 2909, 34*w - 143],
[2917, 2917, w + 1109],
[2917, 2917, w + 1807],
[2927, 2927, w + 924],
[2927, 2927, w + 2002],
[2939, 2939, 28*w + 85],
[2939, 2939, 28*w - 113],
[2953, 2953, w + 1404],
[2953, 2953, w + 1548],
[2957, 2957, w + 575],
[2957, 2957, w + 2381],
[2999, 2999, -10*w - 63],
[2999, 2999, 10*w - 73],
[3019, 3019, -20*w - 49],
[3019, 3019, 20*w - 69],
[3023, 3023, w + 396],
[3023, 3023, w + 2626],
[3037, 3037, w + 623],
[3037, 3037, w + 2413],
[3041, 3041, -14*w - 5],
[3041, 3041, 14*w - 19],
[3083, 3083, w + 675],
[3083, 3083, w + 2407],
[3119, 3119, 24*w - 91],
[3119, 3119, 24*w + 67],
[3121, 3121, -14*w - 1],
[3121, 3121, 14*w - 15],
[3167, 3167, w + 1082],
[3167, 3167, w + 2084],
[3169, 3169, 14*w - 11],
[3169, 3169, 14*w - 3],
[3181, 3181, 14*w - 9],
[3181, 3181, 14*w - 5],
[3187, 3187, w + 252],
[3187, 3187, w + 2934],
[3203, 3203, w + 299],
[3203, 3203, w + 2903],
[3217, 3217, w + 561],
[3217, 3217, w + 2655],
[3221, 3221, 4*w - 61],
[3221, 3221, -4*w - 57],
[3251, 3251, -20*w - 47],
[3251, 3251, 20*w - 67],
[3257, 3257, w + 477],
[3257, 3257, w + 2779],
[3259, 3259, -6*w - 59],
[3259, 3259, 6*w - 65],
[3299, 3299, 2*w - 59],
[3299, 3299, -2*w - 57],
[3301, 3301, 38*w + 123],
[3301, 3301, 38*w - 161],
[3307, 3307, w + 788],
[3307, 3307, w + 2518],
[3313, 3313, w + 81],
[3313, 3313, w + 3231],
[3319, 3319, 16*w - 37],
[3319, 3319, -16*w - 21],
[3323, 3323, w + 1557],
[3323, 3323, w + 1765],
[3329, 3329, 18*w - 53],
[3329, 3329, -18*w - 35],
[3331, 3331, 28*w - 111],
[3331, 3331, 28*w + 83],
[3343, 3343, w + 967],
[3343, 3343, w + 2375],
[3347, 3347, w + 1497],
[3347, 3347, w + 1849],
[3371, 3371, 36*w - 151],
[3371, 3371, 22*w - 117],
[3373, 3373, w + 641],
[3373, 3373, w + 2731],
[3389, 3389, 30*w + 91],
[3389, 3389, 30*w - 121],
[3413, 3413, w + 1165],
[3413, 3413, w + 2247],
[3449, 3449, -8*w - 63],
[3449, 3449, 8*w - 71],
[3461, 3461, -4*w - 59],
[3461, 3461, 4*w - 63],
[3463, 3463, w + 1318],
[3463, 3463, w + 2144],
[3481, 59, -59],
[3511, 3511, 6*w - 67],
[3511, 3511, -6*w - 61],
[3517, 3517, w + 335],
[3517, 3517, w + 3181],
[3539, 3539, 14*w - 89],
[3539, 3539, -14*w - 75],
[3547, 3547, w + 168],
[3547, 3547, w + 3378],
[3557, 3557, w + 1699],
[3557, 3557, w + 1857],
[3559, 3559, 10*w - 77],
[3559, 3559, -10*w - 67],
[3571, 3571, 44*w + 145],
[3571, 3571, 18*w + 85],
[3583, 3583, w + 682],
[3583, 3583, w + 2900],
[3593, 3593, w + 864],
[3593, 3593, w + 2728],
[3607, 3607, w + 502],
[3607, 3607, w + 3104],
[3631, 3631, 16*w - 31],
[3631, 3631, -16*w - 15],
[3673, 3673, w + 1010],
[3673, 3673, w + 2662],
[3677, 3677, w + 1406],
[3677, 3677, w + 2270],
[3691, 3691, -20*w - 43],
[3691, 3691, 20*w - 63],
[3697, 3697, w + 1571],
[3697, 3697, w + 2125],
[3701, 3701, 42*w - 179],
[3701, 3701, 42*w + 137],
[3709, 3709, 4*w - 65],
[3709, 3709, -4*w - 61],
[3719, 3719, -16*w - 13],
[3719, 3719, 16*w - 29],
[3733, 3733, w + 86],
[3733, 3733, w + 3646],
[3761, 3761, 16*w - 97],
[3761, 3761, -16*w - 81],
[3769, 3769, -22*w - 53],
[3769, 3769, 22*w - 75],
[3779, 3779, 2*w - 63],
[3779, 3779, -2*w - 61],
[3803, 3803, w + 610],
[3803, 3803, w + 3192],
[3821, 3821, -18*w - 29],
[3821, 3821, 18*w - 47],
[3833, 3833, w + 1834],
[3833, 3833, w + 1998],
[3851, 3851, -10*w - 69],
[3851, 3851, 10*w - 79],
[3853, 3853, w + 614],
[3853, 3853, w + 3238],
[3863, 3863, w + 838],
[3863, 3863, w + 3024],
[3907, 3907, w + 1883],
[3907, 3907, w + 2023],
[3929, 3929, -40*w + 193],
[3929, 3929, 26*w - 97],
[3947, 3947, w + 1028],
[3947, 3947, w + 2918],
[3967, 3967, w + 1433],
[3967, 3967, w + 2533],
[4001, 4001, 8*w - 75],
[4001, 4001, -8*w - 67],
[4021, 4021, 22*w - 73],
[4021, 4021, -22*w - 51],
[4079, 4079, -16*w - 1],
[4079, 4079, 16*w - 17],
[4091, 4091, 28*w + 79],
[4091, 4091, 28*w - 107],
[4093, 4093, w + 1210],
[4093, 4093, w + 2882],
[4099, 4099, 20*w - 59],
[4099, 4099, -20*w - 39],
[4111, 4111, 16*w - 15],
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[9157, 9157, w + 6273],
[9161, 9161, 8*w - 105],
[9161, 9161, -8*w - 97],
[9173, 9173, w + 4194],
[9173, 9173, w + 4978],
[9181, 9181, 34*w - 115],
[9181, 9181, -34*w - 81],
[9221, 9221, -26*w - 29],
[9221, 9221, 26*w - 55],
[9239, 9239, 24*w - 23],
[9239, 9239, 24*w - 1],
[9277, 9277, w + 953],
[9277, 9277, w + 8323],
[9281, 9281, 66*w - 281],
[9281, 9281, 66*w + 215],
[9293, 9293, w + 1756],
[9293, 9293, w + 7536],
[9311, 9311, 24*w - 19],
[9311, 9311, 24*w - 5],
[9323, 9323, w + 1066],
[9323, 9323, w + 8256],
[9397, 9397, w + 1218],
[9397, 9397, w + 8178],
[9421, 9421, 46*w + 135],
[9421, 9421, 46*w - 181],
[9433, 9433, w + 2210],
[9433, 9433, w + 7222],
[9439, 9439, 64*w - 271],
[9439, 9439, 64*w + 207],
[9461, 9461, 28*w - 163],
[9461, 9461, -28*w - 135],
[9491, 9491, 28*w - 71],
[9491, 9491, -28*w - 43],
[9497, 9497, w + 2522],
[9497, 9497, w + 6974],
[9539, 9539, 2*w - 99],
[9539, 9539, -2*w - 97],
[9547, 9547, w + 817],
[9547, 9547, w + 8729],
[9551, 9551, 48*w - 191],
[9551, 9551, 48*w + 143],
[9587, 9587, w + 3938],
[9587, 9587, w + 5648],
[9613, 9613, w + 1393],
[9613, 9613, w + 8219],
[9619, 9619, -18*w - 113],
[9619, 9619, 18*w - 131],
[9629, 9629, -20*w - 117],
[9629, 9629, 20*w - 137],
[9649, 9649, 50*w + 151],
[9649, 9649, 50*w - 201],
[9677, 9677, w + 3318],
[9677, 9677, w + 6358],
[9689, 9689, 26*w - 49],
[9689, 9689, -26*w - 23],
[9721, 9721, 40*w - 209],
[9721, 9721, 58*w - 241],
[9743, 9743, w + 3061],
[9743, 9743, w + 6681],
[9749, 9749, 44*w - 225],
[9749, 9749, 54*w - 221],
[9787, 9787, w + 523],
[9787, 9787, w + 9263],
[9811, 9811, -14*w - 107],
[9811, 9811, 14*w - 121],
[9817, 9817, w + 560],
[9817, 9817, w + 9256],
[9829, 9829, -26*w - 21],
[9829, 9829, 26*w - 47],
[9833, 9833, w + 280],
[9833, 9833, w + 9552],
[9851, 9851, 44*w - 169],
[9851, 9851, 44*w + 125],
[9871, 9871, -66*w + 317],
[9871, 9871, 40*w - 147],
[9887, 9887, w + 3389],
[9887, 9887, w + 6497],
[9929, 9929, 70*w - 299],
[9929, 9929, 70*w + 229],
[9931, 9931, 28*w - 67],
[9931, 9931, -28*w - 39],
[9941, 9941, -4*w - 99],
[9941, 9941, 4*w - 103],
[9949, 9949, 34*w - 111],
[9949, 9949, -34*w - 77],
[9973, 9973, w + 2785],
[9973, 9973, w + 7187]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [1, 0, 2, 0, -1, 5, 0, 10, -5, 0, 0, -3, 13, 3, 2, -5, 12, -2, 2, -8, 8, -9, -11, 8, 14, -3, 3, 1, -12, 18, 2, -8, -6, 6, 3, 17, 8, -21, 6, 10, -6, 5, 22, 20, -10, -22, 12, 18, 2, 14, -2, 20, -11, 17, -12, -4, 22, 14, -21, 18, 9, -18, 8, 8, 13, -22, -6, -32, -20, 16, -30, 35, -21, 21, -21, 11, -6, -28, -36, 22, 32, -4, -14, -42, -31, 37, 29, 7, -14, 14, 6, -14, -29, 30, 24, 1, -31, 38, -2, -38, 40, 27, 47, -12, -18, 22, -12, 29, -21, -14, 34, -18, 2, -7, 33, 31, 7, -12, -24, -11, -39, 15, 15, -10, 42, 16, -24, 36, -7, 28, 14, 18, 14, -29, 40, 19, 54, -12, 17, -41, 21, -26, -2, -4, -50, -21, -41, -36, -38, 33, -49, 28, -26, 50, -20, -60, 39, 38, -17, -16, 46, -30, 54, 8, -48, 39, -44, 4, -4, -28, 36, 34, 2, 46, 46, -39, -5, -18, 47, -56, 18, -20, -30, -50, 15, 21, -15, -21, 20, -4, 49, -21, 40, -48, -64, 2, -10, 54, -32, 38, 65, 43, 0, -52, -62, 13, 28, 37, -42, -3, 3, -7, 32, -8, -55, -38, 4, -44, -10, 5, 54, -34, -10, 46, -66, -29, 71, -15, -36, -66, 50, 34, -42, 62, 43, -35, 36, -22, -64, 12, 20, -34, -40, 1, 16, -78, 16, -55, -34, -23, 45, -6, 32, 53, 62, 74, 53, 18, -68, 40, -28, -41, 11, 25, 31, -43, -37, -14, -6, 10, -34, -7, -31, -49, -63, -60, -58, -13, -36, -11, -62, 4, 60, 48, -11, -33, 33, 22, 36, -62, 10, 25, -60, -10, -48, -48, -76, 52, 42, -18, -6, 24, 28, 80, -30, 8, -54, 61, -41, -65, -8, -9, 45, -89, -88, -6, 38, -29, 72, -50, 18, 62, -90, 3, -53, 36, -38, 32, 22, -55, -76, -21, -44, 30, -81, -4, -31, 36, -56, 36, 28, -46, 86, 50, -58, -16, 12, 16, 6, 53, 28, 11, 78, -52, 17, 10, -67, 20, -34, 70, 63, 41, -72, -12, -55, -85, 31, -54, 60, 40, 66, 12, 73, 12, -90, -6, -4, 70, 18, 44, 52, -12, -34, -14, 96, -36, -13, -48, 2, 12, 15, -27, 41, -4, -22, -54, 18, 87, 22, 52, -40, -33, -36, 58, -94, -20, 4, 2, 20, -31, 39, 22, -20, -43, -21, 70, -52, 37, 5, 22, 28, -35, -39, 40, -8, 10, 46, 12, 60, 96, 24, 60, -34, -36, -40, 8, -68, -78, -3, 73, -41, -6, -50, -50, -70, 49, 57, 66, 47, 79, -31, -13, -100, 1, 0, 96, -4, -9, -94, 60, 6, -22, 42, 26, 46, -78, 16, 10, -29, 29, -83, -38, -8, -28, -16, 68, 41, 22, 51, -76, 0, 24, -67, 48, 28, -16, -72, 44, 70, -15, 21, -74, 34, -55, 102, -83, 11, 37, 59, -53, 74, -88, 50, -44, 72, -24, -76, -52, 48, -72, -94, -42, 60, 48, 9, -16, 96, -38, -39, -5, -8, 68, -100, -70, -96, 54, 30, 68, 64, -22, 45, 39, 45, -5, -82, -40, -56, -12, 90, -63, -121, -67, -6, -72, -92, -79, -91, -93, 54, 24, 28, 100, 0, -4, -63, 87, -7, 59, -45, 47, 44, 33, 80, -77, -10, 94, 14, 26, 66, -118, 14, -76, -2, -50, -86, -122, 118, -86, 48, 33, -62, 64, 90, 92, 86, -96, 62, 66, 6, -6, -42, -7, 4, -107, -46, 84, -34, 58, -69, 49, -51, -41, 14, 110, -8, -115, 122, 36, -26, -48, 98, -52, -63, 103, 22, 46, -52, -6, 70, 121, -76, 8, -25, 51, -64, -41, -82, 94, 16, -30, -12, -78, 3, -79, -96, -99, -128, 20, -43, 33, -10, -100, -16, 132, 60, 18, 54, -73, -62, 26, 67, -51, 82, -68, -110, 37, -98, -110, 72, 24, 6, 10, -41, -24, -76, -70, -4, -14, 1, -82, -86, -6, -105, 12, -71, 41, 31, -101, -37, -34, 104, -136, -15, 18, 22, -110, 34, 103, -76, 28, -36, -62, 32, -61, -72, -64, 96, 22, 49, -76, -36, -100, 64, 103, -49, 114, -54, -48, 0, -45, 73, -42, -48, 21, 60, 33, 98, -51, 93, -20, 130, -140, -66, -96, -108, -32, -86, -108, -103, 107, -99, 63, 16, -46, 54, -15, -138, -42, 50, -55, 25, -48, -126, 54, 34, -72, -15, 48, -112, 106, -112, 20, 48, 26, -6, 42, 29, -109, 28, -126, -142, -1, 35, 9, 64, 27, -90, -8, -84, 123, -45, -100, 151, 53, -126, 28, 14, 104, -52, 92, -116, 8, -6, 79, -44, -35, -85, -15, 25, 73, -31, -54, -72, -124, -76, -109, -3, 102, -34, 56, -16, -42, -51, -84, -65, -136, 86, 151, -31, -38, -86, 100, 35, 66, -70, 40, 34, 144, 9, -145, -49, 87, -128, 122, -90, 21, 78, -138, -86, 65, 94, -9, 16, -81, 107, 26, -74, -32, -8, 125, 7, 21, -41, 54, -12, -53, 20, 53, 59, 90, -57, 132, 2, 0, 106, -10, -108, 10, 146, 25, 97, -157, -70, -123, 123, -2, 84, -16, -65, 8, -36, 135, 131, 96, -56, 74, -91, -34, 48, -127, 52, 114, 30, -87, 48, 122, -24, -78, -42, -1, 0, -62, -128, -162, -66, -10, -28, 107, 9, 162, 64, -105, -87, 40, 96, 102, -149, 40, 110, 20, -130, 47, 132, -136, -23, -52, 10, 28, -34, -34, 52, 57, 99, 100, -84, 135, 152, 113, -23, 18, -36, -111, 27, -76, 72, -34, -26, -108, -162, -106, -28, 132, -158, -50, -55, 137, -119, -47, 61, -105, -148, -129, 61, -20, 4, -72, -44, -35, -83, -110, 34, -9, 64, -93, 161, 109, -46, -51, -7, -60, 90, -104, -12, -108, 128, 64, 41, -14, -24, -114, -96, 83, 57, -3, 95, -44, 44, -100, 44, 33, -75, 5, 51, -75, 66, 98, 68, -168, 102, -86, -41, 6, -40, -54, -46, 164, 54, -122, 45, 74, -74, -95, 36, -10, -58, 10, -72, 128, -21, 24, -7, 118, -54, 152, -94, 102, 108, 42, 84, -27, 151, -18, -24, 142, -127, -64, -74, -154, 40, 136, -10, 117, -46, -82, 30, 115, -132, -106, -99, 96, 119, -35, -104, 20, -63, 63, -12, 80, -55, -77, -94, 98, 63, -55, -57, -47, -21, -32, -105, -127, -79, 123, 149, -87, -39, -36, 18, -20, 141, -79, 102, 12, 140, -3, -91, -122, -138, 16, -105, 56, -117, -145, -55, 149, -126, -24, 180, -160, 53, 94, 113, -43, 4, 140, -124, 150, -45, 7, -87, 84, -110, 169, -106, 50, 104, 60, -56, 124, -44, 113, 74, -135, -113, 131, -160, -22, 42, 10, -120, -14, 36, 42, 18, -70, 182, -70, 153, 38, -172, 76, 83, -125, -167, 102, -102, -128, -74, 98, -79, 118, 100, 178, 88, 138, -75, 9, -88, 106, 79, -170, -14, -158, 130, -4, 137, -167, 98, 78, 98, -18, 42, 50, 89, 137, 52, 62, -33, 19, 132, 110, 114, -22, 135, -105, -35, -13, 64, -36, 25, 6, 45, -121, 104, -11, -137, -31, -41, -147, 106, -72, -56, 50, -71, 29, 134, 34, -6, 92, 174, 80, -32, -53, -5, -23, -160, 52, 42, -5, -61, 167, 88, 112];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, w + 1}>] := -1;
ALEigenvalues[ideal<ZF | {7, 7, 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;