/*
  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![-24, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {3, 3, 2*w + 9}>;

primesArray := [
[2, 2, 7*w - 38],
[2, 2, -7*w - 31],
[3, 3, 2*w + 9],
[3, 3, 2*w - 11],
[11, 11, -12*w + 65],
[11, 11, -12*w - 53],
[25, 5, 5],
[31, 31, 8*w - 43],
[31, 31, 8*w + 35],
[43, 43, 54*w + 239],
[43, 43, -54*w + 293],
[47, 47, 2*w - 13],
[47, 47, -2*w - 11],
[49, 7, -7],
[53, 53, 4*w + 19],
[53, 53, 4*w - 23],
[61, 61, 2*w - 7],
[61, 61, -2*w - 5],
[73, 73, 22*w + 97],
[73, 73, -22*w + 119],
[79, 79, -30*w + 163],
[79, 79, 30*w + 133],
[89, 89, 6*w + 25],
[89, 89, -6*w + 31],
[97, 97, 2*w - 1],
[101, 101, 20*w - 109],
[101, 101, 20*w + 89],
[103, 103, 64*w + 283],
[103, 103, -64*w + 347],
[109, 109, -50*w + 271],
[109, 109, -50*w - 221],
[113, 113, -18*w + 97],
[113, 113, 18*w + 79],
[151, 151, 6*w + 29],
[151, 151, 6*w - 35],
[163, 163, -4*w - 13],
[163, 163, 4*w - 17],
[167, 167, 144*w - 781],
[167, 167, 134*w - 727],
[169, 13, -13],
[191, 191, 34*w + 151],
[191, 191, 34*w - 185],
[193, 193, 24*w - 131],
[193, 193, -24*w - 107],
[197, 197, 6*w - 29],
[197, 197, -6*w - 23],
[227, 227, 2*w - 19],
[227, 227, -2*w - 17],
[229, 229, 12*w + 55],
[229, 229, 12*w - 67],
[241, 241, 96*w + 425],
[241, 241, -96*w + 521],
[269, 269, -148*w + 803],
[269, 269, 148*w + 655],
[283, 283, 6*w - 37],
[283, 283, 6*w + 31],
[289, 17, -17],
[293, 293, 102*w + 451],
[293, 293, -102*w + 553],
[307, 307, 4*w - 11],
[307, 307, -4*w - 7],
[313, 313, -72*w + 391],
[313, 313, 72*w + 319],
[353, 353, 16*w + 73],
[353, 353, 16*w - 89],
[361, 19, -19],
[379, 379, -4*w - 1],
[379, 379, 4*w - 5],
[389, 389, -6*w - 19],
[389, 389, 6*w - 25],
[397, 397, 14*w + 59],
[397, 397, -14*w + 73],
[419, 419, -62*w + 337],
[419, 419, 62*w + 275],
[421, 421, -38*w + 205],
[421, 421, 38*w + 167],
[431, 431, 14*w + 65],
[431, 431, -14*w + 79],
[449, 449, 32*w - 175],
[449, 449, 32*w + 143],
[461, 461, -18*w + 95],
[461, 461, -18*w - 77],
[463, 463, -8*w + 37],
[463, 463, -8*w - 29],
[467, 467, 12*w + 49],
[467, 467, 12*w - 61],
[479, 479, 2*w - 25],
[479, 479, -2*w - 23],
[487, 487, 42*w + 187],
[487, 487, 42*w - 229],
[491, 491, 10*w + 49],
[491, 491, 10*w - 59],
[503, 503, 168*w + 743],
[503, 503, -168*w + 911],
[509, 509, 30*w + 131],
[509, 509, 30*w - 161],
[521, 521, 176*w + 779],
[521, 521, -176*w + 955],
[529, 23, -23],
[547, 547, -52*w - 229],
[547, 547, -52*w + 281],
[557, 557, 76*w + 337],
[557, 557, -76*w + 413],
[571, 571, -6*w - 35],
[571, 571, 6*w - 41],
[593, 593, -126*w + 683],
[593, 593, -126*w - 557],
[607, 607, 18*w - 101],
[607, 607, 18*w + 83],
[613, 613, 36*w - 197],
[613, 613, 36*w + 161],
[617, 617, -6*w - 13],
[617, 617, 6*w - 19],
[631, 631, 294*w + 1301],
[631, 631, -294*w + 1595],
[643, 643, 66*w + 293],
[643, 643, -66*w + 359],
[647, 647, -48*w + 259],
[647, 647, 48*w + 211],
[661, 661, -10*w - 37],
[661, 661, -10*w + 47],
[673, 673, 24*w + 109],
[673, 673, 24*w - 133],
[677, 677, 6*w - 17],
[677, 677, -6*w - 11],
[683, 683, 12*w - 59],
[683, 683, -12*w - 47],
[691, 691, -114*w + 619],
[691, 691, 114*w + 505],
[701, 701, -4*w - 31],
[701, 701, 4*w - 35],
[727, 727, 6*w - 43],
[727, 727, -6*w - 37],
[733, 733, 12*w + 59],
[733, 733, 12*w - 71],
[743, 743, -24*w - 103],
[743, 743, -24*w + 127],
[751, 751, -80*w - 353],
[751, 751, 80*w - 433],
[773, 773, -6*w - 7],
[773, 773, 6*w - 13],
[787, 787, -28*w + 149],
[787, 787, -28*w - 121],
[809, 809, 6*w - 11],
[809, 809, -6*w - 5],
[811, 811, 220*w + 973],
[811, 811, -220*w + 1193],
[823, 823, 8*w - 31],
[823, 823, -8*w - 23],
[829, 829, -94*w - 415],
[829, 829, 94*w - 509],
[841, 29, -29],
[857, 857, -6*w - 1],
[857, 857, 6*w - 7],
[877, 877, 206*w + 911],
[877, 877, -206*w + 1117],
[881, 881, -104*w + 565],
[881, 881, 104*w + 461],
[937, 937, 122*w - 661],
[937, 937, -122*w - 539],
[967, 967, 136*w - 737],
[967, 967, -136*w - 601],
[971, 971, 374*w + 1655],
[971, 971, -374*w + 2029],
[997, 997, 12*w - 73],
[997, 997, 12*w + 61],
[1013, 1013, 44*w - 241],
[1013, 1013, 44*w + 197],
[1019, 1019, 218*w + 965],
[1019, 1019, -218*w + 1183],
[1031, 1031, 26*w - 145],
[1031, 1031, 26*w + 119],
[1049, 1049, -8*w - 47],
[1049, 1049, 8*w - 55],
[1051, 1051, 20*w + 83],
[1051, 1051, -20*w + 103],
[1061, 1061, 90*w - 487],
[1061, 1061, -90*w - 397],
[1063, 1063, -6*w - 41],
[1063, 1063, 6*w - 47],
[1069, 1069, 180*w + 797],
[1069, 1069, -180*w + 977],
[1091, 1091, -12*w + 55],
[1091, 1091, 12*w + 43],
[1103, 1103, -72*w - 317],
[1103, 1103, -72*w + 389],
[1117, 1117, 84*w - 457],
[1117, 1117, 84*w + 373],
[1129, 1129, -10*w - 31],
[1129, 1129, 10*w - 41],
[1153, 1153, -14*w + 67],
[1153, 1153, -14*w - 53],
[1163, 1163, 74*w - 403],
[1163, 1163, 74*w + 329],
[1213, 1213, -156*w + 847],
[1213, 1213, 156*w + 691],
[1217, 1217, -232*w + 1259],
[1217, 1217, 232*w + 1027],
[1229, 1229, 388*w + 1717],
[1229, 1229, -388*w + 2105],
[1237, 1237, -108*w + 587],
[1237, 1237, 108*w + 479],
[1249, 1249, 48*w - 263],
[1249, 1249, 48*w + 215],
[1259, 1259, 22*w - 125],
[1259, 1259, 22*w + 103],
[1277, 1277, -258*w + 1399],
[1277, 1277, 258*w + 1141],
[1279, 1279, -30*w - 137],
[1279, 1279, 30*w - 167],
[1283, 1283, -12*w - 41],
[1283, 1283, -12*w + 53],
[1297, 1297, -336*w + 1823],
[1297, 1297, 336*w + 1487],
[1327, 1327, -8*w - 11],
[1327, 1327, 8*w - 19],
[1361, 1361, 16*w + 79],
[1361, 1361, 16*w - 95],
[1367, 1367, 58*w + 259],
[1367, 1367, 58*w - 317],
[1369, 37, -37],
[1423, 1423, -32*w + 169],
[1423, 1423, -32*w - 137],
[1433, 1433, 88*w + 391],
[1433, 1433, 88*w - 479],
[1439, 1439, 216*w + 955],
[1439, 1439, -216*w + 1171],
[1447, 1447, -16*w - 61],
[1447, 1447, 16*w - 77],
[1451, 1451, 60*w - 323],
[1451, 1451, 60*w + 263],
[1453, 1453, 82*w + 361],
[1453, 1453, -82*w + 443],
[1459, 1459, 78*w - 425],
[1459, 1459, 78*w + 347],
[1471, 1471, 8*w - 13],
[1471, 1471, -8*w - 5],
[1487, 1487, -146*w + 793],
[1487, 1487, 146*w + 647],
[1499, 1499, -36*w + 191],
[1499, 1499, -36*w - 155],
[1543, 1543, 8*w - 7],
[1543, 1543, 8*w - 1],
[1549, 1549, 12*w - 77],
[1549, 1549, 12*w + 65],
[1553, 1553, -114*w - 503],
[1553, 1553, 114*w - 617],
[1579, 1579, -44*w - 191],
[1579, 1579, 44*w - 235],
[1583, 1583, -48*w - 209],
[1583, 1583, 48*w - 257],
[1601, 1601, 40*w + 181],
[1601, 1601, 40*w - 221],
[1613, 1613, -78*w + 421],
[1613, 1613, 78*w + 343],
[1627, 1627, -6*w - 47],
[1627, 1627, 6*w - 53],
[1637, 1637, -4*w - 43],
[1637, 1637, 4*w - 47],
[1657, 1657, 24*w + 113],
[1657, 1657, 24*w - 137],
[1667, 1667, 2*w - 43],
[1667, 1667, -2*w - 41],
[1681, 41, -41],
[1693, 1693, -468*w + 2539],
[1693, 1693, 468*w + 2071],
[1697, 1697, -8*w - 53],
[1697, 1697, 8*w - 61],
[1699, 1699, 222*w + 983],
[1699, 1699, -222*w + 1205],
[1721, 1721, 160*w + 709],
[1721, 1721, -160*w + 869],
[1747, 1747, 18*w + 89],
[1747, 1747, 18*w - 107],
[1777, 1777, -312*w + 1693],
[1777, 1777, 312*w + 1381],
[1789, 1789, -34*w + 179],
[1789, 1789, -34*w - 145],
[1811, 1811, 12*w - 47],
[1811, 1811, -12*w - 35],
[1831, 1831, 6*w - 55],
[1831, 1831, -6*w - 49],
[1847, 1847, 38*w + 173],
[1847, 1847, 38*w - 211],
[1861, 1861, 70*w + 307],
[1861, 1861, 70*w - 377],
[1867, 1867, -404*w + 2191],
[1867, 1867, 404*w + 1787],
[1879, 1879, -198*w + 1075],
[1879, 1879, 198*w + 877],
[1907, 1907, -274*w + 1487],
[1907, 1907, 274*w + 1213],
[1913, 1913, -282*w + 1529],
[1913, 1913, 282*w + 1247],
[1931, 1931, 10*w - 71],
[1931, 1931, -10*w - 61],
[1949, 1949, 66*w - 355],
[1949, 1949, 66*w + 289],
[1951, 1951, 66*w + 295],
[1951, 1951, 66*w - 361],
[1973, 1973, -42*w - 181],
[1973, 1973, -42*w + 223],
[1987, 1987, 124*w - 671],
[1987, 1987, -124*w - 547],
[1993, 1993, 26*w + 107],
[1993, 1993, -26*w + 133],
[2029, 2029, -46*w + 245],
[2029, 2029, -46*w - 199],
[2039, 2039, -240*w + 1301],
[2039, 2039, -240*w - 1061],
[2053, 2053, 58*w - 311],
[2053, 2053, -58*w - 253],
[2069, 2069, 54*w - 289],
[2069, 2069, -54*w - 235],
[2081, 2081, -18*w - 67],
[2081, 2081, -18*w + 85],
[2087, 2087, 24*w + 97],
[2087, 2087, 24*w - 121],
[2099, 2099, -564*w - 2495],
[2099, 2099, -564*w + 3059],
[2131, 2131, 20*w - 97],
[2131, 2131, -20*w - 77],
[2137, 2137, 96*w + 427],
[2137, 2137, 96*w - 523],
[2143, 2143, 18*w - 109],
[2143, 2143, 18*w + 91],
[2161, 2161, 24*w + 115],
[2161, 2161, 24*w - 139],
[2207, 2207, 2*w - 49],
[2207, 2207, -2*w - 47],
[2213, 2213, 4*w - 53],
[2213, 2213, -4*w - 49],
[2237, 2237, -188*w + 1021],
[2237, 2237, 188*w + 833],
[2239, 2239, 16*w + 55],
[2239, 2239, -16*w + 71],
[2243, 2243, 34*w - 191],
[2243, 2243, 34*w + 157],
[2267, 2267, -12*w - 29],
[2267, 2267, 12*w - 41],
[2281, 2281, -10*w - 7],
[2281, 2281, 10*w - 17],
[2293, 2293, -362*w + 1963],
[2293, 2293, 362*w + 1601],
[2297, 2297, 102*w + 449],
[2297, 2297, -102*w + 551],
[2339, 2339, 130*w + 577],
[2339, 2339, -130*w + 707],
[2371, 2371, 28*w - 143],
[2371, 2371, 28*w + 115],
[2377, 2377, -166*w - 733],
[2377, 2377, 166*w - 899],
[2381, 2381, 18*w + 65],
[2381, 2381, -18*w + 83],
[2389, 2389, -10*w - 1],
[2389, 2389, 10*w - 11],
[2393, 2393, 32*w + 149],
[2393, 2393, 32*w - 181],
[2417, 2417, 8*w - 67],
[2417, 2417, -8*w - 59],
[2423, 2423, -456*w + 2473],
[2423, 2423, 456*w + 2017],
[2437, 2437, -12*w - 71],
[2437, 2437, 12*w - 83],
[2441, 2441, 16*w + 85],
[2441, 2441, 16*w - 101],
[2447, 2447, 302*w + 1337],
[2447, 2447, -302*w + 1639],
[2473, 2473, 264*w + 1169],
[2473, 2473, -264*w + 1433],
[2521, 2521, -22*w - 85],
[2521, 2521, 22*w - 107],
[2531, 2531, -12*w - 25],
[2531, 2531, 12*w - 37],
[2549, 2549, 52*w + 235],
[2549, 2549, 52*w - 287],
[2557, 2557, 194*w - 1051],
[2557, 2557, -194*w - 857],
[2617, 2617, -38*w - 161],
[2617, 2617, -38*w + 199],
[2621, 2621, 20*w - 121],
[2621, 2621, 20*w + 101],
[2663, 2663, 22*w + 109],
[2663, 2663, 22*w - 131],
[2683, 2683, 292*w + 1291],
[2683, 2683, -292*w + 1583],
[2689, 2689, -240*w + 1303],
[2689, 2689, 240*w + 1063],
[2707, 2707, -236*w - 1043],
[2707, 2707, -236*w + 1279],
[2713, 2713, -278*w + 1507],
[2713, 2713, -278*w - 1229],
[2719, 2719, 616*w - 3341],
[2719, 2719, -616*w - 2725],
[2741, 2741, -316*w + 1715],
[2741, 2741, 316*w + 1399],
[2749, 2749, 12*w - 85],
[2749, 2749, -12*w - 73],
[2777, 2777, -42*w + 221],
[2777, 2777, -42*w - 179],
[2789, 2789, 68*w - 373],
[2789, 2789, 68*w + 305],
[2791, 2791, 112*w + 493],
[2791, 2791, -112*w + 605],
[2797, 2797, -36*w + 203],
[2797, 2797, 36*w + 167],
[2801, 2801, 162*w - 877],
[2801, 2801, -162*w - 715],
[2819, 2819, 2*w - 55],
[2819, 2819, -2*w - 53],
[2837, 2837, -628*w + 3407],
[2837, 2837, 628*w + 2779],
[2857, 2857, 192*w + 851],
[2857, 2857, -192*w + 1043],
[2861, 2861, -4*w - 55],
[2861, 2861, 4*w - 59],
[2879, 2879, 50*w - 277],
[2879, 2879, 50*w + 227],
[2909, 2909, 414*w + 1831],
[2909, 2909, -414*w + 2245],
[2953, 2953, 86*w - 463],
[2953, 2953, 86*w + 377],
[2957, 2957, -18*w + 79],
[2957, 2957, -18*w - 61],
[2963, 2963, -12*w - 17],
[2963, 2963, 12*w - 29],
[2971, 2971, -6*w - 59],
[2971, 2971, 6*w - 65],
[2999, 2999, 144*w - 779],
[2999, 2999, -144*w - 635],
[3001, 3001, 576*w - 3125],
[3001, 3001, 602*w - 3265],
[3011, 3011, -108*w + 583],
[3011, 3011, 108*w + 475],
[3019, 3019, -42*w - 193],
[3019, 3019, 42*w - 235],
[3023, 3023, -158*w + 859],
[3023, 3023, 158*w + 701],
[3061, 3061, 420*w + 1859],
[3061, 3061, -420*w + 2279],
[3079, 3079, 40*w + 169],
[3079, 3079, -40*w + 209],
[3137, 3137, 78*w + 341],
[3137, 3137, 78*w - 419],
[3169, 3169, 72*w - 395],
[3169, 3169, 72*w + 323],
[3203, 3203, 12*w - 23],
[3203, 3203, -12*w - 11],
[3209, 3209, 8*w - 73],
[3209, 3209, -8*w - 65],
[3217, 3217, -62*w - 269],
[3217, 3217, 62*w - 331],
[3251, 3251, 372*w + 1645],
[3251, 3251, -372*w + 2017],
[3271, 3271, 54*w + 245],
[3271, 3271, 54*w - 299],
[3299, 3299, -98*w - 437],
[3299, 3299, 98*w - 535],
[3301, 3301, 60*w + 271],
[3301, 3301, 60*w - 331],
[3307, 3307, 138*w - 751],
[3307, 3307, 138*w + 613],
[3323, 3323, 12*w - 19],
[3323, 3323, -12*w - 7],
[3329, 3329, 30*w + 121],
[3329, 3329, 30*w - 151],
[3331, 3331, -140*w + 757],
[3331, 3331, -140*w - 617],
[3347, 3347, -14*w - 83],
[3347, 3347, 14*w - 97],
[3359, 3359, -24*w + 115],
[3359, 3359, -24*w - 91],
[3371, 3371, -12*w - 5],
[3371, 3371, 12*w - 17],
[3373, 3373, 108*w + 481],
[3373, 3373, -108*w + 589],
[3389, 3389, -172*w + 935],
[3389, 3389, 172*w + 763],
[3391, 3391, 306*w + 1355],
[3391, 3391, -306*w + 1661],
[3407, 3407, 82*w + 367],
[3407, 3407, 82*w - 449],
[3413, 3413, -246*w - 1087],
[3413, 3413, 246*w - 1333],
[3449, 3449, 330*w + 1459],
[3449, 3449, -330*w + 1789],
[3457, 3457, -14*w - 29],
[3457, 3457, 14*w - 43],
[3461, 3461, 244*w + 1081],
[3461, 3461, -244*w + 1325],
[3467, 3467, 12*w - 11],
[3467, 3467, 12*w - 1],
[3481, 59, -59],
[3491, 3491, 12*w - 7],
[3491, 3491, 12*w - 5],
[3517, 3517, 708*w + 3133],
[3517, 3517, -708*w + 3841],
[3527, 3527, 96*w + 421],
[3527, 3527, 96*w - 517],
[3539, 3539, -36*w + 185],
[3539, 3539, 36*w + 149],
[3541, 3541, 36*w - 205],
[3541, 3541, 36*w + 169],
[3557, 3557, -186*w - 821],
[3557, 3557, 186*w - 1007],
[3571, 3571, -52*w - 223],
[3571, 3571, -52*w + 275],
[3581, 3581, 4*w - 65],
[3581, 3581, -4*w - 61],
[3583, 3583, 162*w + 719],
[3583, 3583, -162*w + 881],
[3593, 3593, 16*w - 107],
[3593, 3593, -16*w - 91],
[3607, 3607, -16*w - 43],
[3607, 3607, 16*w - 59],
[3613, 3613, -396*w + 2149],
[3613, 3613, 396*w + 1753],
[3637, 3637, -22*w + 101],
[3637, 3637, 22*w + 79],
[3643, 3643, -282*w + 1531],
[3643, 3643, 282*w + 1249],
[3659, 3659, -10*w - 73],
[3659, 3659, 10*w - 83],
[3677, 3677, 44*w + 203],
[3677, 3677, 44*w - 247],
[3697, 3697, 34*w - 173],
[3697, 3697, 34*w + 139],
[3719, 3719, -358*w + 1943],
[3719, 3719, 358*w + 1585],
[3733, 3733, -12*w - 79],
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[8501, 8501, 100*w + 451],
[8501, 8501, 100*w - 551],
[8527, 8527, 18*w - 137],
[8527, 8527, -18*w - 119],
[8537, 8537, 512*w + 2267],
[8537, 8537, -512*w + 2779],
[8539, 8539, 380*w - 2059],
[8539, 8539, -380*w - 1679],
[8563, 8563, 548*w + 2423],
[8563, 8563, -548*w + 2971],
[8597, 8597, 268*w + 1189],
[8597, 8597, -268*w + 1457],
[8609, 8609, 640*w + 2833],
[8609, 8609, -640*w + 3473],
[8627, 8627, -36*w - 133],
[8627, 8627, -36*w + 169],
[8629, 8629, 394*w - 2135],
[8629, 8629, -394*w - 1741],
[8641, 8641, 50*w + 203],
[8641, 8641, 50*w - 253],
[8669, 8669, 412*w + 1825],
[8669, 8669, -412*w + 2237],
[8677, 8677, 58*w + 241],
[8677, 8677, -58*w + 299],
[8681, 8681, 224*w - 1219],
[8681, 8681, 224*w + 995],
[8699, 8699, -84*w - 361],
[8699, 8699, -84*w + 445],
[8719, 8719, 520*w + 2299],
[8719, 8719, -520*w + 2819],
[8731, 8731, -6*w - 95],
[8731, 8731, 6*w - 101],
[8741, 8741, 164*w - 895],
[8741, 8741, -164*w - 731],
[8761, 8761, -422*w - 1865],
[8761, 8761, -422*w + 2287],
[8779, 8779, 138*w - 755],
[8779, 8779, 138*w + 617],
[8783, 8783, 48*w + 193],
[8783, 8783, 48*w - 241],
[8803, 8803, -436*w + 2363],
[8803, 8803, -436*w - 1927],
[8819, 8819, -684*w + 3709],
[8819, 8819, 684*w + 3025],
[8821, 8821, 22*w - 65],
[8821, 8821, -22*w - 43],
[8831, 8831, 34*w - 209],
[8831, 8831, 34*w + 175],
[8839, 8839, -464*w + 2515],
[8839, 8839, -464*w - 2051],
[8849, 8849, -1074*w - 4751],
[8849, 8849, 1074*w - 5825],
[8863, 8863, -88*w + 467],
[8863, 8863, -88*w - 379],
[8893, 8893, 204*w + 907],
[8893, 8893, 204*w - 1111],
[8923, 8923, 44*w - 217],
[8923, 8923, -44*w - 173],
[8933, 8933, 116*w - 637],
[8933, 8933, 116*w + 521],
[8951, 8951, 792*w + 3503],
[8951, 8951, -792*w + 4295],
[8971, 8971, 20*w - 37],
[8971, 8971, -20*w - 17],
[8999, 8999, 22*w - 155],
[8999, 8999, -22*w - 133],
[9013, 9013, 564*w + 2497],
[9013, 9013, -564*w + 3061],
[9029, 9029, 340*w + 1507],
[9029, 9029, -340*w + 1847],
[9043, 9043, 18*w - 139],
[9043, 9043, -18*w - 121],
[9091, 9091, -68*w + 355],
[9091, 9091, 68*w + 287],
[9109, 9109, -886*w + 4805],
[9109, 9109, 886*w + 3919],
[9127, 9127, 6*w - 103],
[9127, 9127, -6*w - 97],
[9151, 9151, 30*w - 191],
[9151, 9151, 30*w + 161],
[9161, 9161, 40*w + 199],
[9161, 9161, 40*w - 239],
[9199, 9199, 256*w - 1385],
[9199, 9199, -256*w - 1129],
[9203, 9203, 46*w - 269],
[9203, 9203, 46*w + 223],
[9209, 9209, 54*w + 221],
[9209, 9209, 54*w - 275],
[9221, 9221, 44*w + 215],
[9221, 9221, 44*w - 259],
[9227, 9227, -36*w - 131],
[9227, 9227, -36*w + 167],
[9239, 9239, -10*w - 103],
[9239, 9239, 10*w - 113],
[9277, 9277, -12*w - 107],
[9277, 9277, 12*w - 119],
[9281, 9281, -30*w - 97],
[9281, 9281, 30*w - 127],
[9311, 9311, 120*w - 643],
[9311, 9311, -120*w - 523],
[9323, 9323, 108*w - 577],
[9323, 9323, -108*w - 469],
[9337, 9337, -26*w - 71],
[9337, 9337, 26*w - 97],
[9343, 9343, -112*w - 487],
[9343, 9343, 112*w - 599],
[9377, 9377, 642*w + 2839],
[9377, 9377, -642*w + 3481],
[9391, 9391, 126*w - 691],
[9391, 9391, 126*w + 565],
[9397, 9397, 202*w + 889],
[9397, 9397, -202*w + 1091],
[9403, 9403, 52*w - 263],
[9403, 9403, 52*w + 211],
[9413, 9413, 4*w - 101],
[9413, 9413, -4*w - 97],
[9421, 9421, 60*w + 281],
[9421, 9421, 60*w - 341],
[9431, 9431, -72*w - 305],
[9431, 9431, -72*w + 377],
[9433, 9433, -22*w - 37],
[9433, 9433, 22*w - 59],
[9463, 9463, -872*w + 4729],
[9463, 9463, 872*w + 3857],
[9473, 9473, -8*w - 101],
[9473, 9473, 8*w - 109],
[9479, 9479, -24*w - 55],
[9479, 9479, 24*w - 79],
[9497, 9497, 1136*w + 5027],
[9497, 9497, -1136*w + 6163],
[9533, 9533, -66*w + 343],
[9533, 9533, 66*w + 277],
[9539, 9539, 180*w + 791],
[9539, 9539, -180*w + 971],
[9587, 9587, -178*w + 971],
[9587, 9587, 178*w + 793],
[9601, 9601, 120*w - 659],
[9601, 9601, 120*w + 539],
[9619, 9619, 20*w - 19],
[9619, 9619, 20*w - 1],
[9689, 9689, 208*w - 1133],
[9689, 9689, 208*w + 925],
[9697, 9697, -46*w - 181],
[9697, 9697, 46*w - 227],
[9733, 9733, 12*w - 121],
[9733, 9733, -12*w - 109],
[9743, 9743, -24*w - 53],
[9743, 9743, 24*w - 77],
[9749, 9749, 668*w + 2957],
[9749, 9749, -668*w + 3625],
[9781, 9781, 36*w + 185],
[9781, 9781, 36*w - 221],
[9791, 9791, -600*w + 3253],
[9791, 9791, 600*w + 2653],
[9803, 9803, 252*w + 1111],
[9803, 9803, -252*w + 1363],
[9829, 9829, 108*w + 487],
[9829, 9829, 108*w - 595],
[9833, 9833, 296*w + 1313],
[9833, 9833, -296*w + 1609],
[9851, 9851, 26*w - 175],
[9851, 9851, -26*w - 149],
[9859, 9859, 30*w - 193],
[9859, 9859, 30*w + 163],
[9883, 9883, -284*w - 1253],
[9883, 9883, 284*w - 1537],
[9929, 9929, -234*w - 1031],
[9929, 9929, -234*w + 1265],
[9941, 9941, -330*w - 1457],
[9941, 9941, 330*w - 1787],
[9967, 9967, 66*w + 307],
[9967, 9967, 66*w - 373],
[9973, 9973, -22*w - 31],
[9973, 9973, 22*w - 53]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [1, -2, 1, 1, -5, -5, -4, 0, 6, 7, -2, 7, -8, -12, -12, -9, 11, -7, 2, -4, 4, 1, 0, 3, 10, 15, 6, 7, -8, 1, -2, -14, 1, 16, -11, -14, -8, 12, -18, -1, 12, -18, 18, -3, -11, -20, -4, 2, 10, -8, -22, -10, -1, 14, 20, -10, -4, -30, -6, -22, -25, 10, -8, -13, 8, -6, 10, 7, -19, 14, 20, 8, -7, 11, 2, 2, 32, -10, -6, -3, -4, -13, -14, 37, -2, -26, 23, 32, 2, -40, -35, 19, 12, -30, 28, -2, 30, 6, 10, -32, 37, 9, 18, -21, 18, -42, 15, -2, -47, 36, -18, 42, -15, 12, 12, -20, 4, -5, 1, -12, 0, -5, -26, -33, -36, -16, 44, -8, -11, -40, 11, -30, -9, 34, -5, -24, -48, -5, 22, 14, -46, 49, -14, -16, 41, -16, -25, 19, -14, 38, -34, -32, 42, 30, -28, 32, -9, 36, 12, 12, -29, -38, -43, 20, 27, 33, 18, -6, -50, -2, -24, -57, 47, -4, 2, -25, 30, -12, 0, 9, -18, -30, -9, -12, -23, -32, 3, 51, 62, 53, 42, 66, -38, -32, 38, 20, 18, 30, -6, 54, -61, 23, 18, 63, -54, 12, -53, -26, -40, -7, -36, 21, -25, 59, -8, -20, -20, -2, 54, 30, 9, 48, -72, 18, 57, 27, -12, -34, 59, -20, 22, -14, -26, -8, -44, 50, 47, 55, -32, 16, -32, 32, -22, 64, 4, -50, 37, -66, 3, -19, -49, 30, -60, 6, -39, 49, 70, 18, 33, -68, 4, 58, -68, -16, -38, -44, -72, 9, 47, 14, -14, 52, -54, 9, -69, -18, -64, 2, 14, 74, 58, -8, -56, 46, -17, -47, 65, -70, -32, 40, 41, 68, 57, -18, -84, 36, -41, 16, -82, -25, -6, -51, -12, 33, -56, 76, -85, 71, -48, -87, 82, -26, 68, -58, -3, 18, -38, -20, 56, 86, -70, 29, 78, -36, -31, -70, -16, -52, -45, -3, 0, 81, -38, 52, 65, -82, -4, 8, 72, -60, 17, -25, 19, 85, 77, 68, -41, 61, 68, -76, 72, -18, 33, -60, -50, 46, -14, -17, 38, -73, -22, 44, -35, -53, -54, 96, 20, 2, -64, -52, -8, 10, 22, -32, 82, -65, -34, -34, 74, 50, -66, 90, 45, 81, -3, 54, 86, -88, -12, -39, -10, 17, 33, 24, -15, -102, 55, -74, 69, 60, 3, -36, -70, -43, -33, 42, -4, 89, -20, -20, 74, -52, -3, 39, 14, 38, 85, -53, 79, 58, -26, -17, -73, -34, -27, 45, 92, -28, -22, -40, -90, -84, -40, 11, -69, 12, 54, 18, -59, -5, 28, -68, 58, -104, 49, 37, 42, -66, 49, 76, -35, -14, -44, 76, -40, 68, -12, -12, 32, -16, 18, 87, 61, 97, -54, 30, -38, 31, -92, 52, -12, 45, -15, 12, -53, -65, 29, -34, -100, -46, -24, -54, 81, 108, 66, -60, -13, 77, 35, -40, -84, 12, -91, 5, 20, -81, -111, 84, 6, 15, 45, 71, 65, 93, 54, -68, -14, -84, 33, 7, -74, 45, -18, -73, -70, -97, 17, -106, 47, 30, -9, -81, -27, 29, -94, 53, -31, 19, -35, 106, -14, 36, -18, 113, 98, -33, 0, -11, 31, 47, -28, 95, 122, -8, -44, -61, -34, -1, 56, 44, -28, -80, 40, -22, 68, -111, -9, 42, 42, 96, 9, -66, 93, 46, -110, -8, 22, -8, -110, -15, -18, -37, -100, 47, 32, -2, -80, -84, 30, -84, 84, 29, -16, 105, -93, 9, 96, 48, -102, -36, -12, 96, -18, 108, 84, 30, 18, -26, 52, 44, -22, -110, 1, -36, 105, -9, -24, 66, 102, 44, -4, 12, -66, 69, 19, -116, -8, -32, 84, 48, -103, -52, -21, 21, -26, -35, 59, -4, 86, 8, 10, 94, -87, 36, 105, -117, -64, 11, 88, 106, 63, 120, 42, 21, -83, -113, -34, 32, -60, -30, 36, -48, 98, 2, -98, 31, -12, -48, -34, -88, 30, 39, -56, -86, 16, -53, -88, -4, -1, 99, -39, 39, 63, 108, -30, 100, 34, 122, -58, -38, 4, 109, -80, -26, 103, 108, -30, -70, -79, 78, 120, -129, 90, -43, 26, -54, 84, -81, -63, -114, -36, -116, -17, -20, 1, -18, 18, -86, -35, -28, 29, 18, -102, 122, -82, 126, -75, 69, -12, 112, -74, 71, 71, -7, 50, -116, 49, -46, -118, -38, 10, -80, 76, 25, -98, 18, 138, -93, 0, 7, -146, -42, -75, 130, 40, 25, -38, -52, 29, 7, 79, 20, 134, -144, -105, -14, 22, 49, -29, 126, 132, 108, -108, -84, 0, 15, -57, 26, -37, -58, -133, -114, -54, 69, 84, -90, -18, -72, -51, 90, 54, 90, -66, 120, 108, 80, 41, -71, -113, 1, 130, 106, -110, 107, 47, 124, -116, -118, -70, -90, -18, 119, 62, -36, -87, 16, 151, -70, -76, -36, -93, 63, 39, -28, 98, -64, -7, -25, -4, -82, -91, -93, 42, 98, 38, -129, 6, 58, -116, -46, 26, 87, -132, -54, -111, -52, -151, 54, -12, 14, 53, 42, -6, -46, -16, -28, 35, -9, 6, 58, 10, 56, -13, 30, 105, -115, -37, 130, -128, 6, 90, -101, -122, -33, 24, -51, 54, -18, -96, -54, -78, -52, -73, -116, 28, -127, -12, 120, -136, 74, -87, 81, -67, 23, -56, 49, 66, 96, -142, -100, 157, -98, 75, 51, -94, 26, -1, -19, 53, 35, -42, -48, -150, -114, -34, -46, -4, 32, -151, 59, 39, 114, 119, 2, 45, 36, 42, 63, 12, -114, -50, 64, -129, -78, 26, -76, 32, 146, -10, -7, -50, 103, -57, -60, -87, 120, -60, 12, -42, 87, -96, 75, -118, 68, 22, 64, -30, -60, -70, 50, -80, -119, 6, 39, 102, -78, -41, -32, -32, -122, -20, -17, 129, 60, 123, -102, 53, -166, 105, 102, 154, 61, 5, 44, -141, 117, -144, -36, -97, -112, -62, 16, -144, 18, 105, -63, -77, -113, -52, 83, 6, -90, -69, 57, -86, -149, -42, 72, 63, 15, -33, -69, 4, 148, -48, 24, 29, 101, -67, 59, 46, -26, 132, -63, -17, -11, 68, 80, 118, 40, -93, 6, 96, -72, 97, -2, -152, 112, -121, 20, 148, 76, -52, -118, -90, -48, -83, 166, -78, 87, -6, 78, 76, 172, -80, -62, 68, -13, 2, 62, -153, 108, -122, -116, 167, -64, -94, 2, -20, -62, 16, -50, 24, 144, 21, 60, -82, 56, -102, 90, -72, 24, -40, 50, 52, -32, -108, 54, -177, -30, 116, -154, -132, -78, -110, 124, -126, 42, -125, 121, -8, -38, 53, 26, -110, 67, 66, 39, -16, -28, -50, 40, -121, -160, -14, -170, 94, -47, 87, 72, 5, 5, 48, 78, 146, 107, 52, -182, -19, -121, 16, 31, -55, 2, 88, -65, -15, 174, 90, -24, -174, -144, -45, -42, -28, 116, 110, -166, -145, 107, -102, -138, -69, 69, 80, -190, -120, 90, 39, -162, 174, -42, -34, -100, 0, 60, -46, 155, 108, 66, 82, -44, 108, -9, -164, -74, -166, 53, -40, -106, 156, 15, 169, -20, -9, -72, 14, -166, -75, -168, 115, -146, 55, 64, -156, -75, 16, -62, -178, -1, -59, 85, -86, -113, 148, 172, -44, -62, -100, -52, 74, -100, 154, 37, -156, -120, 49, -92, 187, 100, -40, 170, 126, -150, 6, 27, 155, -61];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, 2*w + 9}>] := -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;