/*
  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![-18, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {121, 11, -11}>;

primesArray := [
[2, 2, -w - 4],
[2, 2, w - 5],
[3, 3, -4*w - 15],
[3, 3, -4*w + 19],
[19, 19, -6*w - 23],
[19, 19, 6*w - 29],
[23, 23, 14*w - 67],
[23, 23, -14*w - 53],
[25, 5, -5],
[37, 37, 2*w - 7],
[37, 37, -2*w - 5],
[41, 41, 30*w - 143],
[41, 41, 40*w - 191],
[49, 7, -7],
[61, 61, 10*w + 37],
[61, 61, -10*w + 47],
[67, 67, -4*w - 13],
[67, 67, 4*w - 17],
[71, 71, 2*w - 13],
[71, 71, -2*w - 11],
[73, 73, 2*w - 1],
[79, 79, -8*w + 37],
[79, 79, 8*w + 29],
[89, 89, -16*w + 77],
[89, 89, -16*w - 61],
[97, 97, -24*w - 91],
[97, 97, 24*w - 115],
[109, 109, -84*w - 317],
[109, 109, 84*w - 401],
[121, 11, -11],
[127, 127, -6*w + 31],
[127, 127, 6*w + 25],
[137, 137, 18*w + 67],
[137, 137, 18*w - 85],
[149, 149, 4*w + 19],
[149, 149, 4*w - 23],
[169, 13, -13],
[173, 173, -6*w - 19],
[173, 173, 6*w - 25],
[181, 181, 12*w + 47],
[181, 181, -12*w + 59],
[211, 211, 4*w - 11],
[211, 211, -4*w - 7],
[223, 223, 102*w - 487],
[223, 223, 64*w - 305],
[227, 227, -12*w + 55],
[227, 227, -12*w - 43],
[251, 251, 2*w - 19],
[251, 251, -2*w - 17],
[257, 257, -6*w + 23],
[257, 257, -6*w - 17],
[269, 269, 20*w + 77],
[269, 269, -20*w + 97],
[283, 283, -4*w - 1],
[283, 283, 4*w - 5],
[289, 17, -17],
[293, 293, -126*w - 475],
[293, 293, -68*w - 257],
[311, 311, 24*w - 113],
[311, 311, 24*w + 89],
[317, 317, 54*w - 257],
[317, 317, 54*w + 203],
[347, 347, -94*w - 355],
[347, 347, 94*w - 449],
[349, 349, 60*w - 287],
[349, 349, 60*w + 227],
[353, 353, -8*w + 43],
[353, 353, 8*w + 35],
[359, 359, -146*w - 551],
[359, 359, 146*w - 697],
[367, 367, 6*w + 29],
[367, 367, 6*w - 35],
[373, 373, -36*w + 173],
[373, 373, -36*w - 137],
[383, 383, -22*w + 107],
[383, 383, 22*w + 85],
[389, 389, -44*w - 167],
[389, 389, 44*w - 211],
[397, 397, -12*w + 61],
[397, 397, 12*w + 49],
[401, 401, -6*w - 13],
[401, 401, 6*w - 19],
[419, 419, -12*w - 41],
[419, 419, 12*w - 53],
[439, 439, -8*w + 31],
[439, 439, -8*w - 23],
[457, 457, 62*w - 295],
[457, 457, 62*w + 233],
[461, 461, 6*w - 17],
[461, 461, -6*w - 11],
[463, 463, 32*w + 119],
[463, 463, 32*w - 151],
[479, 479, 10*w + 43],
[479, 479, -10*w + 53],
[487, 487, 18*w + 71],
[487, 487, -18*w + 89],
[499, 499, 6*w - 37],
[499, 499, -6*w - 31],
[503, 503, 2*w - 25],
[503, 503, -2*w - 23],
[509, 509, 42*w + 157],
[509, 509, 42*w - 199],
[523, 523, -78*w - 295],
[523, 523, 78*w - 373],
[547, 547, 116*w - 553],
[547, 547, 138*w - 659],
[557, 557, -6*w - 7],
[557, 557, 6*w - 13],
[587, 587, -46*w + 221],
[587, 587, -46*w - 175],
[593, 593, 6*w - 11],
[593, 593, -6*w - 5],
[607, 607, -54*w - 205],
[607, 607, 54*w - 259],
[619, 619, 28*w + 103],
[619, 619, -28*w + 131],
[641, 641, -6*w - 1],
[641, 641, 6*w - 7],
[653, 653, 6*w - 5],
[653, 653, 6*w - 1],
[659, 659, 26*w + 101],
[659, 659, -26*w + 127],
[661, 661, -14*w - 47],
[661, 661, 14*w - 61],
[673, 673, -26*w + 121],
[673, 673, 26*w + 95],
[727, 727, -8*w - 17],
[727, 727, 8*w - 25],
[733, 733, -22*w + 101],
[733, 733, -22*w - 79],
[739, 739, -20*w + 91],
[739, 739, -20*w - 71],
[757, 757, 156*w - 745],
[757, 757, -156*w - 589],
[787, 787, -6*w - 35],
[787, 787, 6*w - 41],
[797, 797, -4*w - 31],
[797, 797, 4*w - 35],
[809, 809, 78*w - 371],
[809, 809, 78*w + 293],
[811, 811, -18*w + 91],
[811, 811, 18*w + 73],
[821, 821, -28*w + 137],
[821, 821, 28*w + 109],
[827, 827, 2*w - 31],
[827, 827, -2*w - 29],
[839, 839, 48*w + 179],
[839, 839, 48*w - 227],
[841, 29, -29],
[853, 853, -12*w + 65],
[853, 853, 12*w + 53],
[857, 857, 8*w - 49],
[857, 857, -8*w - 41],
[877, 877, 58*w + 217],
[877, 877, 58*w - 275],
[911, 911, 96*w + 361],
[911, 911, 96*w - 457],
[937, 937, 24*w + 95],
[937, 937, -24*w + 119],
[941, 941, -148*w - 559],
[941, 941, 148*w - 707],
[947, 947, -12*w + 47],
[947, 947, 12*w + 35],
[953, 953, 16*w + 67],
[953, 953, -16*w + 83],
[961, 31, -31],
[967, 967, 248*w + 935],
[967, 967, -114*w - 431],
[997, 997, -36*w + 175],
[997, 997, 36*w + 139],
[1013, 1013, 18*w - 79],
[1013, 1013, -18*w - 61],
[1019, 1019, 226*w - 1079],
[1019, 1019, 132*w - 629],
[1021, 1021, 86*w - 409],
[1021, 1021, 86*w + 323],
[1031, 1031, -50*w + 241],
[1031, 1031, -50*w - 191],
[1049, 1049, -30*w + 139],
[1049, 1049, 30*w + 109],
[1063, 1063, 56*w - 265],
[1063, 1063, 56*w + 209],
[1087, 1087, 8*w - 13],
[1087, 1087, -8*w - 5],
[1091, 1091, 10*w + 49],
[1091, 1091, -10*w + 59],
[1093, 1093, 12*w + 55],
[1093, 1093, -12*w + 67],
[1097, 1097, 66*w + 247],
[1097, 1097, 66*w - 313],
[1103, 1103, -24*w + 109],
[1103, 1103, -24*w - 85],
[1171, 1171, 90*w + 341],
[1171, 1171, 90*w - 431],
[1187, 1187, -74*w - 281],
[1187, 1187, 74*w - 355],
[1193, 1193, 32*w + 125],
[1193, 1193, -32*w + 157],
[1217, 1217, -42*w + 197],
[1217, 1217, 42*w + 155],
[1223, 1223, 2*w - 37],
[1223, 1223, -2*w - 35],
[1229, 1229, 4*w - 41],
[1229, 1229, -4*w - 37],
[1237, 1237, -132*w - 499],
[1237, 1237, 274*w + 1033],
[1249, 1249, 10*w - 29],
[1249, 1249, -10*w - 19],
[1259, 1259, 12*w - 43],
[1259, 1259, -12*w - 31],
[1277, 1277, -52*w + 251],
[1277, 1277, 52*w + 199],
[1279, 1279, -6*w - 41],
[1279, 1279, 6*w - 47],
[1289, 1289, 18*w - 77],
[1289, 1289, -18*w - 59],
[1291, 1291, -270*w - 1019],
[1291, 1291, 270*w - 1289],
[1381, 1381, -60*w - 229],
[1381, 1381, -60*w + 289],
[1399, 1399, 112*w + 421],
[1399, 1399, 112*w - 533],
[1423, 1423, -16*w - 49],
[1423, 1423, 16*w - 65],
[1433, 1433, -8*w - 47],
[1433, 1433, 8*w - 55],
[1451, 1451, 36*w + 131],
[1451, 1451, -36*w + 167],
[1459, 1459, 6*w - 49],
[1459, 1459, -6*w - 43],
[1483, 1483, 18*w + 77],
[1483, 1483, -18*w + 95],
[1487, 1487, -24*w - 83],
[1487, 1487, 24*w - 107],
[1531, 1531, 30*w + 119],
[1531, 1531, -30*w + 149],
[1549, 1549, -84*w - 319],
[1549, 1549, 84*w - 403],
[1571, 1571, 22*w + 91],
[1571, 1571, -22*w + 113],
[1579, 1579, 210*w - 1003],
[1579, 1579, -210*w - 793],
[1583, 1583, 72*w - 341],
[1583, 1583, 72*w + 269],
[1597, 1597, 12*w + 59],
[1597, 1597, -12*w + 71],
[1607, 1607, 14*w + 65],
[1607, 1607, -14*w + 79],
[1609, 1609, 82*w + 307],
[1609, 1609, 82*w - 389],
[1663, 1663, 42*w + 163],
[1663, 1663, -42*w + 205],
[1667, 1667, -12*w - 25],
[1667, 1667, 12*w - 37],
[1697, 1697, -282*w - 1063],
[1697, 1697, -176*w - 665],
[1733, 1733, -4*w - 43],
[1733, 1733, 4*w - 47],
[1753, 1753, 46*w + 169],
[1753, 1753, -46*w + 215],
[1777, 1777, 22*w - 95],
[1777, 1777, -22*w - 73],
[1787, 1787, 12*w - 35],
[1787, 1787, -12*w - 23],
[1789, 1789, -10*w - 1],
[1789, 1789, 10*w - 11],
[1801, 1801, 110*w - 523],
[1801, 1801, 110*w + 413],
[1823, 1823, -202*w - 763],
[1823, 1823, 202*w - 965],
[1831, 1831, 18*w + 79],
[1831, 1831, -18*w + 97],
[1849, 43, -43],
[1861, 1861, 12*w - 73],
[1861, 1861, -12*w - 61],
[1871, 1871, 38*w + 149],
[1871, 1871, -38*w + 187],
[1873, 1873, -326*w - 1229],
[1873, 1873, -168*w - 635],
[1879, 1879, 80*w + 299],
[1879, 1879, 80*w - 379],
[1889, 1889, 16*w + 73],
[1889, 1889, -16*w + 89],
[1901, 1901, 90*w + 337],
[1901, 1901, 90*w - 427],
[1907, 1907, 358*w - 1709],
[1907, 1907, -358*w - 1351],
[1933, 1933, 26*w - 115],
[1933, 1933, -26*w - 89],
[1973, 1973, -332*w - 1253],
[1973, 1973, 332*w - 1585],
[1979, 1979, 228*w + 859],
[1979, 1979, 228*w - 1087],
[1987, 1987, -28*w - 97],
[1987, 1987, 28*w - 125],
[2003, 2003, 12*w - 31],
[2003, 2003, -12*w - 19],
[2017, 2017, 38*w + 137],
[2017, 2017, -38*w + 175],
[2053, 2053, -34*w - 121],
[2053, 2053, -34*w + 155],
[2063, 2063, -26*w + 133],
[2063, 2063, 26*w + 107],
[2069, 2069, -18*w + 71],
[2069, 2069, 18*w + 53],
[2081, 2081, -8*w - 53],
[2081, 2081, 8*w - 61],
[2099, 2099, -12*w - 17],
[2099, 2099, 12*w - 29],
[2111, 2111, 58*w + 223],
[2111, 2111, -58*w + 281],
[2113, 2113, -72*w + 347],
[2113, 2113, 72*w + 275],
[2129, 2129, -40*w + 197],
[2129, 2129, 40*w + 157],
[2141, 2141, 54*w + 199],
[2141, 2141, -54*w + 253],
[2153, 2153, 152*w - 727],
[2153, 2153, 152*w + 575],
[2209, 47, -47],
[2213, 2213, -66*w + 311],
[2213, 2213, 66*w + 245],
[2239, 2239, -186*w - 703],
[2239, 2239, -352*w - 1327],
[2251, 2251, 402*w - 1919],
[2251, 2251, -402*w - 1517],
[2267, 2267, -12*w - 13],
[2267, 2267, 12*w - 25],
[2269, 2269, 36*w + 143],
[2269, 2269, -36*w + 179],
[2281, 2281, -14*w - 29],
[2281, 2281, 14*w - 43],
[2287, 2287, -246*w - 929],
[2287, 2287, 246*w - 1175],
[2309, 2309, 4*w - 53],
[2309, 2309, -4*w - 49],
[2311, 2311, 136*w - 647],
[2311, 2311, 136*w + 511],
[2333, 2333, 28*w + 115],
[2333, 2333, -28*w + 143],
[2339, 2339, 12*w - 23],
[2339, 2339, -12*w - 11],
[2371, 2371, 76*w - 359],
[2371, 2371, 76*w + 283],
[2377, 2377, -96*w + 461],
[2377, 2377, -96*w - 365],
[2393, 2393, 128*w + 485],
[2393, 2393, 128*w - 613],
[2411, 2411, 396*w + 1493],
[2411, 2411, -178*w - 673],
[2417, 2417, -104*w - 395],
[2417, 2417, 104*w - 499],
[2441, 2441, 126*w - 599],
[2441, 2441, 126*w + 473],
[2447, 2447, -48*w + 223],
[2447, 2447, 48*w + 175],
[2459, 2459, 12*w - 19],
[2459, 2459, -12*w - 7],
[2473, 2473, -48*w + 235],
[2473, 2473, 48*w + 187],
[2531, 2531, 10*w - 71],
[2531, 2531, -10*w - 61],
[2539, 2539, -20*w + 79],
[2539, 2539, 20*w + 59],
[2543, 2543, 96*w + 359],
[2543, 2543, 96*w - 455],
[2549, 2549, -18*w - 49],
[2549, 2549, 18*w - 67],
[2551, 2551, -18*w + 101],
[2551, 2551, 18*w + 83],
[2557, 2557, 106*w + 397],
[2557, 2557, 106*w - 503],
[2579, 2579, -12*w - 1],
[2579, 2579, 12*w - 13],
[2591, 2591, -24*w - 77],
[2591, 2591, 24*w - 101],
[2593, 2593, 74*w + 275],
[2593, 2593, 74*w - 349],
[2609, 2609, -42*w + 193],
[2609, 2609, -42*w - 151],
[2647, 2647, 16*w - 53],
[2647, 2647, -16*w - 37],
[2663, 2663, 144*w + 541],
[2663, 2663, 144*w - 685],
[2677, 2677, 14*w - 37],
[2677, 2677, -14*w - 23],
[2683, 2683, 162*w - 775],
[2683, 2683, 162*w + 613],
[2689, 2689, 264*w - 1261],
[2689, 2689, -264*w - 997],
[2693, 2693, 44*w + 173],
[2693, 2693, -44*w + 217],
[2699, 2699, -60*w + 281],
[2699, 2699, 60*w + 221],
[2707, 2707, 6*w - 61],
[2707, 2707, -6*w - 55],
[2713, 2713, -24*w + 127],
[2713, 2713, 24*w + 103],
[2719, 2719, -90*w + 433],
[2719, 2719, -90*w - 343],
[2749, 2749, -22*w - 67],
[2749, 2749, 22*w - 89],
[2777, 2777, -18*w - 47],
[2777, 2777, 18*w - 65],
[2797, 2797, 134*w - 637],
[2797, 2797, 134*w + 503],
[2801, 2801, 8*w - 67],
[2801, 2801, -8*w - 59],
[2809, 53, -53],
[2843, 2843, 2*w - 55],
[2843, 2843, -2*w - 53],
[2851, 2851, 342*w - 1633],
[2851, 2851, 244*w - 1163],
[2879, 2879, -106*w + 509],
[2879, 2879, -106*w - 403],
[2897, 2897, 16*w + 79],
[2897, 2897, -16*w + 95],
[2917, 2917, -36*w + 181],
[2917, 2917, 36*w + 145],
[2939, 2939, -130*w - 493],
[2939, 2939, 130*w - 623],
[2957, 2957, -4*w - 55],
[2957, 2957, 4*w - 59],
[2969, 2969, -30*w - 101],
[2969, 2969, 30*w - 131],
[2999, 2999, -46*w + 227],
[2999, 2999, 46*w + 181],
[3001, 3001, -14*w - 17],
[3001, 3001, 14*w - 31],
[3011, 3011, 180*w - 857],
[3011, 3011, 180*w + 677],
[3041, 3041, 54*w + 197],
[3041, 3041, -54*w + 251],
[3067, 3067, -404*w - 1523],
[3067, 3067, -222*w - 839],
[3089, 3089, -64*w + 311],
[3089, 3089, 64*w + 247],
[3121, 3121, 26*w - 109],
[3121, 3121, -26*w - 83],
[3137, 3137, -464*w - 1751],
[3137, 3137, 464*w - 2215],
[3163, 3163, -68*w + 319],
[3163, 3163, 68*w + 251],
[3187, 3187, -6*w - 59],
[3187, 3187, 6*w - 65],
[3203, 3203, -36*w - 125],
[3203, 3203, 36*w - 161],
[3209, 3209, 18*w - 61],
[3209, 3209, -18*w - 43],
[3221, 3221, -308*w - 1163],
[3221, 3221, 308*w - 1471],
[3253, 3253, 14*w - 25],
[3253, 3253, -14*w - 11],
[3301, 3301, -12*w - 71],
[3301, 3301, 12*w - 83],
[3323, 3323, 252*w - 1201],
[3323, 3323, 386*w - 1843],
[3331, 3331, 20*w - 73],
[3331, 3331, -20*w - 53],
[3359, 3359, 10*w - 77],
[3359, 3359, -10*w - 67],
[3361, 3361, -72*w + 349],
[3361, 3361, 72*w + 277],
[3407, 3407, 86*w + 329],
[3407, 3407, -86*w + 415],
[3413, 3413, 18*w - 59],
[3413, 3413, -18*w - 41],
[3433, 3433, -14*w - 5],
[3433, 3433, 14*w - 19],
[3449, 3449, -78*w + 367],
[3449, 3449, 78*w + 289],
[3463, 3463, 64*w + 235],
[3463, 3463, -64*w + 299],
[3467, 3467, -206*w - 779],
[3467, 3467, 206*w - 985],
[3469, 3469, -60*w + 293],
[3469, 3469, 60*w + 233],
[3481, 59, -59],
[3527, 3527, 2*w - 61],
[3527, 3527, -2*w - 59],
[3529, 3529, -240*w - 907],
[3529, 3529, -430*w - 1621],
[3539, 3539, -26*w + 139],
[3539, 3539, 26*w + 113],
[3541, 3541, 14*w - 13],
[3541, 3541, 14*w - 1],
[3559, 3559, -32*w - 107],
[3559, 3559, 32*w - 139],
[3571, 3571, 100*w + 373],
[3571, 3571, 100*w - 473],
[3581, 3581, 150*w - 713],
[3581, 3581, 150*w + 563],
[3583, 3583, 16*w - 41],
[3583, 3583, -16*w - 25],
[3593, 3593, 8*w - 73],
[3593, 3593, -8*w - 65],
[3613, 3613, 12*w - 85],
[3613, 3613, -12*w - 73],
[3623, 3623, -24*w + 95],
[3623, 3623, 24*w + 71],
[3631, 3631, 198*w - 947],
[3631, 3631, 198*w + 749],
[3659, 3659, 50*w + 197],
[3659, 3659, -50*w + 247],
[3673, 3673, 34*w - 149],
[3673, 3673, -34*w - 115],
[3677, 3677, 4*w - 65],
[3677, 3677, -4*w - 61],
[3691, 3691, 18*w + 89],
[3691, 3691, -18*w + 107],
[3719, 3719, 120*w - 569],
[3719, 3719, 120*w + 449],
[3727, 3727, -102*w - 389],
[3727, 3727, -102*w + 491],
[3739, 3739, -30*w + 157],
[3739, 3739, 30*w + 127],
[3761, 3761, -232*w - 877],
[3761, 3761, -474*w - 1787],
[3769, 3769, -456*w - 1721],
[3769, 3769, 456*w - 2177],
[3793, 3793, 58*w + 211],
[3793, 3793, -58*w + 269],
[3797, 3797, -18*w - 37],
[3797, 3797, 18*w - 55],
[3821, 3821, -68*w + 331],
[3821, 3821, 68*w + 263],
[3823, 3823, 296*w - 1411],
[3823, 3823, 378*w - 1805],
[3833, 3833, -88*w + 425],
[3833, 3833, 88*w + 337],
[3851, 3851, -110*w - 419],
[3851, 3851, -110*w + 529],
[3853, 3853, -38*w - 131],
[3853, 3853, 38*w - 169],
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[8443, 8443, 258*w + 977],
[8443, 8443, 258*w - 1235],
[8467, 8467, 172*w - 815],
[8467, 8467, 172*w + 643],
[8537, 8537, 32*w + 149],
[8537, 8537, -32*w + 181],
[8539, 8539, 30*w + 143],
[8539, 8539, 30*w - 173],
[8543, 8543, 192*w + 719],
[8543, 8543, 192*w - 911],
[8573, 8573, -148*w - 565],
[8573, 8573, -148*w + 713],
[8623, 8623, -186*w + 893],
[8623, 8623, -186*w - 707],
[8641, 8641, 120*w + 461],
[8641, 8641, -120*w + 581],
[8663, 8663, -24*w - 31],
[8663, 8663, 24*w - 55],
[8669, 8669, 150*w - 709],
[8669, 8669, -150*w - 559],
[8681, 8681, -30*w - 73],
[8681, 8681, 30*w - 103],
[8689, 8689, -22*w - 1],
[8689, 8689, 22*w - 23],
[8693, 8693, 114*w + 421],
[8693, 8693, -114*w + 535],
[8699, 8699, -194*w - 737],
[8699, 8699, 194*w - 931],
[8719, 8719, -234*w - 887],
[8719, 8719, 234*w - 1121],
[8737, 8737, -26*w - 47],
[8737, 8737, 26*w - 73],
[8741, 8741, -66*w - 233],
[8741, 8741, -66*w + 299],
[8761, 8761, -50*w - 167],
[8761, 8761, 50*w - 217],
[8779, 8779, 210*w - 1007],
[8779, 8779, -210*w - 797],
[8783, 8783, -218*w - 827],
[8783, 8783, 218*w - 1045],
[8821, 8821, 12*w - 113],
[8821, 8821, -12*w - 101],
[8831, 8831, -24*w - 29],
[8831, 8831, 24*w - 53],
[8837, 8837, 126*w + 467],
[8837, 8837, -126*w + 593],
[8839, 8839, 114*w + 439],
[8839, 8839, -114*w + 553],
[8849, 8849, -496*w - 1873],
[8849, 8849, 496*w - 2369],
[8887, 8887, 32*w - 115],
[8887, 8887, -32*w - 83],
[8929, 8929, -38*w + 151],
[8929, 8929, 38*w + 113],
[8933, 8933, 292*w + 1105],
[8933, 8933, 292*w - 1397],
[8941, 8941, 230*w - 1093],
[8941, 8941, 230*w + 863],
[8963, 8963, 74*w + 293],
[8963, 8963, -74*w + 367],
[8971, 8971, -18*w - 113],
[8971, 8971, 18*w - 131],
[9011, 9011, -36*w - 103],
[9011, 9011, -36*w + 139],
[9029, 9029, -30*w - 71],
[9029, 9029, 30*w - 101],
[9043, 9043, -102*w + 497],
[9043, 9043, 102*w + 395],
[9049, 9049, 96*w + 373],
[9049, 9049, -96*w + 469],
[9109, 9109, 36*w + 163],
[9109, 9109, -36*w + 199],
[9127, 9127, 402*w + 1519],
[9127, 9127, -664*w - 2503],
[9133, 9133, 314*w - 1495],
[9133, 9133, 314*w + 1181],
[9137, 9137, -726*w - 2737],
[9137, 9137, -368*w - 1391],
[9157, 9157, 94*w + 343],
[9157, 9157, -94*w + 437],
[9161, 9161, 210*w + 787],
[9161, 9161, 210*w - 997],
[9173, 9173, -172*w - 655],
[9173, 9173, -172*w + 827],
[9199, 9199, 200*w - 949],
[9199, 9199, 200*w + 749],
[9221, 9221, 414*w - 1973],
[9221, 9221, 652*w - 3113],
[9239, 9239, 432*w - 2059],
[9239, 9239, 626*w - 2989],
[9277, 9277, 276*w + 1045],
[9277, 9277, 276*w - 1321],
[9283, 9283, 54*w + 223],
[9283, 9283, -54*w + 277],
[9319, 9319, -40*w - 121],
[9319, 9319, -40*w + 161],
[9341, 9341, -20*w - 119],
[9341, 9341, 20*w - 139],
[9343, 9343, 6*w - 103],
[9343, 9343, -6*w - 97],
[9371, 9371, -62*w + 313],
[9371, 9371, 62*w + 251],
[9413, 9413, 268*w - 1283],
[9413, 9413, 268*w + 1015],
[9419, 9419, -14*w - 107],
[9419, 9419, 14*w - 121],
[9421, 9421, -26*w - 41],
[9421, 9421, 26*w - 67],
[9433, 9433, -34*w - 91],
[9433, 9433, 34*w - 125],
[9463, 9463, -18*w - 115],
[9463, 9463, 18*w - 133],
[9467, 9467, 108*w + 397],
[9467, 9467, -108*w + 505],
[9491, 9491, -36*w - 101],
[9491, 9491, -36*w + 137],
[9539, 9539, -394*w - 1489],
[9539, 9539, -708*w - 2669],
[9547, 9547, 28*w - 83],
[9547, 9547, -28*w - 55],
[9551, 9551, 24*w - 43],
[9551, 9551, -24*w - 19],
[9587, 9587, 276*w - 1313],
[9587, 9587, 276*w + 1037],
[9601, 9601, -134*w + 631],
[9601, 9601, 134*w + 497],
[9613, 9613, 252*w - 1207],
[9613, 9613, -252*w - 955],
[9661, 9661, 86*w + 311],
[9661, 9661, -86*w + 397],
[9677, 9677, -244*w - 925],
[9677, 9677, 244*w - 1169],
[9697, 9697, -144*w + 695],
[9697, 9697, 144*w + 551],
[9721, 9721, 72*w + 287],
[9721, 9721, -72*w + 359],
[9733, 9733, 228*w - 1093],
[9733, 9733, -228*w - 865],
[9781, 9781, -540*w - 2039],
[9781, 9781, 540*w - 2579],
[9791, 9791, -120*w + 563],
[9791, 9791, 120*w + 443],
[9817, 9817, -82*w + 377],
[9817, 9817, -82*w - 295],
[9839, 9839, -10*w - 103],
[9839, 9839, 10*w - 113],
[9851, 9851, -156*w - 581],
[9851, 9851, 156*w - 737],
[9857, 9857, -8*w - 101],
[9857, 9857, 8*w - 109],
[9859, 9859, -68*w + 307],
[9859, 9859, -68*w - 239],
[9871, 9871, -80*w + 367],
[9871, 9871, -80*w - 287],
[9887, 9887, 24*w - 37],
[9887, 9887, -24*w - 13],
[9901, 9901, -70*w + 317],
[9901, 9901, -70*w - 247],
[9929, 9929, 64*w + 259],
[9929, 9929, -64*w + 323],
[9931, 9931, -76*w - 271],
[9931, 9931, -76*w + 347]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-2, -2, -1, -1, 0, 0, -1, -1, -9, 3, 3, -8, -8, -10, 12, 12, -7, -7, -3, -3, 4, -10, -10, 15, 15, -7, -7, 10, 10, 1, 8, 8, -7, -7, -10, -10, -10, -6, -6, 7, 7, 12, 12, 19, 19, 18, 18, -23, -23, -2, -2, 10, 10, 4, 4, -30, 24, 24, 12, 12, 13, 13, 28, 28, 30, 30, -21, -21, -20, -20, -17, -17, -26, -26, -1, -1, -15, -15, -2, -2, 2, 2, 20, 20, 40, 40, -12, -12, 12, 12, -11, -11, 20, 20, 23, 23, 20, 20, -26, -26, 15, 15, -16, -16, 8, 8, -2, -2, 28, 28, 44, 44, -22, -22, -25, -25, -33, -33, -41, -41, 10, 10, 37, 37, 14, 14, 3, 3, -36, -36, 50, 50, -22, -22, -32, -32, 53, 53, 0, 0, -38, -38, 22, 22, -52, -52, -5, -5, -58, 14, 14, 8, 8, -12, -12, 12, 12, 8, 8, 42, 42, -27, -27, 34, 34, -13, -32, -32, 38, 38, 39, 39, -10, -10, 22, 22, 32, 32, -55, -55, 44, 44, 8, 8, -58, -58, -51, -51, -42, -42, -51, -51, -3, -3, -12, -12, -21, -21, -42, -42, 14, 14, 60, 60, 18, 18, 40, 40, -25, -25, -47, -47, -15, -15, 0, 0, -8, -8, -68, -68, 60, 60, 29, 29, 54, 54, 52, 52, -20, -20, 49, 49, 58, 58, 32, 32, -15, -15, -28, -28, -30, -30, 34, 34, -32, -32, 33, 33, -10, -10, 4, 4, 48, 48, -42, -42, -6, -6, 34, 34, 8, 8, -57, -57, 10, 10, 52, 52, -56, -56, -43, -43, -50, 62, 62, -3, -3, -6, -6, -35, -35, 70, 70, 77, 77, -52, -52, 54, 54, 79, 79, 30, 30, -22, -22, 4, 4, -17, -17, 84, 84, 24, 24, 70, 70, -18, -18, 35, 35, -38, -38, -86, -86, 20, 20, -58, -58, -26, -26, -30, 4, 4, -70, -70, -48, -48, 93, 93, 25, 25, 7, 7, 38, 38, 60, 60, -13, -13, 59, 59, 10, 10, -28, -28, 3, 3, 54, 54, 62, 62, -22, -22, 42, 42, 3, 3, -50, -50, -11, -11, 57, 57, 0, 0, 34, 34, -20, -20, -98, -98, 13, 13, 20, 20, -58, -58, 14, 14, -30, -30, 38, 38, 39, 39, -7, -7, -16, -16, 5, 5, -41, -41, -55, -55, -17, -17, -56, -56, -70, -70, 50, 50, -42, -42, -42, -42, 52, 52, -70, 4, 4, 2, 2, -40, -40, 38, 38, 88, 88, -50, -50, 3, 3, 70, 70, -80, -80, 27, 27, 62, 62, 42, 42, 13, 13, 25, 25, 22, 22, 8, 8, -26, -26, -2, -2, -6, -6, -10, -10, -103, -103, 74, 74, -73, -73, -76, -76, -43, -43, -45, -45, -88, -88, 18, 18, 9, 9, -66, -66, 40, 40, -96, -96, 38, 38, -85, -85, -93, 18, 18, -35, -35, 20, 20, 42, 42, 0, 0, -28, -28, 32, 32, -96, -96, -26, -26, -26, -26, 4, 4, 32, 32, 30, 30, -76, -76, -62, -62, 92, 92, -55, -55, 23, 23, -60, -60, -88, -88, 40, 40, 34, 34, -82, -82, -3, -3, 84, 84, 19, 19, -73, -73, 74, 74, 54, 54, 58, 58, 7, 7, -22, -22, -57, -57, 0, 0, 54, 54, -96, -96, -92, -92, -46, -46, -32, -32, 54, 54, 15, 15, 22, 22, 28, 28, -123, -123, -60, -60, 92, 92, 62, 62, -25, -25, -36, -36, -42, -42, 85, 85, 88, 88, -63, -63, -68, -68, 64, 64, -16, -16, 70, 70, -113, -113, -53, -53, 39, 39, -60, -60, 55, 55, -117, -117, 72, 72, 42, 42, 83, 83, 102, 102, 78, 78, -82, -82, 8, 8, -36, -36, -10, -10, 112, 112, -112, -112, -36, -36, -8, -8, -52, -52, 89, 89, 18, 18, -80, -80, 64, 64, 114, 114, 17, 17, 22, 22, 30, 30, 129, 129, -48, -48, -101, -101, -36, -36, 105, 105, -132, -132, 5, 5, -126, -126, 115, 115, -42, -42, -52, -52, -1, -1, 28, 28, 40, 40, -3, -3, 15, 15, 93, 93, 2, 2, -32, -32, 29, 29, 15, 15, 48, 48, 28, 28, 97, 97, -60, -60, -80, -80, -18, -18, -96, -96, 63, 63, -70, -70, 57, 57, -22, -22, -76, -76, -126, -126, 23, 23, -11, -11, -15, -15, -2, -2, 109, 109, -3, -3, 128, 128, -95, -95, 44, 44, 48, 48, 22, 22, -122, -122, 104, 104, 64, 64, 60, 60, 4, 4, -57, -57, -125, -125, -110, -110, -146, -146, -72, -72, 19, 19, -30, -30, 25, 25, 104, 104, 92, 92, -1, -1, -78, -78, 100, 100, -66, -66, 80, 80, 33, 33, -68, -68, -27, -27, 73, 73, -63, -63, -101, -101, -42, -42, -26, -26, 48, 48, -8, -8, -16, -16, 57, 57, 94, 94, 52, 52, -86, -86, 113, 113, 90, 90, 4, 4, -38, -38, -62, -62, -128, -128, -115, -115, -30, -30, -132, -132, -132, -132, -23, -23, -76, -76, -86, -86, 60, 60, -12, -12, -75, -75, -22, -22, -28, -28, -37, -37, 105, 105, 52, 52, -91, -91, -18, -18, 117, 117, -3, -3, -80, -80, 68, 68, -98, -98, 130, 130, -28, -28, 48, 48, -50, -50, -122, -122, -95, -95, -38, -38, 40, 40, 15, 15, -83, -83, -120, -120, 7, 7, 117, 117, 104, 104, -132, -132, 64, 64, 122, 122, -130, -90, -90, -88, -88, 38, 38, -72, -72, 30, 30, 8, 8, -128, -128, -101, -101, 23, 23, 164, 164, -62, -62, -60, -60, 120, 120, -72, -72, -145, -145, -73, -73, -142, -142, 162, 162, 25, 25, -110, -110, -21, -21, 138, 138, -25, -25, -150, -150, -80, -80, -88, -88, -43, -43, 78, 78, -87, -87, -42, -42, 24, 24, -92, -92, 109, 109, 64, 64, 40, 40, -33, -33, -130, -130, -70, -70, 84, 84, 107, 107, -51, -51, 92, 92, 28, 28, -21, -21, 58, 58, 14, 14, 60, 60, -36, -36, -147, -147, 20, 20, 74, 74, -30, -30, -48, -48, -136, -136, -26, -26, 137, 137, -75, -75, -46, -46, -42, -42, 77, 77, 112, 112, 123, 123, -130, -130, 107, 107, 169, 169, -17, -17, 99, 99, -26, -26, -72, -72, -117, -117, 84, 84, -92, -92, 168, 168, 110, 110, -60, -60, 107, 107, 54, 54, -122, -122, 13, 13, 145, 145, -126, -126, -146, -146, 74, 74, -108, -108, 124, 124, -130, -130, 132, 132, 90, 90, -86, -86, 135, 135, 40, 40, -2, -2, -108, -108, -13, -13, 55, 55, 69, 69, -78, -78, 157, 157, 3, 3, -70, -70, 90, 90, -62, -62, -40, -40, 109, 109, -133, -133, 129, 129, -128, -128, -38, -38, 115, 115, 144, 144, -130, -130, 85, 85, -12, -12, 79, 79, 78, 78, 63, 63, -138, -138, 24, 24, 80, 80, -103, -103, -60, -60, 43, 43, -166, -166, -20, -20, 42, 42, -151, -151, -78, -78, -166, -166, 0, 0, -78, -78, 94, 94, 184, 184, 8, 8, -108, -108, 0, 0, -2, -2, 27, 27, 18, 18, -173, -173, 64, 64, -13, -13, 138, 138, 178, 178, -178, -178, 49, 49, -18, -18, -108, -108, -102, -102, -45, -45, -48, -48, -117, -117, -85, -85, -128, -128, -117, -117, 102, 102, 0, 0, 57, 57];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {121, 11, -11}>] := -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;