/*
  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![-22, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, -w - 4],
[2, 2, -w + 5],
[5, 5, 4*w - 21],
[5, 5, -4*w - 17],
[9, 3, 3],
[11, 11, 2*w - 11],
[11, 11, -2*w - 9],
[17, 17, -6*w - 25],
[17, 17, -6*w + 31],
[47, 47, 24*w + 101],
[47, 47, 24*w - 125],
[49, 7, -7],
[53, 53, 2*w - 7],
[53, 53, -2*w - 5],
[67, 67, 4*w - 19],
[67, 67, 4*w + 15],
[71, 71, 16*w - 83],
[71, 71, 16*w + 67],
[73, 73, 2*w - 5],
[73, 73, -2*w - 3],
[79, 79, 10*w + 43],
[79, 79, 10*w - 53],
[89, 89, 2*w - 1],
[97, 97, 8*w - 43],
[97, 97, 8*w + 35],
[107, 107, 2*w - 15],
[107, 107, -2*w - 13],
[109, 109, 10*w + 41],
[109, 109, 10*w - 51],
[131, 131, -4*w - 13],
[131, 131, 4*w - 17],
[139, 139, -30*w + 157],
[139, 139, 52*w - 271],
[157, 157, 26*w - 135],
[157, 157, 26*w + 109],
[167, 167, 2*w - 17],
[167, 167, -2*w - 15],
[169, 13, -13],
[173, 173, 4*w - 25],
[173, 173, -4*w - 21],
[179, 179, 12*w + 49],
[179, 179, 12*w - 61],
[199, 199, 8*w - 39],
[199, 199, 8*w + 31],
[223, 223, -6*w - 29],
[223, 223, -6*w + 35],
[227, 227, 44*w + 185],
[227, 227, 58*w + 245],
[233, 233, 16*w + 69],
[233, 233, 16*w - 85],
[251, 251, 20*w + 83],
[251, 251, 20*w - 103],
[257, 257, -8*w - 37],
[257, 257, -8*w + 45],
[263, 263, 14*w - 75],
[263, 263, 14*w + 61],
[269, 269, -4*w - 23],
[269, 269, 4*w - 27],
[271, 271, -38*w + 199],
[271, 271, 38*w + 161],
[277, 277, 12*w - 65],
[277, 277, -12*w - 53],
[283, 283, 28*w - 145],
[283, 283, 28*w + 117],
[307, 307, -4*w - 5],
[307, 307, 4*w - 9],
[311, 311, 2*w - 21],
[311, 311, -2*w - 19],
[317, 317, -6*w - 19],
[317, 317, 6*w - 25],
[331, 331, 4*w - 7],
[331, 331, -4*w - 3],
[347, 347, -4*w - 1],
[347, 347, 4*w - 5],
[361, 19, -19],
[367, 367, 16*w + 65],
[367, 367, 16*w - 81],
[373, 373, 4*w - 29],
[373, 373, -4*w - 25],
[401, 401, 6*w - 23],
[401, 401, -6*w - 17],
[409, 409, 24*w - 127],
[409, 409, 24*w + 103],
[443, 443, -46*w + 241],
[443, 443, 46*w + 195],
[449, 449, -56*w + 293],
[449, 449, 90*w - 469],
[461, 461, 10*w + 37],
[461, 461, 10*w - 47],
[463, 463, 8*w + 27],
[463, 463, -8*w + 35],
[467, 467, 22*w + 95],
[467, 467, 22*w - 117],
[479, 479, -10*w + 57],
[479, 479, -10*w - 47],
[487, 487, 2*w - 25],
[487, 487, -2*w - 23],
[509, 509, 20*w - 107],
[509, 509, 20*w + 87],
[523, 523, 34*w + 145],
[523, 523, -34*w + 179],
[529, 23, -23],
[587, 587, 2*w - 27],
[587, 587, -2*w - 25],
[601, 601, 8*w - 49],
[601, 601, -8*w - 41],
[607, 607, -56*w + 291],
[607, 607, 56*w + 235],
[613, 613, 82*w + 345],
[613, 613, 82*w - 427],
[619, 619, 20*w + 81],
[619, 619, 20*w - 101],
[631, 631, 40*w - 207],
[631, 631, 40*w + 167],
[641, 641, 26*w + 107],
[641, 641, 26*w - 133],
[643, 643, -6*w - 35],
[643, 643, 6*w - 41],
[659, 659, 74*w + 313],
[659, 659, -74*w + 387],
[673, 673, 118*w - 615],
[673, 673, -64*w + 335],
[691, 691, -10*w - 49],
[691, 691, -10*w + 59],
[701, 701, -6*w - 7],
[701, 701, 6*w - 13],
[733, 733, -4*w - 31],
[733, 733, 4*w - 35],
[751, 751, 42*w + 179],
[751, 751, -42*w + 221],
[757, 757, 34*w + 141],
[757, 757, 34*w - 175],
[761, 761, 14*w - 67],
[761, 761, 14*w + 53],
[769, 769, 22*w + 89],
[769, 769, 22*w - 111],
[797, 797, 6*w - 5],
[797, 797, 6*w - 1],
[809, 809, 18*w - 89],
[809, 809, 18*w + 71],
[811, 811, 2*w - 31],
[811, 811, -2*w - 29],
[821, 821, -66*w + 343],
[821, 821, 66*w + 277],
[823, 823, -14*w - 65],
[823, 823, -14*w + 79],
[841, 29, -29],
[881, 881, 40*w - 211],
[881, 881, 40*w + 171],
[907, 907, -62*w + 325],
[907, 907, 62*w + 263],
[911, 911, -10*w + 61],
[911, 911, -10*w - 51],
[929, 929, -10*w - 31],
[929, 929, 10*w - 41],
[937, 937, -72*w + 377],
[937, 937, 72*w + 305],
[947, 947, 36*w - 185],
[947, 947, 36*w + 149],
[961, 31, -31],
[971, 971, 84*w + 353],
[971, 971, 130*w + 549],
[977, 977, -8*w - 45],
[977, 977, 8*w - 53],
[983, 983, -8*w - 17],
[983, 983, 8*w - 25],
[997, 997, 14*w - 65],
[997, 997, 14*w + 51],
[1013, 1013, -4*w - 35],
[1013, 1013, 4*w - 39],
[1019, 1019, 50*w + 213],
[1019, 1019, -50*w + 263],
[1021, 1021, -12*w - 59],
[1021, 1021, -12*w + 71],
[1051, 1051, 44*w + 183],
[1051, 1051, 44*w - 227],
[1063, 1063, 8*w - 23],
[1063, 1063, -8*w - 15],
[1069, 1069, 10*w - 39],
[1069, 1069, -10*w - 29],
[1093, 1093, 36*w + 155],
[1093, 1093, 36*w - 191],
[1117, 1117, 26*w + 105],
[1117, 1117, 26*w - 131],
[1123, 1123, 68*w + 285],
[1123, 1123, 68*w - 353],
[1153, 1153, 38*w - 195],
[1153, 1153, 38*w + 157],
[1193, 1193, -16*w + 91],
[1193, 1193, -16*w - 75],
[1201, 1201, -10*w - 27],
[1201, 1201, 10*w - 37],
[1229, 1229, 100*w + 423],
[1229, 1229, -100*w + 523],
[1237, 1237, 94*w + 395],
[1237, 1237, 148*w + 625],
[1291, 1291, 30*w - 161],
[1291, 1291, 30*w + 131],
[1301, 1301, -20*w - 91],
[1301, 1301, -20*w + 111],
[1303, 1303, 8*w - 15],
[1303, 1303, -8*w - 7],
[1319, 1319, 46*w + 197],
[1319, 1319, 46*w - 243],
[1327, 1327, 58*w + 247],
[1327, 1327, -58*w + 305],
[1367, 1367, 24*w - 119],
[1367, 1367, 24*w + 95],
[1369, 37, -37],
[1399, 1399, -8*w - 1],
[1399, 1399, 8*w - 9],
[1423, 1423, 8*w - 5],
[1423, 1423, 8*w - 3],
[1429, 1429, 62*w + 259],
[1429, 1429, 62*w - 321],
[1433, 1433, 18*w + 67],
[1433, 1433, 18*w - 85],
[1471, 1471, 16*w - 73],
[1471, 1471, 16*w + 57],
[1481, 1481, 130*w + 547],
[1481, 1481, 130*w - 677],
[1493, 1493, 4*w - 45],
[1493, 1493, -4*w - 41],
[1511, 1511, 2*w - 41],
[1511, 1511, -2*w - 39],
[1523, 1523, 12*w - 47],
[1523, 1523, -12*w - 35],
[1531, 1531, -78*w + 409],
[1531, 1531, 78*w + 331],
[1549, 1549, -10*w - 21],
[1549, 1549, 10*w - 31],
[1553, 1553, 22*w + 85],
[1553, 1553, 22*w - 107],
[1597, 1597, -108*w + 565],
[1597, 1597, 166*w - 865],
[1601, 1601, 8*w - 59],
[1601, 1601, -8*w - 51],
[1607, 1607, 184*w - 959],
[1607, 1607, -98*w + 513],
[1613, 1613, 42*w + 173],
[1613, 1613, 42*w - 215],
[1619, 1619, 10*w - 67],
[1619, 1619, -10*w - 57],
[1627, 1627, -18*w + 103],
[1627, 1627, -18*w - 85],
[1657, 1657, -14*w - 45],
[1657, 1657, 14*w - 59],
[1669, 1669, -4*w - 43],
[1669, 1669, 4*w - 47],
[1681, 41, -41],
[1693, 1693, 122*w - 635],
[1693, 1693, -122*w - 513],
[1699, 1699, -6*w - 47],
[1699, 1699, 6*w - 53],
[1709, 1709, -20*w - 93],
[1709, 1709, -20*w + 113],
[1723, 1723, -14*w + 85],
[1723, 1723, -14*w - 71],
[1733, 1733, 18*w + 65],
[1733, 1733, 18*w - 83],
[1741, 1741, -10*w - 17],
[1741, 1741, 10*w - 27],
[1759, 1759, 32*w + 129],
[1759, 1759, 32*w - 161],
[1789, 1789, 36*w + 157],
[1789, 1789, 36*w - 193],
[1801, 1801, 50*w - 257],
[1801, 1801, 50*w + 207],
[1847, 1847, 2*w - 45],
[1847, 1847, -2*w - 43],
[1849, 43, -43],
[1861, 1861, -14*w - 43],
[1861, 1861, 14*w - 57],
[1867, 1867, 44*w + 181],
[1867, 1867, 44*w - 225],
[1871, 1871, -10*w - 59],
[1871, 1871, 10*w - 69],
[1873, 1873, -16*w - 79],
[1873, 1873, -16*w + 95],
[1877, 1877, 28*w - 153],
[1877, 1877, -28*w - 125],
[1879, 1879, 30*w - 163],
[1879, 1879, -30*w - 133],
[1889, 1889, 64*w + 273],
[1889, 1889, -64*w + 337],
[1901, 1901, 10*w - 23],
[1901, 1901, -10*w - 13],
[1913, 1913, 158*w - 823],
[1913, 1913, 158*w + 665],
[1933, 1933, 22*w - 105],
[1933, 1933, 22*w + 83],
[1949, 1949, 58*w + 241],
[1949, 1949, 58*w - 299],
[1979, 1979, -12*w - 29],
[1979, 1979, 12*w - 41],
[1997, 1997, -106*w + 551],
[1997, 1997, 106*w + 445],
[2003, 2003, -106*w + 555],
[2003, 2003, 106*w + 449],
[2011, 2011, 20*w - 93],
[2011, 2011, 20*w + 73],
[2027, 2027, 2*w - 47],
[2027, 2027, -2*w - 45],
[2029, 2029, 10*w - 19],
[2029, 2029, -10*w - 9],
[2039, 2039, 14*w - 87],
[2039, 2039, -14*w - 73],
[2063, 2063, 74*w + 315],
[2063, 2063, -74*w + 389],
[2069, 2069, 30*w - 149],
[2069, 2069, 30*w + 119],
[2081, 2081, -10*w - 7],
[2081, 2081, 10*w - 17],
[2083, 2083, 52*w + 215],
[2083, 2083, 52*w - 267],
[2087, 2087, 62*w - 327],
[2087, 2087, 62*w + 265],
[2089, 2089, 98*w + 411],
[2089, 2089, 98*w - 509],
[2111, 2111, 40*w - 203],
[2111, 2111, 40*w + 163],
[2131, 2131, 10*w - 71],
[2131, 2131, -10*w - 61],
[2137, 2137, 46*w - 235],
[2137, 2137, 46*w + 189],
[2141, 2141, 90*w - 467],
[2141, 2141, 90*w + 377],
[2153, 2153, 82*w + 343],
[2153, 2153, 82*w - 425],
[2161, 2161, 10*w - 13],
[2161, 2161, -10*w - 3],
[2203, 2203, 46*w + 199],
[2203, 2203, 46*w - 245],
[2207, 2207, 24*w - 115],
[2207, 2207, 24*w + 91],
[2221, 2221, 10*w - 3],
[2221, 2221, 10*w - 7],
[2243, 2243, -12*w - 25],
[2243, 2243, 12*w - 37],
[2267, 2267, 36*w + 145],
[2267, 2267, 36*w - 181],
[2269, 2269, 60*w + 257],
[2269, 2269, 60*w - 317],
[2293, 2293, 44*w - 235],
[2293, 2293, 44*w + 191],
[2297, 2297, 8*w - 65],
[2297, 2297, -8*w - 57],
[2309, 2309, -18*w + 79],
[2309, 2309, 18*w + 61],
[2339, 2339, 20*w - 91],
[2339, 2339, 20*w + 71],
[2371, 2371, -116*w + 603],
[2371, 2371, 116*w + 487],
[2381, 2381, 54*w - 277],
[2381, 2381, 54*w + 223],
[2383, 2383, -26*w + 145],
[2383, 2383, -26*w - 119],
[2393, 2393, -104*w + 545],
[2393, 2393, 104*w + 441],
[2399, 2399, -10*w - 63],
[2399, 2399, 10*w - 73],
[2411, 2411, 2*w - 51],
[2411, 2411, -2*w - 49],
[2423, 2423, 48*w - 245],
[2423, 2423, 48*w + 197],
[2437, 2437, -28*w - 127],
[2437, 2437, -28*w + 155],
[2447, 2447, -16*w - 49],
[2447, 2447, 16*w - 65],
[2467, 2467, 38*w + 167],
[2467, 2467, 38*w - 205],
[2503, 2503, -34*w - 151],
[2503, 2503, 34*w - 185],
[2531, 2531, 70*w + 299],
[2531, 2531, 70*w - 369],
[2539, 2539, 100*w - 519],
[2539, 2539, 100*w + 419],
[2549, 2549, -20*w - 97],
[2549, 2549, -20*w + 117],
[2579, 2579, 12*w - 31],
[2579, 2579, -12*w - 19],
[2591, 2591, 160*w - 833],
[2591, 2591, 160*w + 673],
[2617, 2617, 34*w - 169],
[2617, 2617, 34*w + 135],
[2621, 2621, 134*w + 563],
[2621, 2621, -220*w - 929],
[2659, 2659, 20*w + 69],
[2659, 2659, 20*w - 89],
[2671, 2671, 16*w - 63],
[2671, 2671, -16*w - 47],
[2687, 2687, -22*w - 105],
[2687, 2687, -22*w + 127],
[2719, 2719, 54*w + 233],
[2719, 2719, 54*w - 287],
[2741, 2741, 68*w + 291],
[2741, 2741, 68*w - 359],
[2749, 2749, 70*w - 361],
[2749, 2749, 70*w + 291],
[2767, 2767, -102*w + 535],
[2767, 2767, 102*w + 433],
[2777, 2777, -126*w + 655],
[2777, 2777, 126*w + 529],
[2791, 2791, -18*w + 109],
[2791, 2791, -18*w - 91],
[2801, 2801, -8*w - 61],
[2801, 2801, 8*w - 69],
[2803, 2803, 28*w + 107],
[2803, 2803, 28*w - 135],
[2837, 2837, 52*w + 225],
[2837, 2837, 52*w - 277],
[2843, 2843, -12*w - 13],
[2843, 2843, 12*w - 25],
[2857, 2857, 112*w + 475],
[2857, 2857, -112*w + 587],
[2887, 2887, -16*w - 45],
[2887, 2887, 16*w - 61],
[2897, 2897, 152*w + 643],
[2897, 2897, -152*w + 795],
[2903, 2903, 32*w - 157],
[2903, 2903, 32*w + 125],
[2917, 2917, -14*w - 31],
[2917, 2917, 14*w - 45],
[2927, 2927, 64*w - 329],
[2927, 2927, 64*w + 265],
[2939, 2939, 50*w - 267],
[2939, 2939, 50*w + 217],
[2953, 2953, -16*w - 85],
[2953, 2953, -16*w + 101],
[2957, 2957, 86*w + 359],
[2957, 2957, 86*w - 445],
[2969, 2969, 110*w + 461],
[2969, 2969, 110*w - 571],
[2971, 2971, 20*w + 67],
[2971, 2971, -20*w + 87],
[3001, 3001, 94*w - 487],
[3001, 3001, 94*w + 393],
[3037, 3037, 12*w - 85],
[3037, 3037, -12*w - 73],
[3079, 3079, -30*w - 137],
[3079, 3079, -30*w + 167],
[3083, 3083, -12*w - 5],
[3083, 3083, 12*w - 17],
[3119, 3119, 72*w - 371],
[3119, 3119, 72*w + 299],
[3137, 3137, -32*w - 145],
[3137, 3137, -32*w + 177],
[3187, 3187, 42*w - 227],
[3187, 3187, -42*w - 185],
[3203, 3203, 12*w - 7],
[3203, 3203, 12*w - 5],
[3209, 3209, -40*w - 177],
[3209, 3209, 40*w - 217],
[3221, 3221, 34*w + 133],
[3221, 3221, 34*w - 167],
[3229, 3229, 38*w - 189],
[3229, 3229, 38*w + 151],
[3251, 3251, -10*w - 69],
[3251, 3251, 10*w - 79],
[3253, 3253, 76*w - 401],
[3253, 3253, 76*w + 325],
[3257, 3257, -88*w + 463],
[3257, 3257, 88*w + 375],
[3259, 3259, -110*w + 577],
[3259, 3259, 110*w + 467],
[3271, 3271, 80*w + 333],
[3271, 3271, 80*w - 413],
[3301, 3301, 206*w - 1073],
[3301, 3301, 206*w + 867],
[3313, 3313, -24*w - 115],
[3313, 3313, -24*w + 139],
[3329, 3329, 16*w - 103],
[3329, 3329, 16*w + 87],
[3343, 3343, 128*w + 537],
[3343, 3343, 128*w - 665],
[3361, 3361, -120*w + 629],
[3361, 3361, 120*w + 509],
[3371, 3371, 60*w - 307],
[3371, 3371, 60*w + 247],
[3373, 3373, 22*w + 75],
[3373, 3373, 22*w - 97],
[3391, 3391, 170*w + 719],
[3391, 3391, -170*w + 889],
[3407, 3407, 24*w - 109],
[3407, 3407, 24*w + 85],
[3449, 3449, -160*w + 837],
[3449, 3449, 242*w - 1261],
[3461, 3461, 14*w - 37],
[3461, 3461, -14*w - 23],
[3463, 3463, 32*w - 155],
[3463, 3463, 32*w + 123],
[3467, 3467, 98*w + 417],
[3467, 3467, -98*w + 515],
[3469, 3469, -140*w + 733],
[3469, 3469, 278*w - 1449],
[3481, 59, -59],
[3491, 3491, -180*w - 757],
[3491, 3491, 180*w - 937],
[3511, 3511, 2*w - 61],
[3511, 3511, -2*w - 59],
[3539, 3539, 86*w - 453],
[3539, 3539, 86*w + 367],
[3559, 3559, 40*w - 199],
[3559, 3559, 40*w + 159],
[3571, 3571, -20*w - 63],
[3571, 3571, 20*w - 83],
[3581, 3581, -28*w + 159],
[3581, 3581, -28*w - 131],
[3607, 3607, -18*w - 95],
[3607, 3607, 18*w - 113],
[3613, 3613, 4*w - 65],
[3613, 3613, -4*w - 61],
[3617, 3617, 8*w - 75],
[3617, 3617, -8*w - 67],
[3631, 3631, -22*w - 109],
[3631, 3631, -22*w + 131],
[3659, 3659, 68*w - 349],
[3659, 3659, 68*w + 281],
[3671, 3671, 16*w - 53],
[3671, 3671, -16*w - 37],
[3691, 3691, -30*w - 139],
[3691, 3691, -30*w + 169],
[3721, 61, -61],
[3727, 3727, 56*w - 285],
[3727, 3727, 56*w + 229],
[3733, 3733, 62*w + 255],
[3733, 3733, 62*w - 317],
[3739, 3739, -14*w - 83],
[3739, 3739, 14*w - 97],
[3793, 3793, 96*w + 409],
[3793, 3793, -96*w + 505],
[3823, 3823, -6*w - 65],
[3823, 3823, 6*w - 71],
[3847, 3847, 16*w - 51],
[3847, 3847, -16*w - 35],
[3863, 3863, -34*w + 189],
[3863, 3863, -34*w - 155],
[3877, 3877, -14*w - 15],
[3877, 3877, 14*w - 29],
[3907, 3907, 76*w + 315],
[3907, 3907, 76*w - 391],
[3911, 3911, 50*w + 219],
[3911, 3911, 50*w - 269],
[3917, 3917, 234*w + 985],
[3917, 3917, 234*w - 1219],
[3989, 3989, -190*w - 799],
[3989, 3989, 190*w - 989],
[4001, 4001, 40*w - 219],
[4001, 4001, -40*w - 179],
[4003, 4003, -26*w + 151],
[4003, 4003, -26*w - 125],
[4007, 4007, 2*w - 65],
[4007, 4007, -2*w - 63],
[4013, 4013, 44*w - 239],
[4013, 4013, -44*w - 195],
[4021, 4021, -12*w - 79],
[4021, 4021, 12*w - 91],
[4027, 4027, 18*w + 97],
[4027, 4027, 18*w - 115],
[4049, 4049, 70*w - 359],
[4049, 4049, 70*w + 289],
[4073, 4073, -18*w - 47],
[4073, 4073, 18*w - 65],
[4093, 4093, 58*w - 295],
[4093, 4093, 58*w + 237],
[4099, 4099, 6*w - 73],
[4099, 4099, -6*w - 67],
[4111, 4111, 64*w - 327],
[4111, 4111, 64*w + 263],
[4133, 4133, 4*w - 69],
[4133, 4133, -4*w - 65],
[4139, 4139, 20*w - 79],
[4139, 4139, -20*w - 59],
[4201, 4201, -8*w - 71],
[4201, 4201, 8*w - 79],
[4217, 4217, -14*w - 5],
[4217, 4217, 14*w - 19],
[4219, 4219, 28*w + 101],
[4219, 4219, 28*w - 129],
[4261, 4261, -14*w - 3],
[4261, 4261, 14*w - 17],
[4271, 4271, 40*w + 157],
[4271, 4271, 40*w - 197],
[4273, 4273, 264*w + 1115],
[4273, 4273, 182*w + 765],
[4283, 4283, 92*w + 383],
[4283, 4283, 92*w - 475],
[4289, 4289, 80*w - 423],
[4289, 4289, 80*w + 343],
[4297, 4297, -14*w - 1],
[4297, 4297, 14*w - 15],
[4327, 4327, -16*w - 29],
[4327, 4327, 16*w - 45],
[4339, 4339, -148*w + 769],
[4339, 4339, 148*w + 621],
[4357, 4357, 14*w - 9],
[4357, 4357, 14*w - 5],
[4363, 4363, -126*w + 661],
[4363, 4363, 126*w + 535],
[4397, 4397, 92*w + 393],
[4397, 4397, 92*w - 485],
[4441, 4441, -32*w - 149],
[4441, 4441, -32*w + 181],
[4451, 4451, 140*w - 727],
[4451, 4451, 140*w + 587],
[4507, 4507, 78*w - 413],
[4507, 4507, 78*w + 335],
[4517, 4517, 174*w + 731],
[4517, 4517, -292*w - 1233],
[4519, 4519, 200*w - 1041],
[4519, 4519, -200*w - 841],
[4523, 4523, 132*w + 553],
[4523, 4523, 132*w - 685],
[4547, 4547, 116*w - 601],
[4547, 4547, 116*w + 485],
[4549, 4549, 50*w + 201],
[4549, 4549, 50*w - 251],
[4561, 4561, 86*w - 443],
[4561, 4561, 86*w + 357],
[4583, 4583, 146*w + 619],
[4583, 4583, -146*w + 765],
[4603, 4603, 114*w + 485],
[4603, 4603, -114*w + 599],
[4637, 4637, 26*w - 115],
[4637, 4637, 26*w + 89],
[4639, 4639, -186*w + 973],
[4639, 4639, 280*w - 1459],
[4649, 4649, 30*w - 139],
[4649, 4649, 30*w + 109],
[4673, 4673, -176*w + 921],
[4673, 4673, 298*w - 1553],
[4721, 4721, -166*w + 863],
[4721, 4721, 166*w + 697],
[4733, 4733, 218*w + 917],
[4733, 4733, 218*w - 1135],
[4751, 4751, 56*w + 227],
[4751, 4751, 56*w - 283],
[4759, 4759, 80*w - 411],
[4759, 4759, 80*w + 331],
[4789, 4789, -20*w + 127],
[4789, 4789, 20*w + 107],
[4801, 4801, -40*w - 181],
[4801, 4801, -40*w + 221],
[4817, 4817, 8*w - 83],
[4817, 4817, -8*w - 75],
[4831, 4831, -10*w - 79],
[4831, 4831, 10*w - 89],
[4861, 4861, 52*w - 281],
[4861, 4861, -52*w - 229],
[4877, 4877, 74*w + 305],
[4877, 4877, 74*w - 379],
[4903, 4903, 46*w - 251],
[4903, 4903, -46*w - 205],
[4931, 4931, 20*w - 73],
[4931, 4931, -20*w - 53],
[4937, 4937, 88*w + 377],
[4937, 4937, 88*w - 465],
[4967, 4967, 16*w - 35],
[4967, 4967, -16*w - 19],
[4973, 4973, -4*w - 71],
[4973, 4973, 4*w - 75],
[4993, 4993, 22*w - 87],
[4993, 4993, -22*w - 65],
[5009, 5009, 150*w - 779],
[5009, 5009, 150*w + 629],
[5023, 5023, 88*w - 453],
[5023, 5023, 88*w + 365],
[5039, 5039, 48*w - 239],
[5039, 5039, 48*w + 191],
[5051, 5051, 44*w - 217],
[5051, 5051, 44*w + 173],
[5077, 5077, 12*w - 97],
[5077, 5077, -12*w - 85],
[5081, 5081, 210*w - 1093],
[5081, 5081, -210*w - 883],
[5107, 5107, 22*w - 137],
[5107, 5107, -22*w - 115],
[5113, 5113, 144*w + 611],
[5113, 5113, -144*w + 755],
[5153, 5153, 58*w - 293],
[5153, 5153, 58*w + 235],
[5167, 5167, 16*w - 31],
[5167, 5167, -16*w - 15],
[5171, 5171, 10*w - 91],
[5171, 5171, -10*w - 81],
[5179, 5179, 20*w - 71],
[5179, 5179, -20*w - 51],
[5209, 5209, 82*w + 339],
[5209, 5209, 82*w - 421],
[5231, 5231, 122*w + 519],
[5231, 5231, -122*w + 641],
[5233, 5233, 38*w - 183],
[5233, 5233, 38*w + 145],
[5261, 5261, 20*w - 129],
[5261, 5261, 20*w + 109],
[5273, 5273, 18*w - 53],
[5273, 5273, -18*w - 35],
[5323, 5323, -194*w + 1015],
[5323, 5323, 308*w - 1605],
[5351, 5351, -110*w + 579],
[5351, 5351, 110*w + 469],
[5387, 5387, 2*w - 75],
[5387, 5387, -2*w - 73],
[5393, 5393, 42*w - 205],
[5393, 5393, 42*w + 163],
[5407, 5407, 16*w - 25],
[5407, 5407, -16*w - 9],
[5413, 5413, 84*w - 445],
[5413, 5413, 84*w + 361],
[5419, 5419, -20*w - 49],
[5419, 5419, 20*w - 69],
[5431, 5431, 104*w + 433],
[5431, 5431, 104*w - 537],
[5437, 5437, 26*w - 111],
[5437, 5437, -26*w - 85],
[5449, 5449, 50*w - 249],
[5449, 5449, 50*w + 199],
[5471, 5471, -16*w - 7],
[5471, 5471, 16*w - 23],
[5479, 5479, 66*w - 353],
[5479, 5479, 66*w + 287],
[5501, 5501, 90*w + 373],
[5501, 5501, 90*w - 463],
[5507, 5507, -38*w + 213],
[5507, 5507, -38*w - 175],
[5519, 5519, -10*w - 83],
[5519, 5519, 10*w - 93],
[5527, 5527, 16*w - 21],
[5527, 5527, -16*w - 5],
[5557, 5557, 34*w + 125],
[5557, 5557, 34*w - 159],
[5563, 5563, -28*w + 123],
[5563, 5563, 28*w + 95],
[5573, 5573, -4*w - 75],
[5573, 5573, 4*w - 79],
[5591, 5591, -24*w - 73],
[5591, 5591, 24*w - 97],
[5623, 5623, 32*w - 147],
[5623, 5623, 32*w + 115],
[5639, 5639, -14*w - 93],
[5639, 5639, 14*w - 107],
[5641, 5641, -120*w + 631],
[5641, 5641, 120*w + 511],
[5647, 5647, 16*w - 15],
[5647, 5647, 16*w - 1],
[5651, 5651, -20*w - 47],
[5651, 5651, 20*w - 67],
[5657, 5657, 194*w + 815],
[5657, 5657, -328*w - 1385],
[5701, 5701, 108*w - 569],
[5701, 5701, 108*w + 461],
[5717, 5717, 66*w - 335],
[5717, 5717, 66*w + 269],
[5741, 5741, -20*w - 111],
[5741, 5741, 20*w - 131],
[5743, 5743, 128*w + 535],
[5743, 5743, 128*w - 663],
[5749, 5749, 60*w + 263],
[5749, 5749, 60*w - 323],
[5783, 5783, 72*w + 295],
[5783, 5783, 72*w - 367],
[5801, 5801, 8*w - 89],
[5801, 5801, -8*w - 81],
[5807, 5807, 58*w + 255],
[5807, 5807, 58*w - 313],
[5821, 5821, 12*w - 101],
[5821, 5821, -12*w - 89],
[5827, 5827, 44*w - 215],
[5827, 5827, 44*w + 171],
[5849, 5849, -56*w - 247],
[5849, 5849, 56*w - 303],
[5857, 5857, 48*w - 263],
[5857, 5857, -48*w - 215],
[5869, 5869, -22*w - 59],
[5869, 5869, 22*w - 81],
[5879, 5879, -50*w - 223],
[5879, 5879, 50*w - 273],
[5923, 5923, 6*w - 85],
[5923, 5923, -6*w - 79],
[5927, 5927, 24*w - 95],
[5927, 5927, -24*w - 71],
[5953, 5953, 24*w - 149],
[5953, 5953, -24*w - 125],
[5981, 5981, -172*w + 901],
[5981, 5981, 172*w + 729],
[6007, 6007, 78*w - 415],
[6007, 6007, 78*w + 337],
[6043, 6043, -14*w - 95],
[6043, 6043, 14*w - 109],
[6047, 6047, -202*w + 1057],
[6047, 6047, 336*w - 1751],
[6053, 6053, 18*w - 43],
[6053, 6053, -18*w - 25],
[6073, 6073, -178*w + 925],
[6073, 6073, 178*w + 747],
[6091, 6091, -20*w - 43],
[6091, 6091, 20*w - 63],
[6101, 6101, 30*w + 103],
[6101, 6101, 30*w - 133],
[6121, 6121, 34*w + 123],
[6121, 6121, 34*w - 157],
[6131, 6131, 22*w - 141],
[6131, 6131, 22*w + 119],
[6133, 6133, -36*w + 205],
[6133, 6133, -36*w - 169],
[6143, 6143, 32*w - 145],
[6143, 6143, 32*w + 113],
[6151, 6151, 256*w + 1077],
[6151, 6151, 256*w - 1333],
[6163, 6163, 68*w - 345],
[6163, 6163, 68*w + 277],
[6173, 6173, 76*w + 329],
[6173, 6173, 76*w - 405],
[6221, 6221, 170*w + 713],
[6221, 6221, 170*w - 883],
[6229, 6229, -20*w - 113],
[6229, 6229, 20*w - 133],
[6247, 6247, -18*w - 107],
[6247, 6247, 18*w - 125],
[6269, 6269, 122*w - 631],
[6269, 6269, 122*w + 509],
[6277, 6277, 46*w + 179],
[6277, 6277, 46*w - 225],
[6287, 6287, -38*w - 177],
[6287, 6287, -38*w + 215],
[6299, 6299, -20*w - 41],
[6299, 6299, 20*w - 61],
[6301, 6301, 230*w + 967],
[6301, 6301, 308*w + 1301],
[6311, 6311, 2*w - 81],
[6311, 6311, -2*w - 79],
[6317, 6317, 54*w - 269],
[6317, 6317, 54*w + 215],
[6323, 6323, 74*w - 395],
[6323, 6323, 74*w + 321],
[6329, 6329, 162*w - 841],
[6329, 6329, 162*w + 679],
[6337, 6337, 128*w + 545],
[6337, 6337, -128*w + 673],
[6353, 6353, 104*w + 445],
[6353, 6353, 104*w - 549],
[6359, 6359, 40*w - 191],
[6359, 6359, 40*w + 151],
[6361, 6361, 130*w + 543],
[6361, 6361, 130*w - 673],
[6397, 6397, 154*w - 799],
[6397, 6397, 154*w + 645],
[6481, 6481, 160*w + 679],
[6481, 6481, -160*w + 839],
[6529, 6529, 38*w - 179],
[6529, 6529, 38*w + 141],
[6547, 6547, -42*w - 193],
[6547, 6547, -42*w + 235],
[6569, 6569, 30*w + 101],
[6569, 6569, -30*w + 131],
[6577, 6577, -26*w - 79],
[6577, 6577, 26*w - 105],
[6581, 6581, -28*w + 169],
[6581, 6581, -28*w - 141],
[6607, 6607, 138*w + 587],
[6607, 6607, -138*w + 725],
[6653, 6653, -44*w - 201],
[6653, 6653, -44*w + 245],
[6659, 6659, 60*w + 241],
[6659, 6659, 60*w - 301],
[6673, 6673, -22*w - 53],
[6673, 6673, 22*w - 75],
[6679, 6679, -34*w - 163],
[6679, 6679, -34*w + 197],
[6691, 6691, 20*w - 57],
[6691, 6691, -20*w - 37],
[6709, 6709, -180*w + 943],
[6709, 6709, 180*w + 763],
[6719, 6719, 230*w + 973],
[6719, 6719, -230*w + 1203],
[6763, 6763, 66*w + 289],
[6763, 6763, 66*w - 355],
[6781, 6781, 346*w - 1803],
[6781, 6781, -220*w + 1151],
[6803, 6803, 52*w + 205],
[6803, 6803, 52*w - 257],
[6833, 6833, 64*w + 281],
[6833, 6833, 64*w - 345],
[6857, 6857, -8*w - 87],
[6857, 6857, 8*w - 95],
[6863, 6863, -26*w - 135],
[6863, 6863, 26*w - 161],
[6869, 6869, -4*w - 83],
[6869, 6869, 4*w - 87],
[6871, 6871, -30*w - 149],
[6871, 6871, -30*w + 179],
[6889, 83, -83],
[6917, 6917, 52*w - 285],
[6917, 6917, -52*w - 233],
[6947, 6947, 58*w - 315],
[6947, 6947, -58*w - 257],
[6959, 6959, 240*w + 1009],
[6959, 6959, 326*w + 1377],
[6967, 6967, 2*w - 85],
[6967, 6967, -2*w - 83],
[6991, 6991, 10*w - 101],
[6991, 6991, -10*w - 91],
[6997, 6997, -12*w - 95],
[6997, 6997, 12*w - 107],
[7013, 7013, 18*w - 23],
[7013, 7013, -18*w - 5],
[7027, 7027, 284*w - 1479],
[7027, 7027, 284*w + 1195],
[7039, 7039, 40*w - 189],
[7039, 7039, 40*w + 149],
[7103, 7103, 72*w + 293],
[7103, 7103, 72*w - 365],
[7109, 7109, -18*w - 1],
[7109, 7109, 18*w - 19],
[7121, 7121, 112*w - 591],
[7121, 7121, 112*w + 479],
[7129, 7129, 62*w + 249],
[7129, 7129, 62*w - 311],
[7159, 7159, 32*w + 109],
[7159, 7159, 32*w - 141],
[7177, 7177, -206*w + 1071],
[7177, 7177, 206*w + 865],
[7187, 7187, -22*w - 123],
[7187, 7187, 22*w - 145],
[7193, 7193, 18*w - 5],
[7193, 7193, 18*w - 13],
[7207, 7207, 82*w + 355],
[7207, 7207, 82*w - 437],
[7211, 7211, 98*w + 421],
[7211, 7211, 98*w - 519],
[7213, 7213, 4*w - 89],
[7213, 7213, -4*w - 85],
[7219, 7219, 20*w - 51],
[7219, 7219, -20*w - 31],
[7229, 7229, -20*w - 117],
[7229, 7229, 20*w - 137],
[7243, 7243, 28*w - 115],
[7243, 7243, -28*w - 87],
[7253, 7253, 258*w - 1343],
[7253, 7253, -258*w - 1085],
[7297, 7297, 26*w - 101],
[7297, 7297, -26*w - 75],
[7307, 7307, 2*w - 87],
[7307, 7307, -2*w - 85],
[7309, 7309, 118*w - 609],
[7309, 7309, 118*w + 491],
[7351, 7351, 320*w + 1347],
[7351, 7351, 320*w - 1667],
[7369, 7369, 80*w - 427],
[7369, 7369, 80*w + 347],
[7451, 7451, 110*w + 471],
[7451, 7451, 110*w - 581],
[7459, 7459, -26*w - 137],
[7459, 7459, 26*w - 163],
[7477, 7477, 188*w + 797],
[7477, 7477, -188*w + 985],
[7481, 7481, -30*w - 97],
[7481, 7481, 30*w - 127],
[7487, 7487, 24*w - 85],
[7487, 7487, -24*w - 61],
[7523, 7523, -134*w + 705],
[7523, 7523, 134*w + 571],
[7529, 7529, 16*w - 123],
[7529, 7529, -16*w - 107],
[7547, 7547, -238*w + 1245],
[7547, 7547, 356*w - 1855],
[7549, 7549, -44*w - 203],
[7549, 7549, -44*w + 247],
[7561, 7561, 190*w - 987],
[7561, 7561, 190*w + 797],
[7573, 7573, 98*w - 503],
[7573, 7573, 98*w + 405],
[7583, 7583, 48*w - 233],
[7583, 7583, 48*w + 185],
[7607, 7607, 224*w + 941],
[7607, 7607, -382*w - 1613],
[7643, 7643, -28*w - 85],
[7643, 7643, 28*w - 113],
[7649, 7649, 344*w + 1453],
[7649, 7649, 250*w + 1051],
[7699, 7699, 92*w - 471],
[7699, 7699, 92*w + 379],
[7703, 7703, 94*w - 499],
[7703, 7703, 94*w + 405],
[7723, 7723, 52*w - 255],
[7723, 7723, 52*w + 203],
[7727, 7727, 22*w - 147],
[7727, 7727, -22*w - 125],
[7741, 7741, -20*w - 119],
[7741, 7741, 20*w - 139],
[7753, 7753, 142*w - 735],
[7753, 7753, 142*w + 593],
[7759, 7759, 74*w - 397],
[7759, 7759, 74*w + 323],
[7793, 7793, 38*w + 137],
[7793, 7793, 38*w - 175],
[7823, 7823, -166*w + 871],
[7823, 7823, 166*w + 705],
[7841, 7841, -32*w - 159],
[7841, 7841, -32*w + 191],
[7853, 7853, 22*w - 65],
[7853, 7853, -22*w - 43],
[7877, 7877, 46*w + 175],
[7877, 7877, 46*w - 221],
[7879, 7879, -50*w + 277],
[7879, 7879, -50*w - 227],
[7901, 7901, 70*w - 353],
[7901, 7901, 70*w + 283],
[7919, 7919, 120*w + 499],
[7919, 7919, 120*w - 619],
[7937, 7937, 106*w - 545],
[7937, 7937, 106*w + 439],
[7963, 7963, -268*w - 1127],
[7963, 7963, 268*w - 1395],
[7993, 7993, 16*w - 125],
[7993, 7993, -16*w - 109],
[8009, 8009, 176*w + 747],
[8009, 8009, -176*w + 923],
[8011, 8011, 2*w - 91],
[8011, 8011, -2*w - 89],
[8059, 8059, 20*w - 39],
[8059, 8059, -20*w - 19],
[8081, 8081, -40*w - 189],
[8081, 8081, -40*w + 229],
[8089, 8089, 64*w - 347],
[8089, 8089, -64*w - 283],
[8101, 8101, 50*w + 193],
[8101, 8101, 50*w - 243],
[8117, 8117, 66*w + 265],
[8117, 8117, 66*w - 331],
[8167, 8167, -18*w - 115],
[8167, 8167, 18*w - 133],
[8171, 8171, 20*w - 37],
[8171, 8171, -20*w - 17],
[8179, 8179, 10*w - 107],
[8179, 8179, -10*w - 97],
[8209, 8209, -24*w - 133],
[8209, 8209, 24*w - 157],
[8233, 8233, 82*w - 417],
[8233, 8233, 82*w + 335],
[8237, 8237, 54*w - 265],
[8237, 8237, 54*w + 211],
[8243, 8243, 266*w + 1125],
[8243, 8243, -266*w + 1391],
[8269, 8269, 22*w - 61],
[8269, 8269, -22*w - 39],
[8273, 8273, 42*w - 197],
[8273, 8273, 42*w + 155],
[8287, 8287, 118*w + 505],
[8287, 8287, 118*w - 623],
[8293, 8293, -4*w - 91],
[8293, 8293, 4*w - 95],
[8297, 8297, -34*w + 149],
[8297, 8297, 34*w + 115],
[8311, 8311, 136*w - 703],
[8311, 8311, 136*w + 567],
[8317, 8317, 26*w - 95],
[8317, 8317, -26*w - 69],
[8377, 8377, -8*w - 95],
[8377, 8377, 8*w - 103],
[8387, 8387, 36*w + 125],
[8387, 8387, 36*w - 161],
[8419, 8419, -28*w - 81],
[8419, 8419, 28*w - 109],
[8423, 8423, 192*w - 997],
[8423, 8423, 192*w + 805],
[8447, 8447, 144*w - 745],
[8447, 8447, 144*w + 601],
[8527, 8527, 184*w - 955],
[8527, 8527, 184*w + 771],
[8539, 8539, -20*w - 9],
[8539, 8539, 20*w - 29],
[8543, 8543, 152*w - 787],
[8543, 8543, 152*w + 635],
[8597, 8597, -226*w + 1175],
[8597, 8597, 226*w + 949],
[8599, 8599, 160*w - 829],
[8599, 8599, 160*w + 669],
[8623, 8623, 32*w - 135],
[8623, 8623, -32*w - 103],
[8629, 8629, 116*w + 497],
[8629, 8629, 116*w - 613],
[8641, 8641, -26*w - 67],
[8641, 8641, 26*w - 93],
[8669, 8669, 4*w - 97],
[8669, 8669, -4*w - 93],
[8677, 8677, -12*w - 103],
[8677, 8677, 12*w - 115],
[8713, 8713, 322*w + 1355],
[8713, 8713, 322*w - 1677],
[8731, 8731, -20*w - 3],
[8731, 8731, 20*w - 23],
[8747, 8747, 2*w - 95],
[8747, 8747, -2*w - 93],
[8761, 8761, -48*w + 269],
[8761, 8761, -48*w - 221],
[8779, 8779, -20*w - 1],
[8779, 8779, 20*w - 21],
[8803, 8803, -6*w - 95],
[8803, 8803, 6*w - 101],
[8807, 8807, 96*w + 395],
[8807, 8807, 96*w - 491],
[8819, 8819, 20*w - 19],
[8819, 8819, 20*w - 1],
[8821, 8821, -34*w + 147],
[8821, 8821, 34*w + 113],
[8831, 8831, 22*w - 151],
[8831, 8831, -22*w - 129],
[8861, 8861, 100*w + 431],
[8861, 8861, 100*w - 531],
[8969, 8969, 50*w + 191],
[8969, 8969, 50*w - 241],
[8971, 8971, 162*w + 689],
[8971, 8971, -162*w + 851],
[8999, 8999, 194*w + 823],
[8999, 8999, -194*w + 1017],
[9007, 9007, 296*w - 1541],
[9007, 9007, 296*w + 1245],
[9011, 9011, 10*w - 111],
[9011, 9011, -10*w - 101],
[9029, 9029, 210*w + 881],
[9029, 9029, 210*w - 1091],
[9067, 9067, 244*w + 1025],
[9067, 9067, -418*w - 1765],
[9103, 9103, -54*w - 245],
[9103, 9103, -54*w + 299],
[9127, 9127, 2*w - 97],
[9127, 9127, -2*w - 95],
[9133, 9133, -204*w + 1069],
[9133, 9133, 204*w + 865],
[9151, 9151, -42*w + 241],
[9151, 9151, -42*w - 199],
[9157, 9157, 146*w - 755],
[9157, 9157, 146*w + 609],
[9187, 9187, 124*w - 639],
[9187, 9187, 124*w + 515],
[9199, 9199, 6*w - 103],
[9199, 9199, -6*w - 97],
[9203, 9203, 74*w - 399],
[9203, 9203, 74*w + 325],
[9209, 9209, 30*w - 119],
[9209, 9209, -30*w - 89],
[9239, 9239, 48*w + 181],
[9239, 9239, 48*w - 229],
[9257, 9257, -32*w + 195],
[9257, 9257, -32*w - 163],
[9277, 9277, 314*w + 1321],
[9277, 9277, 314*w - 1635],
[9281, 9281, 154*w - 797],
[9281, 9281, 154*w + 643],
[9311, 9311, 70*w - 379],
[9311, 9311, -70*w - 309],
[9323, 9323, 412*w - 2147],
[9323, 9323, -254*w + 1329],
[9337, 9337, 34*w - 145],
[9337, 9337, -34*w - 111],
[9341, 9341, -68*w - 301],
[9341, 9341, 68*w - 369],
[9343, 9343, 448*w - 2335],
[9343, 9343, -234*w + 1225],
[9349, 9349, 430*w - 2241],
[9349, 9349, -244*w + 1277],
[9377, 9377, 186*w - 965],
[9377, 9377, 186*w + 779],
[9413, 9413, 178*w + 745],
[9413, 9413, 178*w - 923],
[9433, 9433, -16*w - 115],
[9433, 9433, 16*w - 131],
[9439, 9439, -10*w - 103],
[9439, 9439, 10*w - 113],
[9473, 9473, 22*w - 47],
[9473, 9473, -22*w - 25],
[9479, 9479, -288*w - 1211],
[9479, 9479, 288*w - 1499],
[9491, 9491, 76*w - 383],
[9491, 9491, 76*w + 307],
[9521, 9521, 160*w + 681],
[9521, 9521, -160*w + 841],
[9533, 9533, 124*w - 655],
[9533, 9533, 124*w + 531],
[9539, 9539, 52*w + 199],
[9539, 9539, 52*w - 251],
[9587, 9587, 44*w - 205],
[9587, 9587, 44*w + 161],
[9601, 9601, -8*w - 101],
[9601, 9601, 8*w - 109],
[9613, 9613, -22*w - 23],
[9613, 9613, 22*w - 45],
[9623, 9623, -34*w + 205],
[9623, 9623, -34*w - 171],
[9629, 9629, 38*w + 131],
[9629, 9629, 38*w - 169],
[9661, 9661, -148*w + 779],
[9661, 9661, 148*w + 631],
[9679, 9679, 40*w - 181],
[9679, 9679, 40*w + 141],
[9697, 9697, 192*w + 815],
[9697, 9697, -192*w + 1007],
[9719, 9719, 72*w + 289],
[9719, 9719, 72*w - 361],
[9721, 9721, 350*w - 1823],
[9721, 9721, 350*w + 1473],
[9733, 9733, 82*w + 333],
[9733, 9733, 82*w - 415],
[9743, 9743, 112*w + 463],
[9743, 9743, 112*w - 575],
[9769, 9769, -40*w + 233],
[9769, 9769, -40*w - 193],
[9781, 9781, 28*w + 151],
[9781, 9781, 28*w - 179],
[9791, 9791, 24*w - 67],
[9791, 9791, -24*w - 43],
[9811, 9811, 220*w + 923],
[9811, 9811, 220*w - 1143],
[9829, 9829, 50*w + 189],
[9829, 9829, 50*w - 239],
[9839, 9839, 170*w + 723],
[9839, 9839, -170*w + 893],
[9857, 9857, 26*w - 85],
[9857, 9857, -26*w - 59],
[9859, 9859, 68*w - 339],
[9859, 9859, 68*w + 271],
[9871, 9871, 90*w + 391],
[9871, 9871, 90*w - 481],
[9883, 9883, 148*w - 765],
[9883, 9883, 148*w + 617],
[9887, 9887, 122*w - 645],
[9887, 9887, 122*w + 523],
[9901, 9901, 106*w + 437],
[9901, 9901, 106*w - 543],
[9923, 9923, 26*w - 171],
[9923, 9923, -26*w - 145],
[9929, 9929, -16*w - 117],
[9929, 9929, 16*w - 133],
[9967, 9967, 22*w - 155],
[9967, 9967, -22*w - 133],
[9973, 9973, -36*w - 179],
[9973, 9973, -36*w + 215]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-2, -2, 1, 1, -5, 1, 1, -2, -2, 8, 8, -10, -6, -6, -7, -7, -3, -3, 4, 4, -10, -10, 15, -7, -7, 18, 18, 10, 10, -18, -18, 10, 10, -7, -7, -12, -12, -10, -6, -6, -15, -15, 0, 0, 19, 19, 18, 18, 24, 24, -23, -23, -2, -2, 14, 14, 10, 10, -28, -28, -2, -2, 4, 4, 8, 8, 12, 12, 13, 13, 7, 7, 28, 28, -38, -17, -17, -26, -26, 2, 2, -30, -30, -11, -11, 35, 35, 12, 12, -11, -11, -27, -27, 20, 20, 23, 23, 15, 15, -16, -16, -45, 28, 28, 2, 2, -22, -22, -16, -16, -25, -25, 7, 7, -33, -33, 29, 29, 10, 10, 14, 14, 17, 17, 2, 2, -36, -36, -23, -23, -22, -22, 12, 12, 20, 20, 53, 53, 0, 0, -38, -38, 22, 22, 39, 39, -58, -43, -43, -12, -12, 12, 12, -30, -30, 8, 8, -27, -27, -13, 47, 47, -27, -27, 39, 39, 38, 38, 39, 39, -10, -10, 22, 22, 2, 2, 44, 44, -20, -20, -51, -51, 48, 48, 24, 24, -31, -31, -21, -21, 2, 2, 60, 60, 18, 18, -8, -8, 27, 27, 39, 39, -30, -30, 68, 68, -72, -72, -65, 60, 60, 29, 29, -70, -70, 54, 54, 22, 22, 32, 32, -36, -36, 37, 37, -41, -41, 32, 32, -15, -15, -56, -56, -32, -32, 2, 2, 33, 33, -6, -6, -20, -20, 78, 78, -2, -2, 50, 50, -18, -6, -6, 40, 40, -45, -45, -46, -46, -6, -6, 17, 17, -40, -40, 10, 10, 52, 52, -52, -52, -50, 62, 62, 28, 28, -3, -3, -6, -6, 18, 18, -35, -35, 70, 70, 77, 77, -36, -36, 54, 54, -40, -40, 30, 30, -72, -72, 4, 4, -13, -13, 63, 63, 45, 45, 60, 60, 24, 24, 70, 70, -18, -18, 89, 89, 48, 48, -10, -10, -38, -38, -68, -68, 73, 73, -58, -58, -26, -26, -13, -13, -1, -1, 48, 48, 22, 22, -56, -56, 93, 93, 25, 25, 29, 29, -57, -57, 60, 60, 10, 10, -28, -28, -18, -18, -36, -36, 54, 54, -75, -75, 62, 62, -31, -31, -82, -82, 3, 3, 3, 3, 14, 14, 57, 57, 0, 0, -20, -20, 20, 20, -58, -58, 18, 18, 22, 22, 40, 40, 72, 72, 23, 23, -70, -70, -58, -58, 50, 50, 48, 48, -42, -42, 42, 42, 52, 52, 44, 44, -62, -62, 4, 4, -82, -82, -57, -57, 38, 38, 54, 54, 88, 88, -72, -72, -50, -50, -86, -86, 3, 3, 70, 70, -53, -53, 27, 27, 13, 13, -20, -20, 29, 29, -90, -90, 8, 8, -2, -2, -6, -6, -10, -10, -103, -103, 70, 70, -48, -48, 74, 74, 58, 58, -60, -60, -3, -3, -73, -73, 4, 4, -100, -100, 44, 44, -88, -88, -103, -103, 4, 4, 92, 92, 18, 18, 40, 40, -38, -38, -96, -96, 38, 38, -85, -85, -93, 17, 17, 12, 12, 20, 20, 0, 0, -28, -28, 32, 32, 58, 58, -26, -26, 63, 63, 32, 32, 30, 30, -78, -78, 92, 92, 22, 23, 23, 114, 114, -60, -60, 34, 34, 84, 84, -42, -42, 54, 54, 58, 58, -22, -22, 12, 12, -57, -57, -90, -90, 102, 102, -46, -46, -32, -32, 54, 54, 22, 22, 28, 28, 25, 25, -31, -31, 94, 94, 20, 20, 62, 62, 4, 4, 20, 20, -98, -98, 33, 33, 10, 10, -113, -113, -53, -53, -41, -41, 39, 39, -60, -60, -62, -62, -107, -107, 55, 55, -117, -117, -6, -6, 108, 108, 42, 42, 102, 102, -82, -82, 8, 8, 5, 5, -36, -36, -12, -12, -10, -10, 112, 112, -36, -36, 89, 89, 18, 18, 20, 20, -80, -80, 114, 114, 22, 22, 129, 129, -48, -48, 30, 30, 110, 110, 77, 77, -132, -132, -68, -68, -88, -88, 93, 93, -126, -126, 32, 32, -42, -42, -52, -52, -1, -1, 44, 44, -45, -45, 54, 54, 15, 15, -48, -48, 8, 8, 132, 132, -32, -32, 29, 29, -111, -111, 28, 28, 97, 97, -60, -60, -50, -50, -18, -18, -96, -96, 87, 87, -21, -21, -76, -76, 77, 77, -2, -2, -11, -11, 138, 138, 14, 14, -110, -110, 82, 82, -2, -2, -5, -5, -3, -3, -95, -95, 27, 27, 48, 48, 0, 0, 8, 8, -122, -122, 134, 134, 104, 104, -33, -33, 64, 64, 60, 60, 17, 17, 113, 113, -48, -48, -57, -57, 77, 77, -72, -72, -8, -8, 19, 19, -30, -30, 104, 104, 102, 102, -92, -92, -78, -78, -12, -12, 80, 80, -137, -137, 10, 10, -55, -55, 59, 59, 123, 123, 94, 94, -18, -18, 8, 8, -101, -101, -42, -42, -26, -26, -26, -26, -8, -8, -48, -48, 97, 97, 57, 57, 94, 94, 4, 4, 52, 52, 59, 59, -86, -86, -128, -128, -115, -115, -102, -102, -30, -30, -132, -132, -132, -132, -30, -30, -23, -23, 12, 12, -147, -147, 89, 89, 50, 50, 38, 38, -86, -86, 60, 60, -138, -138, -22, -22, -18, -18, -80, -80, 68, 68, 130, 130, -72, -72, -93, -93, 48, 48, 69, 69, -95, -95, 4, 4, 40, 40, -83, -83, 60, 60, -120, -120, 29, 29, 7, 7, -11, -11, 24, 24, -127, -127, 64, 64, 85, 85, 122, 122, -130, 38, 38, -72, -72, 30, 30, 8, 8, 142, 142, 23, 23, -126, -126, 23, 23, -60, -60, 24, 24, 15, 15, -3, -3, -145, -145, -105, -105, -142, -142, 153, 153, -6, -6, 78, 78, 162, 162, -66, -66, 25, 25, -110, -110, -21, -21, 59, 59, 138, 138, -7, -7, -25, -25, -48, -48, -80, -80, -148, -148, 135, 135, 138, 138, -43, -43, 78, 78, 24, 24, 145, 145, 123, 123, -45, -45, -138, -138, 109, 109, 64, 64, 38, 38, 159, 159, -130, -130, 50, 50, -51, -51, -151, -151, 33, 33, 92, 92, 19, 19, 55, 55, -21, -21, 14, 14, 17, 17, -96, -96, -122, -122, -165, -165, 127, 127, 20, 20, 18, 18, -136, -136, -26, -26, 85, 85, 137, 137, 130, 130, -18, -18, 135, 135, 77, 77, 168, 168, 123, 123, -28, -28, -130, -130, -35, -35, 169, 169, -17, -17, 99, 99, -100, -100, -26, -26, -72, -72, -6, -6, -117, -117, -38, -38, 18, 18, -92, -92, 168, 168, -60, -60, -156, -156, -152, -152, 148, 148, 145, 145, -126, -126, 78, 78, 80, 80, 74, 74, 145, 145, -108, -108, -130, -130, 98, 98, -61, -61, 82, 82, -32, -32, -13, -13, 55, 55, 124, 124, 48, 48, 50, 50, -78, -78, 157, 157, 12, 12, 15, 15, -128, -128, 25, 25, 153, 153, -38, -38, 115, 115, -147, -147, 14, 14, -12, -12, 79, 79, 102, 102, 63, 63, 58, 58, 80, 80, -166, -166, 120, 120, -60, -60, 88, 88, 43, 43, 132, 132, -13, -13, 134, 134, -67, -67, 42, 42, -151, -151, -100, -100, -142, -142, -166, -166, 94, 94, 65, 65, 114, 114, 60, 60, -108, -108, -78, -78, -36, -36, 0, 0, 18, 18, -173, -173, 64, 64, -61, -61, 25, 25, -13, -13, 30, 30, 178, 178, 110, 110, -178, -178, 49, 49, 164, 164, -5, -5, -18, -18, -108, -108, 62, 62, -80, -80, -45, -45, -117, -117, -85, -85, -128, -128, -81, -81, -117, -117, 102, 102, 49, 49, 0, 0, 43, 43, 4, 4];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, 2*w - 11}>] := -1;
ALEigenvalues[ideal<ZF | {11, 11, -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;