/* 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 := PolynomialRing(Rationals()); g := P![-22, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -3*w + 14], [3, 3, -w + 5], [3, 3, w + 5], [7, 7, 2*w + 9], [7, 7, 2*w - 9], [11, 11, -7*w + 33], [13, 13, -w - 3], [13, 13, -w + 3], [25, 5, -5], [29, 29, 3*w + 13], [29, 29, -3*w + 13], [59, 59, -w - 9], [59, 59, w - 9], [61, 61, 11*w - 51], [61, 61, 25*w - 117], [67, 67, 9*w - 43], [67, 67, -9*w - 43], [79, 79, 2*w - 3], [79, 79, -2*w - 3], [89, 89, 4*w - 21], [89, 89, -4*w - 21], [97, 97, -18*w + 85], [97, 97, 24*w - 113], [101, 101, -9*w - 41], [101, 101, -9*w + 41], [109, 109, 5*w + 21], [109, 109, -5*w + 21], [113, 113, -8*w - 39], [113, 113, 8*w - 39], [127, 127, 4*w - 15], [127, 127, -4*w - 15], [137, 137, 2*w - 15], [137, 137, -2*w - 15], [149, 149, -3*w - 7], [149, 149, 3*w - 7], [151, 151, 20*w - 93], [151, 151, 34*w - 159], [163, 163, 3*w - 19], [163, 163, -3*w - 19], [167, 167, 6*w + 25], [167, 167, 6*w - 25], [173, 173, 3*w - 5], [173, 173, -3*w - 5], [179, 179, -5*w - 27], [179, 179, 5*w - 27], [197, 197, -3*w - 1], [197, 197, 3*w - 1], [239, 239, 60*w - 281], [239, 239, 18*w - 83], [251, 251, 13*w - 63], [251, 251, -85*w + 399], [257, 257, -26*w + 123], [257, 257, 44*w - 207], [263, 263, 6*w - 23], [263, 263, -6*w - 23], [271, 271, -4*w - 9], [271, 271, 4*w - 9], [277, 277, 29*w - 135], [277, 277, 43*w - 201], [289, 17, -17], [293, 293, 21*w - 97], [293, 293, 63*w - 295], [313, 313, -12*w - 59], [313, 313, 12*w - 59], [331, 331, -3*w - 23], [331, 331, 3*w - 23], [349, 349, 7*w - 27], [349, 349, 7*w + 27], [353, 353, 2*w - 21], [353, 353, -2*w - 21], [359, 359, -12*w - 53], [359, 359, 12*w - 53], [361, 19, -19], [373, 373, 89*w - 417], [373, 373, 19*w - 87], [379, 379, 15*w - 73], [379, 379, -15*w - 73], [401, 401, 10*w - 51], [401, 401, -10*w - 51], [419, 419, -w - 21], [419, 419, w - 21], [431, 431, -6*w - 19], [431, 431, 6*w - 19], [433, 433, -6*w - 35], [433, 433, 6*w - 35], [439, 439, 38*w - 177], [439, 439, 52*w - 243], [443, 443, 7*w - 39], [443, 443, -7*w - 39], [449, 449, -14*w - 69], [449, 449, 14*w - 69], [461, 461, 15*w - 67], [461, 461, -15*w - 67], [467, 467, 67*w - 315], [467, 467, -31*w + 147], [479, 479, 30*w - 139], [479, 479, 72*w - 337], [499, 499, 21*w - 101], [499, 499, -105*w + 493], [503, 503, 6*w - 17], [503, 503, -6*w - 17], [521, 521, 58*w - 273], [521, 521, -40*w + 189], [529, 23, -23], [541, 541, -5*w - 3], [541, 541, 5*w - 3], [557, 557, 9*w - 35], [557, 557, 9*w + 35], [577, 577, 6*w - 37], [577, 577, -6*w - 37], [587, 587, -11*w - 57], [587, 587, 11*w - 57], [607, 607, 28*w + 129], [607, 607, -28*w + 129], [613, 613, 23*w - 105], [613, 613, 121*w - 567], [617, 617, -8*w - 45], [617, 617, 8*w - 45], [619, 619, -9*w - 49], [619, 619, 9*w - 49], [641, 641, 2*w - 27], [641, 641, -2*w - 27], [643, 643, -3*w - 29], [643, 643, 3*w - 29], [677, 677, -21*w - 95], [677, 677, 21*w - 95], [683, 683, -23*w - 111], [683, 683, -23*w + 111], [691, 691, -33*w + 157], [691, 691, 93*w - 437], [701, 701, 39*w - 181], [701, 701, 81*w - 379], [733, 733, -17*w - 75], [733, 733, 17*w - 75], [743, 743, -6*w - 7], [743, 743, 6*w - 7], [821, 821, 9*w - 31], [821, 821, -9*w - 31], [853, 853, -7*w - 15], [853, 853, 7*w - 15], [859, 859, 75*w - 353], [859, 859, -51*w + 241], [877, 877, 37*w - 171], [877, 877, 107*w - 501], [881, 881, 32*w - 153], [881, 881, -32*w - 153], [883, 883, 69*w - 325], [883, 883, -57*w + 269], [887, 887, -18*w + 79], [887, 887, 18*w + 79], [907, 907, 21*w - 103], [907, 907, -21*w - 103], [919, 919, -32*w + 147], [919, 919, 32*w + 147], [929, 929, 16*w - 81], [929, 929, -16*w - 81], [941, 941, 9*w - 29], [941, 941, -9*w - 29], [947, 947, -7*w - 45], [947, 947, 7*w - 45], [961, 31, -31], [967, 967, 8*w - 21], [967, 967, -8*w - 21], [971, 971, -5*w - 39], [971, 971, 5*w - 39], [977, 977, -28*w + 135], [977, 977, -28*w - 135], [997, 997, 7*w - 9], [997, 997, -7*w - 9], [1031, 1031, -30*w + 137], [1031, 1031, -30*w - 137], [1049, 1049, 10*w - 57], [1049, 1049, -10*w - 57], [1063, 1063, 14*w + 57], [1063, 1063, 14*w - 57], [1069, 1069, 7*w - 3], [1069, 1069, -7*w - 3], [1117, 1117, 13*w - 51], [1117, 1117, 13*w + 51], [1123, 1123, -27*w - 131], [1123, 1123, 27*w - 131], [1151, 1151, 96*w - 449], [1151, 1151, 54*w - 251], [1153, 1153, 18*w - 91], [1153, 1153, -18*w - 91], [1171, 1171, 3*w - 37], [1171, 1171, -3*w - 37], [1193, 1193, 8*w - 51], [1193, 1193, -8*w - 51], [1223, 1223, -24*w - 107], [1223, 1223, 24*w - 107], [1229, 1229, 15*w + 61], [1229, 1229, 15*w - 61], [1231, 1231, 20*w + 87], [1231, 1231, -20*w + 87], [1259, 1259, -59*w + 279], [1259, 1259, 95*w - 447], [1291, 1291, 15*w - 79], [1291, 1291, -15*w - 79], [1307, 1307, -11*w - 63], [1307, 1307, 11*w - 63], [1319, 1319, 12*w - 43], [1319, 1319, -12*w - 43], [1321, 1321, -12*w - 67], [1321, 1321, 12*w - 67], [1327, 1327, -8*w - 9], [1327, 1327, 8*w - 9], [1369, 37, -37], [1381, 1381, 19*w + 81], [1381, 1381, -19*w + 81], [1399, 1399, -8*w - 3], [1399, 1399, 8*w - 3], [1409, 1409, 136*w - 639], [1409, 1409, -46*w + 219], [1429, 1429, 25*w - 111], [1429, 1429, -25*w - 111], [1433, 1433, 2*w - 39], [1433, 1433, -2*w - 39], [1447, 1447, 74*w - 345], [1447, 1447, 88*w - 411], [1471, 1471, -10*w - 27], [1471, 1471, 10*w - 27], [1483, 1483, -3*w - 41], [1483, 1483, 3*w - 41], [1487, 1487, -12*w - 41], [1487, 1487, 12*w - 41], [1489, 1489, -24*w - 119], [1489, 1489, 24*w - 119], [1493, 1493, 9*w - 17], [1493, 1493, -9*w - 17], [1499, 1499, -w - 39], [1499, 1499, w - 39], [1523, 1523, -7*w - 51], [1523, 1523, 7*w - 51], [1559, 1559, -42*w + 193], [1559, 1559, 42*w + 193], [1571, 1571, 127*w - 597], [1571, 1571, -55*w + 261], [1583, 1583, 66*w - 307], [1583, 1583, 108*w - 505], [1597, 1597, 17*w + 69], [1597, 1597, 17*w - 69], [1609, 1609, 6*w - 49], [1609, 1609, -6*w - 49], [1613, 1613, 9*w - 13], [1613, 1613, -9*w - 13], [1663, 1663, 16*w - 63], [1663, 1663, 16*w + 63], [1669, 1669, 35*w - 159], [1669, 1669, -35*w - 159], [1681, 41, -41], [1693, 1693, 13*w - 45], [1693, 1693, -13*w - 45], [1697, 1697, -64*w + 303], [1697, 1697, 118*w - 555], [1699, 1699, 9*w - 59], [1699, 1699, -9*w - 59], [1721, 1721, -38*w - 183], [1721, 1721, -38*w + 183], [1733, 1733, -9*w - 7], [1733, 1733, 9*w - 7], [1747, 1747, -21*w - 107], [1747, 1747, 21*w - 107], [1753, 1753, -48*w + 229], [1753, 1753, 162*w - 761], [1759, 1759, -10*w - 21], [1759, 1759, 10*w - 21], [1787, 1787, 109*w - 513], [1787, 1787, -73*w + 345], [1789, 1789, 83*w - 387], [1789, 1789, 97*w - 453], [1823, 1823, 114*w - 533], [1823, 1823, 72*w - 335], [1847, 1847, 48*w + 221], [1847, 1847, -48*w + 221], [1849, 43, -43], [1861, 1861, -29*w - 129], [1861, 1861, 29*w - 129], [1873, 1873, -12*w - 71], [1873, 1873, 12*w - 71], [1877, 1877, -27*w - 119], [1877, 1877, 27*w - 119], [1907, 1907, 13*w - 75], [1907, 1907, -13*w - 75], [1933, 1933, 11*w - 27], [1933, 1933, -11*w - 27], [1949, 1949, 75*w - 349], [1949, 1949, 117*w - 547], [1997, 1997, 51*w - 235], [1997, 1997, 177*w - 829], [1999, 1999, 22*w + 93], [1999, 1999, -22*w + 93], [2003, 2003, -w - 45], [2003, 2003, w - 45], [2011, 2011, -3*w - 47], [2011, 2011, 3*w - 47], [2017, 2017, -6*w - 53], [2017, 2017, 6*w - 53], [2027, 2027, -47*w + 225], [2027, 2027, 47*w + 225], [2053, 2053, 157*w - 735], [2053, 2053, 59*w - 273], [2063, 2063, 24*w + 103], [2063, 2063, -24*w + 103], [2083, 2083, 147*w - 691], [2083, 2083, -63*w + 299], [2087, 2087, 18*w - 71], [2087, 2087, 18*w + 71], [2099, 2099, 11*w - 69], [2099, 2099, -11*w - 69], [2111, 2111, -30*w - 133], [2111, 2111, 30*w - 133], [2113, 2113, 36*w - 175], [2113, 2113, -36*w - 175], [2137, 2137, -66*w + 313], [2137, 2137, 144*w - 677], [2141, 2141, -15*w - 53], [2141, 2141, 15*w - 53], [2161, 2161, 12*w - 73], [2161, 2161, -12*w - 73], [2179, 2179, -21*w - 109], [2179, 2179, 21*w - 109], [2203, 2203, 3*w - 49], [2203, 2203, -3*w - 49], [2207, 2207, 12*w - 31], [2207, 2207, -12*w - 31], [2209, 47, -47], [2213, 2213, 123*w - 575], [2213, 2213, 81*w - 377], [2221, 2221, 11*w - 21], [2221, 2221, -11*w - 21], [2239, 2239, 20*w + 81], [2239, 2239, 20*w - 81], [2267, 2267, 31*w - 153], [2267, 2267, -31*w - 153], [2281, 2281, -18*w - 97], [2281, 2281, 18*w - 97], [2287, 2287, -14*w - 45], [2287, 2287, 14*w - 45], [2297, 2297, 182*w - 855], [2297, 2297, -56*w + 267], [2309, 2309, 57*w - 263], [2309, 2309, 183*w - 857], [2347, 2347, -81*w + 383], [2347, 2347, 129*w - 607], [2351, 2351, 84*w - 391], [2351, 2351, 126*w - 589], [2377, 2377, 126*w - 593], [2377, 2377, -84*w + 397], [2383, 2383, 16*w - 57], [2383, 2383, -16*w - 57], [2389, 2389, 17*w - 63], [2389, 2389, -17*w - 63], [2437, 2437, -11*w - 15], [2437, 2437, 11*w - 15], [2467, 2467, 111*w - 523], [2467, 2467, -99*w + 467], [2473, 2473, -102*w + 481], [2473, 2473, 108*w - 509], [2477, 2477, 21*w + 85], [2477, 2477, 21*w - 85], [2503, 2503, 166*w - 777], [2503, 2503, 68*w - 315], [2531, 2531, -65*w + 309], [2531, 2531, 173*w - 813], [2539, 2539, -45*w - 217], [2539, 2539, -45*w + 217], [2543, 2543, -12*w - 25], [2543, 2543, 12*w - 25], [2549, 2549, -15*w - 49], [2549, 2549, 15*w - 49], [2551, 2551, -26*w + 111], [2551, 2551, 26*w + 111], [2579, 2579, -w - 51], [2579, 2579, w - 51], [2591, 2591, -36*w - 161], [2591, 2591, 36*w - 161], [2633, 2633, 28*w - 141], [2633, 2633, -28*w - 141], [2647, 2647, -32*w - 141], [2647, 2647, 32*w - 141], [2689, 2689, -6*w - 59], [2689, 2689, 6*w - 59], [2699, 2699, -5*w - 57], [2699, 2699, 5*w - 57], [2707, 2707, -9*w - 67], [2707, 2707, 9*w - 67], [2719, 2719, 212*w - 993], [2719, 2719, 58*w - 267], [2729, 2729, -74*w + 351], [2729, 2729, 164*w - 771], [2731, 2731, -27*w - 137], [2731, 2731, 27*w - 137], [2741, 2741, 15*w - 47], [2741, 2741, -15*w - 47], [2749, 2749, -235*w + 1101], [2749, 2749, -53*w + 243], [2753, 2753, -32*w - 159], [2753, 2753, 32*w - 159], [2767, 2767, 82*w - 381], [2767, 2767, 152*w - 711], [2777, 2777, -44*w - 213], [2777, 2777, 44*w - 213], [2789, 2789, 93*w - 433], [2789, 2789, 135*w - 631], [2791, 2791, 14*w - 39], [2791, 2791, -14*w - 39], [2803, 2803, 51*w - 245], [2803, 2803, -243*w + 1141], [2809, 53, -53], [2819, 2819, -61*w + 291], [2819, 2819, 205*w - 963], [2837, 2837, 39*w - 175], [2837, 2837, -39*w - 175], [2843, 2843, -13*w - 81], [2843, 2843, 13*w - 81], [2879, 2879, -12*w - 17], [2879, 2879, 12*w - 17], [2897, 2897, 4*w - 57], [2897, 2897, -4*w - 57], [2903, 2903, 18*w + 65], [2903, 2903, 18*w - 65], [2917, 2917, -31*w + 135], [2917, 2917, 31*w + 135], [2953, 2953, 24*w - 125], [2953, 2953, -24*w - 125], [2963, 2963, 11*w - 75], [2963, 2963, -11*w - 75], [2971, 2971, -15*w - 89], [2971, 2971, 15*w - 89], [2999, 2999, -12*w - 13], [2999, 2999, 12*w - 13], [3001, 3001, 30*w - 151], [3001, 3001, -30*w - 151], [3019, 3019, -39*w - 191], [3019, 3019, 39*w - 191], [3041, 3041, -22*w - 117], [3041, 3041, 22*w - 117], [3067, 3067, 21*w - 113], [3067, 3067, -21*w - 113], [3079, 3079, 22*w - 87], [3079, 3079, 22*w + 87], [3083, 3083, -19*w + 105], [3083, 3083, -19*w - 105], [3089, 3089, 196*w - 921], [3089, 3089, -70*w + 333], [3109, 3109, -17*w - 57], [3109, 3109, 17*w - 57], [3119, 3119, 12*w - 7], [3119, 3119, -12*w - 7], [3167, 3167, -12*w - 1], [3167, 3167, 12*w - 1], [3169, 3169, -60*w + 287], [3169, 3169, 60*w + 287], [3181, 3181, 19*w + 69], [3181, 3181, 19*w - 69], [3217, 3217, -42*w - 205], [3217, 3217, 42*w - 205], [3229, 3229, 47*w - 213], [3229, 3229, -47*w - 213], [3253, 3253, 91*w - 423], [3253, 3253, 161*w - 753], [3257, 3257, 14*w - 87], [3257, 3257, -14*w - 87], [3259, 3259, -9*w - 71], [3259, 3259, 9*w - 71], [3319, 3319, -52*w - 237], [3319, 3319, -52*w + 237], [3323, 3323, 187*w - 879], [3323, 3323, -79*w + 375], [3331, 3331, 15*w - 91], [3331, 3331, -15*w - 91], [3343, 3343, -62*w + 285], [3343, 3343, 62*w + 285], [3347, 3347, 49*w - 237], [3347, 3347, -49*w - 237], [3371, 3371, 37*w - 183], [3371, 3371, -37*w - 183], [3373, 3373, -29*w + 123], [3373, 3373, 29*w + 123], [3407, 3407, 18*w - 61], [3407, 3407, -18*w - 61], [3433, 3433, -6*w - 65], [3433, 3433, 6*w - 65], [3457, 3457, 24*w - 127], [3457, 3457, -24*w - 127], [3461, 3461, -21*w - 79], [3461, 3461, 21*w - 79], [3491, 3491, 23*w - 123], [3491, 3491, -23*w - 123], [3499, 3499, 225*w - 1057], [3499, 3499, -69*w + 329], [3511, 3511, 86*w - 399], [3511, 3511, 184*w - 861], [3517, 3517, 119*w - 555], [3517, 3517, 133*w - 621], [3527, 3527, 78*w - 361], [3527, 3527, 204*w - 955], [3529, 3529, -36*w - 179], [3529, 3529, 36*w - 179], [3533, 3533, 39*w - 173], [3533, 3533, -39*w - 173], [3541, 3541, -35*w + 153], [3541, 3541, 35*w + 153], [3547, 3547, 9*w - 73], [3547, 3547, -9*w - 73], [3559, 3559, -28*w + 117], [3559, 3559, 28*w + 117], [3581, 3581, 15*w - 37], [3581, 3581, -15*w - 37], [3583, 3583, 14*w - 27], [3583, 3583, -14*w - 27], [3607, 3607, -16*w - 45], [3607, 3607, 16*w - 45], [3617, 3617, -4*w - 63], [3617, 3617, 4*w - 63], [3637, 3637, -13*w - 9], [3637, 3637, 13*w - 9], [3671, 3671, 30*w + 127], [3671, 3671, -30*w + 127], [3697, 3697, 6*w - 67], [3697, 3697, -6*w - 67], [3709, 3709, 13*w - 3], [3709, 3709, -13*w - 3], [3793, 3793, 216*w - 1015], [3793, 3793, -78*w + 371], [3797, 3797, 63*w - 289], [3797, 3797, 273*w - 1279], [3823, 3823, -34*w + 147], [3823, 3823, 34*w + 147], [3833, 3833, 58*w - 279], [3833, 3833, -292*w + 1371], [3847, 3847, 26*w + 105], [3847, 3847, 26*w - 105], [3851, 3851, 13*w - 87], [3851, 3851, -13*w - 87], [3863, 3863, -36*w + 157], [3863, 3863, 36*w + 157], [3881, 3881, 2*w - 63], [3881, 3881, -2*w - 63], [3911, 3911, 84*w - 389], [3911, 3911, 210*w - 983], [3931, 3931, -33*w - 167], [3931, 3931, 33*w - 167], [3947, 3947, -w - 63], [3947, 3947, w - 63], [3967, 3967, -56*w + 255], [3967, 3967, -56*w - 255], [3989, 3989, -15*w - 31], [3989, 3989, 15*w - 31], [4019, 4019, -71*w + 339], [4019, 4019, 251*w - 1179], [4021, 4021, -61*w - 279], [4021, 4021, -61*w + 279], [4027, 4027, -3*w - 65], [4027, 4027, 3*w - 65], [4049, 4049, 50*w - 243], [4049, 4049, -50*w - 243], [4051, 4051, 207*w - 973], [4051, 4051, -87*w + 413], [4057, 4057, 12*w - 85], [4057, 4057, -12*w - 85], [4073, 4073, 46*w - 225], [4073, 4073, -46*w - 225], [4111, 4111, -16*w - 39], [4111, 4111, 16*w - 39], [4127, 4127, 54*w - 245], [4127, 4127, -54*w - 245], [4129, 4129, -90*w + 427], [4129, 4129, 204*w - 959], [4133, 4133, 159*w - 743], [4133, 4133, 117*w - 545], [4139, 4139, 31*w - 159], [4139, 4139, -31*w - 159], [4157, 4157, 27*w + 109], [4157, 4157, 27*w - 109], [4211, 4211, -5*w - 69], [4211, 4211, 5*w - 69], [4217, 4217, 8*w - 75], [4217, 4217, -8*w - 75], [4231, 4231, 14*w - 9], [4231, 4231, -14*w - 9], [4253, 4253, 69*w + 317], [4253, 4253, -69*w + 317], [4273, 4273, -96*w + 455], [4273, 4273, 198*w - 931], [4283, 4283, 67*w + 321], [4283, 4283, -67*w + 321], [4337, 4337, -14*w - 93], [4337, 4337, 14*w - 93], [4339, 4339, -99*w + 469], [4339, 4339, 195*w - 917], [4373, 4373, 21*w - 73], [4373, 4373, -21*w - 73], [4391, 4391, 24*w - 91], [4391, 4391, -24*w - 91], [4397, 4397, 57*w 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915], [7933, 7933, 19*w - 3], [7933, 7933, -19*w - 3], [7949, 7949, 93*w + 427], [7949, 7949, -93*w + 427], [8009, 8009, -14*w - 111], [8009, 8009, 14*w - 111], [8011, 8011, 195*w - 919], [8011, 8011, -183*w + 863], [8017, 8017, -186*w + 877], [8017, 8017, 192*w - 905], [8069, 8069, -45*w + 191], [8069, 8069, 45*w + 191], [8087, 8087, 138*w - 641], [8087, 8087, 264*w - 1235], [8089, 8089, -30*w - 167], [8089, 8089, 30*w - 167], [8093, 8093, 213*w - 995], [8093, 8093, 171*w - 797], [8117, 8117, 27*w - 89], [8117, 8117, -27*w - 89], [8123, 8123, 71*w - 345], [8123, 8123, -71*w - 345], [8171, 8171, 67*w - 327], [8171, 8171, -67*w - 327], [8191, 8191, 284*w - 1329], [8191, 8191, 130*w - 603], [8209, 8209, -54*w - 269], [8209, 8209, 54*w - 269], [8233, 8233, -6*w - 95], [8233, 8233, 6*w - 95], [8243, 8243, 23*w - 141], [8243, 8243, -23*w - 141], [8263, 8263, -86*w + 393], [8263, 8263, -86*w - 393], [8269, 8269, -65*w - 291], [8269, 8269, 65*w - 291], [8273, 8273, -92*w + 441], [8273, 8273, 92*w + 441], [8293, 8293, 91*w - 417], [8293, 8293, 413*w - 1935], [8297, 8297, 4*w - 93], [8297, 8297, -4*w - 93], [8311, 8311, -26*w - 81], [8311, 8311, 26*w - 81], [8353, 8353, 24*w - 145], [8353, 8353, -24*w - 145], [8363, 8363, 11*w - 105], [8363, 8363, -11*w - 105], [8369, 8369, 266*w - 1251], [8369, 8369, -140*w + 663], [8387, 8387, -41*w - 213], [8387, 8387, 41*w - 213], [8389, 8389, 23*w - 57], [8389, 8389, -23*w - 57], [8419, 8419, -9*w - 101], [8419, 8419, 9*w - 101], [8423, 8423, 72*w - 325], [8423, 8423, -72*w - 325], [8447, 8447, 24*w - 65], [8447, 8447, -24*w - 65], [8461, 8461, -59*w - 261], [8461, 8461, 59*w - 261], [8527, 8527, 46*w + 195], [8527, 8527, -46*w + 195], [8537, 8537, -26*w - 153], [8537, 8537, 26*w - 153], [8539, 8539, -81*w - 391], [8539, 8539, -81*w + 391], [8543, 8543, -84*w + 383], [8543, 8543, -84*w - 383], [8563, 8563, 39*w - 205], [8563, 8563, -39*w - 205], [8597, 8597, 99*w + 455], [8597, 8597, -99*w + 455], [8599, 8599, 28*w - 93], [8599, 8599, -28*w - 93], [8623, 8623, -22*w - 45], [8623, 8623, 22*w - 45], [8627, 8627, -w - 93], [8627, 8627, w - 93], [8663, 8663, -48*w + 205], [8663, 8663, 48*w + 205], [8699, 8699, -19*w - 129], [8699, 8699, 19*w - 129], [8713, 8713, 12*w - 109], [8713, 8713, -12*w - 109], [8719, 8719, -20*w - 9], [8719, 8719, 20*w - 9], [8737, 8737, -36*w - 193], [8737, 8737, 36*w - 193], [8741, 8741, 21*w - 31], [8741, 8741, -21*w - 31], [8761, 8761, -30*w - 169], [8761, 8761, 30*w - 169], [8779, 8779, 351*w - 1649], [8779, 8779, -111*w + 529], [8803, 8803, -33*w - 181], [8803, 8803, 33*w - 181], [8807, 8807, 66*w - 295], [8807, 8807, -66*w - 295], [8821, 8821, 31*w - 111], [8821, 8821, -31*w - 111], [8839, 8839, 32*w - 117], [8839, 8839, 32*w + 117], [8849, 8849, -56*w - 279], [8849, 8849, 56*w - 279], [8861, 8861, -21*w - 29], [8861, 8861, 21*w - 29], [8863, 8863, -52*w + 225], [8863, 8863, 52*w + 225], [8867, 8867, -101*w - 483], [8867, 8867, 101*w - 483], [8887, 8887, 64*w - 285], [8887, 8887, -64*w - 285], [8951, 8951, 24*w - 61], [8951, 8951, -24*w - 61], [8963, 8963, 239*w - 1125], [8963, 8963, -167*w + 789], [8969, 8969, -80*w - 387], [8969, 8969, 80*w - 387], [9001, 9001, 18*w - 127], [9001, 9001, -18*w - 127], [9043, 9043, -87*w + 419], [9043, 9043, -87*w - 419], [9067, 9067, 21*w - 137], [9067, 9067, -21*w - 137], [9091, 9091, 345*w - 1621], [9091, 9091, -117*w + 557], [9103, 9103, 44*w + 183], [9103, 9103, -44*w + 183], [9127, 9127, 22*w - 39], [9127, 9127, -22*w - 39], [9151, 9151, 376*w - 1761], [9151, 9151, 110*w - 507], [9161, 9161, 68*w - 333], [9161, 9161, -68*w - 333], [9173, 9173, 21*w - 23], [9173, 9173, -21*w - 23], [9181, 9181, -85*w - 387], [9181, 9181, 85*w - 387], [9227, 9227, 97*w + 465], [9227, 9227, -97*w + 465], [9239, 9239, 78*w - 353], [9239, 9239, -78*w - 353], [9241, 9241, 342*w - 1607], [9241, 9241, -120*w + 571], [9319, 9319, 100*w + 459], [9319, 9319, -100*w + 459], [9337, 9337, -48*w - 245], [9337, 9337, 48*w - 245], [9341, 9341, -21*w - 19], [9341, 9341, 21*w - 19], [9349, 9349, -43*w + 177], [9349, 9349, 43*w + 177], [9377, 9377, 14*w - 117], [9377, 9377, -14*w - 117], [9391, 9391, 316*w - 1479], [9391, 9391, 134*w - 621], [9403, 9403, -63*w - 311], [9403, 9403, 63*w - 311], [9413, 9413, -21*w - 17], [9413, 9413, 21*w - 17], [9419, 9419, 289*w - 1359], [9419, 9419, -145*w + 687], [9437, 9437, 39*w - 155], [9437, 9437, 39*w + 155], [9479, 9479, -48*w + 203], [9479, 9479, 48*w + 203], [9491, 9491, -53*w - 267], [9491, 9491, 53*w - 267], [9497, 9497, 16*w - 123], [9497, 9497, -16*w - 123], [9511, 9511, 200*w - 933], [9511, 9511, 214*w - 999], [9533, 9533, 21*w - 13], [9533, 9533, -21*w - 13], [9601, 9601, 12*w - 113], [9601, 9601, -12*w - 113], [9613, 9613, 23*w - 45], [9613, 9613, -23*w - 45], [9619, 9619, -21*w - 139], [9619, 9619, 21*w - 139], [9631, 9631, -50*w + 213], [9631, 9631, 50*w + 213], [9677, 9677, 21*w - 5], [9677, 9677, -21*w - 5], [9679, 9679, 28*w - 87], [9679, 9679, -28*w - 87], [9689, 9689, -154*w + 729], [9689, 9689, 280*w - 1317], [9719, 9719, -72*w - 323], [9719, 9719, 72*w - 323], [9739, 9739, 45*w - 233], [9739, 9739, -45*w - 233], [9743, 9743, -36*w - 137], [9743, 9743, 36*w - 137], [9767, 9767, 282*w - 1319], [9767, 9767, 156*w - 725], [9769, 9769, 96*w - 461], [9769, 9769, -450*w + 2113], [9781, 9781, -25*w - 63], [9781, 9781, 25*w - 63], [9817, 9817, 6*w - 103], [9817, 9817, -6*w - 103], [9829, 9829, 167*w - 777], [9829, 9829, 265*w - 1239], [9857, 9857, -106*w + 507], [9857, 9857, 106*w + 507], [9859, 9859, 51*w - 259], [9859, 9859, -51*w - 259], [9883, 9883, -27*w - 161], [9883, 9883, 27*w - 161], [9923, 9923, 271*w - 1275], [9923, 9923, -163*w + 771], [9931, 9931, 327*w - 1537], [9931, 9931, -135*w + 641], [9941, 9941, -111*w + 511], [9941, 9941, 111*w + 511], [9973, 9973, 49*w + 207], [9973, 9973, -49*w + 207]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, 2, 2, 2, 2, 0, -1, -1, -9, -9, -9, 8, 8, -6, -6, 2, 2, 10, 10, -9, -9, -13, -13, 10, 10, 11, 11, -9, -9, 16, 16, -10, -10, -17, -17, 16, 16, -2, -2, -12, -12, -6, -6, 24, 24, 11, 11, -6, -6, -2, -2, 19, 19, 22, 22, -20, -20, -1, -1, -9, -9, -9, 23, 23, -20, -20, 27, 27, -9, -9, 2, 2, -2, -22, -22, -32, -32, 23, 23, 2, 2, -12, -12, 19, 19, -22, -22, -20, -20, -13, -13, -33, -33, 12, 12, 16, 16, 8, 8, 38, 38, 30, 30, -42, -34, -34, 2, 2, -21, -21, -14, -14, 10, 10, -17, -17, -9, -9, 2, 2, -9, -9, -10, -10, 27, 27, 2, 2, 20, 20, -17, -17, -9, -9, 38, 38, 2, 2, -17, -17, 24, 24, 27, 27, 35, 35, -20, -20, 46, 46, 12, 12, -28, -28, -21, -21, 27, 27, -42, -42, -58, -22, -22, 2, 2, -57, -57, -53, -53, -50, -50, -25, -25, -20, -20, -1, -1, 27, 27, 24, 24, 46, 46, -46, -46, 12, 12, -21, -21, -56, -56, -6, -6, 0, 0, 56, 56, -14, -14, -2, -2, 66, 66, 35, 35, 46, 46, -65, -61, -61, 54, 54, -9, -9, 55, 55, -9, -9, 38, 38, -28, -28, 64, 64, -34, -34, 30, 30, 27, 27, -20, -20, 34, 34, -6, -6, 20, 20, 44, 44, -34, -34, 35, 35, -9, -9, -34, -34, -17, -17, -57, -66, -66, -9, -9, 34, 34, 78, 78, -6, -6, -24, -24, 19, 19, 22, 22, 12, 12, 35, 35, -28, -28, 22, 22, -86, 43, 43, -42, -42, -42, -42, -58, -58, -6, -6, 10, 10, 27, 27, 38, 38, -20, -20, -46, -46, -14, -14, 24, 24, -9, -9, -6, -6, 74, 74, -72, -72, 86, 86, 44, 44, 46, 46, 79, 79, 79, 79, -10, -10, 24, 24, 68, 68, 24, 24, -90, -1, -1, 11, 11, 16, 16, -42, -42, 19, 19, -44, -44, 42, 42, -33, -33, 74, 74, -72, -72, -42, -42, 24, 24, -1, -1, 82, 82, -42, -42, 31, 31, -45, -45, 16, 16, -42, -42, -24, -24, -34, -34, 71, 71, -88, -88, -10, -10, 82, 82, -69, -69, -20, -20, -10, -10, -58, -58, 46, 46, -56, -56, -9, -9, 46, 46, -89, -89, -77, -77, 79, 79, -72, -72, -54, -54, 38, 38, -6, -6, -46, -46, -25, -64, -64, 11, 11, 34, 34, -50, -50, -58, -58, -66, -66, 98, 98, -65, -65, 96, 96, -20, -20, -64, -64, 75, 75, -98, -98, 78, 78, -2, -2, 110, 110, 2, 2, -2, -2, -86, -86, 60, 60, 0, 0, -42, -42, 10, 10, 23, 23, -94, -94, 82, 82, 46, 46, 24, 24, 60, 60, -64, -64, 20, 20, 88, 88, -20, -20, -10, -10, 2, 2, -72, -72, -9, -9, -42, -42, 35, 35, 74, 74, 46, 46, 54, 54, -17, -17, -42, -42, -101, -101, 87, 87, -33, -33, -76, -76, -6, -6, -17, -17, -6, -6, -88, -88, -90, -90, -9, -9, -6, -6, -97, -97, 43, 43, -2, -2, -89, -89, -6, -6, -65, -65, 60, 60, -64, -64, -78, -78, -101, -101, 60, 60, -58, -58, -68, -68, -64, -64, 90, 90, -102, -102, -94, -94, -20, -20, 46, 46, -42, -42, -101, -101, 35, 35, 82, 82, 54, 54, -25, -25, -61, -61, -64, -64, -66, -66, 42, 42, -102, -102, 2, 2, 79, 79, -65, -65, 96, 96, 46, 46, -32, -32, -105, -105, -12, -12, -105, -105, -57, -57, -33, -33, -6, -6, 19, 19, 35, 35, -53, -53, -102, -102, 115, 115, -56, -56, 24, 24, -77, -77, -76, -76, 27, 27, -50, -50, 90, 90, -46, -46, 19, 19, 44, 44, 75, 75, -10, -10, -6, -6, -36, -36, 66, 66, 24, 24, 23, 23, -17, -17, 32, 32, 55, 55, 23, 23, 126, 126, -12, -12, -64, -64, 75, 75, 46, 46, -50, -50, 52, 52, -46, -46, 63, 63, -108, -108, 2, 82, 82, -17, -17, 112, 112, -13, -13, 111, 111, -94, -94, -20, -20, -90, -90, -61, -61, -6, -6, 63, 63, -110, -110, -141, -141, -97, -97, -142, 68, 68, -78, -78, 79, 79, 60, 60, -98, -98, -28, -28, 64, 64, -113, -113, 55, 55, 122, 122, -6, -6, 108, 108, -133, -133, 35, 35, -86, -86, -100, -100, -42, -42, -13, -13, -22, -22, -42, -42, 100, 100, -61, -61, 120, 120, 27, 27, -22, -22, 123, 123, 118, 118, 16, 16, -69, -69, 110, 110, -45, -45, -65, -65, 30, 30, 71, 71, -42, -42, 24, 24, 122, 122, 27, 27, 112, 112, -120, -120, 52, 52, -72, -72, 46, 46, 134, 134, 90, 90, 67, 67, -102, -102, 71, 71, 32, 32, -64, -64, 131, 131, 60, 60, -94, -94, 0, 0, 31, 31, -66, -66, -9, -9, -138, -138, -6, -6, -46, -46, -58, -58, 35, 35, -86, -86, 123, 123, -17, -17, 115, 115, 2, 2, -9, -9, 100, 100, -109, -109, 24, 24, -90, -90, -28, -28, 8, 8, -22, -22, 24, 24, -80, -80, 71, 71, 32, 32, -9, -9, -20, -20, 87, 87, -33, -33, -65, -65, 20, 20, -42, -42, 100, 100, -157, -157, 110, 110, 46, 46, -130, -61, -61, 90, 90, -13, -13, 2, 2, 104, 104, -65, -65, -6, -6, 90, 90, 152, 152, -132, -132, 2, 2, -141, -141, -138, -138, 60, 60, 30, 30, 66, 66, 35, 35, 122, 122, 96, 96, -12, -12, 27, 27, -86, -86, -1, -1, -110, -110, 144, 144, 24, 24, 63, 63, 134, 134, 100, 100, 90, 90, -141, -141, -53, -53, 8, 8, 90, 90, -138, -138, 167, 167, -108, -108, -2, -2, 12, 12, -98, -98, 38, 38, 90, 90, 120, 120, 30, 30, -101, -101, 143, 143, 148, 148, 108, 108, -58, -58, 10, 10, 35, 35, -105, -105, -130, -130, 82, 82, -2, -2, 131, 131, 78, 78, -56, -56, 27, 27, -145, -145, -33, -33, 42, 42, -44, -44, 24, 24, 10, 10, -9, -9, -53, -53, 2, 2, -101, -101, 115, 115, -78, -78, -69, -69, 71, 71, -33, -33, 12, 12, 86, 86, 46, 46, 46, 46, -98, -98, 30, 30, 76, 76, -50, -50, 46, 46, -33, -33, -141, -141, 38, 38, 107, 107, 156, 156, -2, -2, 78, 78, 79, 79, 30, 30, 126, 126, -22, -22, 87, 87, -100, -100, -53, -53, 112, 112, 90, 90, -142, -142, 27, 27, -116, -116, 88, 88, -152, -152, 170, 170, -24, -24, -97, -97, 2, 2, -53, -53, -141, -141, -109, -109, -20, -20, -20, -20, 90, 90, -33, -33, -116, -116, 111, 111, 126, 126, -28, -28, 156, 156, 44, 44, -160, -160, -24, -24, -102, -102, 79, 79, 24, 24, 24, 24, -98, -98, -182, -182, -72, -72, -132, -132, 42, 42, 66, 66, 2, 2, 20, 20, 0, 0, 35, 35, 164, 164, 86, 86, -45, -45, 154, 154, -109, -109, -160, -160, 152, 152, 82, 82, 24, 24, -77, -77, 192, 192, -156, -156, -69, -69, -64, -64, -9, -9, 31, 31, -121, -121, 100, 100, -50, -50, -105, -105, 0, 0, -13, -13, -50, -50, -124, -124, 82, 82, -66, -66, -86, -86, -45, -45, 78, 78, 71, 71, -42, -42, -64, -64, 78, 78, 112, 112, 42, 42, 27, 27, 167, 167]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -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;