/* 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![-19, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -3*w - 13], [3, 3, w + 4], [3, 3, w - 4], [5, 5, 2*w + 9], [5, 5, -2*w + 9], [17, 17, w + 6], [17, 17, -w + 6], [19, 19, w], [31, 31, 20*w + 87], [31, 31, 7*w + 30], [49, 7, -7], [59, 59, 6*w + 25], [59, 59, -6*w + 25], [61, 61, -9*w - 40], [61, 61, 9*w - 40], [67, 67, 2*w - 3], [67, 67, -2*w - 3], [71, 71, 3*w + 10], [71, 71, 3*w - 10], [73, 73, 27*w + 118], [73, 73, -12*w - 53], [79, 79, 4*w - 15], [79, 79, -4*w - 15], [101, 101, 5*w + 24], [101, 101, 5*w - 24], [103, 103, 29*w + 126], [103, 103, 16*w + 69], [107, 107, 3*w - 8], [107, 107, -3*w - 8], [121, 11, -11], [127, 127, -8*w - 33], [127, 127, -8*w + 33], [137, 137, -4*w + 21], [137, 137, 4*w + 21], [149, 149, 2*w - 15], [149, 149, -2*w - 15], [151, 151, -5*w - 18], [151, 151, 5*w - 18], [157, 157, 6*w + 29], [157, 157, 6*w - 29], [167, 167, 3*w - 2], [167, 167, -3*w - 2], [169, 13, -13], [179, 179, -15*w - 64], [179, 179, -15*w + 64], [197, 197, -19*w - 84], [197, 197, 46*w + 201], [211, 211, 38*w + 165], [211, 211, 25*w + 108], [223, 223, -4*w - 9], [223, 223, 4*w - 9], [227, 227, 57*w + 248], [227, 227, 18*w + 77], [229, 229, -3*w - 20], [229, 229, 3*w - 20], [233, 233, -28*w - 123], [233, 233, 37*w + 162], [277, 277, -6*w + 31], [277, 277, 6*w + 31], [307, 307, -17*w + 72], [307, 307, -17*w - 72], [313, 313, 3*w - 22], [313, 313, -3*w - 22], [331, 331, 5*w - 12], [331, 331, -5*w - 12], [349, 349, 15*w - 68], [349, 349, -15*w - 68], [353, 353, 68*w + 297], [353, 353, -23*w - 102], [379, 379, -10*w + 39], [379, 379, -10*w - 39], [383, 383, -9*w - 34], [383, 383, -9*w + 34], [389, 389, -13*w + 60], [389, 389, -13*w - 60], [397, 397, 9*w + 44], [397, 397, 9*w - 44], [431, 431, -15*w + 62], [431, 431, -15*w - 62], [439, 439, -5*w - 6], [439, 439, 5*w - 6], [457, 457, 21*w + 94], [457, 457, 21*w - 94], [461, 461, 50*w + 219], [461, 461, -41*w - 180], [487, 487, -8*w - 27], [487, 487, -8*w + 27], [523, 523, -26*w + 111], [523, 523, -26*w - 111], [529, 23, -23], [541, 541, -6*w - 35], [541, 541, 6*w - 35], [547, 547, -22*w - 93], [547, 547, -22*w + 93], [557, 557, -w - 24], [557, 557, w - 24], [563, 563, 6*w - 11], [563, 563, -6*w - 11], [577, 577, 9*w - 46], [577, 577, 9*w + 46], [593, 593, -17*w - 78], [593, 593, -17*w + 78], [599, 599, 75*w + 326], [599, 599, 36*w + 155], [607, 607, -7*w - 18], [607, 607, 7*w - 18], [613, 613, 3*w - 28], [613, 613, -3*w - 28], [617, 617, 11*w + 54], [617, 617, 11*w - 54], [653, 653, 2*w - 27], [653, 653, -2*w - 27], [659, 659, 6*w - 5], [659, 659, -6*w - 5], [683, 683, -6*w - 1], [683, 683, 6*w - 1], [701, 701, 10*w - 51], [701, 701, 10*w + 51], [709, 709, 75*w + 328], [709, 709, -42*w - 185], [733, 733, -18*w + 83], [733, 733, -18*w - 83], [743, 743, 24*w + 101], [743, 743, 141*w + 614], [751, 751, 65*w + 282], [751, 751, 52*w + 225], [757, 757, 66*w + 289], [757, 757, -51*w - 224], [761, 761, -16*w + 75], [761, 761, -16*w - 75], [769, 769, 60*w + 263], [769, 769, -57*w - 250], [787, 787, -7*w - 12], [787, 787, 7*w - 12], [809, 809, -8*w - 45], [809, 809, -8*w + 45], [811, 811, 10*w - 33], [811, 811, -10*w - 33], [821, 821, 5*w - 36], [821, 821, -5*w - 36], [827, 827, -18*w - 73], [827, 827, -18*w + 73], [839, 839, 84*w + 365], [839, 839, 45*w + 194], [841, 29, -29], [853, 853, -3*w - 32], [853, 853, 3*w - 32], [863, 863, 9*w - 26], [863, 863, -9*w - 26], [881, 881, -w - 30], [881, 881, w - 30], [887, 887, 12*w - 43], [887, 887, -12*w - 43], [907, 907, -13*w - 48], [907, 907, -13*w + 48], [911, 911, -15*w - 58], [911, 911, -15*w + 58], [929, 929, -40*w - 177], [929, 929, 103*w + 450], [937, 937, -24*w - 109], [937, 937, 24*w - 109], [971, 971, -30*w - 127], [971, 971, -30*w + 127], [983, 983, -21*w + 86], [983, 983, -21*w - 86], [991, 991, 8*w - 15], [991, 991, -8*w - 15], [997, 997, -6*w - 41], [997, 997, 6*w - 41], [1013, 1013, 2*w - 33], [1013, 1013, -2*w - 33], [1019, 1019, 51*w + 220], [1019, 1019, 90*w + 391], [1033, 1033, 27*w - 122], [1033, 1033, -27*w - 122], [1039, 1039, -20*w - 81], [1039, 1039, -20*w + 81], [1049, 1049, -20*w - 93], [1049, 1049, -20*w + 93], [1061, 1061, 35*w - 156], [1061, 1061, -35*w - 156], [1063, 1063, 44*w + 189], [1063, 1063, 109*w + 474], [1069, 1069, -18*w + 85], [1069, 1069, -18*w - 85], [1091, 1091, -33*w + 140], [1091, 1091, -33*w - 140], [1109, 1109, -58*w - 255], [1109, 1109, 85*w + 372], [1123, 1123, -14*w + 51], [1123, 1123, -14*w - 51], [1171, 1171, -10*w - 27], [1171, 1171, 10*w - 27], [1201, 1201, 15*w + 74], [1201, 1201, 15*w - 74], [1213, 1213, -33*w - 148], [1213, 1213, 33*w - 148], [1217, 1217, -4*w - 39], [1217, 1217, 4*w - 39], [1231, 1231, 175*w + 762], [1231, 1231, 32*w + 135], [1277, 1277, -w - 36], [1277, 1277, w - 36], [1283, 1283, -9*w - 16], [1283, 1283, 9*w - 16], [1289, 1289, -5*w - 42], [1289, 1289, 5*w - 42], [1291, 1291, 83*w + 360], [1291, 1291, 70*w + 303], [1297, 1297, 36*w - 161], [1297, 1297, 36*w + 161], [1301, 1301, 11*w + 60], [1301, 1301, -11*w + 60], [1307, 1307, -27*w - 112], [1307, 1307, -27*w + 112], [1319, 1319, 60*w + 259], [1319, 1319, 99*w + 430], [1367, 1367, -12*w - 37], [1367, 1367, 12*w - 37], [1369, 37, -37], [1373, 1373, -7*w - 48], [1373, 1373, 7*w - 48], [1399, 1399, 11*w - 30], [1399, 1399, -11*w - 30], [1427, 1427, 102*w + 443], [1427, 1427, 63*w + 272], [1429, 1429, 3*w - 40], [1429, 1429, -3*w - 40], [1439, 1439, -9*w - 10], [1439, 1439, 9*w - 10], [1447, 1447, 13*w - 42], [1447, 1447, -13*w - 42], [1453, 1453, -42*w - 187], [1453, 1453, 153*w + 668], [1459, 1459, -10*w - 21], [1459, 1459, 10*w - 21], [1471, 1471, -25*w + 102], [1471, 1471, -25*w - 102], [1489, 1489, -12*w + 65], [1489, 1489, 12*w + 65], [1493, 1493, 107*w + 468], [1493, 1493, -62*w - 273], [1511, 1511, 12*w - 35], [1511, 1511, -12*w - 35], [1523, 1523, 9*w - 4], [1523, 1523, -9*w - 4], [1571, 1571, -15*w + 52], [1571, 1571, -15*w - 52], [1579, 1579, -29*w + 120], [1579, 1579, -29*w - 120], [1597, 1597, -9*w - 56], [1597, 1597, 9*w - 56], [1601, 1601, 89*w + 390], [1601, 1601, -80*w - 351], [1613, 1613, -23*w - 108], [1613, 1613, -23*w + 108], [1621, 1621, 21*w - 100], [1621, 1621, 21*w + 100], [1627, 1627, -22*w + 87], [1627, 1627, -22*w - 87], [1657, 1657, 144*w + 629], [1657, 1657, -51*w - 226], [1663, 1663, -41*w + 174], [1663, 1663, -41*w - 174], [1667, 1667, -18*w - 67], [1667, 1667, -18*w + 67], [1669, 1669, -30*w - 137], [1669, 1669, -30*w + 137], [1681, 41, -41], [1697, 1697, 16*w - 81], [1697, 1697, 16*w + 81], [1699, 1699, -14*w - 45], [1699, 1699, 14*w - 45], [1721, 1721, 4*w - 45], [1721, 1721, -4*w - 45], [1723, 1723, 11*w - 24], [1723, 1723, -11*w - 24], [1733, 1733, -43*w - 192], [1733, 1733, 43*w - 192], [1747, 1747, 127*w + 552], [1747, 1747, 62*w + 267], [1753, 1753, -12*w - 67], [1753, 1753, -12*w + 67], [1823, 1823, 231*w + 1006], [1823, 1823, 36*w + 151], [1849, 43, -43], [1873, 1873, 132*w + 577], [1873, 1873, -63*w - 278], [1901, 1901, -14*w + 75], [1901, 1901, 14*w + 75], [1931, 1931, -18*w - 65], [1931, 1931, 18*w - 65], [1949, 1949, 2*w - 45], [1949, 1949, -2*w - 45], [1951, 1951, -23*w - 90], [1951, 1951, -23*w + 90], [1973, 1973, -13*w + 72], [1973, 1973, -13*w - 72], [1979, 1979, -21*w - 80], [1979, 1979, -21*w + 80], [1993, 1993, 123*w + 538], [1993, 1993, -72*w - 317], [2003, 2003, 39*w + 164], [2003, 2003, 234*w + 1019], [2027, 2027, 78*w + 337], [2027, 2027, 117*w + 508], [2053, 2053, -78*w - 343], [2053, 2053, 117*w + 512], [2069, 2069, -10*w - 63], [2069, 2069, 10*w - 63], [2083, 2083, -34*w - 141], [2083, 2083, -34*w + 141], [2111, 2111, -12*w - 25], [2111, 2111, 12*w - 25], [2113, 2113, -87*w - 382], [2113, 2113, 108*w + 473], [2129, 2129, 47*w - 210], [2129, 2129, -47*w - 210], [2131, 2131, -50*w + 213], [2131, 2131, -50*w - 213], [2137, 2137, 99*w + 434], [2137, 2137, -96*w - 421], [2143, 2143, 136*w + 591], [2143, 2143, 71*w + 306], [2153, 2153, -32*w - 147], [2153, 2153, -32*w + 147], [2179, 2179, -46*w - 195], [2179, 2179, -46*w + 195], [2203, 2203, 14*w - 39], [2203, 2203, -14*w - 39], [2207, 2207, 12*w - 23], [2207, 2207, -12*w - 23], [2209, 47, -47], [2213, 2213, -37*w + 168], [2213, 2213, -37*w - 168], [2221, 2221, -30*w - 139], [2221, 2221, -30*w + 139], [2281, 2281, 24*w + 115], [2281, 2281, 24*w - 115], [2297, 2297, -4*w - 51], [2297, 2297, 4*w - 51], [2311, 2311, 13*w - 30], [2311, 2311, -13*w - 30], [2339, 2339, 15*w - 44], [2339, 2339, -15*w - 44], [2341, 2341, 6*w - 55], [2341, 2341, -6*w - 55], [2347, 2347, 158*w + 687], [2347, 2347, 67*w + 288], [2351, 2351, 177*w + 770], [2351, 2351, 60*w + 257], [2357, 2357, 19*w - 96], [2357, 2357, 19*w + 96], [2371, 2371, 110*w + 477], [2371, 2371, 97*w + 420], [2381, 2381, 151*w + 660], [2381, 2381, -70*w - 309], [2383, 2383, -31*w + 126], [2383, 2383, -31*w - 126], [2417, 2417, 191*w + 834], [2417, 2417, -56*w - 249], [2423, 2423, -36*w + 149], [2423, 2423, -36*w - 149], [2437, 2437, 21*w + 104], [2437, 2437, 21*w - 104], [2441, 2441, -5*w - 54], [2441, 2441, 5*w - 54], [2447, 2447, -12*w - 17], [2447, 2447, 12*w - 17], [2459, 2459, -30*w - 121], [2459, 2459, -30*w + 121], [2477, 2477, -31*w + 144], [2477, 2477, -31*w - 144], [2503, 2503, -19*w + 66], [2503, 2503, -19*w - 66], [2539, 2539, -35*w - 144], [2539, 2539, -35*w + 144], [2557, 2557, 9*w - 64], [2557, 2557, -9*w - 64], [2579, 2579, 129*w + 560], [2579, 2579, 90*w + 389], [2593, 2593, 12*w - 73], [2593, 2593, -12*w - 73], [2609, 2609, -17*w + 90], [2609, 2609, 17*w + 90], [2633, 2633, -88*w - 387], [2633, 2633, 133*w + 582], [2657, 2657, -41*w - 186], [2657, 2657, 41*w - 186], [2659, 2659, -25*w - 96], [2659, 2659, -25*w + 96], [2663, 2663, -24*w + 91], [2663, 2663, -24*w - 91], [2677, 2677, 54*w - 241], [2677, 2677, -54*w - 241], [2687, 2687, 12*w - 7], [2687, 2687, -12*w - 7], [2711, 2711, -12*w - 5], [2711, 2711, 12*w - 5], [2719, 2719, -55*w - 234], [2719, 2719, -55*w + 234], [2731, 2731, -43*w + 180], [2731, 2731, -43*w - 180], [2741, 2741, -106*w - 465], [2741, 2741, 115*w + 504], [2753, 2753, -8*w - 63], [2753, 2753, 8*w - 63], [2767, 2767, 268*w + 1167], [2767, 2767, 47*w + 198], [2797, 2797, -6*w - 59], [2797, 2797, 6*w - 59], [2803, 2803, 119*w + 516], [2803, 2803, 106*w + 459], [2809, 53, -53], [2837, 2837, -14*w - 81], [2837, 2837, 14*w - 81], [2843, 2843, 186*w + 809], [2843, 2843, 69*w + 296], [2857, 2857, 21*w - 106], [2857, 2857, 21*w + 106], [2861, 2861, -10*w - 69], [2861, 2861, 10*w - 69], [2879, 2879, 96*w + 415], [2879, 2879, 135*w + 586], [2887, 2887, 13*w - 18], [2887, 2887, -13*w - 18], [2897, 2897, -w - 54], [2897, 2897, w - 54], [2903, 2903, -21*w - 74], [2903, 2903, 21*w - 74], [2939, 2939, -30*w + 119], [2939, 2939, -30*w - 119], [2963, 2963, -54*w - 229], [2963, 2963, -54*w + 229], [2969, 2969, -35*w + 162], [2969, 2969, -35*w - 162], [3023, 3023, -24*w + 89], [3023, 3023, -24*w - 89], [3037, 3037, 6*w - 61], [3037, 3037, -6*w - 61], [3041, 3041, 55*w + 246], [3041, 3041, 55*w - 246], [3049, 3049, 27*w + 130], [3049, 3049, 27*w - 130], [3067, 3067, 13*w - 12], [3067, 3067, -13*w - 12], [3089, 3089, -40*w + 183], [3089, 3089, -40*w - 183], [3119, 3119, -15*w - 34], [3119, 3119, 15*w - 34], [3121, 3121, -15*w - 86], [3121, 3121, 15*w - 86], [3167, 3167, -57*w + 242], [3167, 3167, -57*w - 242], [3187, 3187, 17*w - 48], [3187, 3187, -17*w - 48], [3191, 3191, 192*w + 835], [3191, 3191, 75*w + 322], [3209, 3209, 155*w + 678], [3209, 3209, -92*w - 405], [3217, 3217, 24*w + 119], [3217, 3217, 24*w - 119], [3251, 3251, 15*w - 32], [3251, 3251, -15*w - 32], [3253, 3253, 18*w + 97], [3253, 3253, -18*w + 97], [3259, 3259, 19*w - 60], [3259, 3259, -19*w - 60], [3271, 3271, 128*w + 555], [3271, 3271, 115*w + 498], [3299, 3299, -39*w - 160], [3299, 3299, -39*w + 160], [3313, 3313, -48*w - 217], [3313, 3313, 48*w - 217], [3319, 3319, 85*w + 366], [3319, 3319, 176*w + 765], [3329, 3329, 28*w - 135], [3329, 3329, 28*w + 135], [3343, 3343, -16*w - 39], [3343, 3343, 16*w - 39], [3347, 3347, 18*w - 53], [3347, 3347, -18*w - 53], [3359, 3359, 105*w + 454], [3359, 3359, 144*w + 625], [3361, 3361, 9*w - 70], [3361, 3361, -9*w - 70], [3371, 3371, 78*w + 335], [3371, 3371, 195*w + 848], [3389, 3389, -110*w - 483], [3389, 3389, 137*w + 600], [3469, 3469, 15*w - 88], [3469, 3469, -15*w - 88], [3491, 3491, -15*w - 28], [3491, 3491, 15*w - 28], [3499, 3499, -14*w - 15], [3499, 3499, 14*w - 15], [3511, 3511, -37*w + 150], [3511, 3511, -37*w - 150], [3527, 3527, 147*w + 638], [3527, 3527, 108*w + 467], [3541, 3541, -6*w - 65], [3541, 3541, 6*w - 65], [3547, 3547, 98*w + 423], [3547, 3547, 163*w + 708], [3557, 3557, 59*w - 264], [3557, 3557, 59*w + 264], [3571, 3571, -22*w - 75], [3571, 3571, -22*w + 75], [3581, 3581, -w - 60], [3581, 3581, w - 60], [3617, 3617, -44*w - 201], [3617, 3617, -44*w + 201], [3623, 3623, 324*w + 1411], [3623, 3623, 51*w + 214], [3631, 3631, -20*w - 63], [3631, 3631, 20*w - 63], [3643, 3643, 14*w - 9], [3643, 3643, -14*w - 9], [3673, 3673, -3*w - 62], [3673, 3673, 3*w - 62], [3697, 3697, 24*w - 121], [3697, 3697, 24*w + 121], [3709, 3709, -90*w - 397], [3709, 3709, 183*w + 800], [3719, 3719, 24*w - 85], [3719, 3719, -24*w - 85], [3727, 3727, 17*w - 42], [3727, 3727, -17*w - 42], [3733, 3733, -42*w - 193], [3733, 3733, -42*w + 193], [3739, 3739, -34*w + 135], [3739, 3739, -34*w - 135], [3769, 3769, -93*w - 410], [3769, 3769, 180*w + 787], [3797, 3797, 26*w + 129], [3797, 3797, 26*w - 129], [3803, 3803, -66*w + 281], [3803, 3803, -66*w - 281], [3851, 3851, -27*w - 100], [3851, 3851, -27*w + 100], [3877, 3877, -99*w - 436], [3877, 3877, 174*w + 761], [3881, 3881, 5*w - 66], [3881, 3881, -5*w - 66], [3907, 3907, 73*w + 312], [3907, 3907, 242*w + 1053], [3943, 3943, -19*w - 54], [3943, 3943, 19*w - 54], [3947, 3947, -18*w - 47], [3947, 3947, 18*w - 47], [3967, 3967, -23*w + 78], [3967, 3967, -23*w - 78], [4001, 4001, 25*w - 126], [4001, 4001, 25*w + 126], [4003, 4003, -49*w + 204], [4003, 4003, -49*w - 204], [4013, 4013, -82*w - 363], [4013, 4013, 217*w + 948], [4019, 4019, -15*w - 16], [4019, 4019, 15*w - 16], [4027, 4027, -38*w - 153], [4027, 4027, -38*w + 153], [4073, 4073, -43*w + 198], [4073, 4073, -43*w - 198], [4079, 4079, 15*w - 14], [4079, 4079, -15*w - 14], [4099, 4099, -65*w + 276], [4099, 4099, -65*w - 276], [4129, 4129, 153*w + 670], [4129, 4129, -120*w - 527], [4153, 4153, -12*w - 83], [4153, 4153, 12*w - 83], [4177, 4177, -129*w - 566], [4177, 4177, 144*w + 631], [4211, 4211, -15*w - 8], [4211, 4211, 15*w - 8], [4229, 4229, -53*w - 240], [4229, 4229, 53*w - 240], [4231, 4231, -35*w - 138], [4231, 4231, -35*w + 138], [4241, 4241, 31*w - 150], [4241, 4241, 31*w + 150], [4253, 4253, -7*w - 72], [4253, 4253, 7*w - 72], [4259, 4259, -15*w - 4], [4259, 4259, 15*w - 4], [4261, 4261, -66*w - 295], [4261, 4261, 66*w - 295], [4271, 4271, 15*w - 2], [4271, 4271, -15*w - 2], [4273, 4273, 33*w - 158], [4273, 4273, 33*w + 158], [4283, 4283, -21*w - 64], [4283, 4283, 21*w - 64], 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-84*w + 355], [8053, 8053, 27*w + 148], [8053, 8053, -27*w + 148], [8059, 8059, 331*w + 1440], [8059, 8059, 110*w + 471], [8081, 8081, -w - 90], [8081, 8081, w - 90], [8087, 8087, 117*w + 502], [8087, 8087, 312*w + 1357], [8101, 8101, 315*w + 1376], [8101, 8101, -114*w - 505], [8117, 8117, 269*w + 1176], [8117, 8117, -134*w - 591], [8123, 8123, -21*w - 16], [8123, 8123, 21*w - 16], [8147, 8147, -33*w + 112], [8147, 8147, -33*w - 112], [8191, 8191, -55*w + 222], [8191, 8191, -55*w - 222], [8209, 8209, 33*w + 170], [8209, 8209, 33*w - 170], [8233, 8233, -117*w - 518], [8233, 8233, 312*w + 1363], [8269, 8269, -15*w - 112], [8269, 8269, 15*w - 112], [8287, 8287, 23*w - 42], [8287, 8287, -23*w - 42], [8293, 8293, -3*w - 92], [8293, 8293, 3*w - 92], [8311, 8311, 200*w + 867], [8311, 8311, 187*w + 810], [8329, 8329, -60*w + 277], [8329, 8329, -60*w - 277], [8363, 8363, 21*w - 4], [8363, 8363, -21*w - 4], [8369, 8369, 260*w + 1137], [8369, 8369, -143*w - 630], [8377, 8377, -24*w - 139], [8377, 8377, 24*w - 139], [8387, 8387, -87*w + 368], [8387, 8387, -87*w - 368], [8419, 8419, -70*w + 291], [8419, 8419, -70*w - 291], [8431, 8431, 32*w - 105], [8431, 8431, -32*w - 105], [8461, 8461, 9*w - 100], [8461, 8461, -9*w - 100], [8467, 8467, -22*w - 27], [8467, 8467, 22*w - 27], [8513, 8513, -28*w + 153], [8513, 8513, 28*w + 153], [8521, 8521, -21*w - 130], [8521, 8521, 21*w - 130], [8527, 8527, -52*w + 207], [8527, 8527, -52*w - 207], [8537, 8537, -56*w - 261], [8537, 8537, -56*w + 261], [8539, 8539, -59*w - 240], [8539, 8539, -59*w + 240], [8543, 8543, -24*w - 49], [8543, 8543, 24*w - 49], [8563, 8563, -74*w + 309], [8563, 8563, -74*w - 309], [8573, 8573, 2*w - 93], [8573, 8573, -2*w - 93], [8597, 8597, -14*w - 111], [8597, 8597, 14*w - 111], [8647, 8647, 157*w + 678], [8647, 8647, 248*w + 1077], [8663, 8663, -69*w - 286], [8663, 8663, -69*w + 286], [8669, 8669, -50*w + 237], [8669, 8669, -50*w - 237], [8681, 8681, 101*w - 450], [8681, 8681, -101*w - 450], [8689, 8689, -63*w + 290], [8689, 8689, -63*w - 290], [8713, 8713, -12*w - 107], [8713, 8713, 12*w - 107], [8731, 8731, 82*w + 345], [8731, 8731, 485*w + 2112], [8737, 8737, 39*w + 194], [8737, 8737, 39*w - 194], [8741, 8741, 5*w - 96], [8741, 8741, -5*w - 96], [8807, 8807, -48*w - 187], [8807, 8807, -48*w + 187], [8819, 8819, -30*w - 91], [8819, 8819, 30*w - 91], [8821, 8821, -30*w + 161], [8821, 8821, 30*w + 161], [8831, 8831, -57*w + 230], [8831, 8831, -57*w - 230], [8861, 8861, 25*w - 144], [8861, 8861, -25*w - 144], [8867, 8867, -42*w - 157], [8867, 8867, -42*w + 157], [8887, 8887, -43*w + 162], [8887, 8887, -43*w - 162], [8893, 8893, 33*w + 172], [8893, 8893, 33*w - 172], [8923, 8923, 29*w - 84], [8923, 8923, -29*w - 84], [8941, 8941, 294*w + 1285], [8941, 8941, -135*w - 596], [8951, 8951, 27*w - 70], [8951, 8951, -27*w - 70], [8963, 8963, 321*w + 1396], [8963, 8963, 126*w + 541], [8969, 8969, 43*w - 210], [8969, 8969, 43*w + 210], [8971, 8971, -22*w - 15], [8971, 8971, 22*w - 15], [8999, 8999, -36*w - 125], [8999, 8999, -36*w + 125], [9013, 9013, -78*w - 353], [9013, 9013, 78*w - 353], [9029, 9029, -22*w - 135], [9029, 9029, 22*w - 135], [9041, 9041, 20*w - 129], [9041, 9041, -20*w - 129], [9043, 9043, 31*w - 96], [9043, 9043, -31*w - 96], [9049, 9049, 45*w + 218], [9049, 9049, 45*w - 218], [9059, 9059, 510*w + 2221], [9059, 9059, 81*w + 340], [9103, 9103, 209*w + 906], [9103, 9103, 196*w + 849], [9137, 9137, 49*w + 234], [9137, 9137, 49*w - 234], [9151, 9151, -23*w - 30], [9151, 9151, 23*w - 30], [9181, 9181, -15*w - 116], [9181, 9181, 15*w - 116], [9187, 9187, 22*w - 3], [9187, 9187, -22*w - 3], [9199, 9199, 305*w + 1326], [9199, 9199, 136*w + 585], [9221, 9221, 371*w + 1620], [9221, 9221, -110*w - 489], [9227, 9227, -27*w - 68], [9227, 9227, 27*w - 68], [9241, 9241, -144*w - 635], [9241, 9241, 285*w + 1246], [9257, 9257, -124*w - 549], [9257, 9257, 331*w + 1446], [9277, 9277, 9*w - 104], [9277, 9277, -9*w - 104], [9281, 9281, -76*w + 345], [9281, 9281, -76*w - 345], [9323, 9323, -66*w - 271], [9323, 9323, -66*w + 271], [9343, 9343, -71*w + 294], [9343, 9343, -71*w - 294], [9349, 9349, -102*w - 455], [9349, 9349, 102*w - 455], [9397, 9397, -69*w + 316], [9397, 9397, -69*w - 316], [9409, 97, -97], [9419, 9419, -51*w + 200], [9419, 9419, -51*w - 200], [9421, 9421, -150*w - 661], [9421, 9421, 279*w + 1220], [9433, 9433, -3*w - 98], [9433, 9433, 3*w - 98], [9439, 9439, -41*w + 150], [9439, 9439, -41*w - 150], [9473, 9473, 7*w - 102], [9473, 9473, -7*w - 102], [9491, 9491, -42*w + 155], [9491, 9491, -42*w - 155], [9497, 9497, -4*w - 99], [9497, 9497, 4*w - 99], [9551, 9551, -48*w + 185], [9551, 9551, -48*w - 185], [9601, 9601, 36*w + 185], [9601, 9601, 36*w - 185], [9643, 9643, 34*w - 111], [9643, 9643, -34*w - 111], [9649, 9649, 15*w - 118], [9649, 9649, -15*w - 118], [9661, 9661, -159*w - 700], [9661, 9661, 270*w + 1181], [9677, 9677, 41*w + 204], [9677, 9677, 41*w - 204], [9679, 9679, -79*w - 330], [9679, 9679, -79*w + 330], [9697, 9697, 9*w - 106], [9697, 9697, -9*w - 106], [9719, 9719, -24*w - 35], [9719, 9719, 24*w - 35], [9733, 9733, 267*w + 1168], [9733, 9733, -162*w - 713], [9743, 9743, 159*w + 686], [9743, 9743, 276*w + 1199], [9787, 9787, 83*w + 348], [9787, 9787, 538*w + 2343], [9803, 9803, -69*w - 284], [9803, 9803, -69*w + 284], [9829, 9829, 3*w - 100], [9829, 9829, -3*w - 100], [9871, 9871, -95*w - 402], [9871, 9871, -95*w + 402], [9883, 9883, 91*w + 384], [9883, 9883, 494*w + 2151], [9907, 9907, -23*w - 12], [9907, 9907, 23*w - 12], [9929, 9929, -5*w - 102], [9929, 9929, 5*w - 102], [9931, 9931, 218*w + 945], [9931, 9931, 205*w + 888], [9941, 9941, -70*w - 321], [9941, 9941, -70*w + 321], [9973, 9973, 18*w - 127], [9973, 9973, -18*w - 127]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, -2, -2, 0, 0, 6, 6, 2, -4, -4, 1, -6, -6, 8, 8, -4, -4, 0, 0, 2, 2, 8, 8, 0, 0, -4, -4, 12, 12, -22, -16, -16, 18, 18, -18, -18, 8, 8, -4, -4, -12, -12, -10, -12, -12, -18, -18, -4, -4, 8, 8, 18, 18, -4, -4, -6, -6, -10, -10, 2, 2, -10, -10, 8, 8, -28, -28, 18, 18, -16, -16, 36, 36, 18, 18, 20, 20, 24, 24, 8, 8, -10, -10, 12, 12, -16, -16, 2, 2, -46, 38, 38, 8, 8, 6, 6, 30, 30, 2, 2, -6, -6, -24, -24, 32, 32, 2, 2, 6, 6, 18, 18, -24, -24, -12, -12, 18, 18, -46, -46, -40, -40, 24, 24, -40, -40, 2, 2, -18, -18, 14, 14, -22, -22, 6, 6, 2, 2, 6, 6, -36, -36, 12, 12, -22, 44, 44, -24, -24, -54, -54, -36, -36, 44, 44, 48, 48, 6, 6, 2, 2, -6, -6, -36, -36, -16, -16, 8, 8, -36, -36, 36, 36, 26, 26, -4, -4, -30, -30, -30, -30, -16, -16, 8, 8, 30, 30, 36, 36, -46, -46, 20, 20, 14, 14, 26, 26, -6, -6, -28, -28, 24, 24, 0, 0, 30, 30, -58, -58, -34, -34, -48, -48, 42, 42, -24, -24, -24, -24, -70, -30, -30, -40, -40, -30, -30, 26, 26, 24, 24, -64, -64, 50, 50, 38, 38, 32, 32, -58, -58, 30, 30, 0, 0, -72, -72, 66, 66, -40, -40, -58, -58, 66, 66, -72, -72, 74, 74, -10, -10, -34, -34, 8, 8, 48, 48, 32, 32, -46, -42, -42, -10, -10, -6, -6, -4, -4, -6, -6, 56, 56, -46, -46, -24, -24, -22, 2, 2, -18, -18, -30, -30, 84, 84, -64, -64, -60, -60, 30, 30, 14, 14, -48, -48, -72, -72, 38, 38, -30, -30, 32, 32, 48, 48, 26, 26, -30, -30, -10, -10, 50, 50, 56, 56, 54, 54, -88, -88, -46, -46, -48, -48, 50, -54, -54, -10, -10, -22, -22, -42, -42, -64, -64, -72, -72, 20, 20, 80, 80, -24, -24, -24, -24, 14, 14, -66, -66, -28, -28, 18, 18, 48, 48, 62, 62, -30, -30, -48, -48, 60, 60, 84, 84, -16, -16, -34, -34, 38, 38, -78, -78, -70, -70, -42, -42, 6, 6, 18, 18, 14, 14, 24, 24, 32, 32, -12, -12, 24, 24, -52, -52, -4, -4, 66, 66, -78, -78, 32, 32, 62, 62, 86, 86, -70, 42, 42, 0, 0, -22, -22, 12, 12, -24, -24, 44, 44, 54, 54, 0, 0, 6, 6, 24, 24, -54, -54, 0, 0, -40, -40, 78, 78, -10, -10, -64, -64, -66, -66, -96, -96, 98, 98, 60, 60, -16, -16, 24, 24, 30, 30, 86, 86, -90, -90, 32, 32, 56, 56, -40, -40, 72, 72, 2, 2, 80, 80, -66, -66, -40, -40, 36, 36, -96, -96, -58, -58, -24, -24, 42, 42, 50, 50, 102, 102, 98, 98, -16, -16, 72, 72, -4, -4, -34, -34, -42, -42, -64, -64, -30, -30, -42, -42, -96, -96, 104, 104, -10, -10, 74, 74, -46, -46, -16, -16, 24, 24, -88, -88, -46, -46, -40, -40, -10, -10, -12, -12, -108, -108, 72, 72, 44, 44, -18, -18, 20, 20, 56, 56, 6, 6, -100, -100, -78, -78, -46, -46, -78, -78, 60, 60, -112, -112, 6, 6, 24, 24, -4, -4, -82, -82, 86, 86, 74, 74, -84, -84, -126, -126, -52, -52, 30, 30, -78, -78, -18, -18, -4, -4, -48, -48, -46, -46, 18, 18, -64, -64, 78, 78, 114, 114, -88, -88, 44, 44, 48, 48, -54, -54, -40, -40, -102, -102, 126, 126, 38, 38, 48, 48, -118, -118, -52, -52, -22, -22, -108, -108, 104, 104, 98, 98, -16, -16, -84, -84, 66, 66, 50, 50, 104, 104, -54, -54, -34, -34, -90, -90, 26, 26, 20, 20, 126, 126, 86, 86, -66, -66, 54, 54, -24, -24, -10, -10, 96, 96, -40, -40, -118, -118, 102, 102, -70, -70, -12, -12, 80, 80, 68, 68, -64, -64, 78, 78, -16, -16, -54, -54, 98, 98, 102, 102, 72, 72, 86, 86, 2, 2, 84, 84, -24, -24, 48, 48, -54, -54, -24, -24, 86, 86, -18, -18, -48, -48, 12, 12, -40, -40, -130, -130, 60, 60, 104, 104, 38, 38, -78, -78, 8, 8, 120, 120, -82, -82, 12, 12, 72, 72, 50, 50, 32, 32, -120, -120, -46, -46, 128, 128, 32, 32, -90, -90, -72, -72, 62, 62, -58, -58, 6, 6, 42, 42, 110, 110, -82, -82, -66, -66, -72, -72, 20, 20, 84, 84, -18, -18, -24, -24, -124, -124, 0, 0, -130, -130, -48, -48, 32, 32, 30, 30, -108, -108, -112, -112, -30, -30, 66, 66, 32, 32, -108, -108, 26, 26, 20, 20, 56, 56, -42, -42, 156, 156, 32, 32, 126, 126, -72, -72, -12, -12, 36, 36, -66, -66, -24, -24, 32, 32, 116, 116, -24, -24, -100, -100, 134, 134, -156, -156, 138, 138, -72, -72, -106, -106, 144, 144, 102, 102, -52, -52, 50, 50, 2, 2, 14, 14, 56, 56, 18, 18, -22, -22, 8, 8, 96, 96, -16, -16, -42, -42, 138, 138, -112, -112, -54, -54, 32, 32, -120, -120, 56, 56, -34, -34, 42, 42, -40, -40, -130, 120, 120, -58, -58, 48, 48, -30, -30, -30, -30, -58, -58, 56, 56, 78, 78, -120, -120, -28, -28, 50, 50, 90, 90, -114, -114, -48, -48, 32, 32, -144, -144, -70, -70, -112, -112, -90, -90, 12, 12, 128, 128, 168, 168, -16, -16, 96, 96, 134, 134, 158, 158, -10, -10, 38, 38, 54, 54, -102, -102, 90, 90, -120, -120, 74, 74, -78, -78, 30, 30, 138, 138, -100, -100, 68, 68, -72, -72, -40, -40, 44, 44, 30, 30, -138, -138, -130, -130, -162, -162, -84, -84, -120, -120, -58, -58, -126, -126, 0, 0, 60, 60, -120, -120, 62, 62, 6, 6, 104, 104, 156, 156, -168, -168, -24, -24, -142, -90, -90, -100, -100, 74, 74, -108, -108, 116, 116, -4, -4, 42, 42, 144, 144, -70, -70, 6, 6, -6, -6, -6, -6, -160, -160, 146, 146, 26, 26, -46, -46, 44, 44, 104, 104, 176, 176, 14, 14, 114, 114, -30, -30, -82, -82, -36, -36, 86, 86, -28, -28, -88, -88, -172, -172, 126, 126, -10, -10, 104, 104, 30, 30, -118, -118, -108, -108, -76, -76, -72, -72, 138, 138, 8, 8, -24, -24, -108, -108, -138, -138, 122, 122, 2, 2, 20, 20, -82, -82, -120, -120, 96, 96, 54, 54, 74, 74, 144, 144, 132, 132, 42, 42, 32, 32, -124, -124, 122, 122, 122, 122, -108, -108, 42, 42, -54, -54, -28, -28, 168, 168, -166, -166, 120, 120, 30, 30, -118, -118, 158, 158, 24, 24, 68, 68, -66, -66, -136, -136, -34, -34, -10, -10, 128, 128, -90, -90, 120, 120, 170, 170, -186, -186, 62, 62, 138, 138, 18, 18, 92, 92, -46, -46, -124, -124, -94, -60, -60, -28, -28, 74, 74, 188, 188, -126, -126, 78, 78, 150, 150, 180, 180, 62, 62, 32, 32, 146, 146, 86, 86, 108, 108, -184, -184, -58, -58, 24, 24, -40, -40, -84, -84, -148, -148, 126, 126, 110, 110, -136, -136, 146, 146, -28, -28, 54, 54, -22, -22, -66, -66, -196, -196]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; 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;