/* 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![-3, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 1], [3, 3, w], [11, 11, -2*w + 1], [11, 11, 2*w + 1], [13, 13, w + 4], [13, 13, -w + 4], [23, 23, -3*w + 2], [23, 23, 3*w + 2], [25, 5, 5], [37, 37, 2*w - 7], [37, 37, -2*w - 7], [47, 47, -4*w - 1], [47, 47, 4*w - 1], [49, 7, -7], [59, 59, 5*w - 4], [59, 59, -5*w - 4], [61, 61, -w - 8], [61, 61, w - 8], [71, 71, 5*w - 2], [71, 71, -5*w - 2], [73, 73, 3*w - 10], [73, 73, -3*w - 10], [83, 83, 6*w - 5], [83, 83, -6*w - 5], [97, 97, -w - 10], [97, 97, w - 10], [107, 107, -6*w - 1], [107, 107, 6*w - 1], [109, 109, 2*w - 11], [109, 109, -2*w - 11], [131, 131, -7*w - 4], [131, 131, 7*w - 4], [157, 157, 2*w - 13], [157, 157, -2*w - 13], [167, 167, 8*w - 5], [167, 167, -8*w - 5], [179, 179, 9*w - 8], [179, 179, -9*w - 8], [181, 181, 5*w - 16], [181, 181, -5*w - 16], [191, 191, -8*w - 1], [191, 191, 8*w - 1], [193, 193, -w - 14], [193, 193, w - 14], [227, 227, 9*w - 4], [227, 227, -9*w - 4], [229, 229, 3*w - 16], [229, 229, -3*w - 16], [239, 239, 9*w - 2], [239, 239, -9*w - 2], [241, 241, 4*w - 17], [241, 241, -4*w - 17], [251, 251, -10*w - 7], [251, 251, 10*w - 7], [263, 263, 11*w - 10], [263, 263, -11*w - 10], [277, 277, 2*w - 17], [277, 277, -2*w - 17], [289, 17, -17], [311, 311, 12*w - 11], [311, 311, -12*w - 11], [313, 313, -4*w - 19], [313, 313, 4*w - 19], [337, 337, 7*w - 22], [337, 337, -7*w - 22], [347, 347, -11*w - 4], [347, 347, 11*w - 4], [349, 349, 2*w - 19], [349, 349, -2*w - 19], [359, 359, 11*w - 2], [359, 359, -11*w - 2], [361, 19, -19], [373, 373, -3*w - 20], [373, 373, 3*w - 20], [383, 383, 12*w - 7], [383, 383, -12*w - 7], [397, 397, -w - 20], [397, 397, w - 20], [409, 409, -5*w - 22], [409, 409, 5*w - 22], [419, 419, 14*w - 13], [419, 419, -14*w - 13], [421, 421, -6*w - 23], [421, 421, 6*w - 23], [431, 431, -12*w - 1], [431, 431, 12*w - 1], [433, 433, -9*w - 26], [433, 433, 9*w - 26], [443, 443, -13*w - 8], [443, 443, 13*w - 8], [457, 457, 3*w - 22], [457, 457, -3*w - 22], [467, 467, 14*w - 11], [467, 467, -14*w - 11], [479, 479, 15*w - 14], [479, 479, -15*w - 14], [491, 491, 13*w - 4], [491, 491, -13*w - 4], [503, 503, 13*w - 2], [503, 503, -13*w - 2], [541, 541, 9*w - 28], [541, 541, -9*w - 28], [563, 563, -14*w - 5], [563, 563, 14*w - 5], [577, 577, 4*w - 25], [577, 577, -4*w - 25], [587, 587, -14*w - 1], [587, 587, 14*w - 1], [599, 599, -16*w - 13], [599, 599, 16*w - 13], [601, 601, 5*w - 26], [601, 601, -5*w - 26], [613, 613, 2*w - 25], [613, 613, -2*w - 25], [647, 647, -16*w - 11], [647, 647, 16*w - 11], [659, 659, -15*w - 4], [659, 659, 15*w - 4], [661, 661, -11*w - 32], [661, 661, 11*w - 32], [673, 673, -w - 26], [673, 673, w - 26], [683, 683, 18*w - 17], [683, 683, -18*w - 17], [709, 709, 5*w - 28], [709, 709, -5*w - 28], [719, 719, -16*w - 7], [719, 719, 16*w - 7], [733, 733, -6*w - 29], [733, 733, 6*w - 29], [743, 743, 16*w - 5], [743, 743, -16*w - 5], [757, 757, 3*w - 28], [757, 757, -3*w - 28], [769, 769, -8*w - 31], [769, 769, 8*w - 31], [827, 827, -19*w - 16], [827, 827, 19*w - 16], [829, 829, 2*w - 29], [829, 829, -2*w - 29], [839, 839, 20*w - 19], [839, 839, -20*w - 19], [841, 29, -29], [853, 853, 6*w - 31], [853, 853, -6*w - 31], [863, 863, 17*w - 2], [863, 863, -17*w - 2], [877, 877, 7*w - 32], [877, 877, -7*w - 32], [887, 887, 19*w - 14], [887, 887, -19*w - 14], [911, 911, 20*w - 17], [911, 911, -20*w - 17], [937, 937, -13*w - 38], [937, 937, 13*w - 38], [947, 947, 18*w - 5], [947, 947, -18*w - 5], [961, 31, -31], [971, 971, -18*w - 1], [971, 971, 18*w - 1], [983, 983, -19*w - 10], [983, 983, 19*w - 10], [997, 997, -3*w - 32], [997, 997, 3*w - 32], [1009, 1009, -7*w - 34], [1009, 1009, 7*w - 34], [1019, 1019, 19*w - 8], [1019, 1019, -19*w - 8], [1021, 1021, -w - 32], [1021, 1021, w - 32], [1031, 1031, 20*w - 13], [1031, 1031, -20*w - 13], [1033, 1033, 8*w - 35], [1033, 1033, -8*w - 35], [1069, 1069, 10*w - 37], [1069, 1069, -10*w - 37], [1091, 1091, -22*w - 19], [1091, 1091, 22*w - 19], [1093, 1093, 13*w - 40], [1093, 1093, -13*w - 40], [1103, 1103, 23*w - 22], [1103, 1103, -23*w - 22], [1117, 1117, -6*w - 35], [1117, 1117, 6*w - 35], [1129, 1129, 3*w - 34], [1129, 1129, -3*w - 34], [1151, 1151, -20*w - 7], [1151, 1151, 20*w - 7], [1153, 1153, -w - 34], [1153, 1153, w - 34], [1163, 1163, 22*w - 17], [1163, 1163, -22*w - 17], [1187, 1187, 23*w - 20], [1187, 1187, -23*w - 20], [1201, 1201, -9*w - 38], [1201, 1201, 9*w - 38], [1213, 1213, 2*w - 35], [1213, 1213, -2*w - 35], [1223, 1223, 21*w - 10], [1223, 1223, -21*w - 10], [1237, 1237, -11*w - 40], [1237, 1237, 11*w - 40], [1249, 1249, 12*w - 41], [1249, 1249, -12*w - 41], [1259, 1259, -21*w - 8], [1259, 1259, 21*w - 8], [1283, 1283, 22*w - 13], [1283, 1283, -22*w - 13], [1297, 1297, -7*w - 38], [1297, 1297, 7*w - 38], [1307, 1307, 21*w - 4], [1307, 1307, -21*w - 4], [1319, 1319, 21*w - 2], [1319, 1319, -21*w - 2], [1321, 1321, 4*w - 37], [1321, 1321, -4*w - 37], [1367, 1367, 24*w - 19], [1367, 1367, -24*w - 19], [1381, 1381, 10*w - 41], [1381, 1381, -10*w - 41], [1427, 1427, -22*w - 5], [1427, 1427, 22*w - 5], [1429, 1429, -13*w - 44], [1429, 1429, 13*w - 44], [1439, 1439, 24*w - 17], [1439, 1439, -24*w - 17], [1451, 1451, -22*w - 1], [1451, 1451, 22*w - 1], [1453, 1453, 7*w - 40], [1453, 1453, -7*w - 40], [1487, 1487, -23*w - 10], [1487, 1487, 23*w - 10], [1489, 1489, 8*w - 41], [1489, 1489, -8*w - 41], [1499, 1499, 26*w - 23], [1499, 1499, -26*w - 23], [1511, 1511, 27*w - 26], [1511, 1511, -27*w - 26], [1523, 1523, -23*w - 8], [1523, 1523, 23*w - 8], [1549, 1549, -10*w - 43], [1549, 1549, 10*w - 43], [1559, 1559, 24*w - 13], [1559, 1559, -24*w - 13], [1571, 1571, -23*w - 4], [1571, 1571, 23*w - 4], [1583, 1583, 23*w - 2], [1583, 1583, -23*w - 2], [1597, 1597, -w - 40], [1597, 1597, w - 40], [1607, 1607, -24*w - 11], [1607, 1607, 24*w - 11], [1609, 1609, 13*w - 46], [1609, 1609, -13*w - 46], [1619, 1619, -25*w - 16], [1619, 1619, 25*w - 16], [1621, 1621, 14*w - 47], [1621, 1621, -14*w - 47], [1657, 1657, 8*w - 43], [1657, 1657, -8*w - 43], [1667, 1667, -26*w - 19], [1667, 1667, 26*w - 19], [1669, 1669, 2*w - 41], [1669, 1669, -2*w - 41], [1681, 41, -41], [1693, 1693, -9*w - 44], [1693, 1693, 9*w - 44], [1741, 1741, 6*w - 43], [1741, 1741, -6*w - 43], [1753, 1753, -11*w - 46], [1753, 1753, 11*w - 46], [1777, 1777, -12*w - 47], [1777, 1777, 12*w - 47], [1787, 1787, 27*w - 20], [1787, 1787, -27*w - 20], [1789, 1789, -7*w - 44], [1789, 1789, 7*w - 44], [1801, 1801, -4*w - 43], [1801, 1801, 4*w - 43], [1811, 1811, 25*w - 8], [1811, 1811, -25*w - 8], [1823, 1823, -28*w - 23], [1823, 1823, 28*w - 23], [1847, 1847, 29*w - 26], [1847, 1847, -29*w - 26], [1849, 43, -43], [1861, 1861, -5*w - 44], [1861, 1861, 5*w - 44], [1871, 1871, 25*w - 2], [1871, 1871, -25*w - 2], [1873, 1873, 9*w - 46], [1873, 1873, -9*w - 46], [1907, 1907, 26*w - 11], [1907, 1907, -26*w - 11], [1931, 1931, 27*w - 16], [1931, 1931, -27*w - 16], [1933, 1933, -w - 44], [1933, 1933, w - 44], [1979, 1979, 26*w - 7], [1979, 1979, -26*w - 7], [1993, 1993, 13*w - 50], [1993, 1993, -13*w - 50], [2003, 2003, 26*w - 5], [2003, 2003, -26*w - 5], [2017, 2017, -8*w - 47], [2017, 2017, 8*w - 47], [2027, 2027, -26*w - 1], [2027, 2027, 26*w - 1], [2029, 2029, -15*w - 52], [2029, 2029, 15*w - 52], [2039, 2039, -29*w - 22], [2039, 2039, 29*w - 22], [2053, 2053, 18*w - 55], [2053, 2053, -18*w - 55], [2063, 2063, -28*w - 17], [2063, 2063, 28*w - 17], [2087, 2087, 27*w - 10], [2087, 2087, -27*w - 10], [2089, 2089, 3*w - 46], [2089, 2089, -3*w - 46], [2099, 2099, -31*w - 28], [2099, 2099, 31*w - 28], [2111, 2111, 32*w - 31], [2111, 2111, -32*w - 31], [2113, 2113, -w - 46], [2113, 2113, w - 46], [2137, 2137, 11*w - 50], [2137, 2137, -11*w - 50], [2161, 2161, -4*w - 47], [2161, 2161, 4*w - 47], [2207, 2207, -31*w - 26], [2207, 2207, 31*w - 26], [2221, 2221, -14*w - 53], [2221, 2221, 14*w - 53], [2243, 2243, 33*w - 32], [2243, 2243, -33*w - 32], [2267, 2267, 29*w - 16], [2267, 2267, -29*w - 16], [2269, 2269, 17*w - 56], [2269, 2269, -17*w - 56], [2281, 2281, -20*w - 59], [2281, 2281, 20*w - 59], [2293, 2293, 6*w - 49], [2293, 2293, -6*w - 49], [2339, 2339, 30*w - 19], [2339, 2339, -30*w - 19], [2341, 2341, -11*w - 52], [2341, 2341, 11*w - 52], [2351, 2351, -28*w - 1], [2351, 2351, 28*w - 1], [2377, 2377, 12*w - 53], [2377, 2377, -12*w - 53], [2389, 2389, 2*w - 49], [2389, 2389, -2*w - 49], [2399, 2399, 31*w - 22], [2399, 2399, -31*w - 22], [2411, 2411, 30*w - 17], [2411, 2411, -30*w - 17], [2423, 2423, -29*w - 10], [2423, 2423, 29*w - 10], [2437, 2437, -14*w - 55], [2437, 2437, 14*w - 55], [2447, 2447, -32*w - 25], [2447, 2447, 32*w - 25], [2459, 2459, -29*w - 8], [2459, 2459, 29*w - 8], [2473, 2473, -3*w - 50], [2473, 2473, 3*w - 50], [2521, 2521, 20*w - 61], [2521, 2521, -20*w - 61], [2531, 2531, -30*w - 13], [2531, 2531, 30*w - 13], [2543, 2543, -32*w - 23], [2543, 2543, 32*w - 23], [2557, 2557, -7*w - 52], [2557, 2557, 7*w - 52], [2579, 2579, -30*w - 11], [2579, 2579, 30*w - 11], [2591, 2591, -33*w - 26], [2591, 2591, 33*w - 26], [2593, 2593, -12*w - 55], [2593, 2593, 12*w - 55], [2617, 2617, -8*w - 53], [2617, 2617, 8*w - 53], [2663, 2663, 36*w - 35], [2663, 2663, -36*w - 35], [2677, 2677, 3*w - 52], [2677, 2677, -3*w - 52], [2687, 2687, 31*w - 14], [2687, 2687, -31*w - 14], [2689, 2689, 15*w - 58], [2689, 2689, -15*w - 58], [2699, 2699, -30*w - 1], [2699, 2699, 30*w - 1], [2711, 2711, 32*w - 19], [2711, 2711, -32*w - 19], [2713, 2713, 16*w - 59], [2713, 2713, -16*w - 59], [2749, 2749, -18*w - 61], [2749, 2749, 18*w - 61], [2797, 2797, 2*w - 53], [2797, 2797, -2*w - 53], [2809, 53, -53], [2819, 2819, -31*w - 8], [2819, 2819, 31*w - 8], [2833, 2833, -8*w - 55], [2833, 2833, 8*w - 55], [2843, 2843, -34*w - 25], [2843, 2843, 34*w - 25], [2857, 2857, -13*w - 58], [2857, 2857, 13*w - 58], [2879, 2879, 31*w - 2], [2879, 2879, -31*w - 2], [2903, 2903, -32*w - 13], [2903, 2903, 32*w - 13], [2917, 2917, 6*w - 55], [2917, 2917, -6*w - 55], [2927, 2927, -36*w - 31], [2927, 2927, 36*w - 31], [2939, 2939, -34*w - 23], [2939, 2939, 34*w - 23], [2953, 2953, 16*w - 61], [2953, 2953, -16*w - 61], [2963, 2963, 38*w - 37], [2963, 2963, -38*w - 37], [2999, 2999, 35*w - 26], [2999, 2999, -35*w - 26], [3001, 3001, 11*w - 58], [3001, 3001, -11*w - 58], [3011, 3011, 33*w - 16], [3011, 3011, -33*w - 16], [3023, 3023, 32*w - 7], [3023, 3023, -32*w - 7], [3037, 3037, 22*w - 67], [3037, 3037, -22*w - 67], [3049, 3049, -12*w - 59], [3049, 3049, 12*w - 59], [3061, 3061, 5*w - 56], [3061, 3061, -5*w - 56], [3083, 3083, 37*w - 32], [3083, 3083, -37*w - 32], [3109, 3109, -3*w - 56], [3109, 3109, 3*w - 56], [3119, 3119, 39*w - 38], [3119, 3119, -39*w - 38], [3121, 3121, -9*w - 58], [3121, 3121, 9*w - 58], [3167, 3167, -33*w - 10], [3167, 3167, 33*w - 10], [3169, 3169, -15*w - 62], [3169, 3169, 15*w - 62], [3181, 3181, -10*w - 59], [3181, 3181, 10*w - 59], [3191, 3191, -35*w - 22], [3191, 3191, 35*w - 22], [3203, 3203, 33*w - 8], [3203, 3203, -33*w - 8], [3217, 3217, -7*w - 58], [3217, 3217, 7*w - 58], [3229, 3229, 17*w - 64], [3229, 3229, -17*w - 64], [3251, 3251, 33*w - 4], [3251, 3251, -33*w - 4], [3253, 3253, 18*w - 65], [3253, 3253, -18*w - 65], [3299, 3299, 34*w - 13], [3299, 3299, -34*w - 13], [3301, 3301, -21*w - 68], [3301, 3301, 21*w - 68], [3313, 3313, 23*w - 70], [3313, 3313, -23*w - 70], [3323, 3323, -37*w - 28], [3323, 3323, 37*w - 28], [3347, 3347, 34*w - 11], [3347, 3347, -34*w - 11], [3359, 3359, -36*w - 23], [3359, 3359, 36*w - 23], [3361, 3361, -w - 58], [3361, 3361, w - 58], [3371, 3371, 38*w - 31], [3371, 3371, -38*w - 31], [3373, 3373, -6*w - 59], [3373, 3373, 6*w - 59], [3407, 3407, 39*w - 34], [3407, 3407, -39*w - 34], [3433, 3433, -4*w - 59], [3433, 3433, 4*w - 59], [3457, 3457, 16*w - 65], [3457, 3457, -16*w - 65], [3467, 3467, -34*w - 1], [3467, 3467, 34*w - 1], [3469, 3469, 2*w - 59], [3469, 3469, -2*w - 59], [3491, 3491, 38*w - 29], [3491, 3491, -38*w - 29], [3517, 3517, 18*w - 67], [3517, 3517, -18*w - 67], [3527, 3527, 36*w - 19], [3527, 3527, -36*w - 19], [3529, 3529, -8*w - 61], [3529, 3529, 8*w - 61], [3539, 3539, -39*w - 32], [3539, 3539, 39*w - 32], [3541, 3541, 19*w - 68], [3541, 3541, -19*w - 68], [3613, 3613, 6*w - 61], [3613, 3613, -6*w - 61], [3623, 3623, 37*w - 22], [3623, 3623, -37*w - 22], [3637, 3637, -14*w - 65], [3637, 3637, 14*w - 65], [3659, 3659, -35*w - 4], [3659, 3659, 35*w - 4], [3671, 3671, 35*w - 2], [3671, 3671, -35*w - 2], [3673, 3673, 4*w - 61], [3673, 3673, -4*w - 61], [3697, 3697, -7*w - 62], [3697, 3697, 7*w - 62], [3709, 3709, 2*w - 61], [3709, 3709, -2*w - 61], [3719, 3719, 36*w - 13], [3719, 3719, -36*w - 13], [3733, 3733, 11*w - 64], [3733, 3733, -11*w - 64], [3767, 3767, 36*w - 11], [3767, 3767, -36*w - 11], [3769, 3769, -5*w - 62], [3769, 3769, 5*w - 62], [3779, 3779, -39*w - 28], [3779, 3779, 39*w - 28], [3793, 3793, 12*w - 65], [3793, 3793, -12*w - 65], [3803, 3803, -38*w - 23], [3803, 3803, 38*w - 23], [3851, 3851, -37*w - 16], [3851, 3851, 37*w - 16], [3853, 3853, 9*w - 64], [3853, 3853, -9*w - 64], [3863, 3863, 36*w - 5], [3863, 3863, -36*w - 5], [3877, 3877, -22*w - 73], [3877, 3877, 22*w - 73], [3889, 3889, -23*w - 74], [3889, 3889, 23*w - 74], [3911, 3911, -37*w - 14], [3911, 3911, 37*w - 14], [3923, 3923, 42*w - 37], [3923, 3923, -42*w - 37], [3947, 3947, -43*w - 40], [3947, 3947, 43*w - 40], [4007, 4007, 37*w - 10], [4007, 4007, -37*w - 10], [4019, 4019, -41*w - 32], [4019, 4019, 41*w - 32], [4021, 4021, -5*w - 64], [4021, 4021, 5*w - 64], [4057, 4057, -12*w - 67], [4057, 4057, 12*w - 67], [4079, 4079, -39*w - 22], [4079, 4079, 39*w - 22], [4091, 4091, 37*w - 4], [4091, 4091, -37*w - 4], [4093, 4093, -w - 64], [4093, 4093, w - 64], [4127, 4127, 44*w - 41], [4127, 4127, -44*w - 41], [4129, 4129, -20*w - 73], [4129, 4129, 20*w - 73], [4139, 4139, 45*w - 44], [4139, 4139, -45*w - 44], [4153, 4153, 21*w - 74], [4153, 4153, -21*w - 74], [4177, 4177, 4*w - 65], [4177, 4177, -4*w - 65], [4201, 4201, 24*w - 77], [4201, 4201, -24*w - 77], [4211, 4211, -38*w - 11], [4211, 4211, 38*w - 11], [4259, 4259, 41*w - 28], [4259, 4259, -41*w - 28], [4261, 4261, -11*w - 68], [4261, 4261, 11*w - 68], [4271, 4271, -40*w - 23], [4271, 4271, 40*w - 23], [4273, 4273, 16*w - 71], [4273, 4273, -16*w - 71], [4283, 4283, -38*w - 7], [4283, 4283, 38*w - 7], [4297, 4297, 8*w - 67], [4297, 4297, -8*w - 67], [4357, 4357, 18*w - 73], [4357, 4357, -18*w - 73], [4391, 4391, -43*w - 34], [4391, 4391, 43*w - 34], [4441, 4441, -4*w - 67], [4441, 4441, 4*w - 67], [4451, 4451, 42*w - 29], [4451, 4451, -42*w - 29], [4463, 4463, -39*w - 10], [4463, 4463, 39*w - 10], [4489, 67, -67], [4513, 4513, 24*w - 79], [4513, 4513, -24*w - 79], [4523, 4523, 43*w - 32], [4523, 4523, -43*w - 32], [4547, 4547, -39*w - 4], [4547, 4547, 39*w - 4], [4549, 4549, 5*w - 68], [4549, 4549, -5*w - 68], [4561, 4561, -16*w - 73], [4561, 4561, 16*w - 73], [4583, 4583, 44*w - 35], [4583, 4583, -44*w - 35], 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8317, -7*w - 92], [8329, 8329, -13*w - 94], [8329, 8329, 13*w - 94], [8353, 8353, 31*w - 106], [8353, 8353, -31*w - 106], [8363, 8363, -53*w - 8], [8363, 8363, 53*w - 8], [8377, 8377, 32*w - 107], [8377, 8377, -32*w - 107], [8387, 8387, 54*w - 19], [8387, 8387, -54*w - 19], [8389, 8389, -5*w - 92], [8389, 8389, 5*w - 92], [8423, 8423, 53*w - 2], [8423, 8423, -53*w - 2], [8447, 8447, 56*w - 31], [8447, 8447, -56*w - 31], [8461, 8461, -w - 92], [8461, 8461, w - 92], [8521, 8521, -19*w - 98], [8521, 8521, 19*w - 98], [8543, 8543, -63*w - 58], [8543, 8543, 63*w - 58], [8581, 8581, -26*w - 103], [8581, 8581, 26*w - 103], [8627, 8627, -54*w - 11], [8627, 8627, 54*w - 11], [8629, 8629, -27*w - 104], [8629, 8629, 27*w - 104], [8641, 8641, 16*w - 97], [8641, 8641, -16*w - 97], [8663, 8663, -61*w - 50], [8663, 8663, 61*w - 50], [8677, 8677, 21*w - 100], [8677, 8677, -21*w - 100], [8689, 8689, -7*w - 94], [8689, 8689, 7*w - 94], [8699, 8699, 54*w - 7], [8699, 8699, -54*w - 7], [8713, 8713, -29*w - 106], [8713, 8713, 29*w - 106], [8737, 8737, 17*w - 98], [8737, 8737, -17*w - 98], [8747, 8747, -54*w - 1], [8747, 8747, 54*w - 1], [8761, 8761, -5*w - 94], [8761, 8761, 5*w - 94], [8783, 8783, -56*w - 25], [8783, 8783, 56*w - 25], [8807, 8807, 64*w - 59], [8807, 8807, -64*w - 59], [8819, 8819, 55*w - 16], [8819, 8819, -55*w - 16], [8821, 8821, -14*w - 97], [8821, 8821, 14*w - 97], [8831, 8831, 65*w - 62], [8831, 8831, -65*w - 62], [8867, 8867, -58*w - 35], [8867, 8867, 58*w - 35], [8893, 8893, 38*w - 115], [8893, 8893, -38*w - 115], [8929, 8929, 15*w - 98], [8929, 8929, -15*w - 98], [8941, 8941, -25*w - 104], [8941, 8941, 25*w - 104], [8951, 8951, -60*w - 43], [8951, 8951, 60*w - 43], [8963, 8963, 57*w - 28], [8963, 8963, -57*w - 28], [8999, 8999, -59*w - 38], [8999, 8999, 59*w - 38], [9001, 9001, 20*w - 101], [9001, 9001, -20*w - 101], [9011, 9011, -55*w - 8], [9011, 9011, 55*w - 8], [9013, 9013, 2*w - 95], [9013, 9013, -2*w - 95], [9049, 9049, 27*w - 106], [9049, 9049, -27*w - 106], [9059, 9059, -55*w - 4], [9059, 9059, 55*w - 4], [9109, 9109, 10*w - 97], [9109, 9109, -10*w - 97], [9133, 9133, 17*w - 100], [9133, 9133, -17*w - 100], [9157, 9157, 22*w - 103], [9157, 9157, -22*w - 103], [9181, 9181, -30*w - 109], [9181, 9181, 30*w - 109], [9203, 9203, 63*w - 52], [9203, 9203, -63*w - 52], [9227, 9227, 61*w - 44], [9227, 9227, -61*w - 44], [9239, 9239, -56*w - 13], [9239, 9239, 56*w - 13], [9241, 9241, -11*w - 98], [9241, 9241, 11*w - 98], [9277, 9277, -33*w - 112], [9277, 9277, 33*w - 112], [9311, 9311, 65*w - 58], [9311, 9311, -65*w - 58], [9323, 9323, 62*w - 47], [9323, 9323, -62*w - 47], [9337, 9337, -36*w - 115], [9337, 9337, 36*w - 115], [9349, 9349, 37*w - 116], [9349, 9349, -37*w - 116], [9371, 9371, -67*w - 64], [9371, 9371, 67*w - 64], [9397, 9397, 2*w - 97], [9397, 9397, -2*w - 97], [9419, 9419, -59*w - 32], [9419, 9419, 59*w - 32], [9421, 9421, 26*w - 107], [9421, 9421, -26*w - 107], [9431, 9431, -60*w - 37], [9431, 9431, 60*w - 37], [9433, 9433, -16*w - 101], [9433, 9433, 16*w - 101], [9467, 9467, -58*w - 25], [9467, 9467, 58*w - 25], [9479, 9479, -64*w - 53], [9479, 9479, 64*w - 53], [9491, 9491, -57*w - 16], [9491, 9491, 57*w - 16], [9539, 9539, 65*w - 56], [9539, 9539, -65*w - 56], [9551, 9551, 57*w - 14], [9551, 9551, -57*w - 14], [9587, 9587, 66*w - 59], [9587, 9587, -66*w - 59], [9601, 9601, -w - 98], [9601, 9601, w - 98], [9613, 9613, 14*w - 101], [9613, 9613, -14*w - 101], [9623, 9623, 67*w - 62], [9623, 9623, -67*w - 62], [9649, 9649, 23*w - 106], [9649, 9649, -23*w - 106], [9661, 9661, -31*w - 112], [9661, 9661, 31*w - 112], [9697, 9697, -32*w - 113], [9697, 9697, 32*w - 113], [9719, 9719, 61*w - 38], [9719, 9719, -61*w - 38], [9721, 9721, 24*w - 107], [9721, 9721, -24*w - 107], [9733, 9733, -19*w - 104], [9733, 9733, 19*w - 104], [9743, 9743, 57*w - 2], [9743, 9743, -57*w - 2], [9767, 9767, -59*w - 26], [9767, 9767, 59*w - 26], [9769, 9769, 12*w - 101], [9769, 9769, -12*w - 101], [9781, 9781, 35*w - 116], [9781, 9781, -35*w - 116], [9791, 9791, 63*w - 46], [9791, 9791, -63*w - 46], [9803, 9803, 58*w - 17], [9803, 9803, -58*w - 17], [9817, 9817, 37*w - 118], [9817, 9817, -37*w - 118], [9829, 9829, -38*w - 119], [9829, 9829, 38*w - 119], [9839, 9839, 60*w - 31], [9839, 9839, -60*w - 31], [9851, 9851, 62*w - 41], [9851, 9851, -62*w - 41], [9887, 9887, -64*w - 49], [9887, 9887, 64*w - 49], [9901, 9901, 10*w - 101], [9901, 9901, -10*w - 101], [9923, 9923, -58*w - 13], [9923, 9923, 58*w - 13], [9949, 9949, -17*w - 104], [9949, 9949, 17*w - 104], [9973, 9973, 3*w - 100], [9973, 9973, -3*w - 100]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 0, 0, 0, 2, -2, 0, 0, -10, -10, 10, 0, 0, -2, 0, 0, 14, -14, 0, 0, -10, -10, 0, 0, -14, -14, 0, 0, 2, -2, 0, 0, -14, 14, 0, 0, 0, 0, -26, 26, 0, 0, -2, -2, 0, 0, -22, 22, 0, 0, 14, 14, 0, 0, 0, 0, -26, 26, 34, 0, 0, 22, 22, -34, -34, 0, 0, -14, 14, 0, 0, 26, -38, 38, 0, 0, 34, -34, -38, -38, 0, 0, 22, -22, 0, 0, 2, 2, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 46, -46, 0, 0, 46, 46, 0, 0, 0, 0, -26, -26, -10, 10, 0, 0, 0, 0, 38, -38, 50, 50, 0, 0, -22, 22, 0, 0, 50, -50, 0, 0, -26, 26, -2, -2, 0, 0, -46, 46, 0, 0, -58, 58, -58, 0, 0, -34, 34, 0, 0, 0, 0, 26, 26, 0, 0, 46, 0, 0, 0, 0, -10, 10, -62, -62, 0, 0, 14, -14, 0, 0, 58, 58, -62, 62, 0, 0, 22, -22, 0, 0, -46, 46, 38, 38, 0, 0, -62, -62, 0, 0, 0, 0, -2, -2, 50, -50, 0, 0, 70, -70, 14, 14, 0, 0, 0, 0, 46, 46, 0, 0, 0, 0, -22, -22, 0, 0, -74, 74, 0, 0, -58, 58, 0, 0, 0, 0, -2, 2, 0, 0, 34, 34, 0, 0, 0, 0, 0, 0, 62, -62, 0, 0, 0, 0, 0, 0, -50, 50, 0, 0, 58, 58, 0, 0, 26, -26, -70, -70, 0, 0, -74, 74, 82, 82, -82, -34, 34, 10, 10, -14, -14, 0, 0, -82, 82, -74, -74, 0, 0, 0, 0, 0, 0, -22, 86, -86, 0, 0, -82, -82, 0, 0, 0, 0, -62, 62, 0, 0, 70, 70, 0, 0, -34, -34, 0, 0, -2, 2, 0, 0, -10, 10, 0, 0, 0, 0, -38, -38, 0, 0, 0, 0, -82, -82, -74, -74, -62, -62, 0, 0, -94, 94, 0, 0, 0, 0, 82, -82, 86, 86, 58, -58, 0, 0, 74, -74, 0, 0, 10, 10, 38, -38, 0, 0, 0, 0, 0, 0, 86, -86, 0, 0, 0, 0, 22, 22, 26, 26, 0, 0, 0, 0, 46, -46, 0, 0, 0, 0, 98, 98, 86, 86, 0, 0, 70, -70, 0, 0, -62, -62, 0, 0, 0, 0, -38, -38, 14, -14, -94, 94, -106, 0, 0, -98, -98, 0, 0, -106, -106, 0, 0, 0, 0, 106, -106, 0, 0, 0, 0, 70, 70, 0, 0, 0, 0, -106, -106, 0, 0, 0, 0, 110, -110, 86, 86, 38, -38, 0, 0, -106, 106, 0, 0, -14, -14, 0, 0, -98, -98, 94, -94, 0, 0, 0, 0, -110, -110, -46, 46, 0, 0, 70, -70, 0, 0, 86, -86, 62, 62, 0, 0, 0, 0, 0, 0, 34, 34, 0, 0, 98, -98, 0, 0, 38, 38, -110, -110, 0, 0, 2, -2, 0, 0, -14, 14, 0, 0, 118, 118, 0, 0, -58, 58, 110, -110, 0, 0, -26, 26, 0, 0, 0, 0, -118, -118, -50, -50, 82, -82, 0, 0, 122, -122, 0, 0, 122, 122, 0, 0, 110, 110, 0, 0, 0, 0, 98, -98, 0, 0, 86, -86, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 122, -122, -26, -26, 0, 0, 0, 0, -50, 50, 0, 0, -98, -98, 0, 0, -122, -122, -34, -34, 86, 86, 0, 0, 0, 0, 106, -106, 0, 0, -130, -130, 0, 0, -70, -70, -10, 10, 0, 0, -74, -74, 0, 0, 0, 0, 122, -50, -50, 0, 0, 0, 0, -86, 86, -94, -94, 0, 0, -134, 134, 34, -34, 0, 0, -130, -130, 0, 0, 0, 0, 0, 0, 58, 58, 0, 0, 0, 0, 134, -134, 0, 0, 2, 2, 130, -130, 46, -46, 0, 0, -94, 94, 0, 0, 0, 0, 118, -118, 0, 0, -50, 50, 0, 0, -26, -26, 130, 130, 0, 0, 0, 0, 0, 0, 134, -134, 0, 0, 0, 0, 98, -98, 70, 70, 0, 0, 0, 0, -130, 130, -118, -118, 0, 0, -142, -142, 0, 0, 94, 94, 0, 0, 0, 0, 0, 0, 0, 0, -22, 22, 146, -146, -122, -122, 0, 0, 0, 0, 0, 0, 0, 0, -146, -146, 0, 0, -70, 70, -82, -82, -34, 34, 0, 0, 0, 0, -58, -58, 0, 0, 38, -38, 134, 134, 74, -74, 0, 0, 86, 86, -122, 122, 0, 0, 0, 0, -46, 46, 0, 0, -14, -14, 0, 0, 98, -98, 0, 0, 106, 106, 0, 0, 0, 0, 0, 0, -130, -130, 0, 0, 0, 0, 154, -154, 0, 0, 122, 122, 154, 154, 0, 0, 58, -58, 0, 0, 0, 0, 134, 134, -154, 154, 142, 0, 0, -106, 106, 0, 0, 0, 0, 146, -146, 0, 0, 0, 0, -46, -46, 0, 0, -154, -154, -10, 10, 14, -14, 122, -122, -22, 22, 82, 82, 0, 0, -146, -146, 0, 0, -118, -118, 0, 0, 130, 130, 0, 0, -34, 34, 0, 0, 74, -74, -158, -158, -38, 38, 0, 0, -98, 98, 0, 0, 146, -146, 0, 0, 122, 122, 0, 0, 0, 0, -158, 158, 134, 134, 0, 0, 0, 0, 0, 0, 0, 0, -118, 118, 0, 0, 14, 14, 0, 0, 0, 0, -166, 166, 0, 0, 0, 0, -146, -146, 142, -142, 0, 0, 0, 0, 0, 0, -154, -154, 0, 0, 74, 74, 0, 0, 0, 0, -158, 158, -170, 170, 0, 0, 0, 0, -130, -130, 0, 0, -62, 62, -166, -166, 0, 0, 170, -170, -118, -118, 142, 142, -170, -170, 0, 0, -166, 166, 0, 0, -82, -82, 0, 0, 0, 0, -50, -50, 0, 0, 14, -14, 0, 0, 134, 134, 70, -70, 0, 0, 0, 0, -22, 22, 0, 0, -26, 26, 146, 146, 0, 0, 0, 0, 74, -74, 0, 0, 142, -142, 58, 58, -34, 34, 0, 0, 62, 62, 0, 0, 0, 0, 0, 0, 178, 178, -178, 170, 170, -94, -94, 0, 0, 122, -122, 0, 0, -166, -166, -106, 106, 0, 0, 0, 0, 0, 0, 14, 14, 0, 0, -98, -98, 0, 0, 178, -178, 0, 0, 22, 22, 0, 0, -158, 158, 0, 0, 182, -182, -110, 110, -182, -182, 178, 178, 0, 0, -134, -134, 0, 0, 182, -182, 0, 0, 0, 0, -62, 62, -122, -122, 0, 0, -182, 182, 0, 0, -154, 154, -46, -46, 0, 0, 170, -170, -178, -178, 0, 0, 182, 182, 50, 50, 0, 0, -86, -86, 0, 0, 0, 0, 0, 0, 134, -134, 0, 0, 0, 0, -178, 178, -142, -142, 158, -158, 0, 0, 0, 0, 0, 0, 154, 154, 0, 0, -122, 122, 182, 182, 0, 0, 38, -38, 190, -190, 106, -106, 82, -82, 0, 0, 0, 0, 0, 0, 166, 166, 46, -46, 0, 0, 0, 0, 70, 70, -86, 86, 0, 0, -154, 154, 0, 0, 194, -194, 0, 0, -10, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 194, 194, 190, -190, 0, 0, 178, 178, 146, -146, 34, 34, 0, 0, 106, 106, 182, -182, 0, 0, 0, 0, -38, -38, 134, -134, 0, 0, 0, 0, 154, 154, 118, -118, 0, 0, 0, 0, 0, 0, 98, -98, 0, 0, -178, 178, 70, -70]; 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;