/* 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, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, 2], [7, 7, -w - 3], [9, 3, 3], [11, 11, w + 5], [13, 13, w + 2], [13, 13, w - 3], [17, 17, w + 1], [17, 17, -w + 2], [19, 19, w], [19, 19, w - 1], [23, 23, w + 6], [23, 23, -w + 7], [25, 5, -5], [37, 37, -w - 7], [37, 37, w - 8], [41, 41, 2*w - 7], [41, 41, -2*w - 5], [53, 53, -w - 8], [53, 53, w - 9], [61, 61, 2*w - 5], [61, 61, -2*w - 3], [67, 67, 2*w - 13], [67, 67, -2*w - 11], [71, 71, -w - 9], [71, 71, w - 10], [73, 73, 2*w - 3], [73, 73, -2*w - 1], [83, 83, 3*w - 11], [83, 83, -3*w - 8], [101, 101, -3*w - 7], [101, 101, 3*w - 10], [113, 113, -w - 11], [113, 113, w - 12], [131, 131, 3*w - 8], [131, 131, -3*w - 5], [137, 137, -w - 12], [137, 137, w - 13], [139, 139, 4*w - 15], [139, 139, -4*w - 11], [163, 163, -w - 13], [163, 163, w - 14], [167, 167, -3*w - 1], [167, 167, 3*w - 4], [173, 173, 3*w - 2], [173, 173, 3*w - 1], [179, 179, 2*w - 17], [179, 179, -2*w - 15], [191, 191, -w - 14], [191, 191, w - 15], [227, 227, 4*w - 11], [227, 227, -4*w - 7], [241, 241, -5*w - 13], [241, 241, 5*w - 18], [271, 271, 5*w - 17], [271, 271, -5*w - 12], [283, 283, 4*w - 7], [283, 283, -4*w - 3], [293, 293, 6*w - 23], [293, 293, -6*w - 17], [307, 307, 4*w - 3], [307, 307, 4*w - 1], [317, 317, -4*w - 23], [317, 317, 4*w - 27], [331, 331, 5*w - 31], [331, 331, -5*w - 26], [349, 349, 5*w - 14], [349, 349, -5*w - 9], [379, 379, 3*w - 25], [379, 379, -3*w - 22], [389, 389, -5*w - 27], [389, 389, 5*w - 32], [401, 401, -w - 20], [401, 401, w - 21], [409, 409, -5*w - 6], [409, 409, 5*w - 11], [421, 421, 4*w - 29], [421, 421, -4*w - 25], [439, 439, 5*w - 9], [439, 439, -5*w - 4], [443, 443, -w - 21], [443, 443, w - 22], [449, 449, 5*w - 33], [449, 449, -5*w - 28], [461, 461, 5*w - 7], [461, 461, -5*w - 2], [463, 463, -7*w - 34], [463, 463, 7*w - 41], [479, 479, 5*w - 4], [479, 479, 5*w - 1], [487, 487, -w - 22], [487, 487, w - 23], [499, 499, 2*w - 25], [499, 499, -2*w - 23], [503, 503, 9*w + 28], [503, 503, 8*w + 23], [523, 523, 7*w - 24], [523, 523, -7*w - 17], [563, 563, 7*w - 23], [563, 563, -7*w - 16], [593, 593, -6*w - 7], [593, 593, 6*w - 13], [599, 599, 2*w - 27], [599, 599, -2*w - 25], [601, 601, -7*w - 15], [601, 601, 7*w - 22], [607, 607, 8*w - 29], [607, 607, -8*w - 21], [617, 617, -8*w - 39], [617, 617, 8*w - 47], [631, 631, -w - 25], [631, 631, w - 26], [641, 641, 5*w - 36], [641, 641, -5*w - 31], [653, 653, 4*w - 33], [653, 653, -4*w - 29], [677, 677, -6*w - 1], [677, 677, 6*w - 7], [683, 683, -w - 26], [683, 683, w - 27], [709, 709, -5*w - 32], [709, 709, 5*w - 37], [733, 733, -7*w - 11], [733, 733, 7*w - 18], [751, 751, -6*w - 35], [751, 751, 6*w - 41], [757, 757, -3*w - 29], [757, 757, 3*w - 32], [761, 761, 7*w - 17], [761, 761, -7*w - 10], [769, 769, 11*w + 34], [769, 769, 10*w + 29], [787, 787, 7*w - 16], [787, 787, -7*w - 9], [811, 811, -7*w - 8], [811, 811, 7*w - 15], [823, 823, 2*w - 31], [823, 823, -2*w - 29], [841, 29, -29], [853, 853, 7*w - 13], [853, 853, -7*w - 6], [857, 857, 9*w - 31], [857, 857, -9*w - 22], [863, 863, 7*w - 46], [863, 863, -7*w - 39], [883, 883, 3*w - 34], [883, 883, -3*w - 31], [887, 887, -7*w - 4], [887, 887, 7*w - 11], [907, 907, 6*w - 43], [907, 907, -6*w - 37], [911, 911, -w - 30], [911, 911, w - 31], [937, 937, 7*w - 6], [937, 937, 7*w - 1], [941, 941, 7*w - 2], [941, 941, 7*w - 5], [947, 947, 2*w - 33], [947, 947, -2*w - 31], [961, 31, -31], [977, 977, 8*w - 51], [977, 977, -8*w - 43], [991, 991, -10*w - 49], [991, 991, 10*w - 59], [997, 997, -11*w - 31], [997, 997, 11*w - 42], [1061, 1061, -4*w - 35], [1061, 1061, 4*w - 39], [1063, 1063, -8*w - 9], [1063, 1063, 8*w - 17], [1069, 1069, 11*w - 41], [1069, 1069, -11*w - 30], [1087, 1087, 3*w - 37], [1087, 1087, -3*w - 34], [1091, 1091, 13*w + 40], [1091, 1091, 12*w + 35], [1093, 1093, -9*w - 47], [1093, 1093, 9*w - 56], [1097, 1097, 9*w - 26], [1097, 1097, -9*w - 17], [1103, 1103, -w - 33], [1103, 1103, w - 34], [1151, 1151, 8*w - 13], [1151, 1151, -8*w - 5], [1171, 1171, -w - 34], [1171, 1171, w - 35], [1213, 1213, 4*w - 41], [1213, 1213, -4*w - 37], [1217, 1217, -9*w - 14], [1217, 1217, 9*w - 23], [1223, 1223, 8*w - 7], [1223, 1223, 8*w - 1], [1231, 1231, 8*w - 5], [1231, 1231, 8*w - 3], [1249, 1249, -10*w - 21], [1249, 1249, 10*w - 31], [1303, 1303, -9*w - 49], [1303, 1303, 9*w - 58], [1319, 1319, 9*w - 20], [1319, 1319, -9*w - 11], [1361, 1361, -13*w - 37], [1361, 1361, 13*w - 50], [1367, 1367, 2*w - 39], [1367, 1367, -2*w - 37], [1373, 1373, -4*w - 39], [1373, 1373, 4*w - 43], [1399, 1399, -11*w - 25], [1399, 1399, 11*w - 36], [1409, 1409, -7*w - 45], [1409, 1409, 7*w - 52], [1423, 1423, 6*w - 49], [1423, 1423, -6*w - 43], [1427, 1427, -9*w - 7], [1427, 1427, 9*w - 16], [1439, 1439, -10*w - 53], [1439, 1439, 10*w - 63], [1447, 1447, -13*w - 36], [1447, 1447, 13*w - 49], [1453, 1453, -12*w - 59], [1453, 1453, 12*w - 71], [1459, 1459, 11*w - 35], [1459, 1459, -11*w - 24], [1487, 1487, 9*w - 13], [1487, 1487, -9*w - 4], [1499, 1499, -5*w - 42], [1499, 1499, 5*w - 47], [1523, 1523, 2*w - 41], [1523, 1523, -2*w - 39], [1531, 1531, -13*w - 35], [1531, 1531, 13*w - 48], [1549, 1549, 3*w - 43], [1549, 1549, -3*w - 40], [1553, 1553, 9*w - 2], [1553, 1553, 9*w - 7], [1559, 1559, 9*w - 4], [1559, 1559, 9*w - 5], [1601, 1601, 10*w - 23], [1601, 1601, -10*w - 13], [1607, 1607, 7*w - 54], [1607, 1607, -7*w - 47], [1613, 1613, -13*w - 34], [1613, 1613, 13*w - 47], [1621, 1621, -w - 40], [1621, 1621, w - 41], [1627, 1627, 11*w - 32], [1627, 1627, -11*w - 21], [1657, 1657, 14*w - 53], [1657, 1657, -14*w - 39], [1669, 1669, -10*w - 11], [1669, 1669, 10*w - 21], [1693, 1693, -13*w - 33], [1693, 1693, 13*w - 46], [1709, 1709, -7*w - 48], [1709, 1709, 7*w - 55], [1747, 1747, -9*w - 53], [1747, 1747, 9*w - 62], [1777, 1777, 11*w - 29], [1777, 1777, -11*w - 18], [1787, 1787, -w - 42], [1787, 1787, w - 43], [1811, 1811, -12*w - 25], [1811, 1811, 12*w - 37], [1823, 1823, -11*w - 17], [1823, 1823, 11*w - 28], [1831, 1831, 11*w - 70], [1831, 1831, -11*w - 59], [1847, 1847, 13*w - 44], [1847, 1847, -13*w - 31], [1849, 43, -43], [1861, 1861, 10*w - 13], [1861, 1861, -10*w - 3], [1867, 1867, 11*w - 27], [1867, 1867, -11*w - 16], [1871, 1871, 5*w - 51], [1871, 1871, -5*w - 46], [1873, 1873, -w - 43], [1873, 1873, w - 44], [1889, 1889, -10*w - 1], [1889, 1889, 10*w - 11], [1901, 1901, 4*w - 49], [1901, 1901, -4*w - 45], [1931, 1931, 12*w - 35], [1931, 1931, -12*w - 23], [1949, 1949, -11*w - 14], [1949, 1949, 11*w - 25], [1979, 1979, 15*w - 56], [1979, 1979, -15*w - 41], [1987, 1987, 11*w - 24], [1987, 1987, -11*w - 13], [1993, 1993, 13*w - 42], [1993, 1993, -13*w - 29], [2003, 2003, -14*w - 69], [2003, 2003, 14*w - 83], [2011, 2011, 6*w - 55], [2011, 2011, -6*w - 49], [2017, 2017, -8*w - 53], [2017, 2017, 8*w - 61], [2027, 2027, -7*w - 51], [2027, 2027, 7*w - 58], [2039, 2039, 2*w - 47], [2039, 2039, -2*w - 45], [2063, 2063, 13*w - 41], [2063, 2063, -13*w - 28], [2069, 2069, 5*w - 53], [2069, 2069, -5*w - 48], [2083, 2083, 3*w - 49], [2083, 2083, -3*w - 46], [2089, 2089, 11*w - 21], [2089, 2089, -11*w - 10], [2131, 2131, -13*w - 27], [2131, 2131, 13*w - 40], [2137, 2137, -7*w - 52], [2137, 2137, 7*w - 59], [2141, 2141, 18*w + 55], [2141, 2141, 17*w + 50], [2143, 2143, -w - 46], [2143, 2143, w - 47], [2179, 2179, -3*w - 47], [2179, 2179, 3*w - 50], [2209, 47, -47], [2237, 2237, -w - 47], [2237, 2237, w - 48], [2239, 2239, -11*w - 4], [2239, 2239, 11*w - 15], [2243, 2243, -12*w - 17], [2243, 2243, 12*w - 29], [2269, 2269, 12*w - 77], [2269, 2269, -12*w - 65], [2273, 2273, 11*w - 13], [2273, 2273, -11*w - 2], [2287, 2287, -11*w - 1], [2287, 2287, 11*w - 12], [2293, 2293, 4*w - 53], [2293, 2293, -4*w - 49], [2297, 2297, -13*w - 68], [2297, 2297, 13*w - 81], [2309, 2309, 11*w - 10], [2309, 2309, 11*w - 1], [2311, 2311, -15*w - 74], [2311, 2311, 15*w - 89], [2333, 2333, -w - 48], [2333, 2333, w - 49], [2347, 2347, -9*w - 58], [2347, 2347, 9*w - 67], [2351, 2351, 15*w - 52], [2351, 2351, -15*w - 37], [2371, 2371, 17*w - 65], [2371, 2371, -17*w - 48], [2377, 2377, 3*w - 52], [2377, 2377, -3*w - 49], [2381, 2381, 5*w - 56], [2381, 2381, -5*w - 51], [2383, 2383, -13*w - 23], [2383, 2383, 13*w - 36], [2393, 2393, 19*w + 58], [2393, 2393, 18*w + 53], [2411, 2411, -12*w - 13], [2411, 2411, 12*w - 25], [2423, 2423, 2*w - 51], [2423, 2423, -2*w - 49], [2441, 2441, -13*w - 22], [2441, 2441, 13*w - 35], [2447, 2447, -13*w - 69], [2447, 2447, 13*w - 82], [2473, 2473, 9*w - 68], [2473, 2473, -9*w - 59], [2477, 2477, -14*w - 29], [2477, 2477, 14*w - 43], [2531, 2531, -w - 50], [2531, 2531, w - 51], [2551, 2551, -13*w - 20], [2551, 2551, 13*w - 33], [2557, 2557, -12*w - 67], [2557, 2557, 12*w - 79], [2593, 2593, 17*w - 63], [2593, 2593, -17*w - 46], [2609, 2609, -15*w - 34], [2609, 2609, 15*w - 49], [2617, 2617, 14*w - 41], [2617, 2617, -14*w - 27], [2633, 2633, -w - 51], [2633, 2633, w - 52], [2659, 2659, 20*w + 61], [2659, 2659, 19*w + 56], [2671, 2671, 6*w - 61], [2671, 2671, -6*w - 55], [2689, 2689, 3*w - 55], [2689, 2689, -3*w - 52], [2699, 2699, -10*w - 63], [2699, 2699, 10*w - 73], [2711, 2711, -5*w - 54], [2711, 2711, 5*w - 59], [2719, 2719, -16*w - 39], [2719, 2719, 16*w - 55], [2731, 2731, 9*w - 70], [2731, 2731, -9*w - 61], [2749, 2749, 14*w - 39], [2749, 2749, -14*w - 25], [2753, 2753, -13*w - 71], [2753, 2753, 13*w - 84], [2789, 2789, 19*w + 55], [2789, 2789, 21*w + 65], [2791, 2791, 13*w - 28], [2791, 2791, -13*w - 15], [2797, 2797, -3*w - 53], [2797, 2797, 3*w - 56], [2833, 2833, -13*w - 14], [2833, 2833, 13*w - 27], [2843, 2843, -w - 53], [2843, 2843, w - 54], [2903, 2903, 16*w - 53], [2903, 2903, -16*w - 37], [2909, 2909, 13*w - 85], [2909, 2909, -13*w - 72], [2917, 2917, -19*w - 54], [2917, 2917, 19*w - 73], [2927, 2927, 11*w - 78], [2927, 2927, -11*w - 67], [2939, 2939, 21*w + 64], [2939, 2939, 20*w + 59], [2963, 2963, -7*w - 59], [2963, 2963, 7*w - 66], [2999, 2999, 15*w - 44], [2999, 2999, -15*w - 29], [3019, 3019, 3*w - 58], [3019, 3019, -3*w - 55], [3061, 3061, -w - 55], [3061, 3061, w - 56], [3067, 3067, 13*w - 86], [3067, 3067, -13*w - 73], [3079, 3079, 16*w - 51], [3079, 3079, -16*w - 35], [3089, 3089, 7*w - 67], [3089, 3089, -7*w - 60], [3121, 3121, 13*w - 18], [3121, 3121, -13*w - 5], [3163, 3163, 13*w - 16], [3163, 3163, -13*w - 3], [3167, 3167, 19*w - 71], [3167, 3167, -19*w - 52], [3181, 3181, 13*w - 15], [3181, 3181, -13*w - 2], [3209, 3209, 15*w - 41], [3209, 3209, -15*w - 26], [3217, 3217, 7*w - 68], [3217, 3217, -7*w - 61], [3221, 3221, 11*w - 80], [3221, 3221, -11*w - 69], [3251, 3251, 13*w - 5], [3251, 3251, 13*w - 8], [3253, 3253, 13*w - 6], [3253, 3253, 13*w - 7], [3257, 3257, -8*w - 63], [3257, 3257, 8*w - 71], [3259, 3259, 10*w - 77], [3259, 3259, -10*w - 67], [3271, 3271, -9*w - 65], [3271, 3271, 9*w - 74], [3301, 3301, -5*w - 59], [3301, 3301, 5*w - 64], [3307, 3307, -17*w - 39], [3307, 3307, 17*w - 56], [3347, 3347, -7*w - 62], [3347, 3347, 7*w - 69], [3371, 3371, 11*w - 81], [3371, 3371, -11*w - 70], [3373, 3373, 14*w - 27], [3373, 3373, -14*w - 13], [3389, 3389, 13*w - 88], [3389, 3389, -13*w - 75], [3407, 3407, 16*w - 47], [3407, 3407, -16*w - 31], [3413, 3413, -4*w - 59], [3413, 3413, 4*w - 63], [3449, 3449, 14*w - 25], [3449, 3449, -14*w - 11], [3461, 3461, 15*w - 37], [3461, 3461, -15*w - 22], [3469, 3469, 12*w - 85], [3469, 3469, -12*w - 73], [3481, 59, -59], [3517, 3517, 14*w - 23], [3517, 3517, -14*w - 9], [3527, 3527, -19*w - 49], [3527, 3527, 19*w - 68], [3529, 3529, 8*w - 73], [3529, 3529, -8*w - 65], [3533, 3533, 18*w - 61], [3533, 3533, -18*w - 43], [3541, 3541, 23*w + 70], [3541, 3541, 22*w + 65], [3557, 3557, 19*w - 112], [3557, 3557, -19*w - 93], [3559, 3559, -16*w - 29], [3559, 3559, 16*w - 45], [3583, 3583, 17*w - 53], [3583, 3583, -17*w - 36], [3613, 3613, 7*w - 71], [3613, 3613, -7*w - 64], [3623, 3623, -14*w - 79], [3623, 3623, 14*w - 93], [3643, 3643, -19*w - 48], [3643, 3643, 19*w - 67], [3659, 3659, -21*w - 59], [3659, 3659, 21*w - 80], [3671, 3671, -17*w - 35], [3671, 3671, 17*w - 52], [3673, 3673, -14*w - 3], [3673, 3673, 14*w - 17], [3677, 3677, 11*w - 83], [3677, 3677, -11*w - 72], [3697, 3697, 9*w - 77], [3697, 3697, -9*w - 68], [3709, 3709, -14*w - 1], [3709, 3709, 14*w - 15], [3719, 3719, -13*w - 77], [3719, 3719, 13*w - 90], [3733, 3733, 3*w - 64], [3733, 3733, -3*w - 61], [3767, 3767, 2*w - 63], [3767, 3767, -2*w - 61], [3769, 3769, 14*w - 9], [3769, 3769, 14*w - 5], [3779, 3779, -15*w - 16], [3779, 3779, 15*w - 31], [3797, 3797, -21*w - 58], [3797, 3797, 21*w - 79], [3833, 3833, -11*w - 73], [3833, 3833, 11*w - 84], [3851, 3851, 10*w - 81], [3851, 3851, -10*w - 71], [3863, 3863, 24*w + 73], [3863, 3863, 23*w + 68], [3911, 3911, -15*w - 13], [3911, 3911, 15*w - 28], [3917, 3917, -4*w - 63], [3917, 3917, 4*w - 67], [3923, 3923, -17*w - 32], [3923, 3923, 17*w - 49], [3931, 3931, -6*w - 65], [3931, 3931, 6*w - 71], [3943, 3943, -17*w - 89], [3943, 3943, 17*w - 106], [3967, 3967, -16*w - 23], [3967, 3967, 16*w - 39], [3989, 3989, 15*w - 26], [3989, 3989, -15*w - 11], [4003, 4003, 17*w - 48], [4003, 4003, -17*w - 31], [4013, 4013, -w - 63], [4013, 4013, w - 64], [4019, 4019, 2*w - 65], [4019, 4019, -2*w - 63], [4021, 4021, 23*w + 67], [4021, 4021, 25*w + 77], [4027, 4027, 7*w - 74], [4027, 4027, -7*w - 67], [4057, 4057, 13*w - 92], [4057, 4057, -13*w - 79], [4091, 4091, -15*w - 8], [4091, 4091, 15*w - 23], [4133, 4133, 22*w - 83], [4133, 4133, -22*w - 61], [4139, 4139, 17*w - 107], [4139, 4139, -17*w - 90], [4157, 4157, 17*w - 46], [4157, 4157, -17*w - 29], [4159, 4159, -10*w - 73], [4159, 4159, 10*w - 83], [4177, 4177, 23*w - 89], [4177, 4177, -23*w - 66], [4211, 4211, 5*w - 71], [4211, 4211, -5*w - 66], [4219, 4219, -20*w - 49], [4219, 4219, 20*w - 69], [4229, 4229, -13*w - 80], [4229, 4229, 13*w - 93], [4231, 4231, 17*w - 45], [4231, 4231, -17*w - 28], [4241, 4241, 15*w - 17], [4241, 4241, -15*w - 2], [4259, 4259, -15*w - 1], [4259, 4259, 15*w - 16], [4271, 4271, -w - 65], [4271, 4271, w - 66], [4289, 4289, 15*w - 14], [4289, 4289, 15*w - 1], [4297, 4297, -19*w - 42], [4297, 4297, 19*w - 61], [4327, 4327, -14*w - 83], [4327, 4327, 14*w - 97], [4337, 4337, 17*w - 108], [4337, 4337, -17*w - 91], [4349, 4349, -5*w - 67], [4349, 4349, 5*w - 72], [4373, 4373, -17*w - 26], [4373, 4373, 17*w - 43], [4441, 4441, 17*w - 42], [4441, 4441, -17*w - 25], [4447, 4447, 9*w - 82], [4447, 4447, -9*w - 73], [4451, 4451, -20*w - 47], [4451, 4451, 20*w - 67], [4457, 4457, -21*w - 53], [4457, 4457, 21*w - 74], [4481, 4481, -16*w - 89], [4481, 4481, 16*w - 105], [4483, 4483, -23*w - 64], [4483, 4483, 23*w - 87], [4507, 4507, -17*w - 24], [4507, 4507, 17*w - 41], [4519, 4519, 3*w - 70], [4519, 4519, -3*w - 67], [4547, 4547, 2*w - 69], [4547, 4547, -2*w - 67], [4549, 4549, 26*w + 79], [4549, 4549, 25*w + 74], [4567, 4567, -16*w - 11], [4567, 4567, 16*w - 27], [4583, 4583, 21*w - 73], [4583, 4583, -21*w - 52], [4597, 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8369, 7*w - 100], [8377, 8377, -23*w - 31], [8377, 8377, 23*w - 54], [8387, 8387, 2*w - 93], [8387, 8387, -2*w - 91], [8389, 8389, -26*w - 55], [8389, 8389, 26*w - 81], [8429, 8429, -20*w - 117], [8429, 8429, 20*w - 137], [8447, 8447, 21*w - 4], [8447, 8447, 21*w - 17], [8461, 8461, -23*w - 30], [8461, 8461, 23*w - 53], [8537, 8537, -w - 92], [8537, 8537, w - 93], [8543, 8543, -23*w - 29], [8543, 8543, 23*w - 52], [8563, 8563, -7*w - 94], [8563, 8563, 7*w - 101], [8599, 8599, 31*w - 115], [8599, 8599, -31*w - 84], [8609, 8609, -25*w - 46], [8609, 8609, 25*w - 71], [8623, 8623, 23*w - 51], [8623, 8623, -23*w - 28], [8641, 8641, -22*w - 15], [8641, 8641, 22*w - 37], [8647, 8647, -18*w - 113], [8647, 8647, 18*w - 131], [8677, 8677, 12*w - 113], [8677, 8677, -12*w - 101], [8707, 8707, 29*w - 101], [8707, 8707, -29*w - 72], [8737, 8737, -13*w - 103], [8737, 8737, 13*w - 116], [8741, 8741, 22*w - 35], [8741, 8741, -22*w - 13], [8753, 8753, -33*w - 94], [8753, 8753, 33*w - 127], [8761, 8761, 16*w - 125], [8761, 8761, -16*w - 109], [8779, 8779, -15*w - 107], [8779, 8779, 15*w - 122], [8803, 8803, 21*w - 142], [8803, 8803, -21*w - 121], [8819, 8819, 36*w + 109], [8819, 8819, 35*w + 104], [8831, 8831, -5*w - 94], [8831, 8831, 5*w - 99], [8839, 8839, -25*w - 44], [8839, 8839, 25*w - 69], [8849, 8849, -25*w - 132], [8849, 8849, 25*w - 157], [8861, 8861, 30*w - 107], [8861, 8861, -30*w - 77], [8923, 8923, -23*w - 24], [8923, 8923, 23*w - 47], [8933, 8933, -29*w - 144], [8933, 8933, 29*w - 173], [8941, 8941, -20*w - 119], [8941, 8941, 20*w - 139], [8951, 8951, -25*w - 43], [8951, 8951, 25*w - 68], [8969, 8969, 8*w - 105], [8969, 8969, -8*w - 97], [8999, 8999, 23*w - 150], [8999, 8999, -23*w - 127], [9013, 9013, -11*w - 101], [9013, 9013, 11*w - 112], [9049, 9049, 29*w - 99], [9049, 9049, -29*w - 70], [9067, 9067, 21*w - 143], [9067, 9067, -21*w - 122], [9103, 9103, -32*w - 87], [9103, 9103, 32*w - 119], [9109, 9109, -12*w - 103], [9109, 9109, 12*w - 115], [9127, 9127, -23*w - 21], [9127, 9127, 23*w - 44], [9157, 9157, -7*w - 97], [9157, 9157, 7*w - 104], [9173, 9173, -22*w - 1], [9173, 9173, 22*w - 23], [9187, 9187, 31*w - 112], [9187, 9187, -31*w - 81], [9199, 9199, -17*w - 113], [9199, 9199, 17*w - 130], [9203, 9203, 27*w - 83], [9203, 9203, -27*w - 56], [9221, 9221, 5*w - 101], [9221, 9221, -5*w - 96], [9227, 9227, 11*w - 113], [9227, 9227, -11*w - 102], [9239, 9239, -24*w - 31], [9239, 9239, 24*w - 55], [9241, 9241, 15*w - 124], [9241, 9241, -15*w - 109], [9257, 9257, 34*w - 131], [9257, 9257, -34*w - 97], [9277, 9277, -23*w - 128], [9277, 9277, 23*w - 151], [9281, 9281, 22*w - 17], [9281, 9281, 22*w - 5], [9293, 9293, -w - 96], [9293, 9293, w - 97], [9311, 9311, 10*w - 111], [9311, 9311, -10*w - 101], [9323, 9323, 37*w + 112], [9323, 9323, 36*w + 107], [9341, 9341, -27*w - 55], [9341, 9341, 27*w - 82], [9371, 9371, 23*w - 40], [9371, 9371, -23*w - 17], [9377, 9377, 8*w - 107], [9377, 9377, -8*w - 99], [9403, 9403, -13*w - 106], [9403, 9403, 13*w - 119], [9409, 97, -97], [9413, 9413, -26*w - 47], [9413, 9413, 26*w - 73], [9419, 9419, -5*w - 97], [9419, 9419, 5*w - 102], [9431, 9431, -25*w - 134], [9431, 9431, 25*w - 159], [9461, 9461, 20*w - 141], [9461, 9461, -20*w - 121], [9467, 9467, 28*w - 89], [9467, 9467, -28*w - 61], [9511, 9511, 32*w - 117], [9511, 9511, -32*w - 85], [9533, 9533, -23*w - 14], [9533, 9533, 23*w - 37], [9539, 9539, 35*w - 136], [9539, 9539, -35*w - 101], [9547, 9547, -29*w - 67], [9547, 9547, 29*w - 96], [9601, 9601, 21*w - 145], [9601, 9601, -21*w - 124], [9619, 9619, 5*w - 103], [9619, 9619, -5*w - 98], [9629, 9629, -13*w - 107], [9629, 9629, 13*w - 120], [9631, 9631, -23*w - 12], [9631, 9631, 23*w - 35], [9649, 9649, -26*w - 45], [9649, 9649, 26*w - 71], [9661, 9661, -11*w - 104], [9661, 9661, 11*w - 115], [9677, 9677, 23*w - 34], [9677, 9677, -23*w - 11], [9679, 9679, 25*w - 61], [9679, 9679, -25*w - 36], [9689, 9689, 17*w - 132], [9689, 9689, -17*w - 115], [9719, 9719, -24*w - 25], [9719, 9719, 24*w - 49], [9721, 9721, -23*w - 10], [9721, 9721, 23*w - 33], [9739, 9739, -10*w - 103], [9739, 9739, 10*w - 113], [9743, 9743, -27*w - 52], [9743, 9743, 27*w - 79], [9769, 9769, -7*w - 100], [9769, 9769, 7*w - 107], [9803, 9803, -23*w - 8], [9803, 9803, 23*w - 31], [9833, 9833, 33*w - 122], [9833, 9833, -33*w - 89], [9839, 9839, -23*w - 130], [9839, 9839, 23*w - 153], [9857, 9857, -13*w - 108], [9857, 9857, 13*w - 121], [9871, 9871, -21*w - 125], [9871, 9871, 21*w - 146], [9923, 9923, 2*w - 101], [9923, 9923, -2*w - 99], [9929, 9929, -30*w - 71], [9929, 9929, 30*w - 101], [9949, 9949, -15*w - 112], [9949, 9949, 15*w - 127], [9973, 9973, -23*w - 3], [9973, 9973, 23*w - 26]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 0, -3, 1, -3, 3, 3, -3, 6, -6, 4, 4, -1, 0, 0, 6, -6, 10, 10, 0, 0, 14, 14, 8, 8, -6, 6, 12, -12, 18, -18, 8, 8, 12, -12, 8, 8, 0, 0, -16, -16, -3, 3, 3, -3, 4, 4, 17, 17, -3, 3, 24, -24, -24, 24, 21, -21, 9, -9, 3, -3, -22, -22, 12, 12, 30, -30, 24, 24, -13, -13, 19, 19, -30, 30, -1, -1, 12, -12, 34, 34, -23, -23, -36, 36, 4, 4, -24, 24, -36, -36, 27, 27, 9, -9, 24, -24, -12, 12, 45, -45, -17, -17, -30, 30, -21, 21, 4, 4, -11, -11, -2, -2, -14, -14, -9, 9, 8, 8, 33, 33, 15, -15, -15, -15, -48, -48, 12, -12, 42, -42, -39, 39, 48, -48, 50, 50, -57, 30, -30, -54, 54, -34, -34, 0, 0, -24, 24, -24, -24, -40, -40, -27, 27, 0, 0, -2, -2, 26, -34, -34, 36, 36, 3, -3, 61, 61, -12, 12, 60, -60, 28, 28, -12, 12, -22, -22, 42, -42, -16, -16, -36, 36, -51, -51, 46, 46, 33, -33, 3, -3, -66, 66, -18, 18, -6, -6, 18, -18, -30, 30, -22, -22, 4, 4, -6, 6, -25, -25, 30, 30, 3, -3, -7, -7, 27, -27, 60, 60, 42, -42, 27, -27, 29, 29, 76, 76, 36, -36, -54, -54, 21, -21, -30, 30, -6, 6, 26, 26, -54, 54, -34, -34, 12, -12, -51, 51, 18, -18, -69, 69, -65, -65, -42, -42, -18, 18, -64, -64, 54, -54, 21, -21, 13, 13, 48, -48, -50, 12, -12, -27, 27, 59, 59, -42, -42, -42, 42, 59, 59, -6, 6, 6, -6, 48, -48, -75, 75, 63, -63, 44, 44, 49, 49, -60, -60, 10, 10, -52, -52, -60, 60, 25, 25, -30, -30, 12, -12, 30, -30, 18, 18, -48, 48, 6, 6, -11, -11, 13, 26, 26, 36, -36, 45, -45, 29, 29, -54, 54, 57, -57, -28, -28, 2, 2, -12, 12, 9, 9, -88, -88, -12, -12, 84, -84, -30, 30, 52, 52, -55, -55, -3, 3, -6, 6, -84, 84, 32, 32, -42, 42, -62, -62, -54, -54, -27, 27, 11, 11, -54, 54, 34, 34, 9, -9, -48, 48, -33, 33, 44, 44, 42, -42, 45, 45, -55, -55, 49, 49, -47, -47, 36, -36, -85, -85, -36, 36, -38, -38, 60, -60, -66, 66, -44, -44, -9, 9, -44, -44, 15, -15, 31, 31, 78, -78, -14, -14, -48, 48, -62, -62, -36, 36, 68, 68, -43, -43, 26, 26, 12, -12, -61, -61, 36, -36, -15, 15, 0, 0, 36, -36, -54, 54, -50, -50, -19, -19, 48, -48, 81, -81, 22, 22, 57, 57, 59, 59, 39, 39, 33, -33, 44, 44, -7, -7, -51, 51, 50, 50, 48, -48, 10, 10, 90, -90, -66, 66, -30, -30, 82, -27, 27, -93, 93, -9, -9, -69, 69, -102, 102, 22, 22, 36, -36, 21, -21, -96, -96, -26, -26, -84, 84, -24, 24, 30, -30, -30, 30, 98, 98, -72, -72, 78, -78, 67, 67, 100, 100, -58, -58, -30, 30, 114, -114, -3, 3, 38, 38, 13, 13, -33, 33, 42, -42, 16, 16, -9, 9, 37, 37, 114, 114, -117, 117, 24, -24, 51, -51, -76, -76, -52, -52, 60, -60, -102, -102, 46, 46, -96, 96, -90, 90, 35, 35, 78, -78, 56, 56, -63, 63, 59, 59, -18, 18, -118, -118, -12, 12, -6, 6, -36, 36, 88, 88, -30, 30, 21, -21, -54, -54, 86, 86, 47, 47, 129, -129, 0, 0, 48, 48, -30, 30, 90, -90, -35, -35, 69, -69, -72, 72, 57, 57, 52, 52, 72, -72, -57, 57, -39, 39, 21, -21, 48, 48, -21, -21, -66, 66, -6, 6, -116, -116, 94, 94, 4, 4, -55, -55, 33, -33, 72, -72, 26, 26, -30, 30, -120, 120, -24, -24, 27, -27, 21, 21, 15, -15, 109, 109, 48, -48, 72, -72, -3, -3, 132, -132, 52, 52, -40, -40, 24, 24, 60, -60, 39, 39, -103, -103, -54, 54, 118, 118, -120, 120, 48, -48, -102, 102, 132, -132, 76, 76, -55, -55, 50, 50, -27, 27, 88, 88, -137, -137, -20, -20, 27, -27, -100, -100, -72, 72, -36, 36, -18, 18, 55, 55, 24, -24, -54, -54, -18, 18, 120, 120, 101, 101, -30, 30, -136, -136, -130, -130, 108, -108, 30, -30, -72, 72, 12, 12, -32, -32, 78, -78, -51, 51, -66, -66, 25, 25, 56, 56, 132, -132, 42, -42, -42, 42, -93, 93, -65, -65, 4, 4, 129, -129, 107, 107, 84, -84, -30, 30, 13, 13, -22, -22, 120, -120, 78, 78, -114, 114, 111, -111, -66, 66, -92, -92, 42, 42, 29, 29, -122, -122, -33, 33, -48, 48, 78, 78, -54, 54, -60, 60, -116, -116, -66, 66, 105, -105, -104, -104, 144, -144, -157, -69, 69, -69, 69, -42, 42, -74, -74, -106, -106, -151, -151, 20, 20, -114, -114, -57, 57, -126, -126, -41, -41, 107, 107, 19, 19, -84, 84, 52, 52, -95, -95, 126, -126, -114, 114, 84, -84, 133, 133, -18, 18, 48, -48, 0, 0, -11, -11, 79, 79, 46, 46, -126, 126, -18, 18, -6, -6, -35, -35, -117, 117, 17, 17, 8, 8, 21, -21, -74, -74, 36, -36, -7, -7, -34, -34, -21, 21, -138, 138, 38, 38, 114, -114, 116, 116, 42, -42, 114, 114, 10, 10, 45, -45, -124, -124, 36, -36, 120, -120, -161, -161, 22, 22, -160, -160, 162, 162, 66, -66, 47, 47, 66, -66, 126, -126, 16, 16, 8, 8, 83, 83, -90, 90, 119, 119, 127, 127, 90, -90, 42, 42, 0, 0, 30, 30, 9, -9, -116, -116, -72, 72, -5, -5, 117, -117, -98, -98, -30, 30, 15, 15, -34, -34, -165, 165, -19, -19, -123, -123, 92, 92, 168, -168, 66, -66, -42, 42, -56, -56, 108, -108, 76, 76, 68, 68, 45, -45, 60, -60, -11, -11, 126, -126, -21, 21, 51, -51, -52, -52, -112, -112, -82, -82, 34, -81, 81, -153, 153, 21, -21, -47, -47, -102, -102, 36, -36, -96, 96, -81, -81, 75, 75, 81, -81, -42, 42, -101, -101, 90, -90, -152, -152, -146, -146, 69, -69, -102, 102, -54, 54, -154, -154, 78, 78, 36, -36, 120, 120, 131, 131, 114, -114, -26, -26, -24, 24, -53, -53, 171, -171, -54, 54, -170, -170, -87, 87, -58, -58, -138, 138, 84, -84, 135, -135, -30, 30, -100, -100, 54, 54, -15, 15, -42, -42, 66, -66, 3, -3, -66, -66, 68, 68, -24, -24, -54, 54, -112, -112, -60, 60, 83, 83, 48, -48, 99, -99, 34, 34, 63, 63, -30, 30, -122, -122, 35, 35, -84, -84, -90, 90, -86, -86, 147, -147, 151, 151, -111, 111, 112, 112, -147, 147, -111, 111, 79, 79, 21, -21, 173, 173, -28, -28, 180, -180, 91, 91, -147, 147, 116, 116, -90, 90, 176, 176, 25, 25, -87, 87, -54, 54, -54, 54, 28, 28, -80, -80, -31, 66, -66, 56, 56, -79, -79, 94, 94, 9, -9, 150, -150, -21, 21, -120, 120, 57, -57, -91, -91, -111, -111, -29, -29, -54, 54, -180, 180, 67, 67, 114, -114, 114, -114, 50, 50, -162, 162, -150, 150, 25, 25, 156, -156, -45, -45, -129, 129, 189, -189, -71, -71, 62, 62, 63, 63, -2, -2, 162, -162, -87, -87, 99, -99]; 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;