/* 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![-5, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [4, 2, 2], [5, 5, w], [5, 5, w - 1], [7, 7, -w - 3], [17, 17, -2*w + 3], [17, 17, -2*w - 1], [37, 37, w + 6], [37, 37, -w + 7], [41, 41, 3*w + 1], [41, 41, -3*w + 4], [43, 43, 3*w + 8], [43, 43, 3*w - 11], [47, 47, 3*w - 2], [47, 47, 3*w - 1], [59, 59, -5*w - 6], [59, 59, -4*w - 3], [67, 67, -w - 8], [67, 67, w - 9], [79, 79, 2*w - 11], [79, 79, -2*w - 9], [83, 83, 4*w - 3], [83, 83, 4*w - 1], [89, 89, -6*w - 7], [89, 89, -5*w - 4], [101, 101, -5*w - 3], [101, 101, 5*w - 8], [109, 109, 5*w - 18], [109, 109, 5*w + 13], [121, 11, -11], [127, 127, -w - 11], [127, 127, w - 12], [131, 131, 5*w - 2], [131, 131, 5*w - 3], [151, 151, -w - 12], [151, 151, w - 13], [163, 163, 3*w - 16], [163, 163, -3*w - 13], [167, 167, -8*w - 9], [167, 167, -7*w - 6], [169, 13, -13], [173, 173, -6*w - 1], [173, 173, 6*w - 7], [193, 193, -3*w - 14], [193, 193, 3*w - 17], [211, 211, 6*w + 17], [211, 211, 6*w - 23], [227, 227, 7*w - 9], [227, 227, -7*w - 2], [251, 251, 7*w - 6], [251, 251, 7*w - 1], [257, 257, 7*w - 3], [257, 257, 7*w - 4], [269, 269, -10*w - 11], [269, 269, -9*w - 8], [277, 277, 4*w - 21], [277, 277, -4*w - 17], [293, 293, -9*w - 7], [293, 293, 9*w - 16], [311, 311, -8*w - 1], [311, 311, 8*w - 9], [331, 331, -5*w - 19], [331, 331, 5*w - 24], [337, 337, -w - 18], [337, 337, w - 19], [353, 353, 9*w - 13], [353, 353, -9*w - 4], [361, 19, -19], [373, 373, 3*w - 22], [373, 373, -3*w - 19], [379, 379, 2*w - 21], [379, 379, -2*w - 19], [383, 383, 9*w - 11], [383, 383, -9*w - 2], [419, 419, 9*w - 2], [419, 419, 9*w - 7], [421, 421, 5*w - 26], [421, 421, -5*w - 21], [457, 457, -w - 21], [457, 457, w - 22], [461, 461, 10*w - 13], [461, 461, -10*w - 3], [463, 463, 2*w - 23], [463, 463, -2*w - 21], [467, 467, -13*w - 14], [467, 467, -12*w - 11], [479, 479, 11*w - 18], [479, 479, -11*w - 7], [487, 487, -6*w - 23], [487, 487, 6*w - 29], [499, 499, 10*w - 37], [499, 499, 10*w + 27], [503, 503, -11*w - 6], [503, 503, 11*w - 17], [509, 509, 10*w - 9], [509, 509, 10*w - 1], [521, 521, 10*w - 3], [521, 521, 10*w - 7], [529, 23, -23], [541, 541, -4*w - 23], [541, 541, 4*w - 27], [547, 547, -w - 23], [547, 547, w - 24], [563, 563, -11*w - 3], [563, 563, 11*w - 14], [571, 571, -5*w - 24], [571, 571, 5*w - 29], [587, 587, 12*w - 19], [587, 587, -12*w - 7], [593, 593, -11*w - 1], [593, 593, 11*w - 12], [613, 613, 7*w - 33], [613, 613, -7*w - 26], [631, 631, 9*w - 37], [631, 631, 10*w - 39], [647, 647, -13*w - 9], [647, 647, 13*w - 22], [673, 673, -7*w - 27], [673, 673, 7*w - 34], [677, 677, -13*w - 8], [677, 677, 13*w - 21], [709, 709, -3*w - 26], [709, 709, 3*w - 29], [719, 719, -16*w - 17], [719, 719, -15*w - 14], [739, 739, -5*w - 27], [739, 739, 5*w - 32], [751, 751, -w - 27], [751, 751, w - 28], [757, 757, -4*w - 27], [757, 757, 4*w - 31], [761, 761, -15*w - 13], [761, 761, -17*w - 19], [773, 773, 14*w - 23], [773, 773, -14*w - 9], [797, 797, 13*w - 16], [797, 797, -13*w - 3], [823, 823, 3*w - 31], [823, 823, -3*w - 28], [839, 839, 15*w - 26], [839, 839, -15*w - 11], [841, 29, -29], [857, 857, 13*w - 12], [857, 857, 13*w - 1], [877, 877, 4*w - 33], [877, 877, -4*w - 29], [881, 881, 13*w - 4], [881, 881, 13*w - 9], [883, 883, -3*w - 29], [883, 883, 3*w - 32], [887, 887, 13*w - 6], [887, 887, 13*w - 7], [907, 907, 9*w - 41], [907, 907, -9*w - 32], [919, 919, 13*w - 49], [919, 919, 13*w + 36], [929, 929, -14*w - 3], [929, 929, 14*w - 17], [941, 941, -15*w - 8], [941, 941, 15*w - 23], [961, 31, -31], [967, 967, 6*w - 37], [967, 967, -6*w - 31], [971, 971, 15*w - 22], [971, 971, -15*w - 7], [983, 983, -16*w - 11], [983, 983, 16*w - 27], [991, 991, 5*w - 36], [991, 991, -5*w - 31], [1009, 1009, 3*w - 34], [1009, 1009, -3*w - 31], [1013, 1013, 14*w - 11], [1013, 1013, 14*w - 3], [1033, 1033, 8*w - 41], [1033, 1033, -8*w - 33], [1049, 1049, -15*w - 4], [1049, 1049, 15*w - 19], [1051, 1051, -w - 32], [1051, 1051, w - 33], [1087, 1087, 14*w + 39], [1087, 1087, 14*w - 53], [1091, 1091, 15*w - 17], [1091, 1091, -15*w - 2], [1093, 1093, 12*w - 49], [1093, 1093, 13*w - 51], [1097, 1097, 17*w - 29], [1097, 1097, -17*w - 12], [1109, 1109, -15*w - 1], [1109, 1109, 15*w - 16], [1117, 1117, -w - 33], [1117, 1117, w - 34], [1129, 1129, 5*w - 38], [1129, 1129, -5*w - 33], [1151, 1151, 15*w - 2], [1151, 1151, 15*w - 13], [1171, 1171, 15*w + 41], [1171, 1171, 15*w - 56], [1181, 1181, 15*w - 7], [1181, 1181, 15*w - 8], [1193, 1193, -19*w - 17], [1193, 1193, -21*w - 23], [1201, 1201, -5*w - 34], [1201, 1201, 5*w - 39], [1213, 1213, 3*w - 37], [1213, 1213, -3*w - 34], [1217, 1217, -18*w - 13], [1217, 1217, 18*w - 31], [1223, 1223, 16*w - 19], [1223, 1223, -16*w - 3], [1259, 1259, -21*w - 22], [1259, 1259, -20*w - 19], [1277, 1277, -17*w - 7], [1277, 1277, 17*w - 24], [1297, 1297, 9*w - 46], [1297, 1297, -9*w - 37], [1301, 1301, -18*w - 11], [1301, 1301, 18*w - 29], [1303, 1303, 7*w - 43], [1303, 1303, -7*w - 36], [1307, 1307, -17*w - 6], [1307, 1307, 17*w - 23], [1319, 1319, 16*w - 3], [1319, 1319, 16*w - 13], [1327, 1327, -w - 36], [1327, 1327, w - 37], [1361, 1361, 17*w - 21], [1361, 1361, -17*w - 4], [1381, 1381, -9*w - 38], [1381, 1381, 9*w - 47], [1423, 1423, 2*w - 39], [1423, 1423, -2*w - 37], [1427, 1427, -17*w - 1], [1427, 1427, 17*w - 18], [1429, 1429, -5*w - 37], [1429, 1429, 5*w - 42], [1433, 1433, -19*w - 12], [1433, 1433, 19*w - 31], [1453, 1453, 17*w + 46], [1453, 1453, 17*w - 63], [1471, 1471, 14*w - 57], [1471, 1471, 15*w - 59], [1487, 1487, 17*w - 14], [1487, 1487, 17*w - 3], [1511, 1511, 17*w - 11], [1511, 1511, 17*w - 6], [1549, 1549, 7*w - 46], [1549, 1549, -7*w - 39], [1553, 1553, 19*w - 28], [1553, 1553, -19*w - 9], [1559, 1559, -21*w - 17], [1559, 1559, -25*w - 29], [1571, 1571, 20*w - 33], [1571, 1571, -20*w - 13], [1579, 1579, 2*w - 41], [1579, 1579, -2*w - 39], [1597, 1597, -4*w - 39], [1597, 1597, 4*w - 43], [1601, 1601, -18*w - 1], [1601, 1601, 18*w - 19], [1613, 1613, -21*w - 16], [1613, 1613, 21*w - 37], [1621, 1621, 11*w - 53], [1621, 1621, -11*w - 42], [1637, 1637, 18*w - 17], [1637, 1637, 18*w - 1], [1663, 1663, 18*w - 67], [1663, 1663, 18*w + 49], [1697, 1697, 18*w - 7], [1697, 1697, 18*w - 11], [1721, 1721, -23*w - 21], [1721, 1721, -25*w - 27], [1723, 1723, -7*w - 41], [1723, 1723, 7*w - 48], [1747, 1747, -6*w - 41], [1747, 1747, 6*w - 47], [1759, 1759, -3*w - 41], [1759, 1759, 3*w - 44], [1789, 1789, 17*w - 66], [1789, 1789, 15*w - 62], [1801, 1801, -w - 42], [1801, 1801, w - 43], [1811, 1811, -20*w - 7], [1811, 1811, 20*w - 27], [1823, 1823, 19*w - 18], [1823, 1823, 19*w - 1], [1831, 1831, 9*w - 52], [1831, 1831, -9*w - 43], [1847, 1847, -27*w - 31], [1847, 1847, -23*w - 19], [1873, 1873, 8*w - 51], [1873, 1873, -8*w - 43], [1889, 1889, 19*w - 7], [1889, 1889, 19*w - 12], [1907, 1907, -23*w - 18], [1907, 1907, 23*w - 41], [1931, 1931, -20*w - 3], [1931, 1931, 20*w - 23], [1933, 1933, 3*w - 46], [1933, 1933, -3*w - 43], [1949, 1949, -26*w - 27], [1949, 1949, -25*w - 24], [1973, 1973, -21*w - 8], [1973, 1973, 21*w - 29], [1979, 1979, -20*w - 1], [1979, 1979, 20*w - 21], [1999, 1999, -7*w - 44], [1999, 1999, 7*w - 51], [2011, 2011, 19*w + 53], [2011, 2011, 19*w - 72], [2017, 2017, -11*w - 46], [2017, 2017, 11*w - 57], [2053, 2053, -12*w - 47], [2053, 2053, 12*w - 59], [2063, 2063, 24*w - 43], [2063, 2063, -24*w - 19], [2083, 2083, 13*w - 61], [2083, 2083, -13*w - 48], [2099, 2099, 20*w - 11], [2099, 2099, 20*w - 9], [2137, 2137, 19*w - 73], [2137, 2137, 19*w + 54], [2141, 2141, -22*w - 9], [2141, 2141, 22*w - 31], [2143, 2143, 17*w - 69], [2143, 2143, 18*w - 71], [2179, 2179, 10*w - 57], [2179, 2179, -10*w - 47], [2221, 2221, 5*w - 51], [2221, 2221, -5*w - 46], [2243, 2243, 21*w - 2], [2243, 2243, 21*w - 19], [2251, 2251, -w - 47], [2251, 2251, w - 48], [2267, 2267, -28*w - 29], [2267, 2267, -27*w - 26], [2269, 2269, 12*w - 61], [2269, 2269, -12*w - 49], [2273, 2273, 21*w - 4], [2273, 2273, 21*w - 17], [2293, 2293, 7*w - 54], [2293, 2293, -7*w - 47], [2309, 2309, 21*w - 13], [2309, 2309, 21*w - 8], [2311, 2311, -6*w - 47], [2311, 2311, 6*w - 53], [2347, 2347, -w - 48], [2347, 2347, w - 49], [2351, 2351, -25*w - 18], [2351, 2351, 25*w - 43], [2357, 2357, 23*w - 32], [2357, 2357, -23*w - 9], [2377, 2377, 21*w - 79], [2377, 2377, 21*w + 58], [2389, 2389, 17*w - 71], [2389, 2389, 20*w - 77], [2393, 2393, -26*w - 21], [2393, 2393, -31*w - 36], [2399, 2399, 24*w - 37], [2399, 2399, -24*w - 13], [2411, 2411, -25*w - 17], [2411, 2411, 25*w - 42], [2437, 2437, -9*w - 49], [2437, 2437, 9*w - 58], [2441, 2441, 22*w - 21], [2441, 2441, 22*w - 1], [2473, 2473, 8*w - 57], [2473, 2473, -8*w - 49], [2477, 2477, 22*w - 3], [2477, 2477, 22*w - 19], [2503, 2503, 3*w - 52], [2503, 2503, -3*w - 49], [2521, 2521, -5*w - 49], [2521, 2521, 5*w - 54], [2557, 2557, -11*w - 51], [2557, 2557, 11*w - 62], [2579, 2579, -25*w - 14], [2579, 2579, 25*w - 39], [2609, 2609, -30*w - 31], [2609, 2609, -29*w - 28], [2621, 2621, -23*w - 1], [2621, 2621, 23*w - 24], [2647, 2647, -w - 51], [2647, 2647, w - 52], [2663, 2663, -24*w - 7], [2663, 2663, 24*w - 31], [2671, 2671, -11*w - 52], [2671, 2671, 11*w - 63], [2683, 2683, 2*w - 53], [2683, 2683, -2*w - 51], [2687, 2687, 23*w - 2], [2687, 2687, 23*w - 21], [2689, 2689, 8*w - 59], [2689, 2689, -8*w - 51], [2693, 2693, -29*w - 27], [2693, 2693, -31*w - 33], [2713, 2713, -7*w - 51], [2713, 2713, 7*w - 58], [2729, 2729, 25*w - 36], [2729, 2729, -25*w - 11], [2731, 2731, 5*w - 56], [2731, 2731, -5*w - 51], [2767, 2767, 9*w - 61], [2767, 2767, -9*w - 52], [2777, 2777, 23*w - 11], [2777, 2777, 23*w - 12], [2789, 2789, -31*w - 32], [2789, 2789, -30*w - 29], [2797, 2797, 21*w - 82], [2797, 2797, 19*w - 78], [2809, 53, -53], [2819, 2819, 25*w - 34], [2819, 2819, -25*w - 9], [2851, 2851, -15*w - 56], [2851, 2851, 15*w - 71], [2857, 2857, -w - 53], [2857, 2857, w - 54], [2861, 2861, 25*w - 33], [2861, 2861, -25*w - 8], [2897, 2897, -27*w - 17], [2897, 2897, 27*w - 44], [2903, 2903, 24*w - 23], [2903, 2903, 24*w - 1], [2939, 2939, 25*w - 31], [2939, 2939, -25*w - 6], [2957, 2957, 27*w - 43], [2957, 2957, -27*w - 16], [2999, 2999, 24*w - 17], [2999, 2999, 24*w - 7], [3019, 3019, -13*w - 56], [3019, 3019, 13*w - 69], [3023, 3023, 24*w - 13], [3023, 3023, 24*w - 11], [3041, 3041, 25*w - 28], [3041, 3041, -25*w - 3], [3049, 3049, 7*w - 61], [3049, 3049, -7*w - 54], [3061, 3061, -5*w - 54], [3061, 3061, 5*w - 59], [3067, 3067, 14*w - 71], [3067, 3067, -14*w - 57], [3083, 3083, -29*w - 22], [3083, 3083, 29*w - 51], [3109, 3109, 15*w - 73], [3109, 3109, -15*w - 58], [3167, 3167, -33*w - 34], [3167, 3167, -32*w - 31], [3187, 3187, -w - 56], [3187, 3187, w - 57], [3191, 3191, 25*w - 3], [3191, 3191, 25*w - 22], [3209, 3209, 25*w - 4], [3209, 3209, 25*w - 21], [3217, 3217, 24*w + 67], [3217, 3217, 24*w - 91], [3229, 3229, 20*w - 83], [3229, 3229, 23*w - 89], [3251, 3251, 25*w - 7], [3251, 3251, 25*w - 18], [3259, 3259, -3*w - 56], [3259, 3259, 3*w - 59], [3271, 3271, -11*w - 57], [3271, 3271, 11*w - 68], [3301, 3301, -w - 57], [3301, 3301, w - 58], [3313, 3313, -17*w - 61], [3313, 3313, 17*w - 78], [3319, 3319, 10*w - 67], [3319, 3319, -10*w - 57], [3323, 3323, -31*w - 26], [3323, 3323, -36*w - 41], [3343, 3343, 2*w - 59], [3343, 3343, -2*w - 57], [3359, 3359, 29*w - 47], [3359, 3359, -29*w - 18], [3361, 3361, 25*w - 94], [3361, 3361, 25*w + 69], [3407, 3407, -27*w - 7], [3407, 3407, 27*w - 34], [3449, 3449, 26*w - 23], [3449, 3449, 26*w - 3], [3461, 3461, -33*w - 31], [3461, 3461, -35*w - 37], [3469, 3469, -12*w - 59], [3469, 3469, 12*w - 71], [3491, 3491, 28*w - 39], [3491, 3491, -28*w - 11], [3511, 3511, 15*w - 76], [3511, 3511, -15*w - 61], [3527, 3527, -37*w - 42], [3527, 3527, -32*w - 27], [3529, 3529, 5*w - 63], [3529, 3529, -5*w - 58], [3533, 3533, 26*w - 17], [3533, 3533, 26*w - 9], [3571, 3571, -10*w - 59], [3571, 3571, 10*w - 69], [3607, 3607, 9*w - 68], [3607, 3607, -9*w - 59], [3613, 3613, -3*w - 59], [3613, 3613, 3*w - 62], [3617, 3617, -27*w - 1], [3617, 3617, 27*w - 28], [3637, 3637, -4*w - 59], [3637, 3637, 4*w - 63], [3659, 3659, -29*w - 13], [3659, 3659, 29*w - 42], [3671, 3671, 27*w - 26], [3671, 3671, 27*w - 1], [3691, 3691, 25*w - 96], [3691, 3691, 25*w + 71], [3697, 3697, -16*w - 63], [3697, 3697, 16*w - 79], [3701, 3701, 30*w - 47], [3701, 3701, -30*w - 17], [3721, 61, -61], [3733, 3733, 12*w - 73], [3733, 3733, -12*w - 61], [3739, 3739, 17*w - 81], [3739, 3739, -17*w - 64], [3779, 3779, -36*w - 37], [3779, 3779, -35*w - 34], [3797, 3797, 27*w - 8], [3797, 3797, 27*w - 19], [3821, 3821, 27*w - 16], [3821, 3821, 27*w - 11], [3823, 3823, 2*w - 63], [3823, 3823, -2*w - 61], [3847, 3847, 26*w - 99], [3847, 3847, 26*w + 73], [3863, 3863, -29*w - 9], [3863, 3863, 29*w - 38], [3881, 3881, -35*w - 33], [3881, 3881, -37*w - 39], [3889, 3889, -8*w - 61], [3889, 3889, 8*w - 69], [3907, 3907, 6*w - 67], [3907, 3907, -6*w - 61], [3911, 3911, -33*w - 26], [3911, 3911, 33*w - 59], [3923, 3923, 31*w - 49], [3923, 3923, -31*w - 18], [3931, 3931, 15*w - 79], [3931, 3931, -15*w - 64], [3943, 3943, 13*w - 76], [3943, 3943, -13*w - 63], [3947, 3947, 28*w - 27], [3947, 3947, 28*w - 1], [3989, 3989, -31*w - 17], [3989, 3989, 31*w - 48], [4007, 4007, 32*w - 53], [4007, 4007, -32*w - 21], [4027, 4027, -w - 63], [4027, 4027, w - 64], [4049, 4049, -30*w - 11], [4049, 4049, 30*w - 41], [4057, 4057, -11*w - 63], [4057, 4057, 11*w - 74], [4073, 4073, 29*w - 33], [4073, 4073, -29*w - 4], [4079, 4079, -39*w - 43], [4079, 4079, -35*w - 31], [4091, 4091, 28*w - 9], [4091, 4091, 28*w - 19], [4099, 4099, -10*w - 63], [4099, 4099, 10*w - 73], [4111, 4111, 19*w - 87], [4111, 4111, -19*w - 68], [4133, 4133, 34*w - 61], [4133, 4133, -34*w - 27], [4153, 4153, -8*w - 63], [4153, 4153, 8*w - 71], [4157, 4157, 33*w - 56], [4157, 4157, -33*w - 23], [4159, 4159, 5*w - 68], [4159, 4159, -5*w - 63], [4201, 4201, 24*w - 97], [4201, 4201, 25*w - 99], [4217, 4217, -38*w - 39], [4217, 4217, -37*w - 36], [4241, 4241, -30*w - 7], [4241, 4241, 30*w - 37], [4243, 4243, 3*w - 67], [4243, 4243, -3*w - 64], [4259, 4259, 29*w - 2], [4259, 4259, 29*w - 27], [4283, 4283, 29*w - 26], [4283, 4283, 29*w - 3], [4289, 4289, -31*w - 12], [4289, 4289, 31*w - 43], [4327, 4327, 29*w + 79], [4327, 4327, 29*w - 108], [4363, 4363, 23*w - 96], [4363, 4363, 27*w - 104], [4373, 4373, 29*w - 21], [4373, 4373, 29*w - 8], [4409, 4409, 29*w - 12], [4409, 4409, 29*w - 17], [4447, 4447, -14*w - 67], [4447, 4447, 14*w - 81], [4451, 4451, -35*w - 27], [4451, 4451, 35*w - 62], [4457, 4457, -33*w - 19], [4457, 4457, 33*w - 52], [4493, 4493, -31*w - 8], [4493, 4493, 31*w - 39], [4519, 4519, 15*w - 83], [4519, 4519, -15*w - 68], [4561, 4561, 5*w - 71], [4561, 4561, -5*w - 66], [4583, 4583, 31*w - 37], [4583, 4583, -31*w - 6], [4603, 4603, 2*w - 69], [4603, 4603, -2*w - 67], [4621, 4621, 11*w - 78], [4621, 4621, -11*w - 67], [4637, 4637, -42*w - 47], [4637, 4637, -37*w - 32], [4657, 4657, 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[8627, 8627, 47*w - 78], [8627, 8627, -47*w - 31], [8647, 8647, 6*w - 97], [8647, 8647, -6*w - 91], [8669, 8669, 41*w - 8], [8669, 8669, 41*w - 33], [8677, 8677, 11*w - 102], [8677, 8677, -11*w - 91], [8689, 8689, 8*w - 99], [8689, 8689, -8*w - 91], [8693, 8693, 41*w - 9], [8693, 8693, 41*w - 32], [8699, 8699, 49*w - 87], [8699, 8699, -49*w - 38], [8719, 8719, -15*w - 92], [8719, 8719, 15*w - 107], [8731, 8731, 41*w + 112], [8731, 8731, 41*w - 153], [8737, 8737, -w - 93], [8737, 8737, w - 94], [8741, 8741, -53*w - 51], [8741, 8741, -55*w - 57], [8753, 8753, 41*w - 12], [8753, 8753, 41*w - 29], [8761, 8761, 40*w + 111], [8761, 8761, 40*w - 151], [8779, 8779, 23*w - 119], [8779, 8779, -23*w - 96], [8783, 8783, 41*w - 27], [8783, 8783, 41*w - 14], [8803, 8803, 38*w - 147], [8803, 8803, 33*w - 137], [8819, 8819, 41*w - 23], [8819, 8819, 41*w - 18], [8821, 8821, 35*w - 141], [8821, 8821, 36*w - 143], [8837, 8837, 43*w - 51], [8837, 8837, -43*w - 8], [8861, 8861, 42*w - 41], [8861, 8861, 42*w - 1], [8863, 8863, 7*w - 99], [8863, 8863, -7*w - 92], [8867, 8867, -57*w - 62], [8867, 8867, -52*w - 47], [8887, 8887, -9*w - 92], [8887, 8887, 9*w - 101], [8929, 8929, 25*w - 123], [8929, 8929, -25*w - 98], [8941, 8941, 4*w - 97], [8941, 8941, -4*w - 93], [8951, 8951, 43*w - 49], [8951, 8951, -43*w - 6], [8963, 8963, -51*w - 43], [8963, 8963, -59*w - 67], [8971, 8971, -14*w - 93], [8971, 8971, 14*w - 107], [9013, 9013, -13*w - 93], [9013, 9013, 13*w - 106], [9029, 9029, -50*w - 39], [9029, 9029, 50*w - 89], [9067, 9067, -11*w - 93], [9067, 9067, 11*w - 104], [9109, 9109, 28*w - 129], [9109, 9109, -28*w - 101], [9151, 9151, -25*w - 99], [9151, 9151, 25*w - 124], [9157, 9157, 29*w - 131], [9157, 9157, -29*w - 102], [9161, 9161, 42*w - 11], [9161, 9161, 42*w - 31], [9173, 9173, -46*w - 21], [9173, 9173, 46*w - 67], [9181, 9181, -5*w - 94], [9181, 9181, 5*w - 99], [9199, 9199, 43*w + 116], [9199, 9199, 43*w - 159], [9203, 9203, -44*w - 9], [9203, 9203, 44*w - 53], [9239, 9239, -56*w - 57], [9239, 9239, -55*w - 54], [9241, 9241, -21*w - 97], [9241, 9241, 21*w - 118], [9257, 9257, 42*w - 19], [9257, 9257, 42*w - 23], [9277, 9277, -9*w - 94], [9277, 9277, 9*w - 103], [9281, 9281, -50*w - 37], [9281, 9281, 50*w - 87], [9283, 9283, -27*w - 101], [9283, 9283, 27*w - 128], [9319, 9319, 35*w - 143], [9319, 9319, 38*w - 149], [9323, 9323, -44*w - 7], [9323, 9323, 44*w - 51], [9341, 9341, -47*w - 24], [9341, 9341, 47*w - 71], [9349, 9349, -20*w - 97], [9349, 9349, 20*w - 117], [9371, 9371, -45*w - 13], [9371, 9371, 45*w - 58], [9391, 9391, -29*w - 103], [9391, 9391, 29*w - 132], [9403, 9403, -17*w - 96], [9403, 9403, 17*w - 113], [9409, 97, -97], [9413, 9413, -49*w - 32], [9413, 9413, 49*w - 81], [9433, 9433, 23*w - 122], [9433, 9433, -23*w - 99], [9467, 9467, 43*w - 6], [9467, 9467, 43*w - 37], [9491, 9491, -53*w - 46], [9491, 9491, -60*w - 67], [9497, 9497, 43*w - 7], [9497, 9497, 43*w - 36], [9533, 9533, -59*w - 64], [9533, 9533, -54*w - 49], [9539, 9539, -44*w - 3], [9539, 9539, 44*w - 47], [9551, 9551, 43*w - 9], [9551, 9551, 43*w - 34], [9601, 9601, 25*w - 126], [9601, 9601, -25*w - 101], [9613, 9613, -17*w - 97], [9613, 9613, 17*w - 114], [9619, 9619, -13*w - 96], [9619, 9619, 13*w - 109], [9623, 9623, 51*w - 89], [9623, 9623, -51*w - 38], [9643, 9643, 7*w - 103], [9643, 9643, -7*w - 96], [9661, 9661, -19*w - 98], [9661, 9661, 19*w - 117], [9677, 9677, 43*w - 16], [9677, 9677, 43*w - 27], [9697, 9697, -w - 98], [9697, 9697, w - 99], [9719, 9719, -59*w - 63], [9719, 9719, -55*w - 51], [9739, 9739, -15*w - 97], [9739, 9739, 15*w - 112], [9743, 9743, -49*w - 29], [9743, 9743, 49*w - 78], [9749, 9749, 51*w - 88], [9749, 9749, -51*w - 37], [9769, 9769, -5*w - 97], [9769, 9769, 5*w - 102], [9781, 9781, -20*w - 99], [9781, 9781, 20*w - 119], [9787, 9787, -14*w - 97], [9787, 9787, 14*w - 111], [9791, 9791, -47*w - 19], [9791, 9791, 47*w - 66], [9803, 9803, 44*w - 41], [9803, 9803, 44*w - 3], [9811, 9811, 6*w - 103], [9811, 9811, -6*w - 97], [9829, 9829, -25*w - 102], [9829, 9829, 25*w - 127], [9833, 9833, -54*w - 47], [9833, 9833, -61*w - 68], [9871, 9871, 11*w - 108], [9871, 9871, -11*w - 97], [9887, 9887, -48*w - 23], [9887, 9887, 48*w - 71], [9907, 9907, -26*w - 103], [9907, 9907, 26*w - 129], [9929, 9929, 45*w - 49], [9929, 9929, -45*w - 4], [9949, 9949, 15*w - 113], [9949, 9949, -15*w - 98]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 1, 1, 1, 4, -6, -6, -2, -2, -6, -6, -8, -8, 12, 12, 12, 12, 8, 8, 8, 8, 0, 0, -6, -6, 6, 6, -2, -2, 10, -4, -4, 12, 12, -8, -8, -16, -16, -12, -12, 26, -18, -18, -14, -14, 4, 4, 24, 24, -12, -12, -6, -6, -18, -18, 14, 14, 6, 6, 0, 0, 28, 28, -14, -14, -6, -6, 26, 14, 14, 4, 4, -12, -12, -12, -12, -10, -10, -38, -38, -18, -18, 20, 20, -24, -24, 0, 0, 4, 4, -20, -20, 12, 12, 30, 30, -6, -6, 34, -34, -34, 8, 8, 0, 0, -4, -4, 0, 0, -6, -6, -34, -34, -40, -40, -12, -12, -14, -14, 6, 6, 22, 22, 0, 0, -4, -4, 16, 16, -2, -2, 42, 42, 6, 6, -18, -18, 44, 44, 24, 24, 10, -30, -30, 38, 38, -30, -30, 16, 16, 36, 36, -32, -32, 16, 16, -30, -30, -18, -18, 50, 20, 20, -60, -60, 60, 60, 16, 16, 2, 2, 54, 54, -22, -22, -6, -6, 44, 44, -52, -52, 12, 12, 14, 14, 18, 18, 54, 54, 22, 22, -22, -22, -24, -24, 52, 52, 30, 30, -30, -30, -14, -14, -10, -10, -6, -6, 12, 12, -36, -36, -18, -18, 50, 50, 54, 54, -20, -20, -48, -48, -24, -24, -68, -68, -30, -30, 22, 22, 4, 4, 24, 24, -26, -26, 66, 66, 22, 22, 16, 16, 60, 60, 0, 0, -50, -50, -54, -54, 72, 72, -36, -36, 20, 20, 22, 22, -78, -78, -66, -66, 38, 38, 54, 54, 68, 68, 42, 42, 42, 42, 8, 8, -8, -8, -8, -8, -50, -50, -38, -38, 60, 60, -12, -12, -40, -40, 36, 36, -14, -14, 66, 66, 24, 24, -12, -12, -10, -10, -66, -66, 6, 6, -84, -84, 56, 56, -4, -4, -62, -62, 46, 46, 36, 36, 16, 16, 36, 36, 58, 58, 30, 30, 52, 52, -52, -52, 46, 46, 48, 48, 44, 44, 48, 48, -34, -34, -54, -54, 14, 14, -90, -90, -8, -8, 64, 64, -24, -24, 6, 6, 74, 74, -10, -10, 66, 66, 0, 0, -12, -12, -82, -82, -6, -6, -86, -86, -18, -18, -52, -52, -86, -86, 38, 38, 84, 84, 18, 18, -66, -66, 52, 52, 12, 12, 64, 64, -88, -88, -84, -84, 98, 98, -42, -42, -86, -86, -54, -54, -52, -52, -4, -4, 18, 18, 6, 6, -10, -10, 106, 36, 36, 52, 52, -22, -22, 78, 78, -54, -54, 12, 12, 60, 60, -18, -18, 24, 24, 20, 20, -60, -60, 18, 18, 10, 10, 86, 86, 32, 32, -72, -72, 86, 86, 12, 12, 8, 8, -96, -96, 42, 42, -94, -94, -34, -34, -36, -36, 52, 52, -40, -40, 22, 22, -14, -14, -80, -80, -24, -24, -76, -76, -96, -96, 82, 82, -36, -36, 90, 90, 102, 102, -50, -50, -84, -84, 8, 8, -12, -12, 10, 10, 30, 30, 116, 116, 4, 4, -74, -74, -54, -54, 14, 14, 12, 12, 48, 48, -100, -100, -62, -62, 54, 54, 74, -34, -34, 4, 4, -12, -12, 102, 102, -114, -114, -92, -92, 4, 4, -36, -36, -6, -6, -62, -62, 56, 56, 96, 96, 0, 0, -100, -100, 28, 28, 96, 96, -42, -42, 36, 36, 112, 112, 18, 18, -22, -22, 114, 114, 0, 0, -12, -12, 76, 76, -64, -64, -90, -90, -22, -22, 30, 30, -88, -88, -70, -70, 66, 66, -30, -30, -64, -64, -108, -108, -24, -24, -30, -30, 52, 52, -8, -8, -42, -42, -102, -102, -20, -20, 60, 60, 66, 66, -66, -66, 16, 16, 2, 2, -36, -36, 88, 88, -50, -50, -18, -18, -94, -94, -4, -4, 24, 24, 36, 36, 66, 66, -22, -22, 72, 72, 4, 4, -72, -72, -74, -74, -30, -30, -26, -26, 64, 64, -48, -48, -66, -66, 90, 90, 46, 46, 24, 24, 12, 12, -8, -8, 22, 22, 78, 78, -46, -46, -80, -80, -120, -120, -124, -124, -48, -48, 130, -130, -130, -54, -54, 108, 108, -36, -36, -40, -40, 8, 8, 92, 92, -84, -84, 10, 10, 50, 50, 42, 42, 30, 30, 98, -42, -42, -48, -48, -90, -90, 90, 90, -2, -2, 114, 114, -28, -28, 64, 64, -138, -138, 30, 30, 20, 20, 24, 24, 0, 0, 52, 52, -62, -62, 110, 110, 0, 0, 76, 76, -98, -98, 6, 6, 120, 120, -90, -90, -38, -38, -10, -10, -4, -4, 32, 32, -6, -6, 6, 6, -98, -98, 8, 8, 96, 96, -120, -120, 122, 122, -126, -126, 16, 16, 132, 132, 36, 36, 126, 126, 148, 148, -108, -108, 104, 104, -36, -36, -42, -42, -134, -134, -54, -54, -20, -20, -84, -84, 94, 94, 126, 126, -64, -64, -140, -140, -134, -134, 78, 78, 90, 90, 108, 108, 60, 60, 62, 62, 126, 126, -94, -94, -52, -52, 24, 24, -68, -68, -44, -44, 54, 54, -58, -58, -32, -32, 20, 20, 86, 86, 114, 114, -40, -40, 48, 48, 58, 58, -150, -150, 82, 82, 120, 120, 4, 4, 22, 22, 30, 30, 134, 134, -14, -14, 64, 64, 68, 68, 96, 96, 138, 138, 90, 90, 40, 40, -84, -84, -96, -96, -130, -130, -70, -70, -108, -108, 136, 136, 16, 16, 24, 24, -28, -28, -12, -12, -102, -102, -2, -2, 102, 102, -36, -36, 56, 56, -46, -46, 36, 36, 10, 10, -72, -72, 52, 52, 140, 140, 126, 126, -96, -96, -48, -48, -130, -130, -2, -2, -138, -138, -40, -40, -110, -110, 154, 154, -78, -78, -60, -60, 44, 44, -82, -82, -150, -150, 30, 30, 0, 0, 72, 72, -134, -134, -78, -78, 32, 32, -60, -60, -64, -64, -24, -24, -30, -30, -154, -154, 82, 82, 20, 20, 36, 36, -84, -84, 56, 56, 108, 108, -22, -22, -78, -78, 54, 54, -66, -66, 80, 80, -66, -66, 118, 118, -54, -54, 152, 152, -62, -62, 120, 120, -140, -140, 6, 6, -30, -30, 10, 10, -74, -74, -72, -72, 72, 72, 72, 72, 112, 112, 144, 144, 26, 26, -18, -18, 46, 46, -6, -6, 156, 156, -136, -136, 70, 70, 34, 34, -120, -120, 88, 88, 60, 60, -136, -136, 54, 54, -118, -118, -4, -4, -60, -60, -64, -64, -162, -162, -30, -30, -120, -120, 116, 116, 30, 30, -146, -146, -62, -62, 6, 6, -132, -132, 40, 40, -20, -20, 82, 82, 6, 6, 138, 138, 58, 58, -28, -28, -108, -108, 16, 16, 132, 132, 22, 22, -42, -42, 126, 126, -172, -172, 24, 24, -44, -44, -142, -142, -98, -98, 48, 48, 96, 96, -100, -100, -146, -146, -90, -90, 32, 32, 134, 134, 112, 112, -82, -82, -102, -102, 54, 54, 142, 142, -40, -40, 0, 0, -72, -72, 26, 26, 18, 18, -58, -58, 18, 18, 16, 16, -160, -160, -72, -72, 30, 30, 166, 166, 132, 132, 112, 112, 56, 56, 146, 54, 54, -70, -70, -96, -96, -36, -36, 114, 114, 126, 126, 132, 132, -24, -24, -142, -142, -74, -74, -116, -116, -84, -84, 184, 184, 14, 14, 126, 126, 50, 50, -120, -120, 20, 20, 132, 132, -42, -42, 154, 154, -58, -58, 112, 112, 120, 120, -72, -72, 20, 20, 22, 22, -30, -30, 64, 64, -84, -84, -152, -152, 42, 42, -50, -50]; 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;