/* 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![8, 1, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w^2 + 2], [2, 2, -w^3 + 2*w^2 + 4*w - 5], [5, 5, -w^3 + 2*w^2 + 3*w - 1], [13, 13, w^3 - 2*w^2 - 3*w + 5], [17, 17, -w^2 - w + 3], [19, 19, -w^3 + 3*w^2 + 2*w - 7], [23, 23, w^3 - 2*w^2 - 3*w + 3], [31, 31, -w^2 + w + 1], [47, 47, -w^3 + w^2 + 4*w - 3], [49, 7, 2*w^3 - 5*w^2 - 7*w + 11], [53, 53, -3*w^3 + 9*w^2 + 10*w - 31], [53, 53, w^3 - w^2 - 4*w + 1], [61, 61, 3*w^3 - 6*w^2 - 13*w + 13], [61, 61, 2*w^2 - 7], [71, 71, w^2 - 3*w - 5], [73, 73, 2*w - 3], [73, 73, -2*w^3 + 6*w^2 + 6*w - 19], [79, 79, 2*w^2 - 5], [81, 3, -3], [101, 101, 2*w^2 - 4*w - 9], [107, 107, 2*w - 1], [109, 109, 2*w^3 - 3*w^2 - 11*w + 7], [109, 109, 2*w^2 - 2*w - 7], [125, 5, -2*w^3 + 8*w^2 - 17], [127, 127, 2*w^2 - 4*w - 7], [127, 127, -w^2 + 3*w + 1], [131, 131, w^3 - w^2 - 4*w - 3], [137, 137, 4*w^3 - 10*w^2 - 14*w + 25], [139, 139, -w^3 + 2*w^2 + w - 5], [139, 139, 2*w^2 - 2*w - 5], [149, 149, -w^3 + 4*w^2 - w - 7], [149, 149, -2*w^3 + 5*w^2 + 9*w - 19], [151, 151, -w^3 + 4*w^2 + w - 11], [151, 151, w^2 + w + 1], [157, 157, -2*w + 5], [163, 163, 4*w^3 - 8*w^2 - 18*w + 19], [163, 163, w^3 - 4*w^2 - w + 9], [167, 167, -w^3 + 3*w^2 + 2*w - 9], [167, 167, w^3 - 2*w^2 - 5*w + 1], [173, 173, w^2 - 3*w - 7], [173, 173, w^3 - 3*w^2 - 4*w + 5], [181, 181, -w^3 + 4*w^2 + w - 13], [191, 191, -w^3 + 2*w^2 + 3*w + 1], [191, 191, -2*w^3 + 3*w^2 + 9*w - 3], [193, 193, -w^3 + 4*w^2 + 3*w - 13], [197, 197, 5*w^3 - 10*w^2 - 21*w + 21], [211, 211, -w^3 + 3*w^2 + 4*w - 13], [227, 227, w^3 - 4*w^2 + w + 5], [229, 229, -w^3 + 5*w^2 - 2*w - 11], [229, 229, -w^3 + 3*w^2 + 4*w - 3], [239, 239, -2*w^3 + 4*w^2 + 8*w - 13], [241, 241, 4*w^3 - 10*w^2 - 16*w + 31], [241, 241, -5*w^3 + 12*w^2 + 21*w - 31], [251, 251, -w^3 + 3*w^2 + 2*w - 11], [263, 263, -2*w^3 + 2*w^2 + 10*w + 1], [263, 263, -2*w - 5], [269, 269, w^2 + w - 5], [269, 269, 4*w^3 - 7*w^2 - 19*w + 15], [271, 271, 2*w^3 - 5*w^2 - 5*w + 11], [271, 271, -w^3 + 7*w + 3], [277, 277, -2*w^3 + 4*w^2 + 6*w - 5], [277, 277, 4*w^2 - 13], [281, 281, w^3 - 5*w^2 + 11], [307, 307, w^3 - 3*w - 3], [313, 313, 5*w^3 - 10*w^2 - 23*w + 27], [313, 313, -2*w^3 + 3*w^2 + 9*w - 7], [317, 317, -3*w^3 + 6*w^2 + 11*w - 15], [347, 347, -w^3 - w^2 + 1], [347, 347, -2*w^3 + 6*w^2 + 6*w - 17], [353, 353, -2*w^3 + 4*w^2 + 8*w - 5], [353, 353, w^3 - 5*w - 1], [359, 359, -2*w^2 + 1], [367, 367, -w^3 + 4*w^2 + 3*w - 17], [373, 373, 2*w^2 - 2*w - 13], [397, 397, 5*w^3 - 13*w^2 - 18*w + 39], [397, 397, 2*w^3 - 5*w^2 - 7*w + 9], [401, 401, -4*w^3 + 12*w^2 + 8*w - 25], [401, 401, -3*w^3 + 7*w^2 + 8*w - 13], [421, 421, -4*w^3 + 11*w^2 + 13*w - 31], [433, 433, 2*w^3 - 12*w - 15], [433, 433, w^3 - w^2 - 6*w + 5], [439, 439, 5*w^3 - 13*w^2 - 12*w + 19], [439, 439, -2*w^3 + 3*w^2 + 7*w + 1], [449, 449, -w^3 + 4*w^2 + 3*w - 11], [461, 461, -4*w^3 + 11*w^2 + 7*w - 19], [461, 461, -w^3 + w^2 + 2*w + 1], [463, 463, -w^3 + 2*w^2 + 7*w - 1], [463, 463, -4*w^3 + 11*w^2 + 7*w - 13], [467, 467, 2*w^2 - 2*w - 3], [467, 467, 2*w^3 - 4*w^2 - 6*w + 11], [487, 487, -3*w^3 + 11*w^2 + 2*w - 23], [487, 487, w^3 - w + 5], [491, 491, w^3 - 2*w^2 - 3*w + 9], [499, 499, w^3 - 3*w^2 - 4*w + 1], [503, 503, -6*w^3 + 11*w^2 + 27*w - 21], [503, 503, -3*w^3 + 8*w^2 + 5*w - 9], [521, 521, 2*w^2 - 4*w - 11], [541, 541, -3*w^3 + 7*w^2 + 8*w - 11], [557, 557, 5*w^3 - 11*w^2 - 18*w + 21], [557, 557, 4*w^3 - 7*w^2 - 17*w + 9], [557, 557, 4*w^3 - 8*w^2 - 16*w + 15], [557, 557, -3*w^3 + 6*w^2 + 11*w - 9], [563, 563, -2*w^3 + 6*w^2 + 8*w - 13], [569, 569, -2*w^3 + 3*w^2 + 7*w - 11], [587, 587, 3*w^3 - 8*w^2 - 9*w + 15], [601, 601, -3*w^2 + 5*w + 9], [607, 607, -w^3 + w^2 - 5], [619, 619, -2*w^3 + 4*w^2 + 6*w - 13], [631, 631, -2*w^3 + 7*w^2 + 5*w - 23], [643, 643, 2*w^3 - 2*w^2 - 12*w - 5], [659, 659, -w^3 + 6*w^2 + 3*w - 11], [659, 659, -3*w^2 + w + 9], [661, 661, -2*w^3 + 4*w^2 + 8*w - 3], [661, 661, -w^3 + 4*w^2 + 5*w - 7], [673, 673, w^2 - w - 9], [673, 673, 2*w^3 - 7*w^2 - 3*w + 17], [677, 677, 9*w^3 - 20*w^2 - 37*w + 51], [701, 701, -7*w^3 + 18*w^2 + 27*w - 57], [709, 709, 4*w^3 - 6*w^2 - 14*w + 17], [709, 709, w^3 + w^2 - 6*w - 17], [719, 719, -2*w^3 + 6*w^2 + 4*w - 15], [719, 719, -2*w^3 + 3*w^2 + 7*w + 3], [727, 727, -w^3 + 3*w^2 + 6*w - 3], [743, 743, -2*w^3 + 2*w^2 + 6*w - 1], [757, 757, 6*w^3 - 16*w^2 - 22*w + 51], [757, 757, w^3 + w^2 - 10*w - 11], [761, 761, w^2 + w - 7], [761, 761, 2*w^2 - 17], [761, 761, 2*w^3 - 6*w^2 - 2*w + 13], [761, 761, 2*w^2 - 4*w - 3], [769, 769, -w^3 - w^2 + 4*w + 7], [769, 769, w^3 + w^2 - 4*w - 1], [787, 787, 3*w^3 - 6*w^2 - 11*w + 3], [809, 809, -w^3 + 3*w^2 + 2*w - 1], [809, 809, 2*w^3 - w^2 - 9*w - 3], [809, 809, 2*w^3 - 7*w^2 - 3*w + 15], [809, 809, 2*w^3 - 2*w^2 - 8*w - 1], [811, 811, -8*w^3 + 19*w^2 + 33*w - 57], [811, 811, -2*w^3 + 2*w^2 + 4*w - 11], [823, 823, 2*w^3 - w^2 - 7*w + 1], [823, 823, -3*w^3 + 8*w^2 + 7*w - 9], [829, 829, -w^3 + 5*w - 1], [853, 853, -w^3 + 5*w^2 - 2*w - 15], [853, 853, -2*w^3 + 4*w^2 + 4*w - 9], [853, 853, 2*w^3 - w^2 - 11*w - 3], [853, 853, 2*w^3 - 5*w^2 - 7*w + 5], [857, 857, -2*w^3 + 5*w^2 + 7*w - 7], [859, 859, w^3 - 2*w^2 - w + 9], [859, 859, w^3 - 2*w^2 - w - 1], [877, 877, 2*w^3 - 6*w^2 - 8*w + 25], [877, 877, -4*w^3 + 8*w^2 + 16*w - 19], [881, 881, -4*w^2 + 6*w + 13], [881, 881, w^3 + 2*w^2 - 7*w - 9], [887, 887, -4*w^2 + 4*w + 11], [887, 887, -w^3 - w^2 + 6*w + 11], [887, 887, w^3 - 6*w^2 - w + 29], [887, 887, 4*w^3 - 9*w^2 - 19*w + 31], [907, 907, -3*w^3 + 9*w^2 + 12*w - 25], [907, 907, -w^3 + 2*w^2 + 7*w - 7], [929, 929, -7*w^3 + 13*w^2 + 32*w - 29], [941, 941, -2*w^3 + 10*w^2 - 2*w - 23], [941, 941, 4*w^3 - 9*w^2 - 13*w + 13], [967, 967, -w^3 + 2*w^2 - w - 1], [977, 977, w^3 - 5*w^2 + 15], [983, 983, w^3 + 2*w^2 - 9*w - 11], [983, 983, 2*w^3 - 5*w^2 - 5*w + 13], [997, 997, -4*w^3 + 10*w^2 + 12*w - 23], [1009, 1009, 2*w^2 + 2*w + 1], [1019, 1019, 2*w^3 - 2*w^2 - 6*w + 3], [1031, 1031, w^3 - 9*w - 7], [1033, 1033, 2*w^3 - 2*w^2 - 8*w + 1], [1063, 1063, -w^3 - w - 7], [1063, 1063, 4*w^3 - 8*w^2 - 12*w + 5], [1091, 1091, 2*w^3 - 2*w^2 - 4*w + 1], [1093, 1093, 3*w^3 - 4*w^2 - 13*w - 1], [1093, 1093, w^3 - w^2 - 4*w - 7], [1097, 1097, w^3 + 3*w^2 - 2*w - 9], [1109, 1109, w^2 + w - 9], [1109, 1109, -2*w^3 + 5*w^2 + 3*w - 11], [1123, 1123, -5*w^3 + 16*w^2 + 7*w - 27], [1123, 1123, 8*w^3 - 17*w^2 - 35*w + 47], [1163, 1163, 3*w^3 - 3*w^2 - 14*w - 5], [1171, 1171, 2*w^3 - w^2 - 7*w - 1], [1171, 1171, -3*w^3 + 6*w^2 + 9*w - 7], [1181, 1181, 3*w^3 - 7*w^2 - 10*w + 21], [1187, 1187, w^3 - 2*w^2 - 7*w + 3], [1201, 1201, -3*w^3 + 5*w^2 + 10*w - 3], [1201, 1201, -3*w^2 + 3*w + 19], [1213, 1213, w^3 - 5*w^2 + 4*w + 7], [1217, 1217, w^3 - 4*w^2 - 5*w + 5], [1223, 1223, -w^3 - w^2 + 2*w + 5], [1223, 1223, w^3 - 5*w^2 + 4*w + 5], [1231, 1231, 7*w^3 - 15*w^2 - 28*w + 33], [1237, 1237, w^3 - 5*w^2 + 2*w + 9], [1249, 1249, -2*w^3 + 2*w^2 + 10*w - 3], [1259, 1259, -4*w^3 + 8*w^2 + 16*w - 13], [1283, 1283, w^3 - 2*w^2 - 3*w - 3], [1289, 1289, -w^3 + w^2 + 4*w - 7], [1291, 1291, 2*w^3 - 5*w^2 - 9*w + 9], [1297, 1297, -10*w^3 + 24*w^2 + 38*w - 63], [1301, 1301, 3*w^3 - 8*w^2 - 11*w + 17], [1301, 1301, 2*w^2 - 4*w - 1], [1307, 1307, w^3 - w^2 + 7], [1327, 1327, -6*w^3 + 12*w^2 + 24*w - 31], [1327, 1327, -w^3 + 2*w^2 - w - 5], [1327, 1327, 8*w^3 - 16*w^2 - 34*w + 35], [1327, 1327, 3*w^3 - 8*w^2 - 3*w + 15], [1367, 1367, -w^3 + 2*w^2 + 7*w - 5], [1369, 37, 7*w^3 - 17*w^2 - 18*w + 19], [1369, 37, w^3 - 7*w^2 + 6*w + 19], [1381, 1381, 6*w^2 - 6*w - 35], [1427, 1427, -2*w^3 + 3*w^2 + 11*w + 1], [1427, 1427, -2*w^3 + 7*w^2 + 7*w - 13], [1427, 1427, 3*w^3 - 8*w^2 - 9*w + 9], [1427, 1427, -12*w^3 + 26*w^2 + 50*w - 65], [1433, 1433, w^3 - 7*w^2 - 4*w + 11], [1439, 1439, -5*w^3 + 12*w^2 + 21*w - 39], [1453, 1453, -w^3 - w^2 - 2*w - 7], [1453, 1453, 2*w^3 - 4*w^2 - 6*w - 1], [1459, 1459, 3*w^2 - 7*w - 15], [1483, 1483, 4*w^3 - 15*w^2 - 9*w + 55], [1483, 1483, w^3 - 3*w - 5], [1493, 1493, -2*w^3 + 8*w^2 + 2*w - 23], [1499, 1499, -w^3 + 9*w + 1], [1499, 1499, w^3 - 4*w^2 + 3*w + 3], [1499, 1499, 5*w^3 - 12*w^2 - 13*w + 17], [1499, 1499, w^3 + w^2 - 8*w - 15], [1511, 1511, -7*w^3 + 17*w^2 + 26*w - 47], [1531, 1531, w^3 - 4*w^2 - 7*w + 13], [1543, 1543, 4*w^3 - 7*w^2 - 21*w + 21], [1543, 1543, 5*w^2 + 3*w - 7], [1579, 1579, 4*w^3 - 9*w^2 - 13*w + 17], [1583, 1583, -2*w^3 + 6*w^2 + 6*w - 23], [1597, 1597, -4*w^3 + 11*w^2 + 13*w - 25], [1609, 1609, 4*w^3 - 11*w^2 - 11*w + 27], [1609, 1609, -4*w^3 + 13*w^2 + 5*w - 21], [1637, 1637, -6*w^3 + 16*w^2 + 20*w - 45], [1667, 1667, 5*w^2 - 3*w - 13], [1667, 1667, -5*w^2 - w + 11], [1697, 1697, -w^3 + 3*w^2 + 8*w + 1], [1699, 1699, -w^3 + 4*w^2 - w - 13], [1699, 1699, 5*w^3 - 14*w^2 - 9*w + 25], [1721, 1721, 2*w^3 - 7*w^2 - w + 17], [1723, 1723, -3*w^3 + 6*w^2 + 9*w - 11], [1733, 1733, w^3 - w^2 - 8*w + 3], [1747, 1747, w^3 - 2*w^2 - 7*w - 5], [1753, 1753, -w^3 + 3*w^2 - 11], [1759, 1759, -w^3 + 6*w^2 - w - 15], [1759, 1759, -4*w^3 + 8*w^2 + 14*w - 19], [1783, 1783, 3*w^3 - 6*w^2 - 7*w + 13], [1783, 1783, -5*w^2 - 3*w + 13], [1787, 1787, -2*w^3 + 7*w^2 + 5*w - 21], [1787, 1787, w^2 - 3*w - 11], [1787, 1787, -2*w - 7], [1787, 1787, -w^3 + 3*w^2 + 4*w - 15], [1789, 1789, w^3 - w^2 - 8*w - 1], [1789, 1789, -4*w^2 + 8*w + 13], [1801, 1801, -3*w^3 + 9*w^2 + 6*w - 11], [1823, 1823, -2*w^3 + 5*w^2 + 5*w - 15], [1831, 1831, 3*w^3 - 5*w^2 - 12*w + 9], [1831, 1831, 3*w^3 - 3*w^2 - 18*w + 1], [1831, 1831, w^3 - 5*w^2 - 2*w + 5], [1831, 1831, w^3 - 5*w^2 - 2*w + 13], [1847, 1847, 2*w^3 - 2*w^2 - 8*w - 9], [1847, 1847, -3*w^3 + 5*w^2 + 14*w - 17], [1849, 43, 3*w^3 - 3*w^2 - 12*w + 5], [1849, 43, 3*w^3 - 2*w^2 - 19*w - 5], [1861, 1861, 4*w^3 - 13*w^2 - 5*w + 27], [1867, 1867, w^3 + 2*w^2 - 5*w - 5], [1871, 1871, -w^3 + 5*w^2 + 2*w - 15], [1871, 1871, 2*w^3 - w^2 - 15*w + 1], [1873, 1873, w^3 - 7*w - 11], [1873, 1873, -4*w^3 + 9*w^2 + 11*w - 17], [1877, 1877, -4*w - 1], [1877, 1877, 4*w + 9], [1889, 1889, -3*w^3 + 7*w^2 + 6*w - 7], [1901, 1901, 2*w^3 - 8*w^2 - 2*w + 17], [1901, 1901, -3*w^2 + 7*w + 7], [1913, 1913, 4*w^3 - 14*w^2 - 2*w + 27], [1931, 1931, w^2 + 3*w - 7], [1931, 1931, 3*w^3 - 3*w^2 - 12*w - 5], [1933, 1933, -2*w^3 + 7*w^2 + 7*w - 17], [1951, 1951, -2*w^3 + 6*w^2 + 4*w - 17], [1951, 1951, -5*w^3 + 13*w^2 + 20*w - 43], [1973, 1973, -2*w^3 + 9*w^2 - w - 19], [1979, 1979, -4*w^3 + 10*w^2 + 18*w - 37], [1979, 1979, -2*w^3 + 6*w^2 + 8*w - 11], [1987, 1987, -w^3 + 3*w^2 + 6*w - 13], [1997, 1997, -4*w^3 + 11*w^2 + 13*w - 29], [1997, 1997, 4*w - 1], [1997, 1997, 9*w^3 - 23*w^2 - 22*w + 35], [1997, 1997, -6*w^3 + 18*w^2 + 8*w - 31], [1999, 1999, 6*w^3 - 19*w^2 - 9*w + 37]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, 0, 0, 4, -8, 4, 8, 4, -12, 4, -6, -6, 8, 12, -8, -8, 8, 0, -2, 12, 4, -14, -14, -4, 8, 16, -20, -22, 4, -12, 6, -6, -16, 0, -12, 12, 4, -4, 12, -24, 0, -16, -8, 0, 2, -26, 4, 8, -12, -20, -24, -16, -4, 0, 24, 8, -12, -24, 20, -28, -22, -10, -26, -8, -4, -16, -18, -12, -12, 20, 28, 8, 8, 10, 8, 20, -34, 14, 10, 16, -36, -12, -40, -30, 12, 40, -16, -24, -12, -20, -16, 32, -12, -20, 28, -20, -28, 30, -14, -14, -14, 18, 32, -10, 36, 4, -8, -20, 8, -16, 12, 20, 40, -40, -14, 2, -28, -34, -26, 38, 16, 28, -12, 16, -42, -22, 4, 24, 40, 20, -16, 12, 44, -10, -10, -6, -6, 32, -40, 0, -16, -12, -40, -36, 8, 8, -8, 16, 12, -50, -34, -24, -40, 36, 48, 32, 28, -52, 36, 8, 48, 36, -48, 30, 8, -12, 52, -30, 20, 56, -6, 36, 44, -4, -38, 26, -28, 56, 24, -44, -40, 36, 12, 20, 20, 60, 30, -62, -34, 12, 36, 8, -32, 20, -40, 0, 12, -54, -52, 30, 32, 36, 4, 48, 40, 24, -32, 48, -22, 10, 6, -12, 48, 0, -40, 24, 16, 46, -14, 0, 56, 12, -58, -4, 44, 28, -20, 12, 16, 32, 76, 12, 24, 18, 48, -64, 24, 60, 36, -44, -64, 0, 56, 44, -22, -28, -72, 0, -48, 4, -16, 68, 28, -12, 28, -2, -18, -54, 0, -16, 0, 16, -32, 40, 0, 10, 70, 60, 72, -40, -16, -30, 46, -42, -22, -32, -14, -18, 42, 8, -12, 18, -56, 76, 24, -20, -88, -44, -14, -18, 50, -18, -40]; 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;