/* 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![41, 9, -14, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, 1/6*w^3 - 7/3*w - 17/6], [4, 2, w - 3], [5, 5, -1/6*w^3 + w^2 + 1/3*w - 25/6], [9, 3, 1/6*w^3 - 7/3*w + 13/6], [9, 3, -w - 2], [11, 11, -1/6*w^3 + 7/3*w + 11/6], [11, 11, -w + 2], [41, 41, -w], [41, 41, 1/6*w^3 - 7/3*w + 1/6], [59, 59, -1/6*w^3 + 1/3*w + 23/6], [59, 59, -1/3*w^3 + 11/3*w + 11/3], [61, 61, -5/3*w^3 - 3*w^2 + 46/3*w + 79/3], [61, 61, 5/6*w^3 + w^2 - 23/3*w - 55/6], [61, 61, w^3 + 2*w^2 - 9*w - 17], [61, 61, -1/6*w^3 + 10/3*w + 35/6], [71, 71, -1/2*w^3 + 2*w^2 + 4*w - 33/2], [71, 71, 1/6*w^3 - w^2 - 4/3*w + 67/6], [79, 79, 1/2*w^3 - 3*w + 1/2], [79, 79, -2/3*w^3 + 19/3*w - 2/3], [89, 89, 1/2*w^3 + w^2 - 5*w - 15/2], [89, 89, 1/3*w^3 - 11/3*w + 7/3], [101, 101, -5/6*w^3 - w^2 + 23/3*w + 61/6], [101, 101, 1/2*w^3 - w^2 - 3*w + 9/2], [109, 109, 1/2*w^3 - w^2 - 4*w + 11/2], [109, 109, -2/3*w^3 - w^2 + 16/3*w + 28/3], [121, 11, -1/2*w^3 + 4*w + 1/2], [131, 131, 2*w - 7], [131, 131, -7/6*w^3 - 2*w^2 + 28/3*w + 89/6], [139, 139, -1/6*w^3 + w^2 + 4/3*w - 31/6], [139, 139, 1/3*w^3 + w^2 - 8/3*w - 29/3], [149, 149, 1/6*w^3 - 2*w^2 + 5/3*w + 37/6], [151, 151, -7/6*w^3 + 3*w^2 + 28/3*w - 133/6], [151, 151, 5/3*w^3 + 3*w^2 - 40/3*w - 67/3], [169, 13, -1/6*w^3 - 2/3*w + 29/6], [169, 13, 7/6*w^3 - 4*w^2 - 19/3*w + 139/6], [181, 181, -2/3*w^3 + 13/3*w + 10/3], [181, 181, 5/6*w^3 - 23/3*w - 19/6], [191, 191, 5/6*w^3 + w^2 - 32/3*w - 97/6], [191, 191, w^3 - 9*w - 5], [191, 191, -w^2 + 4*w - 2], [191, 191, 5/6*w^3 + w^2 - 17/3*w - 43/6], [199, 199, 1/2*w^3 - 2*w - 7/2], [199, 199, 7/6*w^3 - 5*w^2 - 7/3*w + 109/6], [211, 211, -1/6*w^3 + w^2 + 7/3*w - 79/6], [211, 211, 1/2*w^3 + 2*w^2 - 6*w - 37/2], [229, 229, -1/2*w^3 + 5*w - 5/2], [229, 229, 1/2*w^3 - 2*w^2 - 2*w + 23/2], [229, 229, 1/6*w^3 + w^2 - 10/3*w - 53/6], [229, 229, -7/6*w^3 - 2*w^2 + 34/3*w + 107/6], [239, 239, 2*w^3 + 3*w^2 - 22*w - 36], [239, 239, -7/3*w^3 - 3*w^2 + 74/3*w + 113/3], [241, 241, -1/3*w^3 - w^2 + 2/3*w + 11/3], [241, 241, -1/2*w^3 + w^2 + 6*w - 23/2], [271, 271, -5/6*w^3 - w^2 + 20/3*w + 49/6], [271, 271, 1/6*w^3 - 1/3*w - 35/6], [281, 281, -1/6*w^3 - w^2 + 7/3*w + 11/6], [281, 281, -w^3 + 9*w - 1], [281, 281, -1/2*w^3 + 8*w + 19/2], [281, 281, w^3 - 4*w^2 - 4*w + 20], [311, 311, 2*w^2 - 13], [311, 311, -1/3*w^3 - 2*w^2 + 8/3*w + 50/3], [331, 331, 1/2*w^3 + w^2 - 4*w - 5/2], [331, 331, -1/2*w^3 - w^2 + 5*w + 27/2], [331, 331, 1/6*w^3 - 2*w^2 + 2/3*w + 73/6], [331, 331, 5/6*w^3 + 2*w^2 - 26/3*w - 103/6], [349, 349, -1/2*w^3 - w^2 + 4*w + 21/2], [349, 349, 1/6*w^3 + w^2 - 1/3*w - 59/6], [361, 19, 2/3*w^3 - 16/3*w - 7/3], [361, 19, 1/6*w^3 - 4/3*w - 29/6], [379, 379, 1/6*w^3 + w^2 - 10/3*w - 17/6], [379, 379, -5/6*w^3 - 2*w^2 + 23/3*w + 115/6], [379, 379, -1/3*w^3 + 2*w^2 + 5/3*w - 31/3], [379, 379, 1/3*w^3 + w^2 - 14/3*w - 35/3], [389, 389, -1/3*w^3 + 2/3*w + 14/3], [389, 389, 1/2*w^3 + 2*w^2 - 4*w - 35/2], [401, 401, 1/3*w^3 + 2*w^2 - 14/3*w - 38/3], [401, 401, 5/6*w^3 - 4*w^2 - 5/3*w + 101/6], [401, 401, -1/3*w^3 - 2*w^2 + 14/3*w + 50/3], [401, 401, -5/6*w^3 + 2*w^2 + 5/3*w - 29/6], [409, 409, -1/2*w^3 + 6*w - 7/2], [409, 409, 5/6*w^3 + 2*w^2 - 26/3*w - 91/6], [439, 439, 1/3*w^3 + 2*w^2 - 14/3*w - 32/3], [439, 439, 1/3*w^3 + 2*w^2 - 14/3*w - 56/3], [449, 449, -11/6*w^3 - 3*w^2 + 47/3*w + 157/6], [449, 449, -7/6*w^3 + 3*w^2 + 25/3*w - 109/6], [461, 461, 1/6*w^3 + 2*w^2 - 4/3*w - 77/6], [461, 461, 2/3*w^3 + 2*w^2 - 22/3*w - 55/3], [479, 479, 2/3*w^3 - 2*w^2 - 13/3*w + 32/3], [479, 479, -7/6*w^3 - 2*w^2 + 31/3*w + 113/6], [491, 491, 1/3*w^3 + 2*w^2 - 11/3*w - 35/3], [491, 491, 1/6*w^3 + 2*w^2 - 7/3*w - 107/6], [499, 499, 4/3*w^3 + 2*w^2 - 38/3*w - 53/3], [499, 499, 5/6*w^3 - 4*w^2 - 2/3*w + 101/6], [509, 509, -2/3*w^3 + 31/3*w + 40/3], [509, 509, 1/6*w^3 - 19/3*w + 85/6], [521, 521, -2/3*w^3 - 2*w^2 + 19/3*w + 52/3], [521, 521, -w^2 - 2*w + 10], [529, 23, -1/3*w^3 + 2*w^2 + 5/3*w - 49/3], [529, 23, 5/6*w^3 + 2*w^2 - 23/3*w - 79/6], [541, 541, -1/6*w^3 + 10/3*w - 13/6], [541, 541, 1/6*w^3 - 10/3*w - 11/6], [569, 569, 1/2*w^3 + w^2 - 5*w - 9/2], [569, 569, -1/6*w^3 + w^2 + 1/3*w - 61/6], [599, 599, 1/6*w^3 + w^2 - 1/3*w - 65/6], [599, 599, -1/6*w^3 + w^2 + 7/3*w - 25/6], [619, 619, 7/2*w^3 + 6*w^2 - 33*w - 113/2], [619, 619, -7/6*w^3 - 2*w^2 + 31/3*w + 89/6], [631, 631, -1/2*w^3 + 3*w^2 + w - 29/2], [631, 631, -1/6*w^3 + w^2 - 8/3*w + 29/6], [631, 631, -2/3*w^3 + 25/3*w + 16/3], [631, 631, -11/6*w^3 - 3*w^2 + 47/3*w + 145/6], [659, 659, 5/6*w^3 - 3*w^2 - 20/3*w + 155/6], [659, 659, 7/6*w^3 - 2*w^2 - 16/3*w + 49/6], [659, 659, -7/6*w^3 + 34/3*w + 29/6], [659, 659, -5/6*w^3 + w^2 + 14/3*w - 35/6], [661, 661, 1/3*w^3 - 20/3*w + 28/3], [661, 661, 1/3*w^3 - 20/3*w - 26/3], [691, 691, 1/2*w^3 + 2*w^2 - 5*w - 35/2], [691, 691, 2*w^2 - 15], [691, 691, 1/2*w^3 + 2*w^2 - 6*w - 31/2], [691, 691, 2*w^2 - w - 12], [701, 701, 1/3*w^3 + 2*w^2 - 11/3*w - 53/3], [701, 701, 1/6*w^3 + 2*w^2 - 7/3*w - 71/6], [709, 709, -11/6*w^3 - 3*w^2 + 50/3*w + 151/6], [709, 709, 7/6*w^3 + w^2 - 25/3*w - 71/6], [709, 709, 7/6*w^3 - w^2 - 31/3*w + 19/6], [709, 709, 13/6*w^3 + 4*w^2 - 58/3*w - 209/6], [739, 739, 7/6*w^3 - 4*w^2 - 28/3*w + 199/6], [739, 739, 5/3*w^3 - 5*w^2 - 40/3*w + 119/3], [739, 739, 5/6*w^3 - 4*w^2 - 8/3*w + 107/6], [739, 739, -5/6*w^3 + w^2 + 26/3*w - 35/6], [809, 809, 11/6*w^3 + 3*w^2 - 56/3*w - 175/6], [809, 809, -5/6*w^3 + 2*w^2 + 17/3*w - 89/6], [809, 809, -4/3*w^3 + 4*w^2 + 14/3*w - 49/3], [809, 809, -5/6*w^3 + w^2 + 11/3*w - 23/6], [811, 811, -1/3*w^3 + 17/3*w + 17/3], [811, 811, -1/6*w^3 + 13/3*w - 37/6], [821, 821, w^3 - w^2 - 8*w + 3], [821, 821, -2*w^3 - 3*w^2 + 18*w + 28], [821, 821, 7/6*w^3 + w^2 - 28/3*w - 71/6], [821, 821, 1/2*w^3 - w^2 - 7*w + 35/2], [829, 829, 5/6*w^3 + w^2 - 20/3*w - 73/6], [829, 829, 1/6*w^3 + w^2 - 1/3*w - 71/6], [839, 839, 1/2*w^3 - 10*w + 33/2], [839, 839, 1/2*w^3 - 10*w - 31/2], [841, 29, 5/6*w^3 - 20/3*w - 19/6], [841, 29, 1/6*w^3 - 4/3*w - 35/6], [859, 859, -1/6*w^3 + 2*w^2 + 7/3*w - 121/6], [859, 859, -2*w^3 - 4*w^2 + 16*w + 27], [859, 859, 1/2*w^3 - w^2 - 4*w - 1/2], [859, 859, 4/3*w^3 - 4*w^2 - 32/3*w + 97/3], [881, 881, -5/6*w^3 + w^2 + 14/3*w - 11/6], [881, 881, 4/3*w^3 + w^2 - 38/3*w - 38/3], [911, 911, 2*w^2 - 3*w - 4], [911, 911, -2/3*w^3 + 19/3*w - 20/3], [911, 911, 2/3*w^3 + w^2 - 22/3*w - 46/3], [911, 911, -2/3*w^3 + 4*w^2 - 2/3*w - 35/3], [919, 919, -1/6*w^3 + 3*w^2 - 11/3*w - 43/6], [919, 919, -13/6*w^3 - 4*w^2 + 55/3*w + 179/6], [919, 919, 5/6*w^3 - 2*w^2 - 8/3*w + 47/6], [919, 919, 4/3*w^3 - 4*w^2 - 29/3*w + 88/3], [929, 929, -7/6*w^3 + 3*w^2 + 10/3*w - 61/6], [929, 929, -1/2*w^3 + 2*w^2 + 4*w - 25/2], [929, 929, -2/3*w^3 + 22/3*w + 19/3], [929, 929, 2*w^2 - 4*w - 3], [961, 31, 5/6*w^3 - 20/3*w - 13/6], [961, 31, -5/6*w^3 + 20/3*w + 7/6], [971, 971, -7/6*w^3 + 6*w^2 + 4/3*w - 139/6], [971, 971, -2/3*w^3 + 2*w^2 + 1/3*w - 14/3], [991, 991, 7/3*w^3 + 4*w^2 - 59/3*w - 95/3], [991, 991, -1/6*w^3 + w^2 - 11/3*w + 47/6], [1009, 1009, -7/6*w^3 - w^2 + 31/3*w + 35/6], [1009, 1009, -5/6*w^3 + w^2 + 17/3*w - 53/6], [1021, 1021, -11/6*w^3 - 3*w^2 + 44/3*w + 139/6], [1021, 1021, 4/3*w^3 - 3*w^2 - 32/3*w + 64/3], [1021, 1021, -w^3 + 11*w + 3], [1021, 1021, 1/2*w^3 - w - 7/2], [1031, 1031, -w^3 + 6*w + 2], [1031, 1031, 4/3*w^3 - 38/3*w - 5/3], [1049, 1049, -w^3 + 4*w^2 + 3*w - 19], [1049, 1049, -5/2*w^3 - 4*w^2 + 25*w + 79/2], [1049, 1049, -2*w^3 - 4*w^2 + 18*w + 31], [1049, 1049, -w^3 + 4*w^2 + 6*w - 28], [1051, 1051, -3/2*w^3 - 2*w^2 + 13*w + 33/2], [1051, 1051, 1/6*w^3 - w^2 + 5/3*w - 17/6], [1061, 1061, 7/6*w^3 - 5*w^2 - 13/3*w + 157/6], [1061, 1061, -1/2*w^3 + 9*w + 23/2], [1091, 1091, -4/3*w^3 + 2*w^2 + 17/3*w - 16/3], [1091, 1091, -1/6*w^3 + 2*w^2 - 8/3*w - 13/6], [1109, 1109, -1/6*w^3 + 2*w^2 - 2/3*w - 91/6], [1109, 1109, -5/6*w^3 - 2*w^2 + 26/3*w + 85/6], [1151, 1151, -1/2*w^3 + 5*w - 13/2], [1151, 1151, -7/6*w^3 - w^2 + 31/3*w + 59/6], [1171, 1171, 5/6*w^3 - w^2 - 17/3*w + 23/6], [1171, 1171, 7/6*w^3 + w^2 - 31/3*w - 65/6], [1181, 1181, w^2 - 6*w + 10], [1181, 1181, -7/6*w^3 - w^2 + 46/3*w + 143/6], [1229, 1229, -5/3*w^3 - 3*w^2 + 46/3*w + 73/3], [1229, 1229, 1/6*w^3 - w^2 - 13/3*w + 109/6], [1229, 1229, 5/6*w^3 - 3*w^2 - 14/3*w + 119/6], [1229, 1229, -1/6*w^3 + w^2 + 13/3*w + 17/6], [1231, 1231, -2/3*w^3 + 25/3*w - 8/3], [1231, 1231, -1/6*w^3 - 5/3*w - 13/6], [1231, 1231, 1/2*w^3 - w - 3/2], [1231, 1231, w^3 - 11*w - 1], [1249, 1249, 1/3*w^3 + 3*w^2 - 14/3*w - 89/3], [1249, 1249, 7/6*w^3 - 2*w^2 - 25/3*w + 43/6], [1279, 1279, -17/6*w^3 - 5*w^2 + 80/3*w + 265/6], [1279, 1279, 4/3*w^3 - 5*w^2 - 20/3*w + 88/3], [1289, 1289, 5/6*w^3 + 2*w^2 - 23/3*w - 127/6], [1289, 1289, 1/2*w^3 + w^2 - 2*w - 21/2], [1291, 1291, -1/3*w^3 + 26/3*w - 46/3], [1291, 1291, -2/3*w^3 + 34/3*w + 43/3], [1301, 1301, 19/6*w^3 + 5*w^2 - 88/3*w - 275/6], [1301, 1301, -5/3*w^3 + 5*w^2 + 28/3*w - 83/3], [1321, 1321, -2*w^3 - 4*w^2 + 19*w + 34], [1321, 1321, -7/6*w^3 + 2*w^2 + 28/3*w - 97/6], [1321, 1321, 5/3*w^3 + 2*w^2 - 46/3*w - 64/3], [1321, 1321, 5/6*w^3 - 4*w^2 - 11/3*w + 149/6], [1381, 1381, -7/6*w^3 - 3*w^2 + 34/3*w + 167/6], [1381, 1381, 1/3*w^3 - 3*w^2 - 2/3*w + 49/3], [1399, 1399, 7/6*w^3 - 4*w^2 - 22/3*w + 163/6], [1399, 1399, 13/6*w^3 + 4*w^2 - 58/3*w - 191/6], [1409, 1409, -3/2*w^3 + 4*w^2 + 10*w - 45/2], [1409, 1409, 2*w^2 - 9], [1409, 1409, -5/2*w^3 - 4*w^2 + 22*w + 73/2], [1409, 1409, 1/3*w^3 + 2*w^2 - 8/3*w - 62/3], [1439, 1439, 7/6*w^3 - 37/3*w + 7/6], [1439, 1439, -1/2*w^3 - 3*w^2 + 7*w + 33/2], [1459, 1459, 2*w^2 - 2*w - 17], [1459, 1459, -5/6*w^3 + 26/3*w + 55/6], [1459, 1459, 2/3*w^3 + 2*w^2 - 22/3*w - 37/3], [1459, 1459, -1/2*w^3 + 2*w + 19/2], [1511, 1511, 3/2*w^3 - 6*w^2 - 8*w + 75/2], [1511, 1511, -19/6*w^3 - 6*w^2 + 88/3*w + 305/6], [1531, 1531, 5/2*w^3 + 3*w^2 - 29*w - 87/2], [1531, 1531, -13/6*w^3 - 3*w^2 + 61/3*w + 179/6], [1549, 1549, 1/2*w^3 + 2*w^2 - 6*w - 21/2], [1549, 1549, w^3 - 10*w + 2], [1579, 1579, -1/2*w^3 + w^2 + 3*w - 21/2], [1579, 1579, 5/6*w^3 + w^2 - 23/3*w - 25/6], [1601, 1601, 1/3*w^3 + w^2 - 20/3*w - 5/3], [1601, 1601, -7/6*w^3 - 3*w^2 + 31/3*w + 131/6], [1609, 1609, 5/6*w^3 - w^2 - 23/3*w + 23/6], [1609, 1609, -11/6*w^3 - 3*w^2 + 50/3*w + 175/6], [1609, 1609, -5/6*w^3 - w^2 + 17/3*w + 67/6], [1609, 1609, -w^3 + 3*w^2 + 6*w - 15], [1619, 1619, 1/3*w^3 + 2*w^2 - 5/3*w - 53/3], [1619, 1619, 2/3*w^3 - w^2 - 16/3*w + 2/3], [1619, 1619, 3/2*w^3 + 2*w^2 - 12*w - 39/2], [1619, 1619, 1/3*w^3 - 2*w^2 - 14/3*w + 70/3], [1621, 1621, 1/6*w^3 + 3*w^2 - 4/3*w - 113/6], [1621, 1621, -1/3*w^3 - 3*w^2 + 8/3*w + 77/3], [1681, 41, w^3 - 8*w - 4], [1699, 1699, 5/6*w^3 + w^2 - 29/3*w - 37/6], [1699, 1699, -1/6*w^3 + w^2 - 5/3*w - 49/6], [1709, 1709, -1/6*w^3 - 2*w^2 + 13/3*w + 35/6], [1709, 1709, -3/2*w^3 - w^2 + 14*w + 15/2], [1721, 1721, -5/6*w^3 - w^2 + 35/3*w + 103/6], [1721, 1721, 5/3*w^3 + 2*w^2 - 43/3*w - 61/3], [1721, 1721, -7/6*w^3 + 2*w^2 + 25/3*w - 55/6], [1721, 1721, -1/6*w^3 - w^2 + 19/3*w - 19/6], [1741, 1741, 1/6*w^3 - 3*w^2 + 5/3*w + 115/6], [1741, 1741, -1/2*w^3 + 4*w^2 - w - 25/2], [1741, 1741, 7/6*w^3 + 3*w^2 - 37/3*w - 149/6], [1741, 1741, -11/6*w^3 + 6*w^2 + 17/3*w - 131/6], [1789, 1789, 1/6*w^3 - w^2 - 13/3*w + 103/6], [1789, 1789, w^3 + 2*w^2 - 8*w - 18], [1789, 1789, -2*w^3 - 3*w^2 + 18*w + 30], [1789, 1789, -2/3*w^3 + 2*w^2 + 16/3*w - 35/3], [1801, 1801, -7/6*w^3 - w^2 + 28/3*w + 47/6], [1801, 1801, -w^3 + w^2 + 8*w - 7], [1811, 1811, 4/3*w^3 - 29/3*w - 26/3], [1811, 1811, -7/6*w^3 + w^2 + 31/3*w - 43/6], [1831, 1831, -1/3*w^3 + 2/3*w + 32/3], [1831, 1831, -3/2*w^3 - 2*w^2 + 12*w + 33/2], [1849, 43, -7/6*w^3 - 2*w^2 + 31/3*w + 125/6], [1849, 43, -1/2*w^3 + 3*w^2 + 5*w - 43/2], [1871, 1871, -3/2*w^3 - 2*w^2 + 12*w + 29/2], [1871, 1871, 7/6*w^3 - 2*w^2 - 28/3*w + 91/6], [1879, 1879, 3/2*w^3 + 3*w^2 - 13*w - 53/2], [1879, 1879, -5/6*w^3 + 3*w^2 + 17/3*w - 107/6], [1889, 1889, -1/2*w^3 - w^2 + 6*w + 5/2], [1889, 1889, w^2 - 2*w - 12], [1901, 1901, w^2 + 2*w + 2], [1901, 1901, 1/2*w^3 - 3*w^2 + w + 13/2], [1901, 1901, -1/6*w^3 + 4*w^2 - 11/3*w - 73/6], [1901, 1901, 1/6*w^3 - w^2 - 10/3*w + 103/6], [1931, 1931, 7/6*w^3 - w^2 - 28/3*w + 13/6], [1931, 1931, 4/3*w^3 + w^2 - 32/3*w - 38/3], [1931, 1931, -7/6*w^3 - 3*w^2 + 31/3*w + 137/6], [1931, 1931, 1/2*w^3 - 3*w^2 - 3*w + 43/2], [1949, 1949, 4/3*w^3 - 3*w^2 - 32/3*w + 58/3], [1949, 1949, 11/6*w^3 + 3*w^2 - 44/3*w - 151/6], [1951, 1951, 11/6*w^3 - 5*w^2 - 20/3*w + 113/6], [1951, 1951, -4/3*w^3 + 26/3*w - 1/3], [1999, 1999, -1/6*w^3 + 3*w^2 - 2/3*w - 151/6], [1999, 1999, 1/2*w^3 + 2*w^2 - 4*w - 49/2]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-3, 1, 1, -2, -2, 4, -4, 6, -2, -4, -4, -2, -2, -6, 2, 0, 16, 8, -8, -2, -10, 10, 10, 2, 18, -18, -4, 12, -4, 4, -10, 8, 8, -18, -10, -10, 6, 0, 8, -8, -16, -8, -16, -4, -4, 6, 6, 6, -26, -24, 0, 14, -10, -8, -8, 14, 6, -22, 10, -8, 0, 4, 12, -12, 20, 26, 2, -18, 22, -12, -20, 4, 4, 6, -26, 2, -2, -30, -34, -6, -38, -24, -16, 18, 34, -14, -30, 40, 0, 36, 12, -20, -44, 26, 18, -6, -6, 34, 2, 2, 18, -10, -10, -32, 40, -20, 28, 40, -8, -32, -32, 36, -20, -12, 12, -22, -30, -12, 20, 28, -44, -46, -22, -14, -26, -10, 10, -20, 4, -36, -4, 42, 10, -6, 10, -28, -28, -42, 34, 38, -54, -2, -2, -24, 0, -22, 10, 36, 20, 4, -4, -2, 6, -24, 32, -16, 24, -32, -56, -48, 48, -50, -30, 18, 30, -34, 6, 52, 12, -56, -16, 18, -30, -38, -22, 14, -34, 32, 24, 42, -54, 30, -26, 28, 20, -58, 38, 20, -20, 42, -30, 0, -64, 4, 52, -2, -50, 46, 14, -2, -18, -24, 40, -48, -56, -2, -26, 8, -24, -6, 26, -4, -44, -10, -42, -70, 26, -22, 10, -26, 22, 16, 40, -50, -46, 6, 66, 16, 72, 4, 52, 12, -68, 24, 32, 28, -4, 30, 46, -20, 4, -50, 30, 10, -18, -6, -34, 60, -20, -60, -36, 54, -42, 50, 20, 44, 74, 34, -66, -2, -18, 14, 14, -62, 62, 18, 50, -34, 10, -2, -6, -6, 12, -68, -40, -32, 30, 62, 0, 8, -24, 64, 66, -62, 30, 46, 30, -50, 28, -68, 44, -28, 62, -66, 80, -8, -16, -32]; 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;