/* 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^14 - 37*x^12 + 527*x^10 - 3683*x^8 + 13435*x^6 - 24687*x^4 + 18629*x^2 - 1369; K := NumberField(heckePol); heckeEigenvaluesArray := [-5897/36704*e^13 + 343/62*e^11 - 2586463/36704*e^9 + 1884593/4588*e^7 - 40499627/36704*e^5 + 1299314/1147*e^3 - 3380165/36704*e, e, -2869/18352*e^13 + 333/62*e^11 - 1251483/18352*e^9 + 453830/1147*e^7 - 19384047/18352*e^5 + 1233849/1147*e^3 - 1540633/18352*e, 1287/18352*e^13 - 599/248*e^11 + 564659/18352*e^9 - 205599/1147*e^7 + 8808669/18352*e^5 - 4452733/9176*e^3 + 442265/18352*e, -4459/36704*e^13 + 1035/248*e^11 - 1944225/36704*e^9 + 1408841/4588*e^7 - 30019249/36704*e^5 + 7588405/9176*e^3 - 2010811/36704*e, -205/992*e^12 + 220/31*e^10 - 89331/992*e^8 + 64733/124*e^6 - 1380439/992*e^4 + 87493/62*e^2 - 100449/992, -1, 401/992*e^12 - 3445/248*e^10 + 175019/992*e^8 - 127013/124*e^6 + 2715555/992*e^4 - 692227/248*e^2 + 213545/992, -383/992*e^12 + 3295/248*e^10 - 167733/992*e^8 + 122045/124*e^6 - 2616477/992*e^4 + 668485/248*e^2 - 205607/992, -1287/9176*e^13 + 599/124*e^11 - 564659/9176*e^9 + 411198/1147*e^7 - 8808669/9176*e^5 + 4457321/4588*e^3 - 524849/9176*e, -15957/36704*e^13 + 1851/124*e^11 - 6950947/36704*e^9 + 5036687/4588*e^7 - 107529551/36704*e^5 + 13749765/4588*e^3 - 9847249/36704*e, 49/62*e^12 - 1685/62*e^10 + 10711/31*e^8 - 124529/62*e^6 + 166626/31*e^4 - 170027/31*e^2 + 25701/62, 49/62*e^12 - 1685/62*e^10 + 10711/31*e^8 - 124529/62*e^6 + 166626/31*e^4 - 170027/31*e^2 + 25701/62, -107/496*e^12 + 1835/248*e^10 - 46487/496*e^8 + 33593/62*e^6 - 713129/496*e^4 + 359177/248*e^2 - 50597/496, -79/992*e^12 + 679/248*e^10 - 34485/992*e^8 + 24997/124*e^6 - 534125/992*e^4 + 137061/248*e^2 - 40295/992, -143/496*e^12 + 1233/124*e^10 - 62981/496*e^8 + 46071/62*e^6 - 995357/496*e^4 + 256135/124*e^2 - 72487/496, -205/992*e^12 + 220/31*e^10 - 89331/992*e^8 + 64733/124*e^6 - 1380439/992*e^4 + 87555/62*e^2 - 105409/992, -15/62*e^13 + 1031/124*e^11 - 6547/62*e^9 + 37987/62*e^7 - 101289/62*e^5 + 205183/124*e^3 - 6987/62*e, -3437/9176*e^13 + 797/62*e^11 - 1495023/9176*e^9 + 1080982/1147*e^7 - 22952527/9176*e^5 + 5761003/2294*e^3 - 1399157/9176*e, 6003/18352*e^13 - 2787/248*e^11 + 2618423/18352*e^9 - 1898659/2294*e^7 + 40510041/18352*e^5 - 20544101/9176*e^3 + 2994461/18352*e, -1603/36704*e^13 + 375/248*e^11 - 714689/36704*e^9 + 532793/4588*e^7 - 12048025/36704*e^5 + 3528825/9176*e^3 - 3258843/36704*e, -1203/992*e^12 + 10335/248*e^10 - 525057/992*e^8 + 380977/124*e^6 - 8139225/992*e^4 + 2071349/248*e^2 - 630219/992, 225/992*e^12 - 1937/248*e^10 + 98763/992*e^8 - 72113/124*e^6 + 1556659/992*e^4 - 402175/248*e^2 + 128489/992, 1741/18352*e^13 - 813/248*e^11 + 770321/18352*e^9 - 565735/2294*e^7 + 12297735/18352*e^5 - 6392091/9176*e^3 + 965995/18352*e, -4231/36704*e^13 + 987/248*e^11 - 1868469/36704*e^9 + 1371271/4588*e^7 - 29878773/36704*e^5 + 7945369/9176*e^3 - 4112007/36704*e, 347/992*e^12 - 2995/248*e^10 + 153161/992*e^8 - 112171/124*e^6 + 2425761/992*e^4 - 625837/248*e^2 + 190227/992, 23/496*e^12 - 101/62*e^10 + 10605/496*e^8 - 4042/31*e^6 + 185045/496*e^4 - 12769/31*e^2 + 19071/496, 53/248*e^12 - 226/31*e^10 + 22707/248*e^8 - 32387/62*e^6 + 337527/248*e^4 - 83547/62*e^2 + 27465/248, -3459/18352*e^13 + 200/31*e^11 - 1495813/18352*e^9 + 539094/1147*e^7 - 22917465/18352*e^5 + 2948067/2294*e^3 - 2724375/18352*e, -12823/36704*e^13 + 1495/124*e^11 - 5655121/36704*e^9 + 4140889/4588*e^7 - 89633509/36704*e^5 + 11629417/4588*e^3 - 8111259/36704*e, -8055/18352*e^13 + 468/31*e^11 - 3522745/18352*e^9 + 1279548/1147*e^7 - 54703309/18352*e^5 + 6937711/2294*e^3 - 3791451/18352*e, 227/248*e^12 - 3897/124*e^10 + 98863/248*e^8 - 143191/62*e^6 + 1525549/248*e^4 - 773821/124*e^2 + 116041/248, -721/992*e^12 + 773/31*e^10 - 313407/992*e^8 + 226617/124*e^6 - 4820835/992*e^4 + 305415/62*e^2 - 364709/992, 73/1184*e^13 - 17/8*e^11 + 32107/1184*e^9 - 23511/148*e^7 + 513963/1184*e^5 - 142263/296*e^3 + 120713/1184*e, -439/9176*e^13 + 417/248*e^11 - 50577/2294*e^9 + 307149/2294*e^7 - 3490449/9176*e^5 + 3917779/9176*e^3 - 81546/1147*e, 11/62*e^12 - 377/62*e^10 + 2383/31*e^8 - 13725/31*e^6 + 72493/62*e^4 - 73381/62*e^2 + 3495/31, -357/496*e^12 + 767/31*e^10 - 155955/496*e^8 + 113257/62*e^6 - 2421863/496*e^4 + 154045/31*e^2 - 181865/496, 42/31*e^12 - 2893/62*e^10 + 73717/124*e^8 - 214891/62*e^6 + 288563/31*e^4 - 295620/31*e^2 + 90487/124, 65/62*e^12 - 2239/62*e^10 + 57061/124*e^8 - 83168/31*e^6 + 223226/31*e^4 - 456431/62*e^2 + 69389/124, -399/992*e^12 + 1709/124*e^10 - 172873/992*e^8 + 124539/124*e^6 - 2630973/992*e^4 + 329341/124*e^2 - 191955/992, 49/124*e^12 - 1685/124*e^10 + 10711/62*e^8 - 31140/31*e^6 + 333717/124*e^4 - 341139/124*e^2 + 6185/31, 435/18352*e^13 - 103/124*e^11 + 199349/18352*e^9 - 151215/2294*e^7 + 3469713/18352*e^5 - 1041865/4588*e^3 + 1278447/18352*e, 12733/18352*e^13 - 1479/62*e^11 + 5563883/18352*e^9 - 4039913/2294*e^7 + 86310919/18352*e^5 - 10927215/2294*e^3 + 5485673/18352*e, 231/248*e^12 - 1987/62*e^10 + 101109/248*e^8 - 147011/62*e^6 + 1573069/248*e^4 - 200100/31*e^2 + 117743/248, -717/496*e^12 + 6161/124*e^10 - 313083/496*e^8 + 227249/62*e^6 - 4857263/496*e^4 + 1236999/124*e^2 - 377329/496, -10439/18352*e^13 + 2431/124*e^11 - 4588933/18352*e^9 + 1675283/1147*e^7 - 72273933/18352*e^5 + 18747207/4588*e^3 - 7286463/18352*e, 3587/18352*e^13 - 210/31*e^11 + 1599677/18352*e^9 - 1184661/2294*e^7 + 26131945/18352*e^5 - 1757499/1147*e^3 + 3363375/18352*e, 2755/18352*e^13 - 321/62*e^11 + 1213605/18352*e^9 - 888875/2294*e^7 + 19304633/18352*e^5 - 2538587/2294*e^3 + 2159959/18352*e, 1219/9176*e^13 - 143/31*e^11 + 545325/9176*e^9 - 806405/2294*e^7 + 8791081/9176*e^5 - 2246895/2294*e^3 + 282015/9176*e, 8455/36704*e^13 - 983/124*e^11 + 3703945/36704*e^9 - 2697551/4588*e^7 + 57978965/36704*e^5 - 7442939/4588*e^3 + 4695235/36704*e, -367/9176*e^13 + 179/124*e^11 - 180015/9176*e^9 + 286439/2294*e^7 - 3438361/9176*e^5 + 2019263/4588*e^3 - 535521/9176*e, -507/496*e^12 + 8729/248*e^10 - 222355/496*e^8 + 161973/62*e^6 - 3478153/496*e^4 + 1779123/248*e^2 - 264073/496, -687/496*e^12 + 11791/248*e^10 - 299059/496*e^8 + 216551/62*e^6 - 4615005/496*e^4 + 2342685/248*e^2 - 350769/496, 121/248*e^12 - 2089/124*e^10 + 53449/248*e^8 - 78417/62*e^6 + 850867/248*e^4 - 442361/124*e^2 + 72147/248, 2/31*e^12 - 277/124*e^10 + 1781/62*e^8 - 10537/62*e^6 + 14398/31*e^4 - 58917/124*e^2 + 1857/62, 383/992*e^12 - 3295/248*e^10 + 167733/992*e^8 - 122045/124*e^6 + 2615485/992*e^4 - 665261/248*e^2 + 189735/992, -1291/992*e^12 + 11089/248*e^10 - 563185/992*e^8 + 408427/124*e^6 - 8718673/992*e^4 + 2216623/248*e^2 - 679691/992, 251/496*e^12 - 541/31*e^10 + 110541/496*e^8 - 80901/62*e^6 + 1751025/496*e^4 - 113403/31*e^2 + 143055/496, -317/496*e^12 + 1357/62*e^10 - 137215/496*e^8 + 98869/62*e^6 - 2091743/496*e^4 + 262645/62*e^2 - 150461/496, -271/248*e^12 + 4651/124*e^10 - 117927/248*e^8 + 170641/62*e^6 - 1815273/248*e^4 + 919095/124*e^2 - 136809/248, 725/496*e^12 - 3119/62*e^10 + 317575/496*e^8 - 231069/62*e^6 + 4952055/496*e^4 - 631231/62*e^2 + 368581/496, 655/496*e^12 - 2817/62*e^10 + 286733/496*e^8 - 208587/62*e^6 + 4471685/496*e^4 - 570837/62*e^2 + 339943/496, 107/248*e^12 - 1835/124*e^10 + 46487/248*e^8 - 67217/62*e^6 + 714989/248*e^4 - 361967/124*e^2 + 52457/248, -1681/992*e^12 + 1804/31*e^10 - 732415/992*e^8 + 530513/124*e^6 - 11304323/992*e^4 + 716711/62*e^2 - 861477/992, -499/496*e^12 + 4303/124*e^10 - 219785/496*e^8 + 160695/62*e^6 - 3466937/496*e^4 + 891177/124*e^2 - 265443/496, -52097/36704*e^13 + 12105/248*e^11 - 22776395/36704*e^9 + 16548493/4588*e^7 - 354151683/36704*e^5 + 90302979/9176*e^3 - 27596505/36704*e, -1795/9176*e^13 + 1647/248*e^11 - 379613/4588*e^9 + 533643/1147*e^7 - 10804409/9176*e^5 + 9901537/9176*e^3 + 49409/4588*e, -1433/992*e^12 + 12293/248*e^10 - 623171/992*e^8 + 450657/124*e^6 - 9579483/992*e^4 + 2418987/248*e^2 - 695441/992, -1933/992*e^12 + 16625/248*e^10 - 845951/992*e^8 + 615193/124*e^6 - 13181463/992*e^4 + 3365175/248*e^2 - 1028037/992, 32975/36704*e^13 - 1915/62*e^11 + 14407761/36704*e^9 - 10463953/4588*e^7 + 223932653/36704*e^5 - 14315785/2294*e^3 + 19266747/36704*e, -17415/18352*e^13 + 4053/124*e^11 - 7643537/18352*e^9 + 5571837/2294*e^7 - 119779053/18352*e^5 + 30729623/4588*e^3 - 9693155/18352*e, -35913/36704*e^13 + 4167/124*e^11 - 15651503/36704*e^9 + 11338763/4588*e^7 - 241542123/36704*e^5 + 30522433/4588*e^3 - 16535381/36704*e, 85/4588*e^13 - 83/124*e^11 + 20973/2294*e^9 - 67876/1147*e^7 + 854707/4588*e^5 - 1143573/4588*e^3 + 166969/2294*e, 1287/4588*e^13 - 599/62*e^11 + 564659/4588*e^9 - 822396/1147*e^7 + 8806375/4588*e^5 - 4438969/2294*e^3 + 444559/4588*e, -6397/9176*e^13 + 2979/124*e^11 - 2810669/9176*e^9 + 4099839/2294*e^7 - 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140707/62*e^6 + 2999575/496*e^4 - 377813/62*e^2 + 196865/496, -371/124*e^12 + 12749/124*e^10 - 161925/124*e^8 + 235017/31*e^6 - 2512171/124*e^4 + 2561927/124*e^2 - 196037/124, 2003/992*e^12 - 8599/124*e^10 + 872949/992*e^8 - 632467/124*e^6 + 13479305/992*e^4 - 1709083/124*e^2 + 985623/992, 1649/496*e^12 - 14155/124*e^10 + 718291/496*e^8 - 260205/31*e^6 + 11102971/496*e^4 - 2824667/124*e^2 + 870057/496, -597/18352*e^13 + 71/62*e^11 - 277323/18352*e^9 + 215519/2294*e^7 - 5321175/18352*e^5 + 1003445/2294*e^3 - 5235553/18352*e, 7623/18352*e^13 - 442/31*e^11 + 3317873/18352*e^9 - 1200309/1147*e^7 + 50968133/18352*e^5 - 6343799/2294*e^3 + 2280587/18352*e, 2141/992*e^12 - 9205/124*e^10 + 936331/992*e^8 - 680103/124*e^6 + 14528567/992*e^4 - 1841727/124*e^2 + 1069049/992, 1239/248*e^12 - 10635/62*e^10 + 539629/248*e^8 - 390882/31*e^6 + 8333537/248*e^4 - 2115795/62*e^2 + 633571/248, 24627/18352*e^13 - 2855/62*e^11 + 10708701/18352*e^9 - 3870098/1147*e^7 + 164217817/18352*e^5 - 10270952/1147*e^3 + 8918543/18352*e, 70623/36704*e^13 - 8209/124*e^11 + 30911745/36704*e^9 - 22472443/4588*e^7 + 480597437/36704*e^5 - 60779557/4588*e^3 + 29389771/36704*e]; 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;