/* 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![20, 10, -11, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -w + 2], [4, 2, 1/2*w^3 + 1/2*w^2 - 9/2*w - 4], [5, 5, -1/2*w^3 - 3/2*w^2 + 7/2*w + 10], [5, 5, -1/2*w^3 - 1/2*w^2 + 5/2*w + 3], [19, 19, 1/2*w^3 + 1/2*w^2 - 9/2*w - 3], [19, 19, w - 1], [31, 31, 1/2*w^3 + 3/2*w^2 - 7/2*w - 6], [31, 31, -w^3 - 2*w^2 + 7*w + 13], [49, 7, -2*w^3 - 2*w^2 + 15*w + 9], [49, 7, 1/2*w^3 + 3/2*w^2 - 9/2*w - 4], [59, 59, -3/2*w^3 - 3/2*w^2 + 17/2*w + 8], [59, 59, -1/2*w^3 - 5/2*w^2 + 11/2*w + 9], [61, 61, -1/2*w^3 - 3/2*w^2 + 9/2*w + 9], [61, 61, -3/2*w^3 - 5/2*w^2 + 23/2*w + 12], [71, 71, -3/2*w^3 - 1/2*w^2 + 23/2*w - 1], [71, 71, 3/2*w^3 + 3/2*w^2 - 19/2*w - 9], [79, 79, -1/2*w^3 - 3/2*w^2 + 11/2*w + 4], [79, 79, -5/2*w^3 - 7/2*w^2 + 37/2*w + 18], [81, 3, -3], [89, 89, w^3 - 7*w + 1], [89, 89, -2*w^3 - 3*w^2 + 14*w + 17], [89, 89, 1/2*w^3 - 1/2*w^2 - 7/2*w + 2], [89, 89, -5/2*w^3 - 7/2*w^2 + 35/2*w + 18], [101, 101, -2*w^2 + 11], [101, 101, 3*w^3 + 5*w^2 - 21*w - 27], [121, 11, -3/2*w^3 - 3/2*w^2 + 21/2*w + 7], [121, 11, 3/2*w^3 + 3/2*w^2 - 21/2*w - 8], [131, 131, -3/2*w^3 - 5/2*w^2 + 21/2*w + 16], [131, 131, -3/2*w^3 - 5/2*w^2 + 19/2*w + 14], [139, 139, -2*w^3 - w^2 + 16*w + 1], [139, 139, 1/2*w^3 + 1/2*w^2 - 3/2*w - 8], [149, 149, -3/2*w^3 - 7/2*w^2 + 21/2*w + 24], [149, 149, -w^3 + w^2 + 5*w - 3], [149, 149, -5*w^3 - 7*w^2 + 37*w + 39], [149, 149, -3/2*w^3 - 7/2*w^2 + 21/2*w + 14], [181, 181, -5/2*w^3 - 9/2*w^2 + 37/2*w + 27], [181, 181, -5/2*w^3 - 7/2*w^2 + 33/2*w + 19], [181, 181, 3/2*w^3 + 1/2*w^2 - 23/2*w - 2], [181, 181, w^3 + 3*w^2 - 8*w - 13], [191, 191, 1/2*w^3 + 5/2*w^2 - 7/2*w - 14], [191, 191, -5/2*w^3 - 9/2*w^2 + 35/2*w + 24], [199, 199, -1/2*w^3 - 5/2*w^2 + 7/2*w + 16], [199, 199, 5/2*w^3 + 9/2*w^2 - 35/2*w - 22], [211, 211, -2*w^3 - 4*w^2 + 14*w + 19], [211, 211, -w^3 - 3*w^2 + 7*w + 19], [229, 229, 3/2*w^3 + 7/2*w^2 - 21/2*w - 17], [229, 229, -3/2*w^3 - 7/2*w^2 + 21/2*w + 21], [229, 229, -1/2*w^3 - 3/2*w^2 + 13/2*w + 1], [229, 229, 5/2*w^3 + 7/2*w^2 - 41/2*w - 24], [241, 241, -1/2*w^3 - 5/2*w^2 + 7/2*w + 12], [241, 241, 3/2*w^3 + 3/2*w^2 - 23/2*w - 3], [241, 241, w^3 + w^2 - 6*w - 1], [241, 241, -5/2*w^3 - 9/2*w^2 + 35/2*w + 26], [251, 251, 2*w^3 + 2*w^2 - 13*w - 9], [251, 251, -5/2*w^3 - 5/2*w^2 + 37/2*w + 11], [269, 269, -2*w^3 - 3*w^2 + 12*w + 17], [269, 269, 5/2*w^3 + 11/2*w^2 - 37/2*w - 21], [271, 271, 3/2*w^3 + 5/2*w^2 - 23/2*w - 9], [271, 271, 1/2*w^3 + 3/2*w^2 - 9/2*w - 12], [281, 281, 5/2*w^3 + 9/2*w^2 - 39/2*w - 28], [281, 281, 3/2*w^3 + 7/2*w^2 - 25/2*w - 14], [311, 311, 1/2*w^3 + 7/2*w^2 - 5/2*w - 24], [311, 311, 3*w^3 + 5*w^2 - 22*w - 27], [311, 311, 3/2*w^3 + 5/2*w^2 - 17/2*w - 12], [311, 311, 1/2*w^3 + 5/2*w^2 - 9/2*w - 13], [349, 349, 3*w^3 + 4*w^2 - 21*w - 21], [349, 349, -3/2*w^3 - 1/2*w^2 + 21/2*w + 2], [359, 359, -w^3 - 3*w^2 + 7*w + 17], [359, 359, -2*w^3 - 4*w^2 + 14*w + 21], [361, 19, -2*w^3 - 2*w^2 + 14*w + 11], [389, 389, -2*w^3 - 2*w^2 + 13*w + 11], [389, 389, 5/2*w^3 + 5/2*w^2 - 37/2*w - 13], [401, 401, 3/2*w^3 + 7/2*w^2 - 21/2*w - 19], [401, 401, 5/2*w^3 + 9/2*w^2 - 37/2*w - 23], [401, 401, -w^3 - 3*w^2 + 8*w + 17], [431, 431, 5/2*w^3 + 5/2*w^2 - 29/2*w - 13], [431, 431, -15/2*w^3 - 23/2*w^2 + 107/2*w + 61], [439, 439, -3/2*w^3 + 1/2*w^2 + 21/2*w - 6], [439, 439, -9/2*w^3 - 13/2*w^2 + 63/2*w + 32], [449, 449, -w^2 - 2*w + 1], [449, 449, -1/2*w^3 - 3/2*w^2 + 3/2*w + 14], [461, 461, -2*w^3 - 2*w^2 + 12*w + 9], [461, 461, -3*w^3 - 3*w^2 + 23*w + 13], [479, 479, 7/2*w^3 + 9/2*w^2 - 47/2*w - 26], [479, 479, 3/2*w^3 + 5/2*w^2 - 27/2*w - 16], [479, 479, -7/2*w^3 - 9/2*w^2 + 51/2*w + 24], [479, 479, 3/2*w^3 + 1/2*w^2 - 19/2*w - 3], [491, 491, 2*w^3 + 4*w^2 - 15*w - 23], [491, 491, -7/2*w^3 - 7/2*w^2 + 53/2*w + 17], [491, 491, -1/2*w^3 - 5/2*w^2 + 11/2*w + 6], [491, 491, -3/2*w^3 - 7/2*w^2 + 23/2*w + 17], [499, 499, 3*w^2 - 19], [499, 499, 3/2*w^3 + 5/2*w^2 - 19/2*w - 16], [499, 499, -9/2*w^3 - 15/2*w^2 + 63/2*w + 38], [499, 499, -1/2*w^3 + 1/2*w^2 + 9/2*w - 1], [509, 509, 2*w^2 - w - 7], [509, 509, 7/2*w^3 + 11/2*w^2 - 51/2*w - 33], [521, 521, 4*w^3 + 5*w^2 - 32*w - 31], [521, 521, -1/2*w^3 + 1/2*w^2 - 1/2*w + 4], [541, 541, 1/2*w^3 - 7/2*w^2 - 3/2*w + 18], [541, 541, -15/2*w^3 - 23/2*w^2 + 109/2*w + 62], [541, 541, -w^3 - 3*w^2 + 7*w + 11], [541, 541, 2*w^3 + 4*w^2 - 14*w - 27], [569, 569, 1/2*w^3 + 5/2*w^2 - 7/2*w - 21], [569, 569, -5/2*w^3 - 5/2*w^2 + 31/2*w + 14], [571, 571, 2*w^2 + 2*w - 11], [571, 571, 2*w^3 + 4*w^2 - 12*w - 23], [601, 601, w^3 + 3*w^2 - 8*w - 9], [601, 601, 3*w^3 + 2*w^2 - 23*w - 7], [601, 601, -3/2*w^3 - 7/2*w^2 + 23/2*w + 19], [601, 601, -2*w^3 - 4*w^2 + 15*w + 21], [631, 631, -5/2*w^3 - 3/2*w^2 + 33/2*w + 3], [631, 631, -13/2*w^3 - 19/2*w^2 + 101/2*w + 58], [641, 641, 13/2*w^3 + 19/2*w^2 - 97/2*w - 52], [641, 641, -3/2*w^3 - 9/2*w^2 + 23/2*w + 16], [659, 659, 1/2*w^3 - 1/2*w^2 - 1/2*w + 2], [659, 659, 3/2*w^3 + 7/2*w^2 - 15/2*w - 11], [661, 661, -3/2*w^3 - 3/2*w^2 + 25/2*w + 3], [661, 661, -1/2*w^3 - 1/2*w^2 + 3/2*w - 1], [691, 691, -2*w^3 - 7*w^2 + 14*w + 49], [691, 691, -1/2*w^3 - 7/2*w^2 + 13/2*w + 14], [691, 691, -13/2*w^3 - 19/2*w^2 + 101/2*w + 59], [691, 691, -7/2*w^3 - 13/2*w^2 + 49/2*w + 36], [701, 701, 3/2*w^3 + 11/2*w^2 - 21/2*w - 38], [701, 701, -2*w^3 - 2*w^2 + 10*w + 11], [709, 709, 1/2*w^3 + 5/2*w^2 - 9/2*w - 17], [709, 709, -3*w^3 - 3*w^2 + 22*w + 13], [709, 709, -3*w^3 - 5*w^2 + 22*w + 23], [709, 709, -5/2*w^3 - 5/2*w^2 + 33/2*w + 11], [751, 751, -3/2*w^3 + 3/2*w^2 + 15/2*w - 4], [751, 751, 15/2*w^3 + 21/2*w^2 - 111/2*w - 59], [751, 751, 1/2*w^3 - 1/2*w^2 - 9/2*w - 2], [751, 751, -3/2*w^3 - 5/2*w^2 + 19/2*w + 19], [769, 769, -9/2*w^3 - 15/2*w^2 + 65/2*w + 38], [769, 769, -1/2*w^3 - 7/2*w^2 + 9/2*w + 21], [809, 809, -7/2*w^3 - 13/2*w^2 + 51/2*w + 42], [809, 809, -5/2*w^3 - 3/2*w^2 + 27/2*w + 14], [809, 809, -3*w^3 - 5*w^2 + 21*w + 29], [809, 809, 2*w^2 - 9], [811, 811, 3*w^3 + 5*w^2 - 20*w - 27], [811, 811, 1/2*w^3 - 7/2*w^2 - 9/2*w + 31], [811, 811, w^3 - w^2 - 2*w + 1], [811, 811, 3/2*w^3 + 5/2*w^2 - 19/2*w - 18], [821, 821, 9/2*w^3 + 13/2*w^2 - 67/2*w - 34], [821, 821, 3/2*w^3 + 3/2*w^2 - 25/2*w - 1], [829, 829, 1/2*w^3 + 5/2*w^2 - 1/2*w - 21], [829, 829, 4*w^3 + 2*w^2 - 32*w - 3], [839, 839, -1/2*w^3 - 1/2*w^2 + 3/2*w - 2], [839, 839, -3/2*w^3 - 3/2*w^2 + 25/2*w + 2], [841, 29, -5/2*w^3 - 5/2*w^2 + 35/2*w + 11], [841, 29, 5/2*w^3 + 5/2*w^2 - 35/2*w - 14], [881, 881, -9/2*w^3 - 13/2*w^2 + 63/2*w + 36], [881, 881, 1/2*w^3 + 5/2*w^2 - 3/2*w - 21], [911, 911, -1/2*w^3 + 3/2*w^2 + 7/2*w - 7], [911, 911, 7/2*w^3 + 13/2*w^2 - 49/2*w - 38], [911, 911, 7*w^3 + 11*w^2 - 51*w - 61], [911, 911, -7/2*w^3 - 11/2*w^2 + 49/2*w + 31], [919, 919, -1/2*w^3 + 7/2*w^2 + 3/2*w - 16], [919, 919, 15/2*w^3 + 23/2*w^2 - 109/2*w - 64], [929, 929, -3/2*w^3 - 7/2*w^2 + 19/2*w + 23], [929, 929, -5/2*w^3 - 7/2*w^2 + 33/2*w + 21], [941, 941, -1/2*w^3 - 7/2*w^2 + 11/2*w + 18], [941, 941, -5*w^3 - 8*w^2 + 37*w + 43], [961, 31, 5/2*w^3 + 5/2*w^2 - 35/2*w - 13], [971, 971, 2*w^3 + w^2 - 14*w - 3], [971, 971, -5/2*w^3 - 9/2*w^2 + 35/2*w + 31], [971, 971, 7/2*w^3 + 9/2*w^2 - 49/2*w - 22], [971, 971, 3/2*w^3 + 1/2*w^2 - 27/2*w + 6], [991, 991, 5/2*w^3 + 7/2*w^2 - 39/2*w - 16], [991, 991, 5/2*w^3 + 7/2*w^2 - 37/2*w - 14], [1009, 1009, 1/2*w^3 + 7/2*w^2 - 9/2*w - 18], [1009, 1009, 9/2*w^3 + 15/2*w^2 - 65/2*w - 41], [1019, 1019, 2*w^3 + 5*w^2 - 14*w - 21], [1019, 1019, 5/2*w^3 + 11/2*w^2 - 35/2*w - 36], [1021, 1021, 1/2*w^3 + 3/2*w^2 - 7/2*w - 2], [1021, 1021, w^3 + 2*w^2 - 7*w - 17], [1039, 1039, -3/2*w^3 - 5/2*w^2 + 27/2*w + 8], [1039, 1039, -7/2*w^3 - 13/2*w^2 + 51/2*w + 31], [1039, 1039, -3*w^3 - 4*w^2 + 25*w + 19], [1039, 1039, -4*w^3 - 6*w^2 + 29*w + 29], [1051, 1051, 3/2*w^3 + 7/2*w^2 - 27/2*w - 13], [1051, 1051, -7/2*w^3 - 9/2*w^2 + 53/2*w + 22], [1061, 1061, 1/2*w^3 + 3/2*w^2 - 1/2*w - 17], [1061, 1061, -3*w^3 - w^2 + 24*w - 3], [1069, 1069, -7/2*w^3 - 9/2*w^2 + 47/2*w + 23], [1069, 1069, 2*w^3 + 3*w^2 - 14*w - 23], [1129, 1129, 3*w^3 + 4*w^2 - 23*w - 19], [1129, 1129, 5/2*w^3 + 7/2*w^2 - 37/2*w - 13], [1151, 1151, w^3 + w^2 - 4*w - 13], [1151, 1151, -5*w^3 - 8*w^2 + 35*w + 43], [1229, 1229, -3*w^3 - 4*w^2 + 19*w + 23], [1229, 1229, -3/2*w^3 - 9/2*w^2 + 19/2*w + 31], [1249, 1249, 3*w^3 + 6*w^2 - 21*w - 23], [1249, 1249, 7/2*w^3 + 1/2*w^2 - 55/2*w + 7], [1279, 1279, 2*w^3 + 2*w^2 - 17*w - 9], [1279, 1279, -3*w^3 - 6*w^2 + 21*w + 29], [1319, 1319, 3/2*w^3 + 9/2*w^2 - 19/2*w - 22], [1319, 1319, -w^3 + w^2 + 9], [1319, 1319, 5/2*w^3 + 11/2*w^2 - 33/2*w - 33], [1319, 1319, 9*w^3 + 14*w^2 - 65*w - 77], [1321, 1321, 3*w^3 + 3*w^2 - 16*w - 19], [1321, 1321, -3*w^3 - 8*w^2 + 21*w + 53], [1361, 1361, -11*w^3 - 17*w^2 + 79*w + 89], [1361, 1361, -9/2*w^3 - 11/2*w^2 + 59/2*w + 24], [1409, 1409, -3/2*w^3 - 1/2*w^2 + 15/2*w + 3], [1409, 1409, -5*w^3 - 6*w^2 + 39*w + 31], [1409, 1409, 7/2*w^3 + 13/2*w^2 - 51/2*w - 27], [1409, 1409, 1/2*w^3 + 5/2*w^2 - 9/2*w - 21], [1429, 1429, -1/2*w^3 - 1/2*w^2 + 13/2*w - 1], [1429, 1429, 7/2*w^3 + 13/2*w^2 - 49/2*w - 34], [1429, 1429, 1/2*w^3 - 5/2*w^2 - 5/2*w + 17], [1429, 1429, 11/2*w^3 + 17/2*w^2 - 79/2*w - 42], [1439, 1439, -17/2*w^3 - 25/2*w^2 + 125/2*w + 71], [1439, 1439, 9/2*w^3 + 17/2*w^2 - 65/2*w - 51], [1451, 1451, 3/2*w^3 - 3/2*w^2 - 15/2*w + 7], [1451, 1451, -7/2*w^3 - 5/2*w^2 + 47/2*w + 7], [1459, 1459, 3/2*w^3 + 11/2*w^2 - 25/2*w - 18], [1459, 1459, -7/2*w^3 - 9/2*w^2 + 55/2*w + 22], [1459, 1459, 3*w^3 + 5*w^2 - 22*w - 21], [1459, 1459, w^3 + 7*w^2 - 5*w - 53], [1471, 1471, -1/2*w^3 - 9/2*w^2 + 7/2*w + 37], [1471, 1471, -21/2*w^3 - 31/2*w^2 + 151/2*w + 82], [1481, 1481, -11/2*w^3 - 15/2*w^2 + 81/2*w + 38], [1481, 1481, 3/2*w^3 - 1/2*w^2 - 17/2*w + 4], [1489, 1489, -7/2*w^3 - 7/2*w^2 + 45/2*w + 22], [1489, 1489, 9/2*w^3 + 9/2*w^2 - 67/2*w - 26], [1499, 1499, 7/2*w^3 + 9/2*w^2 - 45/2*w - 26], [1499, 1499, 7*w^3 + 11*w^2 - 49*w - 53], [1511, 1511, -7/2*w^3 - 9/2*w^2 + 51/2*w + 19], [1511, 1511, -3/2*w^3 - 1/2*w^2 + 19/2*w - 2], [1511, 1511, -5*w^3 - 3*w^2 + 39*w + 9], [1511, 1511, 1/2*w^3 - 9/2*w^2 - 5/2*w + 26], [1559, 1559, -9/2*w^3 - 13/2*w^2 + 65/2*w + 31], [1559, 1559, -3/2*w^3 - 5/2*w^2 + 27/2*w + 4], [1559, 1559, -6*w^3 - 8*w^2 + 43*w + 37], [1559, 1559, w^3 - w^2 - 6*w + 9], [1571, 1571, 9/2*w^3 + 17/2*w^2 - 65/2*w - 39], [1571, 1571, -2*w^3 - 6*w^2 + 15*w + 39], [1601, 1601, -3*w^3 - 5*w^2 + 24*w + 33], [1601, 1601, -7/2*w^3 - 9/2*w^2 + 51/2*w + 14], [1609, 1609, 4*w^3 + 5*w^2 - 28*w - 23], [1609, 1609, -3/2*w^3 - 3/2*w^2 + 13/2*w + 8], [1609, 1609, -13/2*w^3 - 19/2*w^2 + 91/2*w + 48], [1609, 1609, -5/2*w^3 - 3/2*w^2 + 35/2*w + 4], [1619, 1619, 5/2*w^3 + 7/2*w^2 - 29/2*w - 14], [1619, 1619, -11/2*w^3 - 17/2*w^2 + 75/2*w + 44], [1621, 1621, 2*w^3 + 2*w^2 - 16*w - 3], [1621, 1621, w^3 + w^2 - 5*w + 1], [1669, 1669, -6*w^3 - 6*w^2 + 46*w + 31], [1669, 1669, -1/2*w^3 - 9/2*w^2 + 15/2*w + 11], [1681, 41, 3*w^3 + 3*w^2 - 21*w - 13], [1681, 41, 3*w^3 + 3*w^2 - 21*w - 17], [1699, 1699, 7/2*w^3 + 7/2*w^2 - 39/2*w - 23], [1699, 1699, -7/2*w^3 - 17/2*w^2 + 49/2*w + 56], [1709, 1709, -1/2*w^3 + 1/2*w^2 + 13/2*w - 1], [1709, 1709, 1/2*w^3 + 3/2*w^2 - 1/2*w - 12], [1721, 1721, 5/2*w^3 + 9/2*w^2 - 39/2*w - 19], [1721, 1721, -3*w^3 - 3*w^2 + 23*w + 11], [1741, 1741, 5/2*w^3 + 9/2*w^2 - 31/2*w - 26], [1741, 1741, -3*w^3 - 5*w^2 + 21*w + 31], [1759, 1759, w^3 + 5*w^2 - 6*w - 33], [1759, 1759, 9/2*w^3 + 17/2*w^2 - 61/2*w - 41], [1789, 1789, 5/2*w^3 + 11/2*w^2 - 35/2*w - 28], [1789, 1789, 2*w^3 + 5*w^2 - 14*w - 29], [1801, 1801, w^3 + 3*w^2 - 4*w - 23], [1801, 1801, 3/2*w^3 + 9/2*w^2 - 21/2*w - 26], [1801, 1801, 3*w^3 + 6*w^2 - 21*w - 31], [1801, 1801, 11/2*w^3 + 17/2*w^2 - 79/2*w - 48], [1811, 1811, 1/2*w^3 + 3/2*w^2 - 5/2*w - 1], [1811, 1811, 1/2*w^3 + 3/2*w^2 - 5/2*w - 16], [1831, 1831, -9/2*w^3 - 13/2*w^2 + 63/2*w + 37], [1831, 1831, -w^3 - 3*w^2 + 5*w + 23], [1849, 43, 6*w^3 + 8*w^2 - 44*w - 43], [1849, 43, 1/2*w^3 - 3/2*w^2 + 1/2*w - 3], [1861, 1861, 5/2*w^3 + 7/2*w^2 - 33/2*w - 23], [1861, 1861, 5/2*w^3 + 9/2*w^2 - 33/2*w - 27], [1861, 1861, -2*w^3 + 13*w - 1], [1861, 1861, 2*w^3 + 3*w^2 - 12*w - 7], [1871, 1871, 3/2*w^3 + 5/2*w^2 - 15/2*w - 14], [1871, 1871, 3/2*w^3 + 1/2*w^2 - 27/2*w - 1], [1901, 1901, -4*w^3 - 5*w^2 + 28*w + 27], [1901, 1901, 5/2*w^3 + 3/2*w^2 - 35/2*w - 8], [1931, 1931, -2*w^3 + w^2 + 10*w + 1], [1931, 1931, 17/2*w^3 + 23/2*w^2 - 127/2*w - 66], [1949, 1949, -w^3 - 2*w^2 + 3*w + 17], [1949, 1949, -4*w^2 + 2*w + 17], [1949, 1949, -w^3 - 5*w^2 + 7*w + 27], [1949, 1949, -5*w^3 - 9*w^2 + 35*w + 49], [1951, 1951, -13/2*w^3 - 17/2*w^2 + 103/2*w + 52], [1951, 1951, 11/2*w^3 + 13/2*w^2 - 87/2*w - 41], [1979, 1979, -15/2*w^3 - 19/2*w^2 + 119/2*w + 59], [1979, 1979, 9/2*w^3 + 17/2*w^2 - 61/2*w - 47], [1999, 1999, 3*w^3 + 3*w^2 - 20*w - 17], [1999, 1999, 7/2*w^3 + 7/2*w^2 - 51/2*w - 19]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [3, -1, 2, -1, 4, 4, 0, 0, 2, 2, 4, 12, -10, -2, -8, 0, 0, -8, -14, 10, 14, 6, -6, 6, -10, -10, -2, 4, 4, -4, 4, -22, 14, 6, -6, -18, 2, -14, -2, -8, -24, 16, 0, 12, 20, -6, 26, 10, 2, 2, -2, -26, 18, -28, 12, 18, -14, -8, -8, -10, -10, 16, -16, -8, 0, -10, -2, 0, -8, 6, 22, 30, -14, -6, 10, -8, 8, 32, -40, -14, 26, 34, 2, 24, -24, 8, 0, -12, -12, 20, -28, -12, 20, -28, 20, -14, 34, -10, -10, 2, -30, 18, 2, -30, 42, 28, 20, 42, -6, 30, -26, 40, 8, -22, -46, 12, -12, -10, -18, 44, -20, -12, -20, -30, 2, -10, 10, 14, -6, 40, 8, -48, 32, 30, -50, -46, 26, -38, 26, 12, 12, 28, 20, -38, 10, -2, -50, -24, -56, 42, -38, -18, -50, 40, 40, -24, 56, 24, 16, -30, 34, -38, 42, -58, -36, -12, 28, 36, 0, 16, 30, 14, 36, 36, -14, -62, -40, 0, -16, 8, 4, -28, -58, -42, 14, -18, 22, 22, 8, -56, -26, -18, -26, -2, 40, 0, -32, 24, 24, -8, 22, 38, -18, 62, -30, 14, 14, -6, -62, -38, 34, 42, -40, 56, 36, 36, -4, 52, -52, -36, 40, 24, -18, 38, 10, 50, 60, -60, 0, 0, -72, 72, 16, 16, -40, -56, -60, 36, -50, -18, -34, -42, -58, -42, -12, 36, 6, 38, 18, 10, 10, -70, -28, 60, 50, -6, 6, -58, 22, -74, -8, -24, -82, 22, -58, -10, -18, 14, -52, -52, 32, 72, -34, -58, -6, -38, 70, 38, 32, 16, 66, 34, 28, 60, -18, 62, 10, 66, 72, 48, 60, 60, 64, 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;