/* 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![44, 2, -15, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, w + 2], [4, 2, -1/2*w^3 + 3/2*w^2 + 9/2*w - 10], [9, 3, 1/2*w^3 - 5/2*w^2 - 5/2*w + 17], [9, 3, 3/2*w^3 - 11/2*w^2 - 19/2*w + 28], [11, 11, 1/2*w^3 - 3/2*w^2 - 9/2*w + 11], [11, 11, -w - 3], [25, 5, w^3 - 3*w^2 - 7*w + 15], [29, 29, w + 1], [29, 29, 1/2*w^3 - 3/2*w^2 - 9/2*w + 9], [31, 31, -1/2*w^3 + 5/2*w^2 + 5/2*w - 16], [31, 31, 1/2*w^3 - 3/2*w^2 - 9/2*w + 5], [31, 31, -w + 3], [31, 31, 3/2*w^3 - 11/2*w^2 - 19/2*w + 29], [59, 59, 2*w^2 - w - 13], [59, 59, 9/2*w^3 - 31/2*w^2 - 61/2*w + 85], [61, 61, 2*w^3 - 6*w^2 - 15*w + 31], [61, 61, -3/2*w^3 + 11/2*w^2 + 21/2*w - 34], [71, 71, 3/2*w^3 - 11/2*w^2 - 19/2*w + 32], [71, 71, -1/2*w^3 + 5/2*w^2 + 5/2*w - 13], [79, 79, -3*w^3 + 10*w^2 + 19*w - 51], [79, 79, 15/2*w^3 - 51/2*w^2 - 103/2*w + 139], [101, 101, -2*w^3 + 7*w^2 + 14*w - 37], [101, 101, -1/2*w^3 + 5/2*w^2 + 7/2*w - 16], [109, 109, w^3 - 3*w^2 - 6*w + 19], [109, 109, -w^3 + 4*w^2 + 5*w - 19], [121, 11, -3/2*w^3 + 9/2*w^2 + 21/2*w - 23], [139, 139, -w^3 + 4*w^2 + 7*w - 25], [139, 139, -5/2*w^3 + 17/2*w^2 + 33/2*w - 41], [149, 149, 1/2*w^3 + 1/2*w^2 - 11/2*w - 8], [149, 149, 5/2*w^3 - 19/2*w^2 - 35/2*w + 50], [149, 149, -9/2*w^3 + 31/2*w^2 + 59/2*w - 82], [149, 149, -5/2*w^3 + 19/2*w^2 + 35/2*w - 56], [151, 151, -1/2*w^3 + 3/2*w^2 + 3/2*w - 12], [151, 151, 3/2*w^3 - 9/2*w^2 - 25/2*w + 28], [179, 179, -1/2*w^3 + 3/2*w^2 + 9/2*w - 3], [179, 179, w - 5], [229, 229, -15/2*w^3 + 51/2*w^2 + 105/2*w - 142], [229, 229, 1/2*w^3 - 7/2*w^2 - 7/2*w + 19], [229, 229, -17/2*w^3 + 57/2*w^2 + 119/2*w - 156], [229, 229, -9/2*w^3 + 33/2*w^2 + 55/2*w - 82], [239, 239, -9*w^3 + 31*w^2 + 59*w - 161], [239, 239, w^3 + w^2 - 11*w - 19], [241, 241, 5/2*w^3 - 15/2*w^2 - 37/2*w + 37], [241, 241, -2*w^3 + 6*w^2 + 13*w - 29], [251, 251, -7/2*w^3 + 25/2*w^2 + 45/2*w - 67], [251, 251, 1/2*w^3 - 5/2*w^2 - 13/2*w + 7], [251, 251, -1/2*w^3 + 1/2*w^2 + 13/2*w + 4], [251, 251, -1/2*w^3 + 7/2*w^2 + 3/2*w - 23], [271, 271, -w^3 + 4*w^2 + 5*w - 25], [271, 271, -5/2*w^3 + 17/2*w^2 + 33/2*w - 47], [281, 281, 5/2*w^3 - 15/2*w^2 - 39/2*w + 41], [281, 281, 3/2*w^3 - 7/2*w^2 - 27/2*w + 14], [289, 17, -7/2*w^3 + 25/2*w^2 + 45/2*w - 62], [289, 17, 5/2*w^3 - 15/2*w^2 - 39/2*w + 37], [311, 311, -19/2*w^3 + 65/2*w^2 + 129/2*w - 175], [311, 311, -7/2*w^3 + 25/2*w^2 + 47/2*w - 71], [311, 311, 7/2*w^3 - 25/2*w^2 - 49/2*w + 67], [311, 311, -3/2*w^3 + 11/2*w^2 + 15/2*w - 31], [331, 331, 19/2*w^3 - 65/2*w^2 - 127/2*w + 173], [331, 331, -w^3 - w^2 + 10*w + 15], [349, 349, -5/2*w^3 + 15/2*w^2 + 33/2*w - 35], [349, 349, 3/2*w^3 - 11/2*w^2 - 21/2*w + 36], [361, 19, 2*w^3 - 6*w^2 - 14*w + 29], [361, 19, 2*w^3 - 6*w^2 - 14*w + 31], [379, 379, 1/2*w^3 - 5/2*w^2 - 1/2*w + 20], [379, 379, -1/2*w^3 + 5/2*w^2 + 1/2*w - 9], [401, 401, -7/2*w^3 + 25/2*w^2 + 45/2*w - 69], [401, 401, w^3 - 3*w^2 - 5*w + 19], [409, 409, 3/2*w^3 - 7/2*w^2 - 23/2*w + 13], [409, 409, -11/2*w^3 + 37/2*w^2 + 77/2*w - 100], [409, 409, 7/2*w^3 - 23/2*w^2 - 47/2*w + 58], [409, 409, -w^3 + 3*w^2 + 9*w - 9], [419, 419, 3/2*w^3 - 13/2*w^2 - 17/2*w + 36], [419, 419, -5/2*w^3 + 19/2*w^2 + 31/2*w - 54], [421, 421, -w^3 + 4*w^2 + 5*w - 27], [421, 421, 7/2*w^3 - 23/2*w^2 - 47/2*w + 63], [439, 439, 2*w^3 - 7*w^2 - 14*w + 35], [439, 439, -1/2*w^3 + 5/2*w^2 + 7/2*w - 18], [461, 461, -5/2*w^3 + 17/2*w^2 + 33/2*w - 48], [461, 461, -13/2*w^3 + 45/2*w^2 + 85/2*w - 119], [461, 461, -1/2*w^3 + 3/2*w^2 + 1/2*w - 15], [461, 461, 3/2*w^3 - 11/2*w^2 - 17/2*w + 32], [479, 479, -3*w^3 + 10*w^2 + 19*w - 53], [479, 479, 5/2*w^3 - 17/2*w^2 - 33/2*w + 51], [491, 491, 3*w^2 - 2*w - 23], [491, 491, 13/2*w^3 - 45/2*w^2 - 87/2*w + 120], [499, 499, -3*w^3 + 11*w^2 + 21*w - 63], [499, 499, -5/2*w^3 + 19/2*w^2 + 31/2*w - 50], [499, 499, 11/2*w^3 - 37/2*w^2 - 75/2*w + 101], [499, 499, 9*w^3 - 31*w^2 - 58*w + 159], [509, 509, 3/2*w^3 - 9/2*w^2 - 23/2*w + 29], [509, 509, w^3 - 3*w^2 - 6*w + 21], [521, 521, 2*w^3 - 8*w^2 - 12*w + 43], [521, 521, -3/2*w^3 + 11/2*w^2 + 19/2*w - 36], [521, 521, 2*w^3 - 8*w^2 - 12*w + 47], [521, 521, -7/2*w^3 + 23/2*w^2 + 45/2*w - 58], [541, 541, -1/2*w^3 + 5/2*w^2 + 13/2*w - 6], [541, 541, 9/2*w^3 - 33/2*w^2 - 55/2*w + 80], [541, 541, -15/2*w^3 + 51/2*w^2 + 101/2*w - 134], [541, 541, w^3 - w^2 - 7*w + 1], [571, 571, 13/2*w^3 - 47/2*w^2 - 79/2*w + 116], [571, 571, -5/2*w^3 + 11/2*w^2 + 43/2*w - 21], [599, 599, -3*w^3 + 9*w^2 + 19*w - 47], [599, 599, -15/2*w^3 + 51/2*w^2 + 103/2*w - 137], [599, 599, -w^3 + 2*w^2 + 11*w + 1], [599, 599, 5/2*w^3 - 17/2*w^2 - 35/2*w + 38], [601, 601, -1/2*w^3 + 11/2*w^2 - 5/2*w - 47], [601, 601, -3/2*w^3 + 13/2*w^2 + 17/2*w - 30], [619, 619, -w^3 + 5*w^2 + 5*w - 27], [619, 619, -1/2*w^3 + 1/2*w^2 + 11/2*w - 2], [619, 619, -2*w^3 + 7*w^2 + 12*w - 39], [619, 619, -3*w^3 + 11*w^2 + 19*w - 63], [631, 631, -3/2*w^3 + 13/2*w^2 + 17/2*w - 37], [631, 631, 5/2*w^3 - 19/2*w^2 - 31/2*w + 53], [641, 641, 5*w^3 - 16*w^2 - 35*w + 79], [641, 641, -2*w^3 + 8*w^2 + 12*w - 53], [641, 641, 3/2*w^3 - 11/2*w^2 - 23/2*w + 34], [641, 641, 3/2*w^3 - 11/2*w^2 - 23/2*w + 27], [659, 659, -5/2*w^3 + 19/2*w^2 + 31/2*w - 51], [659, 659, -3/2*w^3 + 13/2*w^2 + 17/2*w - 39], [661, 661, -5/2*w^3 + 17/2*w^2 + 33/2*w - 50], [661, 661, 5*w^3 - 17*w^2 - 30*w + 83], [661, 661, 5/2*w^3 - 17/2*w^2 - 31/2*w + 46], [661, 661, 3/2*w^3 - 5/2*w^2 - 21/2*w + 9], [691, 691, -3/2*w^3 + 13/2*w^2 + 19/2*w - 35], [691, 691, 3*w^3 - 11*w^2 - 20*w + 63], [709, 709, 19/2*w^3 - 65/2*w^2 - 127/2*w + 171], [709, 709, 4*w^3 - 14*w^2 - 28*w + 75], [709, 709, 1/2*w^3 - 5/2*w^2 - 15/2*w + 4], [709, 709, -w^3 + 5*w^2 + 7*w - 31], [739, 739, 11/2*w^3 - 37/2*w^2 - 73/2*w + 95], [739, 739, -3/2*w^3 + 5/2*w^2 + 25/2*w - 5], [739, 739, 27/2*w^3 - 91/2*w^2 - 187/2*w + 247], [739, 739, 4*w^3 - 15*w^2 - 24*w + 73], [751, 751, -5/2*w^3 + 19/2*w^2 + 35/2*w - 49], [751, 751, -2*w^2 + 2*w + 21], [751, 751, -9*w^3 + 31*w^2 + 61*w - 169], [751, 751, 17/2*w^3 - 57/2*w^2 - 117/2*w + 152], [761, 761, 5/2*w^3 - 15/2*w^2 - 33/2*w + 39], [761, 761, -3*w^3 + 9*w^2 + 22*w - 47], [761, 761, 9/2*w^3 - 29/2*w^2 - 61/2*w + 80], [761, 761, 25/2*w^3 - 85/2*w^2 - 169/2*w + 227], [769, 769, 2*w^3 - 8*w^2 - 12*w + 45], [811, 811, 5/2*w^3 - 19/2*w^2 - 27/2*w + 45], [811, 811, 1/2*w^3 - 7/2*w^2 + 1/2*w + 29], [821, 821, 25/2*w^3 - 85/2*w^2 - 173/2*w + 233], [821, 821, 19/2*w^3 - 63/2*w^2 - 135/2*w + 173], [829, 829, -7/2*w^3 + 27/2*w^2 + 43/2*w - 79], [829, 829, 10*w^3 - 34*w^2 - 67*w + 179], [829, 829, -3/2*w^3 + 1/2*w^2 + 27/2*w + 9], [829, 829, 5/2*w^3 - 13/2*w^2 - 41/2*w + 31], [839, 839, -3/2*w^3 + 11/2*w^2 + 25/2*w - 26], [839, 839, 2*w^3 - 7*w^2 - 16*w + 43], [841, 29, 5/2*w^3 - 15/2*w^2 - 35/2*w + 39], [881, 881, -8*w^3 + 27*w^2 + 56*w - 151], [881, 881, -w^3 + 3*w^2 + 4*w - 21], [911, 911, 5/2*w^3 - 19/2*w^2 - 33/2*w + 51], [911, 911, 2*w^3 - 8*w^2 - 13*w + 47], [919, 919, 3/2*w^3 - 9/2*w^2 - 27/2*w + 29], [919, 919, -3*w - 5], [919, 919, 1/2*w^3 - 7/2*w^2 - 5/2*w + 17], [919, 919, -4*w^3 + 14*w^2 + 27*w - 81], [929, 929, 7/2*w^3 - 23/2*w^2 - 49/2*w + 58], [929, 929, 13*w^3 - 44*w^2 - 89*w + 237], [929, 929, 9/2*w^3 - 29/2*w^2 - 59/2*w + 78], [929, 929, -w^3 + 2*w^2 + 7*w - 5], [941, 941, 5/2*w^3 - 19/2*w^2 - 31/2*w + 59], [941, 941, -3/2*w^3 + 13/2*w^2 + 17/2*w - 31], [941, 941, w^3 - 3*w^2 - 10*w + 7], [941, 941, -8*w^3 + 27*w^2 + 56*w - 147], [971, 971, -9/2*w^3 + 29/2*w^2 + 73/2*w - 92], [971, 971, -1/2*w^3 + 5/2*w^2 + 17/2*w - 1], [991, 991, w^3 - 4*w^2 - 3*w + 15], [991, 991, -1/2*w^3 + 1/2*w^2 + 15/2*w + 6], [1019, 1019, 3*w^3 - 9*w^2 - 19*w + 45], [1019, 1019, 3*w^3 - 11*w^2 - 20*w + 53], [1019, 1019, -1/2*w^3 + 1/2*w^2 + 17/2*w + 9], [1019, 1019, -1/2*w^3 + 5/2*w^2 - 3/2*w - 4], [1021, 1021, -3/2*w^3 + 13/2*w^2 + 19/2*w - 37], [1021, 1021, 3*w^3 - 11*w^2 - 20*w + 61], [1049, 1049, w^3 - 4*w^2 - 9*w + 15], [1049, 1049, 5/2*w^3 - 17/2*w^2 - 39/2*w + 54], [1051, 1051, 3/2*w^3 - 11/2*w^2 - 15/2*w + 21], [1051, 1051, -1/2*w^3 + 7/2*w^2 + 7/2*w - 23], [1051, 1051, 9/2*w^3 - 31/2*w^2 - 63/2*w + 83], [1051, 1051, -1/2*w^3 + 1/2*w^2 + 13/2*w + 8], [1061, 1061, 2*w^3 - 5*w^2 - 14*w + 19], [1061, 1061, 9/2*w^3 - 29/2*w^2 - 63/2*w + 72], [1091, 1091, 3/2*w^3 - 5/2*w^2 - 21/2*w + 10], [1091, 1091, 3/2*w^3 - 3/2*w^2 - 19/2*w + 7], [1091, 1091, -13/2*w^3 + 43/2*w^2 + 91/2*w - 116], [1091, 1091, 19/2*w^3 - 63/2*w^2 - 135/2*w + 174], [1109, 1109, -5/2*w^3 + 17/2*w^2 + 33/2*w - 53], [1109, 1109, 4*w^3 - 13*w^2 - 26*w + 67], [1109, 1109, -3/2*w^3 + 11/2*w^2 + 23/2*w - 25], [1109, 1109, 4*w^3 - 13*w^2 - 28*w + 65], [1171, 1171, 1/2*w^3 - 1/2*w^2 + 3/2*w + 12], [1171, 1171, 11/2*w^3 - 35/2*w^2 - 87/2*w + 105], [1181, 1181, 2*w^3 - 8*w^2 - 13*w + 45], [1181, 1181, -5/2*w^3 + 19/2*w^2 + 33/2*w - 53], [1201, 1201, -5/2*w^3 + 15/2*w^2 + 39/2*w - 45], [1201, 1201, -1/2*w^3 + 3/2*w^2 + 13/2*w - 3], [1201, 1201, -w^3 + 3*w^2 + 10*w - 21], [1201, 1201, 3/2*w^3 - 9/2*w^2 - 17/2*w + 29], [1229, 1229, -5*w^3 + 17*w^2 + 35*w - 91], [1229, 1229, 3/2*w^3 - 9/2*w^2 - 25/2*w + 17], [1249, 1249, 5/2*w^3 - 17/2*w^2 - 35/2*w + 40], [1249, 1249, 3/2*w^3 - 3/2*w^2 - 21/2*w + 4], [1249, 1249, w^2 - 13], [1249, 1249, 9*w^3 - 30*w^2 - 63*w + 163], [1289, 1289, 3/2*w^3 - 1/2*w^2 - 23/2*w - 3], [1289, 1289, 11*w^3 - 37*w^2 - 76*w + 201], [1291, 1291, -1/2*w^3 - 7/2*w^2 + 13/2*w + 29], [1291, 1291, 23/2*w^3 - 79/2*w^2 - 155/2*w + 212], [1301, 1301, 11*w^3 - 38*w^2 - 73*w + 201], [1301, 1301, 1/2*w^3 - 5/2*w^2 - 17/2*w + 3], [1319, 1319, 12*w^3 - 40*w^2 - 85*w + 219], [1319, 1319, 3/2*w^3 - 1/2*w^2 - 19/2*w - 1], [1321, 1321, -1/2*w^3 - 1/2*w^2 - 1/2*w - 7], [1321, 1321, -3*w^3 + 11*w^2 + 15*w - 45], [1361, 1361, -3*w^3 + 11*w^2 + 21*w - 57], [1361, 1361, 5/2*w^3 - 15/2*w^2 - 39/2*w + 35], [1381, 1381, -3*w^3 + 9*w^2 + 23*w - 43], [1381, 1381, 2*w^3 - 6*w^2 - 12*w + 27], [1381, 1381, -1/2*w^3 + 3/2*w^2 + 9/2*w - 1], [1381, 1381, w - 7], [1399, 1399, 9/2*w^3 - 33/2*w^2 - 57/2*w + 85], [1399, 1399, -5/2*w^3 + 15/2*w^2 + 41/2*w - 37], [1409, 1409, 3/2*w^3 - 1/2*w^2 - 21/2*w - 1], [1409, 1409, -23/2*w^3 + 77/2*w^2 + 161/2*w - 211], [1409, 1409, 11/2*w^3 - 39/2*w^2 - 71/2*w + 98], [1409, 1409, 1/2*w^3 - 9/2*w^2 - 1/2*w + 37], [1429, 1429, 3/2*w^3 - 5/2*w^2 - 33/2*w - 5], [1429, 1429, 7/2*w^3 - 25/2*w^2 - 37/2*w + 53], [1439, 1439, -7*w^3 + 23*w^2 + 52*w - 135], [1439, 1439, -1/2*w^3 - 1/2*w^2 + 1/2*w - 5], [1451, 1451, 1/2*w^3 + 3/2*w^2 - 13/2*w - 15], [1451, 1451, -13/2*w^3 + 45/2*w^2 + 85/2*w - 120], [1459, 1459, -7/2*w^3 + 27/2*w^2 + 43/2*w - 70], [1459, 1459, 5/2*w^3 - 21/2*w^2 - 29/2*w + 65], [1471, 1471, 5/2*w^3 - 17/2*w^2 - 37/2*w + 43], [1471, 1471, -3*w^3 + 10*w^2 + 21*w - 49], [1481, 1481, -6*w^3 + 21*w^2 + 40*w - 113], [1481, 1481, -16*w^3 + 54*w^2 + 112*w - 299], [1481, 1481, 9*w^3 - 30*w^2 - 63*w + 161], [1481, 1481, 1/2*w^3 - 9/2*w^2 - 3/2*w + 30], [1489, 1489, -8*w^3 + 26*w^2 + 60*w - 151], [1489, 1489, -w^3 + 5*w^2 + 6*w - 35], [1489, 1489, 7/2*w^3 - 25/2*w^2 - 47/2*w + 63], [1489, 1489, -w^3 + w^2 + 3*w - 13], [1499, 1499, 3*w^3 - 10*w^2 - 21*w + 45], [1499, 1499, 9/2*w^3 - 29/2*w^2 - 65/2*w + 75], [1511, 1511, -11/2*w^3 + 35/2*w^2 + 79/2*w - 91], [1511, 1511, -5/2*w^3 + 21/2*w^2 + 33/2*w - 54], [1511, 1511, 7/2*w^3 - 19/2*w^2 - 57/2*w + 42], [1511, 1511, 5/2*w^3 - 15/2*w^2 - 29/2*w + 41], [1531, 1531, -4*w^2 + 2*w + 29], [1531, 1531, 9*w^3 - 31*w^2 - 61*w + 167], [1549, 1549, -1/2*w^3 + 1/2*w^2 + 13/2*w - 5], [1549, 1549, -3/2*w^3 + 11/2*w^2 + 15/2*w - 34], [1559, 1559, 9/2*w^3 - 29/2*w^2 - 63/2*w + 76], [1559, 1559, 2*w^3 - 5*w^2 - 14*w + 23], [1571, 1571, 3*w^3 - 9*w^2 - 20*w + 43], [1571, 1571, 2*w^3 - 7*w^2 - 14*w + 45], [1601, 1601, -27/2*w^3 + 93/2*w^2 + 179/2*w - 247], [1601, 1601, 1/2*w^3 - 1/2*w^2 + 5/2*w + 16], [1609, 1609, -1/2*w^3 + 7/2*w^2 + 1/2*w - 31], [1609, 1609, 3/2*w^3 - 7/2*w^2 - 23/2*w + 25], [1621, 1621, 7/2*w^3 - 25/2*w^2 - 45/2*w + 71], [1621, 1621, -1/2*w^3 + 7/2*w^2 + 3/2*w - 19], [1681, 41, -3*w^3 + 9*w^2 + 21*w - 43], [1681, 41, 3*w^3 - 9*w^2 - 21*w + 47], [1721, 1721, -1/2*w^3 - 3/2*w^2 + 21/2*w + 25], [1721, 1721, -9/2*w^3 + 33/2*w^2 + 49/2*w - 78], [1759, 1759, 4*w^3 - 14*w^2 - 25*w + 75], [1759, 1759, 2*w^3 - 7*w^2 - 12*w + 43], [1831, 1831, -3/2*w^3 + 7/2*w^2 + 29/2*w - 6], [1831, 1831, 15/2*w^3 - 51/2*w^2 - 103/2*w + 136], [1849, 43, -1/2*w^3 + 7/2*w^2 - 3/2*w - 31], [1849, 43, 15/2*w^3 - 51/2*w^2 - 101/2*w + 138], [1871, 1871, 3*w^3 - 9*w^2 - 18*w + 43], [1871, 1871, -7/2*w^3 + 21/2*w^2 + 45/2*w - 51], [1879, 1879, 21/2*w^3 - 71/2*w^2 - 145/2*w + 195], [1879, 1879, -w^3 - w^2 + 8*w + 9], [1889, 1889, 2*w^3 - 4*w^2 - 16*w + 15], [1889, 1889, -6*w^3 + 20*w^2 + 40*w - 105], [1901, 1901, 2*w^2 - 7*w - 29], [1901, 1901, -19/2*w^3 + 65/2*w^2 + 133/2*w - 182], [1901, 1901, 1/2*w^3 - 11/2*w^2 - 7/2*w + 30], [1901, 1901, -1/2*w^3 + 5/2*w^2 + 1/2*w - 24], [1931, 1931, -1/2*w^3 + 9/2*w^2 + 3/2*w - 34], [1931, 1931, -5/2*w^3 + 19/2*w^2 + 35/2*w - 59], [1931, 1931, -5/2*w^3 + 19/2*w^2 + 35/2*w - 47], [1931, 1931, 6*w^3 - 21*w^2 - 40*w + 109], [1949, 1949, -3/2*w^3 + 15/2*w^2 + 19/2*w - 46], [1949, 1949, -11/2*w^3 + 39/2*w^2 + 75/2*w - 105], [1979, 1979, -7/2*w^3 + 21/2*w^2 + 55/2*w - 51], [1979, 1979, 3/2*w^3 - 5/2*w^2 - 29/2*w + 7], [1979, 1979, 9/2*w^3 - 31/2*w^2 - 55/2*w + 81], [1979, 1979, -2*w^3 + 6*w^2 + 11*w - 27]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, 3, 1, -2, -2, -2, 4, 0, -4, -4, -10, 10, 0, -10, 6, -6, 2, -8, 0, -12, 0, 6, -2, 14, -10, -4, -4, 4, -2, 18, -10, -6, -8, -24, -12, -12, -10, 6, 22, 6, 4, 16, 18, 14, 12, 6, -18, 24, -16, -24, 6, -18, 14, -18, 10, -8, 10, -24, -12, -4, 32, 16, 6, -32, 12, -32, -28, -24, -22, 26, -6, 14, 30, 30, -2, -10, 22, -14, 0, -32, -12, 4, 12, 4, -24, 24, 22, 10, 0, -16, 10, -30, 24, -10, -20, -42, 6, 8, -2, -24, 4, 16, 18, -12, 16, -30, -30, 42, -28, -34, 10, -20, -20, -48, 2, -34, 30, -18, 16, 0, -2, -26, 2, 10, -12, -24, 10, 28, 10, -4, 10, 38, 40, 12, 16, 24, 20, -8, 0, -44, 40, 4, 14, 24, -44, 2, 30, 2, 26, -18, 6, -20, -28, 26, -10, -30, 10, -6, -44, -32, -40, 16, -10, -30, 30, -38, -30, 50, 52, -8, -10, -10, -12, -8, -12, 12, 16, 52, -30, 10, -60, -40, 40, 36, -28, 4, -34, 6, 44, 12, -52, 12, 12, 32, 22, 6, 12, -20, -18, -10, 6, -46, 22, 34, -48, 4, 50, 28, 34, -4, 6, -34, 4, 28, -54, -30, -22, 58, -22, 30, 30, -50, -50, -34, 6, 62, -14, -10, -32, -20, 52, 0, 66, 18, -66, 14, -42, -18, 38, -38, 72, 20, -42, -6, 54, -34, 38, -6, -10, 10, 44, -48, 40, 0, 0, 28, 20, -64, 36, 44, -60, -8, -24, -28, 10, -30, 50, 2, 20, -12, 12, -28, 20, 24, 50, -10, 16, 56, -66, 34, -48, 24, 82, -58, 6, 34, 22, 38, 18, -54, -12, 60, -12, 72, 26, -66, 44, 24, 0, 4]; 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;