/* 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^12 + 4*x^11 - 23*x^10 - 109*x^9 + 126*x^8 + 940*x^7 + 140*x^6 - 3015*x^5 - 1835*x^4 + 3250*x^3 + 1835*x^2 - 1350*x - 220; K := NumberField(heckePol); heckeEigenvaluesArray := [0, e, -2932849/122946112*e^11 - 5028135/122946112*e^10 + 42449823/61473056*e^9 + 7478197/6470848*e^8 - 841906049/122946112*e^7 - 1277542819/122946112*e^6 + 3559516939/122946112*e^5 + 1096030283/30736528*e^4 - 339256979/6470848*e^3 - 5059755109/122946112*e^2 + 110986983/3235424*e + 111276015/30736528, 10074419/122946112*e^11 + 22294333/122946112*e^10 - 135391157/61473056*e^9 - 32312799/6470848*e^8 + 2347002283/122946112*e^7 + 5271418953/122946112*e^6 - 7843515241/122946112*e^5 - 4094641757/30736528*e^4 + 532290153/6470848*e^3 + 14906832519/122946112*e^2 - 179289133/3235424*e - 403475125/30736528, 7090775/122946112*e^11 + 12801159/122946112*e^10 - 100996771/61473056*e^9 - 19043907/6470848*e^8 + 1954376237/122946112*e^7 + 3242769517/122946112*e^6 - 8000693859/122946112*e^5 - 1369991299/15368264*e^4 + 745277481/6470848*e^3 + 12020159405/122946112*e^2 - 252692935/3235424*e - 305968123/30736528, -e - 1, 3440467/30736528*e^11 + 429023/1921033*e^10 - 23766381/7684132*e^9 - 10072871/1617712*e^8 + 434529283/15368264*e^7 + 1675126053/30736528*e^6 - 803570219/7684132*e^5 - 2697685115/15368264*e^4 + 257775547/1617712*e^3 + 1329256341/7684132*e^2 - 10639741/101107*e - 67662449/3842066, -2411827/30736528*e^11 - 11503385/61473056*e^10 + 64410851/30736528*e^9 + 8235139/1617712*e^8 - 1098013293/61473056*e^7 - 1319792471/30736528*e^6 + 3532513121/61473056*e^5 + 3958723345/30736528*e^4 - 56318879/808856*e^3 - 6617701055/61473056*e^2 + 85101633/1617712*e + 169493255/15368264, -1924175/30736528*e^11 - 6132361/61473056*e^10 + 53111595/30736528*e^9 + 4538751/1617712*e^8 - 970601165/61473056*e^7 - 750399123/30736528*e^6 + 3544889905/61473056*e^5 + 2352863601/30736528*e^4 - 65885989/808856*e^3 - 4274917839/61473056*e^2 + 67533849/1617712*e + 64981495/15368264, -971721/61473056*e^11 - 3152015/61473056*e^10 + 14208415/30736528*e^9 + 4755437/3235424*e^8 - 285781081/61473056*e^7 - 851058395/61473056*e^6 + 1263301235/61473056*e^5 + 780066623/15368264*e^4 - 136806651/3235424*e^3 - 3754575773/61473056*e^2 + 59040775/1617712*e + 48890231/15368264, -14573257/122946112*e^11 - 35636729/122946112*e^10 + 192634365/61473056*e^9 + 51240285/6470848*e^8 - 3216903059/122946112*e^7 - 8268981683/122946112*e^6 + 9853545373/122946112*e^5 + 3134649285/15368264*e^4 - 551298935/6470848*e^3 - 21147477651/122946112*e^2 + 202990713/3235424*e + 553850677/30736528, 517999/61473056*e^11 + 5266653/61473056*e^10 - 3165821/30736528*e^9 - 7190699/3235424*e^8 - 85420765/61473056*e^7 + 1087757069/61473056*e^6 + 1273216071/61473056*e^5 - 720935459/15368264*e^4 - 211092699/3235424*e^3 + 1422227351/61473056*e^2 + 58049475/1617712*e + 9919491/15368264, 1755/1921033*e^11 + 197721/15368264*e^10 - 219401/7684132*e^9 - 34752/101107*e^8 + 5046673/15368264*e^7 + 5459758/1921033*e^6 - 32964593/15368264*e^5 - 55237721/7684132*e^4 + 3431015/404428*e^3 - 24978617/15368264*e^2 - 5838183/404428*e + 29142699/3842066, 2892139/122946112*e^11 + 9959327/122946112*e^10 - 32475299/61473056*e^9 - 13832519/6470848*e^8 + 323818541/122946112*e^7 + 2085239577/122946112*e^6 + 766850037/122946112*e^5 - 330042617/7684132*e^4 - 341493907/6470848*e^3 + 1886995333/122946112*e^2 + 158582369/3235424*e - 35235675/30736528, 2877473/61473056*e^11 + 7853923/61473056*e^10 - 37002451/30736528*e^9 - 11074405/3235424*e^8 + 577315197/61473056*e^7 + 1728336643/61473056*e^6 - 1456387175/61473056*e^5 - 1209944581/15368264*e^4 + 37719083/3235424*e^3 + 3004443209/61473056*e^2 - 25233475/1617712*e - 43566995/15368264, -218049/7684132*e^11 - 6123819/61473056*e^10 + 21197429/30736528*e^9 + 534419/202214*e^8 - 291651411/61473056*e^7 - 329278061/15368264*e^6 + 524424507/61473056*e^5 + 1787952289/30736528*e^4 - 1275661/1617712*e^3 - 1796631205/61473056*e^2 + 34477451/1617712*e - 157067879/15368264, 20776505/122946112*e^11 + 43907711/122946112*e^10 - 281830551/61473056*e^9 - 63445821/6470848*e^8 + 4968538953/122946112*e^7 + 10273917531/122946112*e^6 - 17114367555/122946112*e^5 - 7831699667/30736528*e^4 + 1217205323/6470848*e^3 + 26743793677/122946112*e^2 - 401507791/3235424*e - 424868183/30736528, 223637/15368264*e^11 + 484003/30736528*e^10 - 6089445/15368264*e^9 - 327253/808856*e^8 + 108719327/30736528*e^7 + 42066317/15368264*e^6 - 382300651/30736528*e^5 - 51328431/15368264*e^4 + 7175599/404428*e^3 - 272384955/30736528*e^2 - 11461863/808856*e + 51051163/7684132, -342963/3842066*e^11 - 4710341/30736528*e^10 + 38008839/15368264*e^9 + 429303/101107*e^8 - 699625005/30736528*e^7 - 278780477/7684132*e^6 + 2610807077/30736528*e^5 + 1692295947/15368264*e^4 - 103845603/808856*e^3 - 2803467755/30736528*e^2 + 61846369/808856*e - 2262437/7684132, 21150843/122946112*e^11 + 51403363/122946112*e^10 - 274229487/61473056*e^9 - 73681399/6470848*e^8 + 4395013617/122946112*e^7 + 11782684921/122946112*e^6 - 12098714191/122946112*e^5 - 4372041629/15368264*e^4 + 479842645/6470848*e^3 + 27680573249/122946112*e^2 - 193732723/3235424*e - 527041975/30736528, -2108127/15368264*e^11 - 2288993/7684132*e^10 + 14441269/3842066*e^9 + 6708091/808856*e^8 - 65049912/1921033*e^7 - 1117800277/15368264*e^6 + 943279647/7684132*e^5 + 1815246995/7684132*e^4 - 150344819/808856*e^3 - 1851684169/7684132*e^2 + 13038143/101107*e + 58164940/1921033, -10999665/122946112*e^11 - 38479215/122946112*e^10 + 130828055/61473056*e^9 + 53886613/6470848*e^8 - 1646979049/122946112*e^7 - 8348520739/122946112*e^6 + 1090823571/122946112*e^5 + 5764341271/30736528*e^4 + 492971037/6470848*e^3 - 13763609533/122946112*e^2 - 108316833/3235424*e + 33062407/30736528, -22675267/122946112*e^11 - 45297145/122946112*e^10 + 311612505/61473056*e^9 + 66603535/6470848*e^8 - 5660259111/122946112*e^7 - 11113851945/122946112*e^6 + 20849291429/122946112*e^5 + 9050476255/30736528*e^4 - 1689705121/6470848*e^3 - 37350827211/122946112*e^2 + 565213225/3235424*e + 1268065577/30736528, -3388779/61473056*e^11 - 6333987/30736528*e^10 + 9400923/7684132*e^9 + 17659183/3235424*e^8 - 91335059/15368264*e^7 - 2716693509/61473056*e^6 - 413369583/30736528*e^5 + 3726621333/30736528*e^4 + 325425273/3235424*e^3 - 2207278695/30736528*e^2 - 45746797/808856*e + 1153630/1921033, -6108569/122946112*e^11 - 8522499/122946112*e^10 + 85744035/61473056*e^9 + 12782013/6470848*e^8 - 1627249197/122946112*e^7 - 2151276203/122946112*e^6 + 6459943495/122946112*e^5 + 1777409457/30736528*e^4 - 571661571/6470848*e^3 - 8177244617/122946112*e^2 + 171434211/3235424*e + 542447875/30736528, -1588403/122946112*e^11 - 10720881/122946112*e^10 + 12856193/61473056*e^9 + 14962975/6470848*e^8 + 74363361/122946112*e^7 - 2331607545/122946112*e^6 - 2209943651/122946112*e^5 + 1637774251/30736528*e^4 + 353591935/6470848*e^3 - 3993758963/122946112*e^2 - 43856655/3235424*e - 128986159/30736528, -2679305/15368264*e^11 - 10037705/30736528*e^10 + 74257551/15368264*e^9 + 7322481/808856*e^8 - 1368491269/30736528*e^7 - 1201771917/15368264*e^6 + 5155922305/30736528*e^5 + 3778502945/15368264*e^4 - 53267465/202214*e^3 - 7098803695/30736528*e^2 + 134848845/808856*e + 125245255/7684132, -27589609/122946112*e^11 - 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996463027/30736528*e^5 - 1592237899/30736528*e^4 + 90419681/3235424*e^3 + 841922693/30736528*e^2 - 16468489/808856*e - 7281220/1921033, 19953951/122946112*e^11 + 49812725/122946112*e^10 - 264682757/61473056*e^9 - 71502683/6470848*e^8 + 4453359163/122946112*e^7 + 11531507293/122946112*e^6 - 13971052689/122946112*e^5 - 8755877687/30736528*e^4 + 855678021/6470848*e^3 + 29862166239/122946112*e^2 - 321452677/3235424*e - 609433925/30736528, 1786117/61473056*e^11 + 1483993/61473056*e^10 - 28137285/30736528*e^9 - 2601033/3235424*e^8 + 631637195/61473056*e^7 + 552181431/61473056*e^6 - 3083578253/61473056*e^5 - 80618510/1921033*e^4 + 318169811/3235424*e^3 + 4441021699/61473056*e^2 - 70090201/1617712*e - 263429429/15368264, -16676175/122946112*e^11 - 31788559/122946112*e^10 + 221828603/61473056*e^9 + 45455387/6470848*e^8 - 3774382757/122946112*e^7 - 7125484789/122946112*e^6 + 12042394187/122946112*e^5 + 2509180243/15368264*e^4 - 711200529/6470848*e^3 - 13149162181/122946112*e^2 + 186072223/3235424*e - 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107894782017/122946112*e^5 - 23426227367/15368264*e^4 + 7728483819/6470848*e^3 + 177457493935/122946112*e^2 - 2332396893/3235424*e - 5693048873/30736528, -13888697/61473056*e^11 - 13287909/61473056*e^10 + 221069001/30736528*e^9 + 21604493/3235424*e^8 - 5054431103/61473056*e^7 - 4151267203/61473056*e^6 + 25707752617/61473056*e^5 + 526472067/1921033*e^4 - 2964728815/3235424*e^3 - 23967106503/61473056*e^2 + 1006447085/1617712*e + 204315593/15368264, 44730573/61473056*e^11 + 89736111/61473056*e^10 - 616932575/30736528*e^9 - 131365889/3235424*e^8 + 11252478209/61473056*e^7 + 21763320743/61473056*e^6 - 41531822115/61473056*e^5 - 17410890453/15368264*e^4 + 3333645951/3235424*e^3 + 67509996237/61473056*e^2 - 1089654815/1617712*e - 1410160647/15368264, 15429965/61473056*e^11 + 44347065/61473056*e^10 - 195399413/30736528*e^9 - 62446481/3235424*e^8 + 2924445915/61473056*e^7 + 9741595663/61473056*e^6 - 6246460109/61473056*e^5 - 1707247689/3842066*e^4 - 99180101/3235424*e^3 + 16853927075/61473056*e^2 + 89501527/1617712*e + 181704731/15368264, 33077013/61473056*e^11 + 34923929/30736528*e^10 - 110947235/7684132*e^9 - 101077329/3235424*e^8 + 1917417875/15368264*e^7 + 16363956059/61473056*e^6 - 12698342251/30736528*e^5 - 24829678899/30736528*e^4 + 1668618249/3235424*e^3 + 20368413197/30736528*e^2 - 262240697/808856*e - 11779530/1921033, 9051379/122946112*e^11 + 27393049/122946112*e^10 - 134487929/61473056*e^9 - 41284287/6470848*e^8 + 2769904615/122946112*e^7 + 7355923961/122946112*e^6 - 12522189893/122946112*e^5 - 6684290551/30736528*e^4 + 1330402513/6470848*e^3 + 32223150251/122946112*e^2 - 466890089/3235424*e - 416874665/30736528, 19290231/61473056*e^11 + 34998205/61473056*e^10 - 266050549/30736528*e^9 - 51370547/3235424*e^8 + 4858068595/61473056*e^7 + 8527914517/61473056*e^6 - 17910516697/61473056*e^5 - 6947575275/15368264*e^4 + 1400361837/3235424*e^3 + 30405883223/61473056*e^2 - 396808133/1617712*e - 1710728965/15368264, 5749025/15368264*e^11 + 8349751/7684132*e^10 - 37586749/3842066*e^9 - 23901093/808856*e^8 + 152695797/1921033*e^7 + 3870645451/15368264*e^6 - 1760504053/7684132*e^5 - 5985453333/7684132*e^4 + 176499925/808856*e^3 + 5467381283/7684132*e^2 - 20303993/101107*e - 179110685/1921033, 35403293/61473056*e^11 + 83693357/61473056*e^10 - 467476785/30736528*e^9 - 120829633/3235424*e^8 + 7816863407/61473056*e^7 + 19548961359/61473056*e^6 - 24235023425/61473056*e^5 - 7388754609/7684132*e^4 + 1474834115/3235424*e^3 + 47423257807/61473056*e^2 - 645945613/1617712*e - 354234025/15368264, 235483/3235424*e^11 + 448879/3235424*e^10 - 3349107/1617712*e^9 - 12348021/3235424*e^8 + 64885373/3235424*e^7 + 107471753/3235424*e^6 - 271762395/3235424*e^5 - 22676939/202214*e^4 + 531039207/3235424*e^3 + 475718581/3235424*e^2 - 221219277/1617712*e - 47137195/808856]; 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;