/* 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, 14, -13, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, w^2 - 2*w - 6], [4, 2, -w^2 + 7], [5, 5, -1/2*w^3 + 2*w^2 + 7/2*w - 14], [5, 5, 1/2*w^3 + 1/2*w^2 - 6*w - 9], [11, 11, 1/2*w^3 - 2*w^2 - 5/2*w + 11], [11, 11, 1/2*w^3 + 1/2*w^2 - 5*w - 7], [31, 31, 1/2*w^2 + 1/2*w - 4], [31, 31, -1/2*w^2 + 3/2*w + 3], [41, 41, 1/2*w^2 + 1/2*w - 6], [41, 41, 2*w^3 - 15/2*w^2 - 25/2*w + 50], [41, 41, 5/2*w^2 - 1/2*w - 17], [41, 41, 1/2*w^2 - 3/2*w - 5], [61, 61, -1/2*w^3 + w^2 + 7/2*w - 1], [61, 61, -1/2*w^3 + 1/2*w^2 + 4*w - 3], [71, 71, 1/2*w^3 + w^2 - 11/2*w - 13], [71, 71, -1/2*w^3 + 5/2*w^2 + 2*w - 17], [81, 3, -3], [89, 89, -w^3 + 3/2*w^2 + 13/2*w - 9], [89, 89, w^3 - 7/2*w^2 - 11/2*w + 20], [89, 89, -w^3 - 1/2*w^2 + 19/2*w + 12], [89, 89, 1/2*w^3 + 2*w^2 - 17/2*w - 19], [109, 109, 1/2*w^3 + 3/2*w^2 - 7*w - 15], [109, 109, -1/2*w^3 + 1/2*w^2 + 4*w - 5], [121, 11, -3/2*w^2 + 3/2*w + 10], [131, 131, 3/2*w^3 - 13/2*w^2 - 9*w + 43], [131, 131, -3/2*w^3 - 2*w^2 + 35/2*w + 29], [139, 139, -3/2*w^3 + 11/2*w^2 + 10*w - 39], [139, 139, 1/2*w^3 + w^2 - 11/2*w - 9], [139, 139, -1/2*w^3 + 5/2*w^2 + 2*w - 13], [139, 139, -3/2*w^3 - w^2 + 33/2*w + 25], [151, 151, -1/2*w^3 + 3/2*w^2 + 3*w - 5], [151, 151, -1/2*w^3 + 9/2*w + 1], [179, 179, 5/2*w^2 - 3/2*w - 20], [179, 179, -w^3 + 5/2*w^2 + 11/2*w - 5], [179, 179, -w^3 + 1/2*w^2 + 15/2*w - 2], [179, 179, w^3 + 1/2*w^2 - 19/2*w - 10], [181, 181, 1/2*w^2 - 3/2*w - 8], [181, 181, 3/2*w^2 - 1/2*w - 13], [181, 181, -3/2*w^2 + 5/2*w + 12], [181, 181, 1/2*w^2 + 1/2*w - 9], [191, 191, w^3 + 1/2*w^2 - 17/2*w - 9], [191, 191, 1/2*w^3 - w^2 - 5/2*w + 8], [199, 199, -1/2*w^3 + 7/2*w^2 + 3*w - 23], [199, 199, -1/2*w^3 - 2*w^2 + 17/2*w + 17], [211, 211, w^3 - 5/2*w^2 - 9/2*w + 3], [211, 211, -w^3 + 1/2*w^2 + 13/2*w - 3], [229, 229, 3/2*w^3 - 2*w^2 - 21/2*w - 6], [229, 229, -w^3 + 3*w^2 + 8*w - 23], [239, 239, -w^3 + 3*w^2 + 4*w - 9], [239, 239, w^3 - 9*w - 7], [239, 239, 1/2*w^3 - 3/2*w^2 - 2*w + 13], [239, 239, -w^3 + 7*w + 3], [241, 241, 1/2*w^3 + 2*w^2 - 15/2*w - 16], [241, 241, -1/2*w^3 + 7/2*w^2 + 2*w - 21], [251, 251, w^3 - 5/2*w^2 - 9/2*w + 10], [251, 251, w^3 - 9*w - 9], [269, 269, w^3 - 1/2*w^2 - 17/2*w - 10], [269, 269, w^3 - 5/2*w^2 - 13/2*w + 18], [281, 281, -1/2*w^3 + 3*w^2 + 3/2*w - 19], [281, 281, 1/2*w^3 + 3/2*w^2 - 6*w - 15], [289, 17, w^3 - 3/2*w^2 - 15/2*w - 1], [289, 17, -w^3 + 3/2*w^2 + 15/2*w - 9], [331, 331, 3/2*w^2 - 1/2*w - 14], [331, 331, -3/2*w^2 + 5/2*w + 13], [349, 349, -w^3 + 3/2*w^2 + 11/2*w - 1], [349, 349, w^3 - 5/2*w^2 - 13/2*w + 14], [359, 359, -w^3 + 5/2*w^2 + 11/2*w - 14], [359, 359, w^3 - 1/2*w^2 - 15/2*w - 7], [361, 19, 2*w^2 - 2*w - 15], [361, 19, -2*w^2 + 2*w + 13], [379, 379, 5/2*w^2 - 3/2*w - 16], [379, 379, 5/2*w^2 - 7/2*w - 15], [389, 389, 7/2*w^2 - 1/2*w - 26], [389, 389, 1/2*w^3 - 3/2*w^2 - 5*w + 5], [389, 389, 1/2*w^3 + 2*w^2 - 11/2*w - 18], [389, 389, 3/2*w^3 - 1/2*w^2 - 13*w - 15], [409, 409, -w^3 + 7/2*w^2 + 9/2*w - 19], [409, 409, w^3 + 1/2*w^2 - 17/2*w - 12], [439, 439, -1/2*w^3 + 5/2*w^2 + w - 15], [439, 439, 1/2*w^3 + w^2 - 9/2*w - 12], [461, 461, 1/2*w^3 + 2*w^2 - 15/2*w - 18], [461, 461, -1/2*w^3 + 7/2*w^2 + 2*w - 23], [479, 479, -1/2*w^3 + 2*w^2 + 7/2*w - 10], [479, 479, 1/2*w^3 - 2*w^2 - 5/2*w + 7], [491, 491, -2*w^3 + 1/2*w^2 + 29/2*w - 1], [491, 491, 2*w^3 - 11/2*w^2 - 19/2*w + 12], [499, 499, -w^3 + 5/2*w^2 + 13/2*w - 13], [499, 499, w^3 - 1/2*w^2 - 17/2*w - 5], [509, 509, -w^3 + 3/2*w^2 + 15/2*w - 2], [509, 509, w^3 - 3/2*w^2 - 15/2*w + 6], [521, 521, -1/2*w^3 + 2*w^2 + 9/2*w - 15], [521, 521, 1/2*w^3 + 1/2*w^2 - 7*w - 9], [529, 23, -w^3 + 2*w^2 + 5*w - 9], [529, 23, -3/2*w^2 + 7/2*w + 12], [541, 541, 3/2*w^3 + 1/2*w^2 - 14*w - 17], [541, 541, -3/2*w^3 + 5*w^2 + 17/2*w - 29], [569, 569, -2*w^3 + 1/2*w^2 + 35/2*w + 18], [569, 569, -w^3 - 5/2*w^2 + 27/2*w + 27], [571, 571, -w^3 + w^2 + 8*w + 7], [571, 571, -1/2*w^3 + 2*w^2 + 3/2*w - 4], [571, 571, 1/2*w^3 + 1/2*w^2 - 4*w - 1], [571, 571, -4*w^3 - 7/2*w^2 + 91/2*w + 68], [599, 599, -1/2*w^3 - 5/2*w^2 + 10*w + 21], [599, 599, 2*w^3 + 5/2*w^2 - 47/2*w - 37], [599, 599, 2*w^3 - 17/2*w^2 - 25/2*w + 56], [599, 599, -1/2*w^3 + 4*w^2 + 7/2*w - 28], [601, 601, -w^3 - 3/2*w^2 + 23/2*w + 18], [601, 601, 1/2*w^3 + 7/2*w^2 - 5*w - 27], [601, 601, 1/2*w^3 - 5*w^2 + 7/2*w + 28], [601, 601, 2*w^3 - 3*w^2 - 13*w + 21], [619, 619, -w^3 + 5/2*w^2 + 9/2*w - 8], [619, 619, w^3 - 1/2*w^2 - 13/2*w - 2], [631, 631, -5/2*w^3 + 1/2*w^2 + 18*w - 1], [631, 631, -5/2*w^3 + 7*w^2 + 23/2*w - 15], [641, 641, 5/2*w^2 - 1/2*w - 20], [641, 641, -5/2*w^2 + 9/2*w + 18], [659, 659, w^3 - 7/2*w^2 - 15/2*w + 28], [659, 659, w^3 + 2*w^2 - 11*w - 21], [661, 661, -w^3 + 15/2*w^2 + 11/2*w - 52], [661, 661, -3/2*w^3 + 1/2*w^2 + 13*w + 5], [661, 661, 3/2*w^3 - 4*w^2 - 19/2*w + 17], [661, 661, -w^3 - 9/2*w^2 + 35/2*w + 40], [691, 691, 2*w^3 - 9/2*w^2 - 21/2*w + 5], [691, 691, -5/2*w^3 - w^2 + 55/2*w + 37], [701, 701, w^3 - 2*w^2 - 7*w + 3], [701, 701, w^3 - w^2 - 8*w + 5], [709, 709, -11/2*w^2 + 3/2*w + 40], [709, 709, -3/2*w^3 - 3/2*w^2 + 16*w + 23], [739, 739, 1/2*w^3 - 9/2*w^2 + 3*w + 25], [739, 739, 1/2*w^3 + 3*w^2 - 9/2*w - 24], [751, 751, -7/2*w^3 + 12*w^2 + 47/2*w - 83], [751, 751, -3/2*w^3 + 23/2*w + 3], [751, 751, -3/2*w^3 + 9/2*w^2 + 7*w - 13], [751, 751, 7/2*w^3 + 3/2*w^2 - 37*w - 51], [761, 761, -3/2*w^3 - 1/2*w^2 + 14*w + 19], [761, 761, -w^3 + 3/2*w^2 + 15/2*w - 4], [761, 761, -3/2*w^3 + 5*w^2 + 17/2*w - 31], [769, 769, -3*w^3 - 5/2*w^2 + 69/2*w + 51], [769, 769, -1/2*w^3 - 4*w^2 + 9/2*w + 27], [769, 769, -w^3 - 5/2*w^2 + 29/2*w + 25], [769, 769, 3*w^3 - 23/2*w^2 - 41/2*w + 80], [809, 809, 1/2*w^3 + 2*w^2 - 13/2*w - 17], [809, 809, 1/2*w^3 - 7/2*w^2 - w + 21], [821, 821, 1/2*w^3 - 1/2*w^2 - 6*w + 3], [821, 821, -1/2*w^3 + w^2 + 11/2*w - 3], [829, 829, -3/2*w^3 + 3*w^2 + 23/2*w - 10], [829, 829, -w^3 + 2*w^2 + 5*w + 1], [829, 829, 7/2*w^2 - 3/2*w - 28], [829, 829, 3/2*w^3 - 3/2*w^2 - 13*w + 3], [839, 839, -w^3 - 7/2*w^2 + 17/2*w + 27], [839, 839, 2*w^3 + 1/2*w^2 - 43/2*w - 27], [839, 839, 2*w^3 - 13/2*w^2 - 29/2*w + 46], [839, 839, -w^3 - 3/2*w^2 + 19/2*w + 14], [841, 29, 5/2*w^2 - 5/2*w - 19], [841, 29, 5/2*w^2 - 5/2*w - 16], [859, 859, 1/2*w^3 + w^2 - 5/2*w - 12], [859, 859, w^3 + 3/2*w^2 - 19/2*w - 19], [859, 859, -w^3 + 9/2*w^2 + 7/2*w - 26], [859, 859, 1/2*w^3 - 5/2*w^2 + w + 13], [881, 881, -1/2*w^3 + 1/2*w^2 + 6*w + 1], [881, 881, 1/2*w^3 - w^2 - 11/2*w + 7], [911, 911, -1/2*w^3 + 5/2*w^2 + 3*w - 13], [911, 911, 1/2*w^3 + w^2 - 13/2*w - 8], [941, 941, -2*w^3 + 7/2*w^2 + 27/2*w - 21], [941, 941, 3/2*w^3 - 7*w^2 - 21/2*w + 51], [941, 941, 3/2*w^3 + 5/2*w^2 - 20*w - 35], [941, 941, -2*w^3 + 13/2*w^2 + 29/2*w - 48], [961, 31, -w^2 + w + 13], [1009, 1009, -5/2*w^2 - 1/2*w + 18], [1009, 1009, 3/2*w^3 - 3/2*w^2 - 12*w - 5], [1021, 1021, w^3 + w^2 - 10*w - 19], [1021, 1021, w^3 - 5/2*w^2 - 9/2*w + 12], [1031, 1031, -1/2*w^3 + 3/2*w^2 + 5*w - 9], [1031, 1031, -w^3 + 3/2*w^2 + 13/2*w - 2], [1049, 1049, -3/2*w^3 - 1/2*w^2 + 16*w + 23], [1049, 1049, -2*w^3 + 9/2*w^2 + 23/2*w - 6], [1049, 1049, -2*w^3 + 3/2*w^2 + 29/2*w - 8], [1049, 1049, 3/2*w^3 - 5*w^2 - 21/2*w + 37], [1051, 1051, -2*w^3 + 11/2*w^2 + 17/2*w - 10], [1051, 1051, -3*w^3 - 1/2*w^2 + 59/2*w + 36], [1051, 1051, -3*w^3 + 19/2*w^2 + 39/2*w - 62], [1051, 1051, 2*w^3 - 1/2*w^2 - 27/2*w + 2], [1061, 1061, 11/2*w^2 - 3/2*w - 38], [1061, 1061, 11/2*w^2 - 19/2*w - 34], [1069, 1069, 1/2*w^3 - 4*w^2 - 5/2*w + 25], [1069, 1069, 2*w^3 - 20*w - 25], [1069, 1069, -2*w^3 + 6*w^2 + 14*w - 43], [1069, 1069, 1/2*w^3 + 5/2*w^2 - 9*w - 19], [1109, 1109, -5/2*w^3 + 13/2*w^2 + 12*w - 13], [1109, 1109, -5/2*w^3 + w^2 + 35/2*w - 3], [1151, 1151, 13/2*w^2 - 3/2*w - 46], [1151, 1151, 1/2*w^3 + 3/2*w^2 - 5*w - 17], [1151, 1151, 1/2*w^3 - 3*w^2 - 1/2*w + 20], [1151, 1151, 13/2*w^2 - 23/2*w - 41], [1171, 1171, w^3 - 3/2*w^2 - 13/2*w + 12], [1171, 1171, 1/2*w^3 - 1/2*w^2 - 6*w + 1], [1171, 1171, -1/2*w^3 + w^2 + 11/2*w - 5], [1171, 1171, -1/2*w^3 + 5*w^2 - 7/2*w - 26], [1201, 1201, 1/2*w^3 + 1/2*w^2 - 6*w - 3], [1201, 1201, -1/2*w^3 + 2*w^2 + 7/2*w - 8], [1229, 1229, 1/2*w^3 + 5/2*w^2 - 8*w - 21], [1229, 1229, -1/2*w^3 + 4*w^2 + 3/2*w - 26], [1249, 1249, -w^3 + 7/2*w^2 + 9/2*w - 12], [1249, 1249, w^3 + 1/2*w^2 - 17/2*w - 5], [1279, 1279, 1/2*w^3 + 3/2*w^2 - 3*w - 15], [1279, 1279, -1/2*w^3 + 3*w^2 - 3/2*w - 16], [1289, 1289, 5/2*w^2 - 9/2*w - 19], [1289, 1289, w^3 - 9/2*w^2 - 15/2*w + 32], [1289, 1289, -w^3 - 3/2*w^2 + 27/2*w + 21], [1289, 1289, 5/2*w^2 - 1/2*w - 21], [1291, 1291, -1/2*w^3 + 5*w^2 - 9/2*w - 23], [1291, 1291, 1/2*w^3 + 7/2*w^2 - 4*w - 23], [1301, 1301, -3/2*w^3 + 2*w^2 + 21/2*w - 10], [1301, 1301, -3/2*w^3 + 5/2*w^2 + 10*w - 1], [1319, 1319, 4*w^3 - 29/2*w^2 - 49/2*w + 93], [1319, 1319, -4*w^3 - 5/2*w^2 + 83/2*w + 58], [1321, 1321, -17/2*w^2 + 31/2*w + 52], [1321, 1321, 17/2*w^2 - 3/2*w - 59], [1361, 1361, -3*w^3 - 7/2*w^2 + 71/2*w + 58], [1361, 1361, -3/2*w^3 - 3*w^2 + 39/2*w + 34], [1369, 37, 3/2*w^3 - 1/2*w^2 - 13*w - 17], [1369, 37, 3/2*w^3 + 1/2*w^2 - 11*w - 3], [1381, 1381, w^3 - 5*w^2 - 2*w + 27], [1381, 1381, w^3 + 2*w^2 - 9*w - 21], [1399, 1399, w^3 + 5/2*w^2 - 19/2*w - 25], [1399, 1399, w^3 - 11/2*w^2 - 3/2*w + 31], [1439, 1439, -2*w^3 + 13/2*w^2 + 23/2*w - 40], [1439, 1439, 4*w^3 + 3*w^2 - 43*w - 65], [1451, 1451, 1/2*w^3 + 9/2*w^2 - 5*w - 35], [1451, 1451, -1/2*w^3 + 6*w^2 - 11/2*w - 35], [1459, 1459, -3/2*w^3 + 9/2*w^2 + 9*w - 25], [1459, 1459, -3/2*w^3 + 1/2*w^2 + 10*w + 7], [1471, 1471, 3/2*w^3 - 11/2*w^2 - 7*w + 31], [1471, 1471, -3/2*w^3 - w^2 + 27/2*w + 20], [1489, 1489, -3*w^3 - 3*w^2 + 34*w + 51], [1489, 1489, 1/2*w^3 + 5/2*w^2 - 6*w - 25], [1489, 1489, -1/2*w^3 + 4*w^2 - 1/2*w - 28], [1489, 1489, 3*w^3 - 12*w^2 - 19*w + 79], [1499, 1499, 1/2*w^3 - 7/2*w^2 + 2*w + 19], [1499, 1499, 1/2*w^3 + 5/2*w^2 - 6*w - 21], [1499, 1499, -1/2*w^3 + 4*w^2 - 1/2*w - 24], [1499, 1499, -w^3 + 5/2*w^2 + 11/2*w - 17], [1511, 1511, -2*w^3 + 15/2*w^2 + 17/2*w - 37], [1511, 1511, -2*w^3 - 3/2*w^2 + 35/2*w + 23], [1531, 1531, 1/2*w^3 - 3*w^2 - 9/2*w + 16], [1531, 1531, -2*w^3 + 9/2*w^2 + 23/2*w - 8], [1531, 1531, -2*w^3 + 3/2*w^2 + 29/2*w - 6], [1531, 1531, w^3 + 5/2*w^2 - 21/2*w - 25], [1549, 1549, -1/2*w^3 + 1/2*w^2 + 3*w - 7], [1549, 1549, 1/2*w^3 - w^2 - 5/2*w - 4], [1579, 1579, w^3 + 3/2*w^2 - 19/2*w - 13], [1579, 1579, -w^3 + 9/2*w^2 + 7/2*w - 20], [1609, 1609, -11/2*w^2 + 1/2*w + 37], [1609, 1609, 11/2*w^2 - 21/2*w - 32], [1619, 1619, 1/2*w^3 - 3/2*w^2 - w + 15], [1619, 1619, w^3 - 3/2*w^2 - 21/2*w + 1], [1621, 1621, -2*w^3 + 15/2*w^2 + 23/2*w - 49], [1621, 1621, 1/2*w^3 + 3/2*w^2 - 9*w - 19], [1669, 1669, 1/2*w^3 + 5/2*w^2 - 7*w - 23], [1669, 1669, 1/2*w^3 - 4*w^2 - 1/2*w + 27], [1699, 1699, -w^3 + 1/2*w^2 + 17/2*w - 2], [1699, 1699, -w^3 + 5/2*w^2 + 13/2*w - 6], [1709, 1709, 1/2*w^3 + 3/2*w^2 - 2*w - 15], [1709, 1709, -1/2*w^3 + 3*w^2 - 5/2*w - 15], [1721, 1721, 5/2*w^3 - w^2 - 45/2*w - 20], [1721, 1721, -5/2*w^3 + 13/2*w^2 + 17*w - 41], [1741, 1741, -w^3 + 5/2*w^2 + 13/2*w - 3], [1741, 1741, -5/2*w^3 - 3*w^2 + 61/2*w + 48], [1741, 1741, -5*w^3 + 35/2*w^2 + 65/2*w - 116], [1741, 1741, -w^3 + 5*w^2 + 6*w - 31], [1759, 1759, 4*w^3 - 16*w^2 - 26*w + 111], [1759, 1759, 3*w^3 + 3/2*w^2 - 61/2*w - 43], [1789, 1789, w^3 + 1/2*w^2 - 17/2*w - 14], [1789, 1789, -w^3 + 7/2*w^2 + 9/2*w - 21], [1801, 1801, w^3 + 5/2*w^2 - 23/2*w - 24], [1801, 1801, w^3 + 11/2*w^2 - 15/2*w - 38], [1831, 1831, 13/2*w^2 - 1/2*w - 46], [1831, 1831, -3/2*w^3 + 8*w^2 + 19/2*w - 53], [1831, 1831, 9/2*w^3 - 33/2*w^2 - 28*w + 109], [1831, 1831, -13/2*w^2 + 25/2*w + 40], [1861, 1861, 5/2*w^2 - 9/2*w - 8], [1861, 1861, 5/2*w^2 - 1/2*w - 10], [1871, 1871, -1/2*w^3 - 1/2*w^2 + 7*w + 15], [1871, 1871, -1/2*w^3 - 4*w^2 + 7/2*w + 26], [1879, 1879, 1/2*w^3 - 9/2*w^2 - w + 25], [1879, 1879, -w^3 - 5/2*w^2 + 23/2*w + 27], [1879, 1879, w^3 - 11/2*w^2 - 7/2*w + 35], [1879, 1879, 1/2*w^3 + 3*w^2 - 17/2*w - 20], [1889, 1889, w^3 + 5/2*w^2 - 25/2*w - 27], [1889, 1889, -w^3 + 11/2*w^2 + 9/2*w - 36], [1901, 1901, -3*w^3 - w^2 + 32*w + 41], [1901, 1901, 3*w^3 - 10*w^2 - 21*w + 69], [1931, 1931, 2*w^3 - 15/2*w^2 - 13/2*w + 26], [1931, 1931, -w^3 + 11/2*w^2 + 5/2*w - 31], [1951, 1951, -w^3 + 1/2*w^2 + 13/2*w + 7], [1951, 1951, 2*w^2 - 19], [1979, 1979, -1/2*w^3 + 4*w^2 + 1/2*w - 25], [1979, 1979, 7/2*w^3 + w^2 - 73/2*w - 47], [1979, 1979, 7/2*w^3 - 23/2*w^2 - 24*w + 79], [1979, 1979, 1/2*w^3 + 5/2*w^2 - 7*w - 21], [1999, 1999, -w^3 + 15/2*w^2 + 11/2*w - 54], [1999, 1999, -w^3 + 15/2*w^2 - 9/2*w - 37]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, 1, -2, -1, -4, 0, 0, -8, 10, 6, 6, -6, -6, -2, 12, 12, -10, 6, -10, 14, 10, 14, -14, -14, -12, 8, -4, -16, 16, 12, -16, -16, -12, -8, -20, -12, -14, 22, -14, 10, 12, 8, 4, -24, -20, 8, 14, 10, 24, 16, 16, -20, 10, 22, 28, -12, -26, 18, 30, -22, 2, -14, 24, -4, 14, -14, -16, -20, 2, 26, -4, -4, -2, -6, 6, -22, -22, 10, 0, 20, -30, 6, -32, 16, 12, -36, -16, 32, 26, 30, -30, -30, 22, -10, -18, -38, -38, 26, -12, -28, 36, 44, -8, 0, 12, 24, 6, 34, 22, -22, 12, -28, 28, -8, 26, -30, -28, -44, 14, -6, -14, 10, 20, 28, 2, 42, 34, -14, -36, -40, -24, 0, -32, -24, -10, 30, 46, 34, 14, -18, 26, 22, -6, -46, -30, -10, -42, 26, 34, -40, 0, 36, -12, -2, 30, 52, -20, 12, 16, 30, 2, 36, -48, -14, 42, -30, 18, 30, -22, -2, -34, -58, 24, 8, 50, 6, 6, -38, -12, 12, 48, -4, -58, 30, -50, 14, -30, -46, 34, 54, -52, -8, 44, 36, -52, -20, 52, 52, -50, 10, -14, -30, 14, -18, -56, -64, 54, -18, 34, -42, -48, -44, -6, -18, -12, -16, -58, -42, 62, 46, -58, -6, 10, -22, -32, -4, 32, -48, -40, -60, 40, 0, 0, 16, 54, -50, -34, 38, 12, 36, -56, 36, -40, -8, -44, -12, -76, 32, 38, -58, 44, -68, -6, 46, -20, -32, 22, 18, 22, -58, -60, -12, -6, 34, 62, -26, 34, 14, -50, -30, 24, 20, 66, 30, 62, 50, 40, -64, -16, -40, -26, 2, -84, 24, -40, -8, -4, -20, 26, 82, -6, 2, -36, 12, -16, 40, 4, -52, -68, -60, 32, 80]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; 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;