/* 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^4 + 4*x^3 + 2*x^2 - 4*x - 1; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1, 1, -e^3 - 3*e^2 + 2, 2*e^3 + 6*e^2 - 2*e - 6, -2*e^2 - 3*e + 5, -e^3 - 4*e^2 - 3*e + 2, e^3 + 4*e^2 + 5*e - 2, e^3 + 4*e^2 + 3*e - 10, -4*e^3 - 11*e^2 + e + 8, -4*e^2 - 6*e + 6, -2*e^3 - 6*e^2 - 4*e + 2, -e^3 + 5*e, 3*e^3 + 13*e^2 + 9*e - 11, -4*e^3 - 11*e^2 + 5*e + 6, 3*e^3 + 13*e^2 + 7*e - 13, 6, 6*e^3 + 17*e^2 + 4*e - 1, e^3 + 6*e^2 + 6*e - 3, 3*e^3 + 12*e^2 + 5*e - 14, 5*e^3 + 11*e^2 - 13*e - 13, 2*e^3 + 8*e^2 + 10*e + 2, 7*e^3 + 22*e^2 + 5*e - 18, 4*e^3 + 8*e^2 - 6*e - 8, e^3 + 9*e^2 + 13*e - 5, -2*e^3 - 3*e^2 + 6*e - 1, -8*e^3 - 27*e^2 - 2*e + 19, -e^3 - 10*e^2 - 14*e + 5, -5*e^3 - 11*e^2 + 2*e - 4, -3*e^3 - 2*e^2 + 18*e + 1, 4*e^3 + 16*e^2 - 20, 4*e^3 + 14*e^2 + 4*e - 12, -10*e^3 - 30*e^2 + e + 17, -6*e^3 - 29*e^2 - 20*e + 17, 7*e^2 + 9*e - 12, -3*e^3 + 14*e - 11, -9*e^3 - 26*e^2 - 7*e + 6, e^3 + 8*e^2 + 8*e - 3, -5*e^3 - 15*e^2 - 4*e, 2*e^3 - 4*e^2 - 22*e + 8, e^3 + 2*e^2 + e - 2, 7*e^3 + 17*e^2 - 9*e - 25, 5*e^3 + 11*e^2 - 15*e - 17, -3*e^3 + 3*e^2 + 27*e - 3, e^2 + 5*e - 10, 7*e^3 + 19*e^2 - 5*e - 21, -e^3 + 3*e + 6, 2*e^3 + 14*e^2 + 11*e - 15, -13*e^3 - 40*e^2 - 4*e + 25, -5*e^3 - 17*e^2 - 2*e + 4, 2*e^3 + 4*e^2 - 10*e - 4, 10*e^3 + 35*e^2 + 5*e - 28, e^3 + 5*e^2 + 9*e - 1, 4*e^3 + 20*e^2 + 18*e - 8, -7*e^3 - 20*e^2 + 11*e + 20, -9*e^2 - 18*e - 5, -10*e^3 - 28*e^2 - 3*e + 15, 6*e^3 + 14*e^2 - 8*e - 10, -9*e^3 - 26*e^2 - 3*e + 6, 2*e^3 + 12*e^2 + 18*e + 2, -7*e^3 - 26*e^2 - e + 26, 6*e^2 - 4*e - 32, -2*e^3 + 8*e^2 + 24*e - 10, 6*e^3 + 24*e^2 + 15*e - 21, 3*e^3 + 10*e^2 - 7*e - 36, 9*e^3 + 33*e^2 + 4*e - 22, -3*e^3 - 13*e^2 - 9*e + 15, -4*e^3 - 12*e^2 - 10*e - 10, -2*e^3 + 7*e^2 + 31*e + 4, -16*e^3 - 50*e^2 - 4*e + 32, 7*e^3 + 32*e^2 + 23*e - 24, 2*e^3 + 24*e^2 + 40*e - 22, 16*e^2 + 40*e - 18, 6*e^3 + 8*e^2 - 26*e - 22, -5*e^3 - 24*e^2 - 9*e + 34, 7*e^3 + 14*e^2 - 19*e - 20, -9*e^3 - 27*e^2 - 11*e + 17, 7*e^3 + 27*e^2 + 12*e - 12, -5*e^3 - 13*e^2 - 8*e, 14*e^3 + 46*e^2 + 17*e - 27, -9*e^3 - 25*e^2 + 21*e + 37, 17*e^3 + 49*e^2 + e - 25, 7*e^3 + 20*e^2 + 2*e - 11, 3*e^3 - 8*e^2 - 31*e + 18, -e^3 + 8*e^2 + 17*e + 2, -6*e^3 - 21*e^2 - 4*e - 1, -3*e^3 - 9*e^2 + 7*e + 29, -9*e^3 - 37*e^2 - 31*e + 23, 4*e^3 + 2*e^2 - 16*e - 6, 6*e^3 + 18*e^2 - 6*e - 24, -e^3 - 19*e^2 - 34*e + 14, -e^3 + 7*e^2 + 16*e - 30, 10*e^3 + 30*e^2 - 3*e - 27, -2*e^3 - 18*e^2 - 14*e + 28, 4*e^3 + 6*e^2 - 17*e - 25, 12*e^3 + 45*e^2 + 27*e - 36, 3*e^3 + 15*e^2 + 9*e + 13, -6*e^3 - 9*e^2 + 11*e - 2, 14*e^3 + 40*e^2 - 4*e - 32, -2*e^3 - 18*e^2 - 16*e + 26, 20*e^3 + 70*e^2 + 18*e - 42, 3*e^3 + 2*e^2 - 2*e + 25, -17*e^3 - 50*e^2 + 5*e + 42, -11*e^3 - 39*e^2 - 3*e + 29, -10*e^3 - 36*e^2 + 4*e + 48, -e^3 - 8*e^2 + 7*e + 28, 3*e^3 + 6*e^2 - 4*e - 17, -16*e^2 - 38*e + 28, -8*e^3 - 4*e^2 + 30*e - 26, -13*e^3 - 36*e^2 - e + 26, -3*e^3 - 26*e^2 - 37*e + 28, -7*e^3 - 23*e^2 - 4*e + 26, 7*e^3 + 24*e^2 + 23*e - 6, e^3 + 2*e^2 - 18*e - 25, 4*e^3 - 22*e + 18, -2*e^3 - 8*e^2 + 4*e + 20, 8*e^3 + 36*e^2 + 16*e - 48, 4*e^3 + 14*e^2 + 20*e + 10, -14*e^3 - 39*e^2 + 2*e + 17, 3*e^3 + 8*e^2 - 4*e - 15, -4*e^3 - 33*e^2 - 48*e + 23, 7*e^3 + 20*e^2 + 2*e - 9, -2*e^3 - 10*e^2 - 3*e + 9, -17*e^3 - 56*e^2 - 27*e + 32, -17*e^3 - 58*e^2 - 20*e + 43, -12*e^3 - 37*e^2 + 12*e + 33, 21*e + 23, e^3 - 10*e^2 - 39*e - 12, -7*e^3 - 35*e^2 - 27*e + 15, -16*e^2 - 30*e + 20, -10*e^3 - 46*e^2 - 32*e + 28, 7*e^3 + 46*e^2 + 47*e - 36, -8*e^3 - 25*e^2 + 23*e + 36, 10*e^3 + 14*e^2 - 38*e - 8, 2*e^3 + 12*e^2 + 22*e + 4, -9*e^3 - 30*e^2 - 14*e + 3, -2*e^2 + 2*e + 8, 7*e^3 + 19*e^2 - 9*e - 29, -3*e^3 - 27*e^2 - 34*e + 24, 14*e^3 + 40*e^2 - 2*e - 16, 5*e^3 + 6*e^2 - 29*e - 6, -12*e^3 - 47*e^2 - 15*e + 42, 11*e^3 + 40*e^2 + 25*e - 48, -8*e^3 - 13*e^2 + 17*e - 14, 10*e^3 + 27*e^2 + 2*e - 21, 4*e^3 + 21*e^2 + 33*e - 8, 14*e^3 + 36*e^2 - 21*e - 47, -3*e^3 - e^2 + 9*e - 29, -15*e^3 - 59*e^2 - 15*e + 55, -e^3 - 11*e^2 - 25*e + 13, 14*e^2 + 32*e - 6, -5*e^3 - 19*e^2 - 9*e - 9, 10*e^3 + 32*e^2 + 8*e - 16, -18*e^3 - 44*e^2 + 26*e + 46, -19*e^3 - 61*e^2 - 5*e + 45, -15*e^3 - 62*e^2 - 52*e + 43, -2*e^3 - 8*e^2 + 2*e + 40, -4*e^3 - 18*e^2 - 24*e + 18, 22*e^3 + 60*e^2 + 2*e - 14, 4*e^3 + 8*e^2 - 20, -21*e^3 - 49*e^2 + 35*e + 65, 3*e^3 + 6*e^2 - 29*e - 36, 21*e^3 + 56*e^2 - 8*e - 41, -4*e^3 - 14*e^2 - 6*e - 32, 3*e^3 + 8*e^2 - 15*e - 18, 2*e^3 + 4*e^2 - 10*e - 16, -7*e^3 - 16*e^2 + 27*e + 14, -3*e^3 + 7*e^2 + 23*e - 21, 14*e^3 + 24*e^2 - 48*e - 42, -6*e^3 - 18*e^2 + 3*e + 21, -19*e^3 - 60*e^2 - 13*e + 40, -2*e^3 - 18*e^2 - 34*e, -8*e^3 - 28*e^2 - 6*e + 16, 9*e^3 + 20*e^2 - 9*e - 36, 10*e^3 + 24*e^2 - 20*e - 42, 3*e^3 + 24*e^2 + 23*e - 50, e^3 + 6*e - 5, 7*e^3 - 13*e^2 - 67*e + 17, e^3 + 3*e^2 + e - 53, -11*e^3 - 45*e^2 - 26*e + 14, 6*e^3 - 6*e^2 - 32*e + 32, -2*e^3 - 18*e^2 - 8*e + 26, -23*e^3 - 74*e^2 - 21*e + 62, -4*e^3 - 26*e^2 - 16*e + 30, -3*e^3 + 2*e^2 + 29*e + 28, -14*e^3 - 22*e^2 + 42*e + 12, -4*e^3 - 8*e^2 + 8*e - 8, -12*e^3 - 36*e^2 + 2*e + 52, -3*e^3 - 22*e^2 - 29*e + 58, -11*e^3 - 49*e^2 - 37*e + 15, -12*e^3 - 46*e^2 - 32*e + 10, -22*e^3 - 72*e^2 - 12*e + 50, 2*e^3 + 12*e^2 - 18*e - 44, 15*e^3 + 34*e^2 - 38*e - 41, -5*e^3 + 33*e - 8, 14*e^3 + 28*e^2 - 40*e - 42, -5*e^3 - 4*e^2 + 15*e - 32, 12*e^3 + 25*e^2 - 34*e - 25, -5*e^3 - 21*e^2 + 6*e + 46, 20*e^3 + 71*e^2 + 36*e - 39, -7*e^3 - 12*e^2 - 11*e - 28, -7*e^3 - 35*e^2 - 44*e + 30, 20*e^3 + 40*e^2 - 36*e - 14, -12*e^3 - 26*e^2 + 23*e + 15, -12*e^3 - 31*e^2 + 9*e + 22, 2*e^2 + 8*e + 14, 19*e^3 + 71*e^2 + 50*e - 52, 17*e^3 + 34*e^2 - 26*e - 7, 9*e^3 + 36*e^2 + 15*e - 10, 24*e^3 + 82*e^2 + 16*e - 46, -4*e^3 + 4*e^2 + 36*e - 22, 18*e^3 + 44*e^2 - 28*e - 46, -17*e^3 - 56*e^2 - 12*e + 25, -10*e^3 - 38*e^2 - 26*e + 30, -4*e^2 - 18*e - 14, -14*e^3 - 18*e^2 + 56*e + 40, 4*e^3 + 10*e^2 - 4*e + 16, 13*e^3 + 40*e^2 - 69, -17*e^3 - 36*e^2 + 41*e + 32, e^3 - 13*e - 26, -3*e^3 - 31*e^2 - 32*e + 38, 8*e^3 + 42*e^2 + 42*e - 8, -18*e^3 - 68*e^2 - 10*e + 66, 15*e^3 + 41*e^2 - 14, 7*e^3 + 11*e^2 - 47*e - 27, -2*e^3 - 14*e^2 - 28*e + 18, 21*e^3 + 58*e^2 - 9*e - 44, 11*e^3 + 38*e^2 + 7*e - 58, 12*e^3 + 22*e^2 - 32*e - 38, -8*e^3 - 39*e^2 - 11*e + 52, 8*e^3 + 32*e^2 + 18*e - 30, -3*e^3 - 9*e^2 + e - 21, 14*e^3 + 36*e^2 + 11*e - 1, -11*e^3 - 39*e^2 - 7*e + 57, 17*e^3 + 29*e^2 - 21*e + 29, -22*e^3 - 76*e^2 + 10*e + 74, 9*e^3 + 38*e^2 + 18*e - 1, -19*e^3 - 57*e^2 + 2*e + 44, -32*e^3 - 88*e^2 + 8*e + 44, -31*e^3 - 91*e^2 - 2*e + 64, -e^3 + 4*e^2 + 25*e + 46, -e^3 - 8*e^2 - 21*e - 6, 8*e^3 + 15*e^2 - 2*e - 19, 2*e^3 + 28*e^2 + 34*e - 30, 20*e^3 + 50*e^2 - 37*e - 53, 23*e^3 + 65*e^2 - 16*e - 82, 41*e^3 + 124*e^2 + 15*e - 80, -15*e^3 - 39*e^2 + 37*e + 71, -6*e^3 - 18*e^2 + 6*e + 28, -14*e^3 - 42*e^2 + 6*e + 12, -9*e^2 - 20*e + 19, -22*e^3 - 62*e^2 + 6, -4*e^3 + 21*e^2 + 68*e - 35, 6*e^3 + 18*e^2 - 16*e + 6, 34*e^3 + 102*e^2 + 11*e - 67, 25*e^3 + 74*e^2 + 3*e - 24, e^3 + 15*e^2 - 40, -15*e^3 - 36*e^2 + 12*e + 33, 7*e^3 - 20*e^2 - 83*e + 28, 12*e^3 + 16*e^2 - 52*e - 34, 10*e^3 + 14*e^2 - 40*e - 54, -2*e^3 - 20*e^2 - 46*e - 14, -24*e^3 - 74*e^2 - 6*e + 28, -2*e^3 + 17*e^2 + 41*e - 44, 9*e^3 + 35*e^2 + 43*e - 5, 40*e^3 + 124*e^2 + 24*e - 62, -14*e^3 - 24*e^2 + 38*e + 8, 28*e^3 + 97*e^2 + 18*e - 51, -5*e^3 - 14*e^2 + e + 26, -15*e^3 - 66*e^2 - 41*e + 42, 2*e^3 + 8*e^2 - 14*e + 12, 7*e^3 + 16*e^2 - 3*e - 20, -8*e^3 + 4*e^2 + 56*e - 10, 21*e^3 + 41*e^2 - 45*e - 29, -2*e^3 + 2*e^2 + 33*e + 29, -4*e^3 - 39*e^2 - 52*e + 49, -20*e^3 - 60*e^2 + 42*e + 72, e^3 - 9*e^2 - 18*e - 10, -16*e^3 - 48*e^2 - 28*e + 4, 8*e^3 + 18*e^2 - 30*e + 4, 24*e^3 + 48*e^2 - 65*e - 79, -3*e^3 - 12*e^2 - 22*e - 1, 14*e^3 + 41*e^2 + 2*e - 35, -5*e^3 - 33*e^2 - 57*e + 43, -4*e^3 - 34*e^2 - 51*e + 27, -12*e^3 - 54*e^2 - 45*e + 39, -6*e^2 - 28*e - 36, -27*e^3 - 75*e^2 - 2*e + 58, 16*e^3 + 54*e^2 + 24*e - 20, 10*e^3 + 18*e^2 - e + 55, 25*e^3 + 99*e^2 + 42*e - 52, -17*e^3 - 70*e^2 - 21*e + 44, -4*e^3 + 7*e^2 + 18*e - 45, -29*e^3 - 74*e^2 + 9*e + 22, -5*e^3 - 33*e^2 - 16*e + 82, -3*e^3 - 29*e^2 - 21*e + 61, -11*e^3 - 35*e^2 + 21*e + 45, 8*e^3 + 29*e^2 - 8*e - 15, 18*e^3 + 78*e^2 + 58*e - 58, e^3 - 20*e^2 - 63*e - 6, -4*e^3 + 6*e^2 + 20*e - 68, 9*e^3 + 27*e^2 + 36*e + 16]; 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;