/* 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^9 + x^8 - 22*x^7 - 20*x^6 + 132*x^5 + 110*x^4 - 206*x^3 - 168*x^2 + 55*x + 45; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1, -1, -7/44*e^8 - 1/22*e^7 + 157/44*e^6 + 31/44*e^5 - 244/11*e^4 - 30/11*e^3 + 1655/44*e^2 + 107/44*e - 529/44, 1/8*e^8 - 11/4*e^6 + 1/4*e^5 + 65/4*e^4 - 5/2*e^3 - 95/4*e^2 + 13/4*e + 65/8, -9/88*e^8 - 1/88*e^7 + 205/88*e^6 - 1/88*e^5 - 1313/88*e^4 + 149/88*e^3 + 2395/88*e^2 - 359/88*e - 465/44, -1/4*e^8 + 11/2*e^6 - 1/2*e^5 - 65/2*e^4 + 5*e^3 + 93/2*e^2 - 9/2*e - 45/4, 39/88*e^8 + 1/11*e^7 - 435/44*e^6 - 51/44*e^5 + 2667/44*e^4 + 38/11*e^3 - 4333/44*e^2 - 357/44*e + 2875/88, 2/11*e^8 + 3/88*e^7 - 351/88*e^6 - 41/88*e^5 + 2069/88*e^4 + 213/88*e^3 - 2961/88*e^2 - 683/88*e + 535/88, -1/88*e^8 + 3/44*e^7 + 3/11*e^6 - 63/44*e^5 - 65/44*e^4 + 323/44*e^3 - 6/11*e^2 - 199/44*e - 85/88, 1/2*e^8 + 1/4*e^7 - 11*e^6 - 17/4*e^5 + 131/2*e^4 + 71/4*e^3 - 98*e^2 - 67/4*e + 25, -61/88*e^8 - 19/88*e^7 + 1343/88*e^6 + 289/88*e^5 - 8007/88*e^4 - 1085/88*e^3 + 12109/88*e^2 + 1363/88*e - 815/22, 15/44*e^8 + 7/88*e^7 - 665/88*e^6 - 81/88*e^5 + 3999/88*e^4 + 57/88*e^3 - 6007/88*e^2 + 445/88*e + 1439/88, -6/11*e^8 - 5/22*e^7 + 133/11*e^6 + 83/22*e^5 - 1615/22*e^4 - 355/22*e^3 + 1296/11*e^2 + 515/22*e - 789/22, 57/88*e^8 + 5/44*e^7 - 629/44*e^6 - 61/44*e^5 + 940/11*e^4 + 201/44*e^3 - 5683/44*e^2 - 625/44*e + 2799/88, -5/8*e^8 + 14*e^6 - 5/4*e^5 - 173/2*e^4 + 25/2*e^3 + 287/2*e^2 - 45/4*e - 371/8, -6/11*e^8 + 1/44*e^7 + 133/11*e^6 - 65/44*e^5 - 802/11*e^4 + 555/44*e^3 + 1252/11*e^2 - 675/44*e - 400/11, -17/88*e^8 - 19/88*e^7 + 375/88*e^6 + 377/88*e^5 - 2287/88*e^4 - 1965/88*e^3 + 4013/88*e^2 + 2155/88*e - 215/11, 53/88*e^8 + 3/22*e^7 - 148/11*e^6 - 41/22*e^5 + 3643/44*e^4 + 68/11*e^3 - 1497/11*e^2 - 116/11*e + 3405/88, 3/44*e^8 + 1/11*e^7 - 18/11*e^6 - 21/11*e^5 + 261/22*e^4 + 115/11*e^3 - 316/11*e^2 - 125/11*e + 915/44, -19/44*e^8 - 7/44*e^7 + 423/44*e^6 + 103/44*e^5 - 2591/44*e^4 - 277/44*e^3 + 4225/44*e^2 - 27/44*e - 615/22, -25/88*e^8 + 7/88*e^7 + 545/88*e^6 - 213/88*e^5 - 3173/88*e^4 + 1553/88*e^3 + 4531/88*e^2 - 2151/88*e - 705/44, -9/22*e^8 - 15/88*e^7 + 809/88*e^6 + 249/88*e^5 - 5087/88*e^4 - 1065/88*e^3 + 9151/88*e^2 + 1435/88*e - 3225/88, -31/88*e^8 - 23/88*e^7 + 689/88*e^6 + 417/88*e^5 - 4217/88*e^4 - 1897/88*e^3 + 6883/88*e^2 + 1995/88*e - 218/11, 53/88*e^8 + 3/22*e^7 - 581/44*e^6 - 71/44*e^5 + 3423/44*e^4 + 24/11*e^3 - 5053/44*e^2 + 9/44*e + 3625/88, 1/4*e^8 - 21/4*e^6 + 1/2*e^5 + 109/4*e^4 - 5*e^3 - 79/4*e^2 + 11/2*e - 25/2, 25/88*e^8 - 9/44*e^7 - 139/22*e^6 + 111/22*e^5 + 420/11*e^4 - 1365/44*e^3 - 664/11*e^2 + 350/11*e + 2235/88, -113/88*e^8 - 6/11*e^7 + 623/22*e^6 + 197/22*e^5 - 7477/44*e^4 - 797/22*e^3 + 5805/22*e^2 + 420/11*e - 6525/88, -19/88*e^8 + 13/44*e^7 + 217/44*e^6 - 295/44*e^5 - 689/22*e^4 + 1693/44*e^3 + 2415/44*e^2 - 1647/44*e - 1945/88, 5/88*e^8 - 1/11*e^7 - 15/11*e^6 + 21/11*e^5 + 435/44*e^4 - 219/22*e^3 - 278/11*e^2 + 217/22*e + 1085/88, -23/22*e^8 - 21/44*e^7 + 254/11*e^6 + 353/44*e^5 - 1528/11*e^4 - 1491/44*e^3 + 2374/11*e^2 + 1767/44*e - 1295/22, -41/88*e^8 - 7/88*e^7 + 907/88*e^6 + 81/88*e^5 - 5451/88*e^4 - 233/88*e^3 + 8273/88*e^2 + 875/88*e - 445/22, -39/88*e^8 - 19/88*e^7 + 881/88*e^6 + 333/88*e^5 - 5609/88*e^4 - 1481/88*e^3 + 10415/88*e^2 + 1715/88*e - 925/22, 5/8*e^8 + 3/8*e^7 - 109/8*e^6 - 53/8*e^5 + 641/8*e^4 + 229/8*e^3 - 975/8*e^2 - 191/8*e + 165/4, -47/88*e^8 - 13/44*e^7 + 130/11*e^6 + 229/44*e^5 - 3165/44*e^4 - 989/44*e^3 + 1335/11*e^2 + 745/44*e - 4215/88, 89/88*e^8 + 19/44*e^7 - 245/11*e^6 - 311/44*e^5 + 5851/44*e^4 + 1195/44*e^3 - 2216/11*e^2 - 747/44*e + 4705/88, 13/22*e^8 - 1/22*e^7 - 145/11*e^6 + 43/22*e^5 + 889/11*e^4 - 313/22*e^3 - 1459/11*e^2 + 235/22*e + 1061/22, -43/88*e^8 - 3/44*e^7 + 483/44*e^6 + 19/44*e^5 - 1513/22*e^4 + 117/44*e^3 + 5381/44*e^2 - 241/44*e - 3985/88, 29/44*e^8 + 13/44*e^7 - 641/44*e^6 - 109/22*e^5 + 970/11*e^4 + 945/44*e^3 - 6231/44*e^2 - 343/11*e + 1915/44, -5/44*e^8 - 7/22*e^7 + 109/44*e^6 + 147/22*e^5 - 639/44*e^4 - 805/22*e^3 + 827/44*e^2 + 875/22*e + 23/11, 1/22*e^8 - 1/44*e^7 - 12/11*e^6 + 21/44*e^5 + 87/11*e^4 - 71/44*e^3 - 207/11*e^2 - 205/44*e + 261/22, -5/8*e^8 - 1/8*e^7 + 111/8*e^6 + 11/8*e^5 - 675/8*e^4 - 7/8*e^3 + 1085/8*e^2 - 15/8*e - 42, 17/22*e^8 + 5/44*e^7 - 761/44*e^6 - 39/44*e^5 + 4717/44*e^4 - 85/44*e^3 - 8059/44*e^2 - 9/44*e + 2725/44, 17/22*e^8 + 4/11*e^7 - 375/22*e^6 - 135/22*e^5 + 2243/22*e^4 + 284/11*e^3 - 3397/22*e^2 - 615/22*e + 475/11, -13/88*e^8 + 3/22*e^7 + 67/22*e^6 - 37/11*e^5 - 669/44*e^4 + 233/11*e^3 + 207/22*e^2 - 551/22*e + 1095/88, 1/22*e^8 + 5/22*e^7 - 12/11*e^6 - 105/22*e^5 + 87/11*e^4 + 575/22*e^3 - 196/11*e^2 - 647/22*e + 85/22, 7/8*e^8 + 1/2*e^7 - 19*e^6 - 35/4*e^5 + 110*e^4 + 77/2*e^3 - 152*e^2 - 157/4*e + 225/8, -19/22*e^8 - 7/22*e^7 + 423/22*e^6 + 239/44*e^5 - 5149/44*e^4 - 287/11*e^3 + 4049/22*e^2 + 1915/44*e - 1965/44, 69/88*e^8 + 1/22*e^7 - 773/44*e^6 + 6/11*e^5 + 1190/11*e^4 - 102/11*e^3 - 7639/44*e^2 + 117/22*e + 3995/88, -19/88*e^8 + 15/88*e^7 + 423/88*e^6 - 337/88*e^5 - 2503/88*e^4 + 1813/88*e^3 + 3169/88*e^2 - 1391/88*e + 375/44, 21/88*e^8 + 3/44*e^7 - 219/44*e^6 - 13/11*e^5 + 1101/44*e^4 + 279/44*e^3 - 563/44*e^2 - 215/22*e - 1075/88, 101/88*e^8 + 4/11*e^7 - 281/11*e^6 - 62/11*e^5 + 6873/44*e^4 + 469/22*e^3 - 2804/11*e^2 - 505/22*e + 7705/88, 79/44*e^8 + 8/11*e^7 - 1753/44*e^6 - 259/22*e^5 + 10655/44*e^4 + 502/11*e^3 - 17031/44*e^2 - 889/22*e + 2615/22, 19/22*e^8 + 25/44*e^7 - 423/22*e^6 - 459/44*e^5 + 2591/22*e^4 + 2171/44*e^3 - 4203/22*e^2 - 2245/44*e + 560/11, -45/88*e^8 - 2/11*e^7 + 124/11*e^6 + 135/44*e^5 - 734/11*e^4 - 164/11*e^3 + 1028/11*e^2 + 1165/44*e - 1779/88, 63/88*e^8 + 5/11*e^7 - 345/22*e^6 - 365/44*e^5 + 1010/11*e^4 + 853/22*e^3 - 1382/11*e^2 - 1785/44*e + 2341/88, -51/88*e^8 - 13/88*e^7 + 1147/88*e^6 + 207/88*e^5 - 7147/88*e^4 - 1055/88*e^3 + 12061/88*e^2 + 2505/88*e - 2195/44, -95/88*e^8 - 3/11*e^7 + 263/11*e^6 + 175/44*e^5 - 1585/11*e^4 - 305/22*e^3 + 4899/22*e^2 + 455/44*e - 5325/88, 73/88*e^8 + 23/44*e^7 - 799/44*e^6 - 203/22*e^5 + 4701/44*e^4 + 1765/44*e^3 - 6773/44*e^2 - 805/22*e + 2949/88, 27/88*e^8 - 19/88*e^7 - 615/88*e^6 + 465/88*e^5 + 3895/88*e^4 - 2933/88*e^3 - 6701/88*e^2 + 3475/88*e + 1527/44, -19/44*e^8 - 9/22*e^7 + 217/22*e^6 + 89/11*e^5 - 700/11*e^4 - 925/22*e^3 + 2525/22*e^2 + 535/11*e - 1285/44, -1/8*e^8 + 1/4*e^7 + 11/4*e^6 - 6*e^5 - 63/4*e^4 + 153/4*e^3 + 75/4*e^2 - 99/2*e - 5/8, -1/8*e^8 - 1/8*e^7 + 21/8*e^6 + 15/8*e^5 - 113/8*e^4 - 23/8*e^3 + 95/8*e^2 - 115/8*e + 53/4, 1/44*e^8 + 4/11*e^7 - 13/44*e^6 - 325/44*e^5 - 1/22*e^4 + 821/22*e^3 + 213/44*e^2 - 1505/44*e - 487/44, 97/88*e^8 + 3/22*e^7 - 269/11*e^6 - 27/44*e^5 + 1634/11*e^4 - 183/22*e^3 - 2641/11*e^2 + 515/44*e + 7255/88, 125/88*e^8 - 1/44*e^7 - 1401/44*e^6 + 131/44*e^5 + 2166/11*e^4 - 1149/44*e^3 - 14479/44*e^2 + 895/44*e + 10515/88, -39/88*e^8 + 3/88*e^7 + 859/88*e^6 - 129/88*e^5 - 5059/88*e^4 + 829/88*e^3 + 6961/88*e^2 + 725/88*e - 255/44, -161/88*e^8 - 23/44*e^7 + 889/22*e^6 + 159/22*e^5 - 2674/11*e^4 - 907/44*e^3 + 4182/11*e^2 + 265/11*e - 10715/88, -37/88*e^8 - 5/22*e^7 + 411/44*e^6 + 47/11*e^5 - 626/11*e^4 - 249/11*e^3 + 3985/44*e^2 + 757/22*e - 1715/88, -9/4*e^8 - 3/4*e^7 + 50*e^6 + 47/4*e^5 - 609/2*e^4 - 185/4*e^3 + 488*e^2 + 253/4*e - 545/4, -115/88*e^8 - 7/44*e^7 + 1281/44*e^6 + 12/11*e^5 - 7827/44*e^4 + 207/44*e^3 + 12695/44*e^2 - 125/11*e - 9335/88, -3/11*e^8 - 4/11*e^7 + 133/22*e^6 + 157/22*e^5 - 835/22*e^4 - 394/11*e^3 + 1615/22*e^2 + 725/22*e - 565/22, -147/88*e^8 - 21/44*e^7 + 408/11*e^6 + 80/11*e^5 - 2474/11*e^4 - 1249/44*e^3 + 7729/22*e^2 + 911/22*e - 8425/88, -35/44*e^8 - 21/44*e^7 + 785/44*e^6 + 375/44*e^5 - 4891/44*e^4 - 1711/44*e^3 + 8363/44*e^2 + 1877/44*e - 1295/22, 5/11*e^8 + 3/11*e^7 - 109/11*e^6 - 52/11*e^5 + 1289/22*e^4 + 224/11*e^3 - 1036/11*e^2 - 210/11*e + 1125/22, 103/88*e^8 + 21/44*e^7 - 287/11*e^6 - 171/22*e^5 + 3529/22*e^4 + 1359/44*e^3 - 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175/2*e^4 - 97/2*e^3 + 95*e^2 + 95/2*e - 107/4, 51/44*e^8 + 37/88*e^7 - 2173/88*e^6 - 579/88*e^5 + 11951/88*e^4 + 2275/88*e^3 - 14035/88*e^2 - 2953/88*e + 1135/88, 53/88*e^8 - 21/88*e^7 - 1195/88*e^6 + 507/88*e^5 + 7539/88*e^4 - 2723/88*e^3 - 13945/88*e^2 + 1569/88*e + 735/11, -207/44*e^8 - 39/22*e^7 + 1143/11*e^6 + 643/22*e^5 - 13741/22*e^4 - 2791/22*e^3 + 10672/11*e^2 + 3775/22*e - 11875/44, -51/88*e^8 - 35/88*e^7 + 1147/88*e^6 + 625/88*e^5 - 7147/88*e^4 - 2397/88*e^3 + 12017/88*e^2 - 245/88*e - 2525/44, 313/88*e^8 + 31/22*e^7 - 1735/22*e^6 - 1027/44*e^5 + 5276/11*e^4 + 1062/11*e^3 - 8550/11*e^2 - 4175/44*e + 19675/88, -13/8*e^8 - 9/8*e^7 + 285/8*e^6 + 159/8*e^5 - 1697/8*e^4 - 691/8*e^3 + 2635/8*e^2 + 649/8*e - 455/4, 119/88*e^8 + 89/88*e^7 - 2669/88*e^6 - 1671/88*e^5 + 16625/88*e^4 + 8079/88*e^3 - 27871/88*e^2 - 8221/88*e + 790/11, -37/44*e^8 - 9/44*e^7 + 389/22*e^6 + 123/44*e^5 - 1032/11*e^4 - 463/44*e^3 + 2049/22*e^2 + 1169/44*e - 175/44, 23/88*e^8 + 19/44*e^7 - 243/44*e^6 - 377/44*e^5 + 338/11*e^4 + 1987/44*e^3 - 1637/44*e^2 - 2705/44*e + 921/88, 37/11*e^8 + 3/44*e^7 - 1655/22*e^6 + 201/44*e^5 + 5096/11*e^4 - 2119/44*e^3 - 16809/22*e^2 + 1055/44*e + 2837/11, 9/11*e^8 + 1/11*e^7 - 399/22*e^6 - 10/11*e^5 + 2373/22*e^4 + 27/11*e^3 - 3349/22*e^2 - 114/11*e + 705/22, 211/88*e^8 + 43/88*e^7 - 4767/88*e^6 - 441/88*e^5 + 30147/88*e^4 - 159/88*e^3 - 53969/88*e^2 + 1665/88*e + 10183/44, 301/88*e^8 + 5/22*e^7 - 837/11*e^6 + 27/22*e^5 + 20379/44*e^4 - 356/11*e^3 - 8149/11*e^2 + 265/11*e + 19821/88, 225/88*e^8 + 9/22*e^7 - 2513/44*e^6 - 191/44*e^5 + 15417/44*e^4 + 94/11*e^3 - 25125/44*e^2 - 1799/44*e + 14505/88, -5/44*e^8 + 2/11*e^7 + 30/11*e^6 - 95/22*e^5 - 413/22*e^4 + 296/11*e^3 + 413/11*e^2 - 819/22*e - 1305/44, 59/22*e^8 + 29/44*e^7 - 2645/44*e^6 - 100/11*e^5 + 4110/11*e^4 + 1201/44*e^3 - 28051/44*e^2 - 575/22*e + 4465/22, 177/88*e^8 + 27/88*e^7 - 3841/88*e^6 - 193/88*e^5 + 22009/88*e^4 - 1031/88*e^3 - 28871/88*e^2 + 3225/88*e + 2985/44, -39/44*e^8 + 29/22*e^7 + 435/22*e^6 - 653/22*e^5 - 1295/11*e^4 + 3709/22*e^3 + 3629/22*e^2 - 3889/22*e - 1885/44, 5/44*e^8 + 7/22*e^7 - 131/44*e^6 - 305/44*e^5 + 567/22*e^4 + 452/11*e^3 - 3797/44*e^2 - 2729/44*e + 2845/44, 51/44*e^8 + 1/22*e^7 - 1147/44*e^6 + 23/22*e^5 + 7213/44*e^4 - 303/22*e^3 - 12897/44*e^2 + 381/22*e + 1235/11, -13/4*e^8 - 3/4*e^7 + 285/4*e^6 + 9*e^5 - 843/2*e^4 - 59/4*e^3 + 2519/4*e^2 + 15/2*e - 685/4, -223/88*e^8 - 35/44*e^7 + 2467/44*e^6 + 537/44*e^5 - 3698/11*e^4 - 2111/44*e^3 + 22313/44*e^2 + 3429/44*e - 12905/88, 27/11*e^8 + 17/22*e^7 - 604/11*e^6 - 247/22*e^5 + 7449/22*e^4 + 657/22*e^3 - 6107/11*e^2 + 251/22*e + 3765/22, 16/11*e^8 + 3/11*e^7 - 1415/44*e^6 - 93/22*e^5 + 8529/44*e^4 + 279/11*e^3 - 13549/44*e^2 - 1905/22*e + 4307/44, -235/88*e^8 - 8/11*e^7 + 1311/22*e^6 + 441/44*e^5 - 8061/22*e^4 - 282/11*e^3 + 13317/22*e^2 - 125/44*e - 17841/88, 35/22*e^8 + 9/44*e^7 - 387/11*e^6 - 23/22*e^5 + 9287/44*e^4 - 395/44*e^3 - 3604/11*e^2 - 130/11*e + 5147/44, -7/22*e^8 + 9/22*e^7 + 347/44*e^6 - 189/22*e^5 - 2579/44*e^4 + 925/22*e^3 + 6093/44*e^2 - 575/22*e - 2983/44, -32/11*e^8 - 6/11*e^7 + 713/11*e^6 + 60/11*e^5 - 8727/22*e^4 + 3/11*e^3 + 7099/11*e^2 - 20/11*e - 4175/22, -19/44*e^8 + 1/11*e^7 + 401/44*e^6 - 75/22*e^5 - 2129/44*e^4 + 313/11*e^3 + 2135/44*e^2 - 723/22*e - 225/11, -29/44*e^8 - 1/22*e^7 + 641/44*e^6 - 23/22*e^5 - 3891/44*e^4 + 501/22*e^3 + 6407/44*e^2 - 1855/22*e - 410/11, -25/88*e^8 - 1/22*e^7 + 75/11*e^6 + 31/44*e^5 - 1093/22*e^4 - 159/22*e^3 + 1368/11*e^2 + 1757/44*e - 7515/88, -1/8*e^8 + 3/8*e^7 + 23/8*e^6 - 57/8*e^5 - 139/8*e^4 + 213/8*e^3 + 133/8*e^2 + 85/8*e + 65/2, 61/88*e^8 + 107/88*e^7 - 1255/88*e^6 - 2137/88*e^5 + 6423/88*e^4 + 11249/88*e^3 - 5729/88*e^2 - 14079/88*e + 215/11]; 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;