/* 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![45, 15, -14, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, 2/3*w^2 + 1/3*w - 5], [5, 5, 2/3*w^2 - 5/3*w - 4], [9, 3, -w + 3], [9, 3, w + 2], [11, 11, 2/3*w^2 + 1/3*w - 4], [16, 2, 2], [19, 19, 1/3*w^3 - 10/3*w - 3], [19, 19, 1/3*w^3 - w^2 - 7/3*w + 6], [29, 29, -1/3*w^3 + 2/3*w^2 + 8/3*w - 2], [29, 29, 1/3*w^3 - 1/3*w^2 - 3*w + 1], [31, 31, -1/3*w^3 + 1/3*w^2 + 3*w + 1], [31, 31, -1/3*w^3 + 2/3*w^2 + 8/3*w - 4], [49, 7, 1/3*w^3 - 10/3*w - 1], [49, 7, -1/3*w^3 + w^2 + 7/3*w - 4], [59, 59, -1/3*w^3 + 5/3*w^2 + 5/3*w - 11], [59, 59, -2/3*w^3 + 5/3*w^2 + 5*w - 13], [61, 61, -1/3*w^3 + 5/3*w^2 + 2/3*w - 9], [61, 61, -1/3*w^3 + 1/3*w^2 + 4*w + 2], [61, 61, 2/3*w^2 - 5/3*w - 1], [61, 61, -1/3*w^2 + 4/3*w + 6], [71, 71, -4/3*w^2 + 7/3*w + 11], [79, 79, -1/3*w^2 + 7/3*w - 3], [79, 79, -2/3*w^3 + 1/3*w^2 + 16/3*w - 4], [101, 101, 2/3*w^2 - 5/3*w - 6], [101, 101, -2/3*w^2 - 1/3*w + 7], [109, 109, -4/3*w^2 + 1/3*w + 9], [109, 109, 1/3*w^3 - 1/3*w^2 - 3*w - 4], [121, 11, -1/3*w^2 + 1/3*w + 6], [131, 131, -2/3*w^3 + 2/3*w^2 + 5*w - 7], [131, 131, 2/3*w^3 - 4/3*w^2 - 13/3*w - 2], [151, 151, -1/3*w^3 - 2/3*w^2 + 5*w + 7], [151, 151, 1/3*w^3 - 10/3*w + 2], [169, 13, 5/3*w^2 + 1/3*w - 11], [169, 13, 5/3*w^2 - 11/3*w - 9], [181, 181, -1/3*w^3 + 4/3*w^2 + 3*w - 8], [181, 181, -1/3*w^3 - 1/3*w^2 + 14/3*w + 4], [191, 191, -2/3*w^3 + w^2 + 17/3*w - 9], [191, 191, 1/3*w^3 - 2*w^2 - 7/3*w + 13], [191, 191, 2/3*w^3 - 1/3*w^2 - 22/3*w - 7], [191, 191, 2/3*w^3 - w^2 - 17/3*w - 3], [199, 199, 1/3*w^3 + w^2 - 13/3*w - 9], [199, 199, 2/3*w^2 - 5/3*w - 7], [199, 199, 1/3*w^3 - w^2 - 4/3*w + 6], [199, 199, -1/3*w^3 + 2*w^2 + 4/3*w - 12], [269, 269, -1/3*w^3 - 4/3*w^2 + 11/3*w + 14], [269, 269, 1/3*w^3 - 7/3*w^2 + 16], [271, 271, 2/3*w^2 + 1/3*w - 9], [271, 271, 2/3*w^2 - 5/3*w - 8], [281, 281, 1/3*w^3 + w^2 - 13/3*w - 11], [281, 281, -2/3*w^3 + 20/3*w + 3], [281, 281, -w^2 + 3*w + 2], [281, 281, -1/3*w^3 + 2*w^2 + 4/3*w - 14], [289, 17, w^3 - 8/3*w^2 - 16/3*w + 4], [289, 17, -w^3 + 1/3*w^2 + 23/3*w - 3], [331, 331, -2/3*w^3 + 2*w^2 + 11/3*w - 7], [331, 331, 2/3*w^3 - 17/3*w - 2], [359, 359, -1/3*w^3 - 1/3*w^2 + 11/3*w - 2], [359, 359, 1/3*w^3 + 4/3*w^2 - 17/3*w - 12], [361, 19, 4/3*w^2 - 4/3*w - 11], [389, 389, 5/3*w^2 - 8/3*w - 13], [389, 389, 5/3*w^2 - 2/3*w - 14], [401, 401, -1/3*w^3 + 5/3*w^2 + 11/3*w - 12], [401, 401, 1/3*w^3 + 2/3*w^2 - 6*w - 7], [409, 409, 1/3*w^3 + 1/3*w^2 - 14/3*w - 3], [409, 409, -1/3*w^3 + 4/3*w^2 + 3*w - 7], [421, 421, -1/3*w^3 + 5/3*w^2 + 11/3*w - 13], [421, 421, 1/3*w^3 + 2/3*w^2 - 6*w - 8], [431, 431, 2/3*w^2 + 4/3*w - 9], [431, 431, 2/3*w^2 - 8/3*w - 7], [449, 449, 2/3*w^3 - 17/3*w + 2], [449, 449, 1/3*w^3 + 4/3*w^2 - 20/3*w - 11], [461, 461, 1/3*w^2 + 5/3*w - 8], [461, 461, 1/3*w^3 + 1/3*w^2 - 17/3*w - 3], [461, 461, -1/3*w^3 + 4/3*w^2 + 4*w - 8], [461, 461, 1/3*w^2 - 7/3*w - 6], [479, 479, -1/3*w^3 + 2/3*w^2 + 5/3*w - 9], [479, 479, 4/3*w^3 + w^2 - 40/3*w - 18], [479, 479, 4/3*w^3 - 5*w^2 - 22/3*w + 29], [479, 479, 1/3*w^3 - 1/3*w^2 - 2*w - 7], [491, 491, 2/3*w^3 - 5/3*w^2 - 5*w + 8], [491, 491, -1/3*w^3 + 3*w^2 + 10/3*w - 24], [491, 491, 1/3*w^3 + 2*w^2 - 25/3*w - 18], [491, 491, 2/3*w^3 - 1/3*w^2 - 19/3*w - 2], [499, 499, -1/3*w^3 + 10/3*w^2 - 4*w - 17], [499, 499, 1/3*w^3 - 7/3*w - 9], [499, 499, -1/3*w^3 + w^2 + 4/3*w - 11], [499, 499, -w^3 + 5/3*w^2 + 22/3*w - 14], [509, 509, w^3 - 14/3*w^2 - 10/3*w + 24], [509, 509, -2/3*w^2 + 5/3*w + 11], [509, 509, 2/3*w^2 + 1/3*w - 12], [509, 509, -2/3*w^3 + 3*w^2 + 8/3*w - 18], [521, 521, 1/3*w^3 - 13/3*w + 1], [521, 521, w^3 + 2/3*w^2 - 32/3*w - 12], [521, 521, 2/3*w^3 + 7/3*w^2 - 13*w - 23], [521, 521, -1/3*w^3 + w^2 + 10/3*w - 3], [541, 541, 2/3*w^3 - 17/3*w - 3], [541, 541, -1/3*w^3 + 4/3*w^2 + 2*w - 14], [541, 541, -1/3*w^3 + 4/3*w^2 + w - 1], [541, 541, -1/3*w^3 + 1/3*w^2 + 5*w - 4], [569, 569, 1/3*w^3 + 11/3*w^2 - 12*w - 26], [569, 569, -w^3 + 10/3*w^2 + 23/3*w - 21], [571, 571, w^3 - 7/3*w^2 - 23/3*w + 17], [571, 571, 2/3*w^3 - 3*w^2 - 11/3*w + 17], [599, 599, 1/3*w^3 + 2/3*w^2 - 4*w - 1], [599, 599, 1/3*w^3 - 5/3*w^2 - 5/3*w + 4], [601, 601, 1/3*w^3 - 7/3*w - 6], [601, 601, -2/3*w^3 + 2*w^2 + 11/3*w - 9], [601, 601, 2/3*w^3 - 17/3*w - 4], [601, 601, -1/3*w^3 + w^2 + 4/3*w - 8], [619, 619, -2/3*w^3 + 10/3*w^2 + 19/3*w - 26], [619, 619, 2/3*w^3 + 4/3*w^2 - 11*w - 17], [631, 631, 1/3*w^3 - 7/3*w^2 - w + 14], [631, 631, 1/3*w^3 + 4/3*w^2 - 14/3*w - 11], [659, 659, -2/3*w^3 + 2/3*w^2 + 6*w + 1], [659, 659, 2/3*w^3 - 4/3*w^2 - 16/3*w + 7], [661, 661, -2/3*w^3 + 13/3*w^2 + 16/3*w - 32], [661, 661, -2/3*w^3 - 7/3*w^2 + 12*w + 23], [691, 691, 1/3*w^3 - 11/3*w^2 - 11/3*w + 29], [691, 691, -2/3*w^3 + 10/3*w^2 + 10/3*w - 23], [701, 701, 2/3*w^3 - 4/3*w^2 - 13/3*w - 3], [701, 701, 2/3*w^3 - 4/3*w^2 - 16/3*w - 1], [701, 701, -2/3*w^3 + 2/3*w^2 + 6*w - 7], [701, 701, -2/3*w^3 + 2/3*w^2 + 5*w - 8], [709, 709, -4/3*w^3 + 31/3*w - 2], [709, 709, -w^3 + 1/3*w^2 + 32/3*w + 4], [719, 719, -2/3*w^3 + w^2 + 17/3*w - 2], [719, 719, 1/3*w^3 - 2*w^2 - 4/3*w + 9], [719, 719, 1/3*w^3 + w^2 - 13/3*w - 6], [719, 719, 2/3*w^3 - w^2 - 17/3*w + 4], [739, 739, 2/3*w^3 + 2/3*w^2 - 19/3*w - 11], [739, 739, -2/3*w^3 + 8/3*w^2 + 3*w - 16], [751, 751, -1/3*w^3 + w^2 + 4/3*w - 9], [751, 751, 1/3*w^3 - 7/3*w - 7], [761, 761, 1/3*w^3 - 16/3*w - 4], [761, 761, w^3 - 1/3*w^2 - 29/3*w - 12], [761, 761, -w^3 + 8/3*w^2 + 22/3*w - 21], [761, 761, -1/3*w^3 + w^2 + 13/3*w - 9], [769, 769, w^3 - 7/3*w^2 - 23/3*w + 8], [769, 769, 1/3*w^3 + 4/3*w^2 - 14/3*w - 14], [769, 769, -1/3*w^3 + 7/3*w^2 + w - 17], [769, 769, 1/3*w^3 - 3*w^2 + 11/3*w + 13], [811, 811, 1/3*w^3 + w^2 - 10/3*w - 12], [811, 811, -1/3*w^3 + 2*w^2 + 1/3*w - 14], [821, 821, -1/3*w^3 + 13/3*w - 3], [821, 821, -1/3*w^3 + w^2 + 10/3*w - 1], [839, 839, -w^3 - 5/3*w^2 + 32/3*w + 18], [839, 839, 1/3*w^3 + 3*w^2 - 7/3*w - 22], [839, 839, 1/3*w^3 - 4*w^2 + 14/3*w + 21], [839, 839, -w^3 + 14/3*w^2 + 13/3*w - 26], [841, 29, -5/3*w^2 + 5/3*w + 11], [859, 859, -w^3 + 5/3*w^2 + 25/3*w - 1], [859, 859, -w^3 + 4/3*w^2 + 26/3*w - 8], [911, 911, -1/3*w^3 + 1/3*w^2 + 5*w + 1], [911, 911, 1/3*w^3 - 2/3*w^2 - 14/3*w + 6], [919, 919, 2/3*w^3 - 2/3*w^2 - 5*w - 7], [919, 919, -2/3*w^3 + 7/3*w^2 + 10/3*w - 7], [919, 919, 2/3*w^3 + 1/3*w^2 - 6*w - 2], [919, 919, -2/3*w^3 + 4/3*w^2 + 13/3*w - 12], [941, 941, 5/3*w^2 + 1/3*w - 9], [941, 941, 5/3*w^2 - 11/3*w - 7], [961, 31, 5/3*w^2 - 5/3*w - 13], [971, 971, -1/3*w^3 + w^2 + 13/3*w - 8], [971, 971, 2/3*w^3 - 2/3*w^2 - 5*w + 2], [971, 971, -2/3*w^3 + 4/3*w^2 + 13/3*w - 3], [971, 971, 1/3*w^3 - 16/3*w - 3], [991, 991, 10/3*w^2 - 22/3*w - 19], [991, 991, 10/3*w^2 + 2/3*w - 23], [1009, 1009, -2*w^2 + 3*w + 16], [1009, 1009, 2*w^2 - w - 17], [1019, 1019, 1/3*w^3 + w^2 - 19/3*w - 14], [1019, 1019, -1/3*w^3 + 2*w^2 + 10/3*w - 19], [1021, 1021, 2/3*w^3 - 3*w^2 - 14/3*w + 19], [1021, 1021, 11/3*w^2 + 4/3*w - 27], [1021, 1021, 11/3*w^2 - 26/3*w - 22], [1021, 1021, -2/3*w^3 - w^2 + 26/3*w + 12], [1031, 1031, -2/3*w^3 + 5/3*w^2 + 4*w + 1], [1031, 1031, -2/3*w^3 + 1/3*w^2 + 16/3*w - 6], [1049, 1049, -1/3*w^3 - 2/3*w^2 + 4*w - 1], [1049, 1049, 1/3*w^3 + 5/3*w^2 - 6*w - 14], [1051, 1051, 1/3*w^3 - 16/3*w - 2], [1051, 1051, -1/3*w^3 + w^2 + 13/3*w - 7], [1061, 1061, 11/3*w^2 + 1/3*w - 27], [1061, 1061, 11/3*w^2 - 23/3*w - 23], [1069, 1069, w^2 + 2*w - 11], [1069, 1069, w^2 - 4*w - 8], [1109, 1109, w^3 - 5/3*w^2 - 19/3*w - 1], [1109, 1109, -w^3 - w^2 + 12*w + 16], [1129, 1129, -w^3 + 5/3*w^2 + 25/3*w - 11], [1129, 1129, -w^3 + 4/3*w^2 + 26/3*w + 2], [1151, 1151, 1/3*w^3 + w^2 - 16/3*w - 7], [1151, 1151, -1/3*w^3 + 2*w^2 + 7/3*w - 11], [1171, 1171, w^3 - 8/3*w^2 - 16/3*w + 1], [1171, 1171, -w^3 + 1/3*w^2 + 23/3*w - 6], [1201, 1201, -1/3*w^3 + 2/3*w^2 + 14/3*w - 4], [1201, 1201, -1/3*w^3 - 4/3*w^2 + 8/3*w + 13], [1201, 1201, 1/3*w^3 - 7/3*w^2 + w + 14], [1201, 1201, 1/3*w^3 - 1/3*w^2 - 5*w + 1], [1231, 1231, -1/3*w^3 + 5/3*w^2 + 8/3*w - 8], [1231, 1231, 1/3*w^3 + 2/3*w^2 - 5*w - 4], [1249, 1249, -4/3*w^3 + 7/3*w^2 + 9*w + 4], [1249, 1249, 4/3*w^2 - 13/3*w - 9], [1249, 1249, 4/3*w^2 + 5/3*w - 12], [1249, 1249, -4/3*w^3 + 5/3*w^2 + 29/3*w - 14], [1259, 1259, 2/3*w^3 - w^2 - 20/3*w + 11], [1259, 1259, -2/3*w^3 + w^2 + 20/3*w + 4], [1279, 1279, -2/3*w^3 - 1/3*w^2 + 5*w + 8], [1279, 1279, 1/3*w^3 - w^2 - 7/3*w + 12], [1279, 1279, -2*w^2 + 3*w + 13], [1279, 1279, -2/3*w^3 + 7/3*w^2 + 7/3*w - 12], [1289, 1289, w^3 + 1/3*w^2 - 28/3*w - 8], [1289, 1289, -2/3*w^3 + 3*w^2 + 8/3*w - 19], [1289, 1289, 2/3*w^3 + w^2 - 20/3*w - 14], [1289, 1289, 5/3*w^3 - 11/3*w^2 - 11*w - 1], [1291, 1291, -w^3 + 2/3*w^2 + 25/3*w - 7], [1291, 1291, w^3 - 7/3*w^2 - 20/3*w + 1], [1301, 1301, w^3 - 10/3*w^2 - 14/3*w + 16], [1301, 1301, -w^3 + 14/3*w^2 + 25/3*w - 34], [1319, 1319, -5/3*w^3 + 6*w^2 + 17/3*w - 14], [1319, 1319, 1/3*w^3 - 2/3*w^2 - 8/3*w - 4], [1319, 1319, -1/3*w^3 + 1/3*w^2 + 3*w - 7], [1319, 1319, -5/3*w^3 - w^2 + 38/3*w + 4], [1361, 1361, w^3 - 4*w^2 - 6*w + 23], [1361, 1361, w^3 + w^2 - 11*w - 14], [1369, 37, -w^3 + 7*w - 4], [1369, 37, -1/3*w^3 - 2*w^2 + 10/3*w + 16], [1381, 1381, -2/3*w^3 + 20/3*w + 1], [1381, 1381, 2/3*w^3 - 2*w^2 - 14/3*w + 7], [1399, 1399, -w^3 + 4/3*w^2 + 20/3*w + 4], [1399, 1399, -w^3 + 5/3*w^2 + 19/3*w - 11], [1429, 1429, -2/3*w^3 + 2/3*w^2 + 4*w - 7], [1429, 1429, w^3 + 1/3*w^2 - 19/3*w + 1], [1451, 1451, w^3 - 4/3*w^2 - 23/3*w + 9], [1451, 1451, -1/3*w^3 + 2*w^2 - 2/3*w - 12], [1451, 1451, -1/3*w^3 - w^2 + 7/3*w + 11], [1451, 1451, 1/3*w^3 - 8/3*w^2 + 10/3*w + 12], [1459, 1459, -2/3*w^3 + 8/3*w^2 + 5*w - 16], [1459, 1459, -1/3*w^3 + 8/3*w^2 + 2/3*w - 16], [1459, 1459, -1/3*w^3 - 5/3*w^2 + 5*w + 13], [1459, 1459, 2/3*w^3 + 2/3*w^2 - 25/3*w - 9], [1471, 1471, -2/3*w^3 + 3*w^2 + 5/3*w - 16], [1471, 1471, -13/3*w^2 - 5/3*w + 33], [1471, 1471, -13/3*w^2 + 31/3*w + 27], [1471, 1471, 2/3*w^3 + w^2 - 17/3*w - 12], [1489, 1489, 1/3*w^3 - 7/3*w^2 - 4*w + 17], [1489, 1489, 1/3*w^3 + 4/3*w^2 - 23/3*w - 11], [1531, 1531, -w^3 + 3*w^2 + 7*w - 17], [1531, 1531, w^3 - 10*w - 8], [1549, 1549, w^3 + 2/3*w^2 - 35/3*w - 13], [1549, 1549, -2/3*w^3 + 5*w^2 + 17/3*w - 36], [1549, 1549, -2/3*w^3 - 3*w^2 + 41/3*w + 26], [1549, 1549, -w^3 + 11/3*w^2 + 22/3*w - 23], [1571, 1571, -w^3 + 3*w^2 + 7*w - 23], [1571, 1571, -1/3*w^3 - 2*w^2 + 4/3*w + 17], [1571, 1571, -1/3*w^3 + 1/3*w^2 + 6*w - 13], [1571, 1571, -2/3*w^3 + 4*w^2 + 8/3*w - 23], [1601, 1601, -w^3 + 4/3*w^2 + 26/3*w - 11], [1601, 1601, -w^3 + 8/3*w^2 + 25/3*w - 24], [1601, 1601, -1/3*w^3 - 7/3*w^2 + 23/3*w + 18], [1601, 1601, -w^3 + 5/3*w^2 + 25/3*w + 2], [1609, 1609, -5/3*w^3 + 2/3*w^2 + 12*w - 8], [1609, 1609, 2/3*w^3 + 4/3*w^2 - 7*w - 17], [1619, 1619, w^3 - 4/3*w^2 - 23/3*w - 3], [1619, 1619, -w^3 + 5/3*w^2 + 22/3*w - 11], [1621, 1621, 7/3*w^2 - 13/3*w - 11], [1621, 1621, -2/3*w^3 + 8/3*w^2 + 6*w - 17], [1669, 1669, 5/3*w^2 + 1/3*w - 16], [1669, 1669, 5/3*w^2 - 11/3*w - 14], [1681, 41, 2*w^2 - 2*w - 17], [1681, 41, 2*w^2 - 2*w - 13], [1699, 1699, 1/3*w^3 - 2*w^2 + 5/3*w + 11], [1699, 1699, -1/3*w^3 - w^2 + 4/3*w + 11], [1721, 1721, 1/3*w^2 + 8/3*w - 11], [1721, 1721, 1/3*w^2 - 10/3*w - 8], [1789, 1789, -2/3*w^2 - 7/3*w - 4], [1789, 1789, -w^3 + 2*w^2 + 8*w - 13], [1789, 1789, 4/3*w^2 - 13/3*w - 8], [1789, 1789, 2/3*w^2 - 11/3*w + 7], [1811, 1811, 2/3*w^2 - 11/3*w - 9], [1811, 1811, 2/3*w^2 + 7/3*w - 12], [1831, 1831, 4/3*w^2 + 5/3*w - 9], [1831, 1831, -w^3 + 5/3*w^2 + 25/3*w - 2], [1861, 1861, 14/3*w^2 + 1/3*w - 33], [1861, 1861, 14/3*w^2 - 29/3*w - 28], [1871, 1871, -w^3 + 2/3*w^2 + 28/3*w - 3], [1871, 1871, 1/3*w^3 + w^2 - 22/3*w - 7], [1871, 1871, -1/3*w^3 + 2*w^2 + 13/3*w - 13], [1871, 1871, -w^3 + 7/3*w^2 + 23/3*w - 6], [1879, 1879, -w^3 + 8/3*w^2 + 22/3*w - 13], [1879, 1879, -w^3 + 1/3*w^2 + 29/3*w + 4], [1889, 1889, -5/3*w^3 + 2*w^2 + 38/3*w - 16], [1889, 1889, 5/3*w^3 - 3*w^2 - 35/3*w - 3], [1901, 1901, w^2 - 3*w - 11], [1901, 1901, w^2 + w - 13], [1931, 1931, 2/3*w^3 + 2*w^2 - 38/3*w - 21], [1931, 1931, w^3 - 2/3*w^2 - 25/3*w - 9], [1931, 1931, -w^3 + 7/3*w^2 + 20/3*w - 17], [1931, 1931, -2/3*w^3 + 4*w^2 + 20/3*w - 31], [1949, 1949, 2/3*w^3 + 5/3*w^2 - 28/3*w - 21], [1949, 1949, -1/3*w^3 + 3*w^2 + 1/3*w - 17], [1951, 1951, -1/3*w^3 + 8/3*w^2 + 2/3*w - 19], [1951, 1951, 1/3*w^3 + 5/3*w^2 - 5*w - 16], [1999, 1999, -5/3*w^3 + 3*w^2 + 38/3*w - 23], [1999, 1999, -1/3*w^3 - 5/3*w^2 + 4*w + 19], [1999, 1999, 1/3*w^3 - 8/3*w^2 + 1/3*w + 21], [1999, 1999, 5/3*w^3 - 2*w^2 - 41/3*w - 9]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, -2, 2, 6, 2, -5, 6, -1, 0, 0, 8, 10, -8, -2, -4, 4, -6, 12, -10, -6, -8, 12, 2, 10, 6, -18, 14, -18, 12, 8, 2, 2, -2, -6, -2, 18, -4, 0, -12, -6, -20, -4, -16, 4, 6, 18, 8, -32, 30, 28, -14, 6, 10, 10, 10, 22, 20, 24, -6, 38, -6, 24, 0, 18, 4, 6, -32, 8, -8, -2, 0, 26, 14, 42, 2, -36, -12, 10, 12, -4, 0, -20, 10, 20, 8, -8, -4, -14, 2, 34, 6, -42, -28, 28, -10, 0, -30, 38, 28, 10, 20, -18, -16, 22, 0, 22, 26, -24, -26, 36, -26, 16, -8, -6, -2, -42, -2, 20, -8, -22, 0, 2, 30, -42, -42, 48, -6, 46, -6, 44, 8, -8, -32, 30, 14, 34, 2, -44, -42, -22, 8, -52, 4, -30, 52, 36, 12, 12, 8, 14, 10, 46, -14, 38, 16, 6, 18, -28, 46, -42, -34, 34, 42, -12, -46, 28, 36, 30, 34, 24, -36, 20, 8, -28, 56, 20, 24, 6, 10, -44, 46, -10, 50, -28, -2, 14, -42, 26, 56, 42, 0, -60, -44, -40, -2, 18, -26, -46, -52, 46, -12, -20, -26, -56, 26, -32, 56, -64, -16, -38, 18, -54, 18, -36, 32, -52, 54, 60, 36, -24, -18, 2, -42, -18, -22, 18, 4, -32, 40, 62, -6, -64, 20, 0, 52, 8, -4, 12, 8, -16, 2, 76, -12, -32, 14, -50, 22, 10, -48, 44, -10, -68, -20, 36, -28, -68, 50, 66, -40, 10, -14, -48, -48, 0, -40, -22, -10, 26, -26, 28, -60, -54, 22, -46, -72, -40, 50, 12, -52, 4, 58, -74, 6, -42, -36, -18, 78, -62, -18, -70, 48, -2, -14, -60, -56, -28, 60, -18, -50, 72, 72, -44, -16, -8, -84]; 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;