/* 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![7, 2, -7, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, w^3 - 3*w^2 - 3*w + 4], [7, 7, w], [7, 7, w^3 - 3*w^2 - 4*w + 5], [9, 3, w^3 - 4*w^2 - w + 9], [17, 17, 2*w^3 - 6*w^2 - 7*w + 8], [17, 17, -w^3 + 3*w^2 + 4*w - 3], [17, 17, -w + 2], [23, 23, 2*w^3 - 7*w^2 - 4*w + 12], [23, 23, -w^2 + 2*w + 3], [31, 31, -2*w^3 + 7*w^2 + 5*w - 12], [31, 31, -w^3 + 4*w^2 + 2*w - 8], [41, 41, 3*w^3 - 10*w^2 - 7*w + 16], [41, 41, 2*w^3 - 7*w^2 - 5*w + 10], [49, 7, 2*w^3 - 6*w^2 - 6*w + 9], [71, 71, w^2 - 2*w - 2], [71, 71, 2*w^3 - 7*w^2 - 4*w + 13], [73, 73, 3*w^3 - 11*w^2 - 5*w + 19], [73, 73, -4*w^3 + 13*w^2 + 10*w - 17], [79, 79, 3*w^3 - 9*w^2 - 10*w + 13], [79, 79, -2*w^3 + 6*w^2 + 5*w - 8], [97, 97, 2*w^3 - 6*w^2 - 8*w + 9], [97, 97, 2*w - 1], [113, 113, 2*w^3 - 6*w^2 - 7*w + 6], [113, 113, -w^3 + 3*w^2 + 2*w - 1], [137, 137, w^3 - 2*w^2 - 5*w + 2], [137, 137, 2*w^3 - 7*w^2 - 3*w + 12], [137, 137, -3*w^3 + 10*w^2 + 7*w - 17], [137, 137, -w^3 + 2*w^2 + 6*w - 2], [167, 167, -2*w^3 + 7*w^2 + 6*w - 10], [167, 167, 3*w^3 - 9*w^2 - 10*w + 10], [169, 13, -2*w^3 + 7*w^2 + 2*w - 9], [169, 13, -2*w^3 + 5*w^2 + 10*w - 4], [193, 193, -2*w^3 + 8*w^2 + 3*w - 13], [193, 193, -3*w^3 + 11*w^2 + 6*w - 22], [199, 199, 5*w^3 - 16*w^2 - 14*w + 26], [199, 199, w^3 - 2*w^2 - 7*w + 3], [223, 223, -3*w^3 + 8*w^2 + 12*w - 8], [223, 223, 3*w^3 - 11*w^2 - 6*w + 17], [223, 223, 2*w^3 - 8*w^2 - 3*w + 18], [223, 223, -4*w^3 + 13*w^2 + 9*w - 18], [233, 233, -3*w^3 + 10*w^2 + 6*w - 16], [233, 233, -2*w^3 + 5*w^2 + 9*w - 6], [241, 241, -3*w^3 + 9*w^2 + 8*w - 10], [241, 241, 4*w^3 - 12*w^2 - 13*w + 15], [257, 257, 2*w^3 - 8*w^2 - w + 16], [257, 257, 4*w^3 - 13*w^2 - 11*w + 18], [257, 257, -w^3 + 5*w^2 - 2*w - 9], [257, 257, -w^3 + 2*w^2 + 4*w + 2], [263, 263, 4*w^3 - 14*w^2 - 9*w + 27], [263, 263, 5*w^3 - 15*w^2 - 17*w + 25], [263, 263, -2*w^3 + 7*w^2 + 6*w - 16], [263, 263, -2*w^3 + 7*w^2 + 6*w - 9], [271, 271, -w - 4], [271, 271, -w^3 + 3*w^2 + 4*w - 9], [271, 271, 5*w^3 - 16*w^2 - 14*w + 24], [271, 271, -4*w^3 + 14*w^2 + 9*w - 24], [281, 281, -2*w^3 + 6*w^2 + 5*w - 4], [281, 281, -3*w^3 + 9*w^2 + 10*w - 9], [281, 281, 3*w^3 - 10*w^2 - 8*w + 12], [281, 281, w^2 - w - 8], [311, 311, w^2 - 5], [311, 311, -4*w^3 + 13*w^2 + 12*w - 20], [313, 313, 6*w^3 - 19*w^2 - 18*w + 31], [313, 313, -2*w^3 + 5*w^2 + 6*w - 6], [337, 337, 3*w^3 - 11*w^2 - 3*w + 15], [337, 337, w^3 - w^2 - 9*w - 5], [353, 353, -w^3 + 2*w^2 + 5*w - 4], [353, 353, w^3 - w^2 - 8*w - 3], [353, 353, 3*w^3 - 10*w^2 - 7*w + 19], [353, 353, -4*w^3 + 14*w^2 + 7*w - 22], [361, 19, -3*w^3 + 11*w^2 + 6*w - 18], [361, 19, 2*w^3 - 8*w^2 - 3*w + 17], [367, 367, -4*w^3 + 12*w^2 + 13*w - 18], [367, 367, 3*w^3 - 9*w^2 - 8*w + 13], [383, 383, 5*w^3 - 17*w^2 - 12*w + 30], [383, 383, 2*w^2 - 3*w - 5], [383, 383, 3*w^3 - 9*w^2 - 11*w + 11], [383, 383, -w^3 + 3*w^2 + w - 1], [401, 401, 3*w^3 - 11*w^2 - 8*w + 23], [401, 401, 4*w^3 - 14*w^2 - 11*w + 22], [431, 431, 2*w^3 - 5*w^2 - 10*w + 8], [431, 431, 6*w^3 - 20*w^2 - 15*w + 29], [433, 433, -4*w^3 + 12*w^2 + 13*w - 17], [433, 433, -3*w^3 + 9*w^2 + 8*w - 12], [433, 433, 3*w^3 - 11*w^2 - 6*w + 20], [433, 433, -2*w^3 + 8*w^2 + 3*w - 15], [439, 439, -3*w^3 + 11*w^2 + 6*w - 19], [439, 439, -2*w^3 + 8*w^2 + 3*w - 16], [449, 449, w^3 - 4*w^2 + w + 11], [449, 449, -w^3 + 2*w^2 + 7*w - 6], [457, 457, w^2 - 4*w - 3], [457, 457, 4*w^3 - 15*w^2 - 6*w + 27], [457, 457, w^2 - 4*w - 2], [457, 457, -2*w^3 + 6*w^2 + 8*w - 15], [479, 479, -4*w^3 + 14*w^2 + 9*w - 29], [479, 479, -w^3 + 5*w^2 - 6], [487, 487, -w^3 + 6*w^2 - 3*w - 16], [487, 487, -5*w^3 + 18*w^2 + 9*w - 29], [503, 503, -2*w^3 + 7*w^2 + 8*w - 15], [503, 503, 4*w^3 - 13*w^2 - 14*w + 20], [521, 521, 3*w^3 - 10*w^2 - 9*w + 10], [521, 521, -2*w^3 + 7*w^2 + 3*w - 16], [529, 23, w^3 - 3*w^2 - 3*w - 1], [599, 599, -4*w^3 + 12*w^2 + 14*w - 15], [599, 599, 2*w^3 - 6*w^2 - 4*w + 5], [599, 599, 4*w^3 - 13*w^2 - 12*w + 19], [599, 599, w^2 - 6], [601, 601, -4*w^3 + 12*w^2 + 11*w - 19], [601, 601, 5*w^3 - 15*w^2 - 16*w + 24], [607, 607, -w^3 + 4*w^2 + w - 13], [607, 607, -w^3 + 4*w^2 + w - 2], [617, 617, -4*w^3 + 13*w^2 + 11*w - 16], [617, 617, 4*w^3 - 13*w^2 - 13*w + 20], [625, 5, -5], [631, 631, -3*w^3 + 8*w^2 + 11*w - 9], [631, 631, 5*w^3 - 16*w^2 - 13*w + 24], [631, 631, w^3 - 3*w^2 - 4*w - 1], [631, 631, w - 6], [641, 641, -3*w^3 + 12*w^2 + w - 17], [641, 641, w^3 - 6*w^2 + 5*w + 18], [673, 673, 4*w^3 - 15*w^2 - 6*w + 26], [673, 673, -2*w^3 + 9*w^2 - 19], [719, 719, 2*w^2 - 4*w - 5], [719, 719, 4*w^3 - 14*w^2 - 8*w + 25], [727, 727, 3*w^3 - 8*w^2 - 11*w + 8], [727, 727, 6*w^3 - 18*w^2 - 19*w + 20], [727, 727, -5*w^3 + 14*w^2 + 19*w - 17], [727, 727, -5*w^3 + 16*w^2 + 13*w - 23], [743, 743, 4*w^3 - 14*w^2 - 7*w + 23], [743, 743, w^3 - w^2 - 8*w - 2], [751, 751, 3*w^3 - 12*w^2 - 3*w + 23], [751, 751, 3*w^3 - 12*w^2 - 3*w + 22], [761, 761, 3*w^3 - 10*w^2 - 6*w + 22], [761, 761, 3*w^3 - 10*w^2 - 6*w + 18], [761, 761, -5*w^3 + 16*w^2 + 17*w - 25], [761, 761, -2*w^3 + 5*w^2 + 9*w - 8], [809, 809, -3*w^3 + 8*w^2 + 13*w - 3], [809, 809, -2*w^3 + 6*w^2 + 3*w - 12], [823, 823, -w^3 + 4*w^2 + 2*w - 2], [823, 823, 2*w^3 - 7*w^2 - 5*w + 18], [839, 839, 4*w^3 - 13*w^2 - 14*w + 27], [839, 839, -7*w^3 + 23*w^2 + 19*w - 41], [857, 857, -3*w^3 + 10*w^2 + 9*w - 12], [857, 857, -w^3 + 4*w^2 + 3*w - 13], [863, 863, 3*w^3 - 11*w^2 - 2*w + 20], [863, 863, -2*w^3 + 4*w^2 + 13*w - 5], [863, 863, 6*w^3 - 19*w^2 - 18*w + 27], [863, 863, -2*w^3 + 5*w^2 + 6*w - 2], [881, 881, -3*w^3 + 10*w^2 + 5*w - 17], [881, 881, -3*w^3 + 8*w^2 + 13*w - 12], [887, 887, -3*w^3 + 9*w^2 + 11*w - 9], [887, 887, w^3 - 3*w^2 - w - 1], [911, 911, 4*w^3 - 14*w^2 - 10*w + 17], [911, 911, 5*w^3 - 17*w^2 - 12*w + 22], [919, 919, -4*w^3 + 15*w^2 + 7*w - 30], [919, 919, -2*w^2 + 7*w + 2], [919, 919, w^3 - 5*w^2 + 4*w + 13], [919, 919, 3*w^3 - 12*w^2 - 4*w + 20], [929, 929, -4*w^3 + 13*w^2 + 9*w - 22], [929, 929, 3*w^3 - 8*w^2 - 12*w + 12], [961, 31, 4*w^3 - 12*w^2 - 12*w + 17], [967, 967, -2*w^3 + 6*w^2 + 7*w - 2], [967, 967, -3*w^3 + 8*w^2 + 15*w - 6], [967, 967, w^3 - 2*w^2 - 5*w - 6], [967, 967, 7*w^3 - 22*w^2 - 21*w + 36], [983, 983, 5*w^3 - 17*w^2 - 13*w + 25], [983, 983, w^3 - 5*w^2 - w + 15], [991, 991, -6*w^3 + 21*w^2 + 13*w - 34], [991, 991, 7*w^3 - 22*w^2 - 21*w + 33], [1009, 1009, w^2 - 2*w + 3], [1009, 1009, -2*w^3 + 7*w^2 + 4*w - 18], [1031, 1031, -4*w^3 + 15*w^2 + 4*w - 26], [1031, 1031, 3*w^2 - 8*w - 9], [1033, 1033, 2*w^3 - 4*w^2 - 13*w + 3], [1033, 1033, -5*w^3 + 15*w^2 + 19*w - 23], [1033, 1033, w^3 - w^2 - 8*w - 10], [1033, 1033, 4*w^3 - 14*w^2 - 7*w + 15], [1049, 1049, -7*w^3 + 24*w^2 + 16*w - 34], [1049, 1049, -5*w^3 + 16*w^2 + 18*w - 24], [1063, 1063, w^3 - 4*w^2 + 2*w + 8], [1063, 1063, w^3 - 2*w^2 - 4*w - 8], [1063, 1063, 7*w^3 - 24*w^2 - 17*w + 39], [1063, 1063, w^3 - 2*w^2 - 7*w - 6], [1103, 1103, w^3 - 5*w^2 + 3*w + 17], [1103, 1103, -w^3 + 5*w^2 - 3*w - 3], [1103, 1103, -w^3 + 5*w^2 - 2*w - 6], [1103, 1103, -2*w^3 + 8*w^2 + w - 19], [1129, 1129, -6*w^3 + 18*w^2 + 20*w - 27], [1129, 1129, -4*w^3 + 12*w^2 + 10*w - 17], [1151, 1151, w^3 - w^2 - 7*w - 1], [1151, 1151, 5*w^3 - 17*w^2 - 11*w + 29], [1153, 1153, -2*w^3 + 8*w^2 + 3*w - 23], [1153, 1153, -6*w^3 + 18*w^2 + 20*w - 25], [1193, 1193, -2*w^3 + 5*w^2 + 9*w - 10], [1193, 1193, 3*w^3 - 10*w^2 - 6*w + 20], [1201, 1201, 6*w^3 - 19*w^2 - 16*w + 24], [1201, 1201, 4*w^3 - 11*w^2 - 14*w + 9], [1217, 1217, 7*w^3 - 22*w^2 - 21*w + 30], [1217, 1217, 7*w^3 - 23*w^2 - 18*w + 39], [1223, 1223, -2*w^3 + 9*w^2 - 15], [1223, 1223, -3*w^3 + 9*w^2 + 13*w - 9], [1223, 1223, -5*w^3 + 17*w^2 + 12*w - 34], [1223, 1223, 4*w^3 - 15*w^2 - 6*w + 30], [1231, 1231, -3*w^3 + 7*w^2 + 12*w - 3], [1231, 1231, -8*w^3 + 26*w^2 + 21*w - 38], [1289, 1289, -4*w^3 + 14*w^2 + 7*w - 24], [1289, 1289, -w^3 + w^2 + 8*w + 1], [1297, 1297, -4*w^3 + 12*w^2 + 11*w - 15], [1297, 1297, -5*w^3 + 15*w^2 + 16*w - 20], [1303, 1303, -3*w^3 + 10*w^2 + 7*w - 23], [1303, 1303, -w^3 + 2*w^2 + 5*w - 8], [1319, 1319, -8*w^3 + 28*w^2 + 15*w - 43], [1319, 1319, w^3 + w^2 - 12*w - 12], [1321, 1321, w^2 - 2*w - 10], [1321, 1321, 2*w^3 - 7*w^2 - 4*w + 5], [1367, 1367, -5*w^3 + 17*w^2 + 14*w - 26], [1367, 1367, 2*w^3 - 8*w^2 - 5*w + 19], [1399, 1399, -2*w^3 + 7*w^2 + 3*w - 4], [1399, 1399, w^3 - 2*w^2 - 6*w - 6], [1409, 1409, w^3 - 2*w^2 - 3*w - 3], [1409, 1409, 3*w^3 - 10*w^2 - 5*w + 18], [1409, 1409, -5*w^3 + 16*w^2 + 15*w - 22], [1409, 1409, -3*w^3 + 8*w^2 + 13*w - 13], [1423, 1423, 7*w^3 - 24*w^2 - 18*w + 44], [1423, 1423, -8*w^3 + 26*w^2 + 21*w - 40], [1433, 1433, -7*w^3 + 24*w^2 + 17*w - 45], [1433, 1433, -3*w^3 + 12*w^2 + 3*w - 29], [1439, 1439, -2*w^3 + 6*w^2 + 4*w - 3], [1439, 1439, -4*w^3 + 12*w^2 + 14*w - 13], [1471, 1471, -w - 6], [1471, 1471, -w^3 + 3*w^2 + 4*w - 11], [1471, 1471, w^3 - 2*w^2 - 8*w + 4], [1471, 1471, w^2 - 5*w - 4], [1489, 1489, -2*w^3 + 9*w^2 + 2*w - 17], [1489, 1489, -6*w^3 + 21*w^2 + 14*w - 38], [1543, 1543, -7*w^3 + 24*w^2 + 19*w - 45], [1543, 1543, 2*w^3 - 3*w^2 - 15*w - 4], [1543, 1543, -5*w^3 + 18*w^2 + 6*w - 26], [1543, 1543, -3*w^3 + 12*w^2 + 7*w - 20], [1553, 1553, -4*w^3 + 14*w^2 + 11*w - 20], [1553, 1553, -3*w^3 + 11*w^2 + 8*w - 25], [1559, 1559, 2*w^2 - 4*w - 3], [1559, 1559, 4*w^3 - 14*w^2 - 8*w + 27], [1567, 1567, -4*w^3 + 15*w^2 + 3*w - 20], [1567, 1567, w^3 - 12*w - 10], [1601, 1601, -3*w^3 + 10*w^2 + 10*w - 12], [1601, 1601, 2*w^3 - 7*w^2 - 7*w + 18], [1607, 1607, -7*w^3 + 23*w^2 + 20*w - 36], [1607, 1607, -6*w^3 + 19*w^2 + 18*w - 26], [1607, 1607, 2*w^2 - w - 9], [1607, 1607, -2*w^3 + 5*w^2 + 6*w - 1], [1609, 1609, -4*w^3 + 15*w^2 + 8*w - 24], [1609, 1609, 4*w^3 - 15*w^2 - 8*w + 31], [1663, 1663, -6*w^3 + 19*w^2 + 13*w - 26], [1663, 1663, -7*w^3 + 20*w^2 + 26*w - 26], [1681, 41, -5*w^3 + 15*w^2 + 15*w - 17], [1697, 1697, -w^3 + 9*w + 6], [1697, 1697, -7*w^3 + 24*w^2 + 15*w - 39], [1721, 1721, -5*w^3 + 18*w^2 + 13*w - 38], [1721, 1721, -w^3 + 6*w^2 - 5*w - 20], [1721, 1721, -w^3 + w^2 + 6*w + 11], [1721, 1721, 6*w^3 - 20*w^2 - 15*w + 24], [1753, 1753, -3*w^3 + 9*w^2 + 13*w - 17], [1753, 1753, w^3 - 3*w^2 - 7*w + 3], [1753, 1753, -6*w^3 + 22*w^2 + 12*w - 45], [1753, 1753, -6*w^3 + 21*w^2 + 16*w - 40], [1759, 1759, -2*w^3 + 6*w^2 + w - 8], [1759, 1759, 7*w^3 - 21*w^2 - 26*w + 33], [1759, 1759, -3*w^3 + 10*w^2 + 10*w - 26], [1759, 1759, -5*w^3 + 17*w^2 + 12*w - 37], [1783, 1783, -5*w^3 + 18*w^2 + 11*w - 29], [1783, 1783, 4*w^3 - 15*w^2 - 7*w + 26], [1783, 1783, 3*w^3 - 12*w^2 - 5*w + 26], [1783, 1783, -3*w^3 + 12*w^2 + 4*w - 24], [1801, 1801, 8*w^3 - 24*w^2 - 27*w + 33], [1801, 1801, 4*w^3 - 12*w^2 - 16*w + 19], [1801, 1801, 4*w - 1], [1801, 1801, 7*w^3 - 21*w^2 - 23*w + 33], [1831, 1831, w^3 - 2*w^2 - 8*w + 2], [1831, 1831, w^2 - 5*w - 2], [1847, 1847, 6*w^3 - 20*w^2 - 15*w + 25], [1847, 1847, w^3 - w^2 - 6*w - 10], [1849, 43, -6*w^3 + 21*w^2 + 14*w - 37], [1849, 43, -2*w^3 + 9*w^2 + 2*w - 18], [1871, 1871, -4*w^3 + 13*w^2 + 8*w - 22], [1871, 1871, 4*w^3 - 13*w^2 - 12*w + 15], [1871, 1871, 4*w^3 - 11*w^2 - 16*w + 17], [1871, 1871, w^2 - 10], [1873, 1873, -w^3 + 5*w^2 - 20], [1873, 1873, -4*w^3 + 14*w^2 + 9*w - 15], [1913, 1913, -2*w^3 + 6*w^2 + 3*w - 4], [1913, 1913, 5*w^3 - 15*w^2 - 18*w + 19], [1913, 1913, 6*w^3 - 20*w^2 - 11*w + 30], [1913, 1913, 9*w^3 - 30*w^2 - 23*w + 45], [1993, 1993, -8*w^3 + 28*w^2 + 17*w - 43], [1993, 1993, 4*w^3 - 13*w^2 - 16*w + 19], [1993, 1993, -6*w^3 + 21*w^2 + 14*w - 36], [1993, 1993, -2*w^3 + 9*w^2 + 2*w - 19], [1999, 1999, -7*w^3 + 22*w^2 + 20*w - 30], [1999, 1999, 4*w^3 - 11*w^2 - 13*w + 10]]; primes := [ideal : I in primesArray]; heckePol := x^4 - 6*x^2 + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [-e^2 + 3, -1, -e^2 + 4, e, e^3 - 3*e, -e^3 + 8*e, 2*e^3 - 9*e, 2*e^3 - 7*e, -2*e, 4, 4, -e^3 + 6*e, -3*e^3 + 11*e, -6, -2*e^3 + 15*e, -e^3 + 3*e, 3*e^2 - 6, -5*e^2 + 14, 3*e^2 - 4, 3*e^2 - 4, 4*e^2 - 6, -4*e^2 + 14, -2*e^3 + 3*e, -e^3 + 10*e, e^3 - 7*e, e^3 - 4*e, 6*e^3 - 29*e, e^3 - 4*e, -3*e^3 + 9*e, 6*e^3 - 23*e, -6*e^2 + 10, 2*e^2 - 10, 2*e^2 + 2, 2*e^2 + 2, -7*e^2 + 28, -5*e^2 + 4, -e^2 - 20, 3*e^2 + 4, 4*e^2 - 8, -4*e^2 + 16, e^3 - 12*e, -3*e^3 + 17*e, 10, 2*e^2 - 14, -5*e^3 + 20*e, 4*e^3 - 31*e, e^3 - 14*e, e^3 - 14*e, 2*e^3 - 12*e, -5*e^3 + 15*e, -3*e^3 + 17*e, -3*e^3 + 17*e, 8*e^2 - 28, -2*e^2 + 16, 8*e^2 - 20, 0, -e^3 + 8*e, -4*e^3 + 25*e, 5*e^3 - 32*e, -8*e^3 + 29*e, e^3 - 19*e, 12*e, 7*e^2 - 2, -6*e^2 + 2, 2*e^2 - 30, 7*e^2 - 14, e^3 - 19*e, 5*e^3 - 13*e, 5*e^3 - 29*e, -7*e^3 + 36*e, 2*e^2 + 18, 6*e^2 - 30, -14*e^2 + 36, 4*e^2 - 28, -6*e^3 + 40*e, 2*e^3 + e, 2*e^3 - 2*e, 2*e^3 - 21*e, -7*e^3 + 32*e, -e^3 - 2*e, -8*e^3 + 38*e, -2*e^3 + 4*e, 6*e^2 - 22, -e^2 - 14, 8*e^2 - 6, -8*e^2 + 34, e^2 - 4, 8*e^2 - 12, -7*e^3 + 47*e, -7*e^3 + 28*e, -8*e^2 + 42, 16*e^2 - 54, e^2 + 10, -3*e^2 + 22, 4*e^3 - 18*e, 2*e^3 - 13*e, -2*e^2 - 24, -6*e^2 + 24, 12*e^3 - 56*e, 2*e^3 + 7*e, 9*e^3 - 31*e, -6*e^3 + 35*e, -7*e^2 + 2, -2*e^3 + 11*e, 6*e, 5*e^3 - 19*e, 10*e^3 - 41*e, -6*e^2 - 14, 4*e^2 + 18, -15*e^2 + 52, 3*e^2 - 12, 10*e^3 - 45*e, e^3 - 13*e, -8*e^2 + 2, -4*e^2 - 8, 10*e^2 - 24, 8*e^2 - 44, 8*e^2 - 44, e^3 - 5*e, e^3 - 5*e, 14*e^2 - 46, 4*e^2 - 2, -e^3 - 13*e, -12*e^3 + 62*e, 10*e^2 - 20, 4*e^2 - 16, 5*e^2 - 28, 4*e^2 - 24, -e^3 + 7*e, 5*e^3 - 27*e, 4*e^2 - 4, -10*e^2 + 12, -6*e^3 + 11*e, -15*e^3 + 57*e, -3*e^3 + 13*e, 3*e^3 - 26*e, 4*e^3 - 23*e, 9*e^3 - 64*e, 8*e^2 - 24, 9*e^2 - 36, -10*e^3 + 52*e, -10*e^3 + 52*e, -8*e^3 + 49*e, 7*e^3 - 36*e, 5*e^3 - 35*e, 5*e^3 - 35*e, 4*e^3 - 16*e, -6*e^3 + 9*e, 9*e^3 - 39*e, -7*e^3 + 58*e, -8*e^3 + 54*e, 10*e^3 - 48*e, 3*e^3 - 5*e, 10*e^3 - 70*e, 4*e^2 - 4, 12*e^2 - 16, -8*e^2 - 4, -12*e^2 + 36, 11*e^3 - 48*e, 9*e^3 - 43*e, -8*e^2 + 18, e^2 - 4, 7*e^2 - 12, 14*e^2 - 20, -4*e^2 - 20, 6*e^3 - 34*e, 14*e^3 - 54*e, 40, 14*e^2 - 52, 14*e^2 - 22, -6, -10*e^3 + 40*e, 2*e^3 - 9*e, 11*e^2 - 38, -16*e^2 + 58, -5*e^2 + 30, -4*e^2 + 18, 7*e^3 - 20*e, 16*e^3 - 71*e, 8*e^2 - 20, -12*e^2 + 32, -8*e^2 + 60, 10*e^2 - 44, 10*e^3 - 67*e, 9*e^3 - 55*e, 2*e^3 - 35*e, -8*e^3 + 66*e, -4*e^2 + 2, 14*e^2 - 62, 17*e^3 - 75*e, 2*e^3 - 28*e, 2*e^2 + 34, -12*e^2 + 50, -2*e^3 + 19*e, -15*e^3 + 61*e, -6*e^2 + 50, -16*e^2 + 18, 11*e^3 - 82*e, 2*e^3 + 7*e, -21*e^3 + 87*e, -10*e^3 + 70*e, -14*e^3 + 61*e, 6*e^3 - 28*e, 4*e^2 + 8, -20*e^2 + 68, -19*e^3 + 90*e, -17*e^3 + 66*e, -16*e^2 + 74, 34, 12*e^2 - 4, 4*e^2 + 16, -2*e^3 + 15*e, -12*e^3 + 78*e, 11*e^2 - 38, -14*e^2 + 34, 11*e^3 - 85*e, 6*e^3 - 44*e, 22*e^2 - 84, 28, 29*e, 17*e^3 - 72*e, 11*e^3 - 46*e, -8*e^3 + 57*e, 2*e^2 + 8, 8*e^2 + 12, 5*e^3 - 5*e, 13*e^3 - 44*e, -2*e^3 + 19*e, 19*e^3 - 81*e, -4*e^2 + 16, 12*e^2 - 24, -8*e^2 + 44, -6*e^2 + 20, -2*e^2 + 58, -22*e^2 + 70, 10*e^2 - 20, -13*e^2 + 68, 16*e^2 - 52, -16*e^2 + 64, -15*e^3 + 66*e, -3*e^3 - 2*e, -16*e^3 + 84*e, 4*e^3 - 4*e, -19*e^2 + 60, -3*e^2 + 20, 17*e^3 - 79*e, 5*e^3 - 49*e, -8*e^3 + 62*e, 6*e^3 - 19*e, -12*e^3 + 72*e, 10*e^3 - 48*e, 34, -6*e^2 - 46, 28*e^2 - 72, 20*e^2 - 52, -5*e^2 + 18, -3*e^3 + 4*e, -13*e^3 + 48*e, -7*e^3 + 36*e, -9*e^3 + 60*e, -16*e^3 + 73*e, 21*e^3 - 86*e, 18, -e^2 + 30, -e^2 + 46, 2*e^2 - 66, -5*e^2 + 12, 4*e^2 - 20, -44, 10*e^2 - 12, 20*e^2 - 56, -7*e^2 + 28, 5*e^2 - 28, 22*e^2 - 92, -2*e^2 - 46, 26*e^2 - 74, -19*e^2 + 86, -26*e^2 + 90, -24*e^2 + 72, 6*e^2 - 60, -12*e^3 + 72*e, -16*e^3 + 82*e, 11*e^2 - 34, -24*e^2 + 82, 4*e^3 + 2*e, -22*e^3 + 114*e, -2*e^3 - 2*e, -10*e^3 + 27*e, -25*e^2 + 86, -10*e^2 + 58, 13*e^3 - 101*e, -3*e^3 + 15*e, -15*e^3 + 58*e, 15*e^3 - 74*e, -14*e^2 - 6, -8*e^2 + 74, -21*e^2 + 42, -4*e^2 + 66, 23*e^2 - 68, -12*e^2 + 48]; 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;