/* 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![18, 0, -12, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -1/3*w^3 - 1/3*w^2 + 3*w + 4], [7, 7, 1/3*w^3 + 1/3*w^2 - 3*w - 3], [7, 7, -1/3*w^2 + w + 1], [7, 7, 1/3*w^2 + w - 1], [7, 7, 1/3*w^3 - 1/3*w^2 - 3*w + 3], [9, 3, w - 3], [41, 41, 1/3*w^3 - 1/3*w^2 - 3*w + 1], [41, 41, -1/3*w^2 + w + 3], [41, 41, 1/3*w^2 + w - 3], [41, 41, -1/3*w^3 - 1/3*w^2 + 3*w + 1], [47, 47, 1/3*w^3 + 1/3*w^2 - 4*w - 1], [47, 47, -1/3*w^3 + 1/3*w^2 + 2*w - 3], [47, 47, 1/3*w^3 + 1/3*w^2 - 2*w - 3], [47, 47, 1/3*w^3 - 1/3*w^2 - 4*w + 1], [89, 89, -1/3*w^3 + 2/3*w^2 + 3*w - 3], [89, 89, 2/3*w^2 + w - 5], [89, 89, 2/3*w^2 - w - 5], [89, 89, 1/3*w^3 + 2/3*w^2 - 3*w - 3], [97, 97, 2/3*w^3 - 1/3*w^2 - 6*w + 5], [97, 97, -w^3 - 5/3*w^2 + 10*w + 15], [97, 97, -5/3*w^3 - 5/3*w^2 + 16*w + 21], [97, 97, -2/3*w^3 - 1/3*w^2 + 6*w + 5], [103, 103, -4/3*w^3 - 4/3*w^2 + 13*w + 17], [103, 103, 1/3*w^3 + w^2 - 5*w - 7], [103, 103, -w^3 - w^2 + 9*w + 11], [103, 103, -2/3*w^3 - 4/3*w^2 + 7*w + 11], [137, 137, -1/3*w^3 - 1/3*w^2 + 3*w - 1], [137, 137, 1/3*w^2 + w - 5], [137, 137, 1/3*w^2 - w - 5], [137, 137, 1/3*w^3 - 1/3*w^2 - 3*w - 1], [151, 151, w^2 - w - 5], [151, 151, 1/3*w^3 + w^2 - 3*w - 7], [151, 151, -1/3*w^3 + w^2 + 3*w - 7], [151, 151, w^2 + w - 5], [191, 191, 1/3*w^3 + 2/3*w^2 - 4*w - 3], [191, 191, 1/3*w^3 + 2/3*w^2 - 2*w - 5], [191, 191, -1/3*w^3 + 2/3*w^2 + 2*w - 5], [191, 191, -1/3*w^3 + 2/3*w^2 + 4*w - 3], [193, 193, 2/3*w^3 + w^2 - 8*w - 7], [193, 193, -1/3*w^3 - 1/3*w^2 + 4*w + 7], [193, 193, -7/3*w^3 - 13/3*w^2 + 26*w + 37], [193, 193, -2/3*w^3 + w^2 + 8*w - 7], [199, 199, 5/3*w^3 + 2*w^2 - 17*w - 23], [199, 199, -2/3*w^3 - 5/3*w^2 + 9*w + 13], [199, 199, -2/3*w^3 - 1/3*w^2 + 5*w + 5], [199, 199, -w^3 - 2*w^2 + 11*w + 17], [233, 233, 1/3*w^3 - 4/3*w^2 - 3*w + 5], [233, 233, 1/3*w^3 - 4/3*w^2 - 5*w + 7], [233, 233, 4/3*w^2 + w - 11], [233, 233, 1/3*w^3 + 4/3*w^2 - 3*w - 5], [239, 239, 1/3*w^3 + w^2 - 4*w - 5], [239, 239, 4/3*w^3 + 4/3*w^2 - 14*w - 17], [239, 239, 4/3*w^3 + 2/3*w^2 - 12*w - 13], [239, 239, -1/3*w^3 + w^2 + 4*w - 5], [241, 241, -2/3*w^3 + 6*w + 1], [241, 241, 2/3*w^3 - 1/3*w^2 - 6*w + 3], [241, 241, -2/3*w^3 - 1/3*w^2 + 6*w + 3], [241, 241, 2/3*w^3 - 6*w + 1], [281, 281, -w^3 - w^2 + 9*w + 13], [281, 281, -2/3*w^3 - 2/3*w^2 + 5*w + 7], [281, 281, -2/3*w^3 + 2/3*w^2 + 5*w - 7], [281, 281, -7/3*w^3 - 3*w^2 + 23*w + 31], [289, 17, w^2 - 5], [289, 17, w^2 - 7], [337, 337, -w^3 + 4/3*w^2 + 8*w - 7], [337, 337, -2*w + 7], [337, 337, 2/3*w^3 - 2*w^2 - 4*w + 11], [337, 337, 1/3*w^3 + 2/3*w^2 - 6*w + 1], [383, 383, 2/3*w^3 + 5/3*w^2 - 8*w - 11], [383, 383, -w^3 - 2/3*w^2 + 10*w + 11], [383, 383, -5/3*w^3 - 4/3*w^2 + 16*w + 19], [383, 383, -4/3*w^3 - 7/3*w^2 + 14*w + 19], [431, 431, -2/3*w^3 + 1/3*w^2 + 4*w - 5], [431, 431, -2/3*w^3 - 1/3*w^2 + 8*w - 1], [431, 431, 2/3*w^3 - 1/3*w^2 - 8*w - 1], [431, 431, 2/3*w^3 + 1/3*w^2 - 4*w - 5], [433, 433, 2/3*w^3 + 7/3*w^2 - 10*w - 15], [433, 433, -3*w^3 - 10/3*w^2 + 30*w + 39], [433, 433, -1/3*w^3 - 4/3*w^2 + 4*w + 9], [433, 433, -4/3*w^3 - 5/3*w^2 + 12*w + 15], [439, 439, -4/3*w^3 - 1/3*w^2 + 11*w + 13], [439, 439, 1/3*w^3 + 2/3*w^2 - 5*w - 5], [439, 439, -3*w^3 - 10/3*w^2 + 29*w + 37], [439, 439, -4/3*w^3 - 7/3*w^2 + 13*w + 19], [479, 479, 1/3*w^2 - 2*w - 5], [479, 479, 2/3*w^3 + 1/3*w^2 - 6*w + 1], [479, 479, 2/3*w^3 - 1/3*w^2 - 6*w - 1], [479, 479, -1/3*w^2 - 2*w + 5], [487, 487, 1/3*w^3 + 4/3*w^2 - 3*w - 7], [487, 487, 4/3*w^2 - w - 9], [487, 487, -4/3*w^2 - w + 9], [487, 487, 1/3*w^3 - 4/3*w^2 - 3*w + 7], [521, 521, 2/3*w^3 - 2/3*w^2 - 5*w - 3], [521, 521, 5/3*w^3 + 7/3*w^2 - 17*w - 21], [521, 521, -7/3*w^3 - 11/3*w^2 + 25*w + 33], [521, 521, -4/3*w^3 - 2/3*w^2 + 13*w + 15], [529, 23, -1/3*w^2 - 3], [529, 23, 1/3*w^2 - 7], [569, 569, 2/3*w^3 + 2/3*w^2 - 5*w + 1], [569, 569, 1/3*w^3 - 2/3*w^2 - 5*w + 9], [569, 569, -1/3*w^3 - 2/3*w^2 + 5*w + 9], [569, 569, -2/3*w^3 + 2/3*w^2 + 5*w + 1], [577, 577, -w^3 + 1/3*w^2 + 8*w + 3], [577, 577, -1/3*w^3 - 1/3*w^2 + 6*w + 7], [577, 577, 1/3*w^3 - 1/3*w^2 - 6*w + 7], [577, 577, w^3 + 1/3*w^2 - 8*w + 3], [617, 617, -2/3*w^3 - 5/3*w^2 + 5*w + 7], [617, 617, -5/3*w^2 + 3*w + 5], [617, 617, -8/3*w^3 - 13/3*w^2 + 27*w + 37], [617, 617, -w^3 - 5/3*w^2 + 9*w + 15], [625, 5, -5], [631, 631, -1/3*w^3 - 5/3*w^2 + 3*w + 7], [631, 631, -5/3*w^2 - w + 13], [631, 631, 5/3*w^2 - w - 13], [631, 631, 1/3*w^3 - 5/3*w^2 - 3*w + 7], [673, 673, 2/3*w^3 + 1/3*w^2 - 8*w - 1], [673, 673, -2/3*w^3 - 1/3*w^2 + 4*w + 3], [673, 673, -2/3*w^3 + 1/3*w^2 + 4*w - 3], [673, 673, 2/3*w^3 - 1/3*w^2 - 8*w + 1], [719, 719, 1/3*w^3 - 1/3*w^2 - 6*w + 9], [719, 719, w^3 - 1/3*w^2 - 8*w - 5], [719, 719, w^3 + 1/3*w^2 - 8*w + 5], [719, 719, -1/3*w^3 - 1/3*w^2 + 6*w + 9], [727, 727, 2/3*w^3 + 1/3*w^2 - 5*w - 1], [727, 727, -1/3*w^3 - 1/3*w^2 + 5*w + 3], [727, 727, 1/3*w^3 - 1/3*w^2 - 5*w + 3], [727, 727, -2/3*w^3 + 1/3*w^2 + 5*w - 1], [761, 761, -2/3*w^3 + 1/3*w^2 + 3*w + 3], [761, 761, 7/3*w^3 + 10/3*w^2 - 23*w - 33], [761, 761, 3*w^3 + 10/3*w^2 - 29*w - 39], [761, 761, -2*w^3 - 5/3*w^2 + 17*w + 21], [769, 769, -4/3*w^3 - w^2 + 12*w + 13], [769, 769, -w^3 - 2/3*w^2 + 10*w + 13], [769, 769, -w^3 - 8/3*w^2 + 12*w + 19], [769, 769, -2*w^3 - 3*w^2 + 22*w + 29], [809, 809, -2/3*w^3 - 2/3*w^2 + 7*w + 3], [809, 809, w^2 + w - 11], [809, 809, w^2 - w - 11], [809, 809, 2/3*w^3 - 2/3*w^2 - 7*w + 3], [823, 823, 1/3*w^3 - 5*w - 1], [823, 823, 2/3*w^3 - 5*w - 1], [823, 823, 2/3*w^3 - 5*w + 1], [823, 823, -1/3*w^3 + 5*w - 1], [857, 857, -5/3*w^2 + w + 15], [857, 857, -1/3*w^3 - 5/3*w^2 + 3*w + 5], [857, 857, 1/3*w^3 - 5/3*w^2 - 3*w + 5], [857, 857, 5/3*w^2 + w - 15], [863, 863, -2/3*w^3 - 5/3*w^2 + 10*w + 15], [863, 863, -5/3*w^3 - 8/3*w^2 + 18*w + 27], [863, 863, -3*w^3 - 14/3*w^2 + 32*w + 45], [863, 863, -5/3*w^2 + 4*w + 9], [911, 911, 1/3*w^3 - 2/3*w^2 - 4*w - 1], [911, 911, -1/3*w^3 + 2/3*w^2 + 2*w - 9], [911, 911, 1/3*w^3 + 2/3*w^2 - 2*w - 9], [911, 911, -1/3*w^3 - 2/3*w^2 + 4*w - 1], [919, 919, w^3 + 1/3*w^2 - 9*w - 1], [919, 919, -w^3 - 2/3*w^2 + 9*w + 7], [919, 919, w^3 - 2/3*w^2 - 9*w + 7], [919, 919, -w^3 + 1/3*w^2 + 9*w - 1], [953, 953, -w^3 - w^2 + 9*w + 5], [953, 953, -w^2 - 3*w + 7], [953, 953, w^2 - 3*w - 7], [953, 953, w^3 - w^2 - 9*w + 5], [961, 31, 4/3*w^2 - 7], [961, 31, 4/3*w^2 - 9], [967, 967, -w^3 - 1/3*w^2 + 9*w + 5], [967, 967, w^3 + 2/3*w^2 - 9*w - 5], [967, 967, -w^3 + 2/3*w^2 + 9*w - 5], [967, 967, w^3 - 1/3*w^2 - 9*w + 5], [1009, 1009, 1/3*w^3 + 5/3*w^2 - 4*w - 9], [1009, 1009, -8/3*w^3 - 10/3*w^2 + 26*w + 33], [1009, 1009, 4/3*w^3 + 8/3*w^2 - 16*w - 21], [1009, 1009, 7/3*w^3 + 5/3*w^2 - 22*w - 27], [1049, 1049, -2/3*w^3 + 2/3*w^2 + 5*w - 9], [1049, 1049, -1/3*w^3 - 2/3*w^2 + 5*w - 1], [1049, 1049, 1/3*w^3 - 2/3*w^2 - 5*w - 1], [1049, 1049, 2/3*w^3 + 2/3*w^2 - 5*w - 9], [1063, 1063, w^3 + 2/3*w^2 - 9*w - 13], [1063, 1063, 2*w^3 + w^2 - 17*w - 19], [1063, 1063, -2/3*w^3 - w^2 + 9*w + 11], [1063, 1063, -3*w^3 - 14/3*w^2 + 31*w + 43], [1097, 1097, -2/3*w^3 + 1/3*w^2 + 7*w + 1], [1097, 1097, 1/3*w^3 + 1/3*w^2 - w - 5], [1097, 1097, -1/3*w^3 + 1/3*w^2 + w - 5], [1097, 1097, 2/3*w^3 + 1/3*w^2 - 7*w + 1], [1103, 1103, 2/3*w^2 + 2*w - 7], [1103, 1103, 2/3*w^3 + 2/3*w^2 - 6*w - 1], [1103, 1103, -2/3*w^3 + 2/3*w^2 + 6*w - 1], [1103, 1103, 2/3*w^2 - 2*w - 7], [1151, 1151, -2/3*w^3 + 4*w - 7], [1151, 1151, 2/3*w^3 - 8*w - 7], [1151, 1151, -2/3*w^3 + 8*w - 7], [1151, 1151, 2/3*w^3 - 4*w - 7], [1153, 1153, -1/3*w^3 + 6*w - 5], [1153, 1153, -w^3 + 8*w + 5], [1153, 1153, -w^3 + 8*w - 5], [1153, 1153, 1/3*w^3 - 6*w - 5], [1193, 1193, -1/3*w^3 - 2/3*w^2 + 5*w - 3], [1193, 1193, 2/3*w^3 + 2/3*w^2 - 5*w - 11], [1193, 1193, -2/3*w^3 + 2/3*w^2 + 5*w - 11], [1193, 1193, 1/3*w^3 - 2/3*w^2 - 5*w - 3], [1201, 1201, -7/3*w^3 - 4/3*w^2 + 20*w + 23], [1201, 1201, -w^3 + 2/3*w^2 + 8*w - 1], [1201, 1201, w^3 + 2/3*w^2 - 8*w - 1], [1201, 1201, 1/3*w^3 + 2/3*w^2 - 6*w - 7], [1249, 1249, -2/3*w^3 + 2*w^2 + 6*w - 11], [1249, 1249, 2*w^2 + 2*w - 13], [1249, 1249, 2*w^2 - 2*w - 13], [1249, 1249, 2/3*w^3 + 2*w^2 - 6*w - 11], [1289, 1289, w^3 - 2/3*w^2 - 11*w + 3], [1289, 1289, 1/3*w^3 + w^2 - 5*w - 11], [1289, 1289, -1/3*w^3 + w^2 + 5*w - 11], [1289, 1289, -w^3 - 2/3*w^2 + 11*w + 3], [1297, 1297, -1/3*w^3 + 4*w - 7], [1297, 1297, -1/3*w^3 + 2*w - 7], [1297, 1297, 1/3*w^3 - 2*w - 7], [1297, 1297, 1/3*w^3 - 4*w - 7], [1303, 1303, 1/3*w^3 - 2/3*w^2 - 5*w + 13], [1303, 1303, -11/3*w^3 - 13/3*w^2 + 37*w + 49], [1303, 1303, -w^3 - 7/3*w^2 + 11*w + 19], [1303, 1303, -1/3*w^3 - 2/3*w^2 + 5*w + 13], [1399, 1399, 4/3*w^3 + 2/3*w^2 - 11*w + 1], [1399, 1399, -1/3*w^3 - 2/3*w^2 + 7*w + 9], [1399, 1399, 1/3*w^3 - 2/3*w^2 - 7*w + 9], [1399, 1399, -4/3*w^3 + 2/3*w^2 + 11*w + 1], [1433, 1433, w^3 - 9*w - 7], [1433, 1433, 4/3*w^3 + 11/3*w^2 - 17*w - 25], [1433, 1433, -4/3*w^3 - 7/3*w^2 + 13*w + 17], [1433, 1433, -w^3 + 9*w - 7], [1439, 1439, -1/3*w^3 - 4/3*w^2 + 6*w + 11], [1439, 1439, -2*w^3 - 3*w^2 + 22*w + 31], [1439, 1439, 4/3*w^3 + 3*w^2 - 16*w - 25], [1439, 1439, -5/3*w^3 - 10/3*w^2 + 20*w + 29], [1447, 1447, 1/3*w^2 + w - 9], [1447, 1447, 1/3*w^3 - 1/3*w^2 - 3*w - 5], [1447, 1447, -1/3*w^3 - 1/3*w^2 + 3*w - 5], [1447, 1447, 1/3*w^2 - w - 9], [1481, 1481, -2/3*w^3 + 5/3*w^2 + 7*w - 9], [1481, 1481, 1/3*w^3 + 5/3*w^2 - w - 11], [1481, 1481, -1/3*w^3 + 5/3*w^2 + w - 11], [1481, 1481, -2/3*w^3 - 5/3*w^2 + 7*w + 9], [1487, 1487, 2/3*w^3 + 8/3*w^2 - 10*w - 15], [1487, 1487, -w^3 - 1/3*w^2 + 10*w + 9], [1487, 1487, -w^3 + 1/3*w^2 + 10*w - 9], [1487, 1487, -2*w^3 - 10/3*w^2 + 20*w + 27], [1489, 1489, -1/3*w^3 + 5/3*w^2 + 2*w - 9], [1489, 1489, -1/3*w^3 + 5/3*w^2 + 4*w - 11], [1489, 1489, 1/3*w^3 + 5/3*w^2 - 4*w - 11], [1489, 1489, 1/3*w^3 + 5/3*w^2 - 2*w - 9], [1543, 1543, -4*w^3 - 13/3*w^2 + 39*w + 49], [1543, 1543, 4*w^3 + 5*w^2 - 41*w - 55], [1543, 1543, 2/3*w^3 - 2/3*w^2 - 9*w + 1], [1543, 1543, 4/3*w^3 + 5/3*w^2 - 15*w - 17], [1583, 1583, -1/3*w^3 + w^2 + 4*w + 1], [1583, 1583, 1/3*w^3 + w^2 - 2*w - 13], [1583, 1583, -1/3*w^3 + w^2 + 2*w - 13], [1583, 1583, -1/3*w^3 - w^2 + 4*w - 1], [1721, 1721, w^3 - 1/3*w^2 - 7*w + 9], [1721, 1721, 2/3*w^3 + 1/3*w^2 - 9*w + 5], [1721, 1721, -2/3*w^3 + 1/3*w^2 + 9*w + 5], [1721, 1721, w^3 + 1/3*w^2 - 7*w - 9], [1777, 1777, -8/3*w^3 - 3*w^2 + 28*w + 37], [1777, 1777, -w^3 - 2/3*w^2 + 8*w + 7], [1777, 1777, -w^3 + 2/3*w^2 + 8*w - 7], [1777, 1777, -2/3*w^3 - 3*w^2 + 10*w + 19], [1783, 1783, -1/3*w^2 - w - 5], [1783, 1783, -1/3*w^3 + 1/3*w^2 + 3*w - 9], [1783, 1783, 1/3*w^3 + 1/3*w^2 - 3*w - 9], [1783, 1783, -1/3*w^2 + w - 5], [1823, 1823, 2/3*w^3 - 4/3*w^2 - 4*w + 9], [1823, 1823, 2/3*w^3 - 4/3*w^2 - 8*w + 7], [1823, 1823, 2/3*w^3 + 4/3*w^2 - 8*w - 7], [1823, 1823, 2/3*w^3 + 4/3*w^2 - 4*w - 9], [1831, 1831, -w - 7], [1831, 1831, -1/3*w^3 + 3*w - 7], [1831, 1831, 1/3*w^3 - 3*w - 7], [1831, 1831, w - 7], [1871, 1871, -4*w^3 - 13/3*w^2 + 38*w + 49], [1871, 1871, -7/3*w^3 - 10/3*w^2 + 22*w + 31], [1871, 1871, -7/3*w^3 - 4/3*w^2 + 20*w + 25], [1871, 1871, 2/3*w^3 + 1/3*w^2 - 4*w - 7], [1873, 1873, -1/3*w^2 + 4*w - 3], [1873, 1873, 4/3*w^3 + 1/3*w^2 - 12*w - 7], [1873, 1873, 4/3*w^3 - 1/3*w^2 - 12*w + 7], [1873, 1873, 1/3*w^2 + 4*w + 3], [1879, 1879, -1/3*w^3 + 7/3*w^2 + 5*w - 11], [1879, 1879, 2/3*w^3 + 7/3*w^2 - 5*w - 17], [1879, 1879, -2/3*w^3 + 7/3*w^2 + 5*w - 17], [1879, 1879, 1/3*w^3 + 7/3*w^2 - 5*w - 11], [1913, 1913, 1/3*w^3 + 1/3*w^2 - w - 7], [1913, 1913, -2/3*w^3 - 1/3*w^2 + 7*w - 3], [1913, 1913, 2/3*w^3 - 1/3*w^2 - 7*w - 3], [1913, 1913, -1/3*w^3 + 1/3*w^2 + w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^2 + x - 1; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -3*e - 2, 2*e - 1, 1, e - 1, 2*e + 1, -5*e + 2, e + 7, 2*e - 7, 4*e + 9, 6*e + 1, -2*e - 1, -7*e - 7, -6*e - 2, -3*e - 14, -8*e - 1, 2*e - 8, -7*e + 4, 6*e - 5, 4*e - 6, 7*e + 9, -8*e + 1, 11*e - 2, 2*e + 5, -5, 2*e - 9, -18, 5*e + 13, -e + 2, -5*e - 5, -9*e - 17, -e - 10, 5*e + 1, 4*e + 10, -6*e - 5, -10*e + 1, -6*e + 2, 4*e + 7, 6*e + 17, -5*e + 10, -9*e - 10, -4*e - 14, 18*e + 9, e - 4, -11*e - 7, -5*e - 11, 12*e + 15, -2*e - 5, -8*e + 3, -11*e - 5, 3*e - 15, -11, 3*e + 4, -18*e - 6, -12*e + 3, -9*e + 9, e + 2, 7*e - 16, -20, 9*e + 25, 23*e + 19, -6*e + 7, -3*e + 5, -3*e - 15, 8*e + 7, 5*e + 21, -18*e + 1, 8*e + 26, -5*e + 26, 10*e + 18, 22*e + 2, -16*e - 10, -e - 16, 9*e - 16, 16*e, -2*e - 21, 6*e - 15, 25*e + 14, 18*e + 17, 10*e + 5, -14*e - 30, -10*e - 5, 10*e - 19, 15*e + 1, 19*e - 2, 3*e + 1, 24*e + 11, -e + 12, -30*e - 22, -20*e - 24, 19*e, -32*e - 13, -35, -6*e + 12, 10*e - 22, 32*e + 11, -13*e - 34, -22*e - 21, -9*e + 11, 5*e - 1, -30*e - 5, 20*e + 23, 17*e - 6, -6*e + 19, -6*e, e - 29, -11*e - 35, 4*e + 35, -11*e - 2, -15*e - 17, -32*e - 8, -2*e - 27, -2*e - 24, -25*e - 25, 12*e - 14, 2*e + 48, -5*e + 23, -9, -24*e - 25, -33*e - 12, 7*e + 10, 9*e + 20, -2*e + 29, 2*e - 32, 20*e - 13, 24*e + 7, 3*e + 11, -4*e + 23, e - 22, -11*e + 13, 4*e - 13, 4*e - 27, -2*e + 7, 35*e + 2, 26*e - 12, 2*e - 2, 12*e + 4, -28*e - 6, 38*e + 26, -29*e - 4, -11*e - 43, -11*e - 43, -14*e - 5, -13*e - 42, e + 34, 29*e + 3, 18*e + 37, 3*e + 51, -6*e - 27, 23*e + 23, -22*e + 21, 20*e + 21, -10*e - 41, -2*e - 1, 7*e + 51, -12*e - 8, -4*e + 28, 9*e - 2, 12*e - 21, -2*e - 41, 12*e + 28, 16*e + 10, -45*e - 28, 34*e + 27, 6*e - 13, -9*e + 32, -8*e - 17, -25*e - 45, 3*e + 16, -2*e + 2, -24*e - 24, -e - 23, 30*e + 29, -e - 6, -8*e - 20, 11*e - 20, 34*e + 36, -35*e - 4, 5*e - 16, 42*e + 17, 10*e + 12, -e - 55, -9*e - 14, 16*e - 3, -22*e + 11, -16*e - 10, 6*e + 17, -29*e + 13, -13*e - 14, 8*e + 5, 27*e - 2, 6*e + 7, 15*e - 17, -5*e - 25, -11*e - 16, -8*e + 18, -15*e + 1, 59*e + 31, -24*e - 11, -28*e - 12, 3*e - 59, -25*e + 25, 43*e + 14, -25*e + 16, 11*e - 4, -18*e - 1, -12*e - 57, 10*e - 4, -e + 8, 28*e + 12, 23*e - 20, -12*e - 5, 42*e + 25, 32*e + 49, -4*e - 5, -6*e + 4, -2*e + 12, -2*e + 63, -2*e - 9, -56, 11*e - 24, -51, 34*e - 12, -4*e + 64, 9*e + 13, -24*e - 31, 40*e + 30, -49*e - 35, -28*e - 13, 17*e - 18, -21*e + 18, -24*e + 22, -9*e + 4, 16*e + 41, -22*e - 29, -10*e - 7, 22*e + 14, -5*e + 28, -19*e - 49, 52*e + 40, -26*e - 20, 18*e + 32, 31*e + 47, -39*e + 1, 24*e - 33, -28*e - 39, 54*e + 38, 47*e + 22, -5*e, 39*e + 19, -11*e - 46, -4*e - 49, -7*e - 59, 31*e, -14*e - 35, 30*e - 5, -49*e - 20, 2*e + 57, 23*e - 9, 19*e - 29, -34*e - 28, 3*e - 18, -12*e + 20, -2*e - 6, 2*e - 23, 28*e + 9, 9*e + 9, 46*e + 42, 28*e + 28, -12*e - 8, 30*e + 19, -53*e - 16, -3*e + 37, 41*e + 10, -15*e - 20, -20*e - 64, -49*e - 22, -5*e - 32, 33*e - 22, -52*e - 29, 52*e + 44, -35*e - 34, -32*e + 14, 2*e - 6, -15*e + 28, -38*e - 43, 9*e + 15, 27*e + 67, -18*e - 57, -42*e - 23, 11*e + 52, -15*e + 17, 42*e + 17]; 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;