/* 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^8 + x^7 - 33*x^6 + 8*x^5 + 305*x^4 - 353*x^3 - 373*x^2 + 354*x + 211; K := NumberField(heckePol); heckeEigenvaluesArray := [-47/4705*e^7 - 42/941*e^6 + 1103/4705*e^5 + 822/941*e^4 - 1678/941*e^3 - 19814/4705*e^2 + 3998/941*e + 16911/4705, e, 103/10351*e^7 + 359/51755*e^6 - 13908/51755*e^5 + 4187/10351*e^4 + 17826/10351*e^3 - 75304/10351*e^2 + 225453/51755*e + 294969/51755, -8/10351*e^7 + 2384/51755*e^6 + 5904/51755*e^5 - 14897/10351*e^4 - 12640/10351*e^3 + 127046/10351*e^2 - 222722/51755*e - 544882/51755, 42/51755*e^7 - 433/51755*e^6 - 4129/51755*e^5 + 1668/10351*e^4 + 13272/10351*e^3 - 40756/51755*e^2 - 274376/51755*e + 12137/51755, 1, 3183/51755*e^7 + 4892/51755*e^6 - 19849/10351*e^5 - 5195/10351*e^4 + 167397/10351*e^3 - 700599/51755*e^2 - 459981/51755*e + 64630/10351, -7321/51755*e^7 - 14972/51755*e^6 + 215236/51755*e^5 + 28654/10351*e^4 - 357097/10351*e^3 + 950243/51755*e^2 + 1391271/51755*e - 175768/51755, -953/51755*e^7 - 3237/51755*e^6 + 28872/51755*e^5 + 15386/10351*e^4 - 52724/10351*e^3 - 477541/51755*e^2 + 698781/51755*e + 476039/51755, 1472/51755*e^7 - 2853/51755*e^6 - 59932/51755*e^5 + 24449/10351*e^4 + 113218/10351*e^3 - 1297781/51755*e^2 - 90841/51755*e + 575216/51755, -2072/51755*e^7 - 2791/51755*e^6 + 65684/51755*e^5 + 520/10351*e^4 - 116500/10351*e^3 + 526986/51755*e^2 + 548828/51755*e - 63957/51755, 4338/51755*e^7 + 8511/51755*e^6 - 124809/51755*e^5 - 11080/10351*e^4 + 201145/10351*e^3 - 941554/51755*e^2 - 469793/51755*e + 599987/51755, -4338/51755*e^7 - 8511/51755*e^6 + 124809/51755*e^5 + 11080/10351*e^4 - 201145/10351*e^3 + 941554/51755*e^2 + 469793/51755*e - 651742/51755, 2072/51755*e^7 + 2791/51755*e^6 - 65684/51755*e^5 - 520/10351*e^4 + 116500/10351*e^3 - 526986/51755*e^2 - 548828/51755*e + 12202/51755, 6416/51755*e^7 + 3431/10351*e^6 - 185168/51755*e^5 - 51287/10351*e^4 + 319541/10351*e^3 + 263127/51755*e^2 - 481281/10351*e - 675516/51755, 1050/10351*e^7 + 18332/51755*e^6 - 133138/51755*e^5 - 60626/10351*e^4 + 199509/10351*e^3 + 171465/10351*e^2 - 1028886/51755*e - 1546771/51755, -4068/51755*e^7 - 3901/51755*e^6 + 136712/51755*e^5 - 4814/10351*e^4 - 250388/10351*e^3 + 1234069/51755*e^2 + 1005348/51755*e - 570761/51755, -589/10351*e^7 - 10796/51755*e^6 + 72397/51755*e^5 + 35349/10351*e^4 - 102540/10351*e^3 - 78587/10351*e^2 + 450933/51755*e - 19751/51755, 502/4705*e^7 + 1322/4705*e^6 - 13683/4705*e^5 - 3454/941*e^4 + 21246/941*e^3 - 6101/4705*e^2 - 77806/4705*e - 55591/4705, 1636/51755*e^7 + 2443/10351*e^6 - 28243/51755*e^5 - 56282/10351*e^4 + 40830/10351*e^3 + 1801177/51755*e^2 - 247920/10351*e - 1571311/51755, -716/10351*e^7 - 4003/51755*e^6 + 114368/51755*e^5 - 3178/10351*e^4 - 189339/10351*e^3 + 232941/10351*e^2 + 23109/51755*e - 531279/51755, 6762/51755*e^7 + 23446/51755*e^6 - 178272/51755*e^5 - 83386/10351*e^4 + 283963/10351*e^3 + 1460309/51755*e^2 - 2325443/51755*e - 1679144/51755, -2547/51755*e^7 - 5534/51755*e^6 + 73688/51755*e^5 + 11230/10351*e^4 - 121686/10351*e^3 + 268276/51755*e^2 + 494342/51755*e + 237711/51755, 3781/51755*e^7 + 1717/10351*e^6 - 106772/51755*e^5 - 16935/10351*e^4 + 170047/10351*e^3 - 524278/51755*e^2 - 94525/10351*e + 344636/51755, -868/10351*e^7 - 10462/51755*e^6 + 123034/51755*e^5 + 24309/10351*e^4 - 201777/10351*e^3 + 69416/10351*e^2 + 915136/51755*e - 67242/51755, 311/10351*e^7 + 481/51755*e^6 - 53551/51755*e^5 + 8522/10351*e^4 + 98042/10351*e^3 - 148988/10351*e^2 - 277183/51755*e + 301733/51755, 2548/10351*e^7 + 27372/51755*e^6 - 379529/51755*e^5 - 63345/10351*e^4 + 641063/10351*e^3 - 198761/10351*e^2 - 3000236/51755*e - 662413/51755, -683/10351*e^7 - 3486/51755*e^6 + 110716/51755*e^5 - 2540/10351*e^4 - 188954/10351*e^3 + 177259/10351*e^2 + 302663/51755*e + 114542/51755, -9313/51755*e^7 - 22531/51755*e^6 + 269112/51755*e^5 + 58968/10351*e^4 - 448317/10351*e^3 + 310279/51755*e^2 + 2592563/51755*e - 29796/51755, -3631/51755*e^7 - 3505/10351*e^6 + 84632/51755*e^5 + 70211/10351*e^4 - 132998/10351*e^3 - 1728447/51755*e^2 + 328848/10351*e + 1569284/51755, -4781/51755*e^7 - 11091/51755*e^6 + 140511/51755*e^5 + 27497/10351*e^4 - 237623/10351*e^3 + 257468/51755*e^2 + 1146228/51755*e + 89972/51755, -1236/51755*e^7 - 5002/51755*e^6 + 31309/51755*e^5 + 18934/10351*e^4 - 48993/10351*e^3 - 338472/51755*e^2 + 527136/51755*e - 153112/51755, 2312/51755*e^7 + 9189/51755*e^6 - 59704/51755*e^5 - 35350/10351*e^4 + 99181/10351*e^3 + 733624/51755*e^2 - 1189537/51755*e - 620833/51755, 1258/10351*e^7 + 18454/51755*e^6 - 172781/51755*e^5 - 56291/10351*e^4 + 279725/10351*e^3 + 87430/10351*e^2 - 1583277/51755*e - 1125967/51755, -1800/10351*e^7 - 22554/51755*e^6 + 262247/51755*e^5 + 64005/10351*e^4 - 452919/10351*e^3 - 35165/10351*e^2 + 3319412/51755*e + 723364/51755, -2141/51755*e^7 - 2634/10351*e^6 + 30324/51755*e^5 + 48056/10351*e^4 - 14092/10351*e^3 - 1040037/51755*e^2 + 32823/10351*e + 330883/51755, 1277/51755*e^7 - 11933/51755*e^6 - 72554/51755*e^5 + 77332/10351*e^4 + 144757/10351*e^3 - 3518861/51755*e^2 + 813369/51755*e + 2610507/51755, -3061/51755*e^7 - 10093/51755*e^6 + 83308/51755*e^5 + 36657/10351*e^4 - 139196/10351*e^3 - 714127/51755*e^2 + 1200354/51755*e + 571731/51755, -6813/51755*e^7 - 26617/51755*e^6 + 169238/51755*e^5 + 94669/10351*e^4 - 269026/10351*e^3 - 1713956/51755*e^2 + 2673401/51755*e + 1647401/51755, -11578/51755*e^7 - 32451/51755*e^6 + 323949/51755*e^5 + 99142/10351*e^4 - 532646/10351*e^3 - 841096/51755*e^2 + 3289728/51755*e + 2340383/51755, -4526/51755*e^7 - 15938/51755*e^6 + 113224/51755*e^5 + 53890/10351*e^4 - 167394/10351*e^3 - 803272/51755*e^2 + 1176459/51755*e + 776588/51755, 8958/51755*e^7 + 12636/51755*e^6 - 55764/10351*e^5 - 3567/10351*e^4 + 470700/10351*e^3 - 2526434/51755*e^2 - 1202558/51755*e + 331116/10351, -2591/51755*e^7 - 17403/51755*e^6 + 44989/51755*e^5 + 74546/10351*e^4 - 52782/10351*e^3 - 2096867/51755*e^2 + 1141604/51755*e + 1679558/51755, -7797/51755*e^7 - 2703/10351*e^6 + 237879/51755*e^5 + 20101/10351*e^4 - 404003/10351*e^3 + 1274131/51755*e^2 + 298435/10351*e + 138673/51755, -5532/51755*e^7 - 13946/51755*e^6 + 32468/10351*e^5 + 42033/10351*e^4 - 278270/10351*e^3 - 213999/51755*e^2 + 2068183/51755*e + 83670/10351, 10233/51755*e^7 + 19454/51755*e^6 - 62349/10351*e^5 - 37218/10351*e^4 + 532017/10351*e^3 - 1360759/51755*e^2 - 2283172/51755*e + 44738/10351, -986/51755*e^7 + 160/10351*e^6 + 25462/51755*e^5 - 19935/10351*e^4 - 21748/10351*e^3 + 1358513/51755*e^2 - 203072/10351*e - 2042136/51755, 8837/51755*e^7 + 23988/51755*e^6 - 246469/51755*e^5 - 67521/10351*e^4 + 401411/10351*e^3 + 117119/51755*e^2 - 2336394/51755*e - 1260168/51755, -942/10351*e^7 - 9112/51755*e^6 + 146593/51755*e^5 + 18487/10351*e^4 - 266942/10351*e^3 + 111157/10351*e^2 + 1944731/51755*e - 692699/51755, -1073/51755*e^7 - 6436/51755*e^6 + 25882/51755*e^5 + 32801/10351*e^4 - 38889/10351*e^3 - 1159601/51755*e^2 + 527463/51755*e + 1025454/51755, 768/4705*e^7 + 406/941*e^6 - 21327/4705*e^5 - 5123/941*e^4 + 34727/941*e^3 - 36814/4705*e^2 - 33315/941*e - 13854/4705, 474/4705*e^7 + 356/4705*e^6 - 15949/4705*e^5 + 1080/941*e^4 + 28395/941*e^3 - 156152/4705*e^2 - 76188/4705*e - 14123/4705, 1668/51755*e^7 + 2027/51755*e^6 - 51598/51755*e^5 + 1180/10351*e^4 + 81995/10351*e^3 - 642644/51755*e^2 - 115341/51755*e + 928584/51755, -18949/51755*e^7 - 44443/51755*e^6 + 109313/10351*e^5 + 106587/10351*e^4 - 905543/10351*e^3 + 1239592/51755*e^2 + 4324964/51755*e + 43103/10351, -2428/10351*e^7 - 21728/51755*e^6 + 373777/51755*e^5 + 38376/10351*e^4 - 637781/10351*e^3 + 342569/10351*e^2 + 2542249/51755*e + 306419/51755, 18902/51755*e^7 + 45174/51755*e^6 - 541698/51755*e^5 - 111411/10351*e^4 + 890691/10351*e^3 - 1028861/51755*e^2 - 4401897/51755*e - 412211/51755, 71/4705*e^7 + 97/4705*e^6 - 2387/4705*e^5 - 541/941*e^4 + 3616/941*e^3 + 25427/4705*e^2 + 4299/4705*e - 93859/4705, 212/51755*e^7 + 10137/51755*e^6 + 12183/51755*e^5 - 56644/10351*e^4 - 15816/10351*e^3 + 2305629/51755*e^2 - 1485991/51755*e - 2273134/51755, 4821/51755*e^7 + 19058/51755*e^6 - 125713/51755*e^5 - 74706/10351*e^4 + 208859/10351*e^3 + 1746807/51755*e^2 - 2455454/51755*e - 2453721/51755, -4652/51755*e^7 - 4288/51755*e^6 + 135962/51755*e^5 - 23571/10351*e^4 - 207210/10351*e^3 + 2683071/51755*e^2 - 567461/51755*e - 2654951/51755, 8123/51755*e^7 + 10647/51755*e^6 - 248733/51755*e^5 + 7633/10351*e^4 + 424211/10351*e^3 - 2812879/51755*e^2 - 1388011/51755*e + 1752664/51755, -3639/51755*e^7 - 4627/51755*e^6 + 129287/51755*e^5 + 9266/10351*e^4 - 259738/10351*e^3 + 344587/51755*e^2 + 2431916/51755*e + 129169/51755, -806/4705*e^7 - 1459/4705*e^6 + 24301/4705*e^5 + 2404/941*e^4 - 40148/941*e^3 + 127908/4705*e^2 + 126157/4705*e - 71018/4705, 853/4705*e^7 + 1669/4705*e^6 - 25404/4705*e^5 - 3226/941*e^4 + 41826/941*e^3 - 108094/4705*e^2 - 146147/4705*e - 54108/4705, 18322/51755*e^7 + 38338/51755*e^6 - 549249/51755*e^5 - 84169/10351*e^4 + 935133/10351*e^3 - 1806741/51755*e^2 - 4453609/51755*e - 115008/51755, -626/4705*e^7 - 1836/4705*e^6 + 17494/4705*e^5 + 5923/941*e^4 - 29377/941*e^3 - 58187/4705*e^2 + 255158/4705*e + 16218/4705, -8553/51755*e^7 - 16072/51755*e^6 + 260446/51755*e^5 + 31481/10351*e^4 - 446230/10351*e^3 + 869129/51755*e^2 + 1855801/51755*e + 1058817/51755, -13223/51755*e^7 - 9654/10351*e^6 + 339029/51755*e^5 + 168375/10351*e^4 - 534916/10351*e^3 - 2781411/51755*e^2 + 837774/10351*e + 2960498/51755, -1645/10351*e^7 - 6638/51755*e^6 + 56484/10351*e^5 - 16120/10351*e^4 - 528900/10351*e^3 + 554276/10351*e^2 + 2518669/51755*e - 270071/10351, 11236/51755*e^7 + 30062/51755*e^6 - 316944/51755*e^5 - 83150/10351*e^4 + 538435/10351*e^3 + 4782/51755*e^2 - 3950846/51755*e - 207833/51755, -6/55*e^7 - 12/55*e^6 + 164/55*e^5 + 10/11*e^4 - 235/11*e^3 + 1878/55*e^2 - 504/55*e - 2342/55, 3629/10351*e^7 + 57168/51755*e^6 - 96758/10351*e^5 - 181400/10351*e^4 + 765340/10351*e^3 + 399052/10351*e^2 - 4690259/51755*e - 736360/10351, -771/51755*e^7 - 12014/51755*e^6 + 4079/51755*e^5 + 64018/10351*e^4 - 15914/10351*e^3 - 2327562/51755*e^2 + 2135522/51755*e + 1411918/51755, 3721/51755*e^7 + 362/10351*e^6 - 128969/51755*e^5 + 17650/10351*e^4 + 244246/10351*e^3 - 1796898/51755*e^2 - 165482/10351*e + 1380142/51755, -19531/51755*e^7 - 10646/10351*e^6 + 537239/51755*e^5 + 147058/10351*e^4 - 851382/10351*e^3 - 132762/51755*e^2 + 747050/10351*e + 2132318/51755, 6241/51755*e^7 + 17234/51755*e^6 - 169689/51755*e^5 - 47886/10351*e^4 + 253890/10351*e^3 - 101858/51755*e^2 - 645562/51755*e + 452202/51755, 5664/51755*e^7 + 28851/51755*e^6 - 138349/51755*e^5 - 128471/10351*e^4 + 237174/10351*e^3 + 3790088/51755*e^2 - 3740843/51755*e - 3211943/51755, 8553/51755*e^7 + 36774/51755*e^6 - 208691/51755*e^5 - 145342/10351*e^4 + 322018/10351*e^3 + 3426536/51755*e^2 - 3781087/51755*e - 2766732/51755, 5947/51755*e^7 + 4053/10351*e^6 - 140786/51755*e^5 - 59562/10351*e^4 + 181688/10351*e^3 + 493964/51755*e^2 + 27292/10351*e - 667857/51755, -3103/51755*e^7 - 1932/10351*e^6 + 87437/51755*e^5 + 34989/10351*e^4 - 131766/10351*e^3 - 518106/51755*e^2 + 25820/10351*e + 1387674/51755, -14404/51755*e^7 - 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1041090/10351*e^3 + 2308879/51755*e^2 + 2611018/51755*e + 782404/51755, 17137/51755*e^7 + 26156/51755*e^6 - 519257/51755*e^5 - 14412/10351*e^4 + 850501/10351*e^3 - 4427561/51755*e^2 - 1759138/51755*e + 2382426/51755, -21898/51755*e^7 - 9758/10351*e^6 + 615412/51755*e^5 + 95937/10351*e^4 - 957592/10351*e^3 + 2866519/51755*e^2 + 392185/10351*e - 187886/51755, -1875/10351*e^7 - 31257/51755*e^6 + 255491/51755*e^5 + 117133/10351*e^4 - 405803/10351*e^3 - 476979/10351*e^2 + 3097161/51755*e + 4146907/51755, 5347/51755*e^7 + 14621/51755*e^6 - 29077/10351*e^5 - 44944/10351*e^4 + 199108/10351*e^3 + 240719/51755*e^2 + 439182/51755*e - 14758/10351, 1472/51755*e^7 - 2853/51755*e^6 - 59932/51755*e^5 + 24449/10351*e^4 + 123569/10351*e^3 - 1246026/51755*e^2 - 815411/51755*e + 2904191/51755, -2283/51755*e^7 + 3574/51755*e^6 + 100968/51755*e^5 - 27083/10351*e^4 - 214229/10351*e^3 + 1520384/51755*e^2 + 1455038/51755*e + 819721/51755, 794/51755*e^7 + 18924/51755*e^6 + 21509/51755*e^5 - 97115/10351*e^4 - 69977/10351*e^3 + 3522718/51755*e^2 - 1362072/51755*e - 463577/51755, 11391/51755*e^7 + 6235/10351*e^6 - 329471/51755*e^5 - 99175/10351*e^4 + 566713/10351*e^3 + 997912/51755*e^2 - 823027/10351*e - 3587/51755, -16636/51755*e^7 - 8401/51755*e^6 + 575732/51755*e^5 - 54414/10351*e^4 - 1033768/10351*e^3 + 6152078/51755*e^2 + 2731613/51755*e - 1711966/51755, 790/10351*e^7 + 2653/51755*e^6 - 137927/51755*e^5 - 1351/10351*e^4 + 244153/10351*e^3 - 88364/10351*e^2 - 1104459/51755*e - 1068729/51755, 50707/51755*e^7 + 116286/51755*e^6 - 1478703/51755*e^5 - 282651/10351*e^4 + 2473953/10351*e^3 - 2941066/51755*e^2 - 12683538/51755*e - 2192151/51755, -14756/51755*e^7 - 6588/51755*e^6 + 494913/51755*e^5 - 68474/10351*e^4 - 853728/10351*e^3 + 7038738/51755*e^2 + 1161334/51755*e - 5623564/51755, 16153/51755*e^7 + 28907/51755*e^6 - 512722/51755*e^5 - 68278/10351*e^4 + 912193/10351*e^3 - 870169/51755*e^2 - 4927756/51755*e - 2976874/51755, 4432/51755*e^7 + 58804/51755*e^6 - 10331/51755*e^5 - 291260/10351*e^4 - 58979/10351*e^3 + 10178349/51755*e^2 - 4508082/51755*e - 8902177/51755, -31558/51755*e^7 - 10542/10351*e^6 + 1006128/51755*e^5 + 95284/10351*e^4 - 1795038/10351*e^3 + 4270129/51755*e^2 + 1958613/10351*e - 112169/51755, 32687/51755*e^7 + 62019/51755*e^6 - 992661/51755*e^5 - 115510/10351*e^4 + 1717060/10351*e^3 - 4424241/51755*e^2 - 9592607/51755*e + 1331813/51755, 101/941*e^7 + 1896/4705*e^6 - 11491/4705*e^5 - 4948/941*e^4 + 18430/941*e^3 - 1668/941*e^2 - 221213/4705*e + 186508/4705, 43218/51755*e^7 + 113397/51755*e^6 - 1226249/51755*e^5 - 322775/10351*e^4 + 2022364/10351*e^3 + 915216/51755*e^2 - 11726711/51755*e - 5066323/51755, 2491/51755*e^7 + 13012/51755*e^6 - 30229/51755*e^5 - 44507/10351*e^4 - 40924/10351*e^3 + 786662/51755*e^2 + 2162514/51755*e + 11782/51755, -5393/10351*e^7 - 7739/10351*e^6 + 812628/51755*e^5 - 3264/10351*e^4 - 1316644/10351*e^3 + 1726740/10351*e^2 + 332542/10351*e - 5791789/51755, 252/51755*e^7 - 2598/51755*e^6 + 3326/10351*e^5 + 41061/10351*e^4 - 96335/10351*e^3 - 2470001/51755*e^2 + 3684509/51755*e + 300252/10351, 15174/51755*e^7 + 21009/51755*e^6 - 465521/51755*e^5 + 5224/10351*e^4 + 799498/10351*e^3 - 4780207/51755*e^2 - 3107817/51755*e + 1557623/51755, -11186/51755*e^7 - 2468/10351*e^6 + 340617/51755*e^5 - 30204/10351*e^4 - 564039/10351*e^3 + 4868353/51755*e^2 + 186491/10351*e - 5078416/51755, 1447/51755*e^7 + 19339/51755*e^6 - 25189/51755*e^5 - 115543/10351*e^4 + 63914/10351*e^3 + 4779349/51755*e^2 - 3927937/51755*e - 6444318/51755]; 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;