/*
  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<x> := PolynomialRing(Rationals());
g := P![1, -6, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, w^3 - w^2 - 5*w - 1}>;

primesArray := [
[4, 2, w^3 - w^2 - 5*w - 2],
[9, 3, w^3 - w^2 - 5*w - 1],
[11, 11, -w^3 + w^2 + 6*w + 2],
[11, 11, w - 1],
[13, 13, w^3 - 2*w^2 - 4*w + 2],
[13, 13, -w^2 + w + 4],
[23, 23, w^2 - 2*w - 2],
[23, 23, w^3 - w^2 - 6*w - 3],
[23, 23, -w^2 + 2*w + 5],
[23, 23, -w + 2],
[37, 37, 2*w^3 - 2*w^2 - 12*w - 1],
[37, 37, w^3 - 2*w^2 - 5*w + 2],
[37, 37, w^3 - 2*w^2 - 5*w + 3],
[37, 37, -w^3 + w^2 + 6*w - 2],
[47, 47, w^2 - 2*w - 1],
[47, 47, w^2 - 2*w - 6],
[59, 59, 2*w - 1],
[59, 59, -2*w^3 + 2*w^2 + 12*w + 3],
[73, 73, -w^3 + w^2 + 7*w + 1],
[83, 83, -w^3 + w^2 + 4*w + 3],
[83, 83, 2*w^3 - 2*w^2 - 11*w - 4],
[107, 107, w^3 - w^2 - 4*w - 5],
[107, 107, -2*w^3 + 2*w^2 + 11*w + 6],
[121, 11, 2*w^3 - 2*w^2 - 10*w - 3],
[131, 131, 2*w^3 - 3*w^2 - 7*w + 2],
[131, 131, 3*w^3 - 2*w^2 - 18*w - 6],
[157, 157, -2*w^3 + 2*w^2 + 9*w],
[157, 157, -2*w^3 + 4*w^2 + 8*w - 3],
[167, 167, -2*w^3 + w^2 + 11*w + 10],
[167, 167, -3*w^3 + 4*w^2 + 14*w + 4],
[169, 13, -2*w^3 + 2*w^2 + 10*w + 7],
[179, 179, -w^3 + 6*w + 2],
[179, 179, 2*w^3 - 3*w^2 - 9*w + 4],
[181, 181, 2*w^2 - w - 8],
[181, 181, 3*w^3 - 5*w^2 - 14*w + 3],
[181, 181, -3*w^3 + 3*w^2 + 18*w + 2],
[181, 181, w^3 - 7*w - 9],
[191, 191, -2*w^3 + 3*w^2 + 10*w + 1],
[191, 191, w^2 - 6],
[193, 193, w^3 - 3*w^2 - 4*w + 3],
[193, 193, 2*w^3 - 4*w^2 - 9*w + 8],
[229, 229, -2*w^3 + 2*w^2 + 13*w + 3],
[229, 229, w^2 - w - 8],
[239, 239, w^3 - 7*w - 2],
[239, 239, -w^3 + 2*w^2 + 3*w - 5],
[241, 241, -3*w^3 + 3*w^2 + 16*w + 2],
[241, 241, 2*w^3 - 2*w^2 - 9*w - 1],
[251, 251, -2*w^3 + 3*w^2 + 8*w - 3],
[251, 251, -2*w^3 + w^2 + 12*w + 4],
[251, 251, w^3 + w^2 - 7*w - 7],
[251, 251, 2*w^3 - 3*w^2 - 10*w - 2],
[263, 263, 3*w^3 - 4*w^2 - 16*w],
[263, 263, w^2 + w - 4],
[277, 277, 2*w^2 - w - 7],
[277, 277, 3*w^3 - 5*w^2 - 14*w + 4],
[313, 313, 2*w^3 - 4*w^2 - 9*w + 3],
[313, 313, -w^3 + 3*w^2 + 4*w - 8],
[337, 337, 2*w^3 - 2*w^2 - 13*w - 1],
[337, 337, w^3 - w^2 - 8*w - 2],
[347, 347, 4*w^3 - 4*w^2 - 22*w + 1],
[347, 347, -2*w^3 + 5*w^2 + 6*w - 10],
[347, 347, -4*w^3 + 5*w^2 + 20*w + 3],
[347, 347, -3*w^2 + 4*w + 9],
[349, 349, -4*w^3 + 5*w^2 + 19*w + 4],
[349, 349, -w^3 + 4*w^2 + w - 11],
[349, 349, 2*w^3 - 3*w^2 - 9*w - 4],
[349, 349, w^3 - 6*w - 10],
[361, 19, -w^3 + 3*w^2 + 4*w - 4],
[361, 19, -2*w^3 + 4*w^2 + 9*w - 7],
[397, 397, 3*w^3 - 3*w^2 - 14*w - 6],
[397, 397, -2*w^3 + 5*w^2 + 6*w - 6],
[397, 397, 3*w^2 - 4*w - 13],
[397, 397, -4*w^3 + 4*w^2 + 21*w + 7],
[409, 409, 2*w^3 - 4*w^2 - 10*w + 7],
[409, 409, -2*w^3 + 4*w^2 + 10*w - 3],
[421, 421, w^3 - w^2 - 6*w - 6],
[421, 421, w - 5],
[431, 431, -3*w^3 + 4*w^2 + 18*w - 4],
[431, 431, 2*w^3 - 3*w^2 - 6*w + 3],
[433, 433, 2*w^3 - 15*w - 12],
[433, 433, w^3 - w^2 - 9*w - 5],
[443, 443, -3*w^3 + 5*w^2 + 14*w - 9],
[443, 443, 2*w^3 - 12*w - 13],
[457, 457, -w^3 + 2*w^2 + 5*w - 8],
[457, 457, -2*w^3 + 4*w^2 + 9*w - 4],
[457, 457, 2*w^3 - 2*w^2 - 14*w - 5],
[457, 457, -w^3 + 3*w^2 + 4*w - 7],
[467, 467, -2*w^3 + 3*w^2 + 8*w - 4],
[467, 467, -2*w^3 + w^2 + 12*w + 3],
[479, 479, w^3 + w^2 - 8*w - 7],
[479, 479, 2*w^3 - 14*w - 9],
[479, 479, -2*w^3 + 4*w^2 + 6*w - 5],
[479, 479, -2*w^3 + 4*w^2 + 7*w - 6],
[491, 491, -2*w^3 + 3*w^2 + 11*w + 2],
[491, 491, -w^3 + 2*w^2 + 6*w - 6],
[541, 541, -4*w^2 + 6*w + 15],
[541, 541, -2*w^3 + 6*w^2 + 4*w - 11],
[563, 563, -2*w^3 + w^2 + 14*w],
[563, 563, -w^2 + 4*w + 9],
[577, 577, w^2 - 4*w - 4],
[577, 577, -2*w^3 + w^2 + 14*w + 5],
[587, 587, w^3 - 4*w^2 + 6],
[587, 587, -w^3 + 3*w^2 + 6*w - 7],
[587, 587, 4*w^3 - 6*w^2 - 21*w + 2],
[587, 587, 3*w^3 - 5*w^2 - 14*w],
[599, 599, 2*w^3 - 2*w^2 - 7*w - 3],
[599, 599, 5*w^3 - 5*w^2 - 28*w - 6],
[601, 601, w^3 + w^2 - 10*w - 10],
[601, 601, 2*w^3 - 3*w^2 - 10*w - 5],
[613, 613, w^2 - 4*w - 3],
[613, 613, 2*w^3 - w^2 - 14*w - 6],
[625, 5, -5],
[647, 647, -3*w^3 + 4*w^2 + 17*w + 1],
[647, 647, -w^3 + 2*w^2 + 7*w - 4],
[659, 659, -5*w^3 + 7*w^2 + 23*w + 3],
[659, 659, w^3 - 2*w^2 - w - 2],
[659, 659, -w^3 + 2*w^2 + 2*w - 6],
[659, 659, -2*w^3 + w^2 + 13*w + 2],
[683, 683, -2*w^3 + 3*w^2 + 8*w - 5],
[683, 683, -2*w^3 + w^2 + 12*w + 2],
[709, 709, w^3 + w^2 - 10*w - 8],
[709, 709, 2*w^2 - 5*w - 7],
[719, 719, w^3 - 5*w - 10],
[719, 719, -3*w^3 + 4*w^2 + 15*w + 5],
[733, 733, -w^3 + w^2 + 4*w - 5],
[733, 733, 2*w^2 - 5*w - 6],
[733, 733, -2*w^3 + 2*w^2 + 11*w - 4],
[733, 733, w^3 + w^2 - 10*w - 9],
[743, 743, -w^3 + 3*w^2 + 2*w - 10],
[743, 743, 2*w^2 - 3*w - 3],
[757, 757, 2*w^3 - 3*w^2 - 8*w - 5],
[757, 757, w^3 - 9*w - 5],
[757, 757, -2*w^3 + w^2 + 12*w + 12],
[757, 757, w^3 - 9*w - 4],
[769, 769, -4*w^3 + 4*w^2 + 21*w + 6],
[769, 769, 3*w^3 - 3*w^2 - 14*w - 5],
[827, 827, -2*w^3 + w^2 + 12*w + 1],
[827, 827, -4*w^3 + 4*w^2 + 22*w + 7],
[827, 827, 2*w^3 - 2*w^2 - 8*w - 5],
[827, 827, -2*w^3 + 3*w^2 + 8*w - 6],
[829, 829, -3*w^3 + w^2 + 17*w + 11],
[829, 829, -4*w^3 + 2*w^2 + 23*w + 16],
[839, 839, w^2 + 2*w - 4],
[839, 839, 2*w^3 + w^2 - 14*w - 13],
[839, 839, -4*w^3 + 7*w^2 + 16*w - 6],
[839, 839, -4*w^3 + 5*w^2 + 22*w + 1],
[863, 863, 2*w^3 - 13*w - 8],
[863, 863, 3*w^3 - 5*w^2 - 12*w + 5],
[887, 887, 2*w^2 + w - 7],
[887, 887, -5*w^3 + 7*w^2 + 26*w - 2],
[911, 911, -3*w^3 + 3*w^2 + 17*w + 7],
[911, 911, 2*w^2 - 9],
[911, 911, w^3 - w^2 - 3*w - 5],
[911, 911, 4*w^3 - 6*w^2 - 20*w + 1],
[937, 937, -4*w^3 + 5*w^2 + 22*w - 7],
[937, 937, 4*w^3 - 4*w^2 - 21*w - 2],
[937, 937, -4*w^2 + 8*w + 13],
[937, 937, 3*w^3 - 3*w^2 - 14*w - 1],
[947, 947, 3*w^3 - 4*w^2 - 12*w - 6],
[947, 947, 3*w^3 - 5*w^2 - 10*w + 4],
[947, 947, -3*w^3 + 4*w^2 + 19*w + 3],
[947, 947, -3*w^3 + 4*w^2 + 19*w - 4],
[971, 971, -3*w^3 + 5*w^2 + 13*w - 7],
[971, 971, w^3 + w^2 - 7*w - 5],
[983, 983, -5*w^3 + 8*w^2 + 23*w - 2],
[983, 983, -w^3 + w^2 + 3*w + 7],
[1009, 1009, w^3 - w^2 - 6*w - 7],
[1009, 1009, w - 6],
[1033, 1033, 4*w^3 - 4*w^2 - 21*w - 5],
[1033, 1033, -3*w^3 + 3*w^2 + 14*w + 4],
[1069, 1069, -w^2 - 3],
[1069, 1069, 2*w^3 - 3*w^2 - 10*w + 8],
[1103, 1103, -w^3 + 2*w^2 + 7*w - 5],
[1103, 1103, 3*w^3 - 3*w^2 - 19*w - 7],
[1103, 1103, 2*w^3 - 11*w - 12],
[1103, 1103, 3*w^3 - 4*w^2 - 17*w - 2],
[1117, 1117, -3*w^3 + 3*w^2 + 14*w + 2],
[1117, 1117, 4*w^3 - 4*w^2 - 21*w - 3],
[1129, 1129, -w^3 + 2*w^2 + 5*w - 9],
[1129, 1129, -2*w^3 + w^2 + 15*w + 2],
[1151, 1151, -3*w^2 + 5*w + 8],
[1151, 1151, 3*w^3 - 3*w^2 - 18*w - 11],
[1153, 1153, 3*w^3 - 3*w^2 - 14*w - 3],
[1153, 1153, -4*w^3 + 4*w^2 + 21*w + 4],
[1163, 1163, -w^3 + 3*w^2 + w - 9],
[1163, 1163, -w^3 - w^2 + 9*w + 5],
[1187, 1187, -3*w^3 + 4*w^2 + 18*w - 6],
[1187, 1187, -2*w^3 + 11*w + 5],
[1187, 1187, 2*w^3 - 3*w^2 - 13*w - 4],
[1187, 1187, 5*w^3 - 7*w^2 - 24*w + 6],
[1201, 1201, -5*w^3 + 5*w^2 + 30*w + 4],
[1201, 1201, -5*w - 1],
[1213, 1213, -6*w^3 + 7*w^2 + 31*w - 2],
[1213, 1213, 2*w^3 - 6*w^2 - 5*w + 14],
[1249, 1249, 3*w^3 - 3*w^2 - 13*w - 1],
[1249, 1249, 3*w^3 - 2*w^2 - 15*w - 4],
[1249, 1249, -5*w^3 + 6*w^2 + 25*w - 1],
[1249, 1249, -5*w^3 + 5*w^2 + 27*w + 3],
[1283, 1283, -w^3 - 3*w^2 + 12*w + 15],
[1283, 1283, -2*w^3 + 6*w^2 + 3*w - 12],
[1297, 1297, 5*w^3 - 7*w^2 - 22*w - 2],
[1297, 1297, 6*w^3 - 8*w^2 - 29*w + 2],
[1307, 1307, -5*w^3 + 6*w^2 + 27*w + 2],
[1307, 1307, -3*w^3 + 2*w^2 + 19*w + 2],
[1319, 1319, 3*w - 4],
[1319, 1319, -3*w^3 + 3*w^2 + 18*w + 7],
[1321, 1321, -5*w^3 + 5*w^2 + 27*w + 5],
[1321, 1321, 4*w^3 - 7*w^2 - 18*w + 7],
[1367, 1367, -3*w^3 + 4*w^2 + 12*w - 4],
[1367, 1367, 4*w^3 - 3*w^2 - 23*w - 4],
[1381, 1381, -2*w^3 + 2*w^2 + 12*w - 5],
[1381, 1381, 2*w^3 - 16*w - 13],
[1381, 1381, -2*w - 7],
[1381, 1381, 2*w^2 - 6*w - 3],
[1427, 1427, 3*w^3 - 5*w^2 - 15*w - 1],
[1427, 1427, -w^3 + 3*w^2 + 5*w - 11],
[1429, 1429, -w^3 + 4*w^2 + 2*w - 10],
[1429, 1429, 2*w^3 - 5*w^2 - 7*w + 8],
[1439, 1439, 4*w^3 - 5*w^2 - 22*w - 2],
[1439, 1439, w^2 + 2*w - 5],
[1451, 1451, 3*w^3 - 5*w^2 - 12*w + 6],
[1451, 1451, 3*w^3 - w^2 - 19*w - 9],
[1451, 1451, 3*w^3 - 5*w^2 - 11*w + 5],
[1451, 1451, 2*w^3 - 13*w - 7],
[1453, 1453, 5*w^3 - 6*w^2 - 25*w - 2],
[1453, 1453, 3*w^3 - 6*w^2 - 13*w + 10],
[1487, 1487, 4*w^3 - 5*w^2 - 20*w - 6],
[1487, 1487, 2*w^3 - w^2 - 10*w - 11],
[1487, 1487, -3*w^3 + 4*w^2 + 16*w + 4],
[1487, 1487, w^2 + w - 8],
[1489, 1489, -4*w^3 + 3*w^2 + 22*w + 7],
[1489, 1489, -4*w^3 + 5*w^2 + 18*w],
[1499, 1499, 3*w^3 - 4*w^2 - 16*w - 6],
[1499, 1499, w^2 + w - 10],
[1511, 1511, -2*w^3 + w^2 + 9*w + 8],
[1511, 1511, -5*w^3 + 6*w^2 + 26*w + 4],
[1523, 1523, 2*w^3 - 5*w^2 - 4*w + 9],
[1523, 1523, 6*w^3 - 6*w^2 - 33*w + 1],
[1559, 1559, 2*w^3 - 14*w - 7],
[1559, 1559, -2*w^3 + 4*w^2 + 6*w - 7],
[1571, 1571, -4*w^3 + 6*w^2 + 23*w - 7],
[1571, 1571, -4*w^3 + 8*w^2 + 15*w - 11],
[1571, 1571, 5*w^3 - 7*w^2 - 23*w - 5],
[1571, 1571, 4*w^3 - w^2 - 28*w - 13],
[1597, 1597, w^2 - 5*w - 4],
[1597, 1597, 5*w^3 - 6*w^2 - 25*w - 1],
[1597, 1597, -3*w^3 + 2*w^2 + 20*w + 6],
[1597, 1597, 3*w^3 - 2*w^2 - 15*w - 6],
[1607, 1607, 3*w^3 - 4*w^2 - 13*w + 6],
[1607, 1607, 4*w^3 - 3*w^2 - 24*w - 4],
[1607, 1607, -3*w^3 + 2*w^2 + 17*w + 1],
[1607, 1607, -2*w^3 + 3*w^2 + 6*w - 5],
[1609, 1609, -2*w^3 + 2*w^2 + 11*w - 5],
[1609, 1609, -w^3 + w^2 + 4*w - 6],
[1609, 1609, w^3 - 6*w^2 + 5*w + 13],
[1609, 1609, -4*w^3 + 3*w^2 + 21*w + 18],
[1619, 1619, -3*w^3 + 4*w^2 + 17*w + 3],
[1619, 1619, -w^3 + 2*w^2 + 7*w - 6],
[1621, 1621, -w^3 + 2*w^2 + 3*w - 10],
[1621, 1621, -w^3 + 7*w - 3],
[1657, 1657, -4*w^3 + 3*w^2 + 22*w + 9],
[1657, 1657, -4*w^3 + 5*w^2 + 18*w + 2],
[1667, 1667, 2*w^3 + w^2 - 14*w - 12],
[1667, 1667, 3*w^3 - 5*w^2 - 15*w - 3],
[1667, 1667, -4*w^3 + 6*w^2 + 17*w - 5],
[1667, 1667, 3*w^3 - w^2 - 18*w - 8],
[1669, 1669, 2*w^3 - 3*w^2 - 15*w - 2],
[1669, 1669, 2*w^3 - 5*w^2 - 8*w + 18],
[1681, 41, 4*w^3 - 3*w^2 - 22*w - 8],
[1681, 41, -4*w^3 + 5*w^2 + 18*w + 1],
[1693, 1693, -4*w^3 + 4*w^2 + 21*w - 5],
[1693, 1693, -3*w^3 + 3*w^2 + 14*w - 6],
[1741, 1741, 2*w^3 - 5*w^2 - 11*w + 4],
[1741, 1741, -5*w^2 + 7*w + 22],
[1787, 1787, 4*w^3 - 6*w^2 - 19*w - 2],
[1787, 1787, 4*w^3 - 4*w^2 - 22*w - 9],
[1787, 1787, 2*w^3 - 2*w^2 - 8*w - 7],
[1787, 1787, w^3 + w^2 - 6*w - 13],
[1801, 1801, 4*w^2 - 4*w - 19],
[1801, 1801, 3*w^3 - 2*w^2 - 19*w - 16],
[1801, 1801, -w^3 + 4*w^2 - 2*w - 8],
[1801, 1801, 4*w^3 - 8*w^2 - 16*w + 5],
[1811, 1811, 2*w^3 + 2*w^2 - 16*w - 17],
[1811, 1811, -4*w^3 + 8*w^2 + 14*w - 9],
[1823, 1823, -3*w^3 + 8*w^2 + 7*w - 14],
[1823, 1823, -4*w^3 + 4*w^2 + 24*w + 7],
[1823, 1823, 4*w - 3],
[1823, 1823, -w^3 - 4*w^2 + 13*w + 19],
[1847, 1847, 3*w^2 - 2*w - 19],
[1847, 1847, -7*w^3 + 8*w^2 + 39*w + 2],
[1861, 1861, -w^3 - 3*w^2 + 8*w + 17],
[1861, 1861, -6*w^3 + 10*w^2 + 27*w - 6],
[1861, 1861, -w^3 + 4*w^2 + 3*w - 8],
[1861, 1861, -3*w^3 + 6*w^2 + 13*w - 9],
[1871, 1871, 2*w^3 + 2*w^2 - 17*w - 17],
[1871, 1871, w^3 - 5*w^2 + w + 9],
[1871, 1871, -5*w^3 + 4*w^2 + 30*w + 12],
[1871, 1871, -w^3 + 5*w^2 - w - 17],
[1873, 1873, -4*w^3 + 7*w^2 + 22*w - 5],
[1873, 1873, 4*w^3 - 2*w^2 - 24*w - 21],
[1873, 1873, 4*w^2 - 5*w - 17],
[1873, 1873, -4*w^3 + 6*w^2 + 16*w + 7],
[1907, 1907, -4*w^3 + 6*w^2 + 15*w + 6],
[1907, 1907, 3*w^3 - 2*w^2 - 14*w - 10],
[1907, 1907, -6*w^3 + 9*w^2 + 28*w - 1],
[1907, 1907, 6*w^3 - 7*w^2 - 31*w - 6],
[1931, 1931, 6*w^3 - 7*w^2 - 34*w - 2],
[1931, 1931, 3*w^3 - 20*w - 14],
[1933, 1933, -3*w^3 + 6*w^2 + 13*w - 6],
[1933, 1933, w^3 - 10*w - 6],
[1933, 1933, -2*w^3 + w^2 + 15*w + 4],
[1933, 1933, -w^3 + 4*w^2 + 3*w - 11],
[1993, 1993, 6*w^3 - 7*w^2 - 31*w],
[1993, 1993, 5*w^3 - 8*w^2 - 25*w + 9]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 28*x^2 + 71;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/10*e^2 - 11/10, 1, e, -e, 1/10*e^2 - 29/10, 1/10*e^2 - 29/10, 1/10*e^3 - 9/10*e, -1/5*e^3 + 24/5*e, -1/10*e^3 + 9/10*e, 1/5*e^3 - 24/5*e, -1/5*e^2 + 4/5, 1/10*e^2 - 69/10, 1/10*e^2 - 69/10, -1/5*e^2 + 4/5, -3/10*e^3 + 77/10*e, 3/10*e^3 - 77/10*e, -3/10*e^3 + 67/10*e, 3/10*e^3 - 67/10*e, -7/5*e^2 + 93/5, -1/5*e^3 + 29/5*e, 1/5*e^3 - 29/5*e, 3/10*e^3 - 57/10*e, -3/10*e^3 + 57/10*e, -17, -1/2*e^3 + 21/2*e, 1/2*e^3 - 21/2*e, 1/5*e^2 + 21/5, 1/5*e^2 + 21/5, -1/2*e^3 + 31/2*e, 1/2*e^3 - 31/2*e, 11/5*e^2 - 154/5, -1/5*e^3 + 14/5*e, 1/5*e^3 - 14/5*e, -3/2*e^2 + 43/2, -3/2*e^2 + 43/2, 2/5*e^2 - 43/5, 2/5*e^2 - 43/5, 3/5*e^3 - 62/5*e, -3/5*e^3 + 62/5*e, -1/10*e^2 - 101/10, -1/10*e^2 - 101/10, -9/10*e^2 - 49/10, -9/10*e^2 - 49/10, -3/10*e^3 + 117/10*e, 3/10*e^3 - 117/10*e, 9/5*e^2 - 136/5, 9/5*e^2 - 136/5, 2/5*e^3 - 23/5*e, -2/5*e^3 + 23/5*e, -1/10*e^3 + 29/10*e, 1/10*e^3 - 29/10*e, 3/10*e^3 - 47/10*e, -3/10*e^3 + 47/10*e, 9/10*e^2 - 131/10, 9/10*e^2 - 131/10, -17/10*e^2 + 223/10, -17/10*e^2 + 223/10, 9/10*e^2 - 321/10, 9/10*e^2 - 321/10, -1/5*e^3 + 44/5*e, 4*e, 1/5*e^3 - 44/5*e, -4*e, 9/10*e^2 - 251/10, 9/10*e^2 - 251/10, 19/10*e^2 - 391/10, 19/10*e^2 - 391/10, 4/5*e^2 - 116/5, 4/5*e^2 - 116/5, -12/5*e^2 + 203/5, -3/5*e^2 + 2/5, -3/5*e^2 + 2/5, -12/5*e^2 + 203/5, 1/5*e^2 - 14/5, 1/5*e^2 - 14/5, -1/2*e^2 + 13/2, -1/2*e^2 + 13/2, -e, e, 1/2*e^2 + 29/2, 1/2*e^2 + 29/2, -9/10*e^3 + 171/10*e, 9/10*e^3 - 171/10*e, -27/10*e^2 + 433/10, 1/5*e^2 - 54/5, -27/10*e^2 + 433/10, 1/5*e^2 - 54/5, -1/10*e^3 + 79/10*e, 1/10*e^3 - 79/10*e, 1/5*e^3 - 14/5*e, 1/5*e^3 - 39/5*e, -1/5*e^3 + 39/5*e, -1/5*e^3 + 14/5*e, 3/5*e^3 - 87/5*e, -3/5*e^3 + 87/5*e, 29/10*e^2 - 351/10, 29/10*e^2 - 351/10, -1/5*e^3 + 29/5*e, 1/5*e^3 - 29/5*e, e^2 - 21, e^2 - 21, 13/10*e^3 - 247/10*e, 6/5*e^3 - 129/5*e, -6/5*e^3 + 129/5*e, -13/10*e^3 + 247/10*e, 1/5*e^3 - 14/5*e, -1/5*e^3 + 14/5*e, 8/5*e^2 - 127/5, 8/5*e^2 - 127/5, 11/5*e^2 - 209/5, 11/5*e^2 - 209/5, -e^2 + 50, 3/5*e^3 - 47/5*e, -3/5*e^3 + 47/5*e, 10*e, -10*e, 4/5*e^3 - 81/5*e, -4/5*e^3 + 81/5*e, -2/5*e^3 + 63/5*e, 2/5*e^3 - 63/5*e, -3*e^2 + 42, -3*e^2 + 42, -11/10*e^3 + 279/10*e, 11/10*e^3 - 279/10*e, -9/5*e^2 + 46/5, 8/5*e^2 - 92/5, -9/5*e^2 + 46/5, 8/5*e^2 - 92/5, 9/10*e^3 - 231/10*e, -9/10*e^3 + 231/10*e, -6/5*e^2 + 99/5, -13/5*e^2 + 197/5, -6/5*e^2 + 99/5, -13/5*e^2 + 197/5, -23/10*e^2 + 447/10, -23/10*e^2 + 447/10, -e^3 + 25*e, -e^3 + 28*e, e^3 - 28*e, e^3 - 25*e, 21/5*e^2 - 294/5, 21/5*e^2 - 294/5, e^3 - 24*e, 2/5*e^3 - 53/5*e, -2/5*e^3 + 53/5*e, -e^3 + 24*e, 9/10*e^3 - 321/10*e, -9/10*e^3 + 321/10*e, 1/5*e^3 + 16/5*e, -1/5*e^3 - 16/5*e, -3/2*e^3 + 59/2*e, -3/5*e^3 + 87/5*e, 3/2*e^3 - 59/2*e, 3/5*e^3 - 87/5*e, -3/5*e^2 - 168/5, 2/5*e^2 + 107/5, -3/5*e^2 - 168/5, 2/5*e^2 + 107/5, 9/5*e^3 - 171/5*e, -9/5*e^3 + 171/5*e, -6/5*e^3 + 154/5*e, 6/5*e^3 - 154/5*e, 1/10*e^3 + 71/10*e, -1/10*e^3 - 71/10*e, -7/10*e^3 + 243/10*e, 7/10*e^3 - 243/10*e, 13/10*e^2 + 193/10, 13/10*e^2 + 193/10, 21/10*e^2 - 159/10, 21/10*e^2 - 159/10, -3/10*e^2 + 467/10, -3/10*e^2 + 467/10, 7/5*e^3 - 208/5*e, 3/5*e^3 - 82/5*e, -3/5*e^3 + 82/5*e, -7/5*e^3 + 208/5*e, 2*e^2 - 71, 2*e^2 - 71, -27/10*e^2 + 453/10, -27/10*e^2 + 453/10, -7/5*e^3 + 218/5*e, 7/5*e^3 - 218/5*e, -19, -19, 7*e, -7*e, -3/5*e^3 + 47/5*e, -6/5*e^3 + 114/5*e, 3/5*e^3 - 47/5*e, 6/5*e^3 - 114/5*e, -9/10*e^2 - 79/10, -9/10*e^2 - 79/10, 6, 6, -17/5*e^2 + 288/5, -13/10*e^2 + 107/10, -13/10*e^2 + 107/10, -17/5*e^2 + 288/5, -e^3 + 22*e, e^3 - 22*e, 27/10*e^2 - 183/10, 27/10*e^2 - 183/10, 23/10*e^3 - 457/10*e, -23/10*e^3 + 457/10*e, e^3 - 29*e, -e^3 + 29*e, -e^2 - 23, -e^2 - 23, 3/10*e^3 - 77/10*e, -3/10*e^3 + 77/10*e, 1/5*e^2 - 229/5, e^2 - 12, 1/5*e^2 - 229/5, e^2 - 12, -3/10*e^3 + 107/10*e, 3/10*e^3 - 107/10*e, -7/2*e^2 + 153/2, -7/2*e^2 + 153/2, 6/5*e^3 - 184/5*e, -6/5*e^3 + 184/5*e, -1/5*e^3 - 31/5*e, -3/5*e^3 + 12/5*e, 3/5*e^3 - 12/5*e, 1/5*e^3 + 31/5*e, -12/5*e^2 + 173/5, -12/5*e^2 + 173/5, -11/10*e^3 + 419/10*e, 11/10*e^3 - 419/10*e, -e^3 + 28*e, e^3 - 28*e, 12/5*e^2 + 7/5, 12/5*e^2 + 7/5, e^3 - 24*e, -e^3 + 24*e, 9/10*e^3 - 211/10*e, -9/10*e^3 + 211/10*e, -8/5*e^3 + 222/5*e, 8/5*e^3 - 222/5*e, -13/10*e^3 + 457/10*e, 13/10*e^3 - 457/10*e, 7/5*e^3 - 203/5*e, -7/5*e^3 + 203/5*e, 1/2*e^3 - 27/2*e, -1/2*e^3 + 27/2*e, -21/5*e^2 + 259/5, 1/2*e^2 - 95/2, -21/5*e^2 + 259/5, 1/2*e^2 - 95/2, -6/5*e^3 + 154/5*e, -1/5*e^3 + 69/5*e, 6/5*e^3 - 154/5*e, 1/5*e^3 - 69/5*e, 11/5*e^2 - 104/5, 11/5*e^2 - 104/5, 3/10*e^2 + 433/10, 3/10*e^2 + 433/10, -4/5*e^3 + 31/5*e, 4/5*e^3 - 31/5*e, -3/2*e^2 - 49/2, -3/2*e^2 - 49/2, -16/5*e^2 + 109/5, -16/5*e^2 + 109/5, -8/5*e^3 + 167/5*e, 8/5*e^3 - 167/5*e, 11/5*e^3 - 234/5*e, -11/5*e^3 + 234/5*e, -7/10*e^2 + 323/10, -7/10*e^2 + 323/10, -5*e^2 + 58, -5*e^2 + 58, 11/10*e^2 + 211/10, 11/10*e^2 + 211/10, 13/10*e^2 + 373/10, 13/10*e^2 + 373/10, 1/2*e^3 - 21/2*e, -9/10*e^3 + 91/10*e, 9/10*e^3 - 91/10*e, -1/2*e^3 + 21/2*e, -2*e^2 - 9, 19/10*e^2 - 271/10, 19/10*e^2 - 271/10, -2*e^2 - 9, -1/2*e^3 + 17/2*e, 1/2*e^3 - 17/2*e, 7*e, 1/2*e^3 + 5/2*e, -1/2*e^3 - 5/2*e, -7*e, 2/5*e^3 - 98/5*e, -2/5*e^3 + 98/5*e, 12/5*e^2 - 253/5, 12/5*e^2 - 253/5, -33/5*e^2 + 477/5, -33/5*e^2 + 477/5, 27/10*e^3 - 563/10*e, 11/5*e^3 - 209/5*e, -27/10*e^3 + 563/10*e, -11/5*e^3 + 209/5*e, 27/10*e^2 - 143/10, -31, 27/10*e^2 - 143/10, -31, -4/5*e^3 + 61/5*e, 2/5*e^3 + 27/5*e, 4/5*e^3 - 61/5*e, -2/5*e^3 - 27/5*e, -9/5*e^3 + 206/5*e, 9/5*e^3 - 206/5*e, -27/10*e^2 + 463/10, -16/5*e^2 + 269/5, -16/5*e^2 + 269/5, -27/10*e^2 + 463/10, -9/5*e^2 + 106/5, -9/5*e^2 + 106/5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, w^3 - w^2 - 5*w - 1}>] := -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;