/*
  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 | {23, 23, w^2 - 2*w - 2}>;

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^6 - 16*x^4 + 25*x^2 - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/10*e^5 + 6/5*e^3 + 23/10*e, -2/5*e^5 + 29/5*e^3 - 9/5*e, -1/5*e^4 + 17/5*e^2 - 12/5, 9/10*e^5 - 69/5*e^3 + 133/10*e, -9/10*e^5 + 69/5*e^3 - 133/10*e, -1, 1/5*e^5 - 17/5*e^3 + 47/5*e, 2/5*e^5 - 29/5*e^3 + 14/5*e, 2/5*e^4 - 29/5*e^2 + 44/5, 1/5*e^4 - 12/5*e^2 - 8/5, -3/10*e^5 + 23/5*e^3 - 61/10*e, -7/10*e^5 + 52/5*e^3 - 49/10*e, 4/5*e^4 - 63/5*e^2 + 28/5, e^4 - 15*e^2 + 16, -e^5 + 15*e^3 - 10*e, -4/5*e^4 + 58/5*e^2 - 8/5, -2/5*e^5 + 29/5*e^3 - 9/5*e, -1/5*e^4 + 17/5*e^2 - 2/5, 2*e^4 - 29*e^2 + 20, 4/5*e^5 - 63/5*e^3 + 68/5*e, -9/5*e^4 + 128/5*e^2 - 88/5, -8/5*e^5 + 121/5*e^3 - 106/5*e, -11/10*e^5 + 86/5*e^3 - 257/10*e, 4/5*e^5 - 58/5*e^3 + 8/5*e, 7/5*e^4 - 114/5*e^2 + 144/5, -3/10*e^5 + 23/5*e^3 - 61/10*e, 3/2*e^5 - 22*e^3 + 21/2*e, 2/5*e^4 - 29/5*e^2 + 44/5, 1/5*e^5 - 12/5*e^3 - 28/5*e, 14/5*e^4 - 213/5*e^2 + 138/5, -2*e^2 + 12, 19/5*e^5 - 293/5*e^3 + 313/5*e, 5/2*e^5 - 39*e^3 + 83/2*e, 1/10*e^5 - 11/5*e^3 + 57/10*e, 2/5*e^4 - 29/5*e^2 + 4/5, 3/5*e^4 - 51/5*e^2 + 36/5, -7/5*e^5 + 104/5*e^3 - 79/5*e, 4*e^4 - 59*e^2 + 44, 3/10*e^5 - 23/5*e^3 + 111/10*e, 1/2*e^5 - 6*e^3 - 31/2*e, -6/5*e^4 + 82/5*e^2 - 62/5, -8/5*e^4 + 116/5*e^2 - 46/5, 2*e^2 - 4, 21/5*e^5 - 317/5*e^3 + 242/5*e, 23/10*e^5 - 178/5*e^3 + 331/10*e, 47/10*e^5 - 357/5*e^3 + 609/10*e, 1/5*e^5 - 12/5*e^3 - 23/5*e, 3*e^4 - 45*e^2 + 44, -2*e^4 + 31*e^2 - 8, 17/5*e^5 - 264/5*e^3 + 289/5*e, e^3 - 19*e, -7/5*e^4 + 99/5*e^2 - 44/5, 1/2*e^5 - 8*e^3 + 13/2*e, -33/10*e^5 + 248/5*e^3 - 411/10*e, 13/10*e^5 - 108/5*e^3 + 421/10*e, 29/10*e^5 - 224/5*e^3 + 553/10*e, 2*e^4 - 30*e^2 + 16, 11/5*e^4 - 172/5*e^2 + 212/5, 12/5*e^4 - 194/5*e^2 + 244/5, -1/5*e^5 + 17/5*e^3 - 72/5*e, -16/5*e^5 + 247/5*e^3 - 247/5*e, 3/5*e^4 - 66/5*e^2 + 156/5, 21/10*e^5 - 166/5*e^3 + 457/10*e, -3/2*e^5 + 23*e^3 - 45/2*e, -e^4 + 14*e^2 - 28, -9/5*e^4 + 128/5*e^2 - 8/5, -5/2*e^5 + 39*e^3 - 83/2*e, 37/10*e^5 - 287/5*e^3 + 599/10*e, -9/10*e^5 + 74/5*e^3 - 333/10*e, -13/10*e^5 + 113/5*e^3 - 551/10*e, -23/10*e^5 + 168/5*e^3 - 51/10*e, -9/2*e^5 + 69*e^3 - 125/2*e, -27/10*e^5 + 207/5*e^3 - 489/10*e, -23/10*e^5 + 193/5*e^3 - 731/10*e, 1/5*e^4 - 27/5*e^2 + 22/5, e^2 - 18, 2/5*e^4 - 39/5*e^2 + 164/5, -24/5*e^5 + 373/5*e^3 - 453/5*e, -16/5*e^4 + 242/5*e^2 - 92/5, 24/5*e^4 - 373/5*e^2 + 348/5, -4/5*e^5 + 63/5*e^3 - 133/5*e, -4/5*e^4 + 58/5*e^2 - 28/5, 26/5*e^4 - 387/5*e^2 + 292/5, 49/10*e^5 - 384/5*e^3 + 893/10*e, -7/5*e^4 + 114/5*e^2 - 124/5, -7/10*e^5 + 52/5*e^3 - 49/10*e, 3*e^5 - 47*e^3 + 58*e, 2*e^4 - 31*e^2 + 12, 3*e^4 - 47*e^2 + 40, -27/5*e^4 + 409/5*e^2 - 304/5, 12/5*e^5 - 174/5*e^3 + 49/5*e, -31/5*e^5 + 472/5*e^3 - 387/5*e, -4*e^5 + 62*e^3 - 59*e, e^4 - 16*e^2 + 20, 3/2*e^5 - 25*e^3 + 89/2*e, -15/2*e^5 + 115*e^3 - 225/2*e, -9/5*e^4 + 153/5*e^2 - 148/5, -17/5*e^5 + 259/5*e^3 - 234/5*e, 4/5*e^4 - 73/5*e^2 + 88/5, 7/5*e^4 - 109/5*e^2 + 84/5, -e^4 + 15*e^2 - 8, -4/5*e^4 + 68/5*e^2 - 28/5, 3/5*e^5 - 31/5*e^3 - 129/5*e, -23/5*e^5 + 341/5*e^3 - 216/5*e, -14/5*e^4 + 208/5*e^2 - 68/5, -21/5*e^5 + 322/5*e^3 - 307/5*e, 14/5*e^4 - 193/5*e^2 + 28/5, -19/5*e^4 + 303/5*e^2 - 328/5, -8/5*e^4 + 116/5*e^2 + 4/5, -16/5*e^4 + 232/5*e^2 - 72/5, 26/5*e^4 - 402/5*e^2 + 362/5, 24/5*e^5 - 368/5*e^3 + 363/5*e, -4/5*e^4 + 78/5*e^2 - 228/5, -11/5*e^5 + 187/5*e^3 - 382/5*e, -13/5*e^4 + 201/5*e^2 - 276/5, 22/5*e^5 - 339/5*e^3 + 289/5*e, -16/5*e^4 + 257/5*e^2 - 252/5, 2/5*e^5 - 29/5*e^3 + 59/5*e, 20, 29/5*e^4 - 443/5*e^2 + 438/5, 26/5*e^4 - 392/5*e^2 + 222/5, 3/5*e^4 - 51/5*e^2 + 56/5, 7*e^5 - 108*e^3 + 110*e, -4/5*e^4 + 63/5*e^2 - 58/5, 26/5*e^4 - 392/5*e^2 + 332/5, 4/5*e^4 - 43/5*e^2 - 2/5, -4*e^4 + 62*e^2 - 52, 12/5*e^5 - 179/5*e^3 + 99/5*e, -4*e^4 + 62*e^2 - 44, 36/5*e^4 - 557/5*e^2 + 472/5, 21/5*e^4 - 297/5*e^2 + 202/5, -17/5*e^4 + 259/5*e^2 - 224/5, 36/5*e^4 - 532/5*e^2 + 362/5, -37/10*e^5 + 277/5*e^3 - 489/10*e, -9/10*e^5 + 79/5*e^3 - 373/10*e, 22/5*e^4 - 324/5*e^2 + 164/5, -e^5 + 17*e^3 - 32*e, 12/5*e^4 - 209/5*e^2 + 344/5, -19/5*e^5 + 308/5*e^3 - 513/5*e, 11/2*e^5 - 85*e^3 + 193/2*e, -11/10*e^5 + 81/5*e^3 - 77/10*e, -5*e^4 + 79*e^2 - 80, -1/5*e^4 + 42/5*e^2 - 212/5, -7/5*e^5 + 94/5*e^3 + 81/5*e, 33/5*e^5 - 501/5*e^3 + 396/5*e, 12/5*e^4 - 174/5*e^2 + 44/5, 9/5*e^5 - 138/5*e^3 + 123/5*e, -36/5*e^4 + 532/5*e^2 - 432/5, -31/5*e^5 + 482/5*e^3 - 522/5*e, -23/5*e^5 + 366/5*e^3 - 581/5*e, -6*e^4 + 86*e^2 - 48, -3*e^4 + 46*e^2 - 24, e^5 - 16*e^3 + 26*e, -6/5*e^4 + 102/5*e^2 - 2/5, -7/10*e^5 + 57/5*e^3 - 49/10*e, -2*e^4 + 30*e^2 + 6, 41/10*e^5 - 306/5*e^3 + 517/10*e, 26/5*e^4 - 412/5*e^2 + 432/5, e^5 - 14*e^3 - 6*e, -31/5*e^5 + 477/5*e^3 - 552/5*e, 2/5*e^4 - 4/5*e^2 - 196/5, -3*e^5 + 47*e^3 - 70*e, -12/5*e^4 + 164/5*e^2 + 16/5, -24/5*e^5 + 368/5*e^3 - 373/5*e, -16, -12/5*e^4 + 184/5*e^2 - 84/5, -4/5*e^4 + 58/5*e^2 + 12/5, 21/10*e^5 - 156/5*e^3 + 227/10*e, -7/2*e^5 + 55*e^3 - 133/2*e, 33/5*e^4 - 501/5*e^2 + 386/5, -23/5*e^4 + 351/5*e^2 - 286/5, 8/5*e^4 - 116/5*e^2 - 64/5, -41/5*e^5 + 632/5*e^3 - 637/5*e, -14/5*e^4 + 193/5*e^2 - 28/5, 3*e^5 - 47*e^3 + 53*e, 63/10*e^5 - 483/5*e^3 + 1041/10*e, -1/10*e^5 + 11/5*e^3 - 87/10*e, -42/5*e^4 + 619/5*e^2 - 554/5, -2/5*e^4 + 49/5*e^2 - 74/5, -14/5*e^4 + 183/5*e^2 + 32/5, -2*e^5 + 30*e^3 - 8*e, -3/10*e^5 + 13/5*e^3 + 279/10*e, -49/10*e^5 + 374/5*e^3 - 653/10*e, 1/5*e^5 + 3/5*e^3 - 218/5*e, -14/5*e^4 + 223/5*e^2 - 28/5, -38/5*e^4 + 606/5*e^2 - 576/5, -11/5*e^4 + 182/5*e^2 - 112/5, -26/5*e^5 + 392/5*e^3 - 247/5*e, -22/5*e^5 + 329/5*e^3 - 149/5*e, 4/5*e^4 - 93/5*e^2 + 318/5, 31/5*e^4 - 447/5*e^2 + 382/5, -19/2*e^5 + 145*e^3 - 253/2*e, -101/10*e^5 + 771/5*e^3 - 1367/10*e, 5/2*e^5 - 41*e^3 + 135/2*e, 33/10*e^5 - 263/5*e^3 + 891/10*e, 37/10*e^5 - 267/5*e^3 + 139/10*e, -57/10*e^5 + 427/5*e^3 - 559/10*e, -13/5*e^4 + 191/5*e^2 - 76/5, -8/5*e^5 + 141/5*e^3 - 451/5*e, 89/10*e^5 - 669/5*e^3 + 1103/10*e, 19/10*e^5 - 154/5*e^3 + 503/10*e, 56/5*e^5 - 857/5*e^3 + 727/5*e, 53/5*e^4 - 796/5*e^2 + 656/5, 29/5*e^4 - 423/5*e^2 + 428/5, -1/5*e^5 + 7/5*e^3 + 138/5*e, 63/10*e^5 - 483/5*e^3 + 981/10*e, 17/2*e^5 - 131*e^3 + 261/2*e, -8/5*e^5 + 116/5*e^3 - 96/5*e, 43/5*e^4 - 651/5*e^2 + 516/5, 4/5*e^4 - 48/5*e^2 - 152/5, 29/5*e^4 - 458/5*e^2 + 338/5, -4/5*e^4 + 88/5*e^2 - 348/5, -19/5*e^4 + 298/5*e^2 - 318/5, 4/5*e^5 - 48/5*e^3 - 162/5*e, 3/5*e^4 - 46/5*e^2 + 196/5, -3/10*e^5 + 43/5*e^3 - 541/10*e, 15/2*e^5 - 113*e^3 + 137/2*e, -33/5*e^5 + 516/5*e^3 - 621/5*e, 18/5*e^4 - 261/5*e^2 + 16/5, 3*e^5 - 49*e^3 + 86*e, 12/5*e^4 - 214/5*e^2 + 284/5, 39/5*e^5 - 603/5*e^3 + 608/5*e, e^4 - 20*e^2 + 28, -13/2*e^5 + 98*e^3 - 129/2*e, -43/10*e^5 + 338/5*e^3 - 1031/10*e, 61/5*e^5 - 942/5*e^3 + 957/5*e, 3/5*e^4 - 21/5*e^2 - 4/5, -41/5*e^5 + 637/5*e^3 - 692/5*e, -33/5*e^4 + 486/5*e^2 - 476/5, -5/2*e^5 + 36*e^3 + 27/2*e, 47/10*e^5 - 347/5*e^3 + 269/10*e, 9/5*e^5 - 143/5*e^3 + 253/5*e, -16/5*e^4 + 252/5*e^2 - 32/5, 13/5*e^4 - 161/5*e^2 + 16/5, -17/5*e^5 + 264/5*e^3 - 254/5*e, -12*e^5 + 183*e^3 - 163*e, 48/5*e^4 - 736/5*e^2 + 596/5, -48/5*e^4 + 731/5*e^2 - 456/5, 27/5*e^5 - 419/5*e^3 + 449/5*e, -33/5*e^4 + 521/5*e^2 - 536/5, 3*e^5 - 50*e^3 + 83*e, 63/5*e^5 - 956/5*e^3 + 856/5*e, 3/5*e^4 - 71/5*e^2 + 116/5, e^4 - 19*e^2 + 86, 43/10*e^5 - 313/5*e^3 + 91/10*e, e^4 - 11*e^2 + 22, -15/2*e^5 + 115*e^3 - 217/2*e, 11*e^5 - 165*e^3 + 122*e, -22/5*e^4 + 344/5*e^2 - 284/5, 22/5*e^4 - 344/5*e^2 + 404/5, 7/5*e^5 - 119/5*e^3 + 349/5*e, -7*e^4 + 105*e^2 - 72, e^4 - 23*e^2 + 80, 33/5*e^4 - 476/5*e^2 + 376/5, 17/5*e^4 - 269/5*e^2 + 124/5, 47/5*e^5 - 714/5*e^3 + 659/5*e, 18/5*e^4 - 286/5*e^2 + 336/5, -42/5*e^4 + 644/5*e^2 - 654/5, -8*e^4 + 115*e^2 - 62, 83/10*e^5 - 628/5*e^3 + 1121/10*e, 67/10*e^5 - 502/5*e^3 + 719/10*e, 32/5*e^4 - 484/5*e^2 + 504/5, -32/5*e^5 + 479/5*e^3 - 409/5*e, 26/5*e^5 - 422/5*e^3 + 662/5*e, 9/5*e^4 - 148/5*e^2 + 188/5, -7*e^4 + 102*e^2 - 108, -23/5*e^4 + 381/5*e^2 - 436/5, 57/10*e^5 - 432/5*e^3 + 809/10*e, -19/10*e^5 + 164/5*e^3 - 783/10*e, 4*e^4 - 54*e^2 + 24, 1/5*e^4 - 22/5*e^2 + 52/5, 3/10*e^5 - 13/5*e^3 - 449/10*e, -21/10*e^5 + 156/5*e^3 - 117/10*e, 31/5*e^5 - 487/5*e^3 + 597/5*e, -52/5*e^5 + 794/5*e^3 - 769/5*e, -11/5*e^4 + 192/5*e^2 - 432/5, 2*e^4 - 33*e^2 - 16, -51/10*e^5 + 381/5*e^3 - 447/10*e, 4*e^4 - 61*e^2 + 52, -52/5*e^4 + 754/5*e^2 - 484/5, -63/10*e^5 + 483/5*e^3 - 961/10*e, 2/5*e^4 - 34/5*e^2 + 244/5, 24/5*e^5 - 388/5*e^3 + 683/5*e, -1/5*e^5 + 7/5*e^3 + 158/5*e, -51/5*e^5 + 787/5*e^3 - 777/5*e, 8/5*e^4 - 106/5*e^2 - 24/5, -34/5*e^4 + 518/5*e^2 - 468/5, 6/5*e^4 - 82/5*e^2 + 252/5, -5*e^5 + 78*e^3 - 91*e, -109/10*e^5 + 824/5*e^3 - 1313/10*e, 29/10*e^5 - 214/5*e^3 + 193/10*e, 31/10*e^5 - 261/5*e^3 + 1057/10*e, -73/10*e^5 + 543/5*e^3 - 651/10*e, 22/5*e^4 - 334/5*e^2 + 284/5, 48/5*e^4 - 721/5*e^2 + 456/5, 26/5*e^5 - 407/5*e^3 + 402/5*e, -6/5*e^5 + 102/5*e^3 - 242/5*e, -9/2*e^5 + 71*e^3 - 207/2*e, -13*e^4 + 197*e^2 - 150, -81/10*e^5 + 601/5*e^3 - 757/10*e, -33/5*e^4 + 501/5*e^2 - 486/5, -43/5*e^4 + 651/5*e^2 - 576/5, 46/5*e^4 - 692/5*e^2 + 452/5, -64/5*e^5 + 973/5*e^3 - 878/5*e, -5*e^5 + 75*e^3 - 72*e, 16/5*e^5 - 267/5*e^3 + 562/5*e, -2/5*e^4 + 29/5*e^2 + 196/5, 5/2*e^5 - 33*e^3 - 53/2*e, -7*e^4 + 102*e^2 - 62, 2*e^4 - 27*e^2 + 18, 11/10*e^5 - 81/5*e^3 + 237/10*e, 119/10*e^5 - 909/5*e^3 + 1793/10*e, -81/10*e^5 + 621/5*e^3 - 1227/10*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, w^2 - 2*w - 2}>] := 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;