/*
  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}>;

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 - 7*x^2 + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^3 - 7/2*e, -e^3 + 5*e, -4, -1/2*e^3 + 3/2*e, 1/2*e^3 - 7/2*e, -2*e^2 + 4, -e^3 + 5*e, e^3 - 5*e, 1, 4, 1/2*e^3 + 1/2*e, -1/2*e^3 + 7/2*e, 2*e^2 - 4, -2*e^2, 2*e^3 - 8*e, 4*e^2 - 16, e^3 - 5*e, 2*e^2 - 10, -4, -2*e^3 + 8*e, -4, -e^3 + e, -1/2*e^3 + 15/2*e, 3*e^3 - 15*e, 2*e^2 - 12, -9/2*e^3 + 39/2*e, 7/2*e^3 - 37/2*e, -20, 4*e^3 - 22*e, -2*e^2 - 6, 6*e^2 - 20, 2*e^3 - 2*e, -13/2*e^3 + 51/2*e, -1/2*e^3 + 7/2*e, 10*e^2 - 32, -8*e^2 + 28, e^3 - e, -2*e^2 - 4, 7/2*e^3 - 33/2*e, 1/2*e^3 - 15/2*e, -4*e^2 + 10, -8*e^2 + 26, -4*e^2, 5*e^3 - 17*e, 9/2*e^3 - 39/2*e, 5/2*e^3 - 19/2*e, -3*e^3 + 19*e, 4*e^2 - 28, 8*e^2 - 40, -4*e^3 + 14*e, -2*e^3 + 14*e, -12*e^2 + 44, -5/2*e^3 + 35/2*e, -5/2*e^3 + 3/2*e, 7/2*e^3 - 17/2*e, -11/2*e^3 + 57/2*e, 8*e^2 - 16, -12*e^2 + 48, 8*e^2 - 36, e^3 - 21*e, -6*e^3 + 20*e, -12*e^2 + 40, -1/2*e^3 - 9/2*e, 1/2*e^3 + 17/2*e, -2*e^2 - 16, -8*e^2 + 44, -5/2*e^3 - 1/2*e, 5/2*e^3 - 51/2*e, -1/2*e^3 + 3/2*e, 11/2*e^3 - 45/2*e, -9/2*e^3 + 39/2*e, 7/2*e^3 - 9/2*e, -5/2*e^3 + 39/2*e, 1/2*e^3 - 15/2*e, -8*e^2 + 22, -12*e^2 + 46, 4*e^2, -2*e^3 + 12*e, -2*e^2 - 4, -8*e^2 + 48, -4*e^3 + 10*e, 12*e^2 - 40, -8*e^2 + 40, 13/2*e^3 - 67/2*e, -2*e^2 + 12, -15/2*e^3 + 65/2*e, e^3 - 13*e, 8, 8*e^2 - 20, -4*e^2 - 12, -8*e^3 + 32*e, 7*e^3 - 35*e, -5*e^3 + 29*e, -12*e^2 + 36, -27/2*e^3 + 113/2*e, -5/2*e^3 + 35/2*e, 4*e^2 - 32, 4*e^3 - 26*e, 14*e^2 - 60, 6*e^2 - 24, -4*e^2, -8*e^2 + 16, -2*e^3 + 6*e, 8*e^3 - 26*e, -6*e^2 + 40, -e^3 + 25*e, 10*e^2 - 20, -4*e^2 + 16, 10*e^2 - 60, 12*e^2 - 20, 12*e^2 - 46, -12*e, 20*e^2 - 76, e^3 - 5*e, 6*e^2, -10*e^3 + 44*e, 4*e^2 - 28, -4*e^3 + 4*e, -8*e^2 + 20, 12*e^2 - 66, 4*e^2 - 26, 14*e^2 - 52, 4*e^3 - 34*e, 30, -2*e^2 + 32, 8*e^2 - 42, 16*e^2 - 68, -6*e^3 + 40*e, 2*e^2 - 20, -4*e^2 + 20, -8*e^2 + 42, -22*e^2 + 72, 14*e^2 - 38, -13/2*e^3 + 91/2*e, -3/2*e^3 + 25/2*e, 12*e^2 - 36, 8*e^3 - 40*e, 14*e^2 - 28, 5*e^3 - 13*e, 1/2*e^3 + 9/2*e, 15/2*e^3 - 61/2*e, -4*e^2 - 4, -12*e^2 + 20, -2*e^3, 3*e^3 - 7*e, 14*e^2 - 24, 2*e^3 + 4*e, 8*e^2 - 32, -5*e^3 + 33*e, -10*e^3 + 56*e, -6*e^2 + 16, 4*e^2 - 44, 12*e^3 - 50*e, 4*e^2 - 26, -3/2*e^3 + 9/2*e, -4*e^2 - 2, -11/2*e^3 + 105/2*e, 20, e^3 - 17*e, 9*e^3 - 37*e, -8*e^2 + 20, -14*e^3 + 58*e, -20, 12*e^3 - 64*e, -4*e^2 + 12, -2*e^2 + 24, -6*e^2 + 4, -5/2*e^3 + 35/2*e, 5/2*e^3 - 55/2*e, -10*e^2 + 18, 8*e^2 + 2, 12*e^2 - 52, -e^3 - 19*e, 22*e^2 - 72, 11*e^3 - 47*e, 35/2*e^3 - 149/2*e, 19/2*e^3 - 69/2*e, -2*e^2 + 6, -10*e^2 + 6, -16*e^2 + 48, -4*e^3 + 36*e, -5/2*e^3 + 63/2*e, -17/2*e^3 + 51/2*e, 4*e^3 - 40*e, 4*e^2 + 4, -6*e^2 + 64, 8*e^2 - 4, -2*e^3 + 38*e, -4*e^3 + 44*e, -4*e^2 - 2, -12*e^2 + 14, -17/2*e^3 + 123/2*e, 23/2*e^3 - 105/2*e, -27/2*e^3 + 105/2*e, 9/2*e^3 - 87/2*e, -21/2*e^3 + 107/2*e, -3/2*e^3 + 65/2*e, 12*e^2, 3*e^3 + e, 37/2*e^3 - 147/2*e, -3/2*e^3 + 69/2*e, -7*e^3 + 15*e, -14*e^2 + 64, -32, 8*e^3 - 16*e, -5/2*e^3 - 21/2*e, 13/2*e^3 - 35/2*e, -7*e^3 + 55*e, -56, 18*e^2 - 48, 4*e^2 - 62, -2*e^2 + 4, -2*e^2 + 42, -12*e^3 + 64*e, 28*e^2 - 96, 1/2*e^3 - 47/2*e, -21/2*e^3 + 99/2*e, -11*e^3 + 39*e, 4*e^2 - 20, -6*e^3 + 38*e, -24*e^2 + 76, 17*e^3 - 77*e, -12*e^2 + 64, -17/2*e^3 + 87/2*e, 5/2*e^3 + 1/2*e, 7*e^3 - 39*e, 8*e^2 - 36, -7*e^3 + 43*e, 4*e^2 - 44, 13/2*e^3 - 83/2*e, 21/2*e^3 - 83/2*e, 9*e^3 - 61*e, -34*e^2 + 120, -12*e^2 + 84, e^3 - 21*e, 5*e^3 - 45*e, 4*e^2 + 20, 56, 10*e^3 - 58*e, 12*e^2 - 72, 12*e^3 - 66*e, -2*e^3 + 24*e, -4*e^2 - 8, -10*e^2 - 18, 3/2*e^3 + 19/2*e, -16*e^2 + 78, 1/2*e^3 - 7/2*e, -3*e^3 + 47*e, 4*e^2 - 32, -12*e^2 + 84, 7*e^3 - 31*e, 4*e^2 + 16, -18*e^2 + 68, -2*e^2, 26*e^2 - 88, -6*e^3 + 48*e, -44, 12*e^2 - 22, -6*e^2 + 42, 25/2*e^3 - 103/2*e, -21/2*e^3 + 75/2*e, 12*e^2 + 8, -2*e^3 + 12*e, 2*e^3 - 6*e, -6*e^2 + 44, -10*e^2 + 28, 12*e^2 - 28, -29/2*e^3 + 115/2*e, -5/2*e^3 + 19/2*e, -30*e^2 + 112, -14*e^2 + 40, 3/2*e^3 - 37/2*e, 3/2*e^3 - 21/2*e, -2*e^3 + 18*e, 17*e^3 - 85*e, 20*e^2 - 44, 2*e^2 - 28, -29/2*e^3 + 127/2*e, 24*e^2 - 88, 6*e^2 - 76, 27/2*e^3 - 153/2*e, -6*e^2 - 24, 8*e^3 - 76*e, e^3 - 17*e, -28*e, -4*e^2 + 12, 24*e^2 - 60, -4*e^2 + 4, 13*e^3 - 69*e, 1/2*e^3 - 47/2*e, 13/2*e^3 - 123/2*e, -5/2*e^3 + 39/2*e, 25/2*e^3 - 79/2*e, 2*e^2 + 36, 36*e^2 - 132, 8*e^3 - 22*e, 17*e^3 - 69*e, 1/2*e^3 + 37/2*e, -18*e^2 + 82, 21/2*e^3 - 103/2*e, -20*e^2 + 66, -24*e^2 + 68, 52, -7*e^3 + 7*e, -10*e^3 + 78*e, 4*e^3 - 6*e, -12*e^2 + 16, -3/2*e^3 + 77/2*e, -12*e^2 + 50, -24*e^2 + 114, -1/2*e^3 + 39/2*e, -9/2*e^3 + 15/2*e, -35/2*e^3 + 141/2*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}>] := -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;