/*
  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 | {13, 13, w^3 - 2*w^2 - 4*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 - 55*x^4 + 718*x^2 - 999;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/615*e^4 - 37/615*e^2 + 676/205, -8/615*e^4 + 319/615*e^2 - 127/205, 1/123*e^5 - 15/41*e^3 + 350/123*e, e, -1, 1/615*e^4 + 37/615*e^2 + 349/205, 2/615*e^5 - 131/615*e^3 + 1274/615*e, -1/205*e^5 + 94/615*e^3 + 139/615*e, 1/123*e^5 - 15/41*e^3 + 473/123*e, 1/615*e^5 + 37/615*e^3 - 676/205*e, 17/615*e^4 - 601/615*e^2 + 1013/205, -2/123*e^4 + 49/123*e^2 + 122/41, 11/205*e^4 - 413/205*e^2 + 2497/205, -26/615*e^4 + 883/615*e^2 - 874/205, -1/615*e^5 - 37/615*e^3 + 471/205*e, -11/615*e^5 + 206/205*e^3 - 7417/615*e, -1/615*e^5 - 37/615*e^3 + 676/205*e, -11/615*e^5 + 206/205*e^3 - 6802/615*e, -9/205*e^4 + 282/205*e^2 - 1018/205, -8/615*e^5 + 524/615*e^3 - 8786/615*e, -1/205*e^5 + 94/615*e^3 - 1091/615*e, -4/615*e^5 + 262/615*e^3 - 3163/615*e, -16/615*e^5 + 281/205*e^3 - 9782/615*e, -16/615*e^4 + 1253/615*e^2 - 4559/205, -3/205*e^5 + 94/205*e^3 - 271/205*e, 2/615*e^5 - 131/615*e^3 + 2504/615*e, e^2 - 14, -34/615*e^4 + 1202/615*e^2 + 1459/205, 22/615*e^5 - 1031/615*e^3 + 2963/205*e, 19/615*e^5 - 937/615*e^3 + 7798/615*e, 8/123*e^4 - 196/123*e^2 - 488/41, 19/615*e^5 - 937/615*e^3 + 8413/615*e, -4/205*e^5 + 376/615*e^3 + 556/615*e, -3/41*e^4 + 94/41*e^2 + 344/41, 83/615*e^4 - 3079/615*e^2 + 4162/205, 49/615*e^4 - 1877/615*e^2 + 1931/205, 49/615*e^4 - 1877/615*e^2 + 1931/205, 1/123*e^5 - 86/123*e^3 + 595/41*e, -8/205*e^5 + 1162/615*e^3 - 12008/615*e, -19/615*e^4 + 527/615*e^2 - 71/205, 16/615*e^4 - 23/615*e^2 - 1181/205, 14/615*e^4 - 97/615*e^2 - 3109/205, -4/123*e^4 + 98/123*e^2 + 244/41, 2/205*e^5 - 131/205*e^3 + 2709/205*e, 23/615*e^5 - 1199/615*e^3 + 15881/615*e, 19/615*e^4 - 1142/615*e^2 + 4171/205, 18/205*e^4 - 564/205*e^2 + 3061/205, -38/615*e^5 + 1669/615*e^3 - 4447/205*e, -29/615*e^5 + 394/205*e^3 - 7813/615*e, 29/615*e^5 - 1387/615*e^3 + 4791/205*e, -16/615*e^5 + 638/615*e^3 - 1484/205*e, 1/3*e^3 - 29/3*e, -11/615*e^5 + 206/205*e^3 - 4957/615*e, 7/123*e^4 - 356/123*e^2 + 1172/41, -16/615*e^4 + 638/615*e^2 + 976/205, 13, -52/615*e^4 + 2381/615*e^2 - 5438/205, -46/615*e^4 + 2603/615*e^2 - 5599/205, -13/205*e^4 + 544/205*e^2 - 5821/205, 2/615*e^5 + 74/615*e^3 - 1967/205*e, 11/205*e^5 - 1649/615*e^3 + 15076/615*e, -16/615*e^5 + 1048/615*e^3 - 16957/615*e, -4/123*e^5 + 139/123*e^3 - 334/123*e, -101/615*e^4 + 3643/615*e^2 - 6344/205, -1/41*e^4 + 45/41*e^2 - 514/41, 38/615*e^4 - 2284/615*e^2 + 5267/205, 22/615*e^4 - 416/615*e^2 - 727/205, 38/615*e^4 - 1669/615*e^2 + 6907/205, 4/123*e^4 + 25/123*e^2 - 1105/41, 61/615*e^4 - 3278/615*e^2 + 8374/205, -24/205*e^4 + 957/205*e^2 - 8318/205, 8/205*e^4 - 319/205*e^2 + 4891/205, -1/15*e^4 + 38/15*e^2 - 34/5, -1/615*e^4 + 578/615*e^2 - 1784/205, 22/205*e^4 - 1236/205*e^2 + 9094/205, -7/41*e^4 + 315/41*e^2 - 2081/41, 91/615*e^4 - 4013/615*e^2 + 6134/205, -49/615*e^5 + 2287/615*e^3 - 22603/615*e, 19/615*e^5 - 937/615*e^3 + 12718/615*e, -32/205*e^4 + 1276/205*e^2 - 6034/205, 10/123*e^4 - 491/123*e^2 + 1768/41, -11/615*e^5 + 206/205*e^3 - 8647/615*e, 4/123*e^5 - 139/123*e^3 + 703/123*e, -37/615*e^4 + 2321/615*e^2 - 4303/205, 12/205*e^4 - 786/205*e^2 + 6619/205, 13/123*e^4 - 503/123*e^2 + 1093/41, 11/615*e^4 + 407/615*e^2 - 5591/205, -12/205*e^5 + 1948/615*e^3 - 23957/615*e, 4/615*e^5 - 262/615*e^3 + 6853/615*e, -16/615*e^5 + 1253/615*e^3 - 7019/205*e, -16/615*e^5 + 281/205*e^3 - 12242/615*e, -22/615*e^5 + 1031/615*e^3 - 3373/205*e, -2/205*e^5 + 188/615*e^3 + 893/615*e, 43/615*e^5 - 1894/615*e^3 + 13861/615*e, -14/615*e^5 + 712/615*e^3 - 2426/205*e, 36/205*e^4 - 1128/205*e^2 + 2227/205, -21/205*e^4 + 863/205*e^2 - 2102/205, -14/205*e^5 + 1931/615*e^3 - 14659/615*e, -4/615*e^5 - 148/615*e^3 + 2499/205*e, -18/205*e^4 + 974/205*e^2 - 7366/205, 83/615*e^4 - 2464/615*e^2 + 1292/205, -1/3*e^3 + 32/3*e, 14/615*e^5 - 917/615*e^3 + 17528/615*e, -14/205*e^5 + 712/205*e^3 - 8098/205*e, -26/615*e^5 + 431/205*e^3 - 17587/615*e, 5/123*e^5 - 266/123*e^3 + 2570/123*e, -2/205*e^5 + 188/615*e^3 - 1567/615*e, -4/205*e^4 + 467/205*e^2 - 3778/205, -2/41*e^4 + 8/41*e^2 + 1309/41, -83/615*e^4 + 2464/615*e^2 - 2317/205, 27/205*e^4 - 846/205*e^2 + 2234/205, -9/205*e^4 + 282/205*e^2 - 8398/205, -2*e, -7/205*e^5 + 863/615*e^3 - 6817/615*e, 17/615*e^5 - 601/615*e^3 + 193/205*e, 28/615*e^5 - 1219/615*e^3 + 6151/615*e, 11/205*e^5 - 1649/615*e^3 + 18151/615*e, -31/615*e^5 + 1313/615*e^3 - 2824/205*e, 23/615*e^5 - 994/615*e^3 + 3517/205*e, 8/615*e^5 - 38/205*e^3 - 3719/615*e, 11/123*e^4 - 454/123*e^2 + 805/41, 18/205*e^4 - 769/205*e^2 + 806/205, -7/615*e^5 + 187/205*e^3 - 13889/615*e, 16/205*e^5 - 843/205*e^3 + 8962/205*e, 6/205*e^4 - 598/205*e^2 + 3617/205, 131/615*e^4 - 5608/615*e^2 + 10664/205, 52/615*e^4 - 2381/615*e^2 - 712/205, 71/615*e^4 - 2293/615*e^2 - 436/205, 2/615*e^5 + 74/615*e^3 - 2377/205*e, -5/123*e^5 + 266/123*e^3 - 3431/123*e, -2*e^2 + 17, 44/205*e^4 - 1447/205*e^2 + 2813/205, -51/205*e^4 + 2008/205*e^2 - 11167/205, -23/615*e^4 - 236/615*e^2 + 4478/205, -17/615*e^4 - 14/615*e^2 - 1013/205, 14/123*e^4 - 589/123*e^2 + 1196/41, -1/41*e^5 + 217/123*e^3 - 3428/123*e, 11/205*e^5 - 1649/615*e^3 + 18151/615*e, -28/615*e^5 + 1424/615*e^3 - 6082/205*e, 7/123*e^5 - 105/41*e^3 + 2327/123*e, -18/205*e^4 + 359/205*e^2 + 2269/205, -71/615*e^4 + 2908/615*e^2 - 6944/205, -44/615*e^5 + 2267/615*e^3 - 28643/615*e, 26/615*e^5 - 883/615*e^3 + 54/205*e, -4/41*e^5 + 180/41*e^3 - 1482/41*e, -14/615*e^5 + 917/615*e^3 - 13223/615*e, 9/205*e^5 - 1666/615*e^3 + 24374/615*e, 71/615*e^5 - 1106/205*e^3 + 29032/615*e, 4/205*e^5 - 262/205*e^3 + 5213/205*e, 11/615*e^5 - 413/615*e^3 + 5777/615*e, -28/615*e^5 + 1219/615*e^3 - 7381/615*e, 58/615*e^5 - 2569/615*e^3 + 21571/615*e, 7/205*e^5 - 356/205*e^3 + 3844/205*e, 52/615*e^5 - 2791/615*e^3 + 32509/615*e, 86/615*e^4 - 2968/615*e^2 - 1966/205, 8/205*e^4 + 91/205*e^2 - 8024/205, 7/41*e^4 - 315/41*e^2 + 2737/41, -113/615*e^4 + 2584/615*e^2 + 6073/205, -31/615*e^5 + 1723/615*e^3 - 19747/615*e, 47/615*e^5 - 2156/615*e^3 + 20099/615*e, 3/205*e^5 - 487/615*e^3 + 2453/615*e, 14/123*e^5 - 671/123*e^3 + 6458/123*e, 37/615*e^5 - 637/205*e^3 + 25619/615*e, 74/615*e^5 - 3617/615*e^3 + 33608/615*e, -4/205*e^5 + 376/615*e^3 - 1289/615*e, 61/615*e^5 - 956/205*e^3 + 24302/615*e, -119/615*e^4 + 3592/615*e^2 - 121/205, 44/615*e^4 - 1447/615*e^2 + 1211/205, 5/123*e^4 - 61/123*e^2 + 269/41, 9/41*e^4 - 405/41*e^2 + 2822/41, 5/123*e^4 - 430/123*e^2 + 802/41, -8/123*e^4 + 442/123*e^2 - 86/41, 31/615*e^5 - 1928/615*e^3 + 7949/205*e, 11/615*e^5 - 411/205*e^3 + 25867/615*e, 2/123*e^5 - 131/123*e^3 + 1889/123*e, -7/615*e^5 - 259/615*e^3 + 4937/205*e, -47/615*e^4 + 3181/615*e^2 - 13328/205, -46/615*e^4 + 143/615*e^2 + 12646/205, 13/615*e^4 - 749/615*e^2 + 1052/205, 13/123*e^4 - 626/123*e^2 + 1831/41, 9/205*e^5 - 1051/615*e^3 + 2234/615*e, 43/615*e^5 - 768/205*e^3 + 26366/615*e, -14/205*e^4 + 712/205*e^2 + 307/205, 34/205*e^4 - 1407/205*e^2 + 8743/205, 3/205*e^5 - 94/205*e^3 - 549/205*e, 11/205*e^5 - 618/205*e^3 + 8237/205*e, -10/123*e^5 + 409/123*e^3 - 2557/123*e, 6/205*e^5 - 769/615*e^3 + 9416/615*e, 7/205*e^5 - 1273/615*e^3 + 19322/615*e, 8/205*e^5 - 1162/615*e^3 + 5243/615*e, -104/615*e^4 + 2917/615*e^2 - 4111/205, -34/615*e^4 + 1817/615*e^2 - 6331/205, 58/615*e^4 - 1544/615*e^2 - 4358/205, 67/615*e^4 - 2441/615*e^2 + 1858/205, 48/205*e^4 - 1504/205*e^2 + 2491/205, 13/205*e^4 - 749/205*e^2 + 2131/205, -61/615*e^4 + 2663/615*e^2 - 1199/205, -29/205*e^4 + 567/205*e^2 + 9817/205, -19/615*e^5 + 244/205*e^3 - 4313/615*e, 29/615*e^5 - 599/205*e^3 + 28723/615*e, 52/615*e^4 - 536/615*e^2 - 9527/205, 26/205*e^4 - 1088/205*e^2 + 5902/205, 13/205*e^5 - 2042/615*e^3 + 25663/615*e, 8/123*e^5 - 401/123*e^3 + 3620/123*e, -23/615*e^5 + 994/615*e^3 - 3722/205*e, 8/123*e^5 - 120/41*e^3 + 2431/123*e, -25/123*e^4 + 797/123*e^2 - 730/41, -24/205*e^4 + 547/205*e^2 + 7672/205, -4/205*e^5 + 262/205*e^3 - 3983/205*e, -7/123*e^5 + 479/123*e^3 - 2484/41*e, 9/205*e^4 + 128/205*e^2 - 13537/205, 2/15*e^4 - 91/15*e^2 + 268/5, 1/205*e^4 + 37/205*e^2 + 227/205, -11/615*e^4 + 208/615*e^2 + 7436/205, 32/615*e^5 - 1276/615*e^3 + 1533/205*e, -8/123*e^5 + 442/123*e^3 - 1767/41*e, -73/205*e^4 + 2424/205*e^2 - 6116/205, 2/205*e^4 - 336/205*e^2 + 12754/205, -28/615*e^5 + 1424/615*e^3 - 5672/205*e, 67/615*e^5 - 1087/205*e^3 + 26279/615*e, 4/41*e^5 - 581/123*e^3 + 5143/123*e, -2/205*e^5 + 803/615*e^3 - 16327/615*e, 17/615*e^5 - 601/615*e^3 - 627/205*e, -6/205*e^5 + 769/615*e^3 - 3266/615*e, -29/205*e^4 + 1182/205*e^2 - 2483/205, -44/205*e^4 + 1652/205*e^2 - 10808/205, 13/205*e^5 - 1222/615*e^3 + 38/615*e, 73/615*e^5 - 3449/615*e^3 + 10307/205*e, 24/205*e^5 - 3281/615*e^3 + 27619/615*e, 1/205*e^5 - 299/615*e^3 + 6421/615*e, 94/615*e^4 - 4517/615*e^2 + 8821/205, 44/615*e^4 - 2062/615*e^2 + 7156/205, -8/205*e^5 + 319/205*e^3 - 996/205*e, 32/615*e^5 - 562/205*e^3 + 17104/615*e, 7/615*e^5 - 187/205*e^3 + 7739/615*e, -16/615*e^5 + 638/615*e^3 + 1386/205*e, -1/41*e^5 + 45/41*e^3 - 473/41*e, -14/615*e^5 + 712/615*e^3 - 3451/205*e, 13/615*e^5 - 113/205*e^3 - 739/615*e, 74/615*e^5 - 3412/615*e^3 + 11886/205*e, -56/615*e^5 + 3053/615*e^3 - 34442/615*e, -11/615*e^5 + 206/205*e^3 - 6802/615*e, 1/615*e^5 - 261/205*e^3 + 21752/615*e, 14/615*e^5 - 97/615*e^3 - 3314/205*e, 18/205*e^4 - 974/205*e^2 + 5726/205, 37/123*e^4 - 1460/123*e^2 + 3278/41, 49/205*e^4 - 2287/205*e^2 + 9278/205, 53/615*e^4 - 2959/615*e^2 + 1072/205, -13/615*e^5 + 134/615*e^3 + 2023/205*e, -43/615*e^5 + 768/205*e^3 - 26981/615*e, -4/41*e^5 + 180/41*e^3 - 1441/41*e, -3/205*e^5 + 692/615*e^3 - 15778/615*e, 2/41*e^4 - 49/41*e^2 - 1145/41, 76/615*e^4 - 2108/615*e^2 + 5819/205, 116/615*e^4 - 4318/615*e^2 + 10349/205, 7/205*e^4 - 151/205*e^2 + 359/205, 23/615*e^5 - 263/205*e^3 + 301/615*e, -1/15*e^5 + 38/15*e^3 - 89/5*e, 7/205*e^4 - 766/205*e^2 + 9584/205, -71/615*e^4 + 4753/615*e^2 - 13504/205, -4/205*e^4 + 877/205*e^2 - 15668/205, -73/615*e^4 + 2834/615*e^2 - 7232/205, 8/205*e^5 - 1367/615*e^3 + 19183/615*e, -56/615*e^5 + 3053/615*e^3 - 42437/615*e, -3/205*e^5 + 299/205*e^3 - 4986/205*e, 2/205*e^5 - 598/615*e^3 + 10382/615*e, 34/615*e^4 + 643/615*e^2 - 13759/205, -257/615*e^4 + 10171/615*e^2 - 17533/205, -79/615*e^4 + 1382/615*e^2 + 7279/205, -154/615*e^4 + 6602/615*e^2 - 6596/205, -61/615*e^4 + 1433/615*e^2 + 8231/205, 203/615*e^4 - 8479/615*e^2 + 16112/205, -43/205*e^4 + 1894/205*e^2 - 10376/205, -53/615*e^4 + 499/615*e^2 + 6718/205, 7/123*e^5 - 479/123*e^3 + 2361/41*e, 11/205*e^5 - 413/205*e^3 + 857/205*e, -8/615*e^5 - 167/205*e^3 + 20939/615*e, 1/41*e^5 - 299/123*e^3 + 6175/123*e, e^2 - 8, -43/123*e^4 + 1484/123*e^2 - 2051/41, -32/205*e^4 + 1276/205*e^2 - 14029/205, 257/615*e^4 - 8326/615*e^2 + 6463/205, 24/205*e^5 - 1162/205*e^3 + 11188/205*e, 16/615*e^5 - 433/615*e^3 - 3953/615*e, 22/615*e^5 - 1031/615*e^3 + 1118/205*e, 32/615*e^5 - 1891/615*e^3 + 7683/205*e, 1/41*e^5 - 176/123*e^3 + 2608/123*e, 11/615*e^5 - 411/205*e^3 + 24022/615*e, 7/123*e^5 - 105/41*e^3 + 3311/123*e, 4/205*e^5 - 262/205*e^3 + 5828/205*e, -24/205*e^4 + 1162/205*e^2 - 4628/205, -25/123*e^4 + 1166/123*e^2 - 3190/41, 167/615*e^4 - 6121/615*e^2 + 5803/205, 16/123*e^4 - 638/123*e^2 - 238/41, -7/615*e^5 + 766/615*e^3 - 17374/615*e, 2/205*e^5 - 188/615*e^3 + 7717/615*e, -7/615*e^5 - 18/205*e^3 + 13171/615*e, -37/615*e^5 + 1501/615*e^3 - 8809/615*e, -23/123*e^4 + 871/123*e^2 - 1795/41, -49/205*e^4 + 2082/205*e^2 - 17068/205, 32/123*e^4 - 1276/123*e^2 + 2312/41, -94/615*e^4 + 3902/615*e^2 - 5746/205, -23/205*e^5 + 994/205*e^3 - 8911/205*e, 62/615*e^5 - 3241/615*e^3 + 11388/205*e, 2/615*e^5 + 93/205*e^3 - 14306/615*e, -7/615*e^5 + 356/615*e^3 - 2238/205*e, -6/205*e^5 + 1589/615*e^3 - 26431/615*e, -11/615*e^5 + 823/615*e^3 - 4044/205*e, -229/615*e^4 + 9362/615*e^2 - 14321/205, 26/205*e^4 - 678/205*e^2 - 7423/205, 4/123*e^4 - 344/123*e^2 + 2134/41, 9/205*e^4 + 128/205*e^2 - 9437/205, -44/205*e^4 + 1242/205*e^2 - 353/205, 88/615*e^4 - 4739/615*e^2 + 2422/205];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, w^3 - 2*w^2 - 4*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;