/*
  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, 8, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[9, 3, w^3 - 5*w + 3],
[9, 3, w^3 - 5*w + 2],
[13, 13, w + 1],
[13, 13, w^3 - 6*w + 4],
[16, 2, 2],
[17, 17, w^2 + w - 2],
[17, 17, w^2 + w - 5],
[23, 23, w^3 - w^2 - 6*w + 6],
[23, 23, w^3 + w^2 - 4*w - 1],
[25, 5, -w^2 + 3],
[25, 5, -w^3 - w^2 + 5*w + 1],
[29, 29, -w^2 - 2*w + 3],
[29, 29, -w^3 + w^2 + 7*w - 7],
[43, 43, 2*w^3 + w^2 - 10*w + 3],
[43, 43, -w^3 + w^2 + 5*w - 7],
[53, 53, w^3 - w^2 - 7*w + 5],
[53, 53, w^2 + 2*w - 5],
[61, 61, 2*w^3 + w^2 - 10*w],
[61, 61, w^3 - 7*w + 3],
[61, 61, 2*w^3 + w^2 - 9*w + 3],
[79, 79, -w^3 + w^2 + 6*w - 5],
[79, 79, w^3 + w^2 - 4*w - 2],
[101, 101, -2*w^3 + 12*w - 7],
[101, 101, -2*w - 1],
[107, 107, -w^3 + 4*w - 5],
[107, 107, 3*w^3 - 17*w + 5],
[107, 107, 2*w^3 - 11*w + 8],
[107, 107, -3*w^3 - w^2 + 17*w - 4],
[127, 127, 3*w^3 - 16*w + 11],
[127, 127, w^3 + w^2 - 7*w + 3],
[127, 127, w^3 - 6*w - 1],
[127, 127, w - 4],
[139, 139, w^2 + 2*w - 6],
[139, 139, w^3 - w^2 - 7*w + 4],
[157, 157, -w^3 + w^2 + 6*w - 4],
[157, 157, w^3 + w^2 - 4*w - 3],
[173, 173, 3*w^3 + w^2 - 15*w + 5],
[173, 173, -2*w^2 - w + 9],
[181, 181, -2*w^3 - w^2 + 10*w - 5],
[181, 181, -w^3 + w^2 + 5*w - 9],
[191, 191, -w^2 - 3*w + 4],
[191, 191, w^3 - w^2 - 7*w + 2],
[211, 211, -2*w^3 + w^2 + 13*w - 10],
[211, 211, w^2 + 3*w - 3],
[233, 233, -3*w^3 + 16*w - 6],
[233, 233, 2*w^3 - 9*w + 3],
[251, 251, w^2 - w - 5],
[251, 251, -2*w^3 - w^2 + 11*w - 4],
[277, 277, w^3 + w^2 - 7*w - 2],
[277, 277, -2*w^3 - w^2 + 12*w - 4],
[283, 283, -w^3 + 8*w - 4],
[283, 283, w^3 + 2*w^2 - 4*w - 4],
[283, 283, 2*w^3 - 13*w + 5],
[283, 283, 2*w^2 + w - 7],
[289, 17, -w^3 + 5*w - 7],
[311, 311, 3*w^3 - 16*w + 8],
[311, 311, 2*w^3 + 2*w^2 - 10*w - 3],
[313, 313, -2*w^3 + 9*w - 4],
[313, 313, -3*w^3 + 16*w - 7],
[337, 337, -2*w^3 + w^2 + 13*w - 6],
[337, 337, 3*w^3 - 17*w + 10],
[347, 347, 3*w^3 - 18*w + 10],
[347, 347, -3*w - 1],
[347, 347, 2*w^3 + 2*w^2 - 11*w - 4],
[347, 347, 4*w^3 + w^2 - 21*w + 5],
[361, 19, 2*w^2 + w - 13],
[361, 19, -2*w^3 + w^2 + 10*w - 12],
[367, 367, -2*w^3 - w^2 + 9*w + 1],
[367, 367, 3*w^3 - w^2 - 18*w + 16],
[367, 367, -2*w^3 + w^2 + 11*w - 6],
[367, 367, 4*w^3 - 22*w + 9],
[373, 373, -w^3 + w^2 + 4*w - 7],
[373, 373, -3*w^3 - w^2 + 16*w - 6],
[389, 389, w^3 - 6*w - 2],
[389, 389, w - 5],
[419, 419, w^3 + 2*w^2 - 6*w - 5],
[419, 419, -2*w^2 - 2*w + 11],
[433, 433, 3*w^3 + w^2 - 15*w - 1],
[433, 433, 2*w^3 - w^2 - 10*w + 3],
[467, 467, -3*w^3 - 2*w^2 + 17*w],
[467, 467, -w^3 + w^2 + 5*w - 12],
[503, 503, w^3 + 2*w^2 - 5*w - 3],
[503, 503, 3*w^3 + w^2 - 18*w + 6],
[503, 503, -3*w^3 - w^2 + 15*w - 9],
[503, 503, w^3 + 2*w^2 - 5*w - 5],
[521, 521, -w^3 - 2*w^2 + 6*w + 2],
[521, 521, 2*w^3 + 2*w^2 - 11*w - 3],
[523, 523, w^3 - 2*w^2 - 9*w + 14],
[523, 523, w^3 - 2*w^2 - 9*w + 6],
[529, 23, 2*w^3 - 10*w + 11],
[547, 547, -2*w^3 - 3*w^2 + 8*w + 5],
[547, 547, -4*w^3 + w^2 + 22*w - 17],
[571, 571, 2*w^3 + 2*w^2 - 12*w - 5],
[571, 571, 4*w^3 + w^2 - 24*w + 8],
[571, 571, -w^3 - w^2 + 9*w],
[571, 571, 2*w^3 + 2*w^2 - 12*w + 3],
[599, 599, -3*w^3 - w^2 + 17*w - 6],
[599, 599, w^2 - 2*w - 4],
[607, 607, 2*w^3 + w^2 - 8*w - 1],
[607, 607, -w^3 - 2*w^2 + 6*w + 3],
[607, 607, 2*w^3 + 2*w^2 - 11*w - 2],
[607, 607, -3*w^3 + w^2 + 17*w - 9],
[641, 641, w^3 - 9*w + 3],
[641, 641, 3*w^3 - 19*w + 9],
[647, 647, w^3 - w^2 - 9*w + 4],
[647, 647, -2*w^3 + w^2 + 14*w - 12],
[653, 653, -w^3 + 9*w - 6],
[653, 653, 3*w^3 - 19*w + 6],
[659, 659, -2*w^3 + w^2 + 13*w - 4],
[659, 659, -w^3 + w^2 + 4*w - 8],
[659, 659, -3*w^3 - w^2 + 16*w - 7],
[659, 659, w^2 + 3*w - 9],
[673, 673, 3*w^3 - w^2 - 19*w + 12],
[673, 673, w^2 + 4*w - 4],
[677, 677, 2*w^3 + w^2 - 9*w - 3],
[677, 677, 2*w^3 - w^2 - 11*w + 4],
[701, 701, w^3 + w^2 - 7*w - 4],
[701, 701, -2*w^3 - w^2 + 12*w - 6],
[751, 751, -2*w^3 + 2*w^2 + 12*w - 11],
[751, 751, -2*w^3 - w^2 + 6*w + 1],
[751, 751, -5*w^3 + w^2 + 29*w - 15],
[751, 751, 2*w^3 + 2*w^2 - 8*w - 3],
[757, 757, w^3 + 2*w^2 - 3*w - 7],
[757, 757, 5*w^3 + 2*w^2 - 27*w + 6],
[757, 757, -w^3 + 2*w^2 + 7*w - 7],
[757, 757, -4*w^3 + w^2 + 23*w - 12],
[797, 797, 3*w^3 - 19*w + 8],
[797, 797, 3*w^3 + 2*w^2 - 16*w + 4],
[797, 797, -w^3 + 9*w - 4],
[797, 797, 2*w^2 - w - 9],
[809, 809, 2*w^3 - 7*w - 1],
[809, 809, -3*w^3 + 2*w^2 + 15*w - 14],
[809, 809, -w^3 + 3*w^2 + 9*w - 17],
[809, 809, 5*w^3 - 28*w + 8],
[823, 823, 2*w^3 + w^2 - 12*w + 7],
[823, 823, w^3 + w^2 - 7*w - 5],
[841, 29, -3*w^3 + 15*w - 8],
[859, 859, -w^3 - w^2 + 4*w - 5],
[859, 859, -w^3 + w^2 + 6*w - 12],
[859, 859, w^3 + 2*w^2 - 2*w - 6],
[859, 859, -2*w^3 + 2*w^2 + 13*w - 11],
[883, 883, w^3 - 4*w - 4],
[883, 883, 2*w^3 - 11*w - 1],
[887, 887, -2*w^3 + w^2 + 14*w - 11],
[887, 887, -w^3 + w^2 + 9*w - 5],
[907, 907, 3*w^3 - 17*w + 13],
[907, 907, -w^3 + 3*w - 7],
[911, 911, 2*w^3 - w^2 - 14*w + 8],
[911, 911, w^3 - 6*w - 3],
[911, 911, -w^3 + w^2 + 9*w - 8],
[911, 911, w - 6],
[953, 953, -2*w^3 - w^2 + 15*w - 4],
[953, 953, 4*w^3 + w^2 - 25*w + 7],
[971, 971, w^2 - 3*w - 3],
[971, 971, -5*w^3 + 30*w - 12],
[971, 971, -3*w^3 + 3*w^2 + 18*w - 19],
[971, 971, 4*w^3 + w^2 - 23*w + 8],
[997, 997, 4*w^3 + w^2 - 20*w + 8],
[997, 997, -3*w^3 + w^2 + 15*w - 12],
[1013, 1013, -2*w^3 - w^2 + 8*w + 2],
[1013, 1013, 3*w^3 - w^2 - 17*w + 8],
[1031, 1031, 2*w^2 - 2*w - 7],
[1031, 1031, 4*w^3 + 2*w^2 - 22*w + 5],
[1039, 1039, w^3 + w^2 - 5*w - 7],
[1039, 1039, -w^2 - 3],
[1063, 1063, w^3 + 3*w^2 - 3*w - 7],
[1063, 1063, 3*w^2 + 2*w - 11],
[1069, 1069, 3*w^3 + w^2 - 19*w],
[1069, 1069, 3*w^3 + 2*w^2 - 15*w - 5],
[1091, 1091, -3*w^3 + 14*w - 9],
[1091, 1091, -4*w^3 + 21*w - 12],
[1109, 1109, 3*w^3 - w^2 - 15*w + 10],
[1109, 1109, 4*w^3 + w^2 - 20*w + 6],
[1117, 1117, w^3 - 2*w^2 - 9*w + 3],
[1117, 1117, w^3 - 2*w^2 - 9*w + 17],
[1117, 1117, -w^3 + 2*w^2 + 7*w - 6],
[1117, 1117, w^3 + 2*w^2 - 3*w - 8],
[1153, 1153, 2*w^3 + 2*w^2 - 9*w - 6],
[1153, 1153, -w^3 + 2*w^2 + 6*w - 5],
[1187, 1187, -w^2 + 2*w + 11],
[1187, 1187, 3*w^3 + w^2 - 17*w + 13],
[1213, 1213, 3*w^3 + w^2 - 16*w + 9],
[1213, 1213, -w^3 + w^2 + 4*w - 10],
[1223, 1223, -w^3 + 2*w^2 + 6*w - 17],
[1223, 1223, 2*w^3 + 2*w^2 - 9*w + 6],
[1223, 1223, -4*w^3 - 3*w^2 + 19*w + 2],
[1223, 1223, -w^3 + 2*w^2 + 5*w - 14],
[1231, 1231, 3*w^3 - 2*w^2 - 14*w + 13],
[1231, 1231, -4*w^3 + w^2 + 22*w - 10],
[1249, 1249, -2*w^3 + 15*w - 10],
[1249, 1249, 3*w^3 - 20*w + 5],
[1283, 1283, -w^3 + w^2 + 7*w - 13],
[1283, 1283, -w^2 - 2*w - 3],
[1291, 1291, -w^3 + w^2 + 3*w - 7],
[1291, 1291, 4*w^3 + w^2 - 22*w + 9],
[1303, 1303, -w^2 - w - 3],
[1303, 1303, w^2 + w - 10],
[1303, 1303, -4*w^3 - w^2 + 24*w - 9],
[1303, 1303, -w^3 - w^2 + 9*w + 1],
[1327, 1327, -3*w^3 - 3*w^2 + 15*w + 5],
[1327, 1327, -3*w^2 + 7],
[1327, 1327, -2*w^3 - 3*w^2 + 10*w + 1],
[1327, 1327, w^3 + 3*w^2 - 5*w - 11],
[1361, 1361, 3*w^3 - 18*w + 13],
[1361, 1361, 3*w^3 - 2*w^2 - 17*w + 13],
[1361, 1361, -3*w - 4],
[1361, 1361, -3*w^3 - 2*w^2 + 13*w + 1],
[1381, 1381, -w^3 + 2*w^2 + 6*w - 3],
[1381, 1381, -2*w^3 + 3*w^2 + 12*w - 14],
[1381, 1381, 2*w^3 + 2*w^2 - 9*w - 8],
[1381, 1381, 3*w^3 + 3*w^2 - 13*w - 4],
[1427, 1427, 3*w^3 + w^2 - 17*w + 8],
[1427, 1427, w^2 - 2*w - 6],
[1429, 1429, 2*w^3 - 15*w + 4],
[1429, 1429, -3*w^3 + 20*w - 11],
[1433, 1433, -w^3 + 2*w^2 + 7*w - 5],
[1433, 1433, w^3 + 2*w^2 - 3*w - 9],
[1447, 1447, -2*w - 7],
[1447, 1447, -2*w^3 + 12*w - 13],
[1453, 1453, -4*w^3 - 2*w^2 + 24*w + 1],
[1453, 1453, 5*w^3 + 2*w^2 - 26*w],
[1459, 1459, 5*w^3 + w^2 - 26*w + 10],
[1459, 1459, -w^2 + w - 4],
[1459, 1459, 3*w^3 - w^2 - 14*w + 11],
[1459, 1459, 2*w^3 + w^2 - 11*w - 5],
[1481, 1481, -w^3 + w^2 + 10*w - 10],
[1481, 1481, 3*w^3 - w^2 - 20*w + 9],
[1499, 1499, -3*w^3 + 13*w - 8],
[1499, 1499, 5*w^3 - 27*w + 14],
[1511, 1511, -3*w^3 + 18*w - 14],
[1511, 1511, -3*w - 5],
[1511, 1511, -w^3 - w^2 + 3*w - 6],
[1511, 1511, -2*w^3 + w^2 + 12*w - 16],
[1531, 1531, 4*w^3 - w^2 - 25*w + 14],
[1531, 1531, w^3 - 7*w - 4],
[1543, 1543, -4*w^3 - w^2 + 21*w - 11],
[1543, 1543, -2*w^3 + w^2 + 9*w - 12],
[1559, 1559, -2*w^3 - 3*w^2 + 9*w + 3],
[1559, 1559, -w^3 + w^2 + 10*w - 4],
[1559, 1559, -3*w^3 + w^2 + 20*w - 15],
[1559, 1559, 3*w^2 + w - 12],
[1609, 1609, -w^3 + 2*w^2 + 7*w - 4],
[1609, 1609, w^3 + 2*w^2 - 3*w - 10],
[1613, 1613, 5*w^3 - 27*w + 11],
[1613, 1613, w^3 + 2*w^2 - 7*w - 7],
[1613, 1613, -3*w^3 - 2*w^2 + 17*w - 5],
[1613, 1613, -3*w^3 + 13*w - 5],
[1621, 1621, 4*w^3 + w^2 - 19*w + 5],
[1621, 1621, 4*w^3 - w^2 - 21*w + 12],
[1637, 1637, 2*w^3 - w^2 - 15*w + 6],
[1637, 1637, -2*w^3 + w^2 + 15*w - 13],
[1663, 1663, 5*w^3 + 3*w^2 - 28*w - 1],
[1663, 1663, 5*w^3 - 26*w + 8],
[1663, 1663, -6*w^3 - w^2 + 32*w - 8],
[1663, 1663, 4*w^3 - 19*w + 5],
[1681, 41, 5*w^3 + 2*w^2 - 26*w - 1],
[1681, 41, -2*w^3 + 2*w^2 + 9*w - 4],
[1699, 1699, -3*w^3 + 19*w - 14],
[1699, 1699, w^3 + 4*w^2 - 2*w - 13],
[1699, 1699, -4*w^3 - w^2 + 22*w + 1],
[1699, 1699, w^3 - 9*w - 2],
[1733, 1733, 3*w^2 + w - 11],
[1733, 1733, 2*w^3 + 3*w^2 - 9*w - 4],
[1733, 1733, 5*w^3 + w^2 - 26*w + 8],
[1733, 1733, 3*w^3 - w^2 - 14*w + 9],
[1741, 1741, -w^3 + 10*w - 6],
[1741, 1741, 4*w^3 - 25*w + 9],
[1759, 1759, -w^3 + 2*w^2 + 10*w - 11],
[1759, 1759, -2*w^3 + 2*w^2 + 15*w - 12],
[1777, 1777, 3*w^3 + 3*w^2 - 15*w - 4],
[1777, 1777, -3*w^2 + 8],
[1811, 1811, -3*w^3 + w^2 + 17*w - 5],
[1811, 1811, 5*w^3 - 27*w + 12],
[1811, 1811, -3*w^3 + 13*w - 6],
[1811, 1811, 2*w^3 + w^2 - 8*w - 5],
[1823, 1823, -5*w^3 - 2*w^2 + 23*w - 11],
[1823, 1823, 5*w^3 - 2*w^2 - 27*w + 25],
[1847, 1847, -3*w^3 - w^2 + 17*w - 9],
[1847, 1847, w^2 - 2*w - 7],
[1849, 43, -4*w^3 + 20*w - 7],
[1871, 1871, -2*w^3 + 2*w^2 + 12*w - 9],
[1871, 1871, 2*w^3 + 2*w^2 - 8*w - 5],
[1871, 1871, -3*w^3 - 2*w^2 + 16*w - 6],
[1871, 1871, 2*w^2 - w - 11],
[1873, 1873, w^3 - 4*w - 5],
[1873, 1873, 2*w^3 - 11*w - 2],
[1889, 1889, -4*w^3 + 22*w - 17],
[1889, 1889, -4*w^2 - 3*w + 9],
[1901, 1901, 3*w^3 - 11*w + 4],
[1901, 1901, -7*w^3 + 39*w - 16],
[1907, 1907, -2*w^3 + 2*w^2 + 15*w - 14],
[1907, 1907, w^3 + 3*w^2 - 6*w - 9],
[1907, 1907, w^3 - 2*w^2 - 10*w + 9],
[1907, 1907, 3*w^3 + 3*w^2 - 16*w],
[1949, 1949, -3*w^3 + 20*w - 10],
[1949, 1949, -2*w^3 + 2*w^2 + 15*w - 13],
[1949, 1949, -w^3 + 2*w^2 + 10*w - 10],
[1949, 1949, 2*w^3 - 15*w + 5],
[1973, 1973, 6*w^3 + 2*w^2 - 31*w + 2],
[1973, 1973, 3*w^3 - 2*w^2 - 14*w + 7],
[1979, 1979, 2*w^3 - 4*w^2 - 12*w + 21],
[1979, 1979, -4*w^3 - 4*w^2 + 18*w + 1],
[1979, 1979, w^3 + 3*w^2 - 2*w - 11],
[1979, 1979, -w^3 + 3*w^2 + 8*w - 10],
[1993, 1993, w^3 + 3*w^2 - 4*w - 8],
[1993, 1993, w^3 + 3*w^2 - 4*w - 7],
[1993, 1993, 6*w^3 + 3*w^2 - 34*w + 6],
[1993, 1993, -w^3 - 3*w^2 + 9*w + 6],
[1999, 1999, 7*w^3 - 39*w + 21],
[1999, 1999, 4*w^3 + w^2 - 19*w],
[1999, 1999, w^3 + 2*w^2 - 8*w - 5],
[1999, 1999, -4*w^3 - 2*w^2 + 23*w - 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 10*x^4 + 20*x^3 + 40*x^2 - 80*x - 80;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, 1/8*e^4 - 3/4*e^3 - 1/2*e^2 + 3*e + 4, 1/8*e^4 - 3/4*e^3 - 1/2*e^2 + 3*e + 4, -1/4*e^4 + 9/4*e^3 - 7/2*e^2 - 6*e + 7, -1/8*e^4 + 3/4*e^3 + e^2 - 6*e - 2, -1/8*e^4 + 3/4*e^3 + e^2 - 6*e - 2, -1/4*e^3 + 3/2*e^2 + e - 6, -1/4*e^3 + 3/2*e^2 + e - 6, -1/4*e^4 + 2*e^3 - 5/2*e^2 - 5*e + 6, -1/4*e^4 + 2*e^3 - 5/2*e^2 - 5*e + 6, 1/4*e^4 - 2*e^3 + 2*e^2 + 7*e, 1/4*e^4 - 2*e^3 + 2*e^2 + 7*e, -1/2*e^3 + 3*e^2 - e - 6, -1/2*e^3 + 3*e^2 - e - 6, -1/2*e^3 + 5*e^2 - 8*e - 16, -1/2*e^3 + 5*e^2 - 8*e - 16, 1/2*e^3 - 9/2*e^2 + 7*e + 12, -1/4*e^4 + 5/2*e^3 - 5*e^2 - 10*e + 12, 1/2*e^3 - 9/2*e^2 + 7*e + 12, -1/8*e^4 + 3/4*e^3 + 3/2*e^2 - 7*e - 10, -1/8*e^4 + 3/4*e^3 + 3/2*e^2 - 7*e - 10, e^3 - 15/2*e^2 + 7*e + 22, e^3 - 15/2*e^2 + 7*e + 22, 1/4*e^4 - 3/2*e^3 - 5/2*e^2 + 9*e + 18, 1/2*e^4 - 5*e^3 + 12*e^2 + 5*e - 22, 1/4*e^4 - 3/2*e^3 - 5/2*e^2 + 9*e + 18, 1/2*e^4 - 5*e^3 + 12*e^2 + 5*e - 22, -e^3 + 13/2*e^2 - 3*e - 22, -e^3 + 13/2*e^2 - 3*e - 22, 7/8*e^4 - 29/4*e^3 + 8*e^2 + 22*e - 2, 7/8*e^4 - 29/4*e^3 + 8*e^2 + 22*e - 2, 5/4*e^3 - 21/2*e^2 + 10*e + 40, 5/4*e^3 - 21/2*e^2 + 10*e + 40, -1/4*e^4 + 7/2*e^3 - 13*e^2 + e + 28, -1/4*e^4 + 7/2*e^3 - 13*e^2 + e + 28, 3/4*e^4 - 19/4*e^3 - 7/2*e^2 + 29*e + 24, 3/4*e^4 - 19/4*e^3 - 7/2*e^2 + 29*e + 24, 3/2*e^3 - 11*e^2 + 10*e + 22, 3/2*e^3 - 11*e^2 + 10*e + 22, 3/8*e^4 - 17/4*e^3 + 11*e^2 + 12*e - 28, 3/8*e^4 - 17/4*e^3 + 11*e^2 + 12*e - 28, 1/2*e^3 - 4*e^2 + 4*e + 2, 1/2*e^3 - 4*e^2 + 4*e + 2, -1/2*e^4 + 17/4*e^3 - 13/2*e^2 - 11*e + 24, -1/2*e^4 + 17/4*e^3 - 13/2*e^2 - 11*e + 24, e^3 - 11*e^2 + 20*e + 32, e^3 - 11*e^2 + 20*e + 32, -3/4*e^3 + 13/2*e^2 - 11*e - 12, -3/4*e^3 + 13/2*e^2 - 11*e - 12, -1/4*e^4 + 4*e^3 - 29/2*e^2 - 3*e + 24, 5/8*e^4 - 23/4*e^3 + 11*e^2 + 12*e - 16, -1/4*e^4 + 4*e^3 - 29/2*e^2 - 3*e + 24, 5/8*e^4 - 23/4*e^3 + 11*e^2 + 12*e - 16, -3/4*e^4 + 21/4*e^3 + 1/2*e^2 - 30*e + 10, -3/8*e^4 + 7/4*e^3 + 6*e^2 - 16*e - 18, -3/8*e^4 + 7/4*e^3 + 6*e^2 - 16*e - 18, 1/8*e^4 - 1/4*e^3 - 11/2*e^2 + 9*e + 34, 1/8*e^4 - 1/4*e^3 - 11/2*e^2 + 9*e + 34, 7/8*e^4 - 31/4*e^3 + 12*e^2 + 24*e - 22, 7/8*e^4 - 31/4*e^3 + 12*e^2 + 24*e - 22, 1/2*e^4 - 11/2*e^3 + 14*e^2 + 7*e - 22, 1/2*e^4 - 11/2*e^3 + 14*e^2 + 7*e - 22, -3/8*e^4 + 15/4*e^3 - 9*e^2 - 4*e + 28, -3/8*e^4 + 15/4*e^3 - 9*e^2 - 4*e + 28, 1/2*e^4 - 5*e^3 + 15*e^2 - 8*e - 38, 1/2*e^4 - 5*e^3 + 15*e^2 - 8*e - 38, -e^2 + 6*e - 12, 1/4*e^4 - 2*e^3 + 4*e^2 - 3*e - 2, -e^2 + 6*e - 12, 1/4*e^4 - 2*e^3 + 4*e^2 - 3*e - 2, -3/8*e^4 + 17/4*e^3 - 14*e^2 + 6*e + 24, -3/8*e^4 + 17/4*e^3 - 14*e^2 + 6*e + 24, -9/8*e^4 + 31/4*e^3 + 1/2*e^2 - 31*e - 40, -9/8*e^4 + 31/4*e^3 + 1/2*e^2 - 31*e - 40, -1/4*e^4 + 1/2*e^3 + 11*e^2 - 26*e - 40, -1/4*e^4 + 1/2*e^3 + 11*e^2 - 26*e - 40, -3/4*e^4 + 11/2*e^3 - 3/2*e^2 - 21*e - 36, -3/4*e^4 + 11/2*e^3 - 3/2*e^2 - 21*e - 36, 1/4*e^4 - e^3 - 7*e^2 + 14*e + 28, 1/4*e^4 - e^3 - 7*e^2 + 14*e + 28, 1/2*e^4 - 5*e^3 + 14*e^2 - 2*e - 26, -1/4*e^4 + 4*e^3 - 19*e^2 + 12*e + 44, -1/4*e^4 + 4*e^3 - 19*e^2 + 12*e + 44, 1/2*e^4 - 5*e^3 + 14*e^2 - 2*e - 26, -1/2*e^3 + 4*e^2 - 18, -1/2*e^3 + 4*e^2 - 18, -e^4 + 13/2*e^3 + 4*e^2 - 39*e - 26, -e^4 + 13/2*e^3 + 4*e^2 - 39*e - 26, 1/2*e^4 - 4*e^3 + 3*e^2 + 13*e + 40, -1/4*e^4 + 3/4*e^3 + 19/2*e^2 - 22*e - 52, -1/4*e^4 + 3/4*e^3 + 19/2*e^2 - 22*e - 52, 3/2*e^4 - 23/2*e^3 + 15/2*e^2 + 37*e + 22, 3/8*e^4 - 19/4*e^3 + 31/2*e^2 + 3*e - 38, 3/8*e^4 - 19/4*e^3 + 31/2*e^2 + 3*e - 38, 3/2*e^4 - 23/2*e^3 + 15/2*e^2 + 37*e + 22, -1/4*e^4 + 14*e^2 - 22*e - 40, -1/4*e^4 + 14*e^2 - 22*e - 40, 5/4*e^4 - 19/2*e^3 + 10*e^2 + 21*e - 2, 9/4*e^3 - 37/2*e^2 + 16*e + 68, 9/4*e^3 - 37/2*e^2 + 16*e + 68, 5/4*e^4 - 19/2*e^3 + 10*e^2 + 21*e - 2, 9/8*e^4 - 45/4*e^3 + 53/2*e^2 + 7*e - 58, 9/8*e^4 - 45/4*e^3 + 53/2*e^2 + 7*e - 58, 1/2*e^4 - 9/2*e^3 + 11*e^2 - 9*e - 22, 1/2*e^4 - 9/2*e^3 + 11*e^2 - 9*e - 22, -1/8*e^4 - 1/4*e^3 + 11*e^2 - 14*e - 56, -1/8*e^4 - 1/4*e^3 + 11*e^2 - 14*e - 56, 5/8*e^4 - 17/4*e^3 + 1/2*e^2 + 11*e + 40, -e^4 + 23/4*e^3 + 21/2*e^2 - 43*e - 50, -e^4 + 23/4*e^3 + 21/2*e^2 - 43*e - 50, 5/8*e^4 - 17/4*e^3 + 1/2*e^2 + 11*e + 40, 7/4*e^4 - 14*e^3 + 11*e^2 + 58*e - 6, 7/4*e^4 - 14*e^3 + 11*e^2 + 58*e - 6, -1/2*e^4 + 7/4*e^3 + 25/2*e^2 - 20*e - 62, -1/2*e^4 + 7/4*e^3 + 25/2*e^2 - 20*e - 62, -3/8*e^4 + 13/4*e^3 - 8*e^2 + 10*e + 22, -3/8*e^4 + 13/4*e^3 - 8*e^2 + 10*e + 22, -1/4*e^4 + 7/2*e^3 - 11*e^2 - 15*e + 42, -1/4*e^4 + 9/4*e^3 - 9/2*e^2 - 8, -1/4*e^4 + 9/4*e^3 - 9/2*e^2 - 8, -1/4*e^4 + 7/2*e^3 - 11*e^2 - 15*e + 42, 1/4*e^4 + 3/2*e^3 - 25*e^2 + 28*e + 98, -5/4*e^4 + 12*e^3 - 24*e^2 - 22*e + 48, 1/4*e^4 + 3/2*e^3 - 25*e^2 + 28*e + 98, -5/4*e^4 + 12*e^3 - 24*e^2 - 22*e + 48, -1/4*e^4 + 15*e^2 - 25*e - 52, 1/8*e^4 - 11/4*e^3 + 11*e^2 + 6*e - 52, -1/4*e^4 + 15*e^2 - 25*e - 52, 1/8*e^4 - 11/4*e^3 + 11*e^2 + 6*e - 52, -11/8*e^4 + 43/4*e^3 - 25/2*e^2 - 29*e + 40, -3/4*e^4 + 3*e^3 + 15*e^2 - 33*e - 60, -3/4*e^4 + 3*e^3 + 15*e^2 - 33*e - 60, -11/8*e^4 + 43/4*e^3 - 25/2*e^2 - 29*e + 40, -e^4 + 13/2*e^3 + 3*e^2 - 37*e - 26, -e^4 + 13/2*e^3 + 3*e^2 - 37*e - 26, 1/2*e^4 - 3*e^3 - e^2 + 9*e + 32, 3/8*e^4 - 19/4*e^3 + 23/2*e^2 + 17*e, 3/8*e^4 - 19/4*e^3 + 23/2*e^2 + 17*e, 5/4*e^4 - 8*e^3 - 7/2*e^2 + 39*e + 50, 5/4*e^4 - 8*e^3 - 7/2*e^2 + 39*e + 50, 3/8*e^4 + 7/4*e^3 - 32*e^2 + 36*e + 104, 3/8*e^4 + 7/4*e^3 - 32*e^2 + 36*e + 104, 5/8*e^4 - 25/4*e^3 + 33/2*e^2 - 3*e - 52, 5/8*e^4 - 25/4*e^3 + 33/2*e^2 - 3*e - 52, -e^4 + 9/2*e^3 + 43/2*e^2 - 61*e - 72, -e^4 + 9/2*e^3 + 43/2*e^2 - 61*e - 72, -9/8*e^4 + 35/4*e^3 - 19/2*e^2 - 15*e - 18, -3/4*e^4 + 29/4*e^3 - 23/2*e^2 - 25*e + 2, -9/8*e^4 + 35/4*e^3 - 19/2*e^2 - 15*e - 18, -3/4*e^4 + 29/4*e^3 - 23/2*e^2 - 25*e + 2, -1/2*e^4 + 9/2*e^3 - 9/2*e^2 - 25*e + 4, -1/2*e^4 + 9/2*e^3 - 9/2*e^2 - 25*e + 4, 3/4*e^4 - 19/4*e^3 + 1/2*e^2 + 15*e - 18, 3/8*e^4 - 19/4*e^3 + 10*e^2 + 28*e - 28, 3/8*e^4 - 19/4*e^3 + 10*e^2 + 28*e - 28, 3/4*e^4 - 19/4*e^3 + 1/2*e^2 + 15*e - 18, -1/4*e^4 + 11/4*e^3 - 25/2*e^2 + 24*e + 38, -1/4*e^4 + 11/4*e^3 - 25/2*e^2 + 24*e + 38, -1/2*e^4 + 5/2*e^3 + 19/2*e^2 - 21*e - 56, -1/2*e^4 + 5/2*e^3 + 19/2*e^2 - 21*e - 56, 5/4*e^4 - 11*e^3 + 18*e^2 + 24*e - 48, 5/4*e^4 - 11*e^3 + 18*e^2 + 24*e - 48, 1/4*e^4 - 1/2*e^3 - 10*e^2 + 11*e + 70, 1/4*e^4 - 1/2*e^3 - 10*e^2 + 11*e + 70, 3/4*e^4 - 6*e^3 + 4*e^2 + 32*e + 4, 3/4*e^4 - 6*e^3 + 4*e^2 + 32*e + 4, 1/8*e^4 - 5/4*e^3 + 7*e^2 - 10*e - 40, 1/8*e^4 - 5/4*e^3 + 7*e^2 - 10*e - 40, -e^4 + 17/2*e^3 - 11*e^2 - 31*e + 22, -e^4 + 17/2*e^3 - 11*e^2 - 31*e + 22, 3/4*e^4 - 17/2*e^3 + 23*e^2 + 6*e - 30, 3/4*e^4 - 17/2*e^3 + 23*e^2 + 6*e - 30, 1/4*e^4 - 3*e^3 + 12*e^2 - 11*e - 32, 1/4*e^4 - 3*e^3 + 12*e^2 - 11*e - 32, 1/2*e^4 - 6*e^3 + 25*e^2 - 23*e - 72, 1/2*e^4 - 6*e^3 + 25*e^2 - 23*e - 72, 1/2*e^4 - 9/2*e^3 + 7*e^2 + 20*e - 6, 1/2*e^4 - 9/2*e^3 + 7*e^2 + 20*e - 6, 2*e^4 - 29/2*e^3 + 3*e^2 + 63*e + 38, 2*e^4 - 29/2*e^3 + 3*e^2 + 63*e + 38, 3/8*e^4 - 15/4*e^3 + 5/2*e^2 + 37*e - 26, 3/8*e^4 - 15/4*e^3 + 5/2*e^2 + 37*e - 26, -3/8*e^4 + 25/4*e^3 - 63/2*e^2 + 29*e + 94, -3/8*e^4 + 25/4*e^3 - 63/2*e^2 + 29*e + 94, e^4 - 17/2*e^3 + 14*e^2 + 19*e - 26, e^4 - 17/2*e^3 + 14*e^2 + 19*e - 26, -5/4*e^4 + 41/4*e^3 - 17/2*e^2 - 51*e + 2, -5/4*e^4 + 41/4*e^3 - 17/2*e^2 - 51*e + 2, -7/4*e^4 + 16*e^3 - 30*e^2 - 34*e + 30, -7/4*e^4 + 16*e^3 - 30*e^2 - 34*e + 30, -7/4*e^4 + 31/2*e^3 - 51/2*e^2 - 43*e + 44, -7/4*e^4 + 31/2*e^3 - 51/2*e^2 - 43*e + 44, -7/8*e^4 + 31/4*e^3 - 31/2*e^2 - 3*e + 42, -7/8*e^4 + 31/4*e^3 - 31/2*e^2 - 3*e + 42, 9/8*e^4 - 35/4*e^3 + 17/2*e^2 + 21*e - 16, 9/8*e^4 - 35/4*e^3 + 17/2*e^2 + 21*e - 16, e^4 - 15/2*e^3 + 15/2*e^2 + 21*e - 6, e^4 - 15/2*e^3 + 15/2*e^2 + 21*e - 6, 3/4*e^4 - 7/2*e^3 - 29/2*e^2 + 45*e + 68, 3/4*e^4 - 7/2*e^3 - 29/2*e^2 + 45*e + 68, -9/8*e^4 + 39/4*e^3 - 27/2*e^2 - 27*e + 28, -9/8*e^4 + 39/4*e^3 - 27/2*e^2 - 27*e + 28, -5/4*e^4 + 43/4*e^3 - 31/2*e^2 - 24*e + 22, -1/2*e^4 + 7*e^3 - 55/2*e^2 - e + 92, -5/4*e^4 + 43/4*e^3 - 31/2*e^2 - 24*e + 22, -1/2*e^4 + 7*e^3 - 55/2*e^2 - e + 92, -3/4*e^4 + 39/4*e^3 - 65/2*e^2 + 82, e^4 - 7*e^3 - 2*e^2 + 30*e + 62, -3/4*e^4 + 39/4*e^3 - 65/2*e^2 + 82, e^4 - 7*e^3 - 2*e^2 + 30*e + 62, 15/8*e^4 - 45/4*e^3 - 21/2*e^2 + 59*e + 98, 15/8*e^4 - 45/4*e^3 - 21/2*e^2 + 59*e + 98, -3/2*e^4 + 41/4*e^3 - 3/2*e^2 - 44*e - 10, -3/2*e^4 + 41/4*e^3 - 3/2*e^2 - 44*e - 10, 3/2*e^4 - 12*e^3 + 11*e^2 + 55*e - 36, 3/2*e^4 - 12*e^3 + 11*e^2 + 55*e - 36, 1/8*e^4 - 3/4*e^3 - 5/2*e^2 + 3*e + 28, 1/8*e^4 - 3/4*e^3 - 5/2*e^2 + 3*e + 28, 1/4*e^4 - 5*e^3 + 30*e^2 - 23*e - 116, 1/4*e^4 - 5*e^3 + 30*e^2 - 23*e - 116, -7/4*e^4 + 21/2*e^3 + 9*e^2 - 60*e - 40, -1/2*e^4 + 15/4*e^3 + 1/2*e^2 - 39*e - 10, -7/4*e^4 + 21/2*e^3 + 9*e^2 - 60*e - 40, -1/2*e^4 + 15/4*e^3 + 1/2*e^2 - 39*e - 10, 7/4*e^4 - 23/2*e^3 - 2*e^2 + 44*e + 82, 7/4*e^4 - 23/2*e^3 - 2*e^2 + 44*e + 82, -7/4*e^4 + 13*e^3 - 7*e^2 - 56*e + 10, -7/4*e^4 + 13*e^3 - 7*e^2 - 56*e + 10, -3/2*e^4 + 31/2*e^3 - 35*e^2 - 29*e + 62, -3/2*e^4 + 31/2*e^3 - 35*e^2 - 29*e + 62, 5/8*e^4 - 7/4*e^3 - 17*e^2 + 36*e + 22, 5/8*e^4 - 7/4*e^3 - 17*e^2 + 36*e + 22, 5/4*e^4 - 11*e^3 + 35/2*e^2 + 31*e - 18, 5/4*e^4 - 11*e^3 + 35/2*e^2 + 31*e - 18, 1/2*e^4 - 5*e^3 + 23/2*e^2 + 9*e - 56, 1/2*e^4 - 5*e^3 + 23/2*e^2 + 9*e - 56, 3/8*e^4 - 5/4*e^3 - 13/2*e^2 + 7*e + 40, 2*e^4 - 37/2*e^3 + 39*e^2 + 25*e - 90, 2*e^4 - 37/2*e^3 + 39*e^2 + 25*e - 90, 3/8*e^4 - 5/4*e^3 - 13/2*e^2 + 7*e + 40, 11/8*e^4 - 29/4*e^3 - 18*e^2 + 72*e + 50, 11/8*e^4 - 29/4*e^3 - 18*e^2 + 72*e + 50, 5/8*e^4 - 15/4*e^3 + e^2 - 2*e + 24, -e^4 + 14*e^3 - 49*e^2 - 5*e + 104, -e^4 + 14*e^3 - 49*e^2 - 5*e + 104, 5/8*e^4 - 15/4*e^3 + e^2 - 2*e + 24, -27/8*e^4 + 115/4*e^3 - 41*e^2 - 70*e + 52, -27/8*e^4 + 115/4*e^3 - 41*e^2 - 70*e + 52, -1/2*e^3 + 3*e^2 - 9*e + 28, -1/2*e^3 + 3*e^2 - 9*e + 28, 15/8*e^4 - 57/4*e^3 + 23/2*e^2 + 49*e + 4, 1/2*e^4 - 13/2*e^3 + 21*e^2 + 2*e - 56, 15/8*e^4 - 57/4*e^3 + 23/2*e^2 + 49*e + 4, 1/2*e^4 - 13/2*e^3 + 21*e^2 + 2*e - 56, -9/4*e^4 + 20*e^3 - 77/2*e^2 - 25*e + 72, -9/4*e^4 + 20*e^3 - 77/2*e^2 - 25*e + 72, -11/8*e^4 + 39/4*e^3 - 1/2*e^2 - 33*e - 60, 7/4*e^4 - 27/2*e^3 + 8*e^2 + 58*e - 30, 7/4*e^4 - 27/2*e^3 + 8*e^2 + 58*e - 30, -11/8*e^4 + 39/4*e^3 - 1/2*e^2 - 33*e - 60, -3/4*e^4 + 8*e^3 - 21*e^2 - 11*e + 44, -3/4*e^4 + 8*e^3 - 21*e^2 - 11*e + 44, 3*e^4 - 25*e^3 + 34*e^2 + 62*e - 36, 3*e^4 - 25*e^3 + 34*e^2 + 62*e - 36, -2*e^4 + 27/2*e^3 + 4*e^2 - 86*e - 8, -2*e^4 + 27/2*e^3 + 4*e^2 - 86*e - 8, 1/2*e^4 - 3*e^3 - 3*e^2 + 27*e - 10, 1/2*e^4 - 3*e^3 - 3*e^2 + 27*e - 10, -13/8*e^4 + 37/4*e^3 + 25/2*e^2 - 53*e - 92, -13/8*e^4 + 37/4*e^3 + 25/2*e^2 - 53*e - 92, -11/8*e^4 + 47/4*e^3 - 10*e^2 - 50*e - 18, 5/4*e^4 - 49/4*e^3 + 55/2*e^2 + 14*e - 28, 5/4*e^4 - 49/4*e^3 + 55/2*e^2 + 14*e - 28, -11/8*e^4 + 47/4*e^3 - 10*e^2 - 50*e - 18, -3/2*e^4 + 23/2*e^3 - 6*e^2 - 43*e - 26, -3/2*e^4 + 23/2*e^3 - 6*e^2 - 43*e - 26, 1/8*e^4 - 21/4*e^3 + 67/2*e^2 - 25*e - 62, 1/8*e^4 - 21/4*e^3 + 67/2*e^2 - 25*e - 62, -5/4*e^4 + 14*e^3 - 48*e^2 + 28*e + 150, -2*e^4 + 14*e^3 - 3*e^2 - 48*e - 68, -2*e^4 + 14*e^3 - 3*e^2 - 48*e - 68, -7/4*e^4 + 31/2*e^3 - 51/2*e^2 - 35*e + 2, -7/4*e^4 + 31/2*e^3 - 51/2*e^2 - 35*e + 2, -2*e^4 + 17*e^3 - 27*e^2 - 32*e + 34, -2*e^4 + 17*e^3 - 27*e^2 - 32*e + 34, -11/4*e^4 + 43/2*e^3 - 22*e^2 - 68*e + 30, -11/4*e^4 + 43/2*e^3 - 22*e^2 - 68*e + 30, 3/2*e^4 - 10*e^3 - 5/2*e^2 + 57*e + 32, 3/2*e^4 - 10*e^3 - 5/2*e^2 + 57*e + 32, -1/4*e^4 + 19/4*e^3 - 41/2*e^2 - 6*e + 48, -5/8*e^4 + 21/4*e^3 - 11/2*e^2 - 9*e - 12, -1/4*e^4 + 19/4*e^3 - 41/2*e^2 - 6*e + 48, -5/8*e^4 + 21/4*e^3 - 11/2*e^2 - 9*e - 12, e^4 - 8*e^3 - e^2 + 73*e, -11/8*e^4 + 41/4*e^3 - 5*e^2 - 48*e - 20, -11/8*e^4 + 41/4*e^3 - 5*e^2 - 48*e - 20, e^4 - 8*e^3 - e^2 + 73*e, 13/8*e^4 - 59/4*e^3 + 29*e^2 + 12*e - 66, 13/8*e^4 - 59/4*e^3 + 29*e^2 + 12*e - 66, -3/2*e^4 + 11*e^3 - 4*e^2 - 56*e + 20, -3/2*e^4 + 11*e^3 - 4*e^2 - 56*e + 20, 1/8*e^4 + 9/4*e^3 - 14*e^2 - 16*e + 40, 1/8*e^4 + 9/4*e^3 - 14*e^2 - 16*e + 40, -e^4 + 2*e^3 + 36*e^2 - 60*e - 106, -e^4 + 2*e^3 + 36*e^2 - 60*e - 106, -3/2*e^4 + 14*e^3 - 25*e^2 - 38*e + 34, -3/2*e^4 + 14*e^3 - 25*e^2 - 38*e + 34, -3/4*e^4 + 19/2*e^3 - 30*e^2 - e + 50, -3/4*e^4 + 19/2*e^3 - 30*e^2 - e + 50, 3/2*e^4 - 33/2*e^3 + 44*e^2 + 17*e - 110, 3/2*e^4 - 33/2*e^3 + 44*e^2 + 17*e - 110];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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