/*
  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![2, 1, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {24, 6, -w^3 + 2*w^2 + 3*w - 1}>;

primesArray := [
[2, 2, -w],
[3, 3, w - 1],
[8, 2, w^3 - 2*w^2 - 5*w + 1],
[13, 13, -w^2 + 2*w + 3],
[17, 17, -w^2 + 3*w + 3],
[17, 17, -w^2 + 2*w + 1],
[27, 3, w^3 - w^2 - 6*w - 5],
[29, 29, w^3 - 2*w^2 - 2*w + 1],
[47, 47, w^3 - 2*w^2 - 4*w - 3],
[49, 7, w^3 - 3*w^2 - 3*w + 1],
[49, 7, -2*w^3 + 5*w^2 + 7*w - 3],
[59, 59, -2*w^3 + 6*w^2 + 5*w - 7],
[61, 61, 2*w^3 - 4*w^2 - 9*w - 3],
[71, 71, w^3 - 3*w^2 - 3*w + 7],
[73, 73, 2*w^3 - 6*w^2 - 3*w + 5],
[73, 73, w^3 - 3*w^2 - 4*w + 5],
[83, 83, -2*w^3 + 5*w^2 + 8*w - 3],
[89, 89, -2*w^3 + 6*w^2 + 4*w - 7],
[89, 89, w^3 - 2*w^2 - 6*w + 3],
[97, 97, -3*w - 1],
[101, 101, w^2 - w - 5],
[101, 101, -w^3 + w^2 + 6*w + 7],
[103, 103, w^3 - 3*w^2 - 2*w - 1],
[107, 107, -w^3 + w^2 + 8*w - 1],
[109, 109, -3*w^3 + 7*w^2 + 11*w - 7],
[131, 131, w^2 - 4*w - 1],
[131, 131, -2*w^2 + 2*w + 1],
[131, 131, w^3 - w^2 - 7*w - 1],
[131, 131, -w^3 + 4*w^2 - 5],
[137, 137, 2*w^3 - 5*w^2 - 9*w + 7],
[139, 139, w^3 - 8*w - 7],
[139, 139, -3*w + 1],
[149, 149, -3*w^3 + 8*w^2 + 8*w - 3],
[149, 149, -2*w^3 + 3*w^2 + 12*w + 3],
[151, 151, 2*w^3 - 4*w^2 - 10*w + 1],
[151, 151, -w^3 + 3*w^2 + w - 5],
[151, 151, -2*w^3 + 6*w^2 + 5*w - 11],
[151, 151, 2*w - 3],
[157, 157, 2*w^3 - 7*w^2 + 1],
[163, 163, -w^3 + w^2 + 6*w + 1],
[173, 173, 3*w^3 - 9*w^2 - 7*w + 9],
[173, 173, -2*w^3 + 7*w^2 + 2*w - 5],
[181, 181, -w^2 + 5*w + 3],
[191, 191, -3*w^3 + 8*w^2 + 10*w - 7],
[193, 193, -2*w^3 + 5*w^2 + 5*w - 3],
[193, 193, 2*w^2 - 4*w - 5],
[223, 223, 2*w^3 - 4*w^2 - 9*w + 3],
[227, 227, -3*w^3 + 7*w^2 + 12*w - 9],
[227, 227, 2*w^3 - 4*w^2 - 7*w - 1],
[241, 241, w^2 - 2*w - 7],
[257, 257, -2*w^3 + 7*w^2 + 2*w - 9],
[257, 257, w^2 - 4*w - 5],
[263, 263, -3*w^3 + 7*w^2 + 12*w - 3],
[269, 269, w^3 - 3*w^2 - 4*w + 7],
[271, 271, 3*w^3 - 10*w^2 - 4*w + 7],
[271, 271, -w^3 + 3*w^2 - 2*w - 1],
[277, 277, -2*w^3 + 6*w^2 + 4*w - 9],
[277, 277, -6*w^3 + 14*w^2 + 22*w - 5],
[281, 281, -2*w^3 + 6*w^2 + 2*w - 5],
[289, 17, w^3 - 2*w^2 + 3],
[293, 293, w^3 - 3*w^2 + w - 1],
[293, 293, 3*w^3 - 8*w^2 - 8*w + 11],
[311, 311, -w^3 + 4*w^2 - 2*w - 5],
[313, 313, 2*w^2 - 4*w - 11],
[313, 313, 3*w^3 - 7*w^2 - 11*w + 1],
[317, 317, -w^3 + w^2 + 5*w - 1],
[317, 317, -2*w^3 + 4*w^2 + 10*w - 7],
[331, 331, w^3 - 5*w^2 - 3*w + 3],
[331, 331, -3*w^3 + 6*w^2 + 14*w - 3],
[337, 337, 3*w^3 - 7*w^2 - 10*w + 1],
[337, 337, w^3 - w^2 - 5*w - 9],
[353, 353, -w^3 + 2*w^2 + 1],
[359, 359, -2*w^2 + 4*w - 1],
[359, 359, 4*w^3 - 10*w^2 - 14*w + 7],
[367, 367, -4*w^3 + 10*w^2 + 13*w - 9],
[379, 379, -2*w^3 + 8*w^2 + w - 17],
[379, 379, w^3 - 4*w^2 - 2*w + 9],
[397, 397, 2*w^3 - 5*w^2 - 9*w + 1],
[401, 401, w^2 - 5],
[401, 401, 2*w^3 - 5*w^2 - 8*w + 1],
[431, 431, -3*w^3 + 3*w^2 + 21*w + 13],
[439, 439, -2*w^2 + 3*w + 13],
[449, 449, 2*w^3 - 4*w^2 - 7*w - 5],
[449, 449, 5*w^3 - 10*w^2 - 22*w - 1],
[457, 457, 2*w^3 - 7*w^2 - 5*w + 5],
[461, 461, 3*w^3 - 5*w^2 - 14*w - 1],
[461, 461, -2*w^3 + 3*w^2 + 11*w + 1],
[463, 463, 2*w^3 - 4*w^2 - 6*w + 1],
[463, 463, 2*w^3 - 5*w^2 - 5*w + 9],
[479, 479, 2*w^2 - 5*w - 1],
[509, 509, -6*w^3 + 10*w^2 + 31*w + 11],
[523, 523, -w^2 + 5*w + 1],
[523, 523, 3*w^3 - 6*w^2 - 14*w - 3],
[547, 547, -3*w^3 + 6*w^2 + 12*w - 1],
[563, 563, 2*w^2 - 6*w - 7],
[563, 563, -w^3 + 2*w^2 + 4*w + 5],
[571, 571, w^2 - 2*w + 3],
[571, 571, 3*w^2 - 8*w - 9],
[577, 577, -3*w^3 + 7*w^2 + 10*w - 3],
[577, 577, -2*w^2 + 5*w - 1],
[601, 601, w^3 - 2*w^2 - 8*w + 7],
[601, 601, 4*w^3 - 9*w^2 - 17*w + 3],
[607, 607, -3*w^3 + 6*w^2 + 12*w + 1],
[613, 613, -2*w^3 + 6*w^2 + 6*w - 3],
[617, 617, 3*w^3 - 9*w^2 - 6*w + 11],
[617, 617, -2*w^3 + 5*w^2 + 7*w + 3],
[619, 619, -3*w^3 + 5*w^2 + 17*w + 1],
[625, 5, -5],
[631, 631, -3*w^2 + 6*w + 11],
[643, 643, -2*w^3 + 7*w^2 + 4*w - 7],
[643, 643, -2*w^3 + 7*w^2 + 3*w - 7],
[653, 653, 3*w^3 - 7*w^2 - 11*w - 1],
[661, 661, 2*w^2 - 7*w - 3],
[661, 661, -w^3 + 5*w^2 - w - 5],
[673, 673, -3*w^3 + 9*w^2 + 8*w - 7],
[677, 677, -4*w^3 + 9*w^2 + 15*w - 3],
[677, 677, -2*w^3 + 7*w^2 - 7],
[683, 683, w^3 - 4*w^2 - 1],
[683, 683, w^3 - 4*w^2 - 2*w + 13],
[709, 709, -3*w^3 + 9*w^2 + 5*w - 9],
[709, 709, 4*w^3 - 12*w^2 - 7*w + 5],
[719, 719, -2*w^3 + 7*w^2 + 4*w - 11],
[719, 719, -5*w^3 + 8*w^2 + 28*w + 9],
[733, 733, -3*w^3 + 8*w^2 + 6*w - 3],
[733, 733, -4*w^3 + 8*w^2 + 20*w + 1],
[739, 739, -4*w^3 + 8*w^2 + 20*w - 7],
[739, 739, -w^2 - 4*w + 1],
[751, 751, 3*w^3 - 7*w^2 - 14*w + 1],
[757, 757, -4*w^3 + 3*w^2 + 29*w + 21],
[761, 761, -w^3 + 5*w^2 - 2*w - 9],
[773, 773, -w^2 + 4*w - 5],
[787, 787, 3*w^3 - 5*w^2 - 17*w - 3],
[787, 787, -w^3 + 3*w^2 - 4*w + 1],
[787, 787, -3*w^3 + 8*w^2 + 10*w - 5],
[787, 787, 4*w^3 - 8*w^2 - 18*w - 3],
[809, 809, -2*w^3 + 7*w^2 + 3*w - 9],
[811, 811, -3*w^2 + 5*w + 15],
[811, 811, -4*w^3 + 11*w^2 + 12*w - 9],
[821, 821, -7*w^3 + 15*w^2 + 29*w - 3],
[823, 823, w^2 + 3],
[827, 827, -2*w^3 + 4*w^2 + 11*w - 3],
[827, 827, 6*w^3 - 12*w^2 - 26*w - 3],
[829, 829, -3*w^3 + 7*w^2 + 9*w - 5],
[859, 859, 3*w^2 - 6*w - 7],
[859, 859, -w^3 + w^2 + 2*w + 3],
[863, 863, w^3 - 3*w^2 - 2*w - 3],
[877, 877, -2*w^3 + w^2 + 14*w + 7],
[877, 877, 5*w^3 - 11*w^2 - 22*w + 3],
[877, 877, 4*w^3 - 12*w^2 - 10*w + 13],
[877, 877, -2*w^3 + 6*w^2 - 5],
[883, 883, 2*w^3 - 4*w^2 - 12*w + 7],
[887, 887, -5*w^3 + 13*w^2 + 15*w - 7],
[947, 947, w^3 + w^2 - 10*w - 9],
[947, 947, -4*w^3 + 13*w^2 + 8*w - 21],
[953, 953, 3*w - 5],
[967, 967, -4*w^3 + 11*w^2 + 11*w - 7],
[967, 967, 5*w^3 - 15*w^2 - 7*w + 7],
[971, 971, -w^3 + w^2 + 9*w + 1],
[977, 977, -6*w^3 + 11*w^2 + 28*w + 7],
[983, 983, 3*w^3 - 9*w^2 - 3*w + 7],
[997, 997, -4*w^3 + 14*w^2 + 5*w - 23],
[1009, 1009, -3*w^3 + 11*w^2 + w - 13],
[1009, 1009, -2*w^3 + 5*w^2 + 11*w - 1],
[1013, 1013, w^3 - 5*w^2 + 3*w + 11],
[1031, 1031, w^3 - 3*w^2 - 3*w - 3],
[1033, 1033, -3*w^3 + 7*w^2 + 6*w - 5],
[1061, 1061, -2*w^3 + 9*w^2 - 3*w - 15],
[1063, 1063, 5*w^3 - 7*w^2 - 30*w - 11],
[1069, 1069, -2*w^3 + 6*w^2 + 2*w - 7],
[1091, 1091, 2*w^2 - 6*w - 9],
[1091, 1091, 4*w^3 - 11*w^2 - 14*w + 17],
[1093, 1093, -2*w^3 + w^2 + 16*w + 7],
[1097, 1097, -4*w^3 + 12*w^2 + 11*w - 15],
[1103, 1103, -2*w^3 + 2*w^2 + 11*w + 9],
[1103, 1103, 3*w^3 - 7*w^2 - 13*w + 1],
[1117, 1117, -5*w^3 + 7*w^2 + 27*w + 15],
[1123, 1123, 2*w^3 - 5*w^2 - 10*w + 5],
[1129, 1129, -5*w^3 + 9*w^2 + 23*w + 7],
[1129, 1129, 6*w^3 - 10*w^2 - 32*w - 11],
[1129, 1129, -3*w^3 + 7*w^2 + 9*w - 3],
[1129, 1129, -5*w^3 + 11*w^2 + 19*w - 9],
[1153, 1153, 4*w^2 - 7*w - 19],
[1163, 1163, -2*w^3 + 3*w^2 + 14*w + 5],
[1193, 1193, -3*w^3 + 9*w^2 + 9*w - 7],
[1201, 1201, -w^2 + 6*w - 1],
[1217, 1217, 2*w^3 - 2*w^2 - 14*w - 3],
[1223, 1223, -w^3 + 3*w^2 + 2*w - 9],
[1229, 1229, 4*w^3 - 9*w^2 - 16*w + 1],
[1231, 1231, 5*w^2 - 10*w - 21],
[1231, 1231, -2*w^3 + 7*w^2 + 5*w - 3],
[1249, 1249, -2*w^3 + 5*w^2 + 3*w - 7],
[1259, 1259, -2*w^3 + 3*w^2 + 10*w - 1],
[1277, 1277, -4*w^3 + 8*w^2 + 16*w + 3],
[1277, 1277, 3*w^3 - 7*w^2 - 8*w + 1],
[1283, 1283, -3*w^3 + 6*w^2 + 16*w + 1],
[1289, 1289, 3*w^3 - 9*w^2 - 6*w + 13],
[1289, 1289, -2*w^2 + 8*w - 1],
[1289, 1289, -5*w^3 + 15*w^2 + 13*w - 21],
[1289, 1289, 2*w^3 - 6*w^2 - 5*w + 13],
[1291, 1291, -7*w^3 + 11*w^2 + 39*w + 13],
[1303, 1303, 4*w^3 - 10*w^2 - 17*w + 13],
[1307, 1307, 2*w^3 - 8*w^2 + 7],
[1319, 1319, -w^3 + 5*w^2 + w - 7],
[1319, 1319, -5*w^3 + 10*w^2 + 24*w - 3],
[1327, 1327, -2*w^3 + 3*w^2 + 11*w - 1],
[1367, 1367, -4*w^3 + 11*w^2 + 8*w - 7],
[1367, 1367, 3*w^3 - 7*w^2 - 12*w - 1],
[1373, 1373, 4*w^3 - 11*w^2 - 7*w + 7],
[1373, 1373, 3*w^2 - 4*w - 9],
[1381, 1381, -w^3 + 5*w^2 - w - 17],
[1381, 1381, -4*w^3 + 8*w^2 + 20*w - 5],
[1399, 1399, w^2 - 2*w - 9],
[1399, 1399, -6*w^3 + 10*w^2 + 30*w + 9],
[1423, 1423, -6*w^3 + 16*w^2 + 16*w - 15],
[1427, 1427, 3*w^3 - 3*w^2 - 21*w - 7],
[1429, 1429, -8*w^3 + 12*w^2 + 43*w + 17],
[1429, 1429, 5*w^3 - 14*w^2 - 10*w + 5],
[1433, 1433, -w^3 + 5*w^2 - 9],
[1433, 1433, -w^3 + w^2 + w + 3],
[1433, 1433, 4*w^3 - 14*w^2 - 3*w + 9],
[1433, 1433, -w^3 + 5*w^2 - 5*w + 3],
[1439, 1439, -2*w^3 + 8*w^2 - w - 13],
[1451, 1451, 3*w^2 - 5*w - 9],
[1459, 1459, -3*w^3 + 13*w^2 + w - 7],
[1459, 1459, -4*w^3 + 11*w^2 + 13*w - 9],
[1471, 1471, -5*w^3 + 8*w^2 + 28*w + 7],
[1481, 1481, 3*w^3 - 6*w^2 - 10*w - 1],
[1483, 1483, w^3 - 8*w - 13],
[1489, 1489, 2*w^3 - 4*w^2 - 2*w + 5],
[1499, 1499, -w^3 + 2*w^2 + 8*w - 1],
[1511, 1511, 3*w^2 - 4*w - 7],
[1511, 1511, -6*w^3 + 15*w^2 + 19*w - 15],
[1543, 1543, 4*w^3 - 3*w^2 - 28*w - 17],
[1543, 1543, 4*w - 5],
[1549, 1549, 3*w^2 - 7*w - 3],
[1553, 1553, 3*w^3 - 10*w^2 - 4*w + 15],
[1553, 1553, -w^3 - 3*w^2 + 14*w + 21],
[1567, 1567, -6*w^3 + 14*w^2 + 22*w - 7],
[1571, 1571, -w^3 - w^2 + 10*w + 17],
[1579, 1579, -w^3 + 6*w + 17],
[1579, 1579, 3*w^3 - 9*w^2 - 7*w + 3],
[1597, 1597, 2*w^3 - w^2 - 18*w - 5],
[1607, 1607, w^3 - 5*w^2 + 17],
[1609, 1609, -3*w^3 + 7*w^2 + 14*w - 5],
[1621, 1621, w^2 - 3*w - 9],
[1627, 1627, 5*w^3 - 13*w^2 - 14*w + 3],
[1627, 1627, -w^3 + 6*w^2 - 4*w + 1],
[1663, 1663, 6*w^3 - 11*w^2 - 31*w - 5],
[1663, 1663, 6*w^3 - 18*w^2 - 15*w + 25],
[1667, 1667, -w^3 + 2*w^2 + 4*w - 7],
[1667, 1667, 4*w^3 - 14*w^2 - 7*w + 31],
[1669, 1669, 3*w^2 - 8*w - 11],
[1669, 1669, -4*w^3 + 11*w^2 + 10*w - 15],
[1681, 41, 6*w^3 - 11*w^2 - 31*w + 1],
[1681, 41, 4*w^3 - 13*w^2 - 8*w + 9],
[1693, 1693, w^3 - w^2 - 7*w - 9],
[1693, 1693, w^3 - 2*w^2 - 3],
[1693, 1693, w^3 + w^2 - 11*w - 5],
[1693, 1693, -2*w^3 + 4*w^2 + 2*w + 1],
[1697, 1697, -4*w^3 + 12*w^2 + w - 7],
[1699, 1699, -7*w^3 + 19*w^2 + 19*w - 17],
[1699, 1699, 6*w^3 - 14*w^2 - 21*w + 3],
[1709, 1709, -w^3 + 8*w + 1],
[1709, 1709, 2*w^3 - 4*w^2 - 6*w - 5],
[1723, 1723, -2*w^3 + 7*w^2 + w - 11],
[1723, 1723, 4*w^3 - 9*w^2 - 14*w + 5],
[1733, 1733, -w^3 + w^2 + 3*w - 5],
[1733, 1733, -w^2 - 5*w + 1],
[1741, 1741, -3*w^3 + 10*w^2 + 6*w - 15],
[1741, 1741, -3*w^3 + 9*w^2 + 4*w - 9],
[1747, 1747, 2*w^3 - 2*w^2 - 13*w - 3],
[1753, 1753, 4*w^2 - 8*w - 15],
[1753, 1753, -w^3 + 7*w^2 - 4*w - 27],
[1759, 1759, -7*w^3 + 17*w^2 + 27*w - 21],
[1801, 1801, -3*w^3 + 7*w^2 + 6*w - 3],
[1811, 1811, -4*w^2 + 7*w + 5],
[1811, 1811, w^3 - w^2 - 4*w - 7],
[1823, 1823, -7*w^3 + 21*w^2 + 18*w - 31],
[1831, 1831, -2*w^2 + 3*w - 3],
[1847, 1847, 5*w^3 - 15*w^2 - 12*w + 17],
[1861, 1861, -4*w^3 + 8*w^2 + 14*w - 5],
[1861, 1861, -4*w^2 + 9*w + 11],
[1871, 1871, -4*w^3 + 10*w^2 + 11*w - 3],
[1873, 1873, -2*w^3 + 7*w^2 + 6*w - 3],
[1879, 1879, w^3 - 7*w^2 + 8*w + 11],
[1879, 1879, -2*w^3 + 19*w + 15],
[1889, 1889, -2*w^3 + 6*w^2 + w - 7],
[1933, 1933, 2*w^3 - 3*w^2 - 12*w + 3],
[1949, 1949, -4*w^3 + 15*w^2 + 4*w - 7],
[1951, 1951, 2*w^2 + 4*w - 5],
[1973, 1973, 6*w^3 - 14*w^2 - 25*w + 7],
[1993, 1993, -4*w^3 + 11*w^2 + 6*w - 5],
[1997, 1997, -4*w^3 + 9*w^2 + 14*w - 3],
[1997, 1997, 5*w^3 - 11*w^2 - 20*w + 9],
[1999, 1999, -3*w^3 + 11*w^2 + 3*w - 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [-1, -1, 1, -4, 4, 0, 2, -6, -4, -6, -10, 4, -10, -8, 6, 12, -6, -14, 10, -12, 2, 4, -8, -12, 10, 16, 0, 8, 4, 2, -18, 20, 6, -20, -8, -8, 0, 0, 18, 14, 16, 26, 10, -18, -12, 6, 8, -8, 2, -14, -18, -18, 6, -22, -14, 16, 14, -2, 0, 6, 6, -30, -18, -16, -30, -12, 30, -20, 28, 28, 30, -32, 24, 6, 24, -10, 10, 12, 0, -10, -36, 26, 18, -26, -4, 30, -30, -26, 16, 8, 12, 4, -4, -38, -44, -20, 44, 0, -10, -16, 10, -18, 16, 20, -2, 22, 42, -46, -6, -8, 28, -30, -18, -8, 32, -46, 34, -20, 14, 2, -34, -18, -36, 22, 6, -40, 52, -32, 26, -34, 30, -24, -32, -20, 8, -56, -12, 20, -54, -24, -32, -48, 34, -56, 28, -36, -6, -52, 38, 8, -28, 16, -12, 12, -26, 32, 4, 54, -32, -22, 50, -22, -34, -4, 32, -60, -38, -36, -50, -32, 24, 58, 42, -44, -20, 2, -28, -14, -22, 32, 10, -36, 24, 42, -24, 2, 40, -6, -32, -56, 8, -28, -30, 26, -16, -14, 40, 4, 40, -28, -36, -12, -32, -24, -22, -60, 22, -58, -18, -22, -28, 32, -8, 4, 34, -2, -4, 18, -20, 58, 74, -16, 0, -4, -44, -4, 54, -28, -68, -68, 12, 36, 4, 30, 4, -18, -14, -56, -60, -12, -60, 42, 68, -2, 60, 44, 20, 64, -4, -14, 20, -62, 34, 66, 34, 30, -62, 54, 30, -26, -44, 14, -8, -14, -14, -40, -32, 46, 14, 42, -14, -44, -54, -66, -10, -44, -20, 20, 40, -12, 16, -10, -46, 62, 8, 8, -14, 46, 50, 70, -58, -4, -50, -18, -8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w - 1}>] := 1;
ALEigenvalues[ideal<ZF | {8, 2, w^3 - 2*w^2 - 5*w + 1}>] := -1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;