/*
  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![6, 0, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {22, 22, -w^3 + w^2 + 3*w - 2}>;

primesArray := [
[2, 2, -w^2 + w + 2],
[3, 3, w^2 - w - 3],
[11, 11, -w^2 + w + 1],
[11, 11, -w^2 - w + 1],
[13, 13, w^3 - 4*w + 1],
[13, 13, -w^3 + 4*w + 1],
[25, 5, -w^2 - 2*w + 1],
[25, 5, w^2 - 2*w - 1],
[37, 37, w^3 - 3*w - 1],
[37, 37, w^3 - 3*w + 1],
[59, 59, w^2 - w - 5],
[59, 59, -w^2 - w + 5],
[61, 61, -w^3 + w^2 + 4*w - 7],
[61, 61, w^3 - 3*w^2 - 6*w + 11],
[73, 73, 2*w^2 - w - 5],
[73, 73, 2*w - 1],
[73, 73, -2*w - 1],
[73, 73, 2*w^2 + w - 5],
[83, 83, 2*w^3 + w^2 - 9*w - 7],
[83, 83, -2*w^3 + w^2 + 7*w + 1],
[107, 107, w^3 + w^2 - 4*w - 1],
[107, 107, -w^3 + w^2 + 4*w - 1],
[109, 109, -w^3 + 2*w^2 + 5*w - 11],
[109, 109, w^3 + 2*w^2 - 5*w - 11],
[121, 11, 2*w^2 - 5],
[131, 131, 3*w^3 - 2*w^2 - 14*w + 11],
[131, 131, 3*w^3 - 12*w + 5],
[157, 157, -3*w^3 + 2*w^2 + 13*w - 11],
[157, 157, -w^3 + 2*w^2 + 7*w - 5],
[167, 167, w^2 + 2*w - 5],
[167, 167, -w^3 + w^2 + w + 1],
[167, 167, 3*w^3 - w^2 - 9*w - 5],
[167, 167, w^2 - 2*w - 5],
[169, 13, -w^2 + 7],
[179, 179, -w^3 + w^2 + 6*w - 1],
[179, 179, w^3 + w^2 - 6*w - 1],
[181, 181, -w^3 + 5*w - 5],
[181, 181, w^3 - 5*w - 5],
[191, 191, -2*w^3 + 2*w^2 + 6*w - 1],
[191, 191, -2*w^2 + 2*w + 7],
[191, 191, -4*w^2 + 4*w + 11],
[191, 191, -2*w^3 + 8*w + 5],
[227, 227, -w^3 + 2*w^2 + 5*w - 5],
[227, 227, w^3 + 2*w^2 - 5*w - 5],
[229, 229, -w^3 + 3*w + 5],
[229, 229, -w^3 - 4*w^2 + 7*w + 13],
[239, 239, -w^3 + w^2 + 5*w - 1],
[239, 239, 2*w^3 - 4*w^2 - 12*w + 13],
[239, 239, 2*w^3 - 2*w^2 - 10*w + 7],
[239, 239, w^3 + w^2 - 5*w - 1],
[241, 241, 2*w^3 - w^2 - 8*w + 5],
[241, 241, -3*w^2 + 4*w + 7],
[241, 241, -2*w^3 + w^2 + 6*w + 1],
[241, 241, -2*w^3 - w^2 + 8*w + 5],
[251, 251, -w^3 + 2*w - 5],
[251, 251, 3*w^3 - 2*w^2 - 12*w + 13],
[263, 263, 2*w^3 - 2*w^2 - 8*w + 11],
[263, 263, -w^3 + w^2 + 3*w - 7],
[263, 263, 3*w^3 - w^2 - 11*w + 11],
[263, 263, -2*w^3 + 6*w - 7],
[277, 277, -3*w^2 - w + 11],
[277, 277, 3*w^2 - w - 11],
[311, 311, -w^3 + 5*w^2 + 7*w - 19],
[311, 311, 2*w^3 - 3*w^2 - 8*w + 17],
[311, 311, -2*w^3 + 6*w - 5],
[311, 311, -2*w^3 - 2*w^2 + 8*w + 5],
[337, 337, -2*w^3 + 8*w + 1],
[337, 337, -w^3 + 3*w^2 + 5*w - 11],
[337, 337, w^3 + 3*w^2 - 5*w - 11],
[337, 337, 2*w^3 - 8*w + 1],
[347, 347, w^3 - 3*w^2 - 6*w + 13],
[347, 347, -w^3 - 3*w^2 + 6*w + 13],
[349, 349, -w^3 + 4*w - 5],
[349, 349, w^3 - 4*w - 5],
[361, 19, 4*w^3 - 2*w^2 - 17*w + 13],
[361, 19, 2*w^3 - 2*w^2 - 11*w + 7],
[373, 373, -2*w^3 - 5*w^2 + 13*w + 17],
[373, 373, 2*w^3 - w^2 - 7*w + 1],
[383, 383, 4*w^3 - w^2 - 16*w + 13],
[383, 383, w^2 + 2*w - 7],
[383, 383, w^2 - 2*w - 7],
[383, 383, 2*w^3 + 3*w^2 - 6*w - 5],
[397, 397, -w^3 + 2*w^2 + 7*w - 7],
[397, 397, w^3 + 2*w^2 - 7*w - 7],
[419, 419, 5*w^2 - 5*w - 11],
[419, 419, 3*w^2 - 3*w - 5],
[421, 421, w^3 + 4*w^2 - 5*w - 17],
[421, 421, -w^3 + 4*w^2 + 5*w - 17],
[433, 433, w^3 - w^2 - 7*w + 1],
[433, 433, 3*w - 1],
[433, 433, -3*w - 1],
[433, 433, 5*w^3 - 3*w^2 - 21*w + 19],
[443, 443, w^3 - 4*w^2 - 6*w + 17],
[443, 443, -3*w^3 + 6*w^2 + 16*w - 25],
[467, 467, -w^3 + w^2 + 2*w - 5],
[467, 467, w^3 + w^2 - 2*w - 5],
[491, 491, 3*w^3 - w^2 - 14*w + 7],
[491, 491, -3*w^3 - w^2 + 14*w + 7],
[529, 23, 3*w^2 - 11],
[529, 23, 3*w^2 - 7],
[541, 541, 3*w^3 + 2*w^2 - 10*w - 1],
[541, 541, 3*w^3 - 4*w^2 - 16*w + 17],
[563, 563, w^3 + 2*w^2 - 2*w - 7],
[563, 563, -w^3 + 2*w^2 + 2*w - 7],
[587, 587, -2*w^3 + w^2 + 5*w - 5],
[587, 587, 2*w^3 + w^2 - 5*w - 5],
[613, 613, 3*w^3 - 13*w - 1],
[613, 613, 3*w^3 - 13*w + 1],
[659, 659, w^3 + 3*w^2 - 4*w - 5],
[659, 659, -w^3 + 3*w^2 + 4*w - 5],
[661, 661, w^3 + 4*w^2 - 9*w - 13],
[661, 661, -3*w^3 + 4*w^2 + 7*w - 5],
[673, 673, -2*w^3 + 7*w + 1],
[673, 673, w^3 + w^2 - 7*w - 5],
[673, 673, -w^3 + w^2 + 7*w - 5],
[673, 673, 2*w^3 - 7*w + 1],
[683, 683, -2*w^3 - w^2 + 9*w + 1],
[683, 683, 2*w^3 - w^2 - 9*w + 1],
[709, 709, -3*w^3 + 2*w^2 + 8*w + 1],
[709, 709, -w^3 + 4*w^2 - 2*w - 7],
[733, 733, 2*w^3 - 5*w^2 - 11*w + 19],
[733, 733, -2*w^3 + w^2 + 7*w - 11],
[743, 743, 2*w^3 + 2*w^2 - 11*w - 5],
[743, 743, 2*w^3 - w^2 - 8*w - 1],
[743, 743, -2*w^3 - w^2 + 8*w - 1],
[743, 743, -2*w^3 + 2*w^2 + 11*w - 5],
[757, 757, 2*w^3 + 3*w^2 - 3*w + 1],
[757, 757, -2*w^3 + 5*w^2 + 11*w - 17],
[827, 827, -5*w^3 + 4*w^2 + 21*w - 23],
[827, 827, -w^3 + w - 7],
[829, 829, 2*w^3 - 3*w^2 - 9*w + 17],
[829, 829, -2*w^3 - 3*w^2 + 9*w + 17],
[841, 29, 2*w^2 - w - 13],
[841, 29, 2*w^2 + w - 13],
[853, 853, -w^3 + 6*w - 7],
[853, 853, -2*w^3 + w^2 + 11*w - 5],
[863, 863, w^3 - 5*w^2 - 9*w + 13],
[863, 863, 2*w^3 + 4*w^2 - 5*w - 11],
[863, 863, -2*w^3 + 4*w^2 + 5*w - 11],
[863, 863, w^3 - 3*w^2 - 7*w + 7],
[877, 877, -w^3 - 3*w^2 + 4*w + 17],
[877, 877, w^3 - 3*w^2 - 4*w + 17],
[887, 887, -2*w^3 - w^2 + 10*w + 1],
[887, 887, 2*w^3 + 2*w^2 - 9*w - 5],
[887, 887, -2*w^3 + 2*w^2 + 9*w - 5],
[887, 887, 2*w^3 - w^2 - 10*w + 1],
[911, 911, -2*w^3 + 11*w - 5],
[911, 911, -w^3 + w^2 + w - 5],
[911, 911, w^3 + w^2 - w - 5],
[911, 911, 2*w^3 - 11*w - 5],
[937, 937, -4*w^3 - 2*w^2 + 16*w + 17],
[937, 937, 3*w^3 - w^2 - 11*w - 1],
[937, 937, -3*w^3 - 5*w^2 + 17*w + 19],
[937, 937, -4*w^2 + 2*w + 13],
[947, 947, -3*w^3 + 2*w^2 + 10*w + 1],
[947, 947, 3*w^3 - 12*w - 7],
[971, 971, w^3 + w^2 - 2*w - 7],
[971, 971, -w^3 + w^2 + 2*w - 7],
[983, 983, -w^3 - 3*w^2 + 7*w + 13],
[983, 983, 2*w^3 - 2*w^2 - 7*w + 1],
[983, 983, -2*w^3 - 2*w^2 + 7*w + 1],
[983, 983, w^3 - 7*w^2 + 3*w + 17],
[997, 997, -w^3 + 2*w^2 + 4*w - 13],
[997, 997, 3*w^3 - 8*w^2 - 18*w + 29]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 6*x^3 - 6*x^2 - 45*x + 27;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -1/3*e^2 - e + 3, e, 1, -1/9*e^3 - 1/3*e^2 + 5/3*e, 2/9*e^3 + 2/3*e^2 - 7/3*e - 3, 2/9*e^3 + 1/3*e^2 - 13/3*e, 2/9*e^3 + 4/3*e^2 + 2/3*e - 9, -4/9*e^3 - 7/3*e^2 + 5/3*e + 6, 2/9*e^3 + e^2 - 4/3*e - 6, 1/3*e^2 - e, e^2 + e - 12, -1/9*e^3 - e^2 + 2/3*e + 3, -1/9*e^3 + e^2 + 14/3*e - 15, -1/3*e^3 - 2*e^2 + 2*e + 5, 2/9*e^3 + 7/3*e^2 + 8/3*e - 15, -4/9*e^3 - 5/3*e^2 + 17/3*e + 9, -e^3 - 14/3*e^2 + 4*e + 14, 1/3*e^2 - 2*e - 6, 1/3*e^3 + 8/3*e^2 + 3*e - 18, 2/3*e^3 + 3*e^2 - 3*e - 3, -1/3*e^3 - 2/3*e^2 + 7*e + 12, 1/3*e^3 + 10/3*e^2 + e - 22, 2/3*e^2 - e - 10, -4/9*e^3 - 7/3*e^2 - 4/3*e + 3, -1/3*e^3 + 1/3*e^2 + 8*e - 3, 2/3*e^3 + 7/3*e^2 - 8*e - 6, 8/9*e^3 + 5*e^2 - 4/3*e - 21, -1/9*e^3 - 3*e^2 - 22/3*e + 18, 1/3*e^3 + 5/3*e^2 - 3*e + 3, 1/3*e^3 + 4/3*e^2 - 5*e - 3, -2/3*e^3 - 14/3*e^2 - e + 21, -1/3*e^3 - 2/3*e^2 + e - 18, -1/3*e^3 - 7/3*e^2 + 11, -1/3*e^3 - 3*e^2 - 3*e + 12, -2/3*e^3 - 3*e^2 + 5*e + 15, -1/9*e^3 - 1/3*e^2 + 20/3*e + 12, -1/9*e^3 - 5/3*e^2 - 13/3*e + 3, -1/3*e^3 + 3*e - 21, 4/3*e^3 + 17/3*e^2 - 7*e - 15, 1/3*e^3 + 4*e^2 + 5*e - 30, 2/3*e^2 + 4*e - 9, -1/3*e^3 - 1/3*e^2 + 8*e - 9, 1/3*e^2 - 2*e - 12, 2/9*e^3 - 2/3*e^2 - 10/3*e + 15, 5/9*e^3 + 5*e^2 + 11/3*e - 27, -e^3 - 5*e^2 + 5*e + 24, -1/3*e^3 - 8/3*e^2 + e + 21, 2/3*e^3 + 4/3*e^2 - 5*e + 15, -1/3*e^3 - 2*e^2 + e + 9, -7/9*e^3 - 4*e^2 + 8/3*e + 3, -1/9*e^3 - 2*e^2 - 10/3*e + 6, -1/9*e^3 + 2/3*e^2 + 14/3*e - 15, 8/9*e^3 + 14/3*e^2 - 16/3*e - 15, 1/3*e^3 + 4/3*e^2 - 7*e - 6, -1/3*e^3 - e^2 + 5*e + 3, e^3 + 5*e^2 - 4*e - 6, 1/3*e^3 + 7/3*e^2 - 2*e - 6, -e^3 - 8/3*e^2 + 14*e + 21, -1/3*e^3 - 2*e^2 + e + 12, -10/9*e^3 - 16/3*e^2 + 14/3*e + 24, -7/9*e^3 - 2*e^2 + 38/3*e + 3, 2/3*e^3 + 7/3*e^2 - 4*e + 3, e^3 + 4*e^2 - 11*e - 27, -e^3 - 16/3*e^2 + 2*e + 24, -7/3*e^2 - 5*e + 27, -1/3*e^3 + 1/3*e^2 + 10*e - 19, -4/9*e^3 - 2*e^2 + 11/3*e + 30, -7/9*e^3 - 5/3*e^2 + 23/3*e - 18, e^3 + 7*e^2 + 4*e - 34, 2/3*e^2 + 8*e + 3, -2/3*e^3 - 14/3*e^2 - 2*e + 27, -1/9*e^3 + e^2 + 11/3*e - 6, -1/9*e^3 + 1/3*e^2 + 8/3*e - 21, 2/9*e^3 + 2*e^2 + 2/3*e - 18, -1/9*e^3 - e^2 + 2/3*e - 3, 11/9*e^3 + 13/3*e^2 - 28/3*e, 2/9*e^3 + 2*e^2 - 19/3*e - 21, -1/3*e^3 - e^2 + 5*e + 3, 2/3*e^3 + 2*e^2 - 11*e - 21, -e^3 - 13/3*e^2 + 9*e + 9, -e^3 - 19/3*e^2 + e + 33, 2/9*e^3 - 2/3*e^2 + 8/3*e + 33, -1/9*e^3 + 4*e^2 + 50/3*e - 24, -1/3*e^3 - 8/3*e^2 + e + 18, 1/3*e^3 + 8/3*e^2 + 4*e - 27, e^3 + 4*e^2 - 3*e - 10, 1/3*e^3 + 10/3*e^2 + 7*e - 7, -1/9*e^3 - e^2 - 4/3*e + 12, 5/9*e^3 - 2/3*e^2 - 43/3*e + 15, -10/9*e^3 - 8/3*e^2 + 32/3*e - 18, -4/9*e^3 - e^2 + 5/3*e - 24, 2/3*e^3 + 3*e^2 - 8*e - 27, -4/3*e^2 + e + 15, -e^3 - 6*e^2 - e + 12, 2/3*e^3 + 20/3*e^2 + 6*e - 45, 1/3*e^3 - 7/3*e^2 - 11*e + 21, 2/3*e^3 + 3*e^2 - e + 9, 5/9*e^3 + 3*e^2 - 10/3*e - 3, 11/9*e^3 + 5/3*e^2 - 58/3*e + 3, -4/3*e^3 - 13/3*e^2 + 12*e + 5, e^3 + 2*e^2 - 18*e - 7, 1/3*e^3 + 4*e^2 + 3*e - 21, -13/3*e^2 - 18*e + 30, 2*e^3 + 10*e^2 - 8*e - 45, e^2 - e - 18, -e^3 - 20/3*e^2 + 7*e + 32, e^2 - 3*e - 25, e^3 - 2/3*e^2 - 24*e + 9, 4/3*e^3 + 17/3*e^2 - 6*e - 3, -5/3*e^3 - 6*e^2 + 17*e + 35, 1/3*e^3 + 2*e^2 - 6*e - 31, -10/9*e^3 - 8*e^2 - 7/3*e + 45, 8/9*e^3 + 22/3*e^2 - 10/3*e - 45, 8/9*e^3 + 3*e^2 - 34/3*e - 15, -4/9*e^3 - e^2 + 38/3*e + 15, 1/3*e^3 + e^2 - 3*e - 3, -7*e^2 - 25*e + 51, 1/3*e^3 + 10/3*e^2 - 5*e - 34, -2/3*e^2 - 4, 2/9*e^3 + 2/3*e^2 - 4/3*e - 27, -7/9*e^3 - 14/3*e^2 - 1/3*e + 24, 2/3*e^3 + e^2 - 13*e - 6, -2/3*e^3 - 16/3*e^2 - 9*e + 24, 1/3*e^3 - 2*e^2 - 14*e + 18, -2/3*e^3 - 23/3*e^2 - 8*e + 36, 5/9*e^3 + 2*e^2 - 25/3*e - 18, 2/9*e^3 + 19/3*e^2 + 41/3*e - 39, -5/3*e^3 - 9*e^2 + e + 30, -2/3*e^3 - 4*e^2 - 5*e + 30, 5/9*e^3 - 5/3*e^2 - 37/3*e + 24, 11/9*e^3 + 6*e^2 - 37/3*e - 36, 11/9*e^3 + 2*e^2 - 49/3*e + 15, 17/9*e^3 + 10*e^2 - 22/3*e - 42, 8/9*e^3 + 13/3*e^2 - 10/3*e - 12, -4/9*e^3 - 25/3*e^2 - 43/3*e + 39, -2*e^3 - 7*e^2 + 19*e + 24, 1/3*e^3 + 9*e^2 + 18*e - 69, -1/3*e^3 - 2/3*e^2 - e - 12, 5/3*e^3 + 22/3*e^2 - 8*e - 21, -10/9*e^3 - 16/3*e^2 + 44/3*e + 33, 17/9*e^3 + 4*e^2 - 85/3*e - 12, -11/3*e^2 - 6*e + 57, -2*e^3 - 22/3*e^2 + 12*e + 3, -7/3*e^3 - 11*e^2 + 14*e + 48, -e^3 + 21*e - 12, -e^3 - 9*e^2 - 9*e + 51, -4/3*e^3 - 16/3*e^2 - 3, 2/3*e^3 - 4/3*e^2 - 18*e + 36, -e^3 + e^2 + 25*e - 18, -7/9*e^3 + 5/3*e^2 + 59/3*e - 21, e^3 + 11/3*e^2 - 5*e - 16, e^3 + 5*e^2 - 15*e - 25, 5/9*e^3 - 3*e^2 - 61/3*e + 27, -2/3*e^3 - 13/3*e^2 + 8*e + 30, -e^3 - 5*e^2 + 12*e + 24, e^3 + 5*e^2 - 6*e - 51, 2*e^3 + 6*e^2 - 13*e + 12, -1/3*e^3 + 5/3*e^2 + 17*e - 6, 2*e^3 + 7*e^2 - 20*e - 30, 5/3*e^3 + 7*e^2 - 20*e - 39, -2/3*e^3 - 5*e^2 - 7*e + 30, e^2 + 2*e - 7, 5/3*e^3 + 16/3*e^2 - 13*e - 19];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,-w^2+w+2}>] := 1;
ALEigenvalues[ideal<ZF | {11,11,-w^2-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;