/*
  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, -2, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, -w^3 + w^2 + 5*w}>;

primesArray := [
[5, 5, -w^3 + 2*w^2 + 3*w],
[7, 7, w - 1],
[11, 11, -w^3 + 2*w^2 + 4*w],
[13, 13, -2*w^3 + 3*w^2 + 10*w - 2],
[16, 2, 2],
[17, 17, -w^3 + 2*w^2 + 5*w - 3],
[17, 17, -w^3 + w^2 + 5*w],
[17, 17, -w^2 + 2*w + 1],
[19, 19, w^2 - w - 2],
[29, 29, w^3 - 2*w^2 - 5*w],
[29, 29, w^2 - w - 3],
[31, 31, -w^3 + 2*w^2 + 3*w - 2],
[43, 43, 2*w^3 - 3*w^2 - 11*w],
[47, 47, w^3 - 7*w - 4],
[53, 53, -2*w^3 + 3*w^2 + 9*w - 1],
[61, 61, -w - 3],
[73, 73, -w^3 + 2*w^2 + 3*w - 3],
[73, 73, w^3 - w^2 - 7*w - 1],
[81, 3, -3],
[83, 83, -2*w^3 + 3*w^2 + 9*w + 1],
[83, 83, -w^3 + 3*w^2 + 2*w - 3],
[89, 89, w - 4],
[103, 103, w^3 - 2*w^2 - 6*w + 3],
[113, 113, 2*w^3 - w^2 - 14*w - 6],
[121, 11, -w^2 + 2*w + 7],
[125, 5, -3*w^3 + 4*w^2 + 15*w + 1],
[139, 139, w^3 - 2*w^2 - 5*w - 3],
[139, 139, w^3 - 5*w - 5],
[151, 151, w^3 - 9*w - 7],
[157, 157, -3*w^3 + 6*w^2 + 13*w - 6],
[163, 163, w^3 - 3*w^2 - 2*w + 9],
[167, 167, -w^3 + 7*w + 3],
[167, 167, -3*w^3 + 4*w^2 + 15*w - 1],
[173, 173, 2*w^3 - 3*w^2 - 7*w - 2],
[181, 181, -w^3 + 3*w^2 + 4*w - 5],
[191, 191, w^3 - w^2 - 4*w - 4],
[193, 193, -w^3 + w^2 + 7*w + 6],
[193, 193, 2*w^3 - 2*w^2 - 13*w - 5],
[199, 199, -2*w^3 + 5*w^2 + 5*w - 5],
[211, 211, w^3 - w^2 - 8*w - 3],
[211, 211, -2*w^2 + 3*w + 8],
[227, 227, -w^3 + 3*w^2 + 4*w - 4],
[227, 227, w^2 - 3*w - 6],
[229, 229, w^2 - 5],
[229, 229, 3*w^3 - 4*w^2 - 17*w - 2],
[233, 233, 3*w^3 - 6*w^2 - 13*w + 3],
[233, 233, 3*w^3 - 4*w^2 - 16*w + 1],
[241, 241, -2*w^3 + 4*w^2 + 8*w + 1],
[251, 251, 2*w^3 - 2*w^2 - 9*w - 3],
[277, 277, 3*w^3 - 4*w^2 - 17*w - 1],
[281, 281, -2*w^2 + 5*w + 1],
[283, 283, -4*w^3 + 6*w^2 + 19*w - 2],
[293, 293, -3*w^3 + 6*w^2 + 12*w - 2],
[307, 307, w^3 - w^2 - 4*w - 5],
[307, 307, w^3 - w^2 - 8*w - 2],
[311, 311, 2*w^2 - 3*w - 7],
[313, 313, -2*w^3 + w^2 + 15*w + 3],
[317, 317, -w^3 + 3*w^2 + w - 5],
[331, 331, 2*w^3 - 3*w^2 - 12*w],
[337, 337, w^3 - w^2 - 8*w - 1],
[343, 7, -w^3 + 6*w + 8],
[347, 347, 2*w^3 - 4*w^2 - 7*w + 5],
[349, 349, 2*w^3 - 4*w^2 - 9*w],
[353, 353, -3*w^3 + 4*w^2 + 16*w + 5],
[353, 353, -3*w^3 + 4*w^2 + 14*w],
[353, 353, -w^3 + 3*w^2 + 4*w - 8],
[353, 353, -w^3 + 2*w^2 + 8*w - 3],
[359, 359, -4*w^3 + 7*w^2 + 19*w - 3],
[367, 367, -w^3 + 2*w^2 + 6*w - 5],
[373, 373, 5*w^3 - 8*w^2 - 26*w + 3],
[379, 379, -3*w^3 + 5*w^2 + 17*w - 4],
[379, 379, -w^3 + 2*w^2 + 4*w - 6],
[383, 383, w^2 - 6],
[389, 389, w^3 - 2*w^2 - 3*w - 4],
[389, 389, w^2 + w - 4],
[397, 397, -4*w^3 + 5*w^2 + 23*w + 5],
[397, 397, -3*w^3 + 5*w^2 + 12*w - 2],
[419, 419, -3*w^3 + 4*w^2 + 17*w + 5],
[419, 419, -3*w^3 + 5*w^2 + 15*w],
[419, 419, 3*w^3 - 5*w^2 - 14*w - 1],
[419, 419, 2*w^3 - 3*w^2 - 10*w - 6],
[431, 431, 2*w^3 - 2*w^2 - 11*w],
[433, 433, w^3 - 8*w - 1],
[433, 433, 2*w^2 - 2*w - 5],
[439, 439, 2*w^3 - 2*w^2 - 9*w - 6],
[443, 443, 2*w^3 - 5*w^2 - 7*w],
[443, 443, 2*w^3 - 5*w^2 - 9*w + 4],
[457, 457, -2*w^3 + 5*w^2 + 7*w - 4],
[457, 457, -w^3 + 2*w^2 - 3],
[461, 461, -4*w^3 + 7*w^2 + 17*w - 7],
[479, 479, 3*w^3 - 5*w^2 - 12*w + 6],
[487, 487, -2*w^3 + 5*w^2 + 9*w - 9],
[487, 487, w^3 - 10*w - 8],
[499, 499, 2*w^3 - 2*w^2 - 12*w + 1],
[503, 503, w^2 - 7],
[503, 503, w^3 + w^2 - 7*w - 8],
[529, 23, -3*w^3 + 5*w^2 + 12*w + 1],
[529, 23, 3*w^3 - 4*w^2 - 14*w - 1],
[541, 541, w^3 - 4*w^2 - 2*w + 11],
[563, 563, -2*w^3 + w^2 + 15*w + 6],
[569, 569, w^3 - 2*w^2 - 4*w - 4],
[601, 601, 3*w^3 - 5*w^2 - 17*w + 1],
[607, 607, -2*w^3 + 5*w^2 + 7*w - 5],
[607, 607, -w^3 + 3*w^2 + 6*w - 5],
[613, 613, -4*w^3 + 5*w^2 + 21*w + 1],
[613, 613, -5*w^3 + 8*w^2 + 23*w - 4],
[619, 619, w^3 + w^2 - 9*w - 5],
[619, 619, -3*w^3 + 4*w^2 + 14*w + 2],
[641, 641, -6*w^3 + 8*w^2 + 31*w + 3],
[643, 643, 3*w^3 - 4*w^2 - 17*w - 6],
[643, 643, -2*w^3 + 5*w^2 + 7*w - 6],
[647, 647, 3*w^3 - 4*w^2 - 13*w - 10],
[653, 653, -3*w^3 + 5*w^2 + 14*w + 2],
[653, 653, w^3 - 2*w^2 - w - 3],
[661, 661, 5*w^3 - 9*w^2 - 24*w + 5],
[683, 683, 2*w^3 - 3*w^2 - 13*w - 2],
[683, 683, 4*w^3 - 7*w^2 - 18*w + 1],
[719, 719, -w^3 + 2*w^2 + 3*w - 7],
[719, 719, 3*w^3 - 5*w^2 - 16*w - 1],
[727, 727, -2*w^3 + 5*w^2 + 8*w - 7],
[727, 727, 2*w^3 - 5*w^2 - 7*w + 13],
[733, 733, -2*w^2 + 7],
[733, 733, -3*w^3 + 5*w^2 + 11*w + 2],
[739, 739, -6*w^3 + 9*w^2 + 29*w - 3],
[739, 739, -2*w^3 + 3*w^2 + 7*w + 5],
[751, 751, 4*w^3 - 5*w^2 - 23*w - 3],
[757, 757, -w^3 + 10*w + 4],
[757, 757, 2*w^3 - w^2 - 14*w - 4],
[769, 769, -w^3 + 4*w^2 + 4*w - 8],
[769, 769, 4*w^3 - 6*w^2 - 21*w - 2],
[787, 787, 3*w^3 - 3*w^2 - 19*w - 6],
[787, 787, -2*w^2 + 6*w + 7],
[839, 839, -3*w^3 + 3*w^2 + 19*w + 5],
[841, 29, 3*w^3 - 4*w^2 - 13*w - 4],
[853, 853, -3*w^3 + 5*w^2 + 14*w + 3],
[853, 853, 2*w^2 - w - 9],
[853, 853, 3*w^3 - 5*w^2 - 11*w + 1],
[853, 853, w - 6],
[857, 857, 2*w^3 - 2*w^2 - 9*w - 7],
[857, 857, -4*w^3 + 8*w^2 + 13*w - 1],
[859, 859, 2*w^3 - 4*w^2 - 13*w + 2],
[859, 859, w^3 - w^2 - 9*w - 2],
[863, 863, -2*w^3 - w^2 + 17*w + 11],
[877, 877, 5*w^3 - 6*w^2 - 28*w - 9],
[877, 877, -4*w^3 + 5*w^2 + 20*w + 6],
[881, 881, 4*w^3 - 5*w^2 - 18*w - 8],
[911, 911, -w^3 + 4*w^2 - w - 8],
[911, 911, 2*w^3 - 3*w^2 - 13*w - 1],
[919, 919, w^3 - w^2 - 7*w + 5],
[929, 929, 2*w^3 - 3*w^2 - 13*w + 1],
[929, 929, -2*w^3 + 5*w^2 + 6*w - 11],
[941, 941, -3*w^3 + 7*w^2 + 12*w - 5],
[953, 953, -2*w^3 + 5*w^2 + 5*w - 7],
[971, 971, w^3 - w^2 - 3*w - 5],
[971, 971, 3*w^3 - 7*w^2 - 12*w + 11],
[977, 977, 3*w^3 - 5*w^2 - 11*w],
[997, 997, -2*w^3 + 3*w^2 + 6*w - 5],
[997, 997, -4*w^3 + 7*w^2 + 16*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 3*x^3 - 9*x^2 - 32*x - 22;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^3 - 2*e^2 + 12*e + 20, e^3 + 2*e^2 - 12*e - 18, e^3 + e^2 - 12*e - 14, -e^2 + 9, -e^3 - 2*e^2 + 12*e + 22, -1, -e^3 - 2*e^2 + 12*e + 22, e^3 + 2*e^2 - 10*e - 18, 2, 2*e^3 + 2*e^2 - 21*e - 22, 3*e^3 + 5*e^2 - 34*e - 48, 3*e^3 + 4*e^2 - 31*e - 42, -e^3 - e^2 + 10*e + 4, e^3 + e^2 - 12*e - 8, -2*e^3 - 4*e^2 + 24*e + 40, e^2 - e - 14, 4*e^3 + 6*e^2 - 44*e - 68, -e^3 - 3*e^2 + 12*e + 26, -5*e^3 - 8*e^2 + 57*e + 80, 2*e^3 + 3*e^2 - 24*e - 30, -e^3 - e^2 + 13*e + 12, 4*e^3 + 7*e^2 - 43*e - 62, 2*e^3 + 5*e^2 - 22*e - 36, -4*e^3 - 6*e^2 + 42*e + 64, 3*e^3 + 4*e^2 - 32*e - 42, -3*e^3 - 2*e^2 + 30*e + 26, -5*e^3 - 6*e^2 + 53*e + 68, -e^2 - e - 4, -e^2 - 3*e + 4, -4*e^3 - 7*e^2 + 46*e + 68, -4*e^3 - 6*e^2 + 43*e + 58, -3*e^3 - 3*e^2 + 32*e + 34, -2*e^3 - 2*e^2 + 18*e + 24, -e^3 - e^2 + 12*e + 22, 5*e^3 + 9*e^2 - 61*e - 94, -4*e^3 - 4*e^2 + 44*e + 58, 2*e^3 + 3*e^2 - 23*e - 32, 9*e^3 + 13*e^2 - 100*e - 138, -2*e^3 - 3*e^2 + 17*e + 26, 8*e^3 + 13*e^2 - 91*e - 134, 2*e^3 + e^2 - 21*e - 18, e^3 + 3*e^2 - 11*e - 36, -4*e^3 - 7*e^2 + 43*e + 72, -3*e^3 - 4*e^2 + 32*e + 40, -3*e^3 - 5*e^2 + 37*e + 54, -e^3 - 2*e^2 + 6*e + 18, -e^3 - 4*e^2 + 11*e + 38, -3*e^3 - 4*e^2 + 38*e + 38, 3*e^3 + 5*e^2 - 34*e - 48, 2*e^3 + 2*e^2 - 23*e - 12, -5*e^3 - 9*e^2 + 56*e + 86, -2*e^3 - e^2 + 22*e + 8, -e^3 - e^2 + 7*e + 12, -5*e^3 - 12*e^2 + 58*e + 106, 3*e^3 + 2*e^2 - 36*e - 30, 5*e^3 + 9*e^2 - 51*e - 90, -7*e^3 - 6*e^2 + 80*e + 84, -3*e^3 - 4*e^2 + 30*e + 44, 9*e^3 + 13*e^2 - 98*e - 136, 2*e^3 + 4*e^2 - 19*e - 58, e^3 + 4*e^2 - 18*e - 48, -7*e^3 - 10*e^2 + 78*e + 96, 5*e^3 + 8*e^2 - 56*e - 74, -7*e^3 - 12*e^2 + 72*e + 124, e^3 + 3*e^2 - 11*e - 28, -3*e^3 - 6*e^2 + 30*e + 56, 6*e^3 + 11*e^2 - 66*e - 102, -2*e^3 + e^2 + 17*e, -3*e^3 + e^2 + 30*e + 2, -9*e^3 - 11*e^2 + 102*e + 128, 6*e^3 + 8*e^2 - 60*e - 84, -5*e^3 - 6*e^2 + 54*e + 78, e^2 + 8, -10*e^3 - 18*e^2 + 110*e + 164, 6*e^3 + 8*e^2 - 77*e - 86, -7*e^3 - 6*e^2 + 78*e + 78, 4*e^3 + 7*e^2 - 48*e - 82, 10*e^3 + 16*e^2 - 110*e - 156, 2*e^3 + 6*e^2 - 22*e - 64, -2*e^3 - 9*e^2 + 22*e + 62, -e^2 + 10*e + 18, -5*e^3 - 4*e^2 + 54*e + 42, -5*e^3 - 8*e^2 + 54*e + 82, -3*e^3 - 4*e^2 + 32*e + 52, -e^3 - 4*e^2 + 18*e + 40, 5*e^3 + 6*e^2 - 52*e - 76, -6*e^3 - 8*e^2 + 66*e + 66, -2*e^3 - 5*e^2 + 23*e + 66, -3*e^3 - 4*e^2 + 39*e + 60, -4*e^3 - 4*e^2 + 44*e + 36, 8*e^3 + 15*e^2 - 88*e - 136, 9*e^3 + 13*e^2 - 94*e - 140, 10*e^3 + 14*e^2 - 114*e - 152, 8*e^3 + 13*e^2 - 86*e - 110, -7*e^3 - 10*e^2 + 70*e + 86, 5*e^3 + 5*e^2 - 54*e - 52, 4*e^3 + 9*e^2 - 41*e - 94, 4*e^2 + 2*e - 12, -10*e^3 - 17*e^2 + 116*e + 170, -6*e^3 - 2*e^2 + 68*e + 54, -6*e^3 - 9*e^2 + 68*e + 74, -3*e^3 + 3*e^2 + 32*e - 6, 3*e^3 + 8*e^2 - 36*e - 64, -7*e^3 - 10*e^2 + 70*e + 94, -9*e^3 - 11*e^2 + 100*e + 122, -13*e^3 - 16*e^2 + 146*e + 186, -6*e^3 - 11*e^2 + 69*e + 94, 3*e^3 - 22*e + 6, 2*e^3 - 28*e - 8, 5*e^3 + 4*e^2 - 57*e - 72, 3*e^3 + 3*e^2 - 22*e - 20, -8*e^3 - 10*e^2 + 90*e + 118, 11*e^3 + 17*e^2 - 117*e - 168, 7*e^3 + 11*e^2 - 77*e - 116, -e^3 + 2*e^2 + 9*e - 10, -12*e^3 - 17*e^2 + 139*e + 182, -4*e^3 - 8*e^2 + 39*e + 66, 8*e^3 + 15*e^2 - 84*e - 146, -4*e^3 - 6*e^2 + 42*e + 78, 3*e^3 + 5*e^2 - 32*e - 22, 2*e^3 + 4*e^2 - 27*e - 42, 5*e^3 + 4*e^2 - 50*e - 42, -7*e^3 - 16*e^2 + 72*e + 134, 3*e^3 + 3*e^2 - 34*e - 12, 3*e^2 + 2*e - 4, -6*e^3 - 10*e^2 + 64*e + 78, 9*e^3 + 14*e^2 - 110*e - 158, -13*e^3 - 18*e^2 + 142*e + 190, 3*e^3 + 6*e^2 - 30*e - 58, -12*e^3 - 16*e^2 + 122*e + 154, 10*e^3 + 19*e^2 - 114*e - 158, e^3 + 8*e^2 - 8*e - 40, -3*e^3 - 2*e^2 + 36*e, -3*e^3 - 5*e^2 + 31*e + 64, -11*e^3 - 17*e^2 + 134*e + 184, -9*e^3 - 17*e^2 + 100*e + 148, 7*e^3 + 13*e^2 - 80*e - 136, 5*e^3 + 6*e^2 - 49*e - 62, 8*e^3 + 14*e^2 - 84*e - 154, -11*e^3 - 17*e^2 + 114*e + 164, 5*e^3 + 8*e^2 - 54*e - 68, -4*e^3 - 6*e^2 + 48*e + 68, 4*e^3 + 3*e^2 - 52*e - 42, 10*e^3 + 13*e^2 - 98*e - 136, -10*e^3 - 18*e^2 + 106*e + 176, -6*e^3 - 9*e^2 + 58*e + 78, 8*e^3 + 13*e^2 - 98*e - 152, 7*e^3 + 4*e^2 - 84*e - 64, -12*e^3 - 14*e^2 + 124*e + 150, 3*e^3 + 10*e^2 - 38*e - 88, -5*e^3 - 9*e^2 + 61*e + 94, 2*e^3 - 20*e - 12, 4*e^3 + 3*e^2 - 46*e - 50, -16*e^3 - 24*e^2 + 173*e + 262, -4*e^3 - 2*e^2 + 38*e, 4*e^3 + 5*e^2 - 42*e - 56, -e^3 - 2*e^2 + 5*e + 16];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, -w^3 + w^2 + 5*w}>] := 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;