/*
  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![-28, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {8, 8, -w + 5}>;

primesArray := [
[2, 2, -w + 6],
[2, 2, w + 5],
[7, 7, 6*w - 35],
[7, 7, -6*w - 29],
[9, 3, 3],
[11, 11, 4*w + 19],
[11, 11, 4*w - 23],
[13, 13, -2*w + 11],
[13, 13, 2*w + 9],
[25, 5, -5],
[31, 31, 2*w - 13],
[31, 31, -2*w - 11],
[41, 41, -8*w - 39],
[41, 41, 8*w - 47],
[53, 53, -26*w - 125],
[53, 53, 26*w - 151],
[61, 61, -14*w + 81],
[61, 61, -14*w - 67],
[83, 83, 2*w - 15],
[83, 83, -2*w - 13],
[97, 97, 2*w - 5],
[97, 97, -2*w - 3],
[109, 109, 2*w - 3],
[109, 109, -2*w - 1],
[113, 113, 2*w - 1],
[127, 127, 8*w - 45],
[127, 127, 8*w + 37],
[131, 131, -50*w + 291],
[131, 131, 50*w + 241],
[139, 139, 6*w - 37],
[139, 139, -6*w - 31],
[149, 149, 20*w + 97],
[149, 149, 20*w - 117],
[157, 157, -12*w - 59],
[157, 157, 12*w - 71],
[163, 163, 4*w - 19],
[163, 163, 4*w + 15],
[173, 173, 4*w + 23],
[173, 173, 4*w - 27],
[211, 211, 2*w - 19],
[211, 211, -2*w - 17],
[227, 227, -4*w - 13],
[227, 227, 4*w - 17],
[233, 233, 6*w + 25],
[233, 233, -6*w + 31],
[239, 239, -14*w - 69],
[239, 239, 14*w - 83],
[241, 241, 46*w + 221],
[241, 241, -46*w + 267],
[251, 251, 22*w - 129],
[251, 251, 22*w + 107],
[257, 257, -34*w + 197],
[257, 257, -34*w - 163],
[277, 277, 4*w - 29],
[277, 277, -4*w - 25],
[283, 283, 4*w - 15],
[283, 283, -4*w - 11],
[289, 17, -17],
[307, 307, 68*w - 395],
[307, 307, 68*w + 327],
[311, 311, 10*w - 61],
[311, 311, -10*w - 51],
[313, 313, -32*w - 155],
[313, 313, 32*w - 187],
[317, 317, -18*w - 85],
[317, 317, 18*w - 103],
[331, 331, -4*w - 9],
[331, 331, 4*w - 13],
[337, 337, -16*w - 79],
[337, 337, 16*w - 95],
[347, 347, -12*w + 67],
[347, 347, 12*w + 55],
[353, 353, 14*w - 79],
[353, 353, 14*w + 65],
[361, 19, -19],
[367, 367, -32*w - 153],
[367, 367, -32*w + 185],
[383, 383, 56*w + 269],
[383, 383, 56*w - 325],
[389, 389, -4*w - 27],
[389, 389, 4*w - 31],
[401, 401, 8*w - 51],
[401, 401, -8*w - 43],
[421, 421, 12*w - 73],
[421, 421, -12*w - 61],
[439, 439, -8*w - 33],
[439, 439, 8*w - 41],
[443, 443, -4*w - 1],
[443, 443, 4*w - 5],
[461, 461, -30*w + 173],
[461, 461, 30*w + 143],
[463, 463, 2*w - 25],
[463, 463, -2*w - 23],
[467, 467, 34*w + 165],
[467, 467, 34*w - 199],
[503, 503, -26*w - 127],
[503, 503, 26*w - 153],
[509, 509, 4*w - 33],
[509, 509, -4*w - 29],
[521, 521, -10*w + 53],
[521, 521, 10*w + 43],
[529, 23, -23],
[547, 547, 14*w - 85],
[547, 547, -14*w - 71],
[557, 557, 66*w + 317],
[557, 557, 66*w - 383],
[563, 563, 2*w - 27],
[563, 563, -2*w - 25],
[569, 569, -64*w + 373],
[569, 569, 64*w + 309],
[587, 587, 12*w + 53],
[587, 587, -12*w + 65],
[593, 593, 8*w + 45],
[593, 593, 8*w - 53],
[601, 601, -26*w - 123],
[601, 601, 26*w - 149],
[617, 617, 6*w - 23],
[617, 617, -6*w - 17],
[647, 647, 24*w - 137],
[647, 647, -24*w - 113],
[653, 653, 28*w - 165],
[653, 653, -28*w - 137],
[677, 677, -22*w - 103],
[677, 677, 22*w - 125],
[691, 691, 20*w - 113],
[691, 691, 20*w + 93],
[709, 709, -10*w - 41],
[709, 709, 10*w - 51],
[719, 719, -8*w + 37],
[719, 719, -8*w - 29],
[727, 727, -22*w - 109],
[727, 727, 22*w - 131],
[739, 739, 46*w - 269],
[739, 739, -46*w - 223],
[761, 761, -6*w - 13],
[761, 761, 6*w - 19],
[769, 769, 110*w - 639],
[769, 769, 110*w + 529],
[773, 773, 4*w - 37],
[773, 773, -4*w - 33],
[787, 787, 2*w - 31],
[787, 787, -2*w - 29],
[809, 809, 56*w + 271],
[809, 809, -56*w + 327],
[821, 821, 6*w - 17],
[821, 821, -6*w - 11],
[823, 823, 38*w - 223],
[823, 823, -38*w - 185],
[827, 827, 132*w - 767],
[827, 827, 132*w + 635],
[841, 29, -29],
[853, 853, -76*w + 443],
[853, 853, 76*w + 367],
[863, 863, 14*w - 87],
[863, 863, -14*w - 73],
[911, 911, 2*w - 33],
[911, 911, -2*w - 31],
[919, 919, -6*w - 41],
[919, 919, 6*w - 47],
[929, 929, 50*w + 239],
[929, 929, 50*w - 289],
[953, 953, 6*w - 11],
[953, 953, -6*w - 5],
[967, 967, -8*w - 25],
[967, 967, 8*w - 33],
[991, 991, 16*w + 71],
[991, 991, -16*w + 87]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - x^5 - 10*x^4 + 7*x^3 + 22*x^2 - 2*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, -1/2*e^5 + 5*e^3 + 1/2*e^2 - 21/2*e - 5/2, -e^5 + e^4 + 9*e^3 - 7*e^2 - 16*e + 2, -1/2*e^5 + e^4 + 5*e^3 - 13/2*e^2 - 23/2*e + 1/2, e^3 + e^2 - 7*e - 3, 1/2*e^5 - e^4 - 5*e^3 + 13/2*e^2 + 21/2*e + 1/2, -e^5 + e^4 + 10*e^3 - 7*e^2 - 22*e + 3, -1/2*e^5 + 4*e^3 + 1/2*e^2 - 9/2*e - 7/2, e^3 - 5*e - 2, -e^5 + e^4 + 10*e^3 - 7*e^2 - 23*e, 3/2*e^5 - 14*e^3 + 1/2*e^2 + 53/2*e + 3/2, -e^5 + 2*e^4 + 12*e^3 - 13*e^2 - 34*e + 2, -2*e^2 + e + 9, -e^5 + 8*e^3 - e^2 - 12*e + 6, 3/2*e^5 - e^4 - 14*e^3 + 17/2*e^2 + 47/2*e - 13/2, -e^5 + 10*e^3 + 2*e^2 - 25*e - 8, e^5 - 10*e^3 + 25*e + 2, -3/2*e^5 + 14*e^3 - 3/2*e^2 - 53/2*e + 7/2, -e^5 + e^4 + 8*e^3 - 6*e^2 - 9*e + 7, 2*e^5 - e^4 - 19*e^3 + 8*e^2 + 33*e - 1, -2*e^5 + 3*e^4 + 21*e^3 - 20*e^2 - 47*e + 3, -e^4 + 7*e^2 - e - 5, 3/2*e^5 - 14*e^3 + 3/2*e^2 + 55/2*e - 25/2, -3*e^5 + 28*e^3 - e^2 - 55*e + 1, -e^5 - e^4 + 12*e^3 + 6*e^2 - 31*e - 5, -7/2*e^5 + 4*e^4 + 36*e^3 - 59/2*e^2 - 165/2*e + 7/2, e^4 + e^3 - 7*e^2 - 5*e - 2, 3*e^5 - 3*e^4 - 31*e^3 + 20*e^2 + 66*e - 3, -3*e^5 + 3*e^4 + 27*e^3 - 21*e^2 - 46*e + 8, -1/2*e^5 + 5*e^3 - 3/2*e^2 - 21/2*e + 23/2, -e^4 - 3*e^3 + 7*e^2 + 19*e, -e^5 - e^4 + 7*e^3 + 10*e^2 - 8*e - 17, 3*e^5 - 5*e^4 - 33*e^3 + 37*e^2 + 81*e - 11, 3/2*e^5 - 2*e^4 - 15*e^3 + 29/2*e^2 + 65/2*e - 23/2, -e^4 + 4*e^2 + 9, 2*e^5 - e^4 - 18*e^3 + 8*e^2 + 36*e - 7, -1/2*e^5 + 2*e^3 - 9/2*e^2 + 11/2*e + 35/2, -e^4 + 7*e^2 + 3*e + 7, e^5 + e^4 - 9*e^3 - 5*e^2 + 12*e + 4, -4*e^5 + 3*e^4 + 41*e^3 - 20*e^2 - 91*e - 5, 2*e^3 - 2*e^2 - 6*e + 10, -3*e^5 + 4*e^4 + 28*e^3 - 29*e^2 - 47*e + 19, 4*e^5 - 4*e^4 - 38*e^3 + 30*e^2 + 71*e - 7, 2*e^3 - 2*e^2 - 5*e + 13, 2*e^4 + 2*e^3 - 17*e^2 - 8*e + 25, -3/2*e^5 + 3*e^4 + 18*e^3 - 43/2*e^2 - 93/2*e + 17/2, -7/2*e^5 + 4*e^4 + 39*e^3 - 55/2*e^2 - 197/2*e + 5/2, -5/2*e^5 + 2*e^4 + 27*e^3 - 33/2*e^2 - 143/2*e + 3/2, -e^5 + e^4 + 14*e^3 - 5*e^2 - 43*e - 18, -3*e^5 + 3*e^4 + 30*e^3 - 19*e^2 - 63*e + 8, -2*e^5 + 4*e^4 + 24*e^3 - 27*e^2 - 70*e + 9, -4*e^4 - 4*e^3 + 27*e^2 + 24*e - 9, e^5 - e^4 - 6*e^3 + 5*e^2 - 5*e, 4*e^5 - 5*e^4 - 36*e^3 + 36*e^2 + 64*e - 13, 3/2*e^5 - e^4 - 16*e^3 + 13/2*e^2 + 73/2*e + 17/2, -2*e^4 - 2*e^3 + 12*e^2 + 8*e, -2*e^5 + e^4 + 18*e^3 - 11*e^2 - 30*e + 10, 2*e^5 + e^4 - 18*e^3 - 6*e^2 + 24*e + 9, 5*e^5 - 7*e^4 - 54*e^3 + 47*e^2 + 139*e - 6, 3*e^5 - 2*e^4 - 26*e^3 + 18*e^2 + 35*e - 24, -5/2*e^5 + e^4 + 20*e^3 - 17/2*e^2 - 55/2*e + 19/2, -3/2*e^5 - 2*e^4 + 15*e^3 + 25/2*e^2 - 57/2*e + 1/2, -1/2*e^5 - e^3 + 7/2*e^2 + 53/2*e - 17/2, -3*e^5 + 2*e^4 + 27*e^3 - 13*e^2 - 53*e, 1/2*e^5 + 3*e^4 + e^3 - 33/2*e^2 - 45/2*e - 35/2, -1/2*e^5 + 2*e^3 + 1/2*e^2 + 17/2*e + 19/2, -2*e^5 + 3*e^4 + 26*e^3 - 24*e^2 - 78*e + 19, -2*e^5 + e^4 + 24*e^3 - 4*e^2 - 69*e - 6, 2*e^5 - 3*e^4 - 20*e^3 + 20*e^2 + 43*e + 14, e^5 + e^4 - 13*e^3 - 2*e^2 + 40*e - 7, 4*e^5 - 5*e^4 - 39*e^3 + 35*e^2 + 79*e - 6, 5*e^5 - 6*e^4 - 47*e^3 + 42*e^2 + 95*e - 9, -5*e^5 + 2*e^4 + 49*e^3 - 16*e^2 - 103*e + 9, 3/2*e^5 - e^4 - 15*e^3 + 21/2*e^2 + 69/2*e - 37/2, -4*e^5 + e^4 + 38*e^3 - 7*e^2 - 74*e + 6, 3*e^5 - 3*e^4 - 26*e^3 + 18*e^2 + 39*e - 7, 7/2*e^5 + e^4 - 33*e^3 - 17/2*e^2 + 135/2*e + 19/2, -e^5 + 2*e^4 + 9*e^3 - 14*e^2 - 26*e + 2, 3*e^5 - 2*e^4 - 32*e^3 + 14*e^2 + 76*e - 3, -3/2*e^5 + 3*e^4 + 14*e^3 - 43/2*e^2 - 59/2*e - 1/2, 7/2*e^5 - e^4 - 33*e^3 + 17/2*e^2 + 133/2*e - 1/2, 1/2*e^5 + e^4 - e^3 - 25/2*e^2 - 21/2*e + 53/2, e^5 - 2*e^4 - 7*e^3 + 11*e^2 + 6*e - 7, 2*e^4 - e^3 - 10*e^2 + 7*e - 4, -e^5 + 9*e^3 + e^2 - 24*e + 7, 3*e^5 - 2*e^4 - 29*e^3 + 13*e^2 + 58*e - 11, -3/2*e^5 + e^4 + 14*e^3 - 19/2*e^2 - 61/2*e + 13/2, e^5 + 2*e^4 - 6*e^3 - 14*e^2 - 5*e + 6, 2*e^5 - 5*e^4 - 22*e^3 + 34*e^2 + 52*e + 1, -e^4 - 2*e^3 + 12*e^2 + 10*e - 5, 3*e^5 - 2*e^4 - 26*e^3 + 18*e^2 + 47*e - 16, 2*e^5 - 2*e^4 - 16*e^3 + 17*e^2 + 10*e - 11, e^5 + 3*e^4 - 13*e^3 - 25*e^2 + 42*e + 32, 9/2*e^5 - 2*e^4 - 47*e^3 + 27/2*e^2 + 217/2*e + 29/2, 5*e^5 - 4*e^4 - 49*e^3 + 28*e^2 + 104*e - 4, 6*e^5 - 6*e^4 - 57*e^3 + 37*e^2 + 107*e + 9, -6*e^4 - 2*e^3 + 42*e^2 + 14*e - 38, -2*e^5 + 2*e^4 + 18*e^3 - 16*e^2 - 24*e + 8, 2*e^5 - 2*e^4 - 17*e^3 + 8*e^2 + 19*e + 20, -2*e^5 + 6*e^4 + 23*e^3 - 36*e^2 - 65*e - 8, 4*e^5 - 5*e^4 - 38*e^3 + 30*e^2 + 79*e + 2, 4*e^5 - 5*e^4 - 37*e^3 + 33*e^2 + 71*e - 10, 3/2*e^5 - 17*e^3 + 7/2*e^2 + 99/2*e + 13/2, 5*e^5 - 3*e^4 - 51*e^3 + 23*e^2 + 109*e - 11, -5/2*e^5 + 23*e^3 - 3/2*e^2 - 67/2*e - 27/2, 2*e^5 - 12*e^3 - e^2 - 2*e + 9, -3*e^5 + 26*e^3 + 2*e^2 - 47*e + 2, -5*e^5 + 5*e^4 + 52*e^3 - 33*e^2 - 115*e - 2, -7*e^5 + 9*e^4 + 72*e^3 - 59*e^2 - 161*e, -e^5 - 5*e^4 + 3*e^3 + 40*e^2 + 14*e - 39, e^4 + 5*e^3 - 5*e^2 - 25*e + 4, e^5 - 5*e^4 - 13*e^3 + 38*e^2 + 36*e - 35, -e^5 + 3*e^4 + 11*e^3 - 20*e^2 - 22*e - 13, 8*e^5 - 8*e^4 - 76*e^3 + 55*e^2 + 152*e - 5, 2*e^5 + 4*e^4 - 20*e^3 - 27*e^2 + 46*e + 9, 5*e^5 - 9*e^4 - 49*e^3 + 61*e^2 + 113*e - 17, -5*e^5 + 3*e^4 + 51*e^3 - 17*e^2 - 113*e + 9, -e^5 - 4*e^4 + 7*e^3 + 29*e^2 - 2*e - 45, -3*e^5 + 6*e^4 + 29*e^3 - 37*e^2 - 54*e + 11, -6*e^5 + 3*e^4 + 55*e^3 - 23*e^2 - 93*e + 2, 8*e^5 - 5*e^4 - 73*e^3 + 35*e^2 + 137*e - 12, -e^5 + 14*e^3 - 45*e - 26, 2*e^5 - 12*e^3 + 7*e^2 - 39, 2*e^5 - 3*e^4 - 25*e^3 + 25*e^2 + 71*e - 18, e^5 + 3*e^4 - 11*e^3 - 18*e^2 + 22*e - 9, 3/2*e^5 - e^4 - 20*e^3 + 3/2*e^2 + 115/2*e + 57/2, -7*e^5 + 12*e^4 + 70*e^3 - 82*e^2 - 156*e + 27, 2*e^5 - 6*e^4 - 25*e^3 + 44*e^2 + 73*e - 20, 17/2*e^5 - 9*e^4 - 77*e^3 + 123/2*e^2 + 261/2*e - 53/2, -4*e^5 + 7*e^4 + 46*e^3 - 45*e^2 - 132*e + 4, -3/2*e^5 + 12*e^3 - 7/2*e^2 - 57/2*e + 59/2, 2*e^5 - 7*e^4 - 20*e^3 + 44*e^2 + 48*e - 19, 7/2*e^5 - 6*e^4 - 32*e^3 + 89/2*e^2 + 125/2*e - 73/2, -15/2*e^5 + 7*e^4 + 66*e^3 - 99/2*e^2 - 227/2*e + 11/2, 3/2*e^5 - 7*e^4 - 14*e^3 + 99/2*e^2 + 59/2*e - 63/2, 3*e^5 - 6*e^4 - 42*e^3 + 38*e^2 + 140*e + 3, -4*e^5 + 34*e^3 - 3*e^2 - 49*e - 10, -2*e^5 + e^4 + 19*e^3 - 12*e^2 - 23*e + 27, 5*e^5 - 3*e^4 - 51*e^3 + 23*e^2 + 118*e - 26, 7/2*e^5 - 5*e^4 - 41*e^3 + 79/2*e^2 + 215/2*e - 41/2, -2*e^5 + 6*e^4 + 17*e^3 - 45*e^2 - 25*e + 33, -3*e^5 + 7*e^4 + 29*e^3 - 53*e^2 - 49*e + 29, -2*e^5 + 9*e^4 + 21*e^3 - 62*e^2 - 58*e + 28, 2*e^5 - 2*e^4 - 26*e^3 + 20*e^2 + 76*e - 36, -5*e^5 + 6*e^4 + 44*e^3 - 43*e^2 - 73*e + 17, -5*e^5 + 5*e^4 + 47*e^3 - 36*e^2 - 86*e + 27, -8*e^5 + 7*e^4 + 73*e^3 - 53*e^2 - 133*e + 18, 3/2*e^5 - 3*e^4 - 17*e^3 + 39/2*e^2 + 83/2*e + 35/2, 4*e^5 - 43*e^3 + 3*e^2 + 113*e + 3, 3*e^5 - 3*e^4 - 30*e^3 + 21*e^2 + 75*e - 8, e^5 + 5*e^4 - 12*e^3 - 34*e^2 + 30*e + 34, 13/2*e^5 - 10*e^4 - 66*e^3 + 129/2*e^2 + 301/2*e - 43/2, -e^5 - 2*e^4 + 5*e^3 + 12*e^2 + 12*e + 6, -4*e^5 + 4*e^4 + 37*e^3 - 19*e^2 - 65*e - 25, -3*e^5 + 3*e^4 + 32*e^3 - 14*e^2 - 85*e - 29, -8*e^5 + 11*e^4 + 84*e^3 - 73*e^2 - 192*e + 2, -2*e^5 + 3*e^4 + 16*e^3 - 24*e^2 - 18*e + 25, 11*e^5 - 7*e^4 - 98*e^3 + 49*e^2 + 167*e - 18, -4*e^5 + 8*e^4 + 43*e^3 - 52*e^2 - 109*e, 2*e^5 + 4*e^4 - 17*e^3 - 30*e^2 + 29*e + 42, 9/2*e^5 - 3*e^4 - 45*e^3 + 27/2*e^2 + 187/2*e + 41/2, -9/2*e^5 + 3*e^4 + 39*e^3 - 43/2*e^2 - 139/2*e + 35/2, -e^5 + 2*e^4 + 19*e^3 - 10*e^2 - 78*e - 4, 4*e^5 - 4*e^4 - 37*e^3 + 39*e^2 + 69*e - 47, 7*e^5 - 2*e^4 - 67*e^3 + 19*e^2 + 128*e + 3, 11*e^5 - 12*e^4 - 101*e^3 + 87*e^2 + 190*e - 31];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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