/*
  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![-4, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 4, w^2 - 7}>;

primesArray := [
[2, 2, -w - 2],
[2, 2, -w - 1],
[5, 5, -w^2 + w + 7],
[5, 5, -w^2 + w + 5],
[17, 17, -w^2 + 3*w + 1],
[23, 23, -w^2 + w + 3],
[27, 3, 3],
[29, 29, -w^2 + 3*w - 1],
[37, 37, w^2 + w + 1],
[41, 41, w^2 - w - 9],
[43, 43, -3*w^2 + 3*w + 17],
[47, 47, 2*w^2 - 2*w - 13],
[47, 47, -2*w + 5],
[53, 53, 3*w^2 - w - 19],
[59, 59, 2*w - 1],
[67, 67, w^2 + w - 5],
[71, 71, -3*w^2 + w + 21],
[79, 79, 3*w^2 - w - 23],
[89, 89, 2*w^2 - 4*w - 7],
[89, 89, -2*w^2 - 2*w + 5],
[89, 89, 2*w - 3],
[97, 97, 2*w^2 - 9],
[103, 103, -2*w^2 + 2*w + 3],
[103, 103, -5*w^2 + 5*w + 27],
[103, 103, -2*w^2 + 6*w + 3],
[107, 107, w^2 - 3*w - 5],
[109, 109, -w^2 + w - 1],
[113, 113, -4*w^2 + 8*w + 9],
[113, 113, 4*w^2 - 2*w - 29],
[113, 113, -2*w^2 + 15],
[127, 127, -w^2 - w + 9],
[131, 131, 2*w^2 - 2*w - 1],
[149, 149, 3*w^2 - 3*w - 19],
[149, 149, -4*w^2 + 4*w + 21],
[149, 149, -3*w^2 + 5*w + 9],
[157, 157, -2*w^2 - 2*w + 1],
[163, 163, 2*w^2 - 2*w - 7],
[167, 167, 2*w^2 - 4*w - 1],
[173, 173, 5*w^2 - 13*w - 5],
[179, 179, -4*w^2 + 6*w + 17],
[179, 179, 4*w + 5],
[179, 179, -2*w - 7],
[181, 181, w^2 - 3*w - 7],
[191, 191, w^2 - w - 11],
[197, 197, -2*w^2 - 4*w + 3],
[199, 199, w^2 - 5*w + 5],
[199, 199, -4*w^2 + 33],
[199, 199, w^2 - 3*w - 9],
[211, 211, w^2 - 3*w + 3],
[211, 211, -5*w^2 + 9*w + 15],
[211, 211, w^2 - 5*w - 5],
[223, 223, w^2 + 7*w + 5],
[227, 227, 4*w^2 - 6*w - 15],
[229, 229, -w^2 - w + 13],
[239, 239, 2*w^2 - 4*w - 9],
[251, 251, -4*w^2 + 2*w + 23],
[271, 271, -2*w^2 + 8*w - 7],
[277, 277, 2*w^2 - 5],
[281, 281, -8*w^2 + 6*w + 51],
[289, 17, 3*w^2 + w - 15],
[331, 331, -5*w^2 + 5*w + 29],
[337, 337, 3*w^2 - 3*w - 13],
[343, 7, -7],
[347, 347, 7*w^2 - 3*w - 51],
[347, 347, -4*w^2 - 4*w + 11],
[347, 347, -2*w^2 + 19],
[349, 349, 2*w^2 - 6*w - 7],
[353, 353, 2*w^2 + 1],
[367, 367, 4*w - 1],
[373, 373, 3*w^2 - 9*w - 1],
[373, 373, 4*w^2 - 4*w - 27],
[373, 373, -3*w^2 - 5*w + 1],
[379, 379, -6*w^2 + 4*w + 43],
[389, 389, w^2 - 5*w + 3],
[397, 397, 2*w^2 - 4*w - 11],
[409, 409, -6*w^2 + 2*w + 43],
[421, 421, -w^2 + 5*w + 15],
[431, 431, 8*w^2 - 6*w - 53],
[439, 439, -9*w^2 + 19*w + 19],
[443, 443, -3*w^2 - w + 11],
[449, 449, 4*w^2 - 23],
[457, 457, 8*w^2 - 4*w - 57],
[461, 461, w^2 - 5*w - 7],
[463, 463, -6*w^2 + 12*w + 17],
[467, 467, -4*w^2 + 8*w + 13],
[479, 479, 2*w^2 - 6*w + 3],
[491, 491, 4*w^2 - 6*w - 7],
[499, 499, 2*w^2 - 8*w + 5],
[499, 499, -4*w^2 + 2*w + 31],
[499, 499, 4*w^2 - 4*w - 19],
[503, 503, -2*w^2 - 2*w + 21],
[509, 509, -2*w - 9],
[529, 23, 3*w^2 - w - 13],
[547, 547, -2*w^2 - 6*w + 1],
[557, 557, 6*w + 7],
[557, 557, 6*w^2 - 16*w - 3],
[557, 557, -5*w^2 + 9*w + 13],
[563, 563, -8*w^2 + 20*w + 11],
[569, 569, 4*w^2 - 2*w - 21],
[571, 571, 2*w^2 + 4*w - 5],
[577, 577, 3*w^2 - 5*w - 15],
[577, 577, 6*w + 11],
[577, 577, -w^2 + 3*w - 5],
[587, 587, -5*w^2 + w + 37],
[607, 607, 3*w^2 - 3*w - 7],
[613, 613, -6*w - 1],
[617, 617, -w^2 - w - 5],
[619, 619, -2*w^2 + 8*w + 3],
[641, 641, -7*w^2 + w + 55],
[643, 643, -6*w^2 + 2*w + 41],
[643, 643, 6*w^2 - 4*w - 35],
[643, 643, 3*w^2 + w - 17],
[653, 653, -w^2 + 7*w + 7],
[659, 659, -3*w^2 - w + 3],
[673, 673, 2*w^2 + 2*w - 11],
[673, 673, w^2 - 7*w + 13],
[673, 673, -7*w^2 + 7*w + 37],
[683, 683, w^2 - 5*w - 13],
[683, 683, 2*w^2 - 6*w - 9],
[683, 683, w^2 + w - 15],
[691, 691, -6*w^2 + 6*w + 31],
[691, 691, 4*w - 5],
[691, 691, -4*w^2 + 35],
[701, 701, 2*w^2 + 4*w + 5],
[719, 719, -w^2 - 3*w - 7],
[727, 727, 4*w^2 - 6*w - 11],
[733, 733, w^2 + 3*w - 7],
[739, 739, 3*w^2 - w - 11],
[743, 743, 5*w^2 - 5*w - 33],
[773, 773, -6*w^2 + 10*w + 23],
[787, 787, -3*w^2 + 5*w + 1],
[809, 809, 4*w^2 - 12*w - 1],
[811, 811, -5*w^2 + 11*w + 5],
[821, 821, -7*w^2 + 5*w + 41],
[829, 829, w^2 + 7*w + 1],
[841, 29, 6*w^2 - 16*w - 9],
[853, 853, -11*w^2 + 9*w + 65],
[857, 857, -9*w^2 + 17*w + 25],
[857, 857, 4*w^2 - 4*w - 17],
[857, 857, 9*w^2 - 7*w - 59],
[859, 859, w^2 + 3*w - 13],
[859, 859, -3*w^2 - w + 25],
[859, 859, 6*w^2 - 8*w - 29],
[877, 877, 3*w^2 - w - 9],
[877, 877, 5*w^2 + 9*w - 7],
[877, 877, -10*w^2 + 20*w + 27],
[881, 881, -3*w^2 - w + 5],
[887, 887, 3*w^2 + w - 7],
[911, 911, 2*w^2 + 2*w - 13],
[937, 937, 2*w^2 - 8*w - 1],
[937, 937, -w^2 + 7*w + 1],
[937, 937, 2*w^2 - 10*w + 11],
[953, 953, -6*w^2 + 8*w + 25],
[953, 953, -5*w^2 + w + 41],
[953, 953, w^2 + 3*w - 11],
[977, 977, 4*w^2 + 2*w - 19],
[983, 983, 3*w^2 + 11*w + 3],
[991, 991, 2*w - 11],
[997, 997, 4*w^2 - 27],
[997, 997, -w^2 - 9*w - 17],
[997, 997, -4*w - 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 2*x^4 - 7*x^3 - 13*x^2 + 4*x + 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, 1/2*e^4 + 1/2*e^3 - 4*e^2 - 5/2*e + 7/2, -1/2*e^4 - 1/2*e^3 + 3*e^2 + 7/2*e + 1/2, -3/2*e^4 - 3/2*e^3 + 11*e^2 + 17/2*e - 13/2, e^4 - 7*e^2 + 6, -e^3 + e^2 + 7*e - 3, 1/2*e^4 + 1/2*e^3 - 4*e^2 - 9/2*e + 11/2, 3/2*e^4 + 1/2*e^3 - 9*e^2 - 5/2*e - 1/2, 1/2*e^4 + 3/2*e^3 - e^2 - 19/2*e - 11/2, -e^4 + e^3 + 8*e^2 - 5*e - 7, e^4 + 3*e^3 - 6*e^2 - 17*e + 3, -2*e^3 + 14*e + 4, 3/2*e^4 + 5/2*e^3 - 10*e^2 - 23/2*e + 15/2, e^4 - 9*e^2 - 2*e + 14, -e^3 - e^2 + 7*e + 7, e^4 - 9*e^2 - 4*e + 12, -e^4 - 2*e^3 + 7*e^2 + 12*e, -1/2*e^4 - 3/2*e^3 + 4*e^2 + 17/2*e - 1/2, 1/2*e^4 + 3/2*e^3 - 5*e^2 - 23/2*e + 17/2, e^4 + 3*e^3 - 8*e^2 - 17*e + 11, -e^4 + 9*e^2 - 4*e - 14, -e^4 - e^3 + 4*e^2 + 5*e + 9, e^4 + e^3 - 10*e^2 - 3*e + 19, -e^4 - 4*e^3 + 5*e^2 + 22*e + 2, -2*e^4 - 2*e^3 + 14*e^2 + 6*e - 12, -1/2*e^4 - 5/2*e^3 + 2*e^2 + 29/2*e + 17/2, 1/2*e^4 - 1/2*e^3 - 3*e^2 + 9/2*e - 15/2, -e^3 + e^2 + 11*e - 1, -5/2*e^4 - 7/2*e^3 + 15*e^2 + 43/2*e - 9/2, -e^4 + e^3 + 6*e^2 - 7*e + 1, -4*e + 8, 3*e^4 + 5*e^3 - 22*e^2 - 31*e + 7, 3/2*e^4 + 5/2*e^3 - 9*e^2 - 29/2*e + 3/2, 3/2*e^4 + 3/2*e^3 - 7*e^2 - 9/2*e - 19/2, 1/2*e^4 - 1/2*e^3 - 5*e^2 + 13/2*e + 9/2, -e^4 + 7*e^2 + 2*e + 4, -4*e^4 - 3*e^3 + 27*e^2 + 21*e - 9, 1/2*e^4 + 9/2*e^3 - e^2 - 51/2*e - 9/2, 2*e^4 + 2*e^3 - 12*e^2 - 10*e + 6, e^4 + 3*e^3 - 6*e^2 - 21*e + 11, e^4 - 11*e^2 - 4*e + 26, -3*e^4 - 4*e^3 + 19*e^2 + 24*e - 18, 2*e^4 + 3*e^3 - 13*e^2 - 21*e + 5, -e^4 - 3*e^3 + 6*e^2 + 9*e - 1, -e^4 + e^3 + 8*e^2 - 3*e - 13, -2*e^4 - 3*e^3 + 13*e^2 + 17*e + 7, -e^4 + e^3 + 10*e^2 - 9*e - 17, -2*e^4 + 10*e^2 - 4*e + 8, -e^4 + 3*e^2 + 10, 5*e^4 + 5*e^3 - 36*e^2 - 33*e + 15, -e^4 + e^3 + 6*e^2 - 9*e - 5, e^4 + 3*e^3 - 6*e^2 - 15*e + 5, -3/2*e^4 - 9/2*e^3 + 9*e^2 + 41/2*e - 3/2, -2*e^4 - e^3 + 13*e^2 - e - 1, -e^4 - e^3 + 10*e^2 + 9*e - 21, -e^4 + 3*e^3 + 8*e^2 - 13*e - 13, -9/2*e^4 - 11/2*e^3 + 31*e^2 + 59/2*e - 41/2, -3/2*e^4 + 1/2*e^3 + 13*e^2 - 15/2*e - 53/2, 5/2*e^4 + 9/2*e^3 - 17*e^2 - 51/2*e - 5/2, -4*e^4 - 4*e^3 + 32*e^2 + 28*e - 16, -2*e^3 - 4*e^2 + 14*e + 10, 4*e^4 + 6*e^3 - 30*e^2 - 30*e + 26, -2*e^4 - 4*e^3 + 18*e^2 + 28*e - 20, -2*e^4 + 2*e^3 + 16*e^2 - 14*e - 14, -e^4 + e^3 + 6*e^2 - 3*e - 7, 5/2*e^4 + 3/2*e^3 - 18*e^2 - 29/2*e + 37/2, -1/2*e^4 - 5/2*e^3 + 3*e^2 + 27/2*e + 9/2, 4*e^4 + 3*e^3 - 23*e^2 - 13*e + 5, 7/2*e^4 + 1/2*e^3 - 22*e^2 + 1/2*e + 31/2, -3/2*e^4 + 5/2*e^3 + 13*e^2 - 39/2*e - 57/2, 2*e^4 + 3*e^3 - 17*e^2 - 17*e + 27, -e^4 + e^3 + 8*e^2 - 11*e - 17, -1/2*e^4 - 7/2*e^3 - 2*e^2 + 37/2*e + 19/2, 2*e^3 + 2*e^2 - 14*e + 8, 7/2*e^4 + 13/2*e^3 - 25*e^2 - 77/2*e + 31/2, -e^4 - 3*e^3 + 6*e^2 + 13*e - 5, -2*e^3 + 6*e - 4, 3*e^4 + 8*e^3 - 21*e^2 - 44*e + 22, -e^4 - e^3 + 10*e^2 + 7*e - 35, 5*e^4 + 5*e^3 - 40*e^2 - 31*e + 27, -5/2*e^4 + 7/2*e^3 + 21*e^2 - 49/2*e - 71/2, -1/2*e^4 - 3/2*e^3 + 5*e^2 + 11/2*e - 5/2, -e^4 - 5*e^3 + 6*e^2 + 19*e - 3, 2*e^4 + 4*e^3 - 10*e^2 - 32*e - 8, -e^4 - e^3 + 10*e^2 + 15*e - 7, -e^4 - 4*e^3 + 11*e^2 + 26*e - 12, -4*e^4 - 2*e^3 + 30*e^2 + 14*e - 10, 4*e^3 - 2*e^2 - 36*e + 6, 5*e^4 + 3*e^3 - 38*e^2 - 7*e + 41, 4*e^2 + 4*e - 32, -3/2*e^4 - 9/2*e^3 + 16*e^2 + 51/2*e - 43/2, 3/2*e^4 - 5/2*e^3 - 13*e^2 + 47/2*e + 33/2, 6*e^4 + 7*e^3 - 43*e^2 - 37*e + 39, 3/2*e^4 - 1/2*e^3 - 13*e^2 - 1/2*e + 21/2, 4*e^4 - 34*e^2 + 28, 1/2*e^4 - 5/2*e^3 + 43/2*e + 5/2, 2*e^4 + 6*e^3 - 14*e^2 - 34*e + 4, 7/2*e^4 + 3/2*e^3 - 26*e^2 - 3/2*e + 33/2, 3*e^4 + 7*e^3 - 20*e^2 - 43*e + 17, 9/2*e^4 + 13/2*e^3 - 30*e^2 - 77/2*e + 7/2, 13/2*e^4 + 13/2*e^3 - 43*e^2 - 91/2*e + 27/2, e^4 + e^3 - 2*e^2 - 7*e - 27, -5*e^4 - 9*e^3 + 34*e^2 + 39*e - 15, -3*e^4 - 3*e^3 + 20*e^2 + 13*e - 27, -5/2*e^4 - 11/2*e^3 + 17*e^2 + 59/2*e - 17/2, -e^4 + 4*e^3 + 11*e^2 - 24*e - 8, -3*e^4 + e^3 + 22*e^2 + 3*e - 11, -2*e^4 - 5*e^3 + 15*e^2 + 31*e + 3, e^4 - 3*e^3 - 2*e^2 + 15*e - 23, -3*e^4 - 3*e^3 + 20*e^2 + 7*e - 17, e^4 + 6*e^3 - 3*e^2 - 34*e - 6, -5*e^4 - 3*e^3 + 38*e^2 + 21*e - 49, -3*e^4 - 9*e^3 + 16*e^2 + 55*e - 7, 1/2*e^4 - 5/2*e^3 - 5*e^2 + 61/2*e + 37/2, -5/2*e^4 + 5/2*e^3 + 25*e^2 - 25/2*e - 101/2, 2*e^4 - 8*e^3 - 14*e^2 + 40*e - 2, 6*e^4 + 6*e^3 - 42*e^2 - 38*e + 24, -e^4 + 3*e^3 + 4*e^2 - 29*e + 3, 5*e^4 + 6*e^3 - 33*e^2 - 40*e + 2, -10*e^4 - 13*e^3 + 67*e^2 + 67*e - 27, -4*e^4 - 4*e^3 + 32*e^2 + 20*e - 32, 5*e^4 + 3*e^3 - 34*e^2 - 21*e + 19, -7/2*e^4 - 3/2*e^3 + 23*e^2 + 37/2*e + 19/2, 2*e^4 + 6*e^3 - 16*e^2 - 34*e + 10, e^4 - 5*e^3 - 4*e^2 + 33*e + 7, 1/2*e^4 - 7/2*e^3 - 11*e^2 + 17/2*e + 71/2, -3*e^4 + 3*e^3 + 24*e^2 - 25*e - 43, -4*e^4 - 8*e^3 + 30*e^2 + 48*e - 26, -6*e^4 - 5*e^3 + 33*e^2 + 31*e + 9, -2*e^4 + e^3 + 13*e^2 - 3*e + 11, -3*e^4 - 12*e^3 + 21*e^2 + 60*e - 20, -e^4 + 2*e^3 + 3*e^2 - 16*e, -9/2*e^4 - 1/2*e^3 + 31*e^2 - 21/2*e - 67/2, -19/2*e^4 - 9/2*e^3 + 68*e^2 + 47/2*e - 111/2, -9/2*e^4 + 1/2*e^3 + 31*e^2 + 11/2*e - 29/2, 1/2*e^4 + 7/2*e^3 - 6*e^2 - 21/2*e + 37/2, 2*e^4 - 5*e^3 - 19*e^2 + 23*e + 17, -1/2*e^4 - 1/2*e^3 + 8*e^2 - 27/2*e - 47/2, 13/2*e^4 + 17/2*e^3 - 50*e^2 - 85/2*e + 79/2, -e^4 - 3*e^3 + 6*e^2 + 29*e + 29, e^4 - 3*e^3 - 6*e^2 + 25*e - 5, -2*e^4 + 4*e^3 + 22*e^2 - 24*e - 36, -15/2*e^4 - 15/2*e^3 + 61*e^2 + 77/2*e - 93/2, 9/2*e^4 + 5/2*e^3 - 23*e^2 - 19/2*e - 49/2, -15/2*e^4 - 11/2*e^3 + 53*e^2 + 73/2*e - 45/2, 3/2*e^4 + 17/2*e^3 - 4*e^2 - 99/2*e - 5/2, 2*e^4 + 6*e^3 - 16*e^2 - 34*e + 2, 4*e^4 + 7*e^3 - 29*e^2 - 33*e + 11, -3/2*e^4 - 5/2*e^3 + 23*e^2 + 41/2*e - 91/2, 7/2*e^4 - 1/2*e^3 - 17*e^2 + 11/2*e - 43/2, -3*e^4 - 4*e^3 + 15*e^2 + 20*e - 14, 5/2*e^4 + 15/2*e^3 - 19*e^2 - 103/2*e + 29/2, e^4 - e^3 - 6*e^2 + 7*e + 21, -7/2*e^4 - 13/2*e^3 + 32*e^2 + 99/2*e - 75/2, 11/2*e^4 + 11/2*e^3 - 40*e^2 - 79/2*e + 93/2, e^4 + 9*e^3 - 6*e^2 - 55*e - 5, 2*e^4 + e^3 - 21*e^2 + e + 57, -7/2*e^4 - 7/2*e^3 + 32*e^2 + 55/2*e - 29/2, 5/2*e^4 - 5/2*e^3 - 23*e^2 + 25/2*e + 49/2, -1/2*e^4 + 5/2*e^3 + 4*e^2 - 51/2*e + 3/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 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;