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

NN := ideal<ZF | {31, 31, 2*w - 3}>;

primesArray := [
[3, 3, w + 1],
[3, 3, w - 1],
[7, 7, w^2 - 2],
[8, 2, 2],
[11, 11, -w^2 + w + 1],
[23, 23, -w - 3],
[29, 29, -w^2 + 2*w + 4],
[31, 31, 2*w - 3],
[41, 41, -2*w^2 + 3*w + 6],
[43, 43, w^2 - 3*w + 3],
[47, 47, w^2 + w - 4],
[49, 7, 2*w^2 - 3*w - 3],
[53, 53, w^2 - 3*w - 2],
[59, 59, 2*w^2 - w - 5],
[59, 59, w^2 - w - 7],
[59, 59, -w^2 - w + 7],
[67, 67, 2*w^2 - 3*w - 7],
[73, 73, -w^2 + 4*w - 5],
[79, 79, w^2 - 8],
[79, 79, w^2 - 5*w + 5],
[79, 79, w^2 - 3*w - 3],
[83, 83, -3*w^2 + w + 9],
[89, 89, 2*w^2 - w - 2],
[97, 97, w^2 - 3*w - 6],
[101, 101, 3*w - 4],
[103, 103, 3*w - 5],
[107, 107, -3*w^2 + 10],
[107, 107, w^2 - 3*w - 5],
[109, 109, w^2 + 2*w - 4],
[121, 11, 3*w^2 - 2*w - 9],
[125, 5, -5],
[127, 127, w^2 + w - 10],
[137, 137, 3*w^2 - 3*w - 8],
[139, 139, -5*w^2 + 3*w + 18],
[149, 149, 3*w^2 - 4*w - 6],
[157, 157, w - 6],
[163, 163, -4*w^2 + 2*w + 13],
[163, 163, -w^2 - 3],
[163, 163, 2*w^2 - 4*w - 11],
[181, 181, 3*w^2 - 11],
[193, 193, 3*w^2 - 3*w - 7],
[193, 193, -3*w^2 + 3*w + 16],
[193, 193, w^2 + 2*w - 7],
[197, 197, 2*w^2 + w - 8],
[211, 211, 4*w^2 - 5*w - 10],
[227, 227, -w - 6],
[229, 229, w^2 - 4*w - 7],
[233, 233, 3*w^2 - 5],
[239, 239, 2*w^2 - 3*w - 13],
[251, 251, 3*w^2 - 4*w - 3],
[257, 257, 3*w^2 - w - 6],
[257, 257, 3*w^2 - 5*w - 9],
[257, 257, w^2 + 3*w - 5],
[263, 263, 2*w^2 + w - 11],
[277, 277, 3*w^2 - 2*w - 6],
[281, 281, 3*w^2 - 3*w - 5],
[281, 281, -w^2 - 4],
[281, 281, 4*w^2 - 3*w - 12],
[293, 293, 2*w^2 - 5*w - 5],
[307, 307, 3*w^2 - 2*w - 3],
[317, 317, 3*w^2 - w - 18],
[317, 317, -w^2 + 2*w - 5],
[317, 317, -4*w^2 + 7*w + 4],
[349, 349, -w^2 + 3*w - 6],
[353, 353, 3*w^2 - 6*w - 7],
[353, 353, w^2 - w - 10],
[353, 353, -5*w - 3],
[367, 367, -4*w^2 - w + 7],
[379, 379, 4*w^2 - 3*w - 11],
[383, 383, 4*w^2 - w - 10],
[389, 389, w^2 - 5*w - 4],
[389, 389, -5*w^2 + 7*w + 11],
[389, 389, -7*w^2 + 4*w + 25],
[397, 397, -w^2 + w - 5],
[397, 397, 5*w^2 - 6*w - 22],
[397, 397, -5*w^2 - w + 11],
[401, 401, 2*w^2 - w - 14],
[401, 401, 5*w^2 - 3*w - 16],
[401, 401, w^2 + 3*w - 8],
[409, 409, w^2 + 3*w - 9],
[431, 431, 2*w - 9],
[433, 433, w^2 - 3*w - 11],
[439, 439, w^2 - 11],
[439, 439, 5*w - 6],
[439, 439, 3*w^2 + 2*w - 9],
[443, 443, -6*w^2 + 3*w + 20],
[443, 443, -w^2 - w - 5],
[443, 443, 2*w^2 - 5*w - 11],
[461, 461, 3*w^2 - 4*w - 18],
[463, 463, 3*w^2 - 6*w - 17],
[463, 463, -4*w^2 - 3*w + 9],
[463, 463, 3*w^2 - 6*w - 8],
[467, 467, 2*w^2 - 5*w - 8],
[487, 487, 2*w^2 + 3*w - 7],
[499, 499, 5*w^2 - 8*w - 8],
[503, 503, 3*w^2 + w - 12],
[509, 509, w^2 - 6*w - 11],
[523, 523, 3*w^2 - 9*w - 2],
[529, 23, -w^2 + 4*w - 8],
[541, 541, 2*w^2 + 2*w - 11],
[569, 569, 4*w^2 - 3*w - 9],
[587, 587, -w^2 - 4*w + 13],
[599, 599, 3*w - 11],
[601, 601, -3*w^2 + 10*w - 9],
[607, 607, 4*w^2 - 6*w - 15],
[613, 613, w - 9],
[619, 619, 5*w^2 - 2*w - 14],
[631, 631, 4*w^2 - 5*w - 4],
[647, 647, 4*w^2 - 4*w - 7],
[647, 647, w^2 - 12],
[647, 647, 4*w^2 - w - 7],
[653, 653, 4*w^2 - 6*w - 17],
[659, 659, 3*w^2 - 6*w - 10],
[659, 659, 3*w^2 + w - 15],
[659, 659, 4*w^2 - 17],
[661, 661, 2*w^2 - 3*w - 15],
[661, 661, -5*w^2 + 5*w + 26],
[661, 661, 5*w^2 - 12*w - 4],
[673, 673, w^2 + w - 13],
[677, 677, w^2 - 3*w - 12],
[683, 683, 5*w^2 - 4*w - 14],
[709, 709, 4*w^2 - 2*w - 7],
[733, 733, 3*w^2 - 6*w - 11],
[739, 739, -2*w - 9],
[743, 743, 6*w - 7],
[751, 751, w^2 - 6*w - 9],
[757, 757, 4*w^2 - 3*w - 5],
[761, 761, -7*w^2 + 8*w + 31],
[769, 769, w^2 - 6*w - 6],
[773, 773, -w - 9],
[773, 773, w^2 - 4*w - 13],
[773, 773, 5*w^2 - 18],
[787, 787, 3*w^2 - 6*w - 14],
[797, 797, w^2 - 6*w - 8],
[811, 811, 2*w^2 + 3*w - 10],
[811, 811, -2*w^2 - 3*w + 16],
[811, 811, -6*w^2 + 11*w + 6],
[823, 823, w^2 + 5*w - 7],
[829, 829, -5*w^2 - 2*w + 14],
[839, 839, 2*w^2 - 6*w - 9],
[841, 29, 2*w^2 - 7*w - 5],
[853, 853, 5*w^2 - 28],
[857, 857, 7*w^2 - 9*w - 23],
[877, 877, -8*w^2 + 3*w + 31],
[881, 881, 6*w^2 - 9*w - 11],
[907, 907, 4*w^2 - 7*w - 13],
[911, 911, 2*w^2 + 3*w - 15],
[911, 911, -w^2 - 7],
[911, 911, 3*w^2 + 3*w - 10],
[919, 919, w^2 - 7*w - 4],
[941, 941, 6*w^2 - 11*w - 12],
[947, 947, 5*w^2 - w - 11],
[947, 947, 3*w^2 - 8*w - 6],
[947, 947, 2*w^2 + 3*w - 12],
[953, 953, -5*w^2 - 4*w + 11],
[953, 953, 4*w^2 - w - 25],
[953, 953, 6*w^2 - 7*w - 15],
[961, 31, 4*w^2 + 2*w - 13],
[977, 977, -2*w^2 + 3*w - 6],
[991, 991, 5*w^2 - 5*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - x^4 - 9*x^3 + 8*x^2 + 12*x - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/4*e^4 - 1/4*e^3 - 9/4*e^2 + e + 3, -1/2*e^4 - 1/2*e^3 + 9/2*e^2 + 3*e - 6, 1/2*e^4 - 1/2*e^3 - 7/2*e^2 + 3*e + 2, -1/4*e^4 - 1/4*e^3 + 11/4*e^2 + 3/2*e - 3, -1/2*e^3 - 1/2*e^2 + 7/2*e + 5, 1/2*e^4 - 1/2*e^3 - 11/2*e^2 + 4*e + 6, 1, 5/4*e^4 + 1/4*e^3 - 39/4*e^2 - 7/2*e + 11, e^4 - 8*e^2 + e + 2, -1/2*e^4 + 1/2*e^3 + 9/2*e^2 - 4*e - 2, e^3 + e^2 - 7*e - 2, 1/4*e^4 - 5/4*e^3 - 5/4*e^2 + 11*e + 1, -3/2*e^4 + e^3 + 11*e^2 - 9/2*e - 3, -3/4*e^4 + 3/4*e^3 + 19/4*e^2 - 2*e + 1, 5/4*e^4 + 5/4*e^3 - 47/4*e^2 - 17/2*e + 17, e^4 + e^3 - 9*e^2 - 6*e + 6, e^4 - 1/2*e^3 - 17/2*e^2 + 13/2*e + 5, 1/2*e^4 + 1/2*e^3 - 7/2*e^2 - 6*e, -2*e^4 - 1/2*e^3 + 39/2*e^2 + 11/2*e - 25, 3/4*e^4 + 1/4*e^3 - 23/4*e^2 + 2*e + 1, 2*e^3 - 11*e, e^2 - 2*e - 8, -e^4 - e^3 + 8*e^2 + 8*e - 12, -3/2*e^4 + 12*e^2 - 7/2*e - 7, -e^4 + e^3 + 9*e^2 - 4*e - 12, -11/4*e^4 - 1/4*e^3 + 87/4*e^2 + e - 19, -1/2*e^4 + 7*e^2 - 1/2*e - 9, -1/2*e^3 + 1/2*e^2 + 3/2*e - 7, 1/2*e^4 + 1/2*e^3 - 3/2*e^2 - 2*e - 6, 1/4*e^4 - 1/4*e^3 - 21/4*e^2 + 4*e + 11, 3/4*e^4 + 7/4*e^3 - 25/4*e^2 - 17/2*e + 3, 1/2*e^4 + 1/2*e^3 - 11/2*e^2 - 10*e + 16, -7/4*e^4 + 9/4*e^3 + 45/4*e^2 - 25/2*e - 1, -3/2*e^4 + 1/2*e^3 + 21/2*e^2 - 5*e - 6, 13/4*e^4 - 3/4*e^3 - 111/4*e^2 + 7/2*e + 29, -1/2*e^4 + 6*e^2 - 5/2*e - 19, -3*e^4 + e^3 + 23*e^2 - 5*e - 20, 3/2*e^4 + 1/2*e^3 - 33/2*e^2 - 4*e + 28, -1/2*e^4 - e^3 + 3*e^2 + 5/2*e + 1, 1/4*e^4 - 5/4*e^3 - 13/4*e^2 + 5*e + 17, 3/2*e^4 - 3/2*e^3 - 31/2*e^2 + 12*e + 16, 3/2*e^4 - 3/2*e^3 - 25/2*e^2 + 2*e + 8, -3/2*e^4 - 1/2*e^3 + 35/2*e^2 + 2*e - 34, -3/2*e^4 + 3/2*e^3 + 19/2*e^2 - 5*e + 8, -2*e^4 - e^3 + 17*e^2 + 3*e - 20, -7/4*e^4 + 1/4*e^3 + 69/4*e^2 + 3/2*e - 25, e^4 - 2*e^3 - 8*e^2 + 13*e + 2, -3/2*e^4 + 1/2*e^3 + 33/2*e^2 - e - 28, -1/4*e^4 + 3/4*e^3 - 1/4*e^2 - 25/2*e + 7, -1/2*e^4 - 1/2*e^3 + 3/2*e^2 + 7*e + 6, -e^4 + 8*e^2 + 8*e - 6, -e^4 - 3*e^3 + 5*e^2 + 22*e - 2, 7/2*e^4 - 5/2*e^3 - 57/2*e^2 + 12*e + 26, 3/4*e^4 - 5/4*e^3 - 17/4*e^2 + 5/2*e - 1, 3/2*e^4 - 5/2*e^3 - 27/2*e^2 + 17*e + 26, -5/4*e^4 + 1/4*e^3 + 57/4*e^2 - 4*e - 31, 3/2*e^4 - 1/2*e^3 - 21/2*e^2 + 2*e, 1/2*e^4 - 1/2*e^3 - 9/2*e^2, -5/4*e^4 + 1/4*e^3 + 49/4*e^2 + 9*e - 25, 2*e^4 + 3/2*e^3 - 37/2*e^2 - 29/2*e + 19, 4*e^4 - 4*e^3 - 36*e^2 + 24*e + 30, -3/2*e^4 + 7/2*e^3 + 29/2*e^2 - 22*e - 26, 5/2*e^4 + 3/2*e^3 - 49/2*e^2 - 9*e + 42, 13/4*e^4 + 11/4*e^3 - 125/4*e^2 - 18*e + 53, -9/2*e^4 + 1/2*e^3 + 79/2*e^2 - 6*e - 34, 3*e^4 + 4*e^3 - 26*e^2 - 28*e + 38, 1/2*e^4 - 1/2*e^3 - 1/2*e^2 + 8*e - 22, 7/2*e^4 - 1/2*e^3 - 65/2*e^2 - 3*e + 48, -1/4*e^4 + 1/4*e^3 + 9/4*e^2 - 4*e + 7, 1/2*e^4 - 2*e^2 - 9/2*e - 3, -3/4*e^4 + 7/4*e^3 + 3/4*e^2 - 15*e + 23, 5/2*e^4 - 3/2*e^3 - 31/2*e^2 + 3*e - 8, -e^4 - e^3 + 5*e^2 + 4*e - 8, -1/4*e^4 + 17/4*e^3 + 25/4*e^2 - 32*e - 7, -5/2*e^4 - 9/2*e^3 + 43/2*e^2 + 27*e - 26, -7/2*e^4 + 1/2*e^3 + 61/2*e^2 - 5*e - 12, -9/2*e^4 + 1/2*e^3 + 83/2*e^2 - 2*e - 34, -e^4 + e^3 + 7*e^2 - 9*e - 22, e^3 - 3*e^2 - 19*e + 20, e^4 + e^3 - 9*e^2 - 10*e + 26, 6*e^4 - 3*e^3 - 47*e^2 + 16*e + 48, 1/4*e^4 + 9/4*e^3 - 35/4*e^2 - 31/2*e + 29, -5*e^4 + 3*e^3 + 41*e^2 - 21*e - 34, 2*e^4 - 18*e^2 + 36, -1/2*e^4 + 7/2*e^3 + 9/2*e^2 - 21*e, 2*e^4 - e^3 - 13*e^2 + 4*e + 8, 3/2*e^4 - 3/2*e^3 - 27/2*e^2 + 2*e + 4, 5/4*e^4 + 11/4*e^3 - 53/4*e^2 - 18*e + 25, 7/4*e^4 + 1/4*e^3 - 63/4*e^2 + 27, -23/4*e^4 + 13/4*e^3 + 197/4*e^2 - 31/2*e - 33, -5*e^4 + 2*e^3 + 40*e^2 - 19*e - 28, -3/2*e^4 - e^3 + 21*e^2 + 21/2*e - 47, 3/2*e^4 - 2*e^3 - 16*e^2 + 31/2*e + 21, -3/4*e^4 - 5/4*e^3 + 47/4*e^2 + 8*e - 33, 5/4*e^4 - 13/4*e^3 - 33/4*e^2 + 24*e + 23, 1/2*e^4 - 11/2*e^3 - 7/2*e^2 + 37*e + 12, -5/2*e^4 - 1/2*e^3 + 47/2*e^2 + 11*e - 34, 11/2*e^4 - 5/2*e^3 - 93/2*e^2 + 19*e + 40, 5/2*e^4 + 3/2*e^3 - 29/2*e^2 - 10*e - 12, 11/2*e^4 - e^3 - 49*e^2 - 3/2*e + 61, e^4 + e^3 - 13*e^2 + 26, -2*e^4 + 4*e^3 + 22*e^2 - 25*e - 26, -e^4 - 3*e^3 + 5*e^2 + 20*e - 10, e^4 - 7/2*e^3 - 19/2*e^2 + 57/2*e + 29, -7/2*e^4 - 3/2*e^3 + 65/2*e^2 + 13*e - 46, 5*e^3 - e^2 - 35*e + 6, -5/4*e^4 + 15/4*e^3 + 47/4*e^2 - 73/2*e - 5, 2*e^4 - 17*e^2 + 10*e + 8, -e^4 - 1/2*e^3 + 3/2*e^2 + 21/2*e + 15, 4*e^4 - 3*e^3 - 29*e^2 + 19*e + 24, -7*e^4 - e^3 + 57*e^2 + 10*e - 62, 11/2*e^4 + 7/2*e^3 - 97/2*e^2 - 19*e + 60, -5/4*e^4 + 15/4*e^3 + 43/4*e^2 - 53/2*e - 1, 7/2*e^4 + 5/2*e^3 - 51/2*e^2 - 24*e + 18, 1/2*e^4 - 3/2*e^3 + 7/2*e^2 + 13*e - 18, e^4 - 3*e^3 - 3*e^2 + 9*e - 20, -7/4*e^4 + 11/4*e^3 + 39/4*e^2 - 23*e + 15, -5*e^4 + 4*e^3 + 35*e^2 - 27*e - 14, -2*e^4 + e^3 + 21*e^2 + 3*e - 32, 5*e^4 + e^3 - 41*e^2 - 12*e + 44, 17/4*e^4 + 3/4*e^3 - 153/4*e^2 + e + 47, 5/4*e^4 + 3/4*e^3 - 37/4*e^2 - 14*e - 15, -1/2*e^4 + 9/2*e^3 + 13/2*e^2 - 34*e - 18, 15/4*e^4 + 21/4*e^3 - 131/4*e^2 - 27*e + 35, -7/2*e^4 - 1/2*e^3 + 67/2*e^2 + 6*e - 50, -5/4*e^4 - 5/4*e^3 + 23/4*e^2 + 9/2*e + 13, 11/2*e^4 - 1/2*e^3 - 105/2*e^2 - 7*e + 74, e^4 + e^3 - 13*e^2 - 22*e + 26, -2*e^4 + 3*e^3 + 17*e^2 - 13*e - 26, -5/4*e^4 - 5/4*e^3 + 31/4*e^2 + 19/2*e - 23, -3/4*e^4 - 11/4*e^3 + 49/4*e^2 + 49/2*e - 7, -5*e^4 + 7/2*e^3 + 79/2*e^2 - 75/2*e - 33, -7/2*e^4 + 5/2*e^3 + 57/2*e^2 - 20*e - 12, -6*e^4 + 48*e^2 - 40, 3/2*e^4 + 3/2*e^3 - 41/2*e^2 - 3*e + 42, 4*e^4 - 4*e^3 - 29*e^2 + 32*e + 20, 9/2*e^4 - 5/2*e^3 - 61/2*e^2 + 13*e - 2, -1/2*e^4 - 2*e^3 + 8*e^2 + 11/2*e - 9, -9/2*e^4 + e^3 + 35*e^2 - 5/2*e - 17, -4*e^4 - 9/2*e^3 + 75/2*e^2 + 51/2*e - 33, 9/2*e^4 - 15/2*e^3 - 83/2*e^2 + 43*e + 44, 9/4*e^4 + 9/4*e^3 - 91/4*e^2 - 17/2*e + 41, e^4 - 6*e^3 - 9*e^2 + 39*e + 4, 5*e^4 + 2*e^3 - 38*e^2 - 9*e + 28, -5/4*e^4 - 3/4*e^3 + 37/4*e^2 + 2*e - 5, -7/2*e^4 - e^3 + 25*e^2 + 11/2*e - 35, 9/2*e^4 + 3/2*e^3 - 83/2*e^2 + 58, -9/4*e^4 - 1/4*e^3 + 91/4*e^2 - 11/2*e - 53, 9/2*e^4 + 3/2*e^3 - 71/2*e^2 - 16*e + 34, -1/4*e^4 + 9/4*e^3 - 23/4*e^2 - 28*e + 31, -2*e^4 + 13*e^2 - 8*e + 24, -1/2*e^4 + 3/2*e^3 + 15/2*e^2 - 5*e - 20, -3/4*e^4 + 21/4*e^3 + 17/4*e^2 - 61/2*e + 3, -9/4*e^4 - 19/4*e^3 + 53/4*e^2 + 34*e + 1, -e^4 - 7/2*e^3 + 5/2*e^2 + 29/2*e + 31, 7/2*e^4 - 5/2*e^3 - 61/2*e^2 + 5*e + 56, 11/2*e^4 + 7/2*e^3 - 97/2*e^2 - 37*e + 74, -1/2*e^4 - 9/2*e^3 + 19/2*e^2 + 25*e - 28, -5*e^4 + 3*e^3 + 45*e^2 - 28*e - 58];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {31, 31, 2*w - 3}>] := -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;