/*
  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, 10, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {1, 1, 1}>;

primesArray := [
[2, 2, -w + 1],
[5, 5, w^3 + w^2 - 4*w + 1],
[7, 7, -w^3 + 6*w - 2],
[8, 2, w^3 - 7*w + 3],
[11, 11, -w^3 + 6*w - 4],
[17, 17, w^3 + w^2 - 6*w - 1],
[17, 17, -w^2 - w + 3],
[31, 31, 2*w^3 + w^2 - 13*w + 3],
[37, 37, -3*w^3 - w^2 + 18*w - 3],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^3 + 2*w^2 - 11*w - 2],
[47, 47, -3*w^3 - w^2 + 19*w - 6],
[59, 59, -2*w^3 + 13*w - 6],
[59, 59, -w^3 - w^2 + 5*w + 2],
[59, 59, w^2 + w - 1],
[59, 59, w^2 + 2*w - 6],
[61, 61, -w^3 + w^2 + 8*w - 7],
[67, 67, w^3 + 2*w^2 - 4*w - 4],
[73, 73, -2*w^3 - w^2 + 12*w - 4],
[81, 3, -3],
[107, 107, 2*w^2 + 2*w - 11],
[109, 109, 2*w^3 + w^2 - 11*w + 3],
[113, 113, w^3 + w^2 - 7*w - 2],
[113, 113, 2*w^3 - 12*w + 9],
[125, 5, -4*w^3 - 3*w^2 + 24*w + 2],
[131, 131, 5*w^3 + 3*w^2 - 31*w + 2],
[151, 151, 4*w^3 + 2*w^2 - 26*w + 1],
[151, 151, w^3 + 2*w^2 - 6*w - 10],
[167, 167, w + 4],
[167, 167, 4*w^3 + w^2 - 26*w + 10],
[173, 173, -2*w^3 + w^2 + 13*w - 11],
[179, 179, 4*w^3 + w^2 - 24*w + 4],
[179, 179, 2*w^3 + 2*w^2 - 12*w - 3],
[179, 179, -w^3 + 8*w - 4],
[179, 179, w^3 + 2*w^2 - 4*w - 6],
[181, 181, 6*w^3 + 3*w^2 - 39*w + 5],
[197, 197, -w^3 + 4*w - 4],
[199, 199, 2*w^3 - 12*w + 3],
[199, 199, 3*w - 2],
[223, 223, w^3 + w^2 - 7*w - 4],
[223, 223, 2*w^3 + w^2 - 10*w + 4],
[227, 227, 7*w^3 + 2*w^2 - 44*w + 12],
[227, 227, -4*w^3 - w^2 + 25*w - 9],
[229, 229, 8*w^3 + 4*w^2 - 51*w + 4],
[229, 229, 3*w^3 + w^2 - 18*w + 5],
[233, 233, -6*w^3 - 3*w^2 + 39*w - 7],
[233, 233, -w^3 - 2*w^2 + 8*w + 2],
[241, 241, 2*w^3 + w^2 - 10*w],
[251, 251, w^3 + w^2 - 7*w - 6],
[257, 257, -6*w^3 - w^2 + 39*w - 17],
[263, 263, -2*w^3 + 13*w - 4],
[263, 263, -w^3 + 6*w - 8],
[269, 269, -8*w^3 - 4*w^2 + 50*w - 7],
[269, 269, -w^3 + w^2 + 7*w - 6],
[277, 277, 2*w^3 + 2*w^2 - 11*w + 4],
[277, 277, w^3 + w^2 - 4*w - 3],
[281, 281, -w^2 - 3*w + 5],
[281, 281, 2*w^3 + 3*w^2 - 9*w - 3],
[283, 283, -3*w^3 - 4*w^2 + 12*w + 2],
[289, 17, -2*w^3 - w^2 + 13*w - 5],
[293, 293, -w^3 - 2*w^2 + 6*w + 6],
[293, 293, w^3 + w^2 - 8*w - 1],
[307, 307, 2*w^3 - 11*w + 12],
[317, 317, 2*w^3 - 14*w + 11],
[317, 317, -2*w^3 - 2*w^2 + 8*w + 1],
[337, 337, 2*w^2 + 4*w - 11],
[343, 7, -5*w^3 - w^2 + 31*w - 10],
[349, 349, 2*w^3 - 11*w + 8],
[349, 349, w^2 + w - 9],
[353, 353, 4*w^3 + 4*w^2 - 23*w - 8],
[373, 373, -2*w - 3],
[373, 373, -4*w^3 + 26*w - 13],
[389, 389, 6*w^3 + 3*w^2 - 37*w + 1],
[389, 389, -2*w^3 - w^2 + 14*w - 6],
[401, 401, 2*w^3 - 12*w + 13],
[421, 421, 2*w^2 + 2*w - 15],
[421, 421, 3*w^3 + 2*w^2 - 20*w + 4],
[431, 431, w^3 + 3*w^2 - 2*w - 9],
[433, 433, 2*w^3 + 2*w^2 - 12*w + 3],
[433, 433, -7*w^3 - 2*w^2 + 44*w - 16],
[433, 433, -3*w^3 + 18*w - 14],
[433, 433, -w^3 + w^2 + 9*w - 10],
[439, 439, 5*w^3 + 3*w^2 - 31*w - 2],
[443, 443, -w^3 - w^2 + 3*w - 4],
[443, 443, -4*w^3 + 27*w - 14],
[449, 449, w^3 + w^2 - 4*w - 5],
[449, 449, w^3 + 3*w^2 - 3*w - 8],
[449, 449, 2*w^3 - 11*w + 6],
[449, 449, -3*w^3 - 2*w^2 + 18*w - 4],
[461, 461, -3*w^3 + w^2 + 20*w - 15],
[463, 463, 2*w^3 - 10*w + 7],
[479, 479, -2*w^3 - 3*w^2 + 11*w + 9],
[491, 491, w^3 + w^2 - 4*w + 5],
[499, 499, -2*w^3 - 2*w^2 + 14*w - 3],
[499, 499, -6*w^3 - 2*w^2 + 39*w - 10],
[499, 499, 4*w^3 + w^2 - 24*w + 14],
[499, 499, -5*w^3 - 3*w^2 + 30*w - 5],
[503, 503, -2*w^3 + w^2 + 14*w - 18],
[509, 509, 4*w^3 + 3*w^2 - 24*w],
[509, 509, 4*w^3 + 2*w^2 - 23*w + 4],
[547, 547, w^3 + 2*w^2 - 2*w - 6],
[547, 547, 6*w^3 + 2*w^2 - 38*w + 7],
[563, 563, -5*w^3 - w^2 + 33*w - 12],
[569, 569, 2*w^3 - w^2 - 15*w + 13],
[569, 569, -4*w^3 - 2*w^2 + 24*w - 7],
[577, 577, -w^2 - 2],
[577, 577, 3*w^3 - 20*w + 8],
[587, 587, 6*w^3 + 4*w^2 - 36*w - 1],
[587, 587, -w^3 + w^2 + 9*w - 8],
[593, 593, 6*w^3 + 3*w^2 - 36*w + 2],
[593, 593, -3*w^3 - w^2 + 21*w - 4],
[599, 599, -2*w^3 - w^2 + 12*w - 10],
[599, 599, -4*w^3 - 2*w^2 + 25*w - 6],
[601, 601, -2*w^3 + 2*w^2 + 15*w - 16],
[607, 607, w^3 - w^2 - 7*w + 4],
[607, 607, -2*w^3 - 2*w^2 + 10*w + 3],
[631, 631, 2*w^2 + 4*w - 7],
[641, 641, -3*w^3 - 2*w^2 + 16*w],
[643, 643, w^3 + w^2 - 3*w - 4],
[643, 643, w^3 + 3*w^2 - 6*w - 13],
[653, 653, -w^3 + 6*w + 2],
[653, 653, -3*w^2 - 5*w + 11],
[659, 659, -2*w^3 + 10*w - 5],
[661, 661, w^3 - w^2 - 8*w + 3],
[673, 673, 3*w^3 + 4*w^2 - 14*w - 2],
[677, 677, 4*w^3 + 2*w^2 - 23*w + 6],
[677, 677, 2*w^3 - 13*w + 14],
[683, 683, 2*w^3 + w^2 - 13*w + 7],
[701, 701, 8*w^3 + 4*w^2 - 51*w + 8],
[701, 701, -4*w^3 - w^2 + 24*w - 12],
[709, 709, -4*w^3 - w^2 + 27*w - 7],
[709, 709, 3*w^3 + 3*w^2 - 18*w - 5],
[709, 709, -3*w^3 + w^2 + 21*w - 14],
[709, 709, w^2 - w - 7],
[719, 719, -9*w^3 - 4*w^2 + 58*w - 8],
[719, 719, 11*w^3 + 5*w^2 - 68*w + 9],
[727, 727, -w^3 + 4*w - 6],
[733, 733, 2*w^3 + w^2 - 13*w + 9],
[739, 739, -2*w^2 - 3*w + 4],
[743, 743, 3*w^3 + 3*w^2 - 19*w - 4],
[751, 751, 2*w^3 + 3*w^2 - 10*w - 2],
[761, 761, -12*w^3 - 5*w^2 + 75*w - 9],
[761, 761, -3*w^3 - 3*w^2 + 17*w],
[769, 769, -6*w^3 - 4*w^2 + 37*w],
[769, 769, -8*w^3 - 3*w^2 + 51*w - 7],
[773, 773, -w^2 - 2*w - 2],
[787, 787, -5*w^3 - 5*w^2 + 28*w + 7],
[811, 811, 7*w^3 + 2*w^2 - 42*w + 6],
[811, 811, -2*w^3 - w^2 + 9*w - 7],
[823, 823, -3*w^3 - 2*w^2 + 14*w - 10],
[829, 829, 4*w^3 + 3*w^2 - 20*w + 6],
[829, 829, -7*w^3 - 2*w^2 + 44*w - 14],
[841, 29, w^3 + 4*w^2 - 2*w - 14],
[841, 29, -w^2 + 10],
[857, 857, -3*w^3 - w^2 + 20*w - 9],
[859, 859, -w^3 + 2*w^2 + 10*w - 18],
[863, 863, 11*w^3 + 5*w^2 - 69*w + 6],
[863, 863, w^3 - 2*w^2 - 10*w + 12],
[877, 877, 3*w^3 - 18*w + 10],
[883, 883, 4*w^3 + 3*w^2 - 22*w],
[887, 887, 3*w^3 + 3*w^2 - 15*w - 2],
[887, 887, w^2 + 4*w - 10],
[887, 887, w^3 + w^2 - 3*w - 8],
[887, 887, w - 6],
[907, 907, 6*w^3 + w^2 - 38*w + 16],
[919, 919, -2*w^3 - 3*w^2 + 11*w + 1],
[919, 919, 5*w^3 - w^2 - 34*w + 27],
[937, 937, -w^3 - w^2 + 8*w + 3],
[941, 941, -4*w^3 + 25*w - 20],
[947, 947, -2*w^3 + w^2 + 8*w - 6],
[967, 967, -7*w^3 - w^2 + 44*w - 23],
[971, 971, 2*w^2 + 2*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 8*x^4 + 10*x^2 - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^4 - 7*e^2 + 5, -e^5 + 8*e^3 - 11*e, e^3 - 6*e, 2*e^5 - 16*e^3 + 20*e, 1, 3*e^5 - 23*e^3 + 21*e, -4*e^5 + 30*e^3 - 26*e, 2*e^5 - 15*e^3 + 14*e, 2*e^5 - 17*e^3 + 22*e, -e^4 + 7*e^2 - 2, -e^5 + 8*e^3 - 9*e, -e^5 + 10*e^3 - 23*e, 2*e^4 - 12*e^2 + 6, -4*e^5 + 30*e^3 - 28*e, -e^4 + 9*e^2 - 8, -e^5 + 7*e^3 - e, 0, -5*e^5 + 38*e^3 - 37*e, -2*e^4 + 16*e^2 - 19, -e^4 + 5*e^2 + 8, 5*e^5 - 39*e^3 + 45*e, -e^4 + 3*e^2 + 15, -e^5 + 9*e^3 - 13*e, e^4 - 9*e^2 + 7, 2*e^4 - 10*e^2 - 8, -3*e^4 + 17*e^2 + 4, -4*e^4 + 24*e^2 - 12, -2, -e^5 + 10*e^3 - 21*e, 8*e^5 - 65*e^3 + 80*e, -2*e^5 + 14*e^3 - 6*e, -2*e^4 + 16*e^2 - 6, 2*e^5 - 10*e^3 - 12*e, -4*e^4 + 26*e^2 - 2, 3*e^5 - 23*e^3 + 19*e, -4*e^4 + 32*e^2 - 21, -2*e^5 + 12*e^3 - 4*e, 2*e^3 - 8*e, -e^4 + 13*e^2 - 16, 8*e^5 - 64*e^3 + 82*e, 4*e^5 - 32*e^3 + 40*e, -6*e^5 + 46*e^3 - 56*e, -3*e^4 + 19*e^2 - 4, e^5 - 3*e^3 - 17*e, e^5 - 5*e^3 - 9*e, -3*e^4 + 23*e^2 - 19, -4*e^4 + 26*e^2 - 25, -e^4 + 5*e^2 + 16, -11*e^5 + 84*e^3 - 85*e, -12*e^5 + 90*e^3 - 90*e, -e^4 + 7*e^2 + 10, -5*e^5 + 37*e^3 - 25*e, -4*e^5 + 35*e^3 - 54*e, -2*e^5 + 16*e^3 - 14*e, -e^4 + 11*e^2 - 29, -10*e^5 + 74*e^3 - 68*e, 4*e^4 - 28*e^2 + 6, 4*e^4 - 30*e^2 + 26, 8*e^5 - 58*e^3 + 50*e, 2*e^4 - 18*e^2 + 22, 2*e^4 - 10*e^2 + 1, 5*e^4 - 31*e^2 + 12, 3*e^5 - 23*e^3 + 19*e, -3*e^4 + 15*e^2 + 1, 5*e^4 - 37*e^2 + 11, 7*e^5 - 56*e^3 + 63*e, -14*e^5 + 106*e^3 - 100*e, -6*e^4 + 44*e^2 - 19, 4*e^4 - 24*e^2 - 9, e^4 + e^2 - 22, 2*e^5 - 21*e^3 + 44*e, 5*e^4 - 39*e^2 + 13, 9*e^5 - 71*e^3 + 91*e, 4*e^4 - 30*e^2 + 8, e^4 - 9*e^2 + 31, -13*e^5 + 97*e^3 - 93*e, -3*e^4 + 15*e^2 - 4, 8*e^5 - 61*e^3 + 74*e, 9*e^5 - 72*e^3 + 75*e, -6*e^5 + 44*e^3 - 26*e, 5*e^5 - 35*e^3 + 15*e, -2*e^4 + 12*e^2 - 16, -4*e^4 + 36*e^2 - 32, -2*e^5 + 20*e^3 - 40*e, 4*e^2 - 11, -6*e^4 + 46*e^2 - 39, 6*e^5 - 50*e^3 + 80*e, 14*e^5 - 112*e^3 + 126*e, 7*e^5 - 59*e^3 + 101*e, -3*e^5 + 26*e^3 - 29*e, e^4 - 9*e^2 + 4, -3*e^4 + 23*e^2 - 16, 14*e^5 - 114*e^3 + 138*e, 8*e^5 - 66*e^3 + 90*e, -9*e^5 + 64*e^3 - 43*e, 4*e^5 - 30*e^3 + 22*e, 2*e^4 - 10*e^2 - 10, 4*e^4 - 24*e^2 + 11, -e^5 + 5*e^3 + 19*e, -6*e^4 + 46*e^2 - 26, -4*e^5 + 28*e^3 - 6*e, 12*e^5 - 94*e^3 + 96*e, 6*e^5 - 43*e^3 + 40*e, -9*e^5 + 67*e^3 - 41*e, 8*e^4 - 66*e^2 + 48, 11*e^5 - 91*e^3 + 115*e, 3*e^4 - 17*e^2 + 22, 2*e^5 - 16*e^3 + 8*e, -3*e^4 + 29*e^2 - 19, 6*e^5 - 41*e^3 + 30*e, 5*e^4 - 43*e^2 + 38, -4*e^5 + 24*e^3 + 2*e, -15*e^5 + 119*e^3 - 137*e, 6*e^5 - 46*e^3 + 40*e, -6*e^4 + 34*e^2 + 6, -12*e^5 + 94*e^3 - 94*e, -4*e^4 + 36*e^2 - 38, 7*e^4 - 51*e^2 + 26, -36, -5*e^4 + 25*e^2 + 1, -6*e^5 + 52*e^3 - 70*e, 9*e^5 - 64*e^3 + 33*e, -3*e^5 + 25*e^3 - 39*e, -7*e^4 + 53*e^2 - 35, -2*e^5 + 19*e^3 - 38*e, 5*e^4 - 33*e^2 + 9, -14*e^5 + 110*e^3 - 138*e, -12*e^5 + 95*e^3 - 108*e, 2*e^5 - 15*e^3 + 20*e, 3*e^5 - 31*e^3 + 63*e, -5*e^4 + 41*e^2 - 53, -6*e^5 + 50*e^3 - 56*e, e^4 - 7*e^2 - 22, 16*e^5 - 126*e^3 + 146*e, 10*e^5 - 76*e^3 + 80*e, -2*e^4 + 20*e^2 - 4, -19*e^5 + 139*e^3 - 119*e, -8*e^5 + 72*e^3 - 116*e, 6*e^4 - 38*e^2 + 22, 5*e^4 - 37*e^2 + 34, 15*e^5 - 117*e^3 + 143*e, 3*e^4 - 23*e^2 + 29, -e^4 + 11*e^2 - 13, 23*e^5 - 175*e^3 + 167*e, 6*e^4 - 46*e^2 + 22, 5*e^4 - 33*e^2 + 34, -7*e^5 + 60*e^3 - 99*e, -8*e^4 + 48*e^2 - 26, 3*e^4 - 25*e^2 + 24, -e^5 + e^3 + 37*e, 16*e^5 - 119*e^3 + 114*e, -3*e^4 + 19*e^2 - 21, 9*e^4 - 55*e^2 + 13, -21*e^5 + 161*e^3 - 159*e, -13*e^4 + 87*e^2 - 60, -2*e^2 - 2, 15*e^5 - 122*e^3 + 171*e, 22*e^5 - 168*e^3 + 170*e, -4*e^4 + 34*e^2 - 16, -2*e^4 + 10*e^2 + 8, e^4 + 5*e^2 - 20, -7*e^4 + 57*e^2 - 44, 2*e^4 - 14*e^2 + 2, -22*e^5 + 168*e^3 - 164*e, -4*e^4 + 32*e^2 + 4, -13*e^5 + 100*e^3 - 107*e, 8*e^4 - 56*e^2 + 46, 2*e^5 - 6*e^3 - 32*e, 3*e^5 - 20*e^3 + 15*e, 18*e^5 - 136*e^3 + 134*e, -5*e^5 + 38*e^3 - 21*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;