/*
  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, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[3, 3, w + 1],
[5, 5, -w^3 + 5*w + 2],
[5, 5, w - 1],
[11, 11, -w^3 + 5*w],
[13, 13, -w^3 + w^2 + 6*w - 3],
[16, 2, 2],
[19, 19, 2*w^3 - w^2 - 11*w + 2],
[25, 5, -w^2 + w + 3],
[27, 3, w^3 - w^2 - 5*w + 4],
[31, 31, -w - 3],
[37, 37, -w^3 - w^2 + 6*w + 4],
[41, 41, w^3 + w^2 - 6*w - 5],
[43, 43, w^3 - 7*w - 2],
[47, 47, -2*w^3 + 11*w + 2],
[61, 61, w^2 - 3],
[67, 67, w^2 + w - 4],
[79, 79, -4*w^3 + 2*w^2 + 22*w - 9],
[97, 97, -3*w^3 + 2*w^2 + 19*w - 6],
[97, 97, -w^3 + 6*w - 3],
[97, 97, -3*w^3 + 16*w],
[97, 97, -w^3 + w^2 + 4*w - 3],
[101, 101, -2*w^3 + w^2 + 11*w - 6],
[103, 103, w^3 - w^2 - 7*w],
[109, 109, -w^3 - w^2 + 6*w + 2],
[121, 11, 2*w^3 - 11*w + 1],
[127, 127, -2*w^3 + w^2 + 13*w - 1],
[127, 127, w^3 - w^2 - 6*w - 2],
[157, 157, -2*w^3 + 2*w^2 + 10*w - 7],
[157, 157, w - 4],
[163, 163, 2*w^3 - w^2 - 13*w + 3],
[173, 173, -2*w^3 + w^2 + 13*w - 6],
[173, 173, 2*w^3 - 12*w + 1],
[173, 173, 2*w^3 - 11*w + 2],
[173, 173, w^2 + w - 5],
[179, 179, -2*w^3 + 10*w + 3],
[181, 181, -w^3 - w^2 + 7*w + 8],
[191, 191, w^3 - 5*w - 5],
[191, 191, 2*w^3 - w^2 - 9*w - 1],
[193, 193, -3*w^3 + 3*w^2 + 19*w - 7],
[197, 197, -w^3 + w^2 + 6*w - 7],
[223, 223, w^3 + w^2 - 6*w + 1],
[227, 227, 6*w^3 - 3*w^2 - 34*w + 9],
[227, 227, w^3 + w^2 - 8*w],
[233, 233, 2*w^3 - 13*w - 3],
[233, 233, -w^3 + 2*w^2 + 4*w - 6],
[239, 239, -2*w^3 + 10*w + 1],
[239, 239, w^3 - 8*w - 2],
[251, 251, -3*w^3 + w^2 + 18*w],
[251, 251, -5*w^3 + 2*w^2 + 30*w - 11],
[263, 263, -2*w^3 + w^2 + 10*w + 2],
[269, 269, 3*w^3 - 18*w - 8],
[271, 271, 2*w^3 + w^2 - 10*w - 2],
[271, 271, -3*w^3 + 19*w],
[277, 277, -w^3 + w^2 + 4*w - 8],
[277, 277, 2*w^3 + w^2 - 14*w - 8],
[281, 281, w^3 - 8*w],
[307, 307, 3*w^3 - w^2 - 16*w + 2],
[307, 307, 2*w^3 - w^2 - 9*w + 1],
[307, 307, -2*w^3 + 2*w^2 + 13*w - 7],
[307, 307, -w^3 + w^2 + 7*w - 6],
[313, 313, -3*w^3 + w^2 + 15*w + 4],
[317, 317, -4*w^3 + w^2 + 22*w - 3],
[317, 317, 2*w^3 - w^2 - 10*w - 3],
[337, 337, w^3 + w^2 - 4*w - 5],
[349, 349, -2*w^3 - w^2 + 11*w + 3],
[349, 349, -4*w^3 + 2*w^2 + 24*w - 11],
[353, 353, 3*w^3 - w^2 - 17*w + 7],
[353, 353, -3*w^3 - 2*w^2 + 17*w + 11],
[367, 367, 2*w^3 - 11*w - 7],
[373, 373, -w^3 + w^2 + 4*w - 5],
[383, 383, -4*w^3 + 3*w^2 + 23*w - 9],
[383, 383, 4*w^3 - w^2 - 22*w],
[389, 389, w^2 - 2*w - 5],
[401, 401, -3*w^3 + 2*w^2 + 16*w - 9],
[401, 401, 2*w^3 - w^2 - 14*w - 1],
[409, 409, 2*w^3 + 2*w^2 - 13*w - 9],
[419, 419, -4*w^3 + 24*w + 7],
[419, 419, 2*w^2 + 2*w - 7],
[419, 419, -w^3 + 2*w^2 + 8*w - 6],
[419, 419, 6*w^3 - 3*w^2 - 33*w + 13],
[431, 431, 2*w^3 - w^2 - 12*w - 5],
[439, 439, -3*w^3 + 2*w^2 + 18*w - 4],
[439, 439, w^3 + 2*w^2 - 7*w - 3],
[443, 443, 6*w^3 - 3*w^2 - 33*w + 7],
[449, 449, -3*w^3 + w^2 + 14*w - 4],
[457, 457, 2*w^3 - 9*w],
[457, 457, 3*w^3 - w^2 - 16*w - 2],
[479, 479, 3*w^3 - w^2 - 19*w + 4],
[487, 487, -w^3 + w^2 + 4*w - 6],
[491, 491, 3*w^3 + w^2 - 17*w - 5],
[499, 499, 5*w^3 - 3*w^2 - 28*w + 8],
[499, 499, 2*w^3 - w^2 - 12*w - 1],
[503, 503, -2*w^3 - w^2 + 12*w + 3],
[503, 503, w^3 - 3*w - 4],
[509, 509, -3*w - 5],
[509, 509, -4*w^3 + 3*w^2 + 25*w - 10],
[523, 523, -2*w^3 + w^2 + 14*w - 6],
[529, 23, 4*w^3 - 2*w^2 - 21*w + 5],
[529, 23, -3*w^3 - w^2 + 15*w + 6],
[557, 557, -2*w^3 + w^2 + 13*w + 2],
[557, 557, 3*w^3 - 2*w^2 - 16*w + 3],
[569, 569, 2*w^2 - w - 8],
[587, 587, w^3 + 2*w^2 - 5*w - 16],
[587, 587, -4*w^3 + w^2 + 25*w - 5],
[599, 599, w^3 + 2*w^2 - 6*w - 5],
[599, 599, -5*w^3 + 3*w^2 + 30*w - 15],
[601, 601, w^3 - 6*w - 6],
[607, 607, -w^3 + w^2 + 3*w - 4],
[607, 607, -w^3 + 2*w - 3],
[613, 613, 6*w^3 - 2*w^2 - 31*w + 11],
[613, 613, w^3 + w^2 - 4*w - 6],
[617, 617, 5*w^3 - w^2 - 30*w + 2],
[617, 617, -2*w^3 + 2*w^2 + 9*w + 3],
[619, 619, -w^3 + 2*w^2 + 6*w - 5],
[619, 619, 2*w^3 - w^2 - 11*w - 3],
[631, 631, -w^3 + 2*w^2 + 5*w - 5],
[631, 631, -3*w^3 - w^2 + 18*w + 5],
[647, 647, 6*w^3 - w^2 - 36*w + 2],
[659, 659, 4*w^3 + w^2 - 24*w - 14],
[661, 661, 2*w^2 - w - 4],
[673, 673, 2*w^3 + w^2 - 14*w - 12],
[677, 677, 3*w^3 + 2*w^2 - 16*w - 7],
[677, 677, -3*w^3 + w^2 + 15*w + 1],
[683, 683, 2*w^2 - 2*w - 9],
[683, 683, 2*w^3 - 14*w + 1],
[691, 691, 5*w^3 - 2*w^2 - 30*w + 6],
[701, 701, -2*w^3 + w^2 + 12*w + 2],
[701, 701, -5*w^3 + 2*w^2 + 26*w],
[719, 719, 3*w^3 - 18*w + 1],
[719, 719, 3*w^3 - w^2 - 19*w + 1],
[727, 727, -3*w^2 - 2*w + 11],
[727, 727, 2*w^3 + w^2 - 13*w - 2],
[733, 733, 2*w^3 - 3*w^2 - 8*w + 5],
[739, 739, -3*w^3 + 16*w - 6],
[743, 743, -4*w^3 - w^2 + 25*w + 8],
[751, 751, w^2 - 2*w - 7],
[751, 751, -3*w^3 + 15*w + 5],
[757, 757, 3*w^3 - 19*w - 2],
[761, 761, 2*w^2 + w - 5],
[769, 769, -2*w^3 + w^2 + 14*w - 9],
[769, 769, 2*w^2 - w - 7],
[773, 773, w^3 - w^2 - 5*w - 4],
[773, 773, w^3 - 3*w - 6],
[787, 787, 3*w^3 - 19*w - 3],
[797, 797, 8*w^3 - 3*w^2 - 47*w + 10],
[809, 809, 4*w^3 - 25*w - 1],
[811, 811, w^3 - 9*w - 1],
[821, 821, 6*w^3 - 3*w^2 - 35*w + 8],
[823, 823, 2*w^3 + w^2 - 10*w + 1],
[823, 823, -2*w^3 + 2*w^2 + 7*w - 4],
[829, 829, -5*w^3 + w^2 + 27*w - 4],
[829, 829, 5*w^3 - 2*w^2 - 26*w + 10],
[839, 839, w^3 + 2*w^2 - 5*w - 7],
[839, 839, 3*w^3 - w^2 - 15*w + 2],
[841, 29, 3*w^3 + w^2 - 14*w - 2],
[841, 29, 4*w^3 - 2*w^2 - 21*w + 10],
[857, 857, -8*w^3 + 4*w^2 + 44*w - 11],
[857, 857, -3*w^3 + 3*w^2 + 20*w - 9],
[859, 859, w^3 + 2*w^2 - 6*w - 6],
[877, 877, w^3 - 3*w^2 - 4*w + 14],
[883, 883, 2*w^3 + w^2 - 10*w - 11],
[887, 887, 3*w^3 + w^2 - 20*w - 2],
[919, 919, 7*w^3 - 3*w^2 - 39*w + 9],
[929, 929, 3*w^3 - 17*w + 2],
[929, 929, -6*w^3 + 2*w^2 + 35*w - 10],
[937, 937, w^3 + w^2 - 3*w - 5],
[937, 937, -2*w^3 + 3*w^2 + 12*w - 12],
[937, 937, -3*w^3 - 2*w^2 + 19*w + 8],
[937, 937, 3*w^3 - 2*w^2 - 13*w + 3],
[941, 941, w^3 + 2*w^2 - 6*w - 8],
[941, 941, 5*w^3 - w^2 - 28*w - 3],
[947, 947, -4*w^3 + 2*w^2 + 25*w - 6],
[953, 953, w^3 - 7*w - 7],
[953, 953, -3*w^3 + 2*w^2 + 18*w - 3],
[967, 967, w^3 - 2*w^2 - 5*w - 3],
[971, 971, w^2 + 2*w - 6],
[977, 977, 2*w^3 - 11*w - 8],
[997, 997, -3*w^3 + w^2 + 17*w + 2],
[997, 997, -2*w^3 + 2*w^2 + 14*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [e, 1, e^3 + 3*e^2 - 2*e - 4, e - 2, e^2 + e - 4, -e - 3, -e^3 - 3*e^2 + e + 4, -e^3 - 4*e^2 + 2, -e^3 - 4*e^2 - 2*e + 6, -2*e^3 - 8*e^2 + 10, -4, 2*e^3 + 5*e^2 - 5*e - 8, -e^3 - e^2 + 6*e + 2, -2*e^2 - 4*e - 2, 2*e^2 + 7*e, -e^3 - 3*e^2 + 6*e + 6, 4*e^2 + 8*e - 10, 2*e^3 + 8*e^2 - 2*e - 16, -e^3 + e^2 + 14*e - 2, 2*e^3 + 4*e^2 - 9*e - 6, -3*e^3 - 8*e^2 + 5*e + 12, e^3 + 3*e^2 - 6*e - 8, -3*e^3 - 8*e^2 + 5*e + 6, e^3 + 5*e^2 + 4*e - 8, 6*e + 10, -e^2 - 2*e + 6, e^3 + 6*e^2 + 3*e - 12, 2*e^3 + 4*e^2 - 2*e, 4*e + 8, -e^3 - 5*e^2 - 4*e + 14, 3*e^3 + 5*e^2 - 18*e - 8, 3*e^3 + 13*e^2 - 22, -e^2 + 2*e + 6, -3*e^3 - 9*e^2 + 8*e + 2, 3*e^2 - e - 16, 5*e^3 + 14*e^2 - 10*e - 24, -e^3 - e^2 - 18, -4*e^3 - 14*e^2 + 5*e + 18, 2*e^3 + 6*e^2 - 3*e - 6, 2*e^3 + 4*e^2 - 12*e - 8, e^3 + 3*e^2 - 4, -3*e^3 - 14*e^2 - 4*e + 22, -6*e^3 - 18*e^2 + 22*e + 34, 2*e^3 - e^2 - 18*e, -6*e^3 - 19*e^2 + 9*e + 16, 3*e^3 + 8*e^2 - 8*e - 16, -5*e^3 - 13*e^2 + 8*e + 4, -2*e^3 - 8*e^2 + 12*e + 30, -4*e^3 - 14*e^2 + 4*e + 32, 6*e^3 + 18*e^2 - 8*e - 26, -e^3 - 2*e^2 + 12*e + 10, 6*e^3 + 18*e^2 - 12*e - 26, -5*e^3 - 15*e^2 + 8*e + 14, -e^3 + 2*e^2 + 2*e - 24, -3*e^2 - 3*e + 10, 4*e^2 + 10*e, -7*e^3 - 21*e^2 + 9*e + 30, -e^2 - 11*e - 8, -2*e^3 - 4*e^2 + 16*e + 22, -4*e^3 - 10*e^2 + 12*e + 2, 3*e^3 + 12*e^2 + 3*e - 8, 2*e^3 + 7*e^2 - 15*e - 26, 4*e^3 + 14*e^2 - 3*e - 16, 2*e^3 + 9*e^2 - 6*e - 12, -2*e^3 - 6*e^2 + 8*e + 10, 2*e^3 + 4*e^2 - 8*e, -3*e^3 - 5*e^2 + 16*e + 6, 4*e^3 + 8*e^2 - 22*e - 14, 2*e^3 - 3*e^2 - 22*e + 6, -9*e^3 - 22*e^2 + 31*e + 42, 10*e^2 + 22*e - 24, 4*e^3 + 8*e^2 - 13*e - 20, -e^2 + 5*e + 18, 3*e^3 + 6*e^2 - 19*e - 28, 4*e^3 + 16*e^2 - 12, 5*e^3 + 18*e^2 - 24, 2*e^3 + 13*e^2 + 9*e - 32, 4*e^3 + 10*e^2 - 12*e - 18, -2*e^3 + 22*e + 8, 6*e^3 + 10*e^2 - 29*e - 12, -5*e^3 - 12*e^2 + 16*e + 14, 3*e^3 + 10*e^2 + 2*e - 6, -4*e^3 - 10*e^2 + 16*e + 30, 2*e^2 + 8*e + 18, e^3 - 5*e^2 - 25*e + 2, -e^3 - 16*e^2 - 23*e + 24, 3*e^3 + 14*e^2 + 2*e - 14, -e^3 - 3*e^2 + 12*e + 8, -e^3 + e^2 - e - 16, 5*e^3 + 15*e^2 - 8*e - 30, 4*e^3 + 6*e^2 - 16*e - 8, 7*e^3 + 22*e^2 - 12*e - 28, 6*e^3 + 16*e^2 - 6*e - 10, -4*e^2 - 10*e + 12, 2*e^3 + 10*e^2 - e - 36, 3*e^2 + 12*e + 12, 4*e^3 + 4*e^2 - 20*e + 6, -8*e^3 - 28*e^2 + 11*e + 26, 7*e^3 + 18*e^2 - 24*e - 36, 3*e^3 + 10*e^2 + 6*e - 4, 2*e^3 + 14*e^2 + 20*e - 28, 7*e^3 + 25*e^2 - 9*e - 24, -6*e^2 - 16*e + 8, -e^3 - 9*e^2 - 20*e + 6, 3*e^3 + 16*e^2 - 4*e - 38, 5*e^3 + 17*e^2 + 4*e - 26, -5*e^3 - 17*e^2 + 20*e + 34, -2*e^3 - 14*e^2 - 16*e + 34, e^3 - 6*e^2 - 9*e + 32, 5*e^3 + 16*e^2 - 10*e - 32, 7*e^3 + 17*e^2 - 6*e + 4, 8*e^3 + 26*e^2 - 14*e - 44, 8*e^3 + 22*e^2 - 28*e - 30, -3*e^3 - 2*e^2 + 5*e - 28, 4*e^3 + 3*e^2 - 23*e - 2, 3*e^3 + 6*e^2 - 12*e - 34, -7*e^3 - 33*e^2 - 6*e + 46, -4*e^3 - 14*e^2 + 8*e + 18, 4*e^3 + 12*e^2 - 10*e - 12, 2*e^2 - 9*e - 6, -15*e^3 - 42*e^2 + 27*e + 50, e^3 + 15*e^2 + 26*e - 18, -e^3 + 4*e^2 + 16*e - 22, 5*e^3 + 20*e^2 + 2*e - 28, -8*e^3 - 25*e^2 + 10*e + 42, 2*e^3 + 10*e^2 + 2*e - 20, 6*e^3 + 18*e^2 - 4*e - 28, 3*e^3 + 4*e^2 - 24*e - 22, -4*e^3 - 16*e^2 + 12*e + 42, -4*e^2 - 18*e - 8, -e^2 - 9*e - 24, -2*e^3 - 18*e^2 - 26*e + 24, -e^3 + 7*e^2 + 13*e - 36, 10*e^3 + 34*e^2 + 2*e - 50, -e^3 - 5*e^2 + 2*e - 6, -10*e^3 - 30*e^2 + 13*e + 28, 9*e^3 + 25*e^2 - 35*e - 58, 2*e^3 + 9*e^2 - 12, 9*e^3 + 28*e^2 - 18*e - 46, -7*e^3 - 23*e^2 - 2*e + 34, 2*e^3 + 9*e^2 + 3*e - 14, -6*e^3 - 20*e^2 - 4*e + 28, -e^3 - 7*e^2 - 2*e + 4, -e^3 + e^2 - 4*e - 22, 2*e^3 - 2*e^2 - 30*e + 10, 4*e^3 + 10*e^2 - 4*e + 20, 6*e^3 + 20*e^2 - 2*e - 20, 5*e^3 + 26*e^2 + 8*e - 52, -e^3 - 13*e^2 - 18*e + 28, 4*e^3 + 8*e^2 - 32*e - 14, 3*e^3 + 10*e^2 + 8*e - 6, 6*e^3 + 16*e^2 - 14*e - 10, 2*e^3 + 7*e^2 + 7*e - 10, -2*e^3 - 7*e^2 + 13*e + 6, 3*e^2 + 21*e - 10, e^3 - 5*e^2 - e + 36, -14*e^3 - 45*e^2 + 7*e + 50, e^3 - e^2 - 18*e - 28, e^3 + 4*e^2 - 6*e + 4, 9*e^3 + 34*e^2 - 16*e - 60, 7*e^3 + 25*e^2 - 18*e - 34, 12*e^3 + 39*e^2 - 16*e - 64, -11*e^3 - 39*e^2 + 15*e + 38, -4*e^3 - 16*e^2 - 6*e + 12, 4*e^3 + 10*e^2 + 2*e - 10, e^3 - 5*e^2 - 22*e - 2, -2*e^3 - 5*e^2 + 7*e + 30, 11*e^3 + 27*e^2 - 38*e - 50, -5*e^3 - 14*e^2 + 25*e + 32, -6*e^3 - 14*e^2 + 12*e + 8, 3*e^3 + 16*e^2 + 10*e - 24, -9*e^3 - 31*e^2 + 20*e + 42, 6*e^3 + 20*e^2 - 38, e^3 - 8*e^2 - 26*e + 22, 8*e^3 + 23*e^2 - 8*e - 12, 12*e^3 + 41*e^2 - 8*e - 46, 3*e^3 + 17*e^2 + 16*e - 22, 4*e^3 + 14*e^2 - 10*e - 42, -2*e^3 - 14*e^2 - 18*e + 38];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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