/*
  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, -2, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 9, w^3 - w^2 - 6*w + 3}>;

primesArray := [
[2, 2, w + 1],
[3, 3, -w^3 + w^2 + 5*w - 2],
[11, 11, -w^3 + 5*w + 1],
[11, 11, -w + 2],
[13, 13, -w^3 + w^2 + 4*w + 1],
[17, 17, -w^3 + w^2 + 6*w - 1],
[17, 17, -w^2 - w + 3],
[23, 23, w^3 - 6*w],
[27, 3, w^3 + w^2 - 5*w - 4],
[41, 41, -w^3 + w^2 + 5*w],
[41, 41, 2*w^3 - 11*w - 4],
[53, 53, 2*w^3 - 2*w^2 - 11*w + 6],
[59, 59, 2*w^3 - 11*w - 2],
[67, 67, w^3 - 7*w - 1],
[71, 71, 3*w^3 - 2*w^2 - 15*w + 1],
[79, 79, -3*w^3 + w^2 + 16*w + 3],
[89, 89, -4*w^3 + 2*w^2 + 22*w - 3],
[101, 101, -2*w^3 + 13*w + 6],
[101, 101, w^3 + w^2 - 6*w - 3],
[101, 101, -3*w^3 + 2*w^2 + 17*w - 3],
[101, 101, w^3 - w^2 - 8*w - 3],
[107, 107, 2*w^3 - 3*w^2 - 6*w],
[109, 109, -w^3 + w^2 + 4*w - 3],
[113, 113, w^3 - 8*w - 4],
[113, 113, 7*w^3 - 4*w^2 - 39*w + 7],
[121, 11, w^3 - w^2 - 6*w - 1],
[127, 127, -2*w^3 + w^2 + 9*w + 3],
[127, 127, 2*w^3 - 13*w],
[131, 131, 6*w^3 - 3*w^2 - 34*w + 6],
[139, 139, w^2 - 2*w - 4],
[149, 149, 3*w^3 - w^2 - 18*w - 3],
[151, 151, -2*w^3 + 2*w^2 + 9*w - 4],
[163, 163, 3*w^3 - 2*w^2 - 16*w + 2],
[163, 163, 2*w^3 - 10*w + 1],
[167, 167, 3*w^3 - 18*w - 4],
[173, 173, -3*w^3 + w^2 + 18*w - 3],
[193, 193, 4*w^3 - 3*w^2 - 23*w + 9],
[197, 197, 3*w^3 - w^2 - 17*w + 2],
[197, 197, -3*w^3 - 2*w^2 + 15*w + 9],
[199, 199, 2*w^3 - w^2 - 8*w - 6],
[223, 223, -6*w^3 + 3*w^2 + 35*w - 7],
[227, 227, 2*w^3 - 12*w - 1],
[227, 227, -w^3 + w^2 + 7*w - 8],
[227, 227, 3*w^3 - w^2 - 18*w - 1],
[227, 227, -4*w^3 - 2*w^2 + 21*w + 14],
[229, 229, -w^3 + 2*w^2 + 3*w - 5],
[233, 233, w^3 - 3*w - 3],
[233, 233, -3*w^3 + 3*w^2 + 15*w - 8],
[241, 241, 4*w^2 - 7*w - 8],
[241, 241, w^3 + w^2 - 4*w - 5],
[251, 251, 2*w^3 - w^2 - 13*w - 1],
[251, 251, 2*w^3 - 9*w - 6],
[257, 257, w^3 - 8*w],
[263, 263, -w^3 + 2*w^2 + 5*w - 7],
[269, 269, 3*w^3 - 3*w^2 - 12*w - 1],
[271, 271, 2*w^3 - w^2 - 12*w - 2],
[271, 271, 9*w^3 - 5*w^2 - 50*w + 9],
[277, 277, w^3 - 2*w^2 - 6*w + 6],
[277, 277, w^3 + w^2 - 7*w - 2],
[281, 281, 4*w^3 - 2*w^2 - 24*w + 5],
[281, 281, -7*w^3 + 4*w^2 + 40*w - 12],
[289, 17, -5*w^3 + 5*w^2 + 26*w - 15],
[293, 293, -w^3 + 2*w^2 + 4*w - 4],
[293, 293, -9*w^3 + 5*w^2 + 51*w - 12],
[307, 307, -2*w^3 + 2*w^2 + 12*w - 7],
[311, 311, -4*w^3 + 24*w + 11],
[313, 313, 2*w^3 - 9*w - 4],
[317, 317, 2*w^3 - 4*w^2 - 2*w - 3],
[317, 317, 2*w^3 - 14*w - 7],
[331, 331, -3*w^3 + 19*w + 3],
[349, 349, -w^3 - 2*w^2 + 8*w + 8],
[349, 349, w^3 + w^2 - 5*w - 8],
[361, 19, -w^3 + w^2 + 7*w - 6],
[361, 19, w^3 + w^2 - 3*w - 4],
[373, 373, 2*w^3 - w^2 - 11*w - 3],
[379, 379, -4*w^3 + 25*w + 14],
[383, 383, -3*w^3 + w^2 + 15*w - 2],
[383, 383, 3*w^3 - 4*w^2 - 10*w],
[401, 401, 4*w^3 - w^2 - 24*w - 4],
[409, 409, w^2 + 5*w + 5],
[419, 419, w^2 - 2*w - 6],
[419, 419, -6*w^3 + 3*w^2 + 34*w - 8],
[433, 433, 3*w^3 - 16*w - 6],
[433, 433, 3*w^3 - w^2 - 16*w - 5],
[443, 443, -w^3 + 2*w^2 + w + 3],
[449, 449, 2*w^2 - 2*w - 5],
[457, 457, -4*w^3 + 4*w^2 + 20*w - 9],
[457, 457, w^2 + w - 7],
[463, 463, w^3 - 7*w - 7],
[463, 463, w^3 + 2*w^2 - 5*w - 9],
[487, 487, w^3 + 2*w^2 - 5*w - 5],
[487, 487, -w^3 + w^2 + 6*w - 7],
[509, 509, -2*w^3 - 2*w^2 + 13*w + 10],
[509, 509, 3*w^3 - 17*w - 3],
[509, 509, -5*w^3 + 3*w^2 + 30*w - 11],
[509, 509, -w^3 - 2*w^2 + 4*w + 6],
[521, 521, -2*w^3 - w^2 + 9*w + 7],
[541, 541, -2*w^3 - 2*w^2 + 12*w + 9],
[541, 541, -2*w^3 + 12*w + 9],
[547, 547, 2*w^3 + w^2 - 7*w - 3],
[547, 547, 5*w^3 - 3*w^2 - 30*w + 3],
[557, 557, w^3 + 2*w^2 - 5*w - 11],
[557, 557, -w^3 + w^2 + 3*w - 4],
[563, 563, 3*w^3 - w^2 - 15*w - 4],
[569, 569, 2*w^2 - w - 8],
[577, 577, 4*w^3 - 22*w - 7],
[593, 593, -3*w^3 + w^2 + 17*w + 4],
[607, 607, -w^3 - 3*w^2 + w + 4],
[613, 613, 3*w^3 - 2*w^2 - 14*w],
[613, 613, w^2 - 3*w - 5],
[619, 619, -w^3 + 2*w^2 + 4*w - 10],
[625, 5, -5],
[641, 641, -3*w^3 - 3*w^2 + 12*w + 7],
[643, 643, 2*w^3 + 2*w^2 - 14*w - 13],
[643, 643, 3*w^3 - 19*w - 7],
[647, 647, w^3 - 5*w - 7],
[653, 653, -3*w^3 + 2*w^2 + 14*w - 2],
[659, 659, -6*w^3 + 2*w^2 + 34*w - 1],
[659, 659, -2*w^3 + w^2 + 17*w + 9],
[683, 683, w^3 + 2*w^2 - 7*w - 7],
[691, 691, 3*w^3 - 15*w - 7],
[701, 701, w^3 + 2*w^2 - 6*w - 10],
[701, 701, -w^3 + 3*w^2 + 6*w - 13],
[701, 701, 5*w^3 - 2*w^2 - 30*w + 4],
[701, 701, 4*w^3 - 2*w^2 - 21*w + 2],
[727, 727, -2*w^3 - 3*w^2 + 9*w + 7],
[743, 743, 3*w - 4],
[751, 751, w^3 - 9*w - 3],
[757, 757, w^3 - 5*w^2 + 4*w + 7],
[757, 757, 4*w^3 - 5*w^2 - 11*w - 5],
[761, 761, 4*w^3 - 5*w^2 - 20*w + 16],
[769, 769, -3*w^3 + 16*w + 12],
[773, 773, 2*w^3 + 2*w^2 - 11*w - 10],
[773, 773, w^3 - 3*w - 5],
[773, 773, -w^3 + 2*w^2 + 9*w + 3],
[773, 773, 5*w^3 - w^2 - 27*w],
[787, 787, -4*w^3 + 3*w^2 + 22*w - 4],
[787, 787, -2*w^3 + w^2 + 14*w - 6],
[797, 797, 2*w^3 - w^2 - 14*w + 2],
[797, 797, -3*w^3 + 2*w^2 + 19*w - 7],
[811, 811, 2*w^3 + w^2 - 14*w - 12],
[811, 811, 2*w^3 + 2*w^2 - 8*w - 3],
[829, 829, -2*w^3 + 14*w + 11],
[857, 857, -3*w^3 + 3*w^2 + 14*w - 7],
[859, 859, 3*w^3 - w^2 - 15*w],
[859, 859, -11*w^3 + 7*w^2 + 60*w - 15],
[863, 863, 2*w^2 - 2*w - 13],
[863, 863, 2*w^3 + w^2 - 12*w - 4],
[877, 877, -6*w^3 - 2*w^2 + 32*w + 19],
[887, 887, 3*w^3 + w^2 - 14*w - 7],
[887, 887, 2*w^3 - w^2 - 15*w - 5],
[911, 911, 3*w^3 - 18*w - 2],
[919, 919, w^3 - w^2 - 6*w - 5],
[919, 919, -4*w^3 + w^2 + 24*w - 2],
[941, 941, 3*w^3 - 3*w^2 - 13*w],
[947, 947, 2*w^3 - 2*w^2 - 10*w - 1],
[947, 947, -3*w^3 + 2*w^2 + 14*w + 6],
[953, 953, -4*w^3 + 3*w^2 + 21*w - 3],
[953, 953, 4*w^3 - w^2 - 23*w + 1],
[961, 31, -2*w^3 + 3*w^2 + 12*w - 8],
[961, 31, 3*w^2 - 4*w - 6],
[967, 967, -5*w^3 - 3*w^2 + 23*w + 14],
[971, 971, 5*w^3 - w^2 - 27*w - 8],
[971, 971, -9*w^3 + 6*w^2 + 50*w - 12],
[983, 983, -w^3 + w^2 + 10*w + 3],
[997, 997, 2*w^3 + w^2 - 11*w - 3],
[997, 997, w^3 + 2*w^2 - 6*w - 8],
[997, 997, -5*w^3 + 4*w^2 + 29*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 72*x^2 + 400;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/80*e^3 - 13/20*e, 0, -1/40*e^3 + 13/10*e, -1/40*e^3 + 23/10*e, -1/80*e^3 + 1/8*e^2 + 13/20*e - 9/2, 1/80*e^3 + 1/8*e^2 - 13/20*e - 9/2, -4, -1/40*e^3 + 23/10*e + 4, 1/4*e^2 - 9, -3/80*e^3 + 1/8*e^2 + 39/20*e - 9/2, 3/80*e^3 - 1/8*e^2 - 39/20*e + 9/2, 1/16*e^3 + 1/8*e^2 - 13/4*e - 9/2, 1/40*e^3 - 23/10*e + 4, 1/40*e^3 - 23/10*e - 4, 1/4*e^2 - 9, 3/40*e^3 - 39/10*e, -1/80*e^3 - 3/8*e^2 + 13/20*e + 27/2, 12, 6, -3/80*e^3 - 1/8*e^2 + 39/20*e + 9/2, -1/16*e^3 + 1/8*e^2 + 13/4*e - 9/2, 1/40*e^3 - 13/10*e, -7/80*e^3 - 1/8*e^2 + 91/20*e + 9/2, 1/20*e^3 - 23/5*e + 4, -11/80*e^3 - 1/8*e^2 + 143/20*e + 9/2, 7/80*e^3 - 3/8*e^2 - 91/20*e + 27/2, -1/40*e^3 + 1/2*e^2 + 13/10*e - 18, -1/40*e^3 + 23/10*e + 8, -3/40*e^3 + 69/10*e, -1/4*e^2 + 9, -3/80*e^3 - 3/8*e^2 + 39/20*e + 27/2, 1/40*e^3 - 1/4*e^2 - 13/10*e + 9, -3/40*e^3 + 1/4*e^2 + 39/10*e - 9, -1/40*e^3 + 23/10*e - 12, -1/20*e^3 + 23/5*e + 4, 1/20*e^3 - 23/5*e - 10, 1/20*e^3 - 23/5*e - 2, 1/20*e^3 - 23/5*e + 12, -1/20*e^3 + 23/5*e - 4, 1/20*e^3 - 23/5*e + 12, 1/20*e^3 - 23/5*e - 4, 1/10*e^3 - 46/5*e - 4, -1/20*e^3 + 23/5*e, -1/40*e^3 + 23/10*e - 12, 1/20*e^3 - 23/5*e + 16, -17/80*e^3 - 1/8*e^2 + 221/20*e + 9/2, -12, -1/80*e^3 - 7/8*e^2 + 13/20*e + 63/2, 7/80*e^3 + 1/8*e^2 - 91/20*e - 9/2, -1/10*e^3 + 46/5*e - 4, -1/20*e^3 + 23/5*e + 8, -1/40*e^3 + 23/10*e - 4, 22, -1/20*e^3 - 3/4*e^2 + 13/5*e + 27, 11/80*e^3 - 3/8*e^2 - 143/20*e + 27/2, -1/10*e^3 + 1/4*e^2 + 26/5*e - 9, -7/40*e^3 + 91/10*e, -1/80*e^3 - 3/8*e^2 + 13/20*e + 27/2, 1/20*e^3 - 23/5*e + 12, -1/20*e^3 + 23/5*e + 10, 1/10*e^3 - 46/5*e + 10, 3/80*e^3 - 3/8*e^2 - 39/20*e + 27/2, -11/80*e^3 - 3/8*e^2 + 143/20*e + 27/2, 1/20*e^3 - 23/5*e + 22, -1/10*e^3 + 46/5*e + 4, 1/20*e^3 + 1/4*e^2 - 13/5*e - 9, -11/80*e^3 + 1/8*e^2 + 143/20*e - 9/2, -1/10*e^3 + 46/5*e + 2, 1/20*e^3 - 23/5*e + 4, 3/40*e^3 - 69/10*e + 8, 9/80*e^3 - 3/8*e^2 - 117/20*e + 27/2, -1/10*e^3 + 46/5*e + 4, -28, -1/20*e^3 + 23/5*e + 4, 3/16*e^3 + 3/8*e^2 - 39/4*e - 27/2, 3/40*e^3 + 1/4*e^2 - 39/10*e - 9, 1/4*e^2 - 9, -1/40*e^3 + 13/10*e, -16, 3/16*e^3 + 3/8*e^2 - 39/4*e - 27/2, e^2 - 36, 1/20*e^3 - 23/5*e - 16, -7/80*e^3 + 1/8*e^2 + 91/20*e - 9/2, -3/80*e^3 - 9/8*e^2 + 39/20*e + 81/2, 3/40*e^3 - 69/10*e + 12, -17/80*e^3 + 3/8*e^2 + 221/20*e - 27/2, -13/80*e^3 + 3/8*e^2 + 169/20*e - 27/2, 1/10*e^3 - 46/5*e + 6, 1/20*e^3 - 23/5*e + 12, 3/40*e^3 + 3/4*e^2 - 39/10*e - 27, 3/40*e^3 - 1/4*e^2 - 39/10*e + 9, 1/40*e^3 - 23/10*e + 24, 1/16*e^3 + 1/8*e^2 - 13/4*e - 9/2, -1/20*e^3 + 23/5*e - 2, -1/20*e^3 + 23/5*e + 20, -23/80*e^3 + 1/8*e^2 + 299/20*e - 9/2, -1/20*e^3 + 23/5*e - 20, 1/80*e^3 + 1/8*e^2 - 13/20*e - 9/2, 26, 1/20*e^3 - 13/5*e, -36, -22, -17/80*e^3 + 1/8*e^2 + 221/20*e - 9/2, 1/20*e^3 + e^2 - 13/5*e - 36, -19/80*e^3 - 1/8*e^2 + 247/20*e + 9/2, -7/80*e^3 + 5/8*e^2 + 91/20*e - 45/2, -17/80*e^3 + 1/8*e^2 + 221/20*e - 9/2, -1/5*e^3 - 1/2*e^2 + 52/5*e + 18, -1/80*e^3 - 11/8*e^2 + 13/20*e + 99/2, -9/80*e^3 - 9/8*e^2 + 117/20*e + 81/2, -1/10*e^3 + 1/2*e^2 + 26/5*e - 18, -1/10*e^3 + 46/5*e + 8, 14, 3/4*e^2 - 27, -1/10*e^3 + 46/5*e + 20, -1/40*e^3 + 23/10*e - 24, -1/80*e^3 + 3/8*e^2 + 13/20*e - 27/2, 1/20*e^3 - 23/5*e + 8, 1/10*e^3 - 46/5*e + 20, 1/5*e^3 - 3/4*e^2 - 52/5*e + 27, 7/40*e^3 - 1/4*e^2 - 91/10*e + 9, 27/80*e^3 + 1/8*e^2 - 351/20*e - 9/2, 22, 3/20*e^3 - 69/5*e, -3/80*e^3 - 5/8*e^2 + 39/20*e + 45/2, -3/40*e^3 + 39/10*e, -1/40*e^3 + 23/10*e - 20, -1/20*e^3 + 23/5*e - 20, -9/80*e^3 - 5/8*e^2 + 117/20*e + 45/2, -2, -13/80*e^3 + 9/8*e^2 + 169/20*e - 81/2, -1/10*e^3 + 46/5*e - 8, 1/80*e^3 - 3/8*e^2 - 13/20*e + 27/2, -22, 9/80*e^3 + 5/8*e^2 - 117/20*e - 45/2, -1/10*e^3 + 46/5*e + 10, 1/40*e^3 + 1/2*e^2 - 13/10*e - 18, -3/40*e^3 + 69/10*e + 28, -1/10*e^3 + 46/5*e - 2, -1/10*e^3 + 46/5*e + 18, -9/40*e^3 + 117/10*e, -1/8*e^3 + 23/2*e - 24, -4, -11/80*e^3 - 9/8*e^2 + 143/20*e + 81/2, 17/40*e^3 + 1/4*e^2 - 221/10*e - 9, -3/40*e^3 + 39/10*e, -9/40*e^3 - 1/2*e^2 + 117/10*e + 18, -1/8*e^3 + 23/2*e - 32, -3/80*e^3 + 3/8*e^2 + 39/20*e - 27/2, 13/40*e^3 + 1/4*e^2 - 169/10*e - 9, -1/20*e^3 + 23/5*e - 20, 1/40*e^3 - 23/10*e + 4, 1/5*e^3 - 92/5*e + 8, 1/40*e^3 - 23/10*e + 52, 7/80*e^3 + 3/8*e^2 - 91/20*e - 27/2, 1/40*e^3 - 13/10*e, 3/20*e^3 + 5/4*e^2 - 39/5*e - 45, 11/80*e^3 - 3/8*e^2 - 143/20*e + 27/2, 1/5*e^3 - 92/5*e - 14, -11/80*e^3 - 7/8*e^2 + 143/20*e + 63/2, 13/80*e^3 - 5/8*e^2 - 169/20*e + 45/2, 3/40*e^3 + 3/2*e^2 - 39/10*e - 54, -1/4*e^3 + 13*e, -1/20*e^3 - e^2 + 13/5*e + 36, 1/20*e^3 - 23/5*e + 4, 3/20*e^3 - 69/5*e + 4, -1/80*e^3 - 9/8*e^2 + 13/20*e + 81/2, -1/20*e^3 + 23/5*e - 16];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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