/*
  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, 3, -1, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6}>;

primesArray := [
[5, 5, -2*w^4 + w^3 + 9*w^2 - 2*w - 3],
[17, 17, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[17, 17, w^2 - 2],
[23, 23, w^3 - 3*w],
[29, 29, 2*w^4 - w^3 - 10*w^2 + 2*w + 4],
[32, 2, 2],
[37, 37, -w^4 + w^3 + 4*w^2 - 2*w],
[41, 41, -3*w^4 + 2*w^3 + 14*w^2 - 5*w - 7],
[43, 43, -3*w^4 + w^3 + 14*w^2 - 3*w - 6],
[47, 47, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],
[53, 53, 2*w^4 - w^3 - 8*w^2 + 2*w + 1],
[53, 53, -2*w^4 + w^3 + 10*w^2 - 4*w - 6],
[59, 59, -w^4 + 4*w^2 + 1],
[59, 59, -3*w^4 + 2*w^3 + 14*w^2 - 6*w - 8],
[61, 61, 4*w^4 - 2*w^3 - 18*w^2 + 5*w + 7],
[61, 61, 3*w^4 - w^3 - 15*w^2 + 2*w + 7],
[73, 73, -4*w^4 + 2*w^3 + 18*w^2 - 7*w - 6],
[83, 83, -2*w^4 + 9*w^2 + 2*w - 4],
[83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6],
[97, 97, -2*w^4 + w^3 + 8*w^2 - 3*w + 1],
[97, 97, -w^3 + w^2 + 4*w - 1],
[101, 101, 2*w^4 - 9*w^2 - w + 5],
[103, 103, 2*w^4 - 2*w^3 - 9*w^2 + 8*w + 2],
[107, 107, -3*w^4 + w^3 + 15*w^2 - 4*w - 8],
[109, 109, -3*w^4 + w^3 + 13*w^2 - 2*w - 4],
[113, 113, 4*w^4 - 3*w^3 - 18*w^2 + 8*w + 8],
[121, 11, -w^4 - w^3 + 5*w^2 + 4*w - 2],
[131, 131, -4*w^4 + w^3 + 18*w^2 - 5],
[139, 139, 2*w^3 - w^2 - 8*w + 1],
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3],
[149, 149, 5*w^4 - 2*w^3 - 23*w^2 + 4*w + 11],
[149, 149, w^4 - 4*w^2 - 3],
[149, 149, 4*w^4 - 3*w^3 - 19*w^2 + 10*w + 8],
[151, 151, 2*w^4 - w^3 - 9*w^2 + 5*w + 4],
[157, 157, -2*w^4 - w^3 + 10*w^2 + 5*w - 4],
[157, 157, 5*w^4 - 3*w^3 - 24*w^2 + 8*w + 13],
[163, 163, -2*w^4 + 11*w^2 + w - 7],
[167, 167, w^4 - 3*w^2 - 2*w - 3],
[167, 167, -3*w^4 + 3*w^3 + 14*w^2 - 10*w - 9],
[169, 13, 3*w^4 - 2*w^3 - 14*w^2 + 6*w + 3],
[191, 191, -3*w^4 + w^3 + 15*w^2 - w - 6],
[193, 193, 3*w^4 - 2*w^3 - 14*w^2 + 4*w + 5],
[199, 199, -5*w^4 + 2*w^3 + 23*w^2 - 3*w - 10],
[223, 223, -4*w^4 + w^3 + 20*w^2 - w - 9],
[223, 223, -w^4 + 3*w^2 + 2],
[233, 233, w^3 - 4*w - 4],
[241, 241, -3*w^4 + 2*w^3 + 15*w^2 - 7*w - 7],
[243, 3, -3],
[251, 251, -3*w^4 + 3*w^3 + 13*w^2 - 8*w - 6],
[257, 257, -4*w^4 + w^3 + 20*w^2 - 2*w - 9],
[257, 257, -3*w^4 + 2*w^3 + 15*w^2 - 5*w - 11],
[269, 269, 2*w^4 - 11*w^2 - w + 6],
[271, 271, w^4 + w^3 - 5*w^2 - 6*w + 2],
[271, 271, 3*w^4 - w^3 - 15*w^2 + 8],
[277, 277, -w^4 - w^3 + 6*w^2 + 3*w - 5],
[283, 283, -5*w^4 + 3*w^3 + 23*w^2 - 10*w - 10],
[293, 293, -w^4 + w^3 + 6*w^2 - 3*w - 8],
[307, 307, -w^4 - w^3 + 6*w^2 + 4*w - 2],
[311, 311, -w^3 + 2*w^2 + 3*w - 2],
[311, 311, 2*w^4 - w^3 - 10*w^2 + 5*w + 7],
[313, 313, -5*w^4 + 4*w^3 + 23*w^2 - 12*w - 11],
[313, 313, -w^4 - w^3 + 5*w^2 + 3*w - 3],
[347, 347, -w^4 + 2*w^3 + 3*w^2 - 7*w + 1],
[353, 353, -3*w^4 + 14*w^2 + 3*w - 7],
[353, 353, -3*w^4 + 3*w^3 + 13*w^2 - 9*w - 7],
[367, 367, -w^4 + w^3 + 5*w^2 - 6*w - 4],
[367, 367, -3*w^4 + w^3 + 15*w^2 - 3*w - 5],
[379, 379, 5*w^4 - 2*w^3 - 23*w^2 + 6*w + 9],
[383, 383, 4*w^4 - 2*w^3 - 19*w^2 + 7*w + 11],
[397, 397, w^4 - 3*w^2 - w - 3],
[397, 397, -2*w^3 + w^2 + 8*w],
[401, 401, 2*w^4 - 2*w^3 - 8*w^2 + 5*w],
[421, 421, -4*w^4 + 2*w^3 + 17*w^2 - 5*w - 6],
[421, 421, -w^4 + 2*w^3 + 6*w^2 - 8*w - 5],
[431, 431, 5*w^4 - 2*w^3 - 22*w^2 + 5*w + 7],
[431, 431, -w^4 - w^3 + 6*w^2 + 5*w - 4],
[439, 439, 4*w^4 - 2*w^3 - 19*w^2 + 7*w + 10],
[443, 443, 4*w^4 - w^3 - 19*w^2 + 3*w + 8],
[457, 457, -3*w^4 + 3*w^3 + 15*w^2 - 10*w - 9],
[461, 461, -w^3 + 6*w],
[461, 461, 3*w^4 - 3*w^3 - 14*w^2 + 9*w + 8],
[463, 463, 5*w^4 - 3*w^3 - 23*w^2 + 7*w + 8],
[499, 499, 6*w^4 - 3*w^3 - 27*w^2 + 7*w + 11],
[509, 509, 2*w^4 - w^3 - 8*w^2 + w - 1],
[509, 509, 3*w^4 - 2*w^3 - 13*w^2 + 6*w + 7],
[521, 521, w^3 - 6*w + 1],
[523, 523, 4*w^4 - 2*w^3 - 18*w^2 + 3*w + 7],
[541, 541, w^2 + w - 5],
[547, 547, 5*w^4 - 2*w^3 - 25*w^2 + 4*w + 12],
[547, 547, -6*w^4 + 3*w^3 + 27*w^2 - 10*w - 9],
[557, 557, -4*w^4 + 3*w^3 + 18*w^2 - 10*w - 9],
[557, 557, -4*w^4 + w^3 + 19*w^2 - 3*w - 6],
[557, 557, 3*w^4 - 2*w^3 - 14*w^2 + 8*w + 7],
[563, 563, -w^4 + 3*w^3 + 4*w^2 - 11*w + 1],
[569, 569, w^2 + 3*w - 2],
[577, 577, -5*w^4 + 2*w^3 + 23*w^2 - 6*w - 8],
[587, 587, -2*w^4 + 10*w^2 + 3*w - 6],
[593, 593, 4*w^4 - 3*w^3 - 19*w^2 + 8*w + 11],
[601, 601, -4*w^4 + w^3 + 18*w^2 - 3*w - 6],
[601, 601, -5*w^4 + 3*w^3 + 24*w^2 - 10*w - 13],
[601, 601, -w^4 + 5*w^2 + 3*w - 3],
[607, 607, -3*w^4 + 2*w^3 + 15*w^2 - 8*w - 9],
[613, 613, 3*w^4 - 14*w^2 - 2*w + 8],
[617, 617, -4*w^4 + w^3 + 19*w^2 - 3*w - 7],
[619, 619, w^3 + w^2 - 3*w - 5],
[625, 5, 4*w^4 - 2*w^3 - 19*w^2 + 3*w + 8],
[631, 631, 3*w^4 - w^3 - 12*w^2 - 1],
[641, 641, -w^4 + 2*w^3 + 3*w^2 - 7*w],
[647, 647, 2*w^2 - w - 5],
[647, 647, -2*w^3 + w^2 + 8*w - 5],
[647, 647, 5*w^4 - 3*w^3 - 23*w^2 + 9*w + 12],
[661, 661, -8*w^4 + 3*w^3 + 39*w^2 - 7*w - 19],
[677, 677, -7*w^4 + 4*w^3 + 34*w^2 - 11*w - 17],
[691, 691, w^2 + 2*w - 4],
[691, 691, -5*w^4 + 2*w^3 + 23*w^2 - 3*w - 8],
[691, 691, -7*w^4 + 5*w^3 + 32*w^2 - 14*w - 15],
[701, 701, w - 4],
[709, 709, -3*w^4 + 3*w^3 + 14*w^2 - 8*w - 7],
[709, 709, 5*w^4 - 2*w^3 - 24*w^2 + 5*w + 9],
[709, 709, -w^4 + w^3 + 5*w^2 - 3*w - 7],
[719, 719, -3*w^4 + 13*w^2 + w - 7],
[719, 719, -3*w^4 + 14*w^2 - 4],
[727, 727, -7*w^4 + 4*w^3 + 32*w^2 - 10*w - 16],
[727, 727, -3*w^4 + w^3 + 14*w^2 - 5*w - 8],
[733, 733, -3*w^4 + 2*w^3 + 15*w^2 - 8*w - 10],
[733, 733, -3*w^4 + 3*w^3 + 13*w^2 - 10*w - 7],
[733, 733, 8*w^4 - 4*w^3 - 38*w^2 + 9*w + 15],
[739, 739, 4*w^4 - 3*w^3 - 20*w^2 + 9*w + 11],
[739, 739, -w^4 + w^3 + 5*w^2 - 3*w + 1],
[743, 743, -5*w^4 + 4*w^3 + 24*w^2 - 13*w - 16],
[743, 743, w^4 - 7*w^2 - 2*w + 5],
[751, 751, -w^4 + 2*w^3 + 3*w^2 - 7*w - 1],
[757, 757, 5*w^4 - 2*w^3 - 22*w^2 + 5*w + 8],
[757, 757, w^4 + w^3 - 6*w^2 - 5*w + 2],
[761, 761, 8*w^4 - 3*w^3 - 38*w^2 + 8*w + 18],
[761, 761, 3*w^4 - w^3 - 14*w^2 + 2*w + 10],
[761, 761, -4*w^4 + 3*w^3 + 19*w^2 - 7*w - 10],
[769, 769, -w^4 + 2*w^3 + 5*w^2 - 8*w - 1],
[773, 773, 9*w^4 - 4*w^3 - 41*w^2 + 9*w + 15],
[773, 773, 4*w^4 - 2*w^3 - 18*w^2 + 3*w + 6],
[773, 773, 2*w^2 - 5],
[787, 787, -w^4 + 6*w^2 - w - 7],
[787, 787, w^4 - w^3 - 6*w^2 + w + 6],
[821, 821, 2*w^4 - w^3 - 9*w^2 + 4*w + 8],
[823, 823, 6*w^4 - w^3 - 27*w^2 - w + 8],
[827, 827, -3*w^4 + 2*w^3 + 16*w^2 - 6*w - 9],
[827, 827, -8*w^4 + 5*w^3 + 37*w^2 - 16*w - 17],
[839, 839, 2*w^4 - 10*w^2 + w + 8],
[853, 853, 2*w^4 - 2*w^3 - 11*w^2 + 7*w + 7],
[857, 857, 5*w^4 - 2*w^3 - 24*w^2 + 7*w + 14],
[877, 877, -w^4 + 2*w^3 + 5*w^2 - 10*w - 6],
[883, 883, 3*w^4 - 3*w^3 - 15*w^2 + 12*w + 10],
[887, 887, -4*w^4 + w^3 + 17*w^2 - w - 7],
[907, 907, 2*w^4 - 11*w^2 - w + 2],
[907, 907, -2*w^4 - w^3 + 10*w^2 + 4*w - 5],
[911, 911, -9*w^4 + 4*w^3 + 43*w^2 - 9*w - 19],
[919, 919, 4*w^4 - w^3 - 20*w^2 + w + 8],
[937, 937, w^4 - 7*w^2 - 2*w + 4],
[947, 947, 2*w^4 + 2*w^3 - 10*w^2 - 9*w + 3],
[953, 953, -2*w^3 + 5*w + 4],
[961, 31, -4*w^4 + w^3 + 19*w^2 - w - 4],
[967, 967, 8*w^4 - 5*w^3 - 37*w^2 + 13*w + 18],
[967, 967, 3*w^4 - w^3 - 13*w^2 - 2*w + 3],
[971, 971, -4*w^4 + 3*w^3 + 21*w^2 - 10*w - 15],
[971, 971, -3*w^4 + 12*w^2 + 3*w - 4],
[971, 971, w^4 - 6*w^2 + 3*w + 6],
[983, 983, -7*w^4 + 2*w^3 + 34*w^2 - 5*w - 16],
[983, 983, w^4 - 7*w^2 - w + 8],
[991, 991, -4*w^4 + 4*w^3 + 19*w^2 - 15*w - 12],
[991, 991, -8*w^4 + 3*w^3 + 37*w^2 - 5*w - 15],
[991, 991, -w^4 + w^3 + 7*w^2 - 3*w - 7],
[997, 997, -8*w^4 + 5*w^3 + 37*w^2 - 16*w - 16]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 5*x^2 - 4*x + 29;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^2 + e + 7, -2/3*e^2 + e + 26/3, e^2 - 12, -4/3*e^2 + 2*e + 34/3, 2*e^2 - 4*e - 15, -4/3*e^2 + 4*e + 22/3, -1/3*e^2 - 2/3, -2*e^2 + 3*e + 16, -1/3*e^2 + 16/3, 4/3*e^2 - 2*e - 28/3, 4/3*e^2 - 2*e - 28/3, -e^2 + 14, 4/3*e^2 - 2*e - 10/3, 1/3*e^2 - 3*e - 1/3, 2/3*e^2 - 50/3, -8, -5/3*e^2 + 2*e + 68/3, 1, -2/3*e^2 + 38/3, -2/3*e^2 + 38/3, 8/3*e^2 - 4*e - 74/3, -1/3*e^2 + 3*e + 7/3, 5/3*e^2 - 7*e - 17/3, -e^2 + e + 17, e^2 - 4, 2/3*e^2 - 4*e - 44/3, -5/3*e^2 + 3*e + 71/3, -4*e, 4/3*e^2 - e - 34/3, -4*e + 10, 11/3*e^2 - 9*e - 71/3, e^2 - 8*e - 2, 2/3*e^2 - 26/3, -6, -3*e^2 + 7*e + 19, 11/3*e^2 - 3*e - 137/3, -2/3*e^2 - 2*e + 38/3, 8/3*e^2 - 7*e - 68/3, 10/3*e^2 - 8*e - 34/3, 8/3*e^2 - 4*e - 50/3, 7/3*e^2 - e - 103/3, -e^2 + 5*e + 1, -8/3*e^2 + 6*e + 32/3, -7/3*e^2 + 3*e + 91/3, -19/3*e^2 + 13*e + 127/3, 4*e^2 - 8*e - 18, 11/3*e^2 - 7*e - 71/3, -8/3*e^2 + 10*e + 44/3, 8/3*e^2 - 5*e - 80/3, -17/3*e^2 + 12*e + 104/3, -4/3*e^2 + e + 34/3, -4/3*e^2 + 58/3, 6*e - 10, 5/3*e^2 - 2/3, 16/3*e^2 - 10*e - 118/3, -4/3*e^2 + 7*e - 2/3, -11/3*e^2 - e + 149/3, 2*e^2 - 13*e - 10, -2*e^2 + 6*e + 20, 1/3*e^2 - 10*e + 20/3, 4*e^2 - 10*e - 16, -11/3*e^2 + 128/3, 3*e^2 - e - 35, -4/3*e^2 + 6*e - 50/3, 17/3*e^2 - 9*e - 125/3, -7*e, -2*e - 18, -8/3*e^2 + 4*e + 56/3, 10/3*e^2 - 10*e - 52/3, -8/3*e^2 + 4*e + 98/3, -2/3*e^2 - 2*e + 2/3, -2/3*e^2 - 2*e + 62/3, 4/3*e^2 - 8*e - 46/3, -7/3*e^2 + 6*e + 46/3, -14/3*e^2 + 4*e + 170/3, 8/3*e^2 - 7*e + 10/3, -4/3*e^2 + 2*e + 46/3, 2/3*e^2 - 4*e - 20/3, -2/3*e^2 + 6*e + 38/3, 14/3*e^2 - 12*e - 50/3, 17/3*e^2 - 5*e - 155/3, 8/3*e^2 - 2*e - 146/3, -4/3*e^2 + 5*e - 56/3, 5/3*e^2 + 2*e - 80/3, -2/3*e^2 + 2*e + 44/3, -3*e^2 + 6*e + 22, -2/3*e^2 - 8*e + 38/3, 19/3*e^2 - 12*e - 136/3, -20/3*e^2 + 8*e + 170/3, 4*e^2 - 4*e - 54, 2*e^2 - 4*e + 18, -16/3*e^2 + 12*e + 148/3, 4*e^2 - 5*e - 42, 6*e^2 - 15*e - 48, 7/3*e^2 + 2*e - 112/3, 2*e^2 - 8*e - 10, -20/3*e^2 + 16*e + 164/3, -13/3*e^2 + 7*e + 139/3, -8/3*e^2 + 44/3, e^2 - 9*e + 5, -2*e^2 - 3*e + 42, 10/3*e^2 + 4*e - 148/3, -26/3*e^2 + 20*e + 134/3, 4*e^2 - 7*e - 56, -10/3*e^2 + 18*e + 28/3, -26/3*e^2 + 19*e + 194/3, 3*e^2 - 2*e - 28, -3*e^2 + 7*e + 17, 2/3*e^2 - 3*e + 40/3, 2*e^2 - 7*e - 38, -8*e^2 + 7*e + 88, -1/3*e^2 - 2*e - 26/3, -5*e^2 + 16*e + 38, 5*e^2 - 11*e - 37, -16/3*e^2 + 4*e + 202/3, 10/3*e^2 - 14*e - 52/3, -2*e^2 + 4*e + 20, -9*e + 26, 8*e^2 - 12*e - 80, -6*e + 18, -28/3*e^2 + 20*e + 184/3, -2*e^2 + 7*e + 20, 2/3*e^2 - e - 2/3, -8/3*e^2 + 4*e + 122/3, 14/3*e^2 - 19*e - 92/3, -4*e^2 + 7*e + 16, 2*e^2 + 4*e - 12, 4*e^2 - 8*e - 12, 4*e^2 - 4*e - 32, -16/3*e^2 + 16*e + 142/3, 8*e^2 - 4*e - 86, -16/3*e^2 + 12*e + 10/3, -22/3*e^2 + 13*e + 208/3, -5*e^2 + 7*e + 39, 2/3*e^2 - 3*e - 110/3, 8/3*e^2 - 4*e - 62/3, 2*e^2 - 4*e - 16, 2/3*e^2 - 8*e + 16/3, -8/3*e^2 + 6*e - 40/3, -2*e - 34, 2*e^2 - 3*e - 4, -38/3*e^2 + 26*e + 230/3, -2*e^2 + 8*e + 26, -22/3*e^2 + 16*e + 106/3, -8/3*e^2 + 8*e + 86/3, 2*e^2 + 8*e - 30, 8/3*e^2 - 20*e - 32/3, -2*e^2 + 5*e - 6, 7/3*e^2 - 19*e - 1/3, 16/3*e^2 - 3*e - 184/3, -19/3*e^2 + 2*e + 184/3, 7/3*e^2 - 4*e - 58/3, 14/3*e^2 - 14*e - 152/3, -14/3*e^2 + 194/3, -8/3*e^2 + 10*e + 56/3, -26/3*e^2 + 15*e + 146/3, 8*e - 12, -2*e^2 - 2*e + 48, -23/3*e^2 + 6*e + 302/3, -10/3*e^2 - 2*e + 166/3, 4/3*e^2 - 4*e + 44/3, 2*e^2 + 6*e - 42, -22/3*e^2 + 15*e + 76/3, 10/3*e^2 - 2*e - 196/3, 3*e^2 - 18*e - 12, 5*e^2 - 14*e - 14, -10*e^2 + 18*e + 88, -4/3*e^2 + 16*e - 38/3, 14/3*e^2 - 17*e - 92/3, 6*e^2 - 16*e - 30, -7/3*e^2 - 7*e + 133/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6}>] := -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;