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

NN := ideal<ZF | {23, 23, -w^2 + w + 2}>;

primesArray := [
[5, 5, -w^4 + w^3 + 4*w^2 - 2*w - 1],
[7, 7, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w - 3],
[19, 19, w^5 - w^4 - 5*w^3 + 2*w^2 + 4*w],
[23, 23, -w^2 + w + 2],
[25, 5, w^5 + w^4 - 6*w^3 - 7*w^2 + 4*w + 2],
[41, 41, 2*w^5 - w^4 - 10*w^3 + 6*w - 1],
[47, 47, w^3 - w^2 - 4*w],
[53, 53, w^5 - 5*w^3 - 3*w^2 + w + 3],
[59, 59, 2*w^5 - 11*w^3 - 4*w^2 + 8*w],
[61, 61, w^2 - 2*w - 2],
[64, 2, -2],
[67, 67, -w^4 + 6*w^2 + w - 3],
[67, 67, -2*w^4 + w^3 + 10*w^2 - 6],
[73, 73, w^5 - 5*w^3 - 2*w^2 + 2*w - 2],
[73, 73, -w^5 - w^4 + 7*w^3 + 6*w^2 - 8*w - 2],
[79, 79, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 1],
[83, 83, 2*w^5 - 11*w^3 - 4*w^2 + 6*w],
[89, 89, -2*w^5 + 11*w^3 + 4*w^2 - 7*w - 2],
[97, 97, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w - 2],
[97, 97, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1],
[97, 97, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 4],
[101, 101, w^5 - 6*w^3 - 3*w^2 + 6*w + 2],
[103, 103, -w^5 - w^4 + 5*w^3 + 8*w^2 - 5],
[107, 107, 2*w^4 - w^3 - 9*w^2 + w + 2],
[109, 109, 2*w^5 - 10*w^3 - 5*w^2 + 4*w + 1],
[113, 113, -w^4 + 6*w^2 + 2*w - 3],
[125, 5, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 6],
[127, 127, 2*w^5 - w^4 - 11*w^3 + 10*w],
[131, 131, 2*w^5 - w^4 - 11*w^3 + 9*w + 1],
[137, 137, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 9*w + 5],
[139, 139, -w^5 - 2*w^4 + 7*w^3 + 11*w^2 - 5*w - 5],
[157, 157, 2*w^5 - 10*w^3 - 5*w^2 + 5*w + 1],
[167, 167, -2*w^5 + 10*w^3 + 5*w^2 - 3*w + 1],
[173, 173, 4*w^5 + 2*w^4 - 22*w^3 - 19*w^2 + 10*w + 5],
[179, 179, 2*w^5 - w^4 - 11*w^3 + 10*w + 1],
[179, 179, 2*w^5 - 11*w^3 - 5*w^2 + 8*w + 1],
[181, 181, -4*w^5 - w^4 + 22*w^3 + 14*w^2 - 12*w - 4],
[191, 191, w^4 + w^3 - 6*w^2 - 4*w + 2],
[191, 191, -3*w^5 + 18*w^3 + 5*w^2 - 15*w - 1],
[193, 193, -w^4 + w^3 + 6*w^2 - 2*w - 7],
[197, 197, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 12*w + 1],
[197, 197, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],
[197, 197, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w + 1],
[211, 211, -2*w^5 + w^4 + 10*w^3 - w^2 - 4*w + 4],
[223, 223, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 4],
[227, 227, w^5 - 7*w^3 - w^2 + 9*w],
[227, 227, -2*w^5 + w^4 + 9*w^3 - 2*w + 1],
[229, 229, w^5 + w^4 - 5*w^3 - 6*w^2 - 2*w - 2],
[233, 233, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 8*w - 4],
[269, 269, 3*w^5 - 16*w^3 - 6*w^2 + 8*w],
[269, 269, -2*w^5 + 12*w^3 + 3*w^2 - 12*w - 1],
[277, 277, 3*w^5 - w^4 - 17*w^3 - w^2 + 16*w],
[289, 17, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 10*w - 3],
[307, 307, w^4 - 5*w^2 - 2*w - 1],
[307, 307, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 12*w + 3],
[307, 307, -w^4 - w^3 + 5*w^2 + 6*w - 2],
[311, 311, -w^4 - w^3 + 7*w^2 + 4*w - 4],
[311, 311, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w + 2],
[313, 313, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w],
[317, 317, 2*w^5 + w^4 - 12*w^3 - 8*w^2 + 7*w],
[331, 331, 2*w^5 - 2*w^4 - 10*w^3 + 5*w^2 + 8*w - 5],
[337, 337, w^5 - 6*w^3 - 3*w^2 + 6*w],
[347, 347, -2*w^5 + 2*w^4 + 10*w^3 - 6*w^2 - 7*w + 2],
[347, 347, 3*w^5 + w^4 - 17*w^3 - 11*w^2 + 9*w + 2],
[349, 349, -2*w^5 + 11*w^3 + 5*w^2 - 7*w - 6],
[367, 367, -w^5 - 2*w^4 + 5*w^3 + 13*w^2 + 2*w - 5],
[367, 367, 2*w^5 + 2*w^4 - 10*w^3 - 16*w^2 - w + 6],
[373, 373, 3*w^5 - w^4 - 17*w^3 - w^2 + 15*w + 1],
[373, 373, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 4],
[379, 379, 2*w^5 + w^4 - 11*w^3 - 10*w^2 + 5*w + 7],
[389, 389, w^2 - 5],
[389, 389, w^5 - 7*w^3 + 8*w],
[397, 397, -2*w^5 - w^4 + 12*w^3 + 9*w^2 - 7*w - 5],
[397, 397, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 14*w - 2],
[397, 397, -2*w^5 - 2*w^4 + 12*w^3 + 13*w^2 - 6*w - 4],
[419, 419, 2*w^5 + 2*w^4 - 11*w^3 - 14*w^2 + 2*w + 5],
[419, 419, -w^5 + 4*w^3 + 3*w^2 + w - 2],
[433, 433, -w^5 + 5*w^3 + 4*w^2 - w - 3],
[433, 433, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 4],
[439, 439, 4*w^5 + 2*w^4 - 22*w^3 - 18*w^2 + 9*w + 4],
[439, 439, w^5 + w^4 - 5*w^3 - 9*w^2 + 6],
[439, 439, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 13*w + 2],
[443, 443, -3*w^5 + w^4 + 15*w^3 + 3*w^2 - 8*w - 2],
[449, 449, -w^5 + w^4 + 5*w^3 - 4*w^2 - 2*w + 3],
[449, 449, 3*w^5 - 16*w^3 - 8*w^2 + 9*w + 4],
[449, 449, w^4 + w^3 - 6*w^2 - 6*w + 5],
[449, 449, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4],
[457, 457, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 6*w + 4],
[457, 457, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 5*w - 4],
[461, 461, -2*w^4 + w^3 + 10*w^2 + w - 5],
[463, 463, -w^5 + w^4 + 6*w^3 - 4*w^2 - 5*w + 4],
[487, 487, -3*w^4 + 2*w^3 + 14*w^2 - w - 7],
[487, 487, 3*w^5 + w^4 - 18*w^3 - 10*w^2 + 14*w + 3],
[491, 491, -3*w^5 + w^4 + 16*w^3 + w^2 - 12*w + 2],
[491, 491, 3*w^5 - w^4 - 16*w^3 - 2*w^2 + 12*w + 3],
[499, 499, 4*w^5 - 22*w^3 - 9*w^2 + 15*w + 4],
[499, 499, w^4 + w^3 - 7*w^2 - 6*w + 6],
[523, 523, w^4 - 2*w^3 - 4*w^2 + 5*w + 2],
[529, 23, -w^5 + 7*w^3 + w^2 - 10*w - 1],
[541, 541, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 9*w + 3],
[557, 557, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 5],
[557, 557, 4*w^5 - 22*w^3 - 8*w^2 + 14*w + 1],
[563, 563, 2*w^5 - w^4 - 12*w^3 + 2*w^2 + 13*w - 1],
[563, 563, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 2*w - 6],
[563, 563, -w^5 + w^4 + 4*w^3 - 2*w^2 + w + 3],
[569, 569, -3*w^5 + w^4 + 16*w^3 + w^2 - 9*w + 1],
[599, 599, -w^4 - w^3 + 7*w^2 + 6*w - 5],
[607, 607, -w^5 + 6*w^3 + 3*w^2 - 7*w - 1],
[613, 613, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 4*w - 7],
[619, 619, 4*w^5 - w^4 - 22*w^3 - 3*w^2 + 16*w],
[653, 653, 2*w^5 - 2*w^4 - 10*w^3 + 4*w^2 + 7*w],
[653, 653, -2*w^5 + w^4 + 12*w^3 - w^2 - 14*w + 1],
[659, 659, -2*w^5 - w^4 + 11*w^3 + 11*w^2 - 5*w - 7],
[661, 661, 2*w^5 - 11*w^3 - 3*w^2 + 7*w],
[661, 661, 2*w^5 - w^4 - 11*w^3 + 11*w + 2],
[673, 673, -w^4 - w^3 + 5*w^2 + 7*w - 1],
[677, 677, 3*w^5 - 16*w^3 - 7*w^2 + 10*w + 3],
[677, 677, w^5 + 3*w^4 - 6*w^3 - 18*w^2 + 12],
[691, 691, w^5 - w^4 - 4*w^3 + 2*w^2 - 4],
[709, 709, -w^4 + w^3 + 3*w^2 - 2*w + 2],
[709, 709, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 6*w - 1],
[719, 719, 2*w^5 + 2*w^4 - 12*w^3 - 14*w^2 + 8*w + 5],
[729, 3, -3],
[739, 739, 4*w^5 - w^4 - 21*w^3 - 5*w^2 + 12*w + 3],
[743, 743, -2*w^5 - w^4 + 10*w^3 + 11*w^2 - w - 6],
[751, 751, -3*w^5 + w^4 + 17*w^3 + 2*w^2 - 15*w - 2],
[751, 751, -2*w^5 + 3*w^4 + 10*w^3 - 11*w^2 - 10*w + 6],
[757, 757, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 7*w - 6],
[761, 761, -w^5 + w^4 + 6*w^3 - 3*w^2 - 6*w - 1],
[787, 787, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 15*w - 2],
[797, 797, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 12*w - 1],
[797, 797, w^2 - 2*w - 5],
[809, 809, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 12*w - 3],
[827, 827, 4*w^5 - 22*w^3 - 8*w^2 + 15*w + 1],
[839, 839, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 10*w + 2],
[839, 839, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 1],
[841, 29, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 7*w - 3],
[853, 853, w^5 - 7*w^3 - w^2 + 7*w + 1],
[857, 857, -2*w^5 + w^4 + 10*w^3 - w^2 - 6*w + 4],
[863, 863, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w],
[877, 877, 2*w^5 - 10*w^3 - 5*w^2 + 3*w + 4],
[887, 887, 2*w^5 + w^4 - 13*w^3 - 8*w^2 + 13*w + 5],
[887, 887, 3*w^5 - 2*w^4 - 15*w^3 + 3*w^2 + 8*w - 4],
[907, 907, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 17*w],
[907, 907, -2*w^5 + 2*w^4 + 10*w^3 - 4*w^2 - 9*w + 1],
[919, 919, -3*w^5 + 17*w^3 + 5*w^2 - 11*w],
[929, 929, -3*w^5 + 17*w^3 + 6*w^2 - 11*w],
[941, 941, 2*w^5 - w^4 - 11*w^3 - w^2 + 9*w + 2],
[947, 947, -w^4 + 6*w^2 + 4*w - 4],
[947, 947, -2*w^5 + w^4 + 11*w^3 - 11*w - 3],
[947, 947, -4*w^5 + w^4 + 22*w^3 + 3*w^2 - 16*w - 1],
[953, 953, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 3*w - 8],
[953, 953, 2*w^5 + 2*w^4 - 13*w^3 - 14*w^2 + 10*w + 6],
[961, 31, 2*w^5 + w^4 - 12*w^3 - 10*w^2 + 9*w + 4],
[967, 967, 2*w^5 - w^4 - 11*w^3 + w^2 + 11*w + 1],
[967, 967, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 13*w - 3],
[983, 983, -4*w^5 + w^4 + 22*w^3 + 4*w^2 - 18*w - 2],
[991, 991, 3*w^5 - 16*w^3 - 7*w^2 + 7*w + 2],
[991, 991, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 11*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [e, 2, -2*e - 4, 1, e^2 + 2*e - 6, e - 4, -4*e - 6, -e^2 + 4, -2*e^2 - 8*e, -e^2 - 6*e - 6, -2*e^2 - 8*e + 3, -2*e^2 - 4*e + 8, 2*e^2 + 8*e - 4, 4*e^2 + 11*e - 12, 2*e^2 + 9*e - 6, -4*e^2 - 8*e + 12, 4*e^2 + 14*e - 4, -2*e^2 - 5*e + 6, e^2 + 2*e - 10, -5*e^2 - 16*e + 6, e^2 + 6*e - 2, 5*e^2 + 14*e - 6, 2*e^2 + 4*e - 6, -4*e^2 - 10*e, -4*e^2 - 11*e, -5*e^2 - 14*e + 6, 3*e^2 + 8*e - 14, 4*e^2 + 12*e - 8, -2*e^2 - 6*e + 8, -3*e^2 - 8*e + 10, 4*e^2 + 6*e - 12, 2*e^2 - 3*e - 20, 4*e^2 + 14*e - 6, 2*e^2 + 13*e + 6, -6*e^2 - 18*e + 12, -2*e^2 - 2*e + 12, -2*e^2 - 5*e, -22, -4*e^2 - 10*e + 16, e^2 - 2*e - 6, 3*e - 8, -7*e^2 - 26*e + 16, 7*e, 2*e^2 + 2*e - 20, 2*e - 16, 2*e^2 + 6*e - 8, -6*e - 20, e^2 - 2, 2*e^2 + 7*e, 5*e^2 + 4*e - 30, 5*e^2 + 24*e - 8, 2*e^2 + 7*e + 14, -e^2 - 6*e + 10, 10*e + 4, -2*e^2 + 4*e + 12, 10*e^2 + 28*e - 16, -6*e^2 - 18*e + 24, -2*e^2 - 2*e + 12, -8*e^2 - 20*e + 10, 2*e^2 + 9*e - 12, -6*e^2 - 20*e + 16, e^2 - 10, 4*e^2 + 18*e + 2, 4*e^2 + 14*e - 12, -4*e^2 - 7*e + 24, 2*e + 10, 2*e^2 + 4*e + 16, 4*e^2 + 8*e - 34, -7*e^2 - 20*e + 8, 2*e^2 + 10*e + 12, 2*e^2 + 5*e - 16, -11*e^2 - 26*e + 24, -6*e^2 - 19*e + 4, -5*e^2 - 8*e + 8, -2*e^2 - 7*e + 16, -2*e^2 - 8*e + 2, 4*e^2 + 16*e - 12, 4*e^2 + 8*e - 26, 4*e^2 + 17*e - 4, 8*e^2 + 22*e - 28, 2*e^2 + 4*e - 4, -4*e^2 - 8*e + 24, -2*e^2 + 2*e + 22, -e^2 + 6*e - 2, -7*e - 4, -e^2 - 4*e + 6, -2*e^2 - 5*e + 12, 4*e^2 + 5*e - 20, 9*e^2 + 20*e - 26, -4*e^2 - 8*e - 6, 2*e^2 + 6*e - 14, -8*e^2 - 22*e + 8, -2*e^2 - 18*e - 8, -4*e^2 - 16*e + 16, 6*e^2 + 26*e + 4, 6*e + 8, -2*e^2 - 2*e + 16, 4*e^2 - 4*e - 30, 2*e^2 + e - 4, 8*e^2 + 13*e - 26, 2*e^2 + 11*e - 16, 3*e^2 - 2*e - 26, 4*e^2 + 12*e - 36, 6*e^2 + 14*e - 16, -4*e^2 + 34, 9*e^2 + 22*e - 2, 4*e^2 + 12*e - 24, 4*e^2 + 6*e - 24, -6*e^2 - 21*e + 12, -2*e^2 - 4*e - 12, -3*e + 6, -3*e^2 - 18*e + 12, 2*e^2 + 6*e - 14, 7*e + 8, 8*e^2 + 29*e - 16, -8*e^2 - 28*e + 26, -6*e^2 - 22*e + 28, 6*e^2 + 23*e + 2, 10*e^2 + 28*e - 38, 11*e^2 + 30*e - 22, 8*e^2 + 19*e - 36, 4*e^2 + 10*e - 28, 8*e^2 + 17*e - 16, -2*e^2 - 12*e + 24, -2*e^2 - 4*e + 4, -6*e^2 - 26*e + 18, 6*e^2 + 22*e - 16, -7*e^2 - 20*e, -12*e^2 - 37*e + 22, -2*e^2 - 6*e + 42, -10*e^2 - 26*e, -4*e^2 - 5*e, 3*e^2 - 4*e - 22, 12*e^2 + 26*e - 24, 2*e^2 + 8*e - 4, 6*e^2 + 28*e - 20, 4*e^2 + 20*e + 2, -8*e^2 - 23*e - 12, 4*e^2 + 15*e - 44, 2*e^2 - 10*e - 24, 4*e^2 + e - 12, 6*e^2 + 28*e - 24, 6*e^2 + 22*e - 10, -10*e^2 - 8*e + 52, 4*e + 44, 6*e^2 + 6*e - 20, -2*e^2 - 17*e + 6, 12*e^2 + 27*e - 38, 10*e^2 + 34*e - 24, -10*e^2 - 26*e + 26, -10*e^2 - 24*e + 24, -8*e^2 - 3*e + 56, -3*e^2 - 8*e - 10, -6*e^2 - 27*e + 10, 10*e^2 + 38*e - 28, 2*e^2 + 8*e - 16, -10*e^2 - 18*e + 36, -12*e + 4, 8*e^2 + 12*e - 4];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -w^2 + 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;