/*
  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![3, -3, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {7, 7, w^3 - w^2 - 4*w + 1}>;

primesArray := [
[3, 3, w],
[5, 5, w - 1],
[7, 7, w^3 - w^2 - 4*w + 1],
[11, 11, -w^2 + w + 2],
[13, 13, -w^3 + w^2 + 3*w - 1],
[16, 2, 2],
[17, 17, -w^3 + 5*w + 2],
[29, 29, -w^2 + w + 1],
[37, 37, -w^3 + 2*w^2 + 4*w - 4],
[41, 41, -w^3 + w^2 + 5*w + 1],
[41, 41, w^2 - 5],
[47, 47, w^3 - w^2 - 5*w - 2],
[47, 47, -w^3 + 2*w^2 + 4*w - 1],
[53, 53, w^3 - 4*w - 4],
[53, 53, 2*w^3 - 2*w^2 - 9*w + 1],
[61, 61, -w^3 + w^2 + 6*w - 2],
[71, 71, w^3 - w^2 - 6*w - 2],
[103, 103, 2*w^3 - 2*w^2 - 8*w + 1],
[103, 103, -3*w^3 + 3*w^2 + 13*w - 5],
[109, 109, 2*w^3 - w^2 - 8*w - 4],
[109, 109, 2*w^3 - w^2 - 9*w - 1],
[113, 113, -2*w^3 + w^2 + 11*w - 1],
[113, 113, 2*w^3 - w^2 - 10*w - 4],
[125, 5, 2*w^3 - 3*w^2 - 10*w + 4],
[131, 131, -2*w^3 + w^2 + 9*w - 1],
[139, 139, 2*w^3 - 4*w^2 - 7*w + 5],
[139, 139, 3*w^3 - w^2 - 16*w - 5],
[149, 149, w^2 + w - 4],
[149, 149, 3*w^3 - 3*w^2 - 13*w + 1],
[151, 151, w^2 - 2*w - 8],
[157, 157, w^3 - w^2 - 7*w - 2],
[157, 157, -2*w^2 + w + 7],
[167, 167, 2*w^3 - 2*w^2 - 7*w + 5],
[173, 173, -w^3 + 6*w - 2],
[173, 173, -w^3 + 2*w^2 + 5*w - 7],
[181, 181, -3*w^3 + 2*w^2 + 14*w + 1],
[191, 191, -w^3 + 2*w^2 + 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 5],
[193, 193, -w^3 + 3*w^2 + 4*w - 4],
[193, 193, 2*w^3 + 3*w^2 - 13*w - 20],
[197, 197, w^3 - 2*w - 2],
[199, 199, 2*w^3 - w^2 - 11*w - 2],
[211, 211, -2*w^3 + 9*w + 5],
[223, 223, -w^3 + w^2 + 4*w - 5],
[223, 223, w^2 - 3*w - 2],
[229, 229, 2*w^3 - 2*w^2 - 7*w + 1],
[241, 241, 3*w^2 - 2*w - 13],
[241, 241, -w^3 + w^2 + 5*w + 4],
[257, 257, -3*w^3 + w^2 + 15*w + 8],
[263, 263, -3*w^3 + 5*w^2 + 9*w - 7],
[263, 263, 2*w^3 - w^2 - 9*w + 2],
[263, 263, -w^3 + 4*w^2 + 2*w - 16],
[263, 263, w^3 - w^2 - 2*w - 2],
[271, 271, -3*w - 1],
[281, 281, 2*w^3 - 3*w^2 - 8*w + 2],
[281, 281, 3*w^3 - 4*w^2 - 12*w + 4],
[283, 283, w^3 - 5*w - 7],
[293, 293, -2*w^3 + w^2 + 10*w - 1],
[307, 307, w^3 - 3*w^2 - 3*w + 8],
[307, 307, 3*w^3 - 15*w - 10],
[311, 311, 2*w^3 - 3*w^2 - 7*w + 7],
[313, 313, -w^3 + 7*w + 2],
[337, 337, 3*w^3 - 17*w - 10],
[337, 337, 2*w^3 - 2*w^2 - 9*w - 5],
[343, 7, -2*w^3 + w^2 + 8*w + 2],
[347, 347, -3*w^3 + 17*w + 14],
[347, 347, -3*w^3 - w^2 + 17*w + 13],
[349, 349, 2*w^3 - w^2 - 8*w - 1],
[349, 349, -w^3 + 3*w^2 + 4*w - 8],
[353, 353, -w^3 - w^2 + 7*w + 4],
[359, 359, -2*w^3 + 9*w + 10],
[367, 367, 3*w^3 - 4*w^2 - 9*w + 4],
[367, 367, -w^3 + w^2 + 4*w + 4],
[367, 367, -w^3 + 3*w^2 + 3*w - 7],
[373, 373, -w^2 + w - 2],
[373, 373, -2*w^3 + 2*w^2 + 11*w - 2],
[379, 379, -2*w^3 + 4*w^2 + 7*w - 8],
[379, 379, -w^2 - 2],
[389, 389, -w^3 - 2*w^2 + 5*w + 11],
[397, 397, -2*w^3 + 4*w^2 + 7*w - 7],
[401, 401, w^3 - 3*w^2 - 2*w + 10],
[401, 401, -2*w^3 + 11*w + 4],
[401, 401, 3*w^3 - 2*w^2 - 14*w - 4],
[401, 401, -3*w^3 + 4*w^2 + 15*w - 5],
[421, 421, -w^3 + w^2 + 7*w - 1],
[421, 421, -2*w^3 + 2*w^2 + 9*w - 7],
[433, 433, w^2 - w - 8],
[439, 439, -w^3 + 3*w^2 + 4*w - 7],
[443, 443, -2*w^3 + w^2 + 10*w - 2],
[449, 449, w^2 + 2*w - 4],
[449, 449, 2*w^3 - 2*w^2 - 10*w - 1],
[461, 461, 2*w^3 - 2*w^2 - 6*w + 5],
[461, 461, -2*w^3 + 4*w^2 + 7*w - 13],
[463, 463, -3*w^3 + 3*w^2 + 12*w - 4],
[463, 463, w - 5],
[467, 467, -3*w^3 + w^2 + 16*w + 8],
[467, 467, 2*w^2 - 3*w - 10],
[479, 479, 2*w^2 - 4*w - 1],
[479, 479, -w^3 + 2*w^2 + 2*w - 7],
[487, 487, -3*w^3 + 5*w^2 + 12*w - 10],
[503, 503, w^3 + 2*w^2 - 8*w - 8],
[509, 509, -3*w^3 + 5*w^2 + 11*w - 5],
[521, 521, 2*w^2 - 3*w - 11],
[523, 523, -2*w^3 - w^2 + 13*w + 11],
[523, 523, -2*w^3 + 4*w^2 + 8*w - 7],
[541, 541, 3*w^3 - 2*w^2 - 13*w + 1],
[563, 563, w^3 - 3*w - 5],
[563, 563, -w^3 + 3*w^2 + 2*w - 11],
[569, 569, -2*w^3 + 4*w^2 + 9*w - 4],
[569, 569, 3*w^2 - w - 14],
[571, 571, -w^3 + 8*w - 1],
[577, 577, 3*w^3 - 2*w^2 - 16*w + 1],
[587, 587, -4*w^3 + 5*w^2 + 18*w - 7],
[587, 587, 2*w^3 + w^2 - 10*w - 7],
[599, 599, 2*w^3 + w^2 - 11*w - 8],
[607, 607, w^3 - 3*w + 4],
[613, 613, 3*w^3 - w^2 - 14*w - 5],
[613, 613, 3*w^2 - w - 11],
[619, 619, 2*w^2 - 5*w - 8],
[631, 631, -2*w^3 - 2*w^2 + 12*w + 19],
[647, 647, 2*w^3 - 2*w^2 - 11*w - 2],
[653, 653, 3*w^3 - 5*w^2 - 8*w + 7],
[677, 677, -2*w^3 + 4*w^2 + 5*w - 8],
[683, 683, 2*w^3 + w^2 - 12*w - 8],
[683, 683, -w^3 + 3*w^2 - 8],
[683, 683, -2*w^3 + 3*w^2 + 11*w - 8],
[683, 683, -2*w^3 + 3*w^2 + 10*w - 2],
[691, 691, w^2 - 4*w - 1],
[709, 709, 3*w^3 - 2*w^2 - 13*w - 2],
[709, 709, w^3 - 2*w^2 - 3*w - 2],
[727, 727, -w^3 + 4*w^2 + 2*w - 7],
[727, 727, -w^3 + 8*w + 5],
[733, 733, 3*w^2 - 2*w - 10],
[733, 733, -w^3 + 5*w + 8],
[757, 757, 3*w^3 - 14*w - 8],
[761, 761, -w^3 + 3*w^2 + 5*w - 10],
[773, 773, -2*w^3 + 3*w^2 + 9*w - 1],
[787, 787, -w^3 + 4*w - 4],
[797, 797, w^2 + 2*w - 5],
[797, 797, w^3 + w^2 - 5*w - 1],
[809, 809, w^3 + 3*w^2 - 9*w - 17],
[811, 811, -4*w^2 + 3*w + 17],
[811, 811, w^3 - 2*w^2 - 8*w + 2],
[823, 823, -4*w^3 + 4*w^2 + 17*w - 4],
[857, 857, 3*w^2 - w - 16],
[881, 881, -2*w^3 + 9*w + 1],
[883, 883, w^3 - w^2 - 4*w - 5],
[887, 887, 3*w^2 + w - 10],
[911, 911, w^3 - 3*w - 7],
[919, 919, -w^3 + w^2 + 3*w - 7],
[919, 919, 3*w^3 - 2*w^2 - 12*w + 2],
[929, 929, 3*w^3 - w^2 - 16*w - 2],
[929, 929, w^3 + w^2 - 6*w - 2],
[941, 941, -3*w^3 + 5*w^2 + 14*w - 14],
[941, 941, -w^3 + 2*w^2 + 6*w - 8],
[947, 947, 3*w^3 - 3*w^2 - 13*w - 5],
[947, 947, 3*w^3 - 4*w^2 - 12*w + 2],
[953, 953, -3*w^3 + 5*w^2 + 10*w - 10],
[961, 31, -w^3 + 4*w^2 + 3*w - 13],
[961, 31, -4*w^3 + 3*w^2 + 17*w - 4],
[967, 967, -w^3 - 4*w^2 + 9*w + 19],
[967, 967, w^3 + 5*w^2 - 9*w - 26],
[971, 971, -3*w^3 + 6*w^2 + 13*w - 7],
[971, 971, w^3 + 2*w^2 - 7*w - 7],
[977, 977, 2*w^3 - 3*w^2 - 4*w - 2],
[983, 983, 3*w^3 - 5*w^2 - 7*w + 7],
[991, 991, 2*w^2 - w - 13],
[991, 991, w^3 - 8*w - 4]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 28*x^4 + 213*x^2 - 236;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [2/5*e^4 - 29/5*e^2 + 32/5, e, -1, -1/5*e^5 + 17/5*e^3 - 46/5*e, -1/5*e^4 + 12/5*e^2 + 14/5, 1/5*e^4 - 17/5*e^2 + 41/5, -1/5*e^5 + 12/5*e^3 + 19/5*e, 1/5*e^5 - 12/5*e^3 - 19/5*e, 6/5*e^4 - 92/5*e^2 + 166/5, -1/5*e^5 + 17/5*e^3 - 41/5*e, -1/5*e^5 + 17/5*e^3 - 41/5*e, -2/5*e^5 + 29/5*e^3 - 32/5*e, -2/5*e^5 + 29/5*e^3 - 32/5*e, e, e, 6/5*e^4 - 92/5*e^2 + 126/5, 2/5*e^5 - 34/5*e^3 + 92/5*e, 7/5*e^4 - 104/5*e^2 + 152/5, 6/5*e^4 - 87/5*e^2 + 76/5, -4/5*e^4 + 63/5*e^2 - 94/5, 2/5*e^4 - 34/5*e^2 + 22/5, 1/5*e^5 - 22/5*e^3 + 91/5*e, -e^3 + 15*e, -3/5*e^5 + 46/5*e^3 - 83/5*e, -e^3 + 14*e, 2*e^2 - 12, 8/5*e^4 - 126/5*e^2 + 228/5, 4/5*e^5 - 63/5*e^3 + 129/5*e, 2/5*e^5 - 29/5*e^3 + 17/5*e, 2/5*e^4 - 29/5*e^2 + 52/5, -2/5*e^4 + 34/5*e^2 - 2/5, -e^4 + 16*e^2 - 42, 1/5*e^5 - 22/5*e^3 + 96/5*e, e^3 - 15*e, -1/5*e^5 + 22/5*e^3 - 91/5*e, -2/5*e^4 + 24/5*e^2 - 42/5, 0, 1/5*e^5 - 7/5*e^3 - 104/5*e, 2*e^4 - 30*e^2 + 42, -17/5*e^4 + 244/5*e^2 - 222/5, 1/5*e^5 - 12/5*e^3 - 39/5*e, -8/5*e^4 + 116/5*e^2 - 168/5, -2/5*e^4 + 24/5*e^2 - 12/5, -1/5*e^4 + 12/5*e^2 - 16/5, 12/5*e^4 - 184/5*e^2 + 272/5, -4/5*e^4 + 48/5*e^2 + 6/5, 2/5*e^4 - 34/5*e^2 + 82/5, 6/5*e^4 - 92/5*e^2 + 146/5, -2/5*e^5 + 29/5*e^3 - 37/5*e, 4*e, -4/5*e^5 + 68/5*e^3 - 204/5*e, 2/5*e^5 - 29/5*e^3 + 52/5*e, -2*e^3 + 28*e, 14/5*e^4 - 218/5*e^2 + 344/5, -1/5*e^5 + 7/5*e^3 + 79/5*e, 2/5*e^5 - 24/5*e^3 - 53/5*e, 8/5*e^4 - 106/5*e^2 - 12/5, 1/5*e^5 - 12/5*e^3 + 1/5*e, -6/5*e^4 + 97/5*e^2 - 136/5, 2/5*e^4 - 24/5*e^2 + 52/5, 2/5*e^5 - 34/5*e^3 + 92/5*e, 12/5*e^4 - 164/5*e^2 + 122/5, 12/5*e^4 - 194/5*e^2 + 402/5, 8/5*e^4 - 116/5*e^2 + 178/5, 12/5*e^4 - 174/5*e^2 + 192/5, 4/5*e^5 - 68/5*e^3 + 194/5*e, -1/5*e^5 + 17/5*e^3 - 86/5*e, 16/5*e^4 - 227/5*e^2 + 146/5, -12/5*e^4 + 184/5*e^2 - 202/5, -e^5 + 17*e^3 - 49*e, -3/5*e^5 + 41/5*e^3 - 28/5*e, -4*e^4 + 62*e^2 - 104, -8/5*e^4 + 96/5*e^2 + 32/5, 8/5*e^4 - 106/5*e^2 + 8/5, 24/5*e^4 - 358/5*e^2 + 494/5, -26/5*e^4 + 392/5*e^2 - 586/5, 7/5*e^4 - 124/5*e^2 + 252/5, -1/5*e^4 + 12/5*e^2 - 116/5, 1/5*e^5 - 22/5*e^3 + 121/5*e, -4/5*e^4 + 38/5*e^2 + 126/5, -1/5*e^5 + 12/5*e^3 + 19/5*e, -1/5*e^5 + 2/5*e^3 + 139/5*e, -2/5*e^5 + 24/5*e^3 + 23/5*e, -2/5*e^5 + 29/5*e^3 + 3/5*e, -8/5*e^4 + 131/5*e^2 - 278/5, -2/5*e^4 + 54/5*e^2 - 242/5, 8/5*e^4 - 126/5*e^2 + 298/5, 6/5*e^4 - 82/5*e^2 - 64/5, 10*e, e^5 - 16*e^3 + 39*e, -4/5*e^5 + 48/5*e^3 + 71/5*e, e^3 - 23*e, 4/5*e^5 - 63/5*e^3 + 149/5*e, -23/5*e^4 + 336/5*e^2 - 448/5, -3/5*e^4 + 36/5*e^2 + 32/5, -3/5*e^5 + 51/5*e^3 - 138/5*e, 2/5*e^5 - 24/5*e^3 - 58/5*e, -8*e, 3/5*e^5 - 46/5*e^3 + 108/5*e, 4/5*e^4 - 68/5*e^2 + 264/5, 2/5*e^5 - 24/5*e^3 - 28/5*e, -e^5 + 16*e^3 - 39*e, -6/5*e^5 + 92/5*e^3 - 141/5*e, 12/5*e^4 - 184/5*e^2 + 332/5, 4/5*e^4 - 58/5*e^2 + 84/5, -4*e^4 + 60*e^2 - 82, -3/5*e^5 + 51/5*e^3 - 138/5*e, 3/5*e^5 - 31/5*e^3 - 122/5*e, 3/5*e^5 - 56/5*e^3 + 223/5*e, 1/5*e^5 - 32/5*e^3 + 231/5*e, 16/5*e^4 - 242/5*e^2 + 276/5, 4/5*e^4 - 38/5*e^2 - 166/5, 3*e^3 - 30*e, 6/5*e^5 - 72/5*e^3 - 94/5*e, -2/5*e^5 + 39/5*e^3 - 172/5*e, 1/5*e^4 - 32/5*e^2 + 176/5, 30, -12/5*e^4 + 179/5*e^2 - 182/5, -2/5*e^4 + 49/5*e^2 - 112/5, -2/5*e^4 + 49/5*e^2 - 212/5, -1/5*e^5 + 22/5*e^3 - 136/5*e, 4/5*e^5 - 68/5*e^3 + 209/5*e, -3/5*e^5 + 46/5*e^3 - 63/5*e, -3*e^3 + 34*e, -6*e, 1/5*e^5 - 22/5*e^3 + 126/5*e, 3/5*e^5 - 51/5*e^3 + 98/5*e, -28/5*e^4 + 396/5*e^2 - 348/5, 16/5*e^4 - 222/5*e^2 + 166/5, -22/5*e^4 + 324/5*e^2 - 402/5, 6/5*e^4 - 77/5*e^2 - 44/5, 2*e^4 - 27*e^2 + 12, -34/5*e^4 + 498/5*e^2 - 554/5, 8/5*e^4 - 121/5*e^2 + 218/5, 2/5*e^4 - 44/5*e^2 + 222/5, 7/5*e^5 - 114/5*e^3 + 247/5*e, -e^5 + 16*e^3 - 27*e, -9/5*e^4 + 108/5*e^2 + 36/5, -3/5*e^5 + 31/5*e^3 + 77/5*e, -e^5 + 14*e^3 - 7*e, 1/5*e^5 - 12/5*e^3 - 9/5*e, -9/5*e^4 + 128/5*e^2 - 4/5, -4/5*e^4 + 48/5*e^2 + 156/5, 6/5*e^4 - 87/5*e^2 - 4/5, 1/5*e^5 - 2/5*e^3 - 89/5*e, 7/5*e^5 - 84/5*e^3 - 133/5*e, -22/5*e^4 + 329/5*e^2 - 552/5, 2/5*e^5 - 39/5*e^3 + 132/5*e, 3/5*e^5 - 46/5*e^3 + 108/5*e, e^4 - 12*e^2 - 40, 2*e^4 - 25*e^2 + 4, -3/5*e^5 + 46/5*e^3 - 133/5*e, e^5 - 14*e^3 + 15*e, 13/5*e^5 - 211/5*e^3 + 493/5*e, -1/5*e^5 + 32/5*e^3 - 251/5*e, -9/5*e^5 + 153/5*e^3 - 434/5*e, 4/5*e^5 - 78/5*e^3 + 334/5*e, 2*e^5 - 32*e^3 + 71*e, 2*e^4 - 30*e^2 + 18, 4*e^4 - 54*e^2 + 18, -4*e^4 + 60*e^2 - 88, 14/5*e^4 - 188/5*e^2 + 104/5, -8/5*e^5 + 131/5*e^3 - 318/5*e, -1/5*e^5 + 2/5*e^3 + 174/5*e, 3*e^3 - 41*e, 6/5*e^5 - 87/5*e^3 + 116/5*e, 18/5*e^4 - 296/5*e^2 + 608/5, 11/5*e^4 - 172/5*e^2 + 336/5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, w^3 - w^2 - 4*w + 1}>] := 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;