/*
  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![9, -1, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[2, 2, -w^2 + w + 4],
[3, 3, -w^2 + w + 3],
[5, 5, w^2 - 4],
[8, 2, w^3 - w^2 - 4*w + 5],
[17, 17, -w^2 + w + 5],
[27, 3, -w^3 + 4*w - 2],
[29, 29, -w^2 + w + 1],
[29, 29, -w^3 + 4*w + 2],
[37, 37, -2*w^3 + 3*w^2 + 9*w - 13],
[41, 41, w^3 - w^2 - 5*w + 2],
[43, 43, -w^3 + w^2 + 3*w - 4],
[61, 61, w^2 - 2*w - 2],
[61, 61, 2*w^2 - w - 8],
[67, 67, -4*w^3 + 5*w^2 + 18*w - 22],
[73, 73, -w^2 - 2*w + 2],
[79, 79, w^2 - 2*w - 4],
[89, 89, w^2 + w - 5],
[89, 89, 2*w^3 - 2*w^2 - 9*w + 8],
[89, 89, w^3 - 5*w - 5],
[89, 89, -w^3 + 2*w^2 + 4*w - 4],
[97, 97, w^3 + w^2 - 5*w - 4],
[101, 101, -2*w^3 + 2*w^2 + 8*w - 11],
[103, 103, w^3 - 5*w + 1],
[109, 109, 2*w - 1],
[109, 109, -w^3 + 2*w^2 + 3*w - 7],
[113, 113, 2*w^2 - 7],
[125, 5, -w^3 + 3*w^2 + 3*w - 8],
[127, 127, 2*w^3 - 4*w^2 - 11*w + 16],
[131, 131, 3*w^2 + 2*w - 8],
[131, 131, -2*w^3 + 4*w^2 + 8*w - 13],
[137, 137, w^3 - 3*w^2 - 7*w + 10],
[149, 149, -w^3 + 4*w - 4],
[149, 149, w^2 - 2*w - 8],
[157, 157, -w - 4],
[163, 163, -w^3 + 2*w^2 + 6*w - 10],
[167, 167, -w^3 + 2*w^2 + 5*w - 11],
[167, 167, -3*w^2 - 2*w + 10],
[167, 167, -w^2 - w + 7],
[167, 167, 2*w^2 - 2*w - 5],
[173, 173, -3*w^3 + 2*w^2 + 13*w - 11],
[179, 179, w^3 - 3*w^2 - 5*w + 10],
[191, 191, -2*w^3 - 3*w^2 + 10*w + 14],
[191, 191, 3*w^2 + w - 7],
[193, 193, w^2 - w + 1],
[193, 193, -w^2 - w - 1],
[227, 227, 4*w^3 - 4*w^2 - 17*w + 20],
[227, 227, -3*w^3 + 4*w^2 + 13*w - 17],
[229, 229, -w^3 + 6*w - 2],
[233, 233, -3*w^3 + 3*w^2 + 13*w - 16],
[241, 241, -w^2 + 2*w - 2],
[263, 263, 2*w^3 - 5*w^2 - 12*w + 16],
[263, 263, -3*w^3 + 4*w^2 + 13*w - 19],
[269, 269, w^3 + 2*w^2 - 5*w - 7],
[269, 269, -w^3 - 2*w^2 + w + 5],
[281, 281, 2*w^3 - 2*w^2 - 8*w + 1],
[281, 281, w^2 - 8],
[289, 17, 2*w^3 - 10*w - 7],
[307, 307, w^3 + 2*w^2 - 6*w - 8],
[307, 307, -2*w^3 + 2*w^2 + 8*w - 7],
[313, 313, 5*w^3 - 6*w^2 - 24*w + 28],
[313, 313, 2*w^3 - 2*w^2 - 10*w + 5],
[337, 337, w^3 - 2*w - 2],
[347, 347, -w^3 + 4*w^2 + w - 13],
[349, 349, -2*w^3 + w^2 + 9*w - 7],
[353, 353, 2*w^3 + w^2 - 7*w + 1],
[359, 359, -3*w^2 - 3*w + 5],
[359, 359, 3*w^3 - 5*w^2 - 13*w + 16],
[367, 367, 3*w^3 - 6*w^2 - 15*w + 23],
[367, 367, -w^3 + 2*w^2 + 6*w - 4],
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],
[367, 367, 2*w^3 - 3*w^2 - 12*w + 16],
[373, 373, -2*w^3 + 4*w^2 + 12*w - 17],
[373, 373, 3*w^2 + 4*w - 8],
[383, 383, -w^3 + 2*w^2 + 3*w - 13],
[383, 383, 2*w^2 - 2*w - 11],
[389, 389, 5*w^3 - 6*w^2 - 24*w + 26],
[389, 389, -w^3 + 3*w - 5],
[401, 401, w^3 + 2*w^2 - 4*w - 10],
[401, 401, 2*w^3 - w^2 - 7*w + 5],
[409, 409, -2*w^2 + 3*w + 10],
[419, 419, w^2 - 3*w - 1],
[419, 419, -w^3 + 5*w^2 + 9*w - 14],
[421, 421, 2*w^3 - 8*w - 7],
[431, 431, -3*w^2 + 16],
[431, 431, -3*w^3 + 11*w - 7],
[433, 433, 4*w^3 - 7*w^2 - 18*w + 26],
[439, 439, 3*w - 2],
[439, 439, -w^3 + w^2 + 5*w - 8],
[443, 443, 2*w^3 - w^2 - 6*w + 4],
[443, 443, w^2 + 3*w - 1],
[449, 449, -2*w^3 + 5*w^2 + 7*w - 17],
[457, 457, 4*w^3 - 6*w^2 - 19*w + 22],
[479, 479, 3*w^2 - 3*w - 13],
[479, 479, -4*w^3 + 8*w^2 + 22*w - 31],
[479, 479, 3*w^3 - 7*w^2 - 17*w + 26],
[479, 479, -6*w^3 + 7*w^2 + 28*w - 32],
[487, 487, -2*w^3 + 4*w^2 + 7*w - 8],
[491, 491, 3*w^2 - w - 11],
[491, 491, 4*w^3 - 5*w^2 - 18*w + 20],
[521, 521, -3*w^3 + 11*w - 5],
[521, 521, 3*w^3 - 3*w^2 - 15*w + 14],
[523, 523, -2*w^3 + 5*w^2 + 8*w - 14],
[529, 23, -2*w^3 + 3*w^2 + 8*w - 14],
[529, 23, 3*w^3 - 2*w^2 - 14*w + 4],
[541, 541, -w^3 + 2*w^2 + 7*w - 11],
[541, 541, -w^3 - w^2 + 5*w - 2],
[547, 547, -3*w - 2],
[547, 547, w^3 - 2*w^2 - 7*w + 7],
[547, 547, w^3 - 4*w^2 - 8*w + 10],
[547, 547, 2*w^3 - 3*w^2 - 7*w + 11],
[563, 563, 2*w^3 - 4*w^2 - 12*w + 13],
[563, 563, -w^3 + 4*w^2 + w - 7],
[571, 571, -w^3 + 2*w^2 + 2*w - 8],
[571, 571, 3*w^2 - 10],
[577, 577, 2*w^3 + w^2 - 7*w - 5],
[599, 599, 2*w^3 - 7*w + 2],
[599, 599, -4*w^3 + 7*w^2 + 21*w - 29],
[617, 617, -2*w^3 + w^2 + 11*w + 1],
[631, 631, 3*w^2 + w - 13],
[631, 631, 3*w^2 - 4*w - 10],
[647, 647, 3*w^3 - 4*w^2 - 13*w + 11],
[677, 677, -2*w^3 + 4*w^2 + 8*w - 17],
[691, 691, -w^3 + 3*w^2 + 3*w - 14],
[733, 733, -w^3 + 4*w^2 + 3*w - 11],
[733, 733, 3*w^3 - w^2 - 15*w - 2],
[733, 733, -w^3 + w^2 + 7*w - 2],
[733, 733, 3*w^3 - 4*w^2 - 16*w + 16],
[739, 739, -3*w^3 + w^2 + 11*w - 8],
[743, 743, -3*w^3 + 2*w^2 + 12*w - 14],
[751, 751, -2*w^3 + w^2 + 8*w - 2],
[761, 761, -2*w^3 + 2*w^2 + 7*w - 10],
[761, 761, 2*w^3 - 4*w^2 - 3*w + 8],
[769, 769, -w^3 + w^2 + 7*w - 4],
[769, 769, 2*w^3 + 2*w^2 - 8*w - 11],
[787, 787, -2*w^3 + 3*w^2 + 6*w - 10],
[787, 787, -2*w^2 - 3*w + 8],
[797, 797, -2*w^3 + w^2 + 7*w - 1],
[797, 797, -2*w^3 - 2*w^2 + 11*w + 10],
[809, 809, -2*w^3 + 5*w^2 + 7*w - 19],
[811, 811, 2*w^3 - 2*w^2 - 7*w - 2],
[811, 811, 3*w^3 - 2*w^2 - 11*w - 1],
[827, 827, 3*w^3 - 2*w^2 - 14*w + 10],
[829, 829, 2*w^3 - 2*w^2 - 9*w + 2],
[829, 829, 4*w^2 + 2*w - 11],
[841, 29, 2*w^3 + w^2 - 6*w - 2],
[863, 863, 2*w^3 - 8*w - 1],
[883, 883, 3*w^3 - 4*w^2 - 15*w + 13],
[887, 887, 3*w^3 - 2*w^2 - 12*w + 10],
[907, 907, 2*w^3 - 3*w^2 - 9*w + 7],
[911, 911, 2*w^3 + w^2 - 11*w - 5],
[919, 919, -2*w^3 + 5*w - 8],
[929, 929, -w^3 + 4*w^2 + 3*w - 13],
[937, 937, w^3 - 5*w - 7],
[937, 937, -2*w^3 + 3*w^2 + 10*w - 8],
[971, 971, -2*w^3 + 4*w^2 + 11*w - 20],
[971, 971, -4*w^3 + 4*w^2 + 19*w - 20],
[977, 977, -4*w^3 + 6*w^2 + 19*w - 28],
[983, 983, -4*w^2 - 4*w + 7],
[991, 991, -5*w^3 + 8*w^2 + 26*w - 32],
[991, 991, -6*w^3 + 6*w^2 + 26*w - 31],
[997, 997, -3*w^3 + 8*w^2 + 18*w - 26]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - x^7 - 16*x^6 + 17*x^5 + 70*x^4 - 76*x^3 - 62*x^2 + 58*x + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -1/14*e^7 + 1/14*e^6 + 9/7*e^5 - 17/14*e^4 - 46/7*e^3 + 37/7*e^2 + 52/7*e - 19/7, 1, 5/14*e^7 + 1/14*e^6 - 38/7*e^5 - 9/14*e^4 + 157/7*e^3 + 11/7*e^2 - 132/7*e - 19/7, -2/7*e^6 + 29/7*e^4 - 5/7*e^3 - 16*e^2 + 36/7*e + 76/7, -1/14*e^7 - 3/14*e^6 + 9/7*e^5 + 41/14*e^4 - 51/7*e^3 - 75/7*e^2 + 81/7*e + 57/7, -1/2*e^7 + 1/14*e^6 + 8*e^5 - 25/14*e^4 - 242/7*e^3 + 9*e^2 + 187/7*e - 19/7, -1/2*e^7 + 1/14*e^6 + 8*e^5 - 25/14*e^4 - 242/7*e^3 + 9*e^2 + 201/7*e - 19/7, -1/14*e^7 + 3/14*e^6 + 9/7*e^5 - 39/14*e^4 - 47/7*e^3 + 65/7*e^2 + 62/7*e - 43/7, 4/7*e^7 + 1/7*e^6 - 65/7*e^5 - 15/7*e^4 + 293/7*e^3 + 68/7*e^2 - 268/7*e - 80/7, 1/14*e^7 - 1/14*e^6 - 9/7*e^5 + 3/14*e^4 + 39/7*e^3 + 19/7*e^2 - 3/7*e - 23/7, -1/2*e^7 + 3/14*e^6 + 8*e^5 - 47/14*e^4 - 243/7*e^3 + 12*e^2 + 211/7*e - 15/7, -2/7*e^6 + 29/7*e^4 - 5/7*e^3 - 13*e^2 + 43/7*e - 8/7, -5/14*e^7 + 3/14*e^6 + 38/7*e^5 - 9/2*e^4 - 152/7*e^3 + 164/7*e^2 + 89/7*e - 71/7, 3/7*e^7 + 1/7*e^6 - 47/7*e^5 - e^4 + 209/7*e^3 - 5/7*e^2 - 223/7*e + 60/7, -1/2*e^7 - 3/14*e^6 + 8*e^5 + 33/14*e^4 - 247/7*e^3 - 4*e^2 + 216/7*e - 27/7, -1/14*e^7 + 5/14*e^6 + 9/7*e^5 - 61/14*e^4 - 41/7*e^3 + 79/7*e^2 + 23/7*e - 11/7, 11/14*e^7 - 1/14*e^6 - 85/7*e^5 + 5/2*e^4 + 354/7*e^3 - 99/7*e^2 - 277/7*e + 33/7, 5/14*e^7 - 1/14*e^6 - 38/7*e^5 + 27/14*e^4 + 151/7*e^3 - 66/7*e^2 - 93/7*e + 5/7, -1/14*e^7 + 1/14*e^6 + 9/7*e^5 - 3/14*e^4 - 53/7*e^3 - 26/7*e^2 + 115/7*e + 37/7, -1/7*e^7 + 3/7*e^6 + 11/7*e^5 - 53/7*e^4 - 17/7*e^3 + 235/7*e^2 - 51/7*e - 114/7, -2/7*e^7 + 3/7*e^6 + 29/7*e^5 - 45/7*e^4 - 108/7*e^3 + 162/7*e^2 + 57/7*e - 16/7, 23/14*e^7 - 1/14*e^6 - 179/7*e^5 + 51/14*e^4 + 760/7*e^3 - 158/7*e^2 - 631/7*e + 61/7, -2/7*e^6 + 29/7*e^4 + 9/7*e^3 - 16*e^2 - 55/7*e + 118/7, 1/14*e^7 - 3/14*e^6 - 9/7*e^5 + 39/14*e^4 + 33/7*e^3 - 72/7*e^2 + 43/7*e + 57/7, 1/14*e^7 - 1/14*e^6 - 9/7*e^5 + 3/14*e^4 + 53/7*e^3 + 19/7*e^2 - 115/7*e - 23/7, -2/7*e^7 + 29/7*e^5 - 5/7*e^4 - 16*e^3 + 43/7*e^2 + 90/7*e - 8, -8/7*e^7 + 2/7*e^6 + 123/7*e^5 - 6*e^4 - 499/7*e^3 + 193/7*e^2 + 338/7*e - 76/7, 3/7*e^7 - 4/7*e^6 - 47/7*e^5 + 62/7*e^4 + 200/7*e^3 - 236/7*e^2 - 22*e + 96/7, 9/14*e^7 + 3/14*e^6 - 74/7*e^5 - 7/2*e^4 + 331/7*e^3 + 122/7*e^2 - 289/7*e - 155/7, 10/7*e^7 - 159/7*e^5 + 11/7*e^4 + 100*e^3 - 75/7*e^2 - 660/7*e + 6, -1/2*e^7 - 3/14*e^6 + 8*e^5 + 47/14*e^4 - 254/7*e^3 - 15*e^2 + 265/7*e + 85/7, 1/2*e^7 + 5/14*e^6 - 8*e^5 - 69/14*e^4 + 253/7*e^3 + 22*e^2 - 269/7*e - 151/7, -6/7*e^6 + 87/7*e^4 - 8/7*e^3 - 46*e^2 + 94/7*e + 172/7, e^7 - 5/7*e^6 - 16*e^5 + 76/7*e^4 + 481/7*e^3 - 38*e^2 - 379/7*e + 36/7, -4/7*e^7 + 1/7*e^6 + 58/7*e^5 - 4*e^4 - 218/7*e^3 + 163/7*e^2 + 113/7*e - 136/7, -1/7*e^7 + 11/7*e^5 - 13/7*e^4 - e^3 + 102/7*e^2 - 130/7*e - 12, -9/7*e^7 + 1/7*e^6 + 141/7*e^5 - 16/7*e^4 - 596/7*e^3 + 29/7*e^2 + 464/7*e + 88/7, -2/7*e^7 - 4/7*e^6 + 29/7*e^5 + 53/7*e^4 - 122/7*e^3 - 181/7*e^2 + 134/7*e + 138/7, e^4 + 2*e^3 - 10*e^2 - 14*e + 12, e^7 + 1/7*e^6 - 16*e^5 + 3/7*e^4 + 482/7*e^3 - 14*e^2 - 368/7*e + 88/7, 2/7*e^7 + 2/7*e^6 - 29/7*e^5 - 31/7*e^4 + 110/7*e^3 + 139/7*e^2 - 6*e - 132/7, 13/14*e^7 - 3/14*e^6 - 103/7*e^5 + 55/14*e^4 + 446/7*e^3 - 110/7*e^2 - 395/7*e + 57/7, -11/7*e^7 - 3/7*e^6 + 170/7*e^5 + 23/7*e^4 - 704/7*e^3 - 19/7*e^2 + 500/7*e + 142/7, e^7 + 1/7*e^6 - 15*e^5 + 3/7*e^4 + 412/7*e^3 - 9*e^2 - 256/7*e - 24/7, 1/7*e^7 - 4/7*e^6 - 18/7*e^5 + 57/7*e^4 + 88/7*e^3 - 214/7*e^2 - 57/7*e + 124/7, -1/2*e^7 - 9/14*e^6 + 8*e^5 + 141/14*e^4 - 265/7*e^3 - 44*e^2 + 347/7*e + 241/7, -2/7*e^7 + 29/7*e^5 + 2/7*e^4 - 17*e^3 - 13/7*e^2 + 146/7*e + 2, 2/7*e^7 + 2/7*e^6 - 36/7*e^5 - 31/7*e^4 + 194/7*e^3 + 118/7*e^2 - 40*e - 34/7, -2/7*e^7 - 6/7*e^6 + 29/7*e^5 + 82/7*e^4 - 113/7*e^3 - 293/7*e^2 + 65/7*e + 172/7, -2/7*e^7 + 29/7*e^5 + 2/7*e^4 - 15*e^3 - 27/7*e^2 + 27/7*e + 4, -1/7*e^7 + 18/7*e^5 - 6/7*e^4 - 11*e^3 + 39/7*e^2 - 18/7*e + 6, -9/14*e^7 + 5/14*e^6 + 74/7*e^5 - 81/14*e^4 - 335/7*e^3 + 144/7*e^2 + 41*e - 11/7, -3/14*e^7 + 1/2*e^6 + 27/7*e^5 - 109/14*e^4 - 17*e^3 + 209/7*e^2 + 15/7*e - 1, e^7 + 3/7*e^6 - 16*e^5 - 26/7*e^4 + 508/7*e^3 + 2*e^2 - 544/7*e + 54/7, -3/14*e^7 - 1/2*e^6 + 27/7*e^5 + 87/14*e^4 - 21*e^3 - 134/7*e^2 + 225/7*e + 11, 4/7*e^7 - 65/7*e^5 + 3/7*e^4 + 46*e^3 - 9/7*e^2 - 467/7*e, -3/7*e^7 + 1/7*e^6 + 47/7*e^5 - 22/7*e^4 - 218/7*e^3 + 124/7*e^2 + 278/7*e - 136/7, -4/7*e^7 + 58/7*e^5 - 10/7*e^4 - 30*e^3 + 65/7*e^2 + 96/7*e + 10, -1/14*e^7 + 1/14*e^6 + 16/7*e^5 - 31/14*e^4 - 130/7*e^3 + 121/7*e^2 + 290/7*e - 159/7, 15/14*e^7 - 1/14*e^6 - 121/7*e^5 + 31/14*e^4 + 536/7*e^3 - 93/7*e^2 - 451/7*e + 89/7, -3/7*e^7 + 4/7*e^6 + 40/7*e^5 - 69/7*e^4 - 116/7*e^3 + 313/7*e^2 - 9*e - 180/7, -12/7*e^7 + 188/7*e^5 - 2/7*e^4 - 112*e^3 - 22/7*e^2 + 540/7*e + 30, 19/14*e^7 + 1/2*e^6 - 150/7*e^5 - 61/14*e^4 + 95*e^3 + 46/7*e^2 - 634/7*e - 13, -2/7*e^7 - 1/7*e^6 + 29/7*e^5 + 20/7*e^4 - 97/7*e^3 - 132/7*e^2 - 32/7*e + 164/7, -3/7*e^7 + 6/7*e^6 + 47/7*e^5 - 12*e^4 - 202/7*e^3 + 264/7*e^2 + 167/7*e - 60/7, -6/7*e^7 + 1/7*e^6 + 94/7*e^5 - 26/7*e^4 - 400/7*e^3 + 129/7*e^2 + 45*e - 52/7, -11/7*e^7 + 1/7*e^6 + 170/7*e^5 - 3*e^4 - 694/7*e^3 + 72/7*e^2 + 442/7*e + 88/7, 6/7*e^7 + 6/7*e^6 - 94/7*e^5 - 79/7*e^4 + 421/7*e^3 + 256/7*e^2 - 69*e - 88/7, 11/7*e^7 + 4/7*e^6 - 177/7*e^5 - 34/7*e^4 + 780/7*e^3 + 40/7*e^2 - 95*e - 96/7, -3/14*e^7 - 9/14*e^6 + 20/7*e^5 + 109/14*e^4 - 55/7*e^3 - 162/7*e^2 - 79/7*e + 129/7, 5/14*e^7 + 15/14*e^6 - 38/7*e^5 - 205/14*e^4 + 164/7*e^3 + 354/7*e^2 - 188/7*e - 159/7, -8/7*e^7 + 130/7*e^5 - 20/7*e^4 - 86*e^3 + 172/7*e^2 + 668/7*e - 24, -4/7*e^7 + 4/7*e^6 + 65/7*e^5 - 61/7*e^4 - 270/7*e^3 + 261/7*e^2 + 108/7*e - 236/7, 3/14*e^7 - 3/14*e^6 - 27/7*e^5 + 51/14*e^4 + 138/7*e^3 - 139/7*e^2 - 184/7*e + 141/7, -3/2*e^7 - 1/2*e^6 + 24*e^5 + 9/2*e^4 - 105*e^3 - 6*e^2 + 91*e + 3, -3/2*e^7 - 9/14*e^6 + 24*e^5 + 99/14*e^4 - 741/7*e^3 - 21*e^2 + 655/7*e + 157/7, 5/14*e^7 - 3/14*e^6 - 45/7*e^5 + 5/2*e^4 + 215/7*e^3 - 38/7*e^2 - 131/7*e - 55/7, 3/14*e^7 - 3/14*e^6 - 20/7*e^5 + 51/14*e^4 + 68/7*e^3 - 118/7*e^2 - 58/7*e + 85/7, -3/7*e^7 + 3/7*e^6 + 47/7*e^5 - 44/7*e^4 - 206/7*e^3 + 159/7*e^2 + 207/7*e - 100/7, -e^7 - 3/7*e^6 + 16*e^5 + 40/7*e^4 - 501/7*e^3 - 23*e^2 + 467/7*e + 156/7, 1/14*e^7 + 11/14*e^6 - 9/7*e^5 - 171/14*e^4 + 40/7*e^3 + 355/7*e^2 + 64/7*e - 279/7, 6/7*e^7 + 6/7*e^6 - 94/7*e^5 - 72/7*e^4 + 400/7*e^3 + 228/7*e^2 - 48*e - 228/7, -2/7*e^7 - 4/7*e^6 + 36/7*e^5 + 74/7*e^4 - 206/7*e^3 - 370/7*e^2 + 344/7*e + 292/7, -e^7 - 2/7*e^6 + 16*e^5 + 22/7*e^4 - 495/7*e^3 - 6*e^2 + 442/7*e + 6/7, -9/7*e^7 - 5/7*e^6 + 134/7*e^5 + 50/7*e^4 - 527/7*e^3 - 118/7*e^2 + 313/7*e + 148/7, -12/7*e^7 + 3/7*e^6 + 181/7*e^5 - 9*e^4 - 710/7*e^3 + 286/7*e^2 + 395/7*e - 128/7, e^7 - 16*e^5 + 3*e^4 + 72*e^3 - 23*e^2 - 72*e + 8, 12/7*e^7 + 3/7*e^6 - 188/7*e^5 - 24/7*e^4 + 802/7*e^3 + 15/7*e^2 - 622/7*e - 44/7, 3/7*e^7 - 47/7*e^5 - 3/7*e^4 + 29*e^3 + 72/7*e^2 - 149/7*e - 26, 6/7*e^7 + 5/7*e^6 - 94/7*e^5 - 61/7*e^4 + 415/7*e^3 + 179/7*e^2 - 423/7*e - 22/7, -8/7*e^6 - e^5 + 109/7*e^4 + 50/7*e^3 - 55*e^2 + 18/7*e + 192/7, 9/7*e^7 - 3/7*e^6 - 141/7*e^5 + 52/7*e^4 + 598/7*e^3 - 197/7*e^2 - 512/7*e - 12/7, -e^7 - 3/7*e^6 + 15*e^5 + 40/7*e^4 - 438/7*e^3 - 19*e^2 + 446/7*e + 16/7, 12/7*e^7 - 188/7*e^5 + 16/7*e^4 + 116*e^3 - 118/7*e^2 - 764/7*e + 4, 3/7*e^7 + 2/7*e^6 - 54/7*e^5 - 25/7*e^4 + 278/7*e^3 + 79/7*e^2 - 339/7*e + 64/7, -1/7*e^6 + 18/7*e^4 - 20/7*e^3 - 13*e^2 + 214/7*e + 52/7, 8/7*e^7 - e^6 - 116/7*e^5 + 118/7*e^4 + 60*e^3 - 487/7*e^2 - 164/7*e + 32, -11/14*e^7 - 1/14*e^6 + 78/7*e^5 - 13/14*e^4 - 290/7*e^3 + 36/7*e^2 + 183/7*e + 229/7, 5/14*e^7 + 17/14*e^6 - 31/7*e^5 - 213/14*e^4 + 93/7*e^3 + 312/7*e^2 - 3/7*e - 127/7, 5/7*e^7 - 2/7*e^6 - 83/7*e^5 + 38/7*e^4 + 401/7*e^3 - 188/7*e^2 - 66*e + 104/7, 25/14*e^7 + 1/14*e^6 - 204/7*e^5 - 1/14*e^4 + 920/7*e^3 + 13/7*e^2 - 862/7*e - 131/7, -11/14*e^7 - 5/14*e^6 + 92/7*e^5 + 87/14*e^4 - 435/7*e^3 - 188/7*e^2 + 499/7*e + 81/7, 13/14*e^7 + 3/14*e^6 - 96/7*e^5 - 25/14*e^4 + 366/7*e^3 + 2/7*e^2 - 190/7*e + 69/7, -3/7*e^7 + 47/7*e^5 - 11/7*e^4 - 25*e^3 + 124/7*e^2 - 26/7*e - 26, -e^7 + 15*e^5 - e^4 - 57*e^3 + 10*e^2 + 22*e - 8, -8/7*e^7 - 4/7*e^6 + 123/7*e^5 + 59/7*e^4 - 500/7*e^3 - 262/7*e^2 + 362/7*e + 236/7, 10/7*e^7 - 5/7*e^6 - 159/7*e^5 + 101/7*e^4 + 677/7*e^3 - 474/7*e^2 - 472/7*e + 232/7, 6/7*e^7 + 1/7*e^6 - 87/7*e^5 - 3/7*e^4 + 342/7*e^3 - 38/7*e^2 - 260/7*e + 88/7, -3/7*e^7 - 3/7*e^6 + 54/7*e^5 + 50/7*e^4 - 298/7*e^3 - 226/7*e^2 + 62*e + 156/7, -1/7*e^7 + e^6 + 25/7*e^5 - 104/7*e^4 - 22*e^3 + 410/7*e^2 + 136/7*e - 40, -12/7*e^7 - 3/7*e^6 + 188/7*e^5 + 3/7*e^4 - 809/7*e^3 + 160/7*e^2 + 692/7*e - 96/7, 9/7*e^7 - 5/7*e^6 - 141/7*e^5 + 88/7*e^4 + 572/7*e^3 - 358/7*e^2 - 42*e + 120/7, 4/7*e^7 - 72/7*e^5 + 3/7*e^4 + 52*e^3 - 44/7*e^2 - 418/7*e + 10, -e^7 - 2/7*e^6 + 16*e^5 + 8/7*e^4 - 495/7*e^3 + 9*e^2 + 400/7*e - 92/7, -4/7*e^7 + 65/7*e^5 + 11/7*e^4 - 42*e^3 - 89/7*e^2 + 264/7*e + 4, 10/7*e^7 - 4/7*e^6 - 152/7*e^5 + 69/7*e^4 + 606/7*e^3 - 278/7*e^2 - 54*e + 110/7, -4/7*e^6 - e^5 + 58/7*e^4 + 81/7*e^3 - 33*e^2 - 215/7*e + 124/7, -1/7*e^7 - 3/7*e^6 + 18/7*e^5 + 55/7*e^4 - 88/7*e^3 - 276/7*e^2 + 92/7*e + 268/7, 10/7*e^7 + 2/7*e^6 - 152/7*e^5 - 32/7*e^4 + 614/7*e^3 + 156/7*e^2 - 486/7*e - 188/7, -17/14*e^7 + 1/14*e^6 + 132/7*e^5 - 29/14*e^4 - 557/7*e^3 + 48/7*e^2 + 419/7*e + 37/7, -1/7*e^7 + 10/7*e^6 + 18/7*e^5 - 158/7*e^4 - 87/7*e^3 + 620/7*e^2 + 26/7*e - 268/7, -3/14*e^7 - 11/14*e^6 + 34/7*e^5 + 159/14*e^4 - 236/7*e^3 - 337/7*e^2 + 415/7*e + 321/7, 9/14*e^7 + 15/14*e^6 - 74/7*e^5 - 223/14*e^4 + 381/7*e^3 + 458/7*e^2 - 607/7*e - 355/7, 5/14*e^7 + 3/14*e^6 - 38/7*e^5 - 3/14*e^4 + 135/7*e^3 - 66/7*e^2 + 81/7*e + 13/7, -1/14*e^7 + 17/14*e^6 + 2/7*e^5 - 249/14*e^4 + 23/7*e^3 + 478/7*e^2 - 71/7*e - 379/7, -e^7 + 1/7*e^6 + 15*e^5 - 32/7*e^4 - 414/7*e^3 + 31*e^2 + 262/7*e - 136/7, -1/7*e^7 - 4/7*e^6 + 18/7*e^5 + 80/7*e^4 - 101/7*e^3 - 437/7*e^2 + 180/7*e + 376/7, -20/7*e^7 + 6/7*e^6 + 311/7*e^5 - 109/7*e^4 - 1280/7*e^3 + 465/7*e^2 + 862/7*e - 172/7, -10/7*e^7 - 1/7*e^6 + 152/7*e^5 - 636/7*e^3 + 33/7*e^2 + 594/7*e + 206/7, -13/14*e^7 - 1/14*e^6 + 96/7*e^5 - 39/14*e^4 - 360/7*e^3 + 173/7*e^2 + 158/7*e - 135/7, -8/7*e^7 - 4/7*e^6 + 123/7*e^5 + 73/7*e^4 - 500/7*e^3 - 395/7*e^2 + 341/7*e + 418/7, -1/2*e^7 + 1/2*e^6 + 8*e^5 - 19/2*e^4 - 34*e^3 + 46*e^2 + 31*e - 29, 19/7*e^7 + 4/7*e^6 - 307/7*e^5 - 8*e^4 + 1396/7*e^3 + 246/7*e^2 - 1410/7*e - 180/7, 1/7*e^7 - 15/7*e^6 - 18/7*e^5 + 206/7*e^4 + 78/7*e^3 - 704/7*e^2 - 48/7*e + 360/7, -23/14*e^7 - 17/14*e^6 + 179/7*e^5 + 31/2*e^4 - 786/7*e^3 - 339/7*e^2 + 758/7*e + 113/7, -5/7*e^7 + 5/7*e^6 + 76/7*e^5 - 85/7*e^4 - 320/7*e^3 + 356/7*e^2 + 254/7*e - 22/7, -11/14*e^7 - 5/14*e^6 + 92/7*e^5 + 73/14*e^4 - 428/7*e^3 - 174/7*e^2 + 408/7*e + 193/7, -4/7*e^7 + 3/7*e^6 + 65/7*e^5 - 29/7*e^4 - 248/7*e^3 + 30/7*e^2 - 5*e + 96/7, 12/7*e^7 - 3/7*e^6 - 188/7*e^5 + 8*e^4 + 787/7*e^3 - 265/7*e^2 - 626/7*e + 184/7, 4/7*e^7 + 4/7*e^6 - 58/7*e^5 - 55/7*e^4 + 213/7*e^3 + 166/7*e^2 - 18*e + 128/7, -15/14*e^7 + 11/14*e^6 + 121/7*e^5 - 155/14*e^4 - 534/7*e^3 + 219/7*e^2 + 438/7*e + 197/7, 10/7*e^7 - 152/7*e^5 + 18/7*e^4 + 92*e^3 - 124/7*e^2 - 618/7*e + 6, 17/7*e^7 + 8/7*e^6 - 271/7*e^5 - 12*e^4 + 1217/7*e^3 + 170/7*e^2 - 1266/7*e - 66/7, 9/7*e^7 - 12/7*e^6 - 141/7*e^5 + 193/7*e^4 + 579/7*e^3 - 750/7*e^2 - 58*e + 232/7, 6/7*e^7 - 5/7*e^6 - 87/7*e^5 + 11*e^4 + 306/7*e^3 - 276/7*e^2 - 96/7*e + 92/7, 4/7*e^7 + 4/7*e^6 - 58/7*e^5 - 62/7*e^4 + 234/7*e^3 + 278/7*e^2 - 36*e - 292/7, -22/7*e^7 + 10/7*e^6 + 354/7*e^5 - 179/7*e^4 - 1550/7*e^3 + 760/7*e^2 + 1272/7*e - 268/7, -6/7*e^7 - 6/7*e^6 + 94/7*e^5 + 72/7*e^4 - 428/7*e^3 - 200/7*e^2 + 76*e + 200/7, 2*e^7 - 2/7*e^6 - 32*e^5 + 43/7*e^4 + 1003/7*e^3 - 26*e^2 - 1007/7*e + 160/7, -15/14*e^7 - 15/14*e^6 + 114/7*e^5 + 187/14*e^4 - 472/7*e^3 - 313/7*e^2 + 56*e + 369/7, -3/2*e^7 + 1/14*e^6 + 23*e^5 - 95/14*e^4 - 641/7*e^3 + 48*e^2 + 313/7*e - 187/7, -4/7*e^7 + e^6 + 65/7*e^5 - 108/7*e^4 - 42*e^3 + 366/7*e^2 + 327/7*e - 2, -3/7*e^6 - e^5 + 33/7*e^4 + 80/7*e^3 - 10*e^2 - 100/7*e - 68/7, -25/7*e^7 + e^6 + 394/7*e^5 - 122/7*e^4 - 238*e^3 + 492/7*e^2 + 1202/7*e - 4, 1/14*e^7 + 19/14*e^6 - 9/7*e^5 - 39/2*e^4 + 85/7*e^3 + 460/7*e^2 - 365/7*e - 179/7, -4/7*e^6 + 44/7*e^4 + 4/7*e^3 - 4*e^2 + 30/7*e - 296/7, 2/7*e^6 + e^5 - 15/7*e^4 - 100/7*e^3 - e^2 + 356/7*e + 36/7, 32/7*e^7 - 2/7*e^6 - 513/7*e^5 + 81/7*e^4 + 2242/7*e^3 - 471/7*e^2 - 1838/7*e + 104/7, 45/14*e^7 + 5/14*e^6 - 370/7*e^5 - 51/14*e^4 + 1695/7*e^3 + 85/7*e^2 - 1656/7*e - 95/7];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, -w^2 + w + 4}>] := -1;
ALEigenvalues[ideal<ZF | {8, 2, w^3 - w^2 - 4*w + 5}>] := -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;