/*
  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![29, 3, -12, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {20, 10, 1/4*w^3 - 1/2*w^2 - 5/2*w + 5/4}>;

primesArray := [
[4, 2, 1/2*w^3 - 2*w^2 - 2*w + 15/2],
[4, 2, -w^2 + w + 8],
[5, 5, -1/4*w^3 + 1/2*w^2 + 1/2*w - 9/4],
[5, 5, 1/2*w^3 - w^2 - 4*w + 9/2],
[19, 19, 1/4*w^3 - 3/2*w^2 - 1/2*w + 21/4],
[19, 19, 1/4*w^3 - 3/2*w^2 - 1/2*w + 41/4],
[29, 29, w],
[29, 29, 1/4*w^3 - 1/2*w^2 - 1/2*w + 1/4],
[29, 29, 1/4*w^3 - 1/2*w^2 - 5/2*w + 9/4],
[29, 29, -1/2*w^3 + w^2 + 4*w - 5/2],
[41, 41, 3/4*w^3 - 5/2*w^2 - 7/2*w + 47/4],
[41, 41, 1/4*w^3 + 1/2*w^2 - 5/2*w - 15/4],
[61, 61, -3/2*w^3 + 3*w^2 + 11*w - 21/2],
[61, 61, w^3 - 2*w^2 - 4*w + 6],
[79, 79, 3/4*w^3 - 5/2*w^2 - 7/2*w + 27/4],
[79, 79, 1/4*w^3 + 1/2*w^2 - 5/2*w - 35/4],
[81, 3, -3],
[89, 89, 7/4*w^3 - 11/2*w^2 - 21/2*w + 119/4],
[89, 89, 1/4*w^3 + 3/2*w^2 - 3/2*w - 23/4],
[109, 109, -1/2*w^3 + 3*w^2 + 2*w - 41/2],
[109, 109, -5/4*w^3 + 7/2*w^2 + 9/2*w - 45/4],
[121, 11, -3/4*w^3 + 3/2*w^2 + 9/2*w - 23/4],
[121, 11, -1/4*w^3 + 1/2*w^2 + 3/2*w - 21/4],
[131, 131, -1/2*w^3 + w^2 + 5*w - 3/2],
[131, 131, 3/4*w^3 - 1/2*w^2 - 7/2*w - 1/4],
[139, 139, -5/4*w^3 + 11/2*w^2 + 9/2*w - 73/4],
[139, 139, -1/4*w^3 + 7/2*w^2 - 3/2*w - 113/4],
[149, 149, -1/4*w^3 + 3/2*w^2 + 1/2*w - 49/4],
[149, 149, 1/4*w^3 - 3/2*w^2 - 1/2*w + 13/4],
[151, 151, 1/2*w^3 - 3*w^2 - w + 23/2],
[151, 151, -1/4*w^3 + 3/2*w^2 + 3/2*w - 45/4],
[151, 151, 3/4*w^3 - 3/2*w^2 - 7/2*w + 11/4],
[151, 151, -2*w^3 + 6*w^2 + 7*w - 20],
[169, 13, -1/4*w^3 + 1/2*w^2 + 1/2*w - 25/4],
[169, 13, 1/2*w^3 - w^2 - 4*w + 17/2],
[179, 179, -1/4*w^3 + 1/2*w^2 + 7/2*w - 21/4],
[179, 179, 1/4*w^3 - 1/2*w^2 - 7/2*w - 3/4],
[181, 181, 1/2*w^3 - 3*w - 7/2],
[181, 181, -5/4*w^3 + 7/2*w^2 + 15/2*w - 57/4],
[191, 191, -3/4*w^3 + 5/2*w^2 + 9/2*w - 39/4],
[191, 191, w^2 - 8],
[199, 199, -3/4*w^3 + 7/2*w^2 + 5/2*w - 43/4],
[199, 199, -1/4*w^3 + 5/2*w^2 - 1/2*w - 81/4],
[199, 199, -3/4*w^3 + 7/2*w^2 + 5/2*w - 75/4],
[199, 199, -1/4*w^3 + 5/2*w^2 - 1/2*w - 49/4],
[229, 229, -1/2*w^3 + 2*w^2 + w - 21/2],
[229, 229, -3/4*w^3 + 5/2*w^2 + 7/2*w - 55/4],
[239, 239, 3/2*w^3 - 4*w^2 - 9*w + 35/2],
[239, 239, 3/2*w^3 - 5*w^2 - 7*w + 45/2],
[241, 241, -5/4*w^3 + 9/2*w^2 + 13/2*w - 93/4],
[241, 241, -1/4*w^3 - 1/2*w^2 - 1/2*w + 3/4],
[251, 251, -w^3 + 3*w^2 + 8*w - 13],
[251, 251, -7/4*w^3 + 13/2*w^2 + 13/2*w - 99/4],
[269, 269, -9/4*w^3 + 13/2*w^2 + 15/2*w - 85/4],
[269, 269, -3/4*w^3 + 5/2*w^2 + 3/2*w - 47/4],
[271, 271, 5/4*w^3 - 5/2*w^2 - 21/2*w + 53/4],
[271, 271, 5/4*w^3 - 1/2*w^2 - 23/2*w - 23/4],
[271, 271, 7/4*w^3 - 11/2*w^2 - 13/2*w + 83/4],
[271, 271, -1/2*w^3 + w^2 - 13/2],
[281, 281, 3/4*w^3 - 1/2*w^2 - 13/2*w - 1/4],
[281, 281, -w^3 + 3*w^2 + 4*w - 13],
[311, 311, -1/4*w^3 + 1/2*w^2 - 1/2*w - 5/4],
[311, 311, 7/4*w^3 - 13/2*w^2 - 19/2*w + 155/4],
[311, 311, -1/4*w^3 + 7/2*w^2 + 1/2*w - 49/4],
[311, 311, -3/4*w^3 + 3/2*w^2 + 13/2*w - 23/4],
[331, 331, -3/4*w^3 + 1/2*w^2 + 15/2*w + 5/4],
[331, 331, -3/4*w^3 + 5/2*w^2 + 3/2*w - 39/4],
[349, 349, -1/4*w^3 + 1/2*w^2 + 7/2*w - 17/4],
[349, 349, 1/4*w^3 - 1/2*w^2 - 7/2*w + 1/4],
[359, 359, -5/4*w^3 - 1/2*w^2 + 25/2*w + 55/4],
[359, 359, w^2 - 2*w - 2],
[359, 359, -9/4*w^3 + 15/2*w^2 + 17/2*w - 113/4],
[359, 359, -1/4*w^3 + 3/2*w^2 - 1/2*w - 45/4],
[361, 19, -w^3 + 2*w^2 + 6*w - 8],
[379, 379, -5/4*w^3 + 5/2*w^2 + 17/2*w - 37/4],
[379, 379, -5/4*w^3 + 7/2*w^2 + 9/2*w - 41/4],
[379, 379, 5/4*w^3 - 3/2*w^2 - 21/2*w - 3/4],
[379, 379, -w^3 + 2*w^2 + 5*w - 7],
[389, 389, 9/4*w^3 - 17/2*w^2 - 15/2*w + 117/4],
[389, 389, -3/4*w^3 - 5/2*w^2 + 21/2*w + 113/4],
[401, 401, -3/4*w^3 - 7/2*w^2 + 23/2*w + 149/4],
[401, 401, 11/4*w^3 - 21/2*w^2 - 19/2*w + 143/4],
[409, 409, -3/4*w^3 + 5/2*w^2 + 9/2*w - 31/4],
[409, 409, w^2 - 10],
[421, 421, 5/4*w^3 - 5/2*w^2 - 19/2*w + 49/4],
[421, 421, -1/4*w^3 + 5/2*w^2 - 1/2*w - 41/4],
[421, 421, 5/4*w^3 - 5/2*w^2 - 7/2*w + 41/4],
[421, 421, 9/4*w^3 - 9/2*w^2 - 35/2*w + 77/4],
[431, 431, 7/4*w^3 - 9/2*w^2 - 11/2*w + 59/4],
[431, 431, 5/4*w^3 - 9/2*w^2 - 13/2*w + 85/4],
[431, 431, 5/4*w^3 - 5/2*w^2 - 11/2*w + 33/4],
[431, 431, -2*w^2 + w + 12],
[439, 439, -1/2*w^3 + w^2 + 5*w - 7/2],
[439, 439, 2*w - 1],
[461, 461, -7/4*w^3 + 13/2*w^2 + 21/2*w - 163/4],
[461, 461, 3/2*w^3 - 5*w^2 - 4*w + 31/2],
[491, 491, 5/4*w^3 - 3/2*w^2 - 17/2*w + 21/4],
[491, 491, -7/4*w^3 + 9/2*w^2 + 19/2*w - 83/4],
[509, 509, -3/2*w^3 + 4*w^2 + 7*w - 27/2],
[509, 509, -3/2*w^3 + 3*w^2 + 10*w - 27/2],
[521, 521, -5/4*w^3 + 3/2*w^2 + 11/2*w - 17/4],
[521, 521, -5/2*w^3 + 6*w^2 + 17*w - 53/2],
[521, 521, -1/2*w^3 + 3*w^2 + w - 31/2],
[541, 541, 9/4*w^3 - 7/2*w^2 - 35/2*w + 37/4],
[541, 541, w^3 - 2*w^2 - 8*w + 12],
[541, 541, -1/4*w^3 - 3/2*w^2 + 7/2*w + 35/4],
[541, 541, -3/4*w^3 + 9/2*w^2 + 5/2*w - 119/4],
[569, 569, -w^3 + 3*w^2 + 6*w - 7],
[569, 569, -7/4*w^3 + 9/2*w^2 + 23/2*w - 75/4],
[599, 599, 1/4*w^3 + 5/2*w^2 - 9/2*w - 95/4],
[599, 599, -7/4*w^3 + 13/2*w^2 + 15/2*w - 91/4],
[619, 619, 9/4*w^3 - 13/2*w^2 - 19/2*w + 97/4],
[619, 619, -7/4*w^3 + 3/2*w^2 + 29/2*w + 9/4],
[619, 619, -1/4*w^3 + 3/2*w^2 - 3/2*w - 45/4],
[619, 619, -1/4*w^3 - 1/2*w^2 + 9/2*w - 1/4],
[631, 631, -9/4*w^3 + 13/2*w^2 + 15/2*w - 93/4],
[631, 631, -13/4*w^3 + 21/2*w^2 + 39/2*w - 233/4],
[631, 631, 7/4*w^3 - 9/2*w^2 - 15/2*w + 67/4],
[631, 631, 1/4*w^3 + 7/2*w^2 - 3/2*w - 51/4],
[641, 641, -1/4*w^3 + 5/2*w^2 + 1/2*w - 57/4],
[641, 641, -w^3 + 4*w^2 + 5*w - 19],
[661, 661, -2*w^3 + 20*w + 19],
[661, 661, 5/4*w^3 - 11/2*w^2 - 13/2*w + 149/4],
[691, 691, -5/4*w^3 + 3/2*w^2 + 13/2*w - 9/4],
[691, 691, -7/4*w^3 + 11/2*w^2 + 23/2*w - 135/4],
[739, 739, -2*w^3 + 7*w^2 + 10*w - 34],
[739, 739, 1/4*w^3 + 5/2*w^2 - 7/2*w - 59/4],
[751, 751, -1/2*w^3 + 3*w^2 + 2*w - 33/2],
[751, 751, -3/4*w^3 + 7/2*w^2 + 7/2*w - 67/4],
[761, 761, -w^3 + 4*w^2 + 5*w - 27],
[761, 761, 7/4*w^3 - 9/2*w^2 - 15/2*w + 63/4],
[769, 769, -5/4*w^3 + 1/2*w^2 + 25/2*w + 19/4],
[769, 769, 3/2*w^3 - 5*w^2 - 4*w + 39/2],
[769, 769, -3*w^3 + 10*w^2 + 11*w - 37],
[769, 769, -7/4*w^3 + 5/2*w^2 + 27/2*w - 31/4],
[811, 811, -w^3 + 3*w^2 + 6*w - 9],
[811, 811, -3/4*w^3 + 7/2*w^2 + 5/2*w - 39/4],
[811, 811, -1/4*w^3 + 5/2*w^2 - 1/2*w - 85/4],
[811, 811, -5/4*w^3 + 7/2*w^2 + 17/2*w - 57/4],
[821, 821, -3/4*w^3 + 3/2*w^2 + 3/2*w - 35/4],
[821, 821, -5/4*w^3 + 11/2*w^2 + 9/2*w - 121/4],
[829, 829, 7/4*w^3 - 13/2*w^2 - 19/2*w + 115/4],
[829, 829, 1/4*w^3 - 7/2*w^2 - 1/2*w + 89/4],
[839, 839, -3/4*w^3 + 5/2*w^2 + 3/2*w - 51/4],
[839, 839, -3/4*w^3 + 1/2*w^2 + 15/2*w - 7/4],
[911, 911, -3/4*w^3 + 1/2*w^2 + 11/2*w - 15/4],
[911, 911, -3/4*w^3 + 7/2*w^2 + 3/2*w - 75/4],
[929, 929, 9/4*w^3 - 9/2*w^2 - 35/2*w + 73/4],
[929, 929, -3/4*w^3 + 7/2*w^2 + 9/2*w - 51/4],
[929, 929, -3/4*w^3 + 7/2*w^2 + 9/2*w - 91/4],
[929, 929, 5/2*w^3 - 4*w^2 - 19*w + 25/2],
[961, 31, -5/4*w^3 + 5/2*w^2 + 15/2*w - 37/4],
[961, 31, -1/2*w^3 + w^2 + 3*w - 19/2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 3*x^6 - 23*x^5 + 69*x^4 + 136*x^3 - 404*x^2 - 192*x + 576;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, 1, -5/288*e^6 + 1/96*e^5 + 151/288*e^4 - 23/96*e^3 - 305/72*e^2 + 97/72*e + 43/6, -35/288*e^6 + 7/96*e^5 + 769/288*e^4 - 161/96*e^3 - 1055/72*e^2 + 463/72*e + 103/6, 1/36*e^6 + 1/12*e^5 - 23/36*e^4 - 17/12*e^3 + 34/9*e^2 + 71/18*e - 8/3, 7/96*e^6 - 3/32*e^5 - 149/96*e^4 + 53/32*e^3 + 193/24*e^2 - 95/24*e - 13/2, -17/144*e^6 + 1/48*e^5 + 391/144*e^4 - 23/48*e^3 - 307/18*e^2 + 13/36*e + 85/3, -7/144*e^6 - 1/48*e^5 + 161/144*e^4 - 1/48*e^3 - 137/18*e^2 + 89/36*e + 47/3, 11/96*e^6 + 1/32*e^5 - 265/96*e^4 - 23/32*e^3 + 443/24*e^2 + 113/24*e - 63/2, -1/24*e^6 + 1/8*e^5 + 35/24*e^4 - 19/8*e^3 - 79/6*e^2 + 28/3*e + 24, 1/48*e^6 - 1/16*e^5 - 47/48*e^4 + 15/16*e^3 + 34/3*e^2 - 35/12*e - 27, 17/288*e^6 - 13/96*e^5 - 283/288*e^4 + 251/96*e^3 + 173/72*e^2 - 625/72*e + 17/6, -19/144*e^6 - 1/48*e^5 + 473/144*e^4 + 23/48*e^3 - 817/36*e^2 - 157/36*e + 113/3, -31/288*e^6 + 11/96*e^5 + 749/288*e^4 - 205/96*e^3 - 1189/72*e^2 + 551/72*e + 131/6, 5/72*e^6 - 7/24*e^5 - 133/72*e^4 + 125/24*e^3 + 511/36*e^2 - 152/9*e - 80/3, -11/96*e^6 - 1/32*e^5 + 265/96*e^4 - 9/32*e^3 - 443/24*e^2 + 151/24*e + 59/2, -35/288*e^6 + 7/96*e^5 + 913/288*e^4 - 113/96*e^3 - 1667/72*e^2 + 67/72*e + 265/6, 25/288*e^6 - 5/96*e^5 - 611/288*e^4 + 163/96*e^3 + 1057/72*e^2 - 593/72*e - 173/6, -35/144*e^6 + 7/48*e^5 + 769/144*e^4 - 161/48*e^3 - 1091/36*e^2 + 427/36*e + 127/3, 91/288*e^6 - 23/96*e^5 - 2129/288*e^4 + 433/96*e^3 + 3373/72*e^2 - 959/72*e - 461/6, 1/6*e^6 - 23/6*e^4 + 1/2*e^3 + 74/3*e^2 - 17/6*e - 46, 1/48*e^6 - 1/16*e^5 - 47/48*e^4 + 15/16*e^3 + 31/3*e^2 - 23/12*e - 19, -19/288*e^6 - 1/96*e^5 + 329/288*e^4 - 73/96*e^3 - 313/72*e^2 + 527/72*e + 77/6, 5/72*e^6 - 7/24*e^5 - 97/72*e^4 + 137/24*e^3 + 169/36*e^2 - 367/18*e - 8/3, -19/288*e^6 + 23/96*e^5 + 401/288*e^4 - 433/96*e^3 - 583/72*e^2 + 1103/72*e + 119/6, 17/72*e^6 - 7/24*e^5 - 373/72*e^4 + 125/24*e^3 + 985/36*e^2 - 128/9*e - 98/3, 1/18*e^6 + 1/6*e^5 - 23/18*e^4 - 17/6*e^3 + 77/9*e^2 + 62/9*e - 22/3, 7/24*e^6 - 1/8*e^5 - 167/24*e^4 + 23/8*e^3 + 545/12*e^2 - 41/6*e - 78, -1/288*e^6 + 5/96*e^5 - 157/288*e^4 - 67/96*e^3 + 713/72*e^2 - 103/72*e - 151/6, 25/144*e^6 - 17/48*e^5 - 575/144*e^4 + 319/48*e^3 + 443/18*e^2 - 827/36*e - 113/3, 1/18*e^6 + 1/6*e^5 - 23/18*e^4 - 23/6*e^3 + 68/9*e^2 + 170/9*e - 22/3, 71/288*e^6 + 5/96*e^5 - 1741/288*e^4 - 19/96*e^3 + 3107/72*e^2 - 355/72*e - 493/6, -41/144*e^6 + 1/48*e^5 + 943/144*e^4 - 47/48*e^3 - 769/18*e^2 + 151/36*e + 241/3, 23/144*e^6 - 7/48*e^5 - 529/144*e^4 + 161/48*e^3 + 373/18*e^2 - 475/36*e - 67/3, -1/96*e^6 + 5/32*e^5 - 13/96*e^4 - 83/32*e^3 + 125/24*e^2 + 125/24*e - 35/2, 7/72*e^6 + 1/24*e^5 - 161/72*e^4 - 23/24*e^3 + 101/9*e^2 + 145/18*e + 2/3, 7/96*e^6 + 5/32*e^5 - 173/96*e^4 - 115/32*e^3 + 331/24*e^2 + 421/24*e - 57/2, -5/36*e^6 + 1/12*e^5 + 97/36*e^4 - 17/12*e^3 - 223/18*e^2 + 41/18*e + 58/3, 41/144*e^6 - 13/48*e^5 - 907/144*e^4 + 251/48*e^3 + 1223/36*e^2 - 565/36*e - 115/3, 29/144*e^6 - 1/48*e^5 - 775/144*e^4 + 23/48*e^3 + 1463/36*e^2 - 145/36*e - 223/3, 11/288*e^6 - 7/96*e^5 - 289/288*e^4 + 161/96*e^3 + 437/72*e^2 - 343/72*e - 31/6, -17/288*e^6 + 13/96*e^5 + 427/288*e^4 - 203/96*e^3 - 713/72*e^2 + 85/72*e + 103/6, 79/288*e^6 - 11/96*e^5 - 1853/288*e^4 + 253/96*e^3 + 2821/72*e^2 - 611/72*e - 323/6, -3/32*e^6 - 3/32*e^5 + 57/32*e^4 + 53/32*e^3 - 57/8*e^2 - 61/8*e + 17/2, 7/144*e^6 + 1/48*e^5 - 161/144*e^4 - 47/48*e^3 + 101/18*e^2 + 307/36*e - 11/3, -23/288*e^6 - 29/96*e^5 + 565/288*e^4 + 523/96*e^3 - 989/72*e^2 - 1361/72*e + 169/6, -1/96*e^6 + 5/32*e^5 + 83/96*e^4 - 51/32*e^3 - 283/24*e^2 - 67/24*e + 67/2, -9/32*e^6 - 1/32*e^5 + 211/32*e^4 + 7/32*e^3 - 329/8*e^2 - 15/8*e + 103/2, 17/144*e^6 - 1/48*e^5 - 319/144*e^4 + 47/48*e^3 + 86/9*e^2 - 139/36*e - 55/3, 5/24*e^6 - 1/8*e^5 - 115/24*e^4 + 19/8*e^3 + 85/3*e^2 - 26/3*e - 30, -1/288*e^6 + 5/96*e^5 + 131/288*e^4 - 67/96*e^3 - 439/72*e^2 + 41/72*e + 143/6, -1/48*e^6 + 1/16*e^5 + 23/48*e^4 - 23/16*e^3 - 23/6*e^2 + 125/12*e + 11, 5/288*e^6 + 23/96*e^5 - 223/288*e^4 - 385/96*e^3 + 575/72*e^2 + 587/72*e - 73/6, -1/4*e^6 + 1/4*e^5 + 25/4*e^4 - 19/4*e^3 - 91/2*e^2 + 14*e + 92, 13/288*e^6 - 17/96*e^5 - 119/288*e^4 + 439/96*e^3 - 161/72*e^2 - 1829/72*e + 49/6, 17/144*e^6 - 1/48*e^5 - 535/144*e^4 + 23/48*e^3 + 613/18*e^2 - 13/36*e - 235/3, 31/144*e^6 - 23/48*e^5 - 713/144*e^4 + 457/48*e^3 + 527/18*e^2 - 1253/36*e - 149/3, 49/288*e^6 + 19/96*e^5 - 1019/288*e^4 - 293/96*e^3 + 1405/72*e^2 + 727/72*e - 209/6, 17/288*e^6 - 37/96*e^5 - 355/288*e^4 + 707/96*e^3 + 299/72*e^2 - 1705/72*e - 1/6, -31/288*e^6 + 35/96*e^5 + 965/288*e^4 - 613/96*e^3 - 1999/72*e^2 + 1235/72*e + 365/6, -1/24*e^6 + 1/8*e^5 - 1/24*e^4 - 23/8*e^3 + 31/3*e^2 + 59/6*e - 32, 1/32*e^6 - 7/32*e^5 - 27/32*e^4 + 113/32*e^3 + 41/8*e^2 - 41/8*e - 19/2, -29/96*e^6 + 1/32*e^5 + 583/96*e^4 - 39/32*e^3 - 671/24*e^2 + 37/24*e + 51/2, -47/144*e^6 - 5/48*e^5 + 1045/144*e^4 + 43/48*e^3 - 1535/36*e^2 - 107/36*e + 229/3, 31/144*e^6 - 11/48*e^5 - 749/144*e^4 + 253/48*e^3 + 1261/36*e^2 - 875/36*e - 149/3, 7/24*e^6 + 1/8*e^5 - 161/24*e^4 - 15/8*e^3 + 125/3*e^2 + 19/6*e - 58, -23/96*e^6 - 5/32*e^5 + 541/96*e^4 + 99/32*e^3 - 875/24*e^2 - 473/24*e + 119/2, 17/72*e^6 - 1/24*e^5 - 355/72*e^4 + 35/24*e^3 + 407/18*e^2 - 20/9*e - 8/3, -11/96*e^6 - 9/32*e^5 + 241/96*e^4 + 175/32*e^3 - 401/24*e^2 - 569/24*e + 69/2, -7/72*e^6 + 5/24*e^5 + 179/72*e^4 - 67/24*e^3 - 575/36*e^2 - 37/18*e + 64/3, 11/96*e^6 - 7/32*e^5 - 241/96*e^4 + 145/32*e^3 + 353/24*e^2 - 427/24*e - 65/2, -5/72*e^6 + 7/24*e^5 + 169/72*e^4 - 113/24*e^3 - 781/36*e^2 + 169/18*e + 122/3, 5/36*e^6 - 1/3*e^5 - 53/18*e^4 + 43/6*e^3 + 509/36*e^2 - 599/18*e - 58/3, 11/96*e^6 - 7/32*e^5 - 241/96*e^4 + 145/32*e^3 + 401/24*e^2 - 523/24*e - 85/2, -3/32*e^6 - 11/32*e^5 + 49/32*e^4 + 173/32*e^3 - 19/8*e^2 - 141/8*e - 19/2, -5/48*e^6 + 1/16*e^5 + 127/48*e^4 - 31/16*e^3 - 227/12*e^2 + 151/12*e + 25, -83/144*e^6 + 19/48*e^5 + 1981/144*e^4 - 365/48*e^3 - 809/9*e^2 + 1009/36*e + 439/3, -13/288*e^6 - 7/96*e^5 + 335/288*e^4 + 209/96*e^3 - 721/72*e^2 - 1123/72*e + 215/6, -31/72*e^6 + 11/24*e^5 + 749/72*e^4 - 229/24*e^3 - 1225/18*e^2 + 713/18*e + 310/3, 1/12*e^6 + 1/4*e^5 - 23/12*e^4 - 21/4*e^3 + 40/3*e^2 + 149/6*e - 24, 127/288*e^6 - 35/96*e^5 - 3029/288*e^4 + 757/96*e^3 + 4939/72*e^2 - 2327/72*e - 623/6, 1/9*e^6 + 1/12*e^5 - 83/36*e^4 - 5/12*e^3 + 373/36*e^2 - 49/18*e - 20/3, -41/144*e^6 + 1/48*e^5 + 943/144*e^4 - 95/48*e^3 - 715/18*e^2 + 439/36*e + 163/3, 1/32*e^6 - 7/32*e^5 - 27/32*e^4 + 145/32*e^3 + 57/8*e^2 - 129/8*e - 39/2, -1/16*e^6 + 3/16*e^5 + 31/16*e^4 - 45/16*e^3 - 18*e^2 + 23/4*e + 41, -1/24*e^6 + 1/8*e^5 + 11/24*e^4 - 27/8*e^3 + 11/6*e^2 + 55/3*e - 2, -43/288*e^6 - 1/96*e^5 + 953/288*e^4 - 25/96*e^3 - 1399/72*e^2 - 181/72*e + 191/6, 59/288*e^6 + 17/96*e^5 - 1321/288*e^4 - 343/96*e^3 + 1799/72*e^2 + 1541/72*e - 139/6, 1/18*e^6 - 1/3*e^5 - 16/9*e^4 + 14/3*e^3 + 307/18*e^2 - 37/9*e - 148/3, 13/72*e^6 + 1/24*e^5 - 353/72*e^4 - 23/24*e^3 + 1433/36*e^2 + 205/18*e - 250/3, -5/16*e^6 + 7/16*e^5 + 123/16*e^4 - 129/16*e^3 - 51*e^2 + 97/4*e + 97, 49/288*e^6 + 19/96*e^5 - 1307/288*e^4 - 293/96*e^3 + 2413/72*e^2 + 655/72*e - 281/6, 5/32*e^6 - 11/32*e^5 - 111/32*e^4 + 205/32*e^3 + 131/8*e^2 - 189/8*e - 13/2, 1/2*e^5 + 1/2*e^4 - 17/2*e^3 - 19/2*e^2 + 25*e + 38, 1/3*e^6 - 1/4*e^5 - 89/12*e^4 + 21/4*e^3 + 511/12*e^2 - 103/6*e - 60, -1/4*e^6 + 11/2*e^4 + 1/2*e^3 - 133/4*e^2 - 29/2*e + 54, -19/36*e^6 + 5/12*e^5 + 455/36*e^4 - 103/12*e^3 - 1499/18*e^2 + 230/9*e + 440/3, 95/288*e^6 + 29/96*e^5 - 2293/288*e^4 - 523/96*e^3 + 3779/72*e^2 + 1865/72*e - 481/6, 19/36*e^6 - 5/12*e^5 - 473/36*e^4 + 97/12*e^3 + 790/9*e^2 - 595/18*e - 398/3, 19/96*e^6 - 7/32*e^5 - 497/96*e^4 + 113/32*e^3 + 895/24*e^2 - 107/24*e - 147/2, 19/288*e^6 + 1/96*e^5 - 761/288*e^4 - 71/96*e^3 + 1933/72*e^2 + 373/72*e - 329/6, 7/36*e^6 - 5/12*e^5 - 161/36*e^4 + 97/12*e^3 + 229/9*e^2 - 475/18*e - 92/3, 43/288*e^6 - 23/96*e^5 - 881/288*e^4 + 529/96*e^3 + 913/72*e^2 - 2159/72*e - 59/6, 7/36*e^6 - 5/12*e^5 - 179/36*e^4 + 91/12*e^3 + 647/18*e^2 - 251/9*e - 224/3, 185/288*e^6 - 61/96*e^5 - 4363/288*e^4 + 1259/96*e^3 + 6911/72*e^2 - 3841/72*e - 847/6, -1/6*e^6 + 23/6*e^4 + 1/2*e^3 - 68/3*e^2 - 37/6*e + 28, 7/32*e^6 + 7/32*e^5 - 133/32*e^4 - 81/32*e^3 + 141/8*e^2 + 25/8*e - 43/2, -31/96*e^6 + 3/32*e^5 + 773/96*e^4 - 37/32*e^3 - 1279/24*e^2 - 37/24*e + 149/2, 19/144*e^6 - 11/48*e^5 - 365/144*e^4 + 205/48*e^3 + 76/9*e^2 - 383/36*e - 23/3, -19/48*e^6 - 1/16*e^5 + 425/48*e^4 + 23/16*e^3 - 661/12*e^2 - 157/12*e + 91, 1/3*e^6 - 23/3*e^4 + 130/3*e^2 + 16/3*e - 48, -211/288*e^6 + 71/96*e^5 + 4961/288*e^4 - 1345/96*e^3 - 7795/72*e^2 + 3431/72*e + 1001/6, 53/144*e^6 - 1/48*e^5 - 1111/144*e^4 + 71/48*e^3 + 1397/36*e^2 - 445/36*e - 97/3, 169/288*e^6 - 5/96*e^5 - 3923/288*e^4 + 115/96*e^3 + 6169/72*e^2 + 235/72*e - 875/6, 103/144*e^6 - 23/48*e^5 - 2441/144*e^4 + 505/48*e^3 + 988/9*e^2 - 1469/36*e - 533/3, -151/288*e^6 + 35/96*e^5 + 3365/288*e^4 - 613/96*e^3 - 4873/72*e^2 + 827/72*e + 611/6, 1/18*e^6 - 1/12*e^5 - 91/36*e^4 + 23/12*e^3 + 1055/36*e^2 - 46/9*e - 220/3, 1/8*e^6 - 1/8*e^5 - 25/8*e^4 + 27/8*e^3 + 103/4*e^2 - 18*e - 68, -9/32*e^6 + 7/32*e^5 + 203/32*e^4 - 161/32*e^3 - 315/8*e^2 + 213/8*e + 137/2, -107/288*e^6 - 17/96*e^5 + 2425/288*e^4 + 295/96*e^3 - 3719/72*e^2 - 689/72*e + 445/6, 1/48*e^6 + 7/16*e^5 - 47/48*e^4 - 129/16*e^3 + 40/3*e^2 + 307/12*e - 37, 35/288*e^6 + 17/96*e^5 - 841/288*e^4 - 247/96*e^3 + 1109/72*e^2 + 149/72*e + 23/6, 35/288*e^6 + 17/96*e^5 - 265/288*e^4 - 247/96*e^3 - 1051/72*e^2 + 293/72*e + 455/6, -31/72*e^6 + 5/24*e^5 + 695/72*e^4 - 79/24*e^3 - 2081/36*e^2 + 55/9*e + 286/3, 19/72*e^6 + 1/24*e^5 - 509/72*e^4 - 35/24*e^3 + 494/9*e^2 + 110/9*e - 310/3, -107/288*e^6 - 17/96*e^5 + 2137/288*e^4 + 199/96*e^3 - 2495/72*e^2 - 833/72*e + 283/6, -65/144*e^6 + 13/48*e^5 + 1531/144*e^4 - 251/48*e^3 - 2489/36*e^2 + 649/36*e + 361/3, 7/36*e^6 + 7/12*e^5 - 143/36*e^4 - 125/12*e^3 + 377/18*e^2 + 334/9*e - 80/3, 85/288*e^6 + 7/96*e^5 - 1631/288*e^4 - 161/96*e^3 + 1747/72*e^2 + 1159/72*e - 185/6, -65/288*e^6 + 13/96*e^5 + 1387/288*e^4 - 299/96*e^3 - 1589/72*e^2 + 1333/72*e + 7/6, -2/9*e^6 + 1/3*e^5 + 101/18*e^4 - 43/6*e^3 - 751/18*e^2 + 521/18*e + 262/3, -31/144*e^6 + 11/48*e^5 + 749/144*e^4 - 205/48*e^3 - 1333/36*e^2 + 119/36*e + 233/3, -1/36*e^6 + 1/6*e^5 - 11/18*e^4 - 17/6*e^3 + 647/36*e^2 + 68/9*e - 178/3, -1/3*e^6 - 1/4*e^5 + 95/12*e^4 + 21/4*e^3 - 625/12*e^2 - 107/6*e + 78, -71/144*e^6 - 5/48*e^5 + 1813/144*e^4 - 5/48*e^3 - 3305/36*e^2 + 265/36*e + 505/3, -7/24*e^6 + 1/8*e^5 + 179/24*e^4 - 19/8*e^3 - 647/12*e^2 + 43/3*e + 104, -5/12*e^6 - 1/4*e^5 + 115/12*e^4 + 13/4*e^3 - 176/3*e^2 - 61/6*e + 82, -25/288*e^6 - 43/96*e^5 + 755/288*e^4 + 653/96*e^3 - 1525/72*e^2 - 1207/72*e + 155/6, 17/24*e^6 - 1/8*e^5 - 367/24*e^4 + 31/8*e^3 + 244/3*e^2 - 67/6*e - 84, -9/32*e^6 - 1/32*e^5 + 179/32*e^4 + 7/32*e^3 - 185/8*e^2 - 23/8*e - 29/2, -101/144*e^6 + 25/48*e^5 + 2143/144*e^4 - 527/48*e^3 - 2831/36*e^2 + 1333/36*e + 307/3, 49/288*e^6 - 5/96*e^5 - 1235/288*e^4 + 211/96*e^3 + 2359/72*e^2 - 1253/72*e - 257/6, 7/96*e^6 - 11/32*e^5 - 173/96*e^4 + 237/32*e^3 + 235/24*e^2 - 863/24*e - 23/2, 71/288*e^6 - 19/96*e^5 - 1957/288*e^4 + 389/96*e^3 + 3845/72*e^2 - 1615/72*e - 517/6, -239/288*e^6 + 91/96*e^5 + 5245/288*e^4 - 1853/96*e^3 - 7397/72*e^2 + 5047/72*e + 949/6, 11/96*e^6 + 1/32*e^5 - 265/96*e^4 - 23/32*e^3 + 443/24*e^2 + 377/24*e - 65/2, -59/144*e^6 - 5/48*e^5 + 1645/144*e^4 + 67/48*e^3 - 1669/18*e^2 - 173/36*e + 577/3, 163/288*e^6 + 1/96*e^5 - 3641/288*e^4 + 121/96*e^3 + 5425/72*e^2 - 563/72*e - 737/6, -2/9*e^6 - 1/6*e^5 + 101/18*e^4 + 17/6*e^3 - 643/18*e^2 - 95/9*e + 166/3, -1/72*e^6 + 5/24*e^5 + 23/72*e^4 - 103/24*e^3 + 19/9*e^2 + 115/9*e - 98/3, 91/288*e^6 + 1/96*e^5 - 2057/288*e^4 + 169/96*e^3 + 2815/72*e^2 - 1247/72*e - 113/6, -53/288*e^6 - 23/96*e^5 + 1039/288*e^4 + 529/96*e^3 - 1199/72*e^2 - 1823/72*e + 91/6];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, 1/2*w^3 - 2*w^2 - 2*w + 15/2}>] := -1;
ALEigenvalues[ideal<ZF | {5, 5, -1/4*w^3 + 1/2*w^2 + 1/2*w - 9/4}>] := -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;