/*
  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, 10, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, -w + 1],
[5, 5, w^3 + w^2 - 4*w + 1],
[7, 7, -w^3 + 6*w - 2],
[8, 2, w^3 - 7*w + 3],
[11, 11, -w^3 + 6*w - 4],
[17, 17, w^3 + w^2 - 6*w - 1],
[17, 17, -w^2 - w + 3],
[31, 31, 2*w^3 + w^2 - 13*w + 3],
[37, 37, -3*w^3 - w^2 + 18*w - 3],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^3 + 2*w^2 - 11*w - 2],
[47, 47, -3*w^3 - w^2 + 19*w - 6],
[59, 59, -2*w^3 + 13*w - 6],
[59, 59, -w^3 - w^2 + 5*w + 2],
[59, 59, w^2 + w - 1],
[59, 59, w^2 + 2*w - 6],
[61, 61, -w^3 + w^2 + 8*w - 7],
[67, 67, w^3 + 2*w^2 - 4*w - 4],
[73, 73, -2*w^3 - w^2 + 12*w - 4],
[81, 3, -3],
[107, 107, 2*w^2 + 2*w - 11],
[109, 109, 2*w^3 + w^2 - 11*w + 3],
[113, 113, w^3 + w^2 - 7*w - 2],
[113, 113, 2*w^3 - 12*w + 9],
[125, 5, -4*w^3 - 3*w^2 + 24*w + 2],
[131, 131, 5*w^3 + 3*w^2 - 31*w + 2],
[151, 151, 4*w^3 + 2*w^2 - 26*w + 1],
[151, 151, w^3 + 2*w^2 - 6*w - 10],
[167, 167, w + 4],
[167, 167, 4*w^3 + w^2 - 26*w + 10],
[173, 173, -2*w^3 + w^2 + 13*w - 11],
[179, 179, 4*w^3 + w^2 - 24*w + 4],
[179, 179, 2*w^3 + 2*w^2 - 12*w - 3],
[179, 179, -w^3 + 8*w - 4],
[179, 179, w^3 + 2*w^2 - 4*w - 6],
[181, 181, 6*w^3 + 3*w^2 - 39*w + 5],
[197, 197, -w^3 + 4*w - 4],
[199, 199, 2*w^3 - 12*w + 3],
[199, 199, 3*w - 2],
[223, 223, w^3 + w^2 - 7*w - 4],
[223, 223, 2*w^3 + w^2 - 10*w + 4],
[227, 227, 7*w^3 + 2*w^2 - 44*w + 12],
[227, 227, -4*w^3 - w^2 + 25*w - 9],
[229, 229, 8*w^3 + 4*w^2 - 51*w + 4],
[229, 229, 3*w^3 + w^2 - 18*w + 5],
[233, 233, -6*w^3 - 3*w^2 + 39*w - 7],
[233, 233, -w^3 - 2*w^2 + 8*w + 2],
[241, 241, 2*w^3 + w^2 - 10*w],
[251, 251, w^3 + w^2 - 7*w - 6],
[257, 257, -6*w^3 - w^2 + 39*w - 17],
[263, 263, -2*w^3 + 13*w - 4],
[263, 263, -w^3 + 6*w - 8],
[269, 269, -8*w^3 - 4*w^2 + 50*w - 7],
[269, 269, -w^3 + w^2 + 7*w - 6],
[277, 277, 2*w^3 + 2*w^2 - 11*w + 4],
[277, 277, w^3 + w^2 - 4*w - 3],
[281, 281, -w^2 - 3*w + 5],
[281, 281, 2*w^3 + 3*w^2 - 9*w - 3],
[283, 283, -3*w^3 - 4*w^2 + 12*w + 2],
[289, 17, -2*w^3 - w^2 + 13*w - 5],
[293, 293, -w^3 - 2*w^2 + 6*w + 6],
[293, 293, w^3 + w^2 - 8*w - 1],
[307, 307, 2*w^3 - 11*w + 12],
[317, 317, 2*w^3 - 14*w + 11],
[317, 317, -2*w^3 - 2*w^2 + 8*w + 1],
[337, 337, 2*w^2 + 4*w - 11],
[343, 7, -5*w^3 - w^2 + 31*w - 10],
[349, 349, 2*w^3 - 11*w + 8],
[349, 349, w^2 + w - 9],
[353, 353, 4*w^3 + 4*w^2 - 23*w - 8],
[373, 373, -2*w - 3],
[373, 373, -4*w^3 + 26*w - 13],
[389, 389, 6*w^3 + 3*w^2 - 37*w + 1],
[389, 389, -2*w^3 - w^2 + 14*w - 6],
[401, 401, 2*w^3 - 12*w + 13],
[421, 421, 2*w^2 + 2*w - 15],
[421, 421, 3*w^3 + 2*w^2 - 20*w + 4],
[431, 431, w^3 + 3*w^2 - 2*w - 9],
[433, 433, 2*w^3 + 2*w^2 - 12*w + 3],
[433, 433, -7*w^3 - 2*w^2 + 44*w - 16],
[433, 433, -3*w^3 + 18*w - 14],
[433, 433, -w^3 + w^2 + 9*w - 10],
[439, 439, 5*w^3 + 3*w^2 - 31*w - 2],
[443, 443, -w^3 - w^2 + 3*w - 4],
[443, 443, -4*w^3 + 27*w - 14],
[449, 449, w^3 + w^2 - 4*w - 5],
[449, 449, w^3 + 3*w^2 - 3*w - 8],
[449, 449, 2*w^3 - 11*w + 6],
[449, 449, -3*w^3 - 2*w^2 + 18*w - 4],
[461, 461, -3*w^3 + w^2 + 20*w - 15],
[463, 463, 2*w^3 - 10*w + 7],
[479, 479, -2*w^3 - 3*w^2 + 11*w + 9],
[491, 491, w^3 + w^2 - 4*w + 5],
[499, 499, -2*w^3 - 2*w^2 + 14*w - 3],
[499, 499, -6*w^3 - 2*w^2 + 39*w - 10],
[499, 499, 4*w^3 + w^2 - 24*w + 14],
[499, 499, -5*w^3 - 3*w^2 + 30*w - 5],
[503, 503, -2*w^3 + w^2 + 14*w - 18],
[509, 509, 4*w^3 + 3*w^2 - 24*w],
[509, 509, 4*w^3 + 2*w^2 - 23*w + 4],
[547, 547, w^3 + 2*w^2 - 2*w - 6],
[547, 547, 6*w^3 + 2*w^2 - 38*w + 7],
[563, 563, -5*w^3 - w^2 + 33*w - 12],
[569, 569, 2*w^3 - w^2 - 15*w + 13],
[569, 569, -4*w^3 - 2*w^2 + 24*w - 7],
[577, 577, -w^2 - 2],
[577, 577, 3*w^3 - 20*w + 8],
[587, 587, 6*w^3 + 4*w^2 - 36*w - 1],
[587, 587, -w^3 + w^2 + 9*w - 8],
[593, 593, 6*w^3 + 3*w^2 - 36*w + 2],
[593, 593, -3*w^3 - w^2 + 21*w - 4],
[599, 599, -2*w^3 - w^2 + 12*w - 10],
[599, 599, -4*w^3 - 2*w^2 + 25*w - 6],
[601, 601, -2*w^3 + 2*w^2 + 15*w - 16],
[607, 607, w^3 - w^2 - 7*w + 4],
[607, 607, -2*w^3 - 2*w^2 + 10*w + 3],
[631, 631, 2*w^2 + 4*w - 7],
[641, 641, -3*w^3 - 2*w^2 + 16*w],
[643, 643, w^3 + w^2 - 3*w - 4],
[643, 643, w^3 + 3*w^2 - 6*w - 13],
[653, 653, -w^3 + 6*w + 2],
[653, 653, -3*w^2 - 5*w + 11],
[659, 659, -2*w^3 + 10*w - 5],
[661, 661, w^3 - w^2 - 8*w + 3],
[673, 673, 3*w^3 + 4*w^2 - 14*w - 2],
[677, 677, 4*w^3 + 2*w^2 - 23*w + 6],
[677, 677, 2*w^3 - 13*w + 14],
[683, 683, 2*w^3 + w^2 - 13*w + 7],
[701, 701, 8*w^3 + 4*w^2 - 51*w + 8],
[701, 701, -4*w^3 - w^2 + 24*w - 12],
[709, 709, -4*w^3 - w^2 + 27*w - 7],
[709, 709, 3*w^3 + 3*w^2 - 18*w - 5],
[709, 709, -3*w^3 + w^2 + 21*w - 14],
[709, 709, w^2 - w - 7],
[719, 719, -9*w^3 - 4*w^2 + 58*w - 8],
[719, 719, 11*w^3 + 5*w^2 - 68*w + 9],
[727, 727, -w^3 + 4*w - 6],
[733, 733, 2*w^3 + w^2 - 13*w + 9],
[739, 739, -2*w^2 - 3*w + 4],
[743, 743, 3*w^3 + 3*w^2 - 19*w - 4],
[751, 751, 2*w^3 + 3*w^2 - 10*w - 2],
[761, 761, -12*w^3 - 5*w^2 + 75*w - 9],
[761, 761, -3*w^3 - 3*w^2 + 17*w],
[769, 769, -6*w^3 - 4*w^2 + 37*w],
[769, 769, -8*w^3 - 3*w^2 + 51*w - 7],
[773, 773, -w^2 - 2*w - 2],
[787, 787, -5*w^3 - 5*w^2 + 28*w + 7],
[811, 811, 7*w^3 + 2*w^2 - 42*w + 6],
[811, 811, -2*w^3 - w^2 + 9*w - 7],
[823, 823, -3*w^3 - 2*w^2 + 14*w - 10],
[829, 829, 4*w^3 + 3*w^2 - 20*w + 6],
[829, 829, -7*w^3 - 2*w^2 + 44*w - 14],
[841, 29, w^3 + 4*w^2 - 2*w - 14],
[841, 29, -w^2 + 10],
[857, 857, -3*w^3 - w^2 + 20*w - 9],
[859, 859, -w^3 + 2*w^2 + 10*w - 18],
[863, 863, 11*w^3 + 5*w^2 - 69*w + 6],
[863, 863, w^3 - 2*w^2 - 10*w + 12],
[877, 877, 3*w^3 - 18*w + 10],
[883, 883, 4*w^3 + 3*w^2 - 22*w],
[887, 887, 3*w^3 + 3*w^2 - 15*w - 2],
[887, 887, w^2 + 4*w - 10],
[887, 887, w^3 + w^2 - 3*w - 8],
[887, 887, w - 6],
[907, 907, 6*w^3 + w^2 - 38*w + 16],
[919, 919, -2*w^3 - 3*w^2 + 11*w + 1],
[919, 919, 5*w^3 - w^2 - 34*w + 27],
[937, 937, -w^3 - w^2 + 8*w + 3],
[941, 941, -4*w^3 + 25*w - 20],
[947, 947, -2*w^3 + w^2 + 8*w - 6],
[967, 967, -7*w^3 - w^2 + 44*w - 23],
[971, 971, 2*w^2 + 2*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 3*x^7 - 9*x^6 + 30*x^5 + 19*x^4 - 89*x^3 + 7*x^2 + 80*x - 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/4*e^7 + 1/4*e^6 + 13/4*e^5 - 3*e^4 - 53/4*e^3 + 45/4*e^2 + 67/4*e - 12, -1, 1/4*e^7 - 3/4*e^6 - 9/4*e^5 + 13/2*e^4 + 23/4*e^3 - 57/4*e^2 - 17/4*e + 9, 3/4*e^7 - 5/4*e^6 - 31/4*e^5 + 21/2*e^4 + 93/4*e^3 - 91/4*e^2 - 83/4*e + 13, 5/4*e^7 - 9/4*e^6 - 53/4*e^5 + 19*e^4 + 173/4*e^3 - 165/4*e^2 - 175/4*e + 22, -e^7 + 3/2*e^6 + 11*e^5 - 27/2*e^4 - 73/2*e^3 + 35*e^2 + 73/2*e - 27, -3/2*e^7 + 5/2*e^6 + 35/2*e^5 - 24*e^4 - 125/2*e^3 + 131/2*e^2 + 131/2*e - 49, 1/2*e^7 - e^6 - 9/2*e^5 + 15/2*e^4 + 11*e^3 - 21/2*e^2 - 8*e + 1, -1/4*e^7 + 1/4*e^6 + 9/4*e^5 - 2*e^4 - 21/4*e^3 + 17/4*e^2 + 19/4*e + 1, e^5 - e^4 - 8*e^3 + 7*e^2 + 11*e - 10, 3/2*e^7 - 3/2*e^6 - 37/2*e^5 + 14*e^4 + 139/2*e^3 - 75/2*e^2 - 155/2*e + 32, 1/2*e^7 - 1/2*e^6 - 11/2*e^5 + 4*e^4 + 33/2*e^3 - 15/2*e^2 - 17/2*e + 3, -5/4*e^7 + 3/4*e^6 + 61/4*e^5 - 11/2*e^4 - 227/4*e^3 + 49/4*e^2 + 249/4*e - 16, -3/4*e^7 + 5/4*e^6 + 31/4*e^5 - 19/2*e^4 - 93/4*e^3 + 59/4*e^2 + 83/4*e + 2, 1/4*e^7 + 5/4*e^6 - 21/4*e^5 - 23/2*e^4 + 107/4*e^3 + 99/4*e^2 - 133/4*e - 6, 1/4*e^7 - 1/4*e^6 - 17/4*e^5 + 4*e^4 + 81/4*e^3 - 69/4*e^2 - 103/4*e + 18, 1/2*e^7 - 1/2*e^6 - 15/2*e^5 + 6*e^4 + 69/2*e^3 - 39/2*e^2 - 93/2*e + 15, 1/4*e^7 + 1/4*e^6 - 13/4*e^5 - 5/2*e^4 + 47/4*e^3 + 11/4*e^2 - 49/4*e + 14, -1/2*e^7 + 2*e^6 + 7/2*e^5 - 33/2*e^4 - 4*e^3 + 57/2*e^2 - 3*e, 7/4*e^7 - 9/4*e^6 - 83/4*e^5 + 41/2*e^4 + 301/4*e^3 - 199/4*e^2 - 315/4*e + 29, -9/4*e^7 + 21/4*e^6 + 85/4*e^5 - 45*e^4 - 229/4*e^3 + 389/4*e^2 + 183/4*e - 44, -e^7 + 5/2*e^6 + 10*e^5 - 43/2*e^4 - 65/2*e^3 + 49*e^2 + 73/2*e - 31, -9/4*e^7 + 13/4*e^6 + 109/4*e^5 - 32*e^4 - 401/4*e^3 + 369/4*e^2 + 419/4*e - 80, -1/4*e^7 + 1/4*e^6 + 17/4*e^5 - 3*e^4 - 77/4*e^3 + 25/4*e^2 + 79/4*e + 7, -5/2*e^7 + 9/2*e^6 + 49/2*e^5 - 36*e^4 - 135/2*e^3 + 135/2*e^2 + 105/2*e - 25, -3/4*e^7 + 5/4*e^6 + 39/4*e^5 - 25/2*e^4 - 165/4*e^3 + 143/4*e^2 + 235/4*e - 30, 15/4*e^7 - 21/4*e^6 - 171/4*e^5 + 93/2*e^4 + 597/4*e^3 - 455/4*e^2 - 611/4*e + 78, 1/4*e^7 + 1/4*e^6 - 13/4*e^5 - 7/2*e^4 + 39/4*e^3 + 71/4*e^2 + 7/4*e - 22, -3*e^7 + 4*e^6 + 34*e^5 - 35*e^4 - 119*e^3 + 91*e^2 + 128*e - 76, -3/2*e^7 + 3*e^6 + 35/2*e^5 - 53/2*e^4 - 66*e^3 + 119/2*e^2 + 81*e - 31, -9/4*e^7 + 11/4*e^6 + 97/4*e^5 - 45/2*e^4 - 303/4*e^3 + 181/4*e^2 + 269/4*e - 16, -3*e^7 + 5*e^6 + 36*e^5 - 48*e^4 - 134*e^3 + 129*e^2 + 149*e - 98, 2*e^5 - 3*e^4 - 16*e^3 + 15*e^2 + 28*e - 2, e^7 - 2*e^6 - 10*e^5 + 18*e^4 + 29*e^3 - 46*e^2 - 30*e + 32, -7/4*e^7 + 7/4*e^6 + 87/4*e^5 - 15*e^4 - 335/4*e^3 + 143/4*e^2 + 377/4*e - 26, -17/4*e^7 + 25/4*e^6 + 181/4*e^5 - 54*e^4 - 561/4*e^3 + 489/4*e^2 + 487/4*e - 70, 5*e^7 - 9*e^6 - 51*e^5 + 76*e^4 + 154*e^3 - 166*e^2 - 138*e + 93, 1/2*e^7 - 1/2*e^6 - 13/2*e^5 + 5*e^4 + 45/2*e^3 - 19/2*e^2 - 27/2*e, -1/2*e^7 - 3/2*e^6 + 23/2*e^5 + 11*e^4 - 127/2*e^3 - 31/2*e^2 + 171/2*e - 9, 9/4*e^7 - 15/4*e^6 - 89/4*e^5 + 61/2*e^4 + 247/4*e^3 - 253/4*e^2 - 181/4*e + 41, 13/4*e^7 - 15/4*e^6 - 149/4*e^5 + 65/2*e^4 + 527/4*e^3 - 345/4*e^2 - 581/4*e + 86, 1/4*e^7 + 1/4*e^6 - 5/4*e^5 - 11/2*e^4 - 21/4*e^3 + 103/4*e^2 + 75/4*e - 28, 2*e^7 - 5*e^6 - 18*e^5 + 42*e^4 + 42*e^3 - 85*e^2 - 22*e + 34, 7/4*e^7 - 11/4*e^6 - 79/4*e^5 + 24*e^4 + 275/4*e^3 - 227/4*e^2 - 285/4*e + 45, 5/2*e^7 - 5*e^6 - 53/2*e^5 + 91/2*e^4 + 87*e^3 - 229/2*e^2 - 94*e + 85, -3/4*e^7 + 3/4*e^6 + 39/4*e^5 - 10*e^4 - 155/4*e^3 + 143/4*e^2 + 205/4*e - 27, -2*e^7 + 5/2*e^6 + 23*e^5 - 43/2*e^4 - 169/2*e^3 + 54*e^2 + 197/2*e - 46, 1/4*e^7 + 1/4*e^6 - 9/4*e^5 - 3/2*e^4 - 1/4*e^3 - 9/4*e^2 + 91/4*e + 12, -3*e^7 + 5*e^6 + 33*e^5 - 43*e^4 - 112*e^3 + 103*e^2 + 114*e - 71, 6*e^7 - 10*e^6 - 64*e^5 + 87*e^4 + 204*e^3 - 205*e^2 - 192*e + 130, 2*e^7 - 3*e^6 - 23*e^5 + 30*e^4 + 78*e^3 - 86*e^2 - 75*e + 65, -3/2*e^7 + 2*e^6 + 37/2*e^5 - 39/2*e^4 - 72*e^3 + 109/2*e^2 + 82*e - 36, -1/2*e^7 + e^6 + 7/2*e^5 - 11/2*e^4 - 2*e^3 - 13/2*e^2 - 8*e + 26, -17/4*e^7 + 27/4*e^6 + 181/4*e^5 - 111/2*e^4 - 591/4*e^3 + 473/4*e^2 + 609/4*e - 66, -15/4*e^7 + 19/4*e^6 + 179/4*e^5 - 43*e^4 - 663/4*e^3 + 431/4*e^2 + 729/4*e - 70, 7/2*e^7 - 9/2*e^6 - 87/2*e^5 + 45*e^4 + 335/2*e^3 - 267/2*e^2 - 373/2*e + 124, 5/4*e^7 - 11/4*e^6 - 41/4*e^5 + 45/2*e^4 + 79/4*e^3 - 193/4*e^2 - 49/4*e + 29, -3/2*e^7 + 5/2*e^6 + 29/2*e^5 - 18*e^4 - 75/2*e^3 + 39/2*e^2 + 39/2*e + 19, 7/4*e^7 - 5/4*e^6 - 91/4*e^5 + 23/2*e^4 + 377/4*e^3 - 143/4*e^2 - 471/4*e + 39, 15/4*e^7 - 25/4*e^6 - 167/4*e^5 + 119/2*e^4 + 573/4*e^3 - 651/4*e^2 - 607/4*e + 125, -3*e^7 + 11/2*e^6 + 31*e^5 - 91/2*e^4 - 197/2*e^3 + 97*e^2 + 199/2*e - 62, 9/4*e^7 - 7/4*e^6 - 113/4*e^5 + 29/2*e^4 + 431/4*e^3 - 133/4*e^2 - 477/4*e + 27, e^7 - 3/2*e^6 - 14*e^5 + 29/2*e^4 + 125/2*e^3 - 39*e^2 - 155/2*e + 28, -15/4*e^7 + 15/4*e^6 + 183/4*e^5 - 32*e^4 - 691/4*e^3 + 323/4*e^2 + 777/4*e - 71, -3*e^7 + 9/2*e^6 + 34*e^5 - 79/2*e^4 - 245/2*e^3 + 94*e^2 + 285/2*e - 62, 3*e^7 - 4*e^6 - 36*e^5 + 37*e^4 + 134*e^3 - 101*e^2 - 143*e + 82, 1/4*e^7 - 3/4*e^6 - 9/4*e^5 + 11/2*e^4 + 31/4*e^3 - 25/4*e^2 - 69/4*e + 4, 17/4*e^7 - 17/4*e^6 - 209/4*e^5 + 40*e^4 + 773/4*e^3 - 445/4*e^2 - 823/4*e + 93, 3*e^7 - 9/2*e^6 - 35*e^5 + 85/2*e^4 + 249/2*e^3 - 115*e^2 - 247/2*e + 96, 17/4*e^7 - 19/4*e^6 - 205/4*e^5 + 85/2*e^4 + 763/4*e^3 - 433/4*e^2 - 853/4*e + 87, -7/4*e^7 + 11/4*e^6 + 79/4*e^5 - 27*e^4 - 275/4*e^3 + 315/4*e^2 + 277/4*e - 54, -5/2*e^7 + 5*e^6 + 41/2*e^5 - 79/2*e^4 - 36*e^3 + 153/2*e^2 - 4*e - 26, 19/4*e^7 - 39/4*e^6 - 187/4*e^5 + 85*e^4 + 547/4*e^3 - 791/4*e^2 - 517/4*e + 118, -1/2*e^7 + 1/2*e^6 + 21/2*e^5 - 9*e^4 - 111/2*e^3 + 71/2*e^2 + 149/2*e - 36, 1/2*e^6 + 4*e^5 - 25/2*e^4 - 61/2*e^3 + 63*e^2 + 99/2*e - 70, 2*e^7 - 9/2*e^6 - 21*e^5 + 85/2*e^4 + 137/2*e^3 - 113*e^2 - 139/2*e + 85, -9/4*e^7 + 19/4*e^6 + 73/4*e^5 - 73/2*e^4 - 127/4*e^3 + 269/4*e^2 - 11/4*e - 22, 7/2*e^7 - 5*e^6 - 75/2*e^5 + 91/2*e^4 + 117*e^3 - 231/2*e^2 - 102*e + 74, 3/4*e^7 + 7/4*e^6 - 51/4*e^5 - 31/2*e^4 + 253/4*e^3 + 109/4*e^2 - 367/4*e + 4, -11/2*e^7 + 17/2*e^6 + 121/2*e^5 - 77*e^4 - 403/2*e^3 + 399/2*e^2 + 395/2*e - 158, -1/4*e^7 + 5/4*e^6 - 3/4*e^5 - 8*e^4 + 71/4*e^3 + 9/4*e^2 - 125/4*e + 32, 7/2*e^7 - 9/2*e^6 - 87/2*e^5 + 44*e^4 + 337/2*e^3 - 245/2*e^2 - 383/2*e + 100, 29/4*e^7 - 51/4*e^6 - 305/4*e^5 + 221/2*e^4 + 967/4*e^3 - 1041/4*e^2 - 953/4*e + 181, 1/4*e^7 + 9/4*e^6 - 37/4*e^5 - 23/2*e^4 + 231/4*e^3 - 81/4*e^2 - 361/4*e + 72, 4*e^7 - 11/2*e^6 - 44*e^5 + 93/2*e^4 + 291/2*e^3 - 105*e^2 - 269/2*e + 53, 11/4*e^7 - 19/4*e^6 - 123/4*e^5 + 47*e^4 + 407/4*e^3 - 519/4*e^2 - 373/4*e + 78, -5/2*e^7 + 3/2*e^6 + 71/2*e^5 - 19*e^4 - 309/2*e^3 + 149/2*e^2 + 381/2*e - 90, -3/2*e^7 + 3/2*e^6 + 35/2*e^5 - 10*e^4 - 131/2*e^3 + 33/2*e^2 + 157/2*e - 9, -7/2*e^7 + 5*e^6 + 81/2*e^5 - 97/2*e^4 - 145*e^3 + 285/2*e^2 + 163*e - 118, -3/4*e^7 - 7/4*e^6 + 39/4*e^5 + 41/2*e^4 - 157/4*e^3 - 217/4*e^2 + 187/4*e + 12, 3*e^7 - 6*e^6 - 33*e^5 + 59*e^4 + 110*e^3 - 161*e^2 - 110*e + 110, -6*e^7 + 11*e^6 + 62*e^5 - 95*e^4 - 189*e^3 + 216*e^2 + 175*e - 118, -1/2*e^7 + 5/2*e^6 + 3/2*e^5 - 21*e^4 + 23/2*e^3 + 91/2*e^2 - 47/2*e - 18, -1/4*e^7 - 1/4*e^6 + 5/4*e^5 + 11/2*e^4 + 29/4*e^3 - 103/4*e^2 - 115/4*e + 28, 11/4*e^7 - 25/4*e^6 - 103/4*e^5 + 105/2*e^4 + 289/4*e^3 - 455/4*e^2 - 279/4*e + 59, -29/4*e^7 + 43/4*e^6 + 341/4*e^5 - 201/2*e^4 - 1247/4*e^3 + 1073/4*e^2 + 1381/4*e - 215, -3*e^7 + 3*e^6 + 37*e^5 - 27*e^4 - 143*e^3 + 68*e^2 + 171*e - 54, -1/4*e^7 + 1/4*e^6 - 3/4*e^5 + 5*e^4 + 79/4*e^3 - 167/4*e^2 - 129/4*e + 53, 7/4*e^7 - 7/4*e^6 - 79/4*e^5 + 20*e^4 + 255/4*e^3 - 283/4*e^2 - 233/4*e + 62, -5/4*e^7 + 3/4*e^6 + 53/4*e^5 - 5/2*e^4 - 151/4*e^3 - 11/4*e^2 + 53/4*e - 4, -19/4*e^7 + 25/4*e^6 + 223/4*e^5 - 115/2*e^4 - 813/4*e^3 + 619/4*e^2 + 955/4*e - 133, 1/2*e^7 - 3/2*e^6 - 15/2*e^5 + 20*e^4 + 77/2*e^3 - 165/2*e^2 - 141/2*e + 85, 3*e^7 - 13/2*e^6 - 33*e^5 + 125/2*e^4 + 229/2*e^3 - 166*e^2 - 257/2*e + 124, 15/4*e^7 - 11/4*e^6 - 199/4*e^5 + 29*e^4 + 795/4*e^3 - 375/4*e^2 - 893/4*e + 95, 3*e^7 - 4*e^6 - 36*e^5 + 37*e^4 + 137*e^3 - 100*e^2 - 162*e + 91, -5*e^7 + 13/2*e^6 + 56*e^5 - 115/2*e^4 - 377/2*e^3 + 139*e^2 + 387/2*e - 94, -e^6 - e^5 + 17*e^4 + 7*e^3 - 70*e^2 - 16*e + 56, -3/4*e^7 + 13/4*e^6 - 1/4*e^5 - 49/2*e^4 + 163/4*e^3 + 119/4*e^2 - 325/4*e + 34, 8*e^7 - 27/2*e^6 - 82*e^5 + 227/2*e^4 + 485/2*e^3 - 243*e^2 - 411/2*e + 124, 17/4*e^7 - 37/4*e^6 - 161/4*e^5 + 78*e^4 + 429/4*e^3 - 649/4*e^2 - 347/4*e + 72, 11/4*e^7 - 13/4*e^6 - 131/4*e^5 + 63/2*e^4 + 457/4*e^3 - 363/4*e^2 - 443/4*e + 84, 11/2*e^7 - 13/2*e^6 - 133/2*e^5 + 58*e^4 + 499/2*e^3 - 295/2*e^2 - 575/2*e + 125, -5/4*e^7 + 21/4*e^6 + 17/4*e^5 - 37*e^4 + 99/4*e^3 + 177/4*e^2 - 265/4*e + 31, -2*e^7 + e^6 + 24*e^5 - 8*e^4 - 88*e^3 + 31*e^2 + 86*e - 68, -7/2*e^7 + 17/2*e^6 + 65/2*e^5 - 74*e^4 - 165/2*e^3 + 333/2*e^2 + 105/2*e - 86, -6*e^7 + 7*e^6 + 71*e^5 - 61*e^4 - 258*e^3 + 155*e^2 + 277*e - 121, -2*e^7 + 3*e^6 + 22*e^5 - 28*e^4 - 70*e^3 + 79*e^2 + 48*e - 70, -13/4*e^7 + 23/4*e^6 + 137/4*e^5 - 103/2*e^4 - 411/4*e^3 + 509/4*e^2 + 301/4*e - 90, -6*e^7 + 6*e^6 + 76*e^5 - 58*e^4 - 298*e^3 + 176*e^2 + 350*e - 174, -19/4*e^7 + 35/4*e^6 + 199/4*e^5 - 76*e^4 - 603/4*e^3 + 707/4*e^2 + 493/4*e - 118, -11/4*e^7 + 25/4*e^6 + 111/4*e^5 - 115/2*e^4 - 349/4*e^3 + 563/4*e^2 + 371/4*e - 69, 7/4*e^7 - 13/4*e^6 - 75/4*e^5 + 53/2*e^4 + 241/4*e^3 - 167/4*e^2 - 207/4*e - 2, -1/2*e^6 - 5/2*e^4 + 15/2*e^3 + 32*e^2 - 51/2*e - 25, -1/2*e^7 + 15/2*e^5 - 3/2*e^4 - 29*e^3 + 9/2*e^2 + 16*e - 7, -3*e^7 + 15/2*e^6 + 29*e^5 - 133/2*e^4 - 161/2*e^3 + 157*e^2 + 129/2*e - 98, 11/2*e^7 - 12*e^6 - 113/2*e^5 + 211/2*e^4 + 176*e^3 - 489/2*e^2 - 177*e + 153, -3/2*e^7 + 3/2*e^6 + 39/2*e^5 - 17*e^4 - 171/2*e^3 + 125/2*e^2 + 257/2*e - 62, 3/4*e^7 + 5/4*e^6 - 47/4*e^5 - 12*e^4 + 179/4*e^3 + 129/4*e^2 - 105/4*e - 24, -4*e^7 + 7/2*e^6 + 43*e^5 - 45/2*e^4 - 273/2*e^3 + 31*e^2 + 247/2*e - 20, -3/2*e^7 + 2*e^6 + 43/2*e^5 - 43/2*e^4 - 95*e^3 + 149/2*e^2 + 120*e - 100, -15/2*e^7 + 14*e^6 + 161/2*e^5 - 249/2*e^4 - 265*e^3 + 605/2*e^2 + 270*e - 196, e^7 - 15*e^5 + 5*e^4 + 66*e^3 - 44*e^2 - 82*e + 79, 9/4*e^7 - 15/4*e^6 - 93/4*e^5 + 57/2*e^4 + 275/4*e^3 - 149/4*e^2 - 245/4*e - 11, -1/2*e^7 + 5/2*e^6 + 9/2*e^5 - 20*e^4 - 27/2*e^3 + 65/2*e^2 + 25/2*e - 4, 3/4*e^7 - 9/4*e^6 - 35/4*e^5 + 31/2*e^4 + 165/4*e^3 - 71/4*e^2 - 295/4*e - 4, -5/4*e^7 + 7/4*e^6 + 49/4*e^5 - 29/2*e^4 - 131/4*e^3 + 129/4*e^2 + 61/4*e - 26, 9/4*e^7 - 29/4*e^6 - 73/4*e^5 + 59*e^4 + 153/4*e^3 - 409/4*e^2 - 111/4*e + 3, 2*e^6 - 6*e^5 - 16*e^4 + 44*e^3 + 26*e^2 - 56*e - 2, 1/4*e^7 - 7/4*e^6 - 9/4*e^5 + 31/2*e^4 + 27/4*e^3 - 121/4*e^2 - 57/4*e + 15, 17/4*e^7 - 19/4*e^6 - 189/4*e^5 + 71/2*e^4 + 619/4*e^3 - 249/4*e^2 - 541/4*e + 8, 4*e^7 - 17/2*e^6 - 38*e^5 + 151/2*e^4 + 199/2*e^3 - 185*e^2 - 155/2*e + 116, -6*e^7 + 21/2*e^6 + 69*e^5 - 193/2*e^4 - 495/2*e^3 + 245*e^2 + 555/2*e - 184, -3*e^7 + 13/2*e^6 + 24*e^5 - 103/2*e^4 - 73/2*e^3 + 95*e^2 - 35/2*e - 49, -15/4*e^7 + 23/4*e^6 + 147/4*e^5 - 43*e^4 - 407/4*e^3 + 291/4*e^2 + 309/4*e - 17, 4*e^7 - 8*e^6 - 35*e^5 + 59*e^4 + 80*e^3 - 92*e^2 - 47*e + 1, 5/4*e^7 - 7/4*e^6 - 73/4*e^5 + 39/2*e^4 + 327/4*e^3 - 253/4*e^2 - 409/4*e + 60, 5*e^6 - 9*e^5 - 43*e^4 + 63*e^3 + 86*e^2 - 80*e - 6, -2*e^7 + 2*e^6 + 25*e^5 - 17*e^4 - 95*e^3 + 35*e^2 + 106*e - 14, -3/2*e^7 + 3/2*e^6 + 39/2*e^5 - 12*e^4 - 151/2*e^3 + 53/2*e^2 + 153/2*e - 37, 1/2*e^7 - e^6 - 11/2*e^5 + 17/2*e^4 + 14*e^3 - 27/2*e^2 + 4*e - 15, 3/2*e^7 + e^6 - 45/2*e^5 - 15/2*e^4 + 100*e^3 - 5/2*e^2 - 120*e + 46, -5/4*e^7 - 3/4*e^6 + 69/4*e^5 + 5*e^4 - 269/4*e^3 + 37/4*e^2 + 255/4*e - 47, 19/4*e^7 - 31/4*e^6 - 207/4*e^5 + 69*e^4 + 675/4*e^3 - 711/4*e^2 - 597/4*e + 139, -9/2*e^7 + 9*e^6 + 93/2*e^5 - 167/2*e^4 - 136*e^3 + 411/2*e^2 + 99*e - 124, -17/2*e^7 + 31/2*e^6 + 183/2*e^5 - 137*e^4 - 607/2*e^3 + 657/2*e^2 + 627/2*e - 230, -7*e^7 + 10*e^6 + 77*e^5 - 90*e^4 - 252*e^3 + 228*e^2 + 236*e - 174, -3*e^7 + 3*e^6 + 38*e^5 - 24*e^4 - 155*e^3 + 55*e^2 + 196*e - 62, -11/2*e^7 + 19/2*e^6 + 127/2*e^5 - 85*e^4 - 467/2*e^3 + 409/2*e^2 + 537/2*e - 134, 1/4*e^7 + 1/4*e^6 - 17/4*e^5 - 1/2*e^4 + 83/4*e^3 - 17/4*e^2 - 121/4*e - 5, -13/4*e^7 + 11/4*e^6 + 157/4*e^5 - 45/2*e^4 - 575/4*e^3 + 209/4*e^2 + 541/4*e - 29, 8*e^7 - 14*e^6 - 85*e^5 + 123*e^4 + 275*e^3 - 294*e^2 - 280*e + 191, 15/4*e^7 - 17/4*e^6 - 159/4*e^5 + 61/2*e^4 + 481/4*e^3 - 203/4*e^2 - 363/4*e + 10, -1/4*e^7 + 7/4*e^6 - 11/4*e^5 - 27/2*e^4 + 145/4*e^3 + 81/4*e^2 - 279/4*e - 5, -3/4*e^7 - 7/4*e^6 + 63/4*e^5 + 21/2*e^4 - 325/4*e^3 - 5/4*e^2 + 411/4*e - 31, -3*e^7 + 5*e^6 + 31*e^5 - 40*e^4 - 93*e^3 + 71*e^2 + 69*e - 16, 3/2*e^7 - 1/2*e^6 - 35/2*e^5 + 3*e^4 + 109/2*e^3 - 13/2*e^2 - 63/2*e + 35, -7/4*e^7 + 13/4*e^6 + 51/4*e^5 - 45/2*e^4 - 29/4*e^3 + 79/4*e^2 - 165/4*e + 44, 3/4*e^7 - 13/4*e^6 - 11/4*e^5 + 53/2*e^4 - 59/4*e^3 - 215/4*e^2 + 161/4*e - 4, -11/2*e^7 + 17/2*e^6 + 123/2*e^5 - 79*e^4 - 415/2*e^3 + 413/2*e^2 + 397/2*e - 149, 11/4*e^7 - 21/4*e^6 - 111/4*e^5 + 85/2*e^4 + 321/4*e^3 - 291/4*e^2 - 271/4*e + 7, -19/2*e^7 + 35/2*e^6 + 191/2*e^5 - 149*e^4 - 557/2*e^3 + 669/2*e^2 + 455/2*e - 196];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, -w^3 + 6*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;