/*
  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![11, 0, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {49, 7, -w^3 + w^2 + 4*w - 1}>;

primesArray := [
[4, 2, -w^2 + w + 3],
[5, 5, w + 1],
[5, 5, -w^3 + w^2 + 4*w - 4],
[11, 11, w],
[29, 29, w^3 - 2*w^2 - 3*w + 7],
[29, 29, -w^3 - 2*w^2 + 3*w + 7],
[31, 31, -w^3 + w^2 + 4*w - 2],
[31, 31, -w^3 - w^2 + 4*w + 2],
[41, 41, w^3 + 2*w^2 - 4*w - 6],
[41, 41, w^3 - 5*w + 2],
[49, 7, -w^3 + w^2 + 4*w - 1],
[49, 7, w^3 + w^2 - 4*w - 1],
[59, 59, -3*w^2 - w + 10],
[59, 59, -3*w^2 + w + 10],
[61, 61, -2*w^3 + 2*w^2 + 7*w - 5],
[61, 61, 2*w^3 + w^2 - 8*w - 6],
[71, 71, 2*w^2 + w - 9],
[71, 71, 2*w^2 - w - 9],
[81, 3, -3],
[101, 101, 2*w^2 - w - 10],
[101, 101, 2*w^2 + w - 10],
[109, 109, 2*w^3 + w^2 - 8*w - 3],
[109, 109, -2*w^3 + w^2 + 8*w - 3],
[121, 11, -w^2 + 7],
[131, 131, 4*w^2 - 2*w - 15],
[131, 131, 2*w^2 + 2*w - 3],
[131, 131, 2*w^2 - 2*w - 3],
[131, 131, -w^3 - w^2 + 3*w + 7],
[139, 139, w^3 + 4*w^2 - 5*w - 16],
[139, 139, w^3 + 3*w^2 - 6*w - 10],
[139, 139, 2*w^3 - w^2 - 8*w + 1],
[139, 139, -4*w^2 - w + 12],
[149, 149, 2*w^3 - w^2 - 8*w + 4],
[149, 149, 2*w^3 + w^2 - 8*w - 4],
[151, 151, 3*w^2 - 3*w - 8],
[151, 151, 2*w^3 - 4*w^2 - 8*w + 17],
[151, 151, w^2 - 2*w - 7],
[151, 151, w^3 + 2*w^2 - 6*w - 8],
[179, 179, w^3 - w^2 - 5*w + 1],
[179, 179, -w^3 - w^2 + 5*w + 1],
[191, 191, 2*w^3 - w^2 - 7*w + 2],
[191, 191, 3*w^3 - 2*w^2 - 10*w + 2],
[199, 199, w^3 - 4*w^2 - 3*w + 13],
[199, 199, -w^3 - 4*w^2 + 3*w + 13],
[211, 211, -w^3 + w^2 + 6*w - 5],
[211, 211, w^3 - 5*w - 5],
[211, 211, -w^3 + 5*w - 5],
[211, 211, w^3 + w^2 - 6*w - 5],
[229, 229, w^3 - 4*w^2 - 2*w + 14],
[229, 229, 2*w^3 + 3*w^2 - 7*w - 12],
[229, 229, 2*w^3 - 3*w^2 - 7*w + 12],
[229, 229, 2*w^3 + w^2 - 7*w - 8],
[241, 241, w^3 + w^2 - 6*w - 4],
[241, 241, -w^3 + w^2 + 6*w - 4],
[251, 251, 2*w^2 - w - 2],
[251, 251, 2*w^2 + w - 2],
[271, 271, 2*w^3 - 2*w^2 - 8*w + 5],
[271, 271, -4*w^2 - w + 14],
[271, 271, -4*w^2 + w + 14],
[271, 271, 2*w^3 + 2*w^2 - 8*w - 5],
[281, 281, -2*w^3 - w^2 + 9*w + 2],
[281, 281, 2*w^3 - w^2 - 9*w + 2],
[311, 311, -w^3 + 3*w - 5],
[311, 311, w^3 - 3*w - 5],
[331, 331, -3*w^3 + 3*w^2 + 12*w - 13],
[331, 331, 3*w^3 + 3*w^2 - 12*w - 13],
[349, 349, 2*w^3 + 2*w^2 - 7*w - 10],
[349, 349, -2*w^3 + 2*w^2 + 7*w - 10],
[361, 19, 4*w^2 - 15],
[361, 19, -4*w^2 + 13],
[379, 379, -w^3 + 3*w^2 + 3*w - 13],
[379, 379, w^3 + 3*w^2 - 3*w - 13],
[409, 409, 3*w^3 - 2*w^2 - 11*w + 2],
[409, 409, w^3 + 3*w^2 - 6*w - 9],
[419, 419, -w^3 + w^2 + 6*w + 2],
[419, 419, -3*w^3 + 5*w^2 + 13*w - 19],
[421, 421, -w^3 + 3*w^2 + 7*w - 13],
[421, 421, 2*w^3 - 2*w^2 - 9*w + 6],
[421, 421, -2*w^3 - 2*w^2 + 9*w + 6],
[421, 421, w^3 + 3*w^2 - 7*w - 13],
[439, 439, 3*w^2 + 2*w - 13],
[439, 439, 3*w^3 + 2*w^2 - 12*w - 10],
[439, 439, -3*w^3 + 2*w^2 + 12*w - 10],
[439, 439, 3*w^2 - 2*w - 13],
[449, 449, 2*w^3 - 4*w^2 - 9*w + 14],
[449, 449, -3*w^3 + 3*w^2 + 11*w - 9],
[449, 449, 3*w^3 + 3*w^2 - 11*w - 9],
[449, 449, w^3 - 2*w^2 - 5*w + 2],
[461, 461, -w - 5],
[461, 461, w - 5],
[499, 499, w^3 - 5*w^2 - w + 17],
[499, 499, -w^3 - 5*w^2 + w + 17],
[509, 509, w^3 + 5*w^2 - 4*w - 16],
[509, 509, w^2 - 2*w - 10],
[509, 509, w^2 + 2*w - 10],
[509, 509, w^3 - w^2 - 5*w + 9],
[541, 541, 3*w^3 - 10*w - 6],
[541, 541, -2*w^3 + 5*w^2 + 5*w - 16],
[569, 569, 3*w^3 + 2*w^2 - 13*w - 6],
[569, 569, -2*w^3 + 4*w^2 + 7*w - 10],
[599, 599, w^3 + 2*w^2 - 7*w - 10],
[599, 599, -w^3 + 2*w^2 + 7*w - 10],
[601, 601, -3*w^3 - w^2 + 12*w + 6],
[601, 601, 3*w^3 - w^2 - 12*w + 6],
[619, 619, 3*w^3 - 12*w + 2],
[619, 619, -3*w^3 + 12*w + 2],
[631, 631, -w^3 + w^2 + 2*w - 8],
[631, 631, w^3 + w^2 - 2*w - 8],
[641, 641, w^3 + w^2 - 7*w - 1],
[641, 641, -3*w^3 - 2*w^2 + 12*w + 4],
[641, 641, 3*w^3 - 2*w^2 - 12*w + 4],
[641, 641, w^3 - w^2 - 7*w + 1],
[659, 659, -2*w^3 + w^2 + 10*w - 1],
[659, 659, -w^3 + 5*w^2 + 4*w - 19],
[659, 659, w^3 + 5*w^2 - 4*w - 19],
[659, 659, 2*w^3 + w^2 - 10*w - 1],
[691, 691, 3*w^2 - 2*w - 14],
[691, 691, 3*w^2 + 2*w - 14],
[701, 701, w^3 + 4*w^2 - 7*w - 12],
[701, 701, 3*w^3 - w^2 - 12*w - 1],
[719, 719, -w^3 + 7*w - 3],
[719, 719, w^3 - 7*w - 3],
[739, 739, 3*w^3 + w^2 - 12*w - 3],
[739, 739, w^3 + 2*w^2 - 3*w - 1],
[739, 739, -w^3 + 2*w^2 + 3*w - 1],
[739, 739, -3*w^3 + w^2 + 12*w - 3],
[751, 751, -2*w^3 + 6*w^2 + 7*w - 20],
[751, 751, 2*w^3 + 6*w^2 - 7*w - 20],
[761, 761, 2*w^3 + 6*w^2 - 8*w - 21],
[761, 761, -2*w^3 + 6*w^2 + 8*w - 21],
[769, 769, -2*w^3 + 6*w^2 + 8*w - 23],
[769, 769, w^2 + 2*w + 3],
[809, 809, 2*w^3 + 3*w^2 - 7*w - 14],
[809, 809, -2*w^3 + 3*w^2 + 7*w - 14],
[821, 821, -3*w^3 - w^2 + 11*w + 1],
[821, 821, 3*w^3 - w^2 - 11*w + 1],
[829, 829, 3*w^3 - w^2 - 12*w + 4],
[829, 829, w^3 - 4*w^2 - 6*w + 12],
[829, 829, -w^3 - 4*w^2 + 6*w + 12],
[829, 829, -3*w^3 - w^2 + 12*w + 4],
[839, 839, -w^3 + 3*w - 6],
[839, 839, w^3 - 3*w - 6],
[841, 29, 5*w^2 - 19],
[859, 859, w^3 + 2*w^2 - 7*w - 9],
[859, 859, -w^3 + 2*w^2 + 7*w - 9],
[911, 911, -2*w^3 + 4*w^2 + 9*w - 13],
[911, 911, 2*w^3 + 4*w^2 - 9*w - 13],
[919, 919, 2*w^3 + 4*w^2 - 7*w - 17],
[919, 919, 3*w^3 + 2*w^2 - 11*w - 6],
[919, 919, 3*w^3 - 2*w^2 - 11*w + 6],
[919, 919, 2*w^3 - 4*w^2 - 7*w + 17],
[941, 941, -2*w^3 + 2*w^2 + 7*w - 15],
[941, 941, 2*w^3 + 2*w^2 - 7*w - 15],
[961, 31, -2*w^2 + 13],
[971, 971, w^3 + 2*w^2 - 7*w - 7],
[971, 971, -w^3 + 2*w^2 + 7*w - 7],
[991, 991, 3*w^3 + 2*w^2 - 11*w - 7],
[991, 991, -3*w^3 + 2*w^2 + 11*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 6*x^5 + 3*x^4 - 34*x^3 - 41*x^2 + 30*x + 25;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 2/5*e^5 + 7/5*e^4 - 14/5*e^3 - 43/5*e^2 + 28/5*e + 5, -4/5*e^5 - 14/5*e^4 + 23/5*e^3 + 76/5*e^2 - 36/5*e - 8, 4/5*e^5 + 14/5*e^4 - 18/5*e^3 - 71/5*e^2 + 6/5*e + 5, 1/5*e^5 + 6/5*e^4 + 3/5*e^3 - 34/5*e^2 - 26/5*e + 8, 7/5*e^5 + 22/5*e^4 - 44/5*e^3 - 128/5*e^2 + 53/5*e + 16, -6/5*e^5 - 21/5*e^4 + 32/5*e^3 + 114/5*e^2 - 44/5*e - 15, 7/5*e^5 + 22/5*e^4 - 44/5*e^3 - 118/5*e^2 + 68/5*e + 4, -4/5*e^5 - 14/5*e^4 + 23/5*e^3 + 86/5*e^2 - 16/5*e - 14, 2/5*e^5 + 7/5*e^4 - 14/5*e^3 - 48/5*e^2 + 18/5*e + 4, 1, 6/5*e^5 + 21/5*e^4 - 32/5*e^3 - 104/5*e^2 + 44/5*e + 3, 2/5*e^5 + 2/5*e^4 - 24/5*e^3 - 3/5*e^2 + 88/5*e + 1, 2/5*e^5 + 12/5*e^4 + 1/5*e^3 - 68/5*e^2 - 47/5*e + 6, -2/5*e^5 - 7/5*e^4 + 4/5*e^3 + 33/5*e^2 + 22/5*e - 8, -9/5*e^5 - 24/5*e^4 + 68/5*e^3 + 136/5*e^2 - 131/5*e - 14, -11/5*e^5 - 36/5*e^4 + 77/5*e^3 + 214/5*e^2 - 154/5*e - 26, -11/5*e^5 - 36/5*e^4 + 57/5*e^3 + 184/5*e^2 - 54/5*e - 16, 6/5*e^5 + 26/5*e^4 - 22/5*e^3 - 139/5*e^2 + 14/5*e + 15, 6/5*e^5 + 21/5*e^4 - 32/5*e^3 - 99/5*e^2 + 54/5*e, 2*e^5 + 6*e^4 - 14*e^3 - 34*e^2 + 27*e + 12, 2*e^5 + 5*e^4 - 16*e^3 - 30*e^2 + 30*e + 20, -14/5*e^5 - 44/5*e^4 + 88/5*e^3 + 251/5*e^2 - 106/5*e - 27, 4/5*e^5 + 9/5*e^4 - 38/5*e^3 - 56/5*e^2 + 86/5*e - 1, 6/5*e^5 + 26/5*e^4 - 17/5*e^3 - 154/5*e^2 - 66/5*e + 20, -16/5*e^5 - 51/5*e^4 + 102/5*e^3 + 289/5*e^2 - 164/5*e - 35, 11/5*e^5 + 36/5*e^4 - 72/5*e^3 - 224/5*e^2 + 109/5*e + 36, 22/5*e^5 + 72/5*e^4 - 124/5*e^3 - 393/5*e^2 + 128/5*e + 38, 6/5*e^5 + 11/5*e^4 - 62/5*e^3 - 69/5*e^2 + 174/5*e + 8, 8/5*e^5 + 33/5*e^4 - 26/5*e^3 - 187/5*e^2 - 48/5*e + 19, -28/5*e^5 - 98/5*e^4 + 151/5*e^3 + 552/5*e^2 - 137/5*e - 64, 8/5*e^5 + 28/5*e^4 - 56/5*e^3 - 172/5*e^2 + 122/5*e + 24, -19/5*e^5 - 64/5*e^4 + 113/5*e^3 + 366/5*e^2 - 176/5*e - 52, 2/5*e^5 + 12/5*e^4 + 1/5*e^3 - 78/5*e^2 - 37/5*e + 16, 2*e^5 + 7*e^4 - 12*e^3 - 36*e^2 + 24*e + 3, -14/5*e^5 - 54/5*e^4 + 58/5*e^3 + 286/5*e^2 - 21/5*e - 30, 2*e^5 + 8*e^4 - 9*e^3 - 48*e^2 + e + 32, -2*e^4 - 6*e^3 + 11*e^2 + 30*e - 7, 8/5*e^5 + 38/5*e^4 - 26/5*e^3 - 237/5*e^2 - 48/5*e + 34, 18/5*e^5 + 63/5*e^4 - 86/5*e^3 - 342/5*e^2 + 32/5*e + 34, 4/5*e^5 + 4/5*e^4 - 43/5*e^3 - 6/5*e^2 + 116/5*e - 10, 19/5*e^5 + 74/5*e^4 - 83/5*e^3 - 406/5*e^2 + 36/5*e + 40, 7/5*e^5 + 22/5*e^4 - 39/5*e^3 - 98/5*e^2 + 38/5*e - 4, 4/5*e^5 + 24/5*e^4 - 8/5*e^3 - 146/5*e^2 - 4/5*e + 22, 19/5*e^5 + 54/5*e^4 - 143/5*e^3 - 306/5*e^2 + 276/5*e + 20, -21/5*e^5 - 66/5*e^4 + 132/5*e^3 + 374/5*e^2 - 194/5*e - 46, -2*e^5 - 8*e^4 + 7*e^3 + 40*e^2 + 7*e - 18, -9/5*e^5 - 24/5*e^4 + 78/5*e^3 + 156/5*e^2 - 186/5*e - 14, 1/5*e^5 + 6/5*e^4 + 13/5*e^3 - 14/5*e^2 - 61/5*e - 2, 8/5*e^5 + 28/5*e^4 - 46/5*e^3 - 162/5*e^2 + 32/5*e + 14, -7/5*e^5 - 22/5*e^4 + 49/5*e^3 + 108/5*e^2 - 128/5*e - 6, -12/5*e^5 - 47/5*e^4 + 54/5*e^3 + 253/5*e^2 - 18/5*e - 16, -2/5*e^5 - 2/5*e^4 + 24/5*e^3 - 2/5*e^2 - 103/5*e + 12, 12/5*e^5 + 32/5*e^4 - 84/5*e^3 - 178/5*e^2 + 138/5*e + 24, 1/5*e^5 + 6/5*e^4 - 17/5*e^3 - 64/5*e^2 + 64/5*e + 20, -34/5*e^5 - 119/5*e^4 + 188/5*e^3 + 656/5*e^2 - 216/5*e - 65, -22/5*e^5 - 82/5*e^4 + 114/5*e^3 + 468/5*e^2 - 78/5*e - 53, 2*e^5 + 6*e^4 - 13*e^3 - 28*e^2 + 24*e - 8, 2*e^5 + 7*e^4 - 10*e^3 - 36*e^2 + 14*e + 12, 2*e^5 + 6*e^4 - 12*e^3 - 30*e^2 + 16*e + 8, -37/5*e^5 - 132/5*e^4 + 199/5*e^3 + 738/5*e^2 - 208/5*e - 78, -14/5*e^5 - 54/5*e^4 + 83/5*e^3 + 336/5*e^2 - 111/5*e - 54, 3/5*e^5 + 8/5*e^4 - 21/5*e^3 - 52/5*e^2 + 22/5*e + 16, -14/5*e^5 - 54/5*e^4 + 58/5*e^3 + 281/5*e^2 - 6/5*e - 20, -4*e^5 - 14*e^4 + 21*e^3 + 72*e^2 - 25*e - 22, -3/5*e^5 - 18/5*e^4 - 9/5*e^3 + 72/5*e^2 + 63/5*e - 6, 1/5*e^5 - 4/5*e^4 - 37/5*e^3 - 4/5*e^2 + 184/5*e + 8, 2/5*e^5 - 3/5*e^4 - 24/5*e^3 + 37/5*e^2 + 98/5*e + 1, -27/5*e^5 - 102/5*e^4 + 139/5*e^3 + 578/5*e^2 - 168/5*e - 84, e^5 + 4*e^4 - 2*e^3 - 22*e^2 - 12*e + 22, -17/5*e^5 - 42/5*e^4 + 134/5*e^3 + 238/5*e^2 - 263/5*e - 26, 13/5*e^5 + 48/5*e^4 - 81/5*e^3 - 312/5*e^2 + 112/5*e + 54, 11/5*e^5 + 36/5*e^4 - 62/5*e^3 - 184/5*e^2 + 69/5*e - 2, 32/5*e^5 + 112/5*e^4 - 184/5*e^3 - 613/5*e^2 + 248/5*e + 41, -32/5*e^5 - 112/5*e^4 + 159/5*e^3 + 578/5*e^2 - 143/5*e - 36, 32/5*e^5 + 107/5*e^4 - 174/5*e^3 - 593/5*e^2 + 158/5*e + 76, -2/5*e^5 + 3/5*e^4 + 24/5*e^3 - 32/5*e^2 - 88/5*e - 13, 3*e^5 + 10*e^4 - 18*e^3 - 52*e^2 + 24*e + 8, 18/5*e^5 + 48/5*e^4 - 146/5*e^3 - 272/5*e^2 + 312/5*e + 22, -4/5*e^5 + 1/5*e^4 + 68/5*e^3 - 14/5*e^2 - 246/5*e + 11, 12/5*e^5 + 37/5*e^4 - 84/5*e^3 - 243/5*e^2 + 98/5*e + 41, -3*e^4 - 10*e^3 + 16*e^2 + 42*e - 5, -27/5*e^5 - 102/5*e^4 + 149/5*e^3 + 618/5*e^2 - 158/5*e - 86, 3/5*e^5 + 18/5*e^4 - 11/5*e^3 - 142/5*e^2 + 47/5*e + 54, 4/5*e^5 + 14/5*e^4 - 23/5*e^3 - 116/5*e^2 + 31/5*e + 52, -22/5*e^5 - 82/5*e^4 + 114/5*e^3 + 473/5*e^2 - 148/5*e - 76, 18/5*e^5 + 58/5*e^4 - 116/5*e^3 - 337/5*e^2 + 212/5*e + 39, -13/5*e^5 - 48/5*e^4 + 66/5*e^3 + 272/5*e^2 - 52/5*e - 44, -17/5*e^5 - 72/5*e^4 + 64/5*e^3 + 378/5*e^2 - 48/5*e - 34, 16/5*e^5 + 66/5*e^4 - 62/5*e^3 - 344/5*e^2 + 14/5*e + 30, -38/5*e^5 - 118/5*e^4 + 266/5*e^3 + 692/5*e^2 - 472/5*e - 74, -8*e^5 - 30*e^4 + 37*e^3 + 164*e^2 - 12*e - 90, -32/5*e^5 - 97/5*e^4 + 224/5*e^3 + 543/5*e^2 - 428/5*e - 36, 58/5*e^5 + 198/5*e^4 - 316/5*e^3 - 1062/5*e^2 + 352/5*e + 84, -2*e^5 - 6*e^4 + 20*e^3 + 37*e^2 - 66*e - 10, -21/5*e^5 - 76/5*e^4 + 117/5*e^3 + 434/5*e^2 - 119/5*e - 48, 16/5*e^5 + 56/5*e^4 - 92/5*e^3 - 304/5*e^2 + 134/5*e + 30, -2*e^5 - 6*e^4 + 15*e^3 + 42*e^2 - 21*e - 58, -34/5*e^5 - 129/5*e^4 + 158/5*e^3 + 741/5*e^2 - 76/5*e - 97, 2*e^5 + 8*e^4 - 4*e^3 - 36*e^2 - 16*e, 9*e^5 + 30*e^4 - 55*e^3 - 174*e^2 + 85*e + 120, -7/5*e^5 - 12/5*e^4 + 64/5*e^3 + 88/5*e^2 - 78/5*e - 26, 10*e^5 + 37*e^4 - 52*e^3 - 213*e^2 + 46*e + 133, 31/5*e^5 + 106/5*e^4 - 167/5*e^3 - 554/5*e^2 + 209/5*e + 46, 19/5*e^5 + 74/5*e^4 - 83/5*e^3 - 406/5*e^2 + 16/5*e + 32, 44/5*e^5 + 149/5*e^4 - 268/5*e^3 - 851/5*e^2 + 406/5*e + 107, 28/5*e^5 + 98/5*e^4 - 146/5*e^3 - 517/5*e^2 + 152/5*e + 21, -42/5*e^5 - 147/5*e^4 + 234/5*e^3 + 818/5*e^2 - 268/5*e - 84, 2*e^5 + 7*e^4 - 12*e^3 - 46*e^2 + 18*e + 68, -42/5*e^5 - 132/5*e^4 + 279/5*e^3 + 718/5*e^2 - 573/5*e - 68, 3*e^5 + 10*e^4 - 19*e^3 - 58*e^2 + 30*e + 18, e^4 + 6*e^3 + 2*e^2 - 28*e - 27, 16/5*e^5 + 46/5*e^4 - 102/5*e^3 - 264/5*e^2 + 134/5*e + 58, -14/5*e^5 - 54/5*e^4 + 78/5*e^3 + 321/5*e^2 - 136/5*e - 62, -8/5*e^5 - 13/5*e^4 + 66/5*e^3 + 22/5*e^2 - 212/5*e + 21, -27/5*e^5 - 92/5*e^4 + 164/5*e^3 + 538/5*e^2 - 243/5*e - 66, 3/5*e^5 + 18/5*e^4 - 16/5*e^3 - 142/5*e^2 - 33/5*e + 26, -18/5*e^5 - 58/5*e^4 + 106/5*e^3 + 307/5*e^2 - 132/5*e - 47, -2/5*e^5 - 7/5*e^4 + 4/5*e^3 + 48/5*e^2 + 62/5*e - 23, -22/5*e^5 - 57/5*e^4 + 174/5*e^3 + 348/5*e^2 - 318/5*e - 53, 24/5*e^5 + 104/5*e^4 - 98/5*e^3 - 606/5*e^2 + 46/5*e + 92, 3/5*e^5 + 8/5*e^4 - 41/5*e^3 - 72/5*e^2 + 177/5*e + 4, 4*e^5 + 14*e^4 - 19*e^3 - 74*e^2 + 22*e + 50, 19/5*e^5 + 74/5*e^4 - 73/5*e^3 - 376/5*e^2 + 11/5*e + 22, 18/5*e^5 + 58/5*e^4 - 111/5*e^3 - 312/5*e^2 + 197/5*e + 14, 31/5*e^5 + 106/5*e^4 - 157/5*e^3 - 564/5*e^2 + 109/5*e + 38, 26/5*e^5 + 91/5*e^4 - 142/5*e^3 - 494/5*e^2 + 174/5*e + 45, 14/5*e^5 + 49/5*e^4 - 68/5*e^3 - 256/5*e^2 - 24/5*e - 1, 2/5*e^5 + 22/5*e^4 + 41/5*e^3 - 148/5*e^2 - 267/5*e + 28, -14/5*e^5 - 54/5*e^4 + 98/5*e^3 + 346/5*e^2 - 206/5*e - 45, -3*e^5 - 10*e^4 + 23*e^3 + 56*e^2 - 66*e - 30, 8/5*e^5 + 28/5*e^4 - 36/5*e^3 - 152/5*e^2 + 22/5*e + 14, -22/5*e^5 - 57/5*e^4 + 144/5*e^3 + 263/5*e^2 - 208/5*e + 4, 18/5*e^5 + 43/5*e^4 - 166/5*e^3 - 262/5*e^2 + 392/5*e + 19, -2*e^5 - 4*e^4 + 12*e^3 + 14*e^2 - 5*e - 2, 16/5*e^5 + 56/5*e^4 - 92/5*e^3 - 334/5*e^2 + 84/5*e + 56, -8*e^5 - 27*e^4 + 44*e^3 + 140*e^2 - 66*e - 55, 9/5*e^5 + 34/5*e^4 - 28/5*e^3 - 196/5*e^2 - 74/5*e + 32, 16/5*e^5 + 51/5*e^4 - 102/5*e^3 - 259/5*e^2 + 224/5*e + 3, 48/5*e^5 + 178/5*e^4 - 241/5*e^3 - 1012/5*e^2 + 167/5*e + 124, 6*e^5 + 18*e^4 - 46*e^3 - 108*e^2 + 101*e + 70, 36/5*e^5 + 116/5*e^4 - 252/5*e^3 - 659/5*e^2 + 514/5*e + 48, 2/5*e^5 + 2/5*e^4 - 24/5*e^3 - 18/5*e^2 + 38/5*e + 3, -26/5*e^5 - 81/5*e^4 + 192/5*e^3 + 494/5*e^2 - 424/5*e - 63, 6*e^5 + 22*e^4 - 31*e^3 - 126*e^2 + 32*e + 70, 10*e^5 + 34*e^4 - 56*e^3 - 184*e^2 + 70*e + 78, -e^5 + 12*e^3 - 8*e^2 - 29*e + 38, -38/5*e^5 - 133/5*e^4 + 196/5*e^3 + 752/5*e^2 - 122/5*e - 99, -48/5*e^5 - 163/5*e^4 + 236/5*e^3 + 832/5*e^2 - 162/5*e - 54, 14/5*e^5 + 39/5*e^4 - 88/5*e^3 - 216/5*e^2 + 126/5*e + 57, -64/5*e^5 - 224/5*e^4 + 318/5*e^3 + 1186/5*e^2 - 266/5*e - 102, 7*e^5 + 22*e^4 - 48*e^3 - 132*e^2 + 84*e + 92, 6/5*e^5 + 16/5*e^4 - 32/5*e^3 - 54/5*e^2 + 4/5*e - 40, 8/5*e^5 + 18/5*e^4 - 96/5*e^3 - 132/5*e^2 + 307/5*e + 6, 18/5*e^5 + 53/5*e^4 - 116/5*e^3 - 272/5*e^2 + 142/5*e + 17, 19/5*e^5 + 54/5*e^4 - 113/5*e^3 - 236/5*e^2 + 146/5*e, -8*e^5 - 30*e^4 + 40*e^3 + 173*e^2 - 20*e - 112, -2*e^5 - 11*e^4 + 6*e^3 + 70*e^2 - 8*e - 87];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {49, 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;