/*
  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![2, 0, -6, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {1, 1, 1}>;

primesArray := [
[2, 2, w],
[5, 5, -w^3 + 3*w^2 + 2*w - 1],
[5, 5, w - 1],
[11, 11, w^3 - 2*w^2 - 5*w - 1],
[13, 13, -w^3 + 3*w^2 + 3*w - 1],
[17, 17, 2*w^3 - 5*w^2 - 9*w + 5],
[23, 23, -w^3 + 3*w^2 + 4*w - 5],
[25, 5, -w^2 + 3*w + 1],
[29, 29, -2*w^3 + 6*w^2 + 5*w - 1],
[31, 31, -w^2 + 2*w + 1],
[43, 43, -w^3 + 3*w^2 + 2*w - 3],
[43, 43, -w^3 + 2*w^2 + 6*w - 3],
[59, 59, 2*w^3 - 6*w^2 - 7*w + 7],
[73, 73, 3*w^3 - 6*w^2 - 16*w - 5],
[79, 79, -5*w^3 + 14*w^2 + 20*w - 17],
[81, 3, -3],
[83, 83, -w - 3],
[83, 83, w^2 - 3*w - 7],
[89, 89, w^2 - 2*w - 7],
[101, 101, -2*w^3 + 5*w^2 + 8*w - 3],
[101, 101, 5*w^3 - 14*w^2 - 19*w + 17],
[103, 103, -w^3 + w^2 + 8*w + 3],
[103, 103, 2*w^3 - 5*w^2 - 7*w - 1],
[109, 109, 3*w^3 - 7*w^2 - 14*w + 1],
[109, 109, -w^2 + 4*w + 1],
[127, 127, -3*w^3 + 8*w^2 + 11*w - 7],
[127, 127, -w^3 + 2*w^2 + 7*w + 1],
[137, 137, -2*w^3 + 5*w^2 + 9*w - 1],
[139, 139, -w^3 + 3*w^2 + 2*w - 5],
[149, 149, w^3 - 3*w^2 - w - 1],
[149, 149, -2*w^3 + 4*w^2 + 10*w + 1],
[163, 163, w^3 - 4*w^2 - w + 11],
[163, 163, 2*w^3 - 4*w^2 - 10*w + 1],
[173, 173, -4*w^3 + 11*w^2 + 15*w - 13],
[179, 179, 2*w^2 - 6*w - 3],
[179, 179, -w^3 + 2*w^2 + 7*w - 1],
[181, 181, w^3 - 10*w - 9],
[181, 181, -3*w^3 + 8*w^2 + 14*w - 7],
[199, 199, w^3 - 8*w - 9],
[211, 211, 2*w - 3],
[223, 223, -w^3 + 4*w^2 + w - 5],
[227, 227, w^3 - 2*w^2 - 5*w - 5],
[227, 227, -3*w^3 + 7*w^2 + 14*w - 5],
[227, 227, -4*w^3 + 10*w^2 + 19*w - 7],
[227, 227, w^2 - 4*w - 5],
[239, 239, -2*w^3 + 5*w^2 + 7*w - 3],
[251, 251, -w^3 + 4*w^2 + 2*w - 11],
[251, 251, 2*w^3 - 6*w^2 - 6*w + 1],
[257, 257, -3*w^3 + 6*w^2 + 16*w + 3],
[263, 263, -w^3 + 3*w^2 + 3*w - 7],
[263, 263, w^3 - 4*w^2 + 7],
[269, 269, -2*w^3 + 6*w^2 + 6*w - 11],
[269, 269, -w^3 + 4*w^2 - w - 5],
[271, 271, -3*w^3 + 9*w^2 + 11*w - 11],
[271, 271, w^2 - 5*w - 3],
[277, 277, w^3 - 4*w^2 - 2*w + 7],
[277, 277, -2*w^3 + 7*w^2 + 8*w - 9],
[281, 281, 2*w^2 - 4*w - 5],
[281, 281, 2*w^3 - 4*w^2 - 11*w + 1],
[283, 283, -3*w^3 + 8*w^2 + 14*w - 11],
[293, 293, w^3 - 2*w^2 - 6*w + 5],
[307, 307, -2*w^3 + 5*w^2 + 7*w - 1],
[313, 313, 2*w^3 - 4*w^2 - 9*w + 3],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 3],
[317, 317, -2*w^3 + 5*w^2 + 9*w + 1],
[347, 347, 4*w^3 - 9*w^2 - 21*w + 3],
[347, 347, 4*w^3 - 10*w^2 - 18*w + 5],
[349, 349, -2*w^3 + 6*w^2 + 7*w - 3],
[353, 353, -w^3 + 12*w + 5],
[353, 353, w^2 - w - 7],
[359, 359, -2*w^3 + 5*w^2 + 6*w - 7],
[359, 359, -w^3 + 4*w^2 - 11],
[373, 373, -w^3 + 2*w^2 + 6*w + 5],
[379, 379, -3*w^3 + 8*w^2 + 14*w - 5],
[397, 397, 6*w^3 - 14*w^2 - 30*w + 7],
[401, 401, 3*w^3 - 5*w^2 - 20*w - 7],
[409, 409, 3*w^3 - 7*w^2 - 12*w - 3],
[409, 409, 2*w^2 - 4*w - 7],
[421, 421, w^3 - 11*w - 11],
[421, 421, -w^3 + 3*w^2 + w - 7],
[431, 431, 2*w^3 - 3*w^2 - 14*w - 3],
[431, 431, -4*w^3 + 11*w^2 + 15*w - 15],
[433, 433, 2*w^3 - 5*w^2 - 6*w + 1],
[439, 439, -4*w^3 + 11*w^2 + 13*w - 9],
[439, 439, -2*w^3 + 6*w^2 + 8*w - 11],
[443, 443, 2*w - 5],
[443, 443, -6*w^3 + 17*w^2 + 24*w - 21],
[449, 449, w^3 - 2*w^2 - 8*w + 3],
[449, 449, 4*w^3 - 12*w^2 - 15*w + 17],
[457, 457, -2*w^3 + 4*w^2 + 9*w + 5],
[461, 461, -3*w^3 + 6*w^2 + 15*w - 1],
[461, 461, -3*w^3 + 7*w^2 + 17*w + 1],
[463, 463, -4*w^3 + 10*w^2 + 17*w - 7],
[463, 463, -3*w^3 + 10*w^2 + 9*w - 17],
[467, 467, 4*w^3 - 8*w^2 - 21*w - 7],
[467, 467, w^3 - 6*w^2 + 3*w + 19],
[479, 479, -w^3 + w^2 + 8*w + 1],
[487, 487, -4*w^3 + 11*w^2 + 14*w - 15],
[487, 487, 3*w^3 - 6*w^2 - 15*w - 5],
[491, 491, w^2 - 5],
[503, 503, 2*w^3 - 5*w^2 - 11*w + 3],
[523, 523, -3*w^3 + 7*w^2 + 13*w + 1],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 3],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 9],
[557, 557, -3*w^3 + 9*w^2 + 9*w - 11],
[557, 557, -6*w^3 + 15*w^2 + 27*w - 13],
[563, 563, -3*w^3 + 7*w^2 + 14*w - 7],
[571, 571, -5*w^3 + 11*w^2 + 26*w - 3],
[593, 593, -6*w^3 + 15*w^2 + 26*w - 7],
[599, 599, 7*w^3 - 18*w^2 - 29*w + 17],
[599, 599, -2*w^3 + 6*w^2 + 8*w - 3],
[601, 601, -7*w^3 + 20*w^2 + 28*w - 23],
[601, 601, 3*w^3 - 7*w^2 - 12*w + 5],
[607, 607, 5*w^3 - 12*w^2 - 24*w + 9],
[607, 607, w^3 - 5*w^2 + 15],
[613, 613, -w^3 + 5*w^2 - 2*w - 9],
[617, 617, w^3 + w^2 - 13*w - 15],
[617, 617, 2*w^3 - 4*w^2 - 8*w - 1],
[619, 619, 5*w^3 - 10*w^2 - 27*w - 7],
[619, 619, 2*w^3 - 4*w^2 - 10*w + 3],
[643, 643, w^3 - 5*w^2 + 5*w + 5],
[643, 643, 4*w^3 - 9*w^2 - 20*w - 1],
[643, 643, 3*w^3 - 7*w^2 - 14*w - 3],
[643, 643, -3*w^3 + 7*w^2 + 13*w - 3],
[647, 647, -3*w^3 + 8*w^2 + 12*w - 3],
[653, 653, -w^3 + 9*w + 13],
[653, 653, -w^3 + 4*w^2 + 3*w - 9],
[653, 653, -2*w^3 + 7*w^2 + 4*w - 7],
[653, 653, w^2 - 4*w - 9],
[673, 673, -w^3 + 5*w^2 + w - 13],
[683, 683, w^3 - 3*w^2 - 3*w - 3],
[701, 701, 5*w^3 - 13*w^2 - 23*w + 9],
[709, 709, 2*w^3 - w^2 - 19*w - 15],
[719, 719, -5*w^3 + 15*w^2 + 20*w - 19],
[727, 727, -w - 5],
[751, 751, -4*w^3 + 10*w^2 + 15*w - 9],
[751, 751, 6*w^3 - 17*w^2 - 22*w + 21],
[757, 757, 3*w^3 - 6*w^2 - 18*w - 1],
[757, 757, -w^3 + 4*w^2 + 2*w + 1],
[769, 769, w^2 - 2*w + 3],
[773, 773, 5*w^3 - 12*w^2 - 23*w + 9],
[787, 787, 2*w^3 - 4*w^2 - 5*w + 1],
[797, 797, -3*w^3 + 8*w^2 + 10*w - 3],
[809, 809, 2*w^2 - 5*w - 1],
[809, 809, 4*w^3 - 6*w^2 - 28*w - 11],
[827, 827, 4*w^3 - 8*w^2 - 22*w - 1],
[827, 827, 2*w^2 - 6*w - 9],
[829, 829, -w^3 + w^2 + 10*w + 1],
[839, 839, -2*w^3 + 2*w^2 + 16*w + 7],
[877, 877, 6*w^3 - 16*w^2 - 25*w + 13],
[877, 877, w^3 - w^2 - 7*w + 1],
[881, 881, 2*w^3 - 2*w^2 - 15*w - 7],
[881, 881, -4*w^3 + 7*w^2 + 26*w + 9],
[887, 887, 2*w^3 - 4*w^2 - 13*w - 1],
[907, 907, 5*w^3 - 11*w^2 - 26*w - 1],
[907, 907, 5*w^3 - 13*w^2 - 22*w + 9],
[907, 907, w^3 - 5*w^2 + 2*w + 13],
[907, 907, -2*w^3 + 7*w^2 + 5*w - 7],
[911, 911, -3*w^3 + 8*w^2 + 9*w - 5],
[919, 919, -2*w^3 + 7*w^2 + 6*w - 3],
[919, 919, -3*w^3 + 8*w^2 + 15*w - 7],
[947, 947, -2*w^3 + 5*w^2 + 12*w - 9],
[947, 947, -w^2 + w - 3],
[947, 947, 3*w^3 - 6*w^2 - 14*w - 7],
[947, 947, -w^3 + 5*w^2 - w - 17],
[953, 953, 3*w^3 - 9*w^2 - 9*w + 13],
[953, 953, -4*w^3 + 13*w^2 + 11*w - 23],
[967, 967, 8*w^3 - 20*w^2 - 33*w + 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 30*x^6 + 260*x^4 - 600*x^2 + 400;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-3/200*e^7 + 2/5*e^5 - 27/10*e^3 + 2*e, e, 1/10*e^6 - 14/5*e^4 + 21*e^2 - 26, -9/100*e^7 + 13/5*e^5 - 101/5*e^3 + 26*e, 1/50*e^7 - 3/5*e^5 + 23/5*e^3 - 3*e, -9/100*e^7 + 13/5*e^5 - 101/5*e^3 + 27*e, -1/10*e^6 + 14/5*e^4 - 20*e^2 + 20, -7/100*e^7 + 2*e^5 - 78/5*e^3 + 23*e, 1/10*e^7 - 14/5*e^5 + 21*e^3 - 27*e, 3/25*e^7 - 17/5*e^5 + 128/5*e^3 - 32*e, -1/5*e^4 + 2*e^2 + 8, 3/100*e^7 - 4/5*e^5 + 27/5*e^3 - 6*e, -1/10*e^6 + 13/5*e^4 - 18*e^2 + 20, -9/100*e^7 + 12/5*e^5 - 81/5*e^3 + 11*e, -1/10*e^6 + 16/5*e^4 - 28*e^2 + 44, 2/5*e^4 - 7*e^2 + 14, 3/10*e^6 - 43/5*e^4 + 66*e^2 - 84, -1/5*e^6 + 27/5*e^4 - 38*e^2 + 40, 2/5*e^4 - 7*e^2 + 22, -1/50*e^7 + 3/5*e^5 - 28/5*e^3 + 17*e, -1/10*e^6 + 16/5*e^4 - 27*e^2 + 30, -9/50*e^7 + 5*e^5 - 182/5*e^3 + 40*e, 8/25*e^7 - 9*e^5 + 338/5*e^3 - 84*e, -e^3 + 13*e, -9/50*e^7 + 5*e^5 - 182/5*e^3 + 41*e, 7/50*e^7 - 19/5*e^5 + 136/5*e^3 - 32*e, 1/10*e^7 - 14/5*e^5 + 20*e^3 - 16*e, -9/100*e^7 + 13/5*e^5 - 96/5*e^3 + 11*e, 1/10*e^6 - 3*e^4 + 26*e^2 - 36, 3/10*e^6 - 44/5*e^4 + 69*e^2 - 90, 3/50*e^7 - 9/5*e^5 + 74/5*e^3 - 27*e, -3/10*e^6 + 43/5*e^4 - 66*e^2 + 84, -17/100*e^7 + 5*e^5 - 203/5*e^3 + 62*e, 1/5*e^6 - 29/5*e^4 + 44*e^2 - 50, 1/100*e^7 - 1/5*e^5 + 9/5*e^3 - 14*e, -3/100*e^7 + 4/5*e^5 - 27/5*e^3 + 2*e, 1/5*e^6 - 29/5*e^4 + 44*e^2 - 42, 2/25*e^7 - 12/5*e^5 + 97/5*e^3 - 27*e, 1/10*e^6 - 2*e^4 + 8*e^2 - 4, 1/10*e^6 - 17/5*e^4 + 30*e^2 - 36, 1/10*e^6 - 14/5*e^4 + 20*e^2 - 20, 2/5*e^6 - 11*e^4 + 82*e^2 - 112, 7/20*e^7 - 10*e^5 + 77*e^3 - 102*e, 31/100*e^7 - 9*e^5 + 349/5*e^3 - 86*e, -3/5*e^4 + 10*e^2 - 24, 7/50*e^7 - 21/5*e^5 + 176/5*e^3 - 64*e, -1/10*e^6 + 11/5*e^4 - 10*e^2 - 4, -21/100*e^7 + 6*e^5 - 229/5*e^3 + 62*e, -11/100*e^7 + 16/5*e^5 - 124/5*e^3 + 31*e, -1/10*e^6 + 14/5*e^4 - 20*e^2 + 12, 16, -1/5*e^6 + 27/5*e^4 - 36*e^2 + 30, -1/5*e^6 + 27/5*e^4 - 36*e^2 + 30, 2/5*e^6 - 56/5*e^4 + 84*e^2 - 104, 17/50*e^7 - 49/5*e^5 + 376/5*e^3 - 92*e, -1/2*e^6 + 14*e^4 - 103*e^2 + 118, -3/5*e^6 + 84/5*e^4 - 124*e^2 + 150, 3/5*e^6 - 88/5*e^4 + 137*e^2 - 162, -1/20*e^7 + 7/5*e^5 - 11*e^3 + 19*e, -1/2*e^6 + 69/5*e^4 - 102*e^2 + 124, -1/2*e^6 + 14*e^4 - 103*e^2 + 134, 19/100*e^7 - 26/5*e^5 + 181/5*e^3 - 26*e, -21/100*e^7 + 29/5*e^5 - 209/5*e^3 + 47*e, -11/100*e^7 + 3*e^5 - 104/5*e^3 + 15*e, 7/50*e^7 - 4*e^5 + 151/5*e^3 - 31*e, 7/100*e^7 - 2*e^5 + 83/5*e^3 - 38*e, -27/100*e^7 + 38/5*e^5 - 293/5*e^3 + 94*e, 1/25*e^7 - 7/5*e^5 + 66/5*e^3 - 19*e, 31/100*e^7 - 9*e^5 + 354/5*e^3 - 101*e, -3/5*e^6 + 87/5*e^4 - 136*e^2 + 178, 7/10*e^6 - 96/5*e^4 + 140*e^2 - 180, -7/10*e^6 + 20*e^4 - 152*e^2 + 196, -1/5*e^6 + 31/5*e^4 - 52*e^2 + 70, -19/100*e^7 + 27/5*e^5 - 201/5*e^3 + 42*e, -11/50*e^7 + 31/5*e^5 - 233/5*e^3 + 53*e, 63/100*e^7 - 89/5*e^5 + 672/5*e^3 - 169*e, 19/100*e^7 - 26/5*e^5 + 186/5*e^3 - 45*e, 2/5*e^6 - 52/5*e^4 + 69*e^2 - 66, 1/10*e^6 - 12/5*e^4 + 17*e^2 - 34, 1/5*e^6 - 24/5*e^4 + 28*e^2 - 26, 3/5*e^7 - 17*e^5 + 128*e^3 - 156*e, 1/10*e^6 - 16/5*e^4 + 28*e^2 - 28, 21/100*e^7 - 31/5*e^5 + 244/5*e^3 - 61*e, -17/25*e^7 + 97/5*e^5 - 742/5*e^3 + 188*e, -3/10*e^6 + 42/5*e^4 - 64*e^2 + 76, -2/5*e^6 + 53/5*e^4 - 74*e^2 + 72, 1/5*e^4 - 2*e^2 - 16, -33/100*e^7 + 48/5*e^5 - 382/5*e^3 + 111*e, 3/5*e^6 - 86/5*e^4 + 133*e^2 - 170, 1/5*e^6 - 33/5*e^4 + 56*e^2 - 54, -21/50*e^7 + 12*e^5 - 458/5*e^3 + 117*e, -11/50*e^7 + 31/5*e^5 - 223/5*e^3 + 33*e, 13/50*e^7 - 37/5*e^5 + 284/5*e^3 - 80*e, 1/10*e^6 - 14/5*e^4 + 20*e^2 - 28, -11/20*e^7 + 78/5*e^5 - 119*e^3 + 158*e, -1/5*e^6 + 27/5*e^4 - 42*e^2 + 72, -1/10*e^7 + 16/5*e^5 - 28*e^3 + 44*e, -8, -11/50*e^7 + 32/5*e^5 - 258/5*e^3 + 88*e, -3/5*e^6 + 17*e^4 - 126*e^2 + 152, -3/25*e^7 + 17/5*e^5 - 118/5*e^3 + 8*e, 1/4*e^7 - 7*e^5 + 51*e^3 - 42*e, -17/100*e^7 + 23/5*e^5 - 163/5*e^3 + 34*e, -2/5*e^6 + 51/5*e^4 - 70*e^2 + 104, -1/2*e^6 + 66/5*e^4 - 91*e^2 + 110, -1/50*e^7 + 4/5*e^5 - 38/5*e^3 + 5*e, -49/100*e^7 + 71/5*e^5 - 551/5*e^3 + 142*e, 51/100*e^7 - 73/5*e^5 + 559/5*e^3 - 134*e, -61/100*e^7 + 87/5*e^5 - 659/5*e^3 + 159*e, -27/50*e^7 + 77/5*e^5 - 586/5*e^3 + 136*e, -3/50*e^7 + 9/5*e^5 - 74/5*e^3 + 28*e, 2/5*e^4 - 7*e^2 + 38, -17/100*e^7 + 23/5*e^5 - 163/5*e^3 + 43*e, 31/50*e^7 - 88/5*e^5 + 668/5*e^3 - 168*e, -1/5*e^6 + 26/5*e^4 - 36*e^2 + 32, -1/2*e^6 + 68/5*e^4 - 99*e^2 + 134, -1/5*e^6 + 26/5*e^4 - 36*e^2 + 42, 1/20*e^7 - 6/5*e^5 + 7*e^3 - 9*e, 9/20*e^7 - 64/5*e^5 + 99*e^3 - 146*e, -9/100*e^7 + 11/5*e^5 - 61/5*e^3 - 6*e, 39/100*e^7 - 54/5*e^5 + 401/5*e^3 - 106*e, 39/100*e^7 - 54/5*e^5 + 401/5*e^3 - 106*e, -41/100*e^7 + 59/5*e^5 - 459/5*e^3 + 134*e, -23/100*e^7 + 32/5*e^5 - 227/5*e^3 + 30*e, 8/25*e^7 - 9*e^5 + 338/5*e^3 - 76*e, -1/2*e^6 + 14*e^4 - 107*e^2 + 142, 1/5*e^6 - 29/5*e^4 + 44*e^2 - 66, -1/10*e^6 + 18/5*e^4 - 35*e^2 + 54, -1/5*e^6 + 28/5*e^4 - 44*e^2 + 46, -3*e^2 + 38, -2/5*e^6 + 57/5*e^4 - 86*e^2 + 104, 37/50*e^7 - 106/5*e^5 + 816/5*e^3 - 215*e, 7/10*e^6 - 98/5*e^4 + 145*e^2 - 186, 1/2*e^6 - 76/5*e^4 + 124*e^2 - 156, 3/5*e^6 - 84/5*e^4 + 124*e^2 - 136, -1/5*e^5 + 4*e^3 - 24*e, -4/5*e^6 + 114/5*e^4 - 176*e^2 + 232, -16/25*e^7 + 91/5*e^5 - 696/5*e^3 + 189*e, 1/10*e^6 - 14/5*e^4 + 21*e^2 - 18, -1/5*e^6 + 6*e^4 - 51*e^2 + 94, 14/25*e^7 - 16*e^5 + 614/5*e^3 - 155*e, 1/100*e^7 - 3/5*e^5 + 49/5*e^3 - 46*e, -1/5*e^7 + 27/5*e^5 - 36*e^3 + 17*e, -31/100*e^7 + 44/5*e^5 - 339/5*e^3 + 99*e, 39/100*e^7 - 56/5*e^5 + 441/5*e^3 - 129*e, -9/20*e^7 + 13*e^5 - 101*e^3 + 134*e, 4/5*e^6 - 109/5*e^4 + 158*e^2 - 168, 9/25*e^7 - 10*e^5 + 374/5*e^3 - 103*e, 1/5*e^7 - 28/5*e^5 + 42*e^3 - 48*e, -27/25*e^7 + 152/5*e^5 - 1142/5*e^3 + 281*e, 1/25*e^7 - 6/5*e^5 + 61/5*e^3 - 55*e, -23/100*e^7 + 33/5*e^5 - 262/5*e^3 + 79*e, 37/100*e^7 - 52/5*e^5 + 388/5*e^3 - 97*e, 7/50*e^7 - 4*e^5 + 146/5*e^3 - 24*e, 13/100*e^7 - 17/5*e^5 + 107/5*e^3 + 2*e, -83/100*e^7 + 118/5*e^5 - 897/5*e^3 + 234*e, 2/5*e^6 - 57/5*e^4 + 90*e^2 - 144, -1/5*e^6 + 31/5*e^4 - 46*e^2 + 16, 4/25*e^7 - 24/5*e^5 + 204/5*e^3 - 76*e, 11/50*e^7 - 31/5*e^5 + 238/5*e^3 - 76*e, -11/25*e^7 + 64/5*e^5 - 506/5*e^3 + 136*e, 1/10*e^6 - 17/5*e^4 + 30*e^2 - 44, 1/10*e^6 - 17/5*e^4 + 30*e^2 - 44, 1/5*e^6 - 29/5*e^4 + 50*e^2 - 112, 4/5*e^6 - 113/5*e^4 + 166*e^2 - 192, -e^6 + 143/5*e^4 - 216*e^2 + 266, 4/5*e^6 - 112/5*e^4 + 165*e^2 - 186, -12/25*e^7 + 14*e^5 - 542/5*e^3 + 124*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;