/*
  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, -4, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {27, 3, 3}>;

primesArray := [
[2, 2, -w],
[2, 2, w - 1],
[11, 11, w^2 - w - 1],
[17, 17, -w^2 - w + 3],
[19, 19, w^2 - w + 1],
[23, 23, 2*w - 3],
[27, 3, 3],
[29, 29, 2*w + 1],
[31, 31, 2*w^2 - 2*w - 9],
[37, 37, 2*w^2 - 2*w - 5],
[41, 41, 2*w^2 - 9],
[43, 43, w^2 + w - 5],
[43, 43, -3*w^2 + w + 15],
[43, 43, -2*w^2 + 2*w + 11],
[53, 53, w^2 - w - 7],
[61, 61, 4*w^2 - 2*w - 15],
[67, 67, -5*w^2 + 3*w + 23],
[73, 73, 2*w^2 - 3],
[73, 73, -3*w^2 - w + 7],
[73, 73, -6*w^2 + 4*w + 25],
[79, 79, w^2 - 5*w + 1],
[79, 79, 3*w^2 - 3*w - 11],
[83, 83, 2*w^2 - 2*w - 3],
[109, 109, w^2 - 3*w - 3],
[113, 113, 3*w^2 - 3*w - 13],
[121, 11, 3*w^2 - w - 9],
[125, 5, -5],
[131, 131, -6*w^2 + 2*w + 25],
[137, 137, -w^2 + w - 3],
[149, 149, 2*w - 7],
[151, 151, w^2 - 3*w - 5],
[157, 157, 3*w^2 - 3*w - 7],
[163, 163, 2*w^2 - 13],
[167, 167, -4*w + 5],
[173, 173, -7*w^2 + 3*w + 27],
[179, 179, 4*w^2 - 4*w - 11],
[181, 181, 4*w^2 - 15],
[181, 181, w^2 - 3*w + 5],
[181, 181, w^2 + 3*w - 5],
[193, 193, 2*w^2 - 4*w - 5],
[197, 197, 2*w^2 + 2*w - 7],
[211, 211, 4*w^2 - 2*w - 13],
[211, 211, -7*w^2 + 5*w + 29],
[211, 211, w^2 - w - 9],
[223, 223, 5*w^2 - w - 23],
[227, 227, -3*w^2 + 3*w + 17],
[227, 227, 3*w^2 - w - 7],
[227, 227, 3*w^2 + w - 11],
[229, 229, 5*w^2 - w - 21],
[233, 233, 3*w^2 - 5*w - 7],
[239, 239, -w^2 + 7*w - 5],
[257, 257, -4*w - 3],
[257, 257, w^2 - 5*w - 1],
[257, 257, -3*w^2 + w + 17],
[263, 263, 3*w^2 - 3*w - 5],
[271, 271, 4*w^2 - 4*w - 17],
[271, 271, 3*w^2 - w - 3],
[271, 271, w^2 + 3*w - 7],
[283, 283, 6*w - 1],
[289, 17, w^2 + 3*w - 9],
[293, 293, 3*w^2 - w - 5],
[307, 307, 2*w^2 + 2*w - 9],
[307, 307, -2*w^2 + 2*w - 3],
[307, 307, 4*w^2 - 17],
[311, 311, w^2 + w - 11],
[311, 311, 2*w^2 - 4*w - 7],
[311, 311, -2*w^2 - 2*w + 13],
[313, 313, -2*w - 7],
[313, 313, 4*w^2 - 19],
[313, 313, -5*w^2 - w + 13],
[331, 331, 2*w^2 - 4*w - 11],
[331, 331, 6*w^2 - 4*w - 21],
[331, 331, -4*w^2 + 2*w + 21],
[343, 7, -7],
[347, 347, -2*w^2 + 8*w - 1],
[349, 349, w^2 - 7*w + 1],
[353, 353, -6*w^2 + 2*w + 21],
[361, 19, -5*w^2 + 3*w + 25],
[367, 367, 3*w^2 + w - 15],
[373, 373, 2*w^2 - 4*w - 9],
[383, 383, -9*w^2 + 3*w + 37],
[401, 401, 4*w^2 - 2*w - 11],
[409, 409, 4*w^2 - 4*w - 9],
[431, 431, -5*w^2 + 5*w + 19],
[439, 439, 2*w - 9],
[443, 443, w^2 - 5*w - 3],
[443, 443, 5*w^2 - 5*w - 23],
[443, 443, 3*w^2 - 5*w - 17],
[449, 449, -4*w^2 + 4*w + 1],
[461, 461, 5*w^2 - 7*w - 9],
[467, 467, 5*w^2 + w - 17],
[479, 479, 2*w^2 + 4*w - 7],
[487, 487, 5*w^2 - 5*w - 13],
[499, 499, -w^2 + 3*w - 7],
[509, 509, -7*w^2 + w + 27],
[521, 521, w^2 - w - 11],
[523, 523, 4*w^2 - 6*w - 11],
[529, 23, 4*w^2 + 2*w - 13],
[547, 547, -11*w^2 + 5*w + 43],
[563, 563, 4*w^2 - 7],
[569, 569, -6*w - 1],
[569, 569, 4*w^2 - 5],
[569, 569, w^2 - 5*w - 9],
[571, 571, 5*w^2 - 3*w - 15],
[577, 577, w^2 + 5*w - 7],
[599, 599, 4*w^2 - 4*w - 7],
[601, 601, -12*w^2 + 8*w + 51],
[613, 613, 4*w^2 + 1],
[643, 643, 6*w^2 - 2*w - 31],
[647, 647, 2*w^2 - 2*w - 15],
[647, 647, 4*w^2 - 4*w - 23],
[647, 647, w^2 - 5*w - 7],
[661, 661, -8*w^2 + 6*w + 39],
[673, 673, 3*w^2 - 5*w - 15],
[683, 683, -w^2 + 9*w - 7],
[701, 701, 4*w^2 - 4*w - 5],
[709, 709, 4*w^2 - 2*w - 7],
[719, 719, 3*w^2 - 5*w - 13],
[727, 727, w^2 - 7*w - 1],
[733, 733, 2*w^2 + 4*w - 9],
[733, 733, 3*w^2 + 3*w - 11],
[733, 733, w^2 + w - 13],
[743, 743, 2*w^2 - 6*w - 5],
[751, 751, 6*w^2 - 2*w - 19],
[761, 761, -12*w^2 + 6*w + 47],
[761, 761, 3*w^2 - 5*w - 21],
[761, 761, 2*w^2 + 6*w - 7],
[769, 769, -6*w^2 + 6*w + 23],
[773, 773, -2*w^2 + 2*w - 5],
[773, 773, -13*w^2 + 5*w + 57],
[773, 773, -11*w^2 + 7*w + 51],
[787, 787, 5*w^2 - w - 13],
[797, 797, -2*w^2 + 8*w - 11],
[811, 811, -5*w^2 + 5*w + 1],
[821, 821, -7*w^2 + 3*w + 35],
[821, 821, -4*w^2 + 2*w + 23],
[821, 821, 5*w^2 + w - 19],
[823, 823, w^2 + 5*w - 13],
[823, 823, -12*w^2 + 4*w + 49],
[823, 823, -7*w^2 + w + 29],
[829, 829, -w^2 + w - 7],
[839, 839, 4*w^2 + 2*w - 15],
[841, 29, 4*w^2 - 6*w - 13],
[853, 853, 7*w^2 - 7*w - 25],
[863, 863, 7*w^2 - 5*w - 35],
[877, 877, -6*w - 5],
[877, 877, w^2 + 5*w - 11],
[877, 877, 5*w^2 - 3*w - 13],
[881, 881, w^2 - 3*w - 13],
[887, 887, -9*w^2 + 5*w + 33],
[907, 907, w^2 - 9*w + 1],
[911, 911, -6*w^2 + 6*w + 25],
[919, 919, 5*w^2 - 3*w - 27],
[929, 929, 2*w - 11],
[937, 937, -7*w^2 + w + 31],
[941, 941, -6*w^2 + 4*w + 31],
[941, 941, -8*w^2 + 2*w + 27],
[941, 941, 3*w^2 + 5*w - 9],
[947, 947, -10*w + 7],
[947, 947, 2*w^2 + 8*w - 7],
[947, 947, 2*w^2 + 4*w - 11],
[961, 31, -14*w^2 + 8*w + 55],
[967, 967, -w^2 - w - 7],
[971, 971, -10*w^2 + 8*w + 39],
[977, 977, -10*w^2 + 6*w + 37],
[983, 983, 3*w^2 + 3*w - 13],
[983, 983, -4*w - 11],
[983, 983, -10*w^2 + 4*w + 37],
[997, 997, w^2 + 9*w - 7],
[997, 997, 5*w^2 - 7*w - 27],
[997, 997, 6*w^2 - 8*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 4*x^7 - 6*x^6 + 37*x^5 - 8*x^4 - 88*x^3 + 62*x^2 + 40*x - 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/4*e^7 - 1/2*e^6 - 5/2*e^5 + 17/4*e^4 + 13/2*e^3 - 8*e^2 - 5/2*e + 1, e^7 - 3*e^6 - 9*e^5 + 27*e^4 + 20*e^3 - 60*e^2 - 4*e + 24, -3/2*e^7 + 4*e^6 + 14*e^5 - 73/2*e^4 - 34*e^3 + 86*e^2 + 15*e - 42, -5/2*e^7 + 13/2*e^6 + 24*e^5 - 117/2*e^4 - 123/2*e^3 + 131*e^2 + 29*e - 54, -e^7 + 3*e^6 + 9*e^5 - 28*e^4 - 19*e^3 + 66*e^2 - 28, 1, -1/2*e^7 + e^6 + 5*e^5 - 19/2*e^4 - 14*e^3 + 26*e^2 + 11*e - 18, -3/2*e^7 + 7/2*e^6 + 15*e^5 - 63/2*e^4 - 83/2*e^3 + 71*e^2 + 25*e - 30, 3*e^7 - 15/2*e^6 - 29*e^5 + 68*e^4 + 147/2*e^3 - 155*e^2 - 32*e + 68, -2*e^7 + 5*e^6 + 20*e^5 - 45*e^4 - 56*e^3 + 100*e^2 + 34*e - 38, -1/2*e^7 + 3/2*e^6 + 5*e^5 - 29/2*e^4 - 27/2*e^3 + 35*e^2 + 5*e - 14, 3/2*e^7 - 4*e^6 - 14*e^5 + 75/2*e^4 + 33*e^3 - 92*e^2 - 13*e + 46, e^7 - 5/2*e^6 - 9*e^5 + 23*e^4 + 37/2*e^3 - 56*e^2 + 2*e + 28, -1/2*e^7 + 2*e^6 + 3*e^5 - 37/2*e^4 + 2*e^3 + 44*e^2 - 17*e - 18, e^7 - 5/2*e^6 - 9*e^5 + 21*e^4 + 39/2*e^3 - 41*e^2 - 4*e + 12, -3/2*e^7 + 4*e^6 + 15*e^5 - 73/2*e^4 - 41*e^3 + 82*e^2 + 23*e - 34, -5*e^7 + 13*e^6 + 47*e^5 - 116*e^4 - 115*e^3 + 258*e^2 + 46*e - 108, -e^7 + 5/2*e^6 + 10*e^5 - 22*e^4 - 53/2*e^3 + 46*e^2 + 8*e - 10, 2*e^7 - 11/2*e^6 - 18*e^5 + 51*e^4 + 77/2*e^3 - 121*e^2 - 8*e + 52, e^7 - 5/2*e^6 - 9*e^5 + 22*e^4 + 39/2*e^3 - 50*e^2 - 2*e + 24, 3*e^7 - 15/2*e^6 - 30*e^5 + 68*e^4 + 167/2*e^3 - 152*e^2 - 52*e + 64, -e^7 + 2*e^6 + 9*e^5 - 15*e^4 - 19*e^3 + 22*e^2, -e^7 + 7/2*e^6 + 7*e^5 - 30*e^4 - 13/2*e^3 + 62*e^2 - 8*e - 14, 2*e^7 - 6*e^6 - 18*e^5 + 54*e^4 + 40*e^3 - 122*e^2 - 10*e + 58, 4*e^7 - 23/2*e^6 - 37*e^5 + 105*e^4 + 177/2*e^3 - 242*e^2 - 36*e + 114, e^7 - 3*e^6 - 10*e^5 + 29*e^4 + 29*e^3 - 74*e^2 - 20*e + 46, 4*e^7 - 11*e^6 - 37*e^5 + 99*e^4 + 89*e^3 - 224*e^2 - 36*e + 96, -5/2*e^7 + 7*e^6 + 24*e^5 - 127/2*e^4 - 63*e^3 + 142*e^2 + 39*e - 62, e^5 - 7*e^3 - 4*e^2 + 6*e + 10, 6*e^7 - 29/2*e^6 - 58*e^5 + 129*e^4 + 301/2*e^3 - 286*e^2 - 78*e + 120, -3*e^7 + 15/2*e^6 + 29*e^5 - 67*e^4 - 151/2*e^3 + 146*e^2 + 40*e - 50, -1/2*e^6 + 4*e^4 - 1/2*e^3 - 4*e^2 + 8*e - 4, 2*e^7 - 6*e^6 - 19*e^5 + 55*e^4 + 50*e^3 - 128*e^2 - 28*e + 60, -e^7 + e^6 + 12*e^5 - 8*e^4 - 42*e^3 + 14*e^2 + 38*e - 2, e^7 - 2*e^6 - 10*e^5 + 16*e^4 + 29*e^3 - 28*e^2 - 18*e + 4, -1/2*e^6 + 6*e^4 - 1/2*e^3 - 22*e^2 + 4*e + 26, 6*e^7 - 16*e^6 - 57*e^5 + 142*e^4 + 147*e^3 - 310*e^2 - 78*e + 128, -6*e^7 + 31/2*e^6 + 58*e^5 - 140*e^4 - 303/2*e^3 + 315*e^2 + 84*e - 136, 3*e^7 - 7*e^6 - 29*e^5 + 60*e^4 + 77*e^3 - 122*e^2 - 50*e + 40, 8*e^7 - 21*e^6 - 77*e^5 + 190*e^4 + 200*e^3 - 434*e^2 - 104*e + 202, 9/2*e^7 - 10*e^6 - 46*e^5 + 177/2*e^4 + 133*e^3 - 196*e^2 - 91*e + 90, -4*e^7 + 19/2*e^6 + 39*e^5 - 85*e^4 - 205/2*e^3 + 196*e^2 + 52*e - 100, 15/2*e^7 - 39/2*e^6 - 72*e^5 + 357/2*e^4 + 363/2*e^3 - 415*e^2 - 79*e + 186, -1/2*e^7 + e^6 + 5*e^5 - 17/2*e^4 - 11*e^3 + 18*e^2 - 9*e - 10, 9*e^7 - 24*e^6 - 87*e^5 + 220*e^4 + 224*e^3 - 514*e^2 - 100*e + 240, e^7 - 2*e^6 - 11*e^5 + 18*e^4 + 36*e^3 - 42*e^2 - 36*e + 20, -3*e^7 + 8*e^6 + 30*e^5 - 76*e^4 - 81*e^3 + 186*e^2 + 40*e - 96, -4*e^7 + 21/2*e^6 + 38*e^5 - 95*e^4 - 191/2*e^3 + 213*e^2 + 48*e - 88, -8*e^7 + 22*e^6 + 75*e^5 - 201*e^4 - 182*e^3 + 466*e^2 + 70*e - 210, -5*e^7 + 12*e^6 + 51*e^5 - 107*e^4 - 147*e^3 + 236*e^2 + 94*e - 100, 15/2*e^7 - 19*e^6 - 74*e^5 + 343/2*e^4 + 202*e^3 - 390*e^2 - 117*e + 178, -6*e^7 + 16*e^6 + 56*e^5 - 145*e^4 - 133*e^3 + 330*e^2 + 48*e - 138, 7/2*e^7 - 9*e^6 - 34*e^5 + 161/2*e^4 + 89*e^3 - 182*e^2 - 41*e + 82, 4*e^7 - 12*e^6 - 35*e^5 + 109*e^4 + 72*e^3 - 248*e^2 - 8*e + 96, 1/2*e^7 - 1/2*e^6 - 5*e^5 + 1/2*e^4 + 37/2*e^3 + 13*e^2 - 31*e - 14, -7*e^7 + 37/2*e^6 + 65*e^5 - 165*e^4 - 313/2*e^3 + 368*e^2 + 64*e - 160, -5/2*e^7 + 13/2*e^6 + 23*e^5 - 115/2*e^4 - 107/2*e^3 + 125*e^2 + 21*e - 38, -5/2*e^7 + 11/2*e^6 + 27*e^5 - 107/2*e^4 - 165/2*e^3 + 137*e^2 + 57*e - 78, 2*e^7 - 11/2*e^6 - 19*e^5 + 50*e^4 + 97/2*e^3 - 117*e^2 - 22*e + 68, 4*e^7 - 11*e^6 - 37*e^5 + 97*e^4 + 91*e^3 - 214*e^2 - 46*e + 98, 3/2*e^6 - e^5 - 18*e^4 + 13/2*e^3 + 66*e^2 - 12*e - 52, -2*e^7 + 11/2*e^6 + 18*e^5 - 54*e^4 - 69/2*e^3 + 140*e^2 - 10*e - 68, 2*e^7 - 6*e^6 - 19*e^5 + 59*e^4 + 46*e^3 - 154*e^2 - 18*e + 84, -4*e^7 + 11*e^6 + 37*e^5 - 97*e^4 - 89*e^3 + 206*e^2 + 40*e - 80, -8*e^7 + 22*e^6 + 75*e^5 - 198*e^4 - 183*e^3 + 440*e^2 + 76*e - 184, -2*e^7 + 5*e^6 + 21*e^5 - 43*e^4 - 65*e^3 + 84*e^2 + 44*e - 24, -8*e^7 + 43/2*e^6 + 76*e^5 - 199*e^4 - 379/2*e^3 + 476*e^2 + 84*e - 230, -3*e^7 + 6*e^6 + 33*e^5 - 54*e^4 - 104*e^3 + 118*e^2 + 76*e - 42, 2*e^7 - 11/2*e^6 - 16*e^5 + 50*e^4 + 37/2*e^3 - 116*e^2 + 32*e + 54, -1/2*e^7 + 3/2*e^6 + 5*e^5 - 27/2*e^4 - 25/2*e^3 + 31*e^2 - 7*e - 22, -1/2*e^7 + 1/2*e^6 + 4*e^5 - 1/2*e^4 - 15/2*e^3 - 11*e^2 + 3*e + 6, -2*e^7 + 6*e^6 + 19*e^5 - 59*e^4 - 46*e^3 + 158*e^2 + 14*e - 92, -8*e^7 + 21*e^6 + 77*e^5 - 192*e^4 - 198*e^3 + 450*e^2 + 104*e - 216, -11*e^7 + 29*e^6 + 106*e^5 - 264*e^4 - 274*e^3 + 610*e^2 + 138*e - 292, 10*e^7 - 49/2*e^6 - 97*e^5 + 218*e^4 + 509/2*e^3 - 478*e^2 - 144*e + 194, 9/2*e^7 - 12*e^6 - 45*e^5 + 223/2*e^4 + 123*e^3 - 264*e^2 - 67*e + 134, e^7 - 7/2*e^6 - 10*e^5 + 33*e^4 + 63/2*e^3 - 77*e^2 - 32*e + 32, 15/2*e^7 - 21*e^6 - 70*e^5 + 385/2*e^4 + 167*e^3 - 448*e^2 - 59*e + 210, 5*e^7 - 27/2*e^6 - 47*e^5 + 124*e^4 + 223/2*e^3 - 295*e^2 - 24*e + 140, 2*e^7 - 5*e^6 - 22*e^5 + 48*e^4 + 73*e^3 - 118*e^2 - 72*e + 56, 5/2*e^7 - 5*e^6 - 25*e^5 + 81/2*e^4 + 71*e^3 - 78*e^2 - 55*e + 38, -9*e^7 + 49/2*e^6 + 83*e^5 - 221*e^4 - 387/2*e^3 + 501*e^2 + 66*e - 220, 5*e^7 - 12*e^6 - 48*e^5 + 104*e^4 + 123*e^3 - 216*e^2 - 56*e + 68, -e^7 + e^6 + 11*e^5 - 5*e^4 - 34*e^3 - 6*e^2 + 22*e + 16, -6*e^7 + 17*e^6 + 56*e^5 - 156*e^4 - 137*e^3 + 368*e^2 + 68*e - 188, 8*e^7 - 21*e^6 - 81*e^5 + 195*e^4 + 227*e^3 - 456*e^2 - 132*e + 204, -7*e^7 + 17*e^6 + 69*e^5 - 156*e^4 - 183*e^3 + 362*e^2 + 96*e - 156, 11/2*e^7 - 13*e^6 - 56*e^5 + 229/2*e^4 + 163*e^3 - 246*e^2 - 117*e + 110, 2*e^7 - 5*e^6 - 21*e^5 + 49*e^4 + 59*e^3 - 128*e^2 - 24*e + 82, -e^7 + 3*e^6 + 7*e^5 - 26*e^4 - 5*e^3 + 54*e^2 - 10*e - 20, -5*e^7 + 12*e^6 + 49*e^5 - 106*e^4 - 136*e^3 + 230*e^2 + 104*e - 92, 10*e^7 - 55/2*e^6 - 96*e^5 + 250*e^4 + 493/2*e^3 - 564*e^2 - 118*e + 232, -5/2*e^7 + 9/2*e^6 + 28*e^5 - 79/2*e^4 - 193/2*e^3 + 85*e^2 + 91*e - 26, 15/2*e^7 - 17*e^6 - 75*e^5 + 299/2*e^4 + 211*e^3 - 326*e^2 - 141*e + 146, -11/2*e^7 + 15*e^6 + 51*e^5 - 269/2*e^4 - 116*e^3 + 296*e^2 + 19*e - 122, 13/2*e^7 - 19*e^6 - 57*e^5 + 345/2*e^4 + 119*e^3 - 398*e^2 - 19*e + 178, -4*e^7 + 25/2*e^6 + 37*e^5 - 115*e^4 - 183/2*e^3 + 268*e^2 + 40*e - 118, -2*e^7 + 9/2*e^6 + 19*e^5 - 38*e^4 - 97/2*e^3 + 74*e^2 + 16*e - 4, 3*e^7 - 8*e^6 - 27*e^5 + 68*e^4 + 62*e^3 - 130*e^2 - 24*e + 40, 21/2*e^7 - 29*e^6 - 99*e^5 + 533/2*e^4 + 243*e^3 - 624*e^2 - 105*e + 302, 11/2*e^7 - 16*e^6 - 52*e^5 + 295/2*e^4 + 131*e^3 - 350*e^2 - 55*e + 178, -3*e^7 + 8*e^6 + 27*e^5 - 69*e^4 - 61*e^3 + 144*e^2 + 16*e - 46, 9*e^7 - 26*e^6 - 82*e^5 + 239*e^4 + 192*e^3 - 564*e^2 - 78*e + 276, 7*e^7 - 39/2*e^6 - 65*e^5 + 176*e^4 + 311/2*e^3 - 405*e^2 - 60*e + 208, 3*e^7 - 4*e^6 - 33*e^5 + 31*e^4 + 99*e^3 - 42*e^2 - 60*e - 16, -13*e^7 + 69/2*e^6 + 124*e^5 - 313*e^4 - 629/2*e^3 + 719*e^2 + 146*e - 328, -5*e^7 + 27/2*e^6 + 48*e^5 - 124*e^4 - 243/2*e^3 + 284*e^2 + 56*e - 110, -11*e^7 + 57/2*e^6 + 104*e^5 - 252*e^4 - 517/2*e^3 + 540*e^2 + 122*e - 196, 2*e^7 - 3*e^6 - 23*e^5 + 25*e^4 + 75*e^3 - 40*e^2 - 52*e - 4, 6*e^7 - 17*e^6 - 53*e^5 + 152*e^4 + 112*e^3 - 342*e^2 - 28*e + 156, 3*e^7 - 8*e^6 - 29*e^5 + 77*e^4 + 67*e^3 - 188*e^2 - 8*e + 64, 4*e^7 - 25/2*e^6 - 36*e^5 + 115*e^4 + 173/2*e^3 - 272*e^2 - 56*e + 142, -5*e^7 + 16*e^6 + 45*e^5 - 150*e^4 - 104*e^3 + 366*e^2 + 44*e - 186, 3*e^7 - 7*e^6 - 32*e^5 + 66*e^4 + 96*e^3 - 154*e^2 - 58*e + 52, 21/2*e^7 - 24*e^6 - 105*e^5 + 431/2*e^4 + 285*e^3 - 478*e^2 - 157*e + 190, e^7 - e^6 - 11*e^5 + 8*e^4 + 35*e^3 - 22*e^2 - 30*e + 48, 9*e^7 - 26*e^6 - 85*e^5 + 239*e^4 + 213*e^3 - 556*e^2 - 92*e + 244, 9*e^7 - 49/2*e^6 - 86*e^5 + 222*e^4 + 441/2*e^3 - 512*e^2 - 106*e + 248, -8*e^7 + 45/2*e^6 + 75*e^5 - 206*e^4 - 373/2*e^3 + 488*e^2 + 92*e - 254, -6*e^7 + 13*e^6 + 64*e^5 - 118*e^4 - 195*e^3 + 264*e^2 + 130*e - 104, e^7 - 9/2*e^6 - 10*e^5 + 43*e^4 + 69/2*e^3 - 103*e^2 - 48*e + 56, -9*e^7 + 24*e^6 + 87*e^5 - 217*e^4 - 229*e^3 + 486*e^2 + 132*e - 208, -1/2*e^7 + 7/2*e^6 + 2*e^5 - 81/2*e^4 + 19/2*e^3 + 135*e^2 - 23*e - 94, -5/2*e^7 + 5*e^6 + 25*e^5 - 87/2*e^4 - 66*e^3 + 96*e^2 + 33*e - 58, 29/2*e^7 - 37*e^6 - 141*e^5 + 679/2*e^4 + 362*e^3 - 786*e^2 - 163*e + 342, 16*e^7 - 40*e^6 - 153*e^5 + 354*e^4 + 393*e^3 - 774*e^2 - 208*e + 310, -9*e^7 + 23*e^6 + 90*e^5 - 211*e^4 - 247*e^3 + 480*e^2 + 136*e - 192, -9*e^7 + 23*e^6 + 88*e^5 - 209*e^4 - 227*e^3 + 472*e^2 + 90*e - 194, 3/2*e^7 - 3*e^6 - 18*e^5 + 67/2*e^4 + 66*e^3 - 104*e^2 - 71*e + 58, 1/2*e^7 - 6*e^6 + 2*e^5 + 119/2*e^4 - 38*e^3 - 156*e^2 + 61*e + 86, 12*e^7 - 59/2*e^6 - 117*e^5 + 264*e^4 + 615/2*e^3 - 582*e^2 - 158*e + 220, e^7 + e^6 - 12*e^5 - 16*e^4 + 40*e^3 + 68*e^2 - 36*e - 62, e^7 - 5*e^6 - 5*e^5 + 47*e^4 - 10*e^3 - 122*e^2 + 46*e + 76, 3/2*e^7 - 4*e^6 - 14*e^5 + 83/2*e^4 + 31*e^3 - 120*e^2 - 9*e + 66, 3*e^7 - 8*e^6 - 27*e^5 + 77*e^4 + 49*e^3 - 194*e^2 + 26*e + 86, -17/2*e^7 + 22*e^6 + 82*e^5 - 401/2*e^4 - 207*e^3 + 460*e^2 + 79*e - 218, -17/2*e^7 + 21*e^6 + 84*e^5 - 383/2*e^4 - 225*e^3 + 444*e^2 + 121*e - 198, -8*e^7 + 22*e^6 + 73*e^5 - 197*e^4 - 168*e^3 + 442*e^2 + 50*e - 184, 7/2*e^7 - 11*e^6 - 28*e^5 + 201/2*e^4 + 43*e^3 - 236*e^2 + 5*e + 98, 17*e^7 - 93/2*e^6 - 159*e^5 + 420*e^4 + 777/2*e^3 - 951*e^2 - 158*e + 424, -2*e^7 + 8*e^6 + 15*e^5 - 82*e^4 - 13*e^3 + 226*e^2 - 40*e - 120, -3*e^7 + 19/2*e^6 + 28*e^5 - 89*e^4 - 131/2*e^3 + 199*e^2 + 18*e - 52, -3*e^7 + 19/2*e^6 + 24*e^5 - 80*e^4 - 75/2*e^3 + 162*e^2 - 32*e - 70, 10*e^7 - 24*e^6 - 101*e^5 + 213*e^4 + 288*e^3 - 464*e^2 - 192*e + 184, 2*e^7 - 9*e^6 - 14*e^5 + 86*e^4 + 13*e^3 - 212*e^2 + 30*e + 84, -11*e^7 + 61/2*e^6 + 106*e^5 - 275*e^4 - 549/2*e^3 + 605*e^2 + 138*e - 244, -6*e^7 + 35/2*e^6 + 56*e^5 - 165*e^4 - 263/2*e^3 + 402*e^2 + 32*e - 178, -9/2*e^7 + 11*e^6 + 45*e^5 - 203/2*e^4 - 124*e^3 + 246*e^2 + 83*e - 138, -7*e^7 + 17*e^6 + 68*e^5 - 146*e^4 - 178*e^3 + 292*e^2 + 96*e - 96, -2*e^7 + 19/2*e^6 + 11*e^5 - 86*e^4 + 21/2*e^3 + 194*e^2 - 54*e - 76, 2*e^7 - 8*e^6 - 17*e^5 + 76*e^4 + 39*e^3 - 184*e^2 - 20*e + 72, 9/2*e^7 - 31/2*e^6 - 40*e^5 + 293/2*e^4 + 165/2*e^3 - 351*e^2 + 3*e + 166, 13*e^7 - 35*e^6 - 125*e^5 + 318*e^4 + 321*e^3 - 720*e^2 - 142*e + 298, 5*e^7 - 27/2*e^6 - 46*e^5 + 119*e^4 + 219/2*e^3 - 265*e^2 - 38*e + 120, -5*e^7 + 12*e^6 + 49*e^5 - 111*e^4 - 131*e^3 + 276*e^2 + 72*e - 146, -1/2*e^7 + e^6 + 6*e^5 - 15/2*e^4 - 21*e^3 + 14*e^2 + 3*e - 22, 19/2*e^7 - 25*e^6 - 94*e^5 + 465/2*e^4 + 255*e^3 - 556*e^2 - 151*e + 270, -8*e^7 + 19*e^6 + 80*e^5 - 170*e^4 - 215*e^3 + 372*e^2 + 108*e - 140, -9*e^7 + 25*e^6 + 82*e^5 - 224*e^4 - 184*e^3 + 500*e^2 + 40*e - 224, 12*e^7 - 31*e^6 - 114*e^5 + 277*e^4 + 286*e^3 - 614*e^2 - 132*e + 248, 2*e^7 - 11/2*e^6 - 18*e^5 + 48*e^4 + 83/2*e^3 - 107*e^2 - 28*e + 48, 25/2*e^7 - 63/2*e^6 - 126*e^5 + 583/2*e^4 + 711/2*e^3 - 685*e^2 - 221*e + 310, -4*e^7 + 11*e^6 + 36*e^5 - 100*e^4 - 75*e^3 + 232*e^2 + 2*e - 108, 43/2*e^7 - 57*e^6 - 204*e^5 + 1035/2*e^4 + 506*e^3 - 1182*e^2 - 197*e + 498, 6*e^7 - 17*e^6 - 55*e^5 + 155*e^4 + 125*e^3 - 354*e^2 - 36*e + 152, -7*e^7 + 19*e^6 + 64*e^5 - 171*e^4 - 143*e^3 + 378*e^2 + 28*e - 132, 13*e^7 - 37*e^6 - 121*e^5 + 336*e^4 + 291*e^3 - 758*e^2 - 102*e + 308, 3*e^7 - 19/2*e^6 - 27*e^5 + 84*e^4 + 119/2*e^3 - 175*e^2 - 10*e + 52, -9*e^7 + 47/2*e^6 + 91*e^5 - 213*e^4 - 519/2*e^3 + 476*e^2 + 160*e - 186, 5*e^7 - 23/2*e^6 - 52*e^5 + 107*e^4 + 303/2*e^3 - 251*e^2 - 110*e + 108];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {27, 3, 3}>] := -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;