/*
  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![-28, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, 3}>;

primesArray := [
[2, 2, -w + 6],
[2, 2, w + 5],
[7, 7, 6*w - 35],
[7, 7, -6*w - 29],
[9, 3, 3],
[11, 11, 4*w + 19],
[11, 11, 4*w - 23],
[13, 13, -2*w + 11],
[13, 13, 2*w + 9],
[25, 5, -5],
[31, 31, 2*w - 13],
[31, 31, -2*w - 11],
[41, 41, -8*w - 39],
[41, 41, 8*w - 47],
[53, 53, -26*w - 125],
[53, 53, 26*w - 151],
[61, 61, -14*w + 81],
[61, 61, -14*w - 67],
[83, 83, 2*w - 15],
[83, 83, -2*w - 13],
[97, 97, 2*w - 5],
[97, 97, -2*w - 3],
[109, 109, 2*w - 3],
[109, 109, -2*w - 1],
[113, 113, 2*w - 1],
[127, 127, 8*w - 45],
[127, 127, 8*w + 37],
[131, 131, -50*w + 291],
[131, 131, 50*w + 241],
[139, 139, 6*w - 37],
[139, 139, -6*w - 31],
[149, 149, 20*w + 97],
[149, 149, 20*w - 117],
[157, 157, -12*w - 59],
[157, 157, 12*w - 71],
[163, 163, 4*w - 19],
[163, 163, 4*w + 15],
[173, 173, 4*w + 23],
[173, 173, 4*w - 27],
[211, 211, 2*w - 19],
[211, 211, -2*w - 17],
[227, 227, -4*w - 13],
[227, 227, 4*w - 17],
[233, 233, 6*w + 25],
[233, 233, -6*w + 31],
[239, 239, -14*w - 69],
[239, 239, 14*w - 83],
[241, 241, 46*w + 221],
[241, 241, -46*w + 267],
[251, 251, 22*w - 129],
[251, 251, 22*w + 107],
[257, 257, -34*w + 197],
[257, 257, -34*w - 163],
[277, 277, 4*w - 29],
[277, 277, -4*w - 25],
[283, 283, 4*w - 15],
[283, 283, -4*w - 11],
[289, 17, -17],
[307, 307, 68*w - 395],
[307, 307, 68*w + 327],
[311, 311, 10*w - 61],
[311, 311, -10*w - 51],
[313, 313, -32*w - 155],
[313, 313, 32*w - 187],
[317, 317, -18*w - 85],
[317, 317, 18*w - 103],
[331, 331, -4*w - 9],
[331, 331, 4*w - 13],
[337, 337, -16*w - 79],
[337, 337, 16*w - 95],
[347, 347, -12*w + 67],
[347, 347, 12*w + 55],
[353, 353, 14*w - 79],
[353, 353, 14*w + 65],
[361, 19, -19],
[367, 367, -32*w - 153],
[367, 367, -32*w + 185],
[383, 383, 56*w + 269],
[383, 383, 56*w - 325],
[389, 389, -4*w - 27],
[389, 389, 4*w - 31],
[401, 401, 8*w - 51],
[401, 401, -8*w - 43],
[421, 421, 12*w - 73],
[421, 421, -12*w - 61],
[439, 439, -8*w - 33],
[439, 439, 8*w - 41],
[443, 443, -4*w - 1],
[443, 443, 4*w - 5],
[461, 461, -30*w + 173],
[461, 461, 30*w + 143],
[463, 463, 2*w - 25],
[463, 463, -2*w - 23],
[467, 467, 34*w + 165],
[467, 467, 34*w - 199],
[503, 503, -26*w - 127],
[503, 503, 26*w - 153],
[509, 509, 4*w - 33],
[509, 509, -4*w - 29],
[521, 521, -10*w + 53],
[521, 521, 10*w + 43],
[529, 23, -23],
[547, 547, 14*w - 85],
[547, 547, -14*w - 71],
[557, 557, 66*w + 317],
[557, 557, 66*w - 383],
[563, 563, 2*w - 27],
[563, 563, -2*w - 25],
[569, 569, -64*w + 373],
[569, 569, 64*w + 309],
[587, 587, 12*w + 53],
[587, 587, -12*w + 65],
[593, 593, 8*w + 45],
[593, 593, 8*w - 53],
[601, 601, -26*w - 123],
[601, 601, 26*w - 149],
[617, 617, 6*w - 23],
[617, 617, -6*w - 17],
[647, 647, 24*w - 137],
[647, 647, -24*w - 113],
[653, 653, 28*w - 165],
[653, 653, -28*w - 137],
[677, 677, -22*w - 103],
[677, 677, 22*w - 125],
[691, 691, 20*w - 113],
[691, 691, 20*w + 93],
[709, 709, -10*w - 41],
[709, 709, 10*w - 51],
[719, 719, -8*w + 37],
[719, 719, -8*w - 29],
[727, 727, -22*w - 109],
[727, 727, 22*w - 131],
[739, 739, 46*w - 269],
[739, 739, -46*w - 223],
[761, 761, -6*w - 13],
[761, 761, 6*w - 19],
[769, 769, 110*w - 639],
[769, 769, 110*w + 529],
[773, 773, 4*w - 37],
[773, 773, -4*w - 33],
[787, 787, 2*w - 31],
[787, 787, -2*w - 29],
[809, 809, 56*w + 271],
[809, 809, -56*w + 327],
[821, 821, 6*w - 17],
[821, 821, -6*w - 11],
[823, 823, 38*w - 223],
[823, 823, -38*w - 185],
[827, 827, 132*w - 767],
[827, 827, 132*w + 635],
[841, 29, -29],
[853, 853, -76*w + 443],
[853, 853, 76*w + 367],
[863, 863, 14*w - 87],
[863, 863, -14*w - 73],
[911, 911, 2*w - 33],
[911, 911, -2*w - 31],
[919, 919, -6*w - 41],
[919, 919, 6*w - 47],
[929, 929, 50*w + 239],
[929, 929, 50*w - 289],
[953, 953, 6*w - 11],
[953, 953, -6*w - 5],
[967, 967, -8*w - 25],
[967, 967, 8*w - 33],
[991, 991, 16*w + 71],
[991, 991, -16*w + 87]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 + x^9 - 15*x^8 - 11*x^7 + 76*x^6 + 36*x^5 - 150*x^4 - 44*x^3 + 98*x^2 + 20*x - 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, 3/4*e^9 + 1/4*e^8 - 45/4*e^7 - 3/4*e^6 + 55*e^5 - 10*e^4 - 189/2*e^3 + 33*e^2 + 73/2*e - 14, 3/4*e^9 + 1/4*e^8 - 45/4*e^7 - 3/4*e^6 + 55*e^5 - 10*e^4 - 189/2*e^3 + 33*e^2 + 73/2*e - 14, -1, -e^9 - e^8 + 14*e^7 + 9*e^6 - 64*e^5 - 16*e^4 + 106*e^3 - 6*e^2 - 43*e + 6, -e^9 - e^8 + 14*e^7 + 9*e^6 - 64*e^5 - 16*e^4 + 106*e^3 - 6*e^2 - 43*e + 6, -1/2*e^9 + 1/2*e^8 + 9*e^7 - 15/2*e^6 - 105/2*e^5 + 36*e^4 + 107*e^3 - 59*e^2 - 52*e + 19, -1/2*e^9 + 1/2*e^8 + 9*e^7 - 15/2*e^6 - 105/2*e^5 + 36*e^4 + 107*e^3 - 59*e^2 - 52*e + 19, 1/4*e^9 + 1/4*e^8 - 13/4*e^7 - 7/4*e^6 + 27/2*e^5 - e^4 - 39/2*e^3 + 14*e^2 + 7/2*e - 1, 1/2*e^7 + e^6 - 11/2*e^5 - 9*e^4 + 16*e^3 + 17*e^2 - 9*e - 3, 1/2*e^7 + e^6 - 11/2*e^5 - 9*e^4 + 16*e^3 + 17*e^2 - 9*e - 3, -3/2*e^9 + 1/2*e^8 + 49/2*e^7 - 23/2*e^6 - 131*e^5 + 74*e^4 + 246*e^3 - 146*e^2 - 105*e + 54, -3/2*e^9 + 1/2*e^8 + 49/2*e^7 - 23/2*e^6 - 131*e^5 + 74*e^4 + 246*e^3 - 146*e^2 - 105*e + 54, -1/2*e^9 - 1/2*e^8 + 13/2*e^7 + 7/2*e^6 - 27*e^5 + 2*e^4 + 42*e^3 - 28*e^2 - 22*e + 12, -1/2*e^9 - 1/2*e^8 + 13/2*e^7 + 7/2*e^6 - 27*e^5 + 2*e^4 + 42*e^3 - 28*e^2 - 22*e + 12, 3/4*e^9 - 1/4*e^8 - 47/4*e^7 + 27/4*e^6 + 121/2*e^5 - 46*e^4 - 221/2*e^3 + 90*e^2 + 97/2*e - 31, 3/4*e^9 - 1/4*e^8 - 47/4*e^7 + 27/4*e^6 + 121/2*e^5 - 46*e^4 - 221/2*e^3 + 90*e^2 + 97/2*e - 31, 3/2*e^9 - 1/2*e^8 - 49/2*e^7 + 23/2*e^6 + 131*e^5 - 74*e^4 - 246*e^3 + 144*e^2 + 105*e - 50, 3/2*e^9 - 1/2*e^8 - 49/2*e^7 + 23/2*e^6 + 131*e^5 - 74*e^4 - 246*e^3 + 144*e^2 + 105*e - 50, 2*e^9 + e^8 - 30*e^7 - 7*e^6 + 148*e^5 - 2*e^4 - 262*e^3 + 45*e^2 + 110*e - 19, 2*e^9 + e^8 - 30*e^7 - 7*e^6 + 148*e^5 - 2*e^4 - 262*e^3 + 45*e^2 + 110*e - 19, -1/4*e^9 + 3/4*e^8 + 21/4*e^7 - 37/4*e^6 - 67/2*e^5 + 34*e^4 + 143/2*e^3 - 37*e^2 - 67/2*e + 1, -1/4*e^9 + 3/4*e^8 + 21/4*e^7 - 37/4*e^6 - 67/2*e^5 + 34*e^4 + 143/2*e^3 - 37*e^2 - 67/2*e + 1, 2*e^9 + e^8 - 28*e^7 - 3*e^6 + 126*e^5 - 40*e^4 - 198*e^3 + 130*e^2 + 72*e - 48, -2*e^9 - e^8 + 30*e^7 + 6*e^6 - 148*e^5 + 14*e^4 + 262*e^3 - 82*e^2 - 106*e + 43, -2*e^9 - e^8 + 30*e^7 + 6*e^6 - 148*e^5 + 14*e^4 + 262*e^3 - 82*e^2 - 106*e + 43, 3/2*e^9 - 3/2*e^8 - 53/2*e^7 + 45/2*e^6 + 151*e^5 - 108*e^4 - 298*e^3 + 176*e^2 + 139*e - 58, 3/2*e^9 - 3/2*e^8 - 53/2*e^7 + 45/2*e^6 + 151*e^5 - 108*e^4 - 298*e^3 + 176*e^2 + 139*e - 58, -1/4*e^9 - 5/4*e^8 + 9/4*e^7 + 63/4*e^6 - 5/2*e^5 - 61*e^4 - 29/2*e^3 + 79*e^2 + 49/2*e - 25, -1/4*e^9 - 5/4*e^8 + 9/4*e^7 + 63/4*e^6 - 5/2*e^5 - 61*e^4 - 29/2*e^3 + 79*e^2 + 49/2*e - 25, 1/2*e^9 - 1/2*e^8 - 19/2*e^7 + 11/2*e^6 + 57*e^5 - 16*e^4 - 114*e^3 + 12*e^2 + 43*e + 2, 1/2*e^9 - 1/2*e^8 - 19/2*e^7 + 11/2*e^6 + 57*e^5 - 16*e^4 - 114*e^3 + 12*e^2 + 43*e + 2, -3*e^9 - 2*e^8 + 43*e^7 + 13*e^6 - 202*e^5 + 16*e^4 + 346*e^3 - 124*e^2 - 154*e + 57, -3*e^9 - 2*e^8 + 43*e^7 + 13*e^6 - 202*e^5 + 16*e^4 + 346*e^3 - 124*e^2 - 154*e + 57, -5/4*e^9 - 1/4*e^8 + 77/4*e^7 - 1/4*e^6 - 193/2*e^5 + 17*e^4 + 335/2*e^3 - 38*e^2 - 123/2*e + 7, -5/4*e^9 - 1/4*e^8 + 77/4*e^7 - 1/4*e^6 - 193/2*e^5 + 17*e^4 + 335/2*e^3 - 38*e^2 - 123/2*e + 7, 2*e^8 + 5*e^7 - 22*e^6 - 52*e^5 + 68*e^4 + 140*e^3 - 62*e^2 - 83*e + 10, 2*e^8 + 5*e^7 - 22*e^6 - 52*e^5 + 68*e^4 + 140*e^3 - 62*e^2 - 83*e + 10, 11/4*e^9 + 1/4*e^8 - 173/4*e^7 + 17/4*e^6 + 223*e^5 - 54*e^4 - 809/2*e^3 + 120*e^2 + 321/2*e - 42, 11/4*e^9 + 1/4*e^8 - 173/4*e^7 + 17/4*e^6 + 223*e^5 - 54*e^4 - 809/2*e^3 + 120*e^2 + 321/2*e - 42, -9/2*e^9 - 3/2*e^8 + 135/2*e^7 + 5/2*e^6 - 333*e^5 + 80*e^4 + 590*e^3 - 248*e^2 - 243*e + 108, -9/2*e^9 - 3/2*e^8 + 135/2*e^7 + 5/2*e^6 - 333*e^5 + 80*e^4 + 590*e^3 - 248*e^2 - 243*e + 108, -1/2*e^9 - 5/2*e^8 + 7/2*e^7 + 59/2*e^6 + 4*e^5 - 104*e^4 - 43*e^3 + 124*e^2 + 41*e - 48, -1/2*e^9 - 5/2*e^8 + 7/2*e^7 + 59/2*e^6 + 4*e^5 - 104*e^4 - 43*e^3 + 124*e^2 + 41*e - 48, -e^8 - 2*e^7 + 13*e^6 + 21*e^5 - 56*e^4 - 58*e^3 + 92*e^2 + 35*e - 36, -e^8 - 2*e^7 + 13*e^6 + 21*e^5 - 56*e^4 - 58*e^3 + 92*e^2 + 35*e - 36, e^9 - e^8 - 17*e^7 + 17*e^6 + 96*e^5 - 91*e^4 - 200*e^3 + 160*e^2 + 114*e - 53, e^9 - e^8 - 17*e^7 + 17*e^6 + 96*e^5 - 91*e^4 - 200*e^3 + 160*e^2 + 114*e - 53, 1/2*e^9 - 1/2*e^8 - 17/2*e^7 + 19/2*e^6 + 49*e^5 - 58*e^4 - 109*e^3 + 120*e^2 + 71*e - 44, 1/2*e^9 - 1/2*e^8 - 17/2*e^7 + 19/2*e^6 + 49*e^5 - 58*e^4 - 109*e^3 + 120*e^2 + 71*e - 44, 2*e^9 + 2*e^8 - 28*e^7 - 16*e^6 + 129*e^5 + 10*e^4 - 220*e^3 + 74*e^2 + 101*e - 36, 2*e^9 + 2*e^8 - 28*e^7 - 16*e^6 + 129*e^5 + 10*e^4 - 220*e^3 + 74*e^2 + 101*e - 36, -13/4*e^9 - 11/4*e^8 + 183/4*e^7 + 89/4*e^6 - 210*e^5 - 21*e^4 + 695/2*e^3 - 62*e^2 - 279/2*e + 20, -13/4*e^9 - 11/4*e^8 + 183/4*e^7 + 89/4*e^6 - 210*e^5 - 21*e^4 + 695/2*e^3 - 62*e^2 - 279/2*e + 20, 11/4*e^9 + 9/4*e^8 - 157/4*e^7 - 79/4*e^6 + 183*e^5 + 34*e^4 - 617/2*e^3 + 10*e^2 + 257/2*e - 18, 11/4*e^9 + 9/4*e^8 - 157/4*e^7 - 79/4*e^6 + 183*e^5 + 34*e^4 - 617/2*e^3 + 10*e^2 + 257/2*e - 18, 7/2*e^9 + 7/2*e^8 - 48*e^7 - 57/2*e^6 + 429/2*e^5 + 27*e^4 - 351*e^3 + 86*e^2 + 148*e - 23, 5/4*e^9 + 5/4*e^8 - 69/4*e^7 - 43/4*e^6 + 157/2*e^5 + 16*e^4 - 275/2*e^3 + 11*e^2 + 151/2*e + 1, 5/4*e^9 + 5/4*e^8 - 69/4*e^7 - 43/4*e^6 + 157/2*e^5 + 16*e^4 - 275/2*e^3 + 11*e^2 + 151/2*e + 1, 5*e^9 + 3*e^8 - 73*e^7 - 19*e^6 + 351*e^5 - 28*e^4 - 613*e^3 + 190*e^2 + 256*e - 76, 5*e^9 + 3*e^8 - 73*e^7 - 19*e^6 + 351*e^5 - 28*e^4 - 613*e^3 + 190*e^2 + 256*e - 76, -7/4*e^9 - 7/4*e^8 + 99/4*e^7 + 61/4*e^6 - 229/2*e^5 - 21*e^4 + 389/2*e^3 - 37*e^2 - 193/2*e + 31, -7/4*e^9 - 7/4*e^8 + 99/4*e^7 + 61/4*e^6 - 229/2*e^5 - 21*e^4 + 389/2*e^3 - 37*e^2 - 193/2*e + 31, 5/2*e^9 + 1/2*e^8 - 73/2*e^7 + 13/2*e^6 + 173*e^5 - 96*e^4 - 287*e^3 + 234*e^2 + 107*e - 84, 5/2*e^9 + 1/2*e^8 - 73/2*e^7 + 13/2*e^6 + 173*e^5 - 96*e^4 - 287*e^3 + 234*e^2 + 107*e - 84, 17/4*e^9 + 7/4*e^8 - 255/4*e^7 - 29/4*e^6 + 315*e^5 - 53*e^4 - 1127/2*e^3 + 194*e^2 + 503/2*e - 74, 17/4*e^9 + 7/4*e^8 - 255/4*e^7 - 29/4*e^6 + 315*e^5 - 53*e^4 - 1127/2*e^3 + 194*e^2 + 503/2*e - 74, 3/2*e^9 - 3/2*e^8 - 27*e^7 + 41/2*e^6 + 311/2*e^5 - 86*e^4 - 307*e^3 + 113*e^2 + 140*e - 29, 3/2*e^9 - 3/2*e^8 - 27*e^7 + 41/2*e^6 + 311/2*e^5 - 86*e^4 - 307*e^3 + 113*e^2 + 140*e - 29, -1/2*e^9 - 1/2*e^8 + 13/2*e^7 + 7/2*e^6 - 27*e^5 + 42*e^3 - 14*e^2 - 22*e + 2, -1/2*e^9 - 1/2*e^8 + 13/2*e^7 + 7/2*e^6 - 27*e^5 + 42*e^3 - 14*e^2 - 22*e + 2, 5*e^9 + e^8 - 77*e^7 + 3*e^6 + 388*e^5 - 98*e^4 - 690*e^3 + 266*e^2 + 282*e - 106, 5*e^9 + e^8 - 77*e^7 + 3*e^6 + 388*e^5 - 98*e^4 - 690*e^3 + 266*e^2 + 282*e - 106, -2*e^9 + e^8 + 32*e^7 - 21*e^6 - 166*e^5 + 127*e^4 + 294*e^3 - 231*e^2 - 102*e + 90, 5/4*e^9 - 3/4*e^8 - 77/4*e^7 + 69/4*e^6 + 193/2*e^5 - 112*e^4 - 343/2*e^3 + 221*e^2 + 155/2*e - 79, 5/4*e^9 - 3/4*e^8 - 77/4*e^7 + 69/4*e^6 + 193/2*e^5 - 112*e^4 - 343/2*e^3 + 221*e^2 + 155/2*e - 79, 3/2*e^9 - 3/2*e^8 - 55/2*e^7 + 41/2*e^6 + 161*e^5 - 86*e^4 - 322*e^3 + 112*e^2 + 150*e - 22, 3/2*e^9 - 3/2*e^8 - 55/2*e^7 + 41/2*e^6 + 161*e^5 - 86*e^4 - 322*e^3 + 112*e^2 + 150*e - 22, 6*e^9 + 3*e^8 - 88*e^7 - 17*e^6 + 422*e^5 - 44*e^4 - 721*e^3 + 224*e^2 + 287*e - 114, 6*e^9 + 3*e^8 - 88*e^7 - 17*e^6 + 422*e^5 - 44*e^4 - 721*e^3 + 224*e^2 + 287*e - 114, -5*e^9 - 4*e^8 + 71*e^7 + 32*e^6 - 328*e^5 - 28*e^4 + 541*e^3 - 98*e^2 - 215*e + 48, -5*e^9 - 4*e^8 + 71*e^7 + 32*e^6 - 328*e^5 - 28*e^4 + 541*e^3 - 98*e^2 - 215*e + 48, -9/2*e^9 + 3/2*e^8 + 72*e^7 - 71/2*e^6 - 755/2*e^5 + 231*e^4 + 699*e^3 - 457*e^2 - 298*e + 165, -9/2*e^9 + 3/2*e^8 + 72*e^7 - 71/2*e^6 - 755/2*e^5 + 231*e^4 + 699*e^3 - 457*e^2 - 298*e + 165, 7*e^9 + 2*e^8 - 105*e^7 + 2*e^6 + 516*e^5 - 157*e^4 - 902*e^3 + 439*e^2 + 378*e - 168, 7*e^9 + 2*e^8 - 105*e^7 + 2*e^6 + 516*e^5 - 157*e^4 - 902*e^3 + 439*e^2 + 378*e - 168, -4*e^9 - 2*e^8 + 59*e^7 + 10*e^6 - 284*e^5 + 46*e^4 + 481*e^3 - 202*e^2 - 174*e + 94, -4*e^9 - 2*e^8 + 59*e^7 + 10*e^6 - 284*e^5 + 46*e^4 + 481*e^3 - 202*e^2 - 174*e + 94, -3/2*e^9 - 3/2*e^8 + 43/2*e^7 + 29/2*e^6 - 99*e^5 - 32*e^4 + 155*e^3 + 2*e^2 - 49*e + 30, -3/2*e^9 - 3/2*e^8 + 43/2*e^7 + 29/2*e^6 - 99*e^5 - 32*e^4 + 155*e^3 + 2*e^2 - 49*e + 30, 23/4*e^9 + 19/4*e^8 - 327/4*e^7 - 157/4*e^6 + 759/2*e^5 + 39*e^4 - 1273/2*e^3 + 132*e^2 + 525/2*e - 85, 23/4*e^9 + 19/4*e^8 - 327/4*e^7 - 157/4*e^6 + 759/2*e^5 + 39*e^4 - 1273/2*e^3 + 132*e^2 + 525/2*e - 85, 7/2*e^9 + 11/2*e^8 - 89/2*e^7 - 105/2*e^6 + 179*e^5 + 110*e^4 - 254*e^3 + 16*e^2 + 71*e - 60, 7/2*e^9 + 11/2*e^8 - 89/2*e^7 - 105/2*e^6 + 179*e^5 + 110*e^4 - 254*e^3 + 16*e^2 + 71*e - 60, -3/2*e^9 + 3/2*e^8 + 53/2*e^7 - 45/2*e^6 - 154*e^5 + 108*e^4 + 321*e^3 - 182*e^2 - 167*e + 60, -3/2*e^9 + 3/2*e^8 + 53/2*e^7 - 45/2*e^6 - 154*e^5 + 108*e^4 + 321*e^3 - 182*e^2 - 167*e + 60, 3/2*e^9 - 1/2*e^8 - 51/2*e^7 + 19/2*e^6 + 142*e^5 - 56*e^4 - 279*e^3 + 108*e^2 + 126*e - 38, 3/2*e^9 - 1/2*e^8 - 51/2*e^7 + 19/2*e^6 + 142*e^5 - 56*e^4 - 279*e^3 + 108*e^2 + 126*e - 38, 15/2*e^9 + 3/2*e^8 - 233/2*e^7 + 3/2*e^6 + 592*e^5 - 114*e^4 - 1060*e^3 + 308*e^2 + 432*e - 120, 15/2*e^9 + 3/2*e^8 - 233/2*e^7 + 3/2*e^6 + 592*e^5 - 114*e^4 - 1060*e^3 + 308*e^2 + 432*e - 120, -1/4*e^9 - 7/4*e^8 + 3/4*e^7 + 81/4*e^6 + 12*e^5 - 67*e^4 - 93/2*e^3 + 59*e^2 + 81/2*e + 24, -3*e^9 + e^8 + 49*e^7 - 21*e^6 - 264*e^5 + 125*e^4 + 516*e^3 - 224*e^2 - 272*e + 69, -3*e^9 + e^8 + 49*e^7 - 21*e^6 - 264*e^5 + 125*e^4 + 516*e^3 - 224*e^2 - 272*e + 69, -5*e^9 + 79*e^7 - 16*e^6 - 413*e^5 + 150*e^4 + 781*e^3 - 334*e^2 - 358*e + 126, -5*e^9 + 79*e^7 - 16*e^6 - 413*e^5 + 150*e^4 + 781*e^3 - 334*e^2 - 358*e + 126, -13/2*e^9 - 7/2*e^8 + 191/2*e^7 + 41/2*e^6 - 460*e^5 + 48*e^4 + 798*e^3 - 250*e^2 - 342*e + 86, -13/2*e^9 - 7/2*e^8 + 191/2*e^7 + 41/2*e^6 - 460*e^5 + 48*e^4 + 798*e^3 - 250*e^2 - 342*e + 86, 3*e^9 + 3*e^8 - 41*e^7 - 25*e^6 + 179*e^5 + 28*e^4 - 271*e^3 + 62*e^2 + 90*e - 26, 3*e^9 + 3*e^8 - 41*e^7 - 25*e^6 + 179*e^5 + 28*e^4 - 271*e^3 + 62*e^2 + 90*e - 26, 6*e^9 + e^8 - 91*e^7 + 11*e^6 + 454*e^5 - 170*e^4 - 810*e^3 + 410*e^2 + 341*e - 148, 6*e^9 + e^8 - 91*e^7 + 11*e^6 + 454*e^5 - 170*e^4 - 810*e^3 + 410*e^2 + 341*e - 148, -1/2*e^9 + 3/2*e^8 + 23/2*e^7 - 33/2*e^6 - 75*e^5 + 50*e^4 + 147*e^3 - 40*e^2 - 46*e + 16, -1/2*e^9 + 3/2*e^8 + 23/2*e^7 - 33/2*e^6 - 75*e^5 + 50*e^4 + 147*e^3 - 40*e^2 - 46*e + 16, -11/4*e^9 + 11/4*e^8 + 193/4*e^7 - 169/4*e^6 - 272*e^5 + 209*e^4 + 1041/2*e^3 - 358*e^2 - 421/2*e + 132, -11/4*e^9 + 11/4*e^8 + 193/4*e^7 - 169/4*e^6 - 272*e^5 + 209*e^4 + 1041/2*e^3 - 358*e^2 - 421/2*e + 132, -4*e^9 - 3*e^8 + 57*e^7 + 23*e^6 - 264*e^5 - 10*e^4 + 433*e^3 - 108*e^2 - 156*e + 58, -4*e^9 - 3*e^8 + 57*e^7 + 23*e^6 - 264*e^5 - 10*e^4 + 433*e^3 - 108*e^2 - 156*e + 58, 17/2*e^9 + 7/2*e^8 - 255/2*e^7 - 33/2*e^6 + 626*e^5 - 86*e^4 - 1088*e^3 + 352*e^2 + 420*e - 154, 17/2*e^9 + 7/2*e^8 - 255/2*e^7 - 33/2*e^6 + 626*e^5 - 86*e^4 - 1088*e^3 + 352*e^2 + 420*e - 154, 1/2*e^9 + 3/2*e^8 - 11/2*e^7 - 37/2*e^6 + 16*e^5 + 70*e^4 - 10*e^3 - 80*e^2 + 2*e - 8, 1/2*e^9 + 3/2*e^8 - 11/2*e^7 - 37/2*e^6 + 16*e^5 + 70*e^4 - 10*e^3 - 80*e^2 + 2*e - 8, 7/2*e^9 + 5/2*e^8 - 101/2*e^7 - 35/2*e^6 + 241*e^5 - 12*e^4 - 426*e^3 + 156*e^2 + 201*e - 70, 7/2*e^9 + 5/2*e^8 - 101/2*e^7 - 35/2*e^6 + 241*e^5 - 12*e^4 - 426*e^3 + 156*e^2 + 201*e - 70, -1/4*e^9 - 15/4*e^8 - 17/4*e^7 + 161/4*e^6 + 63*e^5 - 115*e^4 - 361/2*e^3 + 79*e^2 + 237/2*e - 18, -1/4*e^9 - 15/4*e^8 - 17/4*e^7 + 161/4*e^6 + 63*e^5 - 115*e^4 - 361/2*e^3 + 79*e^2 + 237/2*e - 18, 13/4*e^9 - 5/4*e^8 - 215/4*e^7 + 107/4*e^6 + 292*e^5 - 165*e^4 - 1119/2*e^3 + 311*e^2 + 475/2*e - 104, 13/4*e^9 - 5/4*e^8 - 215/4*e^7 + 107/4*e^6 + 292*e^5 - 165*e^4 - 1119/2*e^3 + 311*e^2 + 475/2*e - 104, -3/2*e^9 + 3/2*e^8 + 49/2*e^7 - 49/2*e^6 - 128*e^5 + 124*e^4 + 226*e^3 - 198*e^2 - 92*e + 58, -3/2*e^9 + 3/2*e^8 + 49/2*e^7 - 49/2*e^6 - 128*e^5 + 124*e^4 + 226*e^3 - 198*e^2 - 92*e + 58, -e^9 - 2*e^8 + 23/2*e^7 + 21*e^6 - 75/2*e^5 - 58*e^4 + 32*e^3 + 38*e^2 + 11*e + 5, -e^9 - 2*e^8 + 23/2*e^7 + 21*e^6 - 75/2*e^5 - 58*e^4 + 32*e^3 + 38*e^2 + 11*e + 5, 2*e^9 + 5*e^8 - 22*e^7 - 53*e^6 + 66*e^5 + 153*e^4 - 44*e^3 - 125*e^2 - 22*e + 28, 2*e^9 + 5*e^8 - 22*e^7 - 53*e^6 + 66*e^5 + 153*e^4 - 44*e^3 - 125*e^2 - 22*e + 28, 3/2*e^9 + 9/2*e^8 - 33/2*e^7 - 103/2*e^6 + 53*e^5 + 172*e^4 - 65*e^3 - 174*e^2 + 44*e + 20, 3/2*e^9 + 9/2*e^8 - 33/2*e^7 - 103/2*e^6 + 53*e^5 + 172*e^4 - 65*e^3 - 174*e^2 + 44*e + 20, e^6 + 2*e^5 - 11*e^4 - 22*e^3 + 27*e^2 + 48*e - 7, e^6 + 2*e^5 - 11*e^4 - 22*e^3 + 27*e^2 + 48*e - 7, -3/2*e^9 - 5/2*e^8 + 41/2*e^7 + 59/2*e^6 - 89*e^5 - 108*e^4 + 126*e^3 + 150*e^2 - 9*e - 68, -3/2*e^9 - 5/2*e^8 + 41/2*e^7 + 59/2*e^6 - 89*e^5 - 108*e^4 + 126*e^3 + 150*e^2 - 9*e - 68, -5*e^9 - 3*e^8 + 73*e^7 + 18*e^6 - 352*e^5 + 39*e^4 + 622*e^3 - 217*e^2 - 264*e + 81, -5*e^9 - 3*e^8 + 73*e^7 + 18*e^6 - 352*e^5 + 39*e^4 + 622*e^3 - 217*e^2 - 264*e + 81, 9/2*e^9 + 3/2*e^8 - 133/2*e^7 + 3/2*e^6 + 325*e^5 - 118*e^4 - 580*e^3 + 332*e^2 + 248*e - 132, 9/2*e^9 + 3/2*e^8 - 133/2*e^7 + 3/2*e^6 + 325*e^5 - 118*e^4 - 580*e^3 + 332*e^2 + 248*e - 132, -e^9 + e^8 + 16*e^7 - 19*e^6 - 84*e^5 + 108*e^4 + 160*e^3 - 192*e^2 - 89*e + 62, -e^9 + e^8 + 16*e^7 - 19*e^6 - 84*e^5 + 108*e^4 + 160*e^3 - 192*e^2 - 89*e + 62, -2*e^9 + e^8 + 32*e^7 - 21*e^6 - 166*e^5 + 131*e^4 + 300*e^3 - 255*e^2 - 136*e + 84, -2*e^9 + e^8 + 32*e^7 - 21*e^6 - 166*e^5 + 131*e^4 + 300*e^3 - 255*e^2 - 136*e + 84, 3/2*e^9 - 1/2*e^8 - 47/2*e^7 + 31/2*e^6 + 122*e^5 - 116*e^4 - 227*e^3 + 254*e^2 + 102*e - 74, 3/2*e^9 - 1/2*e^8 - 47/2*e^7 + 31/2*e^6 + 122*e^5 - 116*e^4 - 227*e^3 + 254*e^2 + 102*e - 74, -8*e^9 - 6*e^8 + 114*e^7 + 45*e^6 - 530*e^5 - 11*e^4 + 882*e^3 - 231*e^2 - 350*e + 145, 4*e^9 - 66*e^7 + 7*e^6 + 358*e^5 - 65*e^4 - 690*e^3 + 143*e^2 + 322*e - 51, 4*e^9 - 66*e^7 + 7*e^6 + 358*e^5 - 65*e^4 - 690*e^3 + 143*e^2 + 322*e - 51, e^9 - 5*e^8 - 24*e^7 + 63*e^6 + 163*e^5 - 242*e^4 - 355*e^3 + 298*e^2 + 183*e - 66, e^9 - 5*e^8 - 24*e^7 + 63*e^6 + 163*e^5 - 242*e^4 - 355*e^3 + 298*e^2 + 183*e - 66, -7*e^9 + e^8 + 111*e^7 - 37*e^6 - 580*e^5 + 274*e^4 + 1085*e^3 - 544*e^2 - 491*e + 168, -7*e^9 + e^8 + 111*e^7 - 37*e^6 - 580*e^5 + 274*e^4 + 1085*e^3 - 544*e^2 - 491*e + 168, -9*e^9 - 3*e^8 + 139*e^7 + 10*e^6 - 708*e^5 + 113*e^4 + 1288*e^3 - 389*e^2 - 536*e + 171, -9*e^9 - 3*e^8 + 139*e^7 + 10*e^6 - 708*e^5 + 113*e^4 + 1288*e^3 - 389*e^2 - 536*e + 171, -2*e^9 + 3*e^8 + 38*e^7 - 43*e^6 - 228*e^5 + 202*e^4 + 457*e^3 - 340*e^2 - 181*e + 120, -2*e^9 + 3*e^8 + 38*e^7 - 43*e^6 - 228*e^5 + 202*e^4 + 457*e^3 - 340*e^2 - 181*e + 120, -3/2*e^9 - 5/2*e^8 + 41/2*e^7 + 59/2*e^6 - 93*e^5 - 104*e^4 + 165*e^3 + 108*e^2 - 92*e - 6, -3/2*e^9 - 5/2*e^8 + 41/2*e^7 + 59/2*e^6 - 93*e^5 - 104*e^4 + 165*e^3 + 108*e^2 - 92*e - 6, -10*e^9 - 3*e^8 + 301/2*e^7 - e^6 - 1487/2*e^5 + 220*e^4 + 1310*e^3 - 639*e^2 - 525*e + 273, -10*e^9 - 3*e^8 + 301/2*e^7 - e^6 - 1487/2*e^5 + 220*e^4 + 1310*e^3 - 639*e^2 - 525*e + 273, e^9 - e^8 - 35/2*e^7 + 16*e^6 + 195/2*e^5 - 81*e^4 - 174*e^3 + 135*e^2 + 29*e - 57, e^9 - e^8 - 35/2*e^7 + 16*e^6 + 195/2*e^5 - 81*e^4 - 174*e^3 + 135*e^2 + 29*e - 57];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 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;