/*
  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 | {11, 11, 4*w + 19}>;

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^9 + 4*x^8 - 6*x^7 - 35*x^6 + x^5 + 90*x^4 + 31*x^3 - 67*x^2 - 29*x - 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [13/9*e^8 + 17/3*e^7 - 28/3*e^6 - 452/9*e^5 + 8*e^4 + 133*e^3 + 223/9*e^2 - 965/9*e - 70/3, e, -19/9*e^8 - 23/3*e^7 + 46/3*e^6 + 608/9*e^5 - 25*e^4 - 175*e^3 - 94/9*e^2 + 1211/9*e + 73/3, 13/9*e^8 + 17/3*e^7 - 28/3*e^6 - 452/9*e^5 + 8*e^4 + 132*e^3 + 214/9*e^2 - 929/9*e - 64/3, 7/3*e^8 + 9*e^7 - 16*e^6 - 242/3*e^5 + 20*e^4 + 216*e^3 + 82/3*e^2 - 521/3*e - 36, 1, -16/9*e^8 - 20/3*e^7 + 37/3*e^6 + 539/9*e^5 - 15*e^4 - 161*e^3 - 226/9*e^2 + 1178/9*e + 91/3, 4/9*e^8 + 5/3*e^7 - 10/3*e^6 - 146/9*e^5 + 5*e^4 + 49*e^3 + 43/9*e^2 - 407/9*e - 28/3, -16/9*e^8 - 20/3*e^7 + 37/3*e^6 + 530/9*e^5 - 17*e^4 - 154*e^3 - 136/9*e^2 + 1070/9*e + 70/3, 4/3*e^8 + 5*e^7 - 9*e^6 - 128/3*e^5 + 11*e^4 + 103*e^3 + 43/3*e^2 - 206/3*e - 20, 22/3*e^8 + 27*e^7 - 51*e^6 - 710/3*e^5 + 69*e^4 + 611*e^3 + 211/3*e^2 - 1400/3*e - 98, 1/9*e^8 + 2/3*e^7 - 4/3*e^6 - 77/9*e^5 + 4*e^4 + 32*e^3 + 4/9*e^2 - 311/9*e - 16/3, -1/3*e^8 - 2*e^7 + e^6 + 59/3*e^5 + 8*e^4 - 61*e^3 - 85/3*e^2 + 170/3*e + 16, -28/9*e^8 - 38/3*e^7 + 58/3*e^6 + 1022/9*e^5 - 11*e^4 - 306*e^3 - 598/9*e^2 + 2264/9*e + 175/3, -37/9*e^8 - 47/3*e^7 + 82/3*e^6 + 1247/9*e^5 - 27*e^4 - 364*e^3 - 607/9*e^2 + 2597/9*e + 202/3, -8/9*e^8 - 10/3*e^7 + 20/3*e^6 + 283/9*e^5 - 11*e^4 - 90*e^3 - 50/9*e^2 + 679/9*e + 32/3, -13/3*e^8 - 18*e^7 + 26*e^6 + 485/3*e^5 - 7*e^4 - 436*e^3 - 325/3*e^2 + 1064/3*e + 78, -52/9*e^8 - 65/3*e^7 + 118/3*e^6 + 1700/9*e^5 - 50*e^4 - 483*e^3 - 496/9*e^2 + 3293/9*e + 205/3, -61/9*e^8 - 77/3*e^7 + 139/3*e^6 + 2042/9*e^5 - 56*e^4 - 591*e^3 - 775/9*e^2 + 4049/9*e + 298/3, -28/3*e^8 - 34*e^7 + 66*e^6 + 899/3*e^5 - 95*e^4 - 781*e^3 - 229/3*e^2 + 1826/3*e + 121, 17/3*e^8 + 19*e^7 - 44*e^6 - 499/3*e^5 + 86*e^4 + 424*e^3 + 17/3*e^2 - 934/3*e - 75, 91/9*e^8 + 116/3*e^7 - 205/3*e^6 - 3092/9*e^5 + 78*e^4 + 908*e^3 + 1201/9*e^2 - 6494/9*e - 439/3, 32/9*e^8 + 46/3*e^7 - 56/3*e^6 - 1231/9*e^5 - 14*e^4 + 374*e^3 + 1127/9*e^2 - 2914/9*e - 248/3, -71/9*e^8 - 91/3*e^7 + 158/3*e^6 + 2398/9*e^5 - 59*e^4 - 689*e^3 - 869/9*e^2 + 4756/9*e + 308/3, 29/9*e^8 + 43/3*e^7 - 53/3*e^6 - 1153/9*e^5 - 4*e^4 + 346*e^3 + 800/9*e^2 - 2602/9*e - 170/3, -35/9*e^8 - 52/3*e^7 + 62/3*e^6 + 1408/9*e^5 + 15*e^4 - 427*e^3 - 1256/9*e^2 + 3190/9*e + 227/3, 1/3*e^8 + 3*e^7 + e^6 - 95/3*e^5 - 25*e^4 + 108*e^3 + 175/3*e^2 - 341/3*e - 17, 41/9*e^8 + 55/3*e^7 - 86/3*e^6 - 1474/9*e^5 + 19*e^4 + 439*e^3 + 812/9*e^2 - 3220/9*e - 254/3, 19/9*e^8 + 26/3*e^7 - 37/3*e^6 - 680/9*e^5 + 3*e^4 + 195*e^3 + 409/9*e^2 - 1382/9*e - 76/3, 11/3*e^8 + 15*e^7 - 24*e^6 - 409/3*e^5 + 22*e^4 + 371*e^3 + 188/3*e^2 - 904/3*e - 63, -41/9*e^8 - 55/3*e^7 + 86/3*e^6 + 1483/9*e^5 - 17*e^4 - 445*e^3 - 893/9*e^2 + 3292/9*e + 242/3, 88/9*e^8 + 110/3*e^7 - 202/3*e^6 - 2942/9*e^5 + 83*e^4 + 872*e^3 + 1090/9*e^2 - 6389/9*e - 442/3, 112/9*e^8 + 137/3*e^7 - 262/3*e^6 - 3629/9*e^5 + 119*e^4 + 1050*e^3 + 1123/9*e^2 - 7256/9*e - 520/3, 11/3*e^8 + 13*e^7 - 27*e^6 - 343/3*e^5 + 47*e^4 + 296*e^3 + 32/3*e^2 - 670/3*e - 45, -44/3*e^8 - 53*e^7 + 105*e^6 + 1402/3*e^5 - 158*e^4 - 1218*e^3 - 332/3*e^2 + 2848/3*e + 190, -92/9*e^8 - 124/3*e^7 + 188/3*e^6 + 3313/9*e^5 - 29*e^4 - 985*e^3 - 2051/9*e^2 + 7192/9*e + 518/3, -77/9*e^8 - 94/3*e^7 + 185/3*e^6 + 2527/9*e^5 - 93*e^4 - 748*e^3 - 614/9*e^2 + 5317/9*e + 341/3, 41/9*e^8 + 52/3*e^7 - 92/3*e^6 - 1402/9*e^5 + 32*e^4 + 419*e^3 + 605/9*e^2 - 3013/9*e - 200/3, -77/9*e^8 - 100/3*e^7 + 170/3*e^6 + 2680/9*e^5 - 57*e^4 - 796*e^3 - 1181/9*e^2 + 5731/9*e + 389/3, 7/9*e^8 + 5/3*e^7 - 28/3*e^6 - 143/9*e^5 + 36*e^4 + 41*e^3 - 350/9*e^2 - 224/9*e - 49/3, -131/9*e^8 - 172/3*e^7 + 281/3*e^6 + 4570/9*e^5 - 78*e^4 - 1336*e^3 - 2207/9*e^2 + 9502/9*e + 647/3, -20/9*e^8 - 25/3*e^7 + 47/3*e^6 + 640/9*e^5 - 27*e^4 - 174*e^3 + 1/9*e^2 + 1090/9*e + 62/3, -103/9*e^8 - 134/3*e^7 + 226/3*e^6 + 3620/9*e^5 - 70*e^4 - 1092*e^3 - 1690/9*e^2 + 8129/9*e + 538/3, -4/3*e^8 - 6*e^7 + 9*e^6 + 179/3*e^5 - 8*e^4 - 183*e^3 - 79/3*e^2 + 515/3*e + 25, -2/3*e^8 - 4*e^7 + e^6 + 109/3*e^5 + 22*e^4 - 99*e^3 - 164/3*e^2 + 241/3*e + 2, -12*e^8 - 44*e^7 + 86*e^6 + 388*e^5 - 132*e^4 - 1010*e^3 - 91*e^2 + 789*e + 177, 59/9*e^8 + 70/3*e^7 - 143/3*e^6 - 1834/9*e^5 + 77*e^4 + 518*e^3 + 299/9*e^2 - 3436/9*e - 254/3, 1/3*e^8 + 2*e^7 - e^6 - 56/3*e^5 - 9*e^4 + 55*e^3 + 115/3*e^2 - 185/3*e - 32, -88/9*e^8 - 116/3*e^7 + 193/3*e^6 + 3104/9*e^5 - 66*e^4 - 920*e^3 - 1261/9*e^2 + 6659/9*e + 457/3, -10/3*e^8 - 12*e^7 + 23*e^6 + 311/3*e^5 - 31*e^4 - 263*e^3 - 73/3*e^2 + 590/3*e + 31, 7*e^8 + 25*e^7 - 51*e^6 - 216*e^5 + 88*e^4 + 542*e^3 + 15*e^2 - 398*e - 75, 89/9*e^8 + 115/3*e^7 - 194/3*e^6 - 3055/9*e^5 + 57*e^4 + 891*e^3 + 1544/9*e^2 - 6304/9*e - 437/3, 38/9*e^8 + 46/3*e^7 - 86/3*e^6 - 1189/9*e^5 + 36*e^4 + 337*e^3 + 341/9*e^2 - 2404/9*e - 125/3, -149/9*e^8 - 193/3*e^7 + 320/3*e^6 + 5101/9*e^5 - 89*e^4 - 1481*e^3 - 2504/9*e^2 + 10456/9*e + 728/3, 10*e^8 + 34*e^7 - 78*e^6 - 301*e^5 + 156*e^4 + 780*e^3 - 14*e^2 - 589*e - 106, 50/9*e^8 + 64/3*e^7 - 116/3*e^6 - 1717/9*e^5 + 53*e^4 + 508*e^3 + 515/9*e^2 - 3598/9*e - 284/3, 92/9*e^8 + 112/3*e^7 - 221/3*e^6 - 2980/9*e^5 + 117*e^4 + 873*e^3 + 521/9*e^2 - 6265/9*e - 371/3, 44/9*e^8 + 55/3*e^7 - 101/3*e^6 - 1480/9*e^5 + 43*e^4 + 441*e^3 + 383/9*e^2 - 3127/9*e - 143/3, -61/3*e^8 - 75*e^7 + 143*e^6 + 2006/3*e^5 - 194*e^4 - 1776*e^3 - 643/3*e^2 + 4247/3*e + 291, -2/9*e^8 + 8/3*e^7 + 35/3*e^6 - 188/9*e^5 - 76*e^4 + 54*e^3 + 1063/9*e^2 - 449/9*e - 22/3, 14/3*e^8 + 19*e^7 - 29*e^6 - 502/3*e^5 + 20*e^4 + 441*e^3 + 251/3*e^2 - 1090/3*e - 65, -32/3*e^8 - 42*e^7 + 70*e^6 + 1126/3*e^5 - 67*e^4 - 1003*e^3 - 497/3*e^2 + 2443/3*e + 155, -13/9*e^8 - 23/3*e^7 + 16/3*e^6 + 614/9*e^5 + 20*e^4 - 180*e^3 - 601/9*e^2 + 1172/9*e + 58/3, -7/9*e^8 - 5/3*e^7 + 25/3*e^6 + 125/9*e^5 - 28*e^4 - 27*e^3 + 179/9*e^2 + 62/9*e + 61/3, -68/9*e^8 - 79/3*e^7 + 176/3*e^6 + 2140/9*e^5 - 115*e^4 - 641*e^3 - 2/9*e^2 + 4615/9*e + 263/3, 133/9*e^8 + 164/3*e^7 - 316/3*e^6 - 4409/9*e^5 + 151*e^4 + 1308*e^3 + 1261/9*e^2 - 9431/9*e - 622/3, -2/9*e^8 + 2/3*e^7 + 11/3*e^6 - 71/9*e^5 - 18*e^4 + 29*e^3 + 298/9*e^2 - 278/9*e - 67/3, 44/9*e^8 + 61/3*e^7 - 89/3*e^6 - 1624/9*e^5 + 14*e^4 + 473*e^3 + 914/9*e^2 - 3208/9*e - 233/3, 125/9*e^8 + 172/3*e^7 - 251/3*e^6 - 4585/9*e^5 + 30*e^4 + 1353*e^3 + 2948/9*e^2 - 9787/9*e - 710/3, -17/3*e^8 - 22*e^7 + 39*e^6 + 592/3*e^5 - 49*e^4 - 525*e^3 - 212/3*e^2 + 1243/3*e + 103, -5/3*e^8 - 5*e^7 + 13*e^6 + 121/3*e^5 - 29*e^4 - 91*e^3 + 64/3*e^2 + 178/3*e - 19, -92/9*e^8 - 112/3*e^7 + 209/3*e^6 + 2917/9*e^5 - 85*e^4 - 831*e^3 - 1088/9*e^2 + 5797/9*e + 440/3, -26/9*e^8 - 28/3*e^7 + 74/3*e^6 + 805/9*e^5 - 57*e^4 - 264*e^3 + 175/9*e^2 + 2164/9*e + 41/3, -52/9*e^8 - 65/3*e^7 + 121/3*e^6 + 1745/9*e^5 - 52*e^4 - 519*e^3 - 667/9*e^2 + 3761/9*e + 262/3, -22*e^8 - 86*e^7 + 143*e^6 + 761*e^5 - 131*e^4 - 2002*e^3 - 345*e^2 + 1582*e + 323, 11/9*e^8 + 10/3*e^7 - 35/3*e^6 - 253/9*e^5 + 34*e^4 + 60*e^3 - 172/9*e^2 - 154/9*e - 74/3, 11*e^8 + 43*e^7 - 71*e^6 - 381*e^5 + 60*e^4 + 1009*e^3 + 189*e^2 - 808*e - 160, 59/9*e^8 + 79/3*e^7 - 128/3*e^6 - 2122/9*e^5 + 40*e^4 + 633*e^3 + 911/9*e^2 - 4696/9*e - 290/3, -18*e^8 - 71*e^7 + 114*e^6 + 630*e^5 - 78*e^4 - 1668*e^3 - 349*e^2 + 1326*e + 285, 4/9*e^8 - 7/3*e^7 - 25/3*e^6 + 313/9*e^5 + 50*e^4 - 157*e^3 - 929/9*e^2 + 1771/9*e + 137/3, -92/9*e^8 - 115/3*e^7 + 209/3*e^6 + 3016/9*e^5 - 87*e^4 - 860*e^3 - 998/9*e^2 + 5851/9*e + 431/3, 274/9*e^8 + 353/3*e^7 - 607/3*e^6 - 9407/9*e^5 + 210*e^4 + 2762*e^3 + 3967/9*e^2 - 19658/9*e - 1348/3, -5/9*e^8 - 1/3*e^7 + 20/3*e^6 + 7/9*e^5 - 27*e^4 + 3*e^3 + 340/9*e^2 - 29/9*e - 52/3, 137/9*e^8 + 184/3*e^7 - 287/3*e^6 - 4933/9*e^5 + 64*e^4 + 1474*e^3 + 2681/9*e^2 - 10891/9*e - 758/3, 11*e^8 + 41*e^7 - 78*e^6 - 365*e^5 + 113*e^4 + 960*e^3 + 88*e^2 - 745*e - 133, 25/3*e^8 + 33*e^7 - 53*e^6 - 884/3*e^5 + 37*e^4 + 787*e^3 + 490/3*e^2 - 1880/3*e - 137, -74/9*e^8 - 85/3*e^7 + 185/3*e^6 + 2215/9*e^5 - 110*e^4 - 619*e^3 - 251/9*e^2 + 3943/9*e + 275/3, 91/9*e^8 + 125/3*e^7 - 184/3*e^6 - 3353/9*e^5 + 25*e^4 + 1006*e^3 + 2092/9*e^2 - 7547/9*e - 535/3, 11*e^8 + 43*e^7 - 74*e^6 - 391*e^5 + 74*e^4 + 1076*e^3 + 199*e^2 - 908*e - 207, -13/3*e^8 - 23*e^7 + 15*e^6 + 614/3*e^5 + 70*e^4 - 552*e^3 - 706/3*e^2 + 1379/3*e + 100, 32/3*e^8 + 39*e^7 - 77*e^6 - 1036/3*e^5 + 123*e^4 + 901*e^3 + 167/3*e^2 - 2074/3*e - 128, 19/3*e^8 + 25*e^7 - 39*e^6 - 659/3*e^5 + 19*e^4 + 577*e^3 + 430/3*e^2 - 1385/3*e - 113, 103/9*e^8 + 140/3*e^7 - 211/3*e^6 - 3710/9*e^5 + 39*e^4 + 1084*e^3 + 2068/9*e^2 - 7832/9*e - 514/3, 200/9*e^8 + 250/3*e^7 - 464/3*e^6 - 6697/9*e^5 + 199*e^4 + 1975*e^3 + 2402/9*e^2 - 14122/9*e - 1007/3, -5/3*e^8 - 7*e^7 + 11*e^6 + 196/3*e^5 - 5*e^4 - 178*e^3 - 182/3*e^2 + 415/3*e + 62, 3*e^8 + 13*e^7 - 18*e^6 - 120*e^5 + 3*e^4 + 336*e^3 + 86*e^2 - 286*e - 78, 43/9*e^8 + 53/3*e^7 - 91/3*e^6 - 1367/9*e^5 + 18*e^4 + 382*e^3 + 1045/9*e^2 - 2501/9*e - 271/3, 80/9*e^8 + 97/3*e^7 - 191/3*e^6 - 2614/9*e^5 + 92*e^4 + 785*e^3 + 770/9*e^2 - 5836/9*e - 419/3, -10/3*e^8 - 9*e^7 + 29*e^6 + 215/3*e^5 - 80*e^4 - 147*e^3 + 260/3*e^2 + 176/3*e - 35, -155/9*e^8 - 208/3*e^7 + 320/3*e^6 + 5545/9*e^5 - 62*e^4 - 1647*e^3 - 3077/9*e^2 + 12151/9*e + 791/3, 262/9*e^8 + 326/3*e^7 - 595/3*e^6 - 8573/9*e^5 + 240*e^4 + 2459*e^3 + 3046/9*e^2 - 16979/9*e - 1183/3, -20/3*e^8 - 22*e^7 + 52*e^6 + 574/3*e^5 - 108*e^4 - 488*e^3 + 76/3*e^2 + 1096/3*e + 64, 22*e^8 + 88*e^7 - 141*e^6 - 789*e^5 + 110*e^4 + 2126*e^3 + 406*e^2 - 1744*e - 367, -4/9*e^8 - 14/3*e^7 - 8/3*e^6 + 398/9*e^5 + 39*e^4 - 129*e^3 - 754/9*e^2 + 947/9*e + 40/3, 86/3*e^8 + 106*e^7 - 203*e^6 - 2821/3*e^5 + 293*e^4 + 2468*e^3 + 731/3*e^2 - 5800/3*e - 383, 46/3*e^8 + 56*e^7 - 107*e^6 - 1460/3*e^5 + 148*e^4 + 1235*e^3 + 424/3*e^2 - 2747/3*e - 196, 29*e^8 + 110*e^7 - 197*e^6 - 976*e^5 + 234*e^4 + 2579*e^3 + 355*e^2 - 2054*e - 420, 67/3*e^8 + 80*e^7 - 161*e^6 - 2108/3*e^5 + 252*e^4 + 1817*e^3 + 415/3*e^2 - 4169/3*e - 271, 107/9*e^8 + 151/3*e^7 - 212/3*e^6 - 4090/9*e^5 + 14*e^4 + 1236*e^3 + 2786/9*e^2 - 9247/9*e - 662/3, 17*e^8 + 62*e^7 - 119*e^6 - 545*e^5 + 169*e^4 + 1420*e^3 + 125*e^2 - 1101*e - 204, 20*e^8 + 72*e^7 - 147*e^6 - 638*e^5 + 243*e^4 + 1663*e^3 + 113*e^2 - 1272*e - 267, 293/9*e^8 + 379/3*e^7 - 638/3*e^6 - 10078/9*e^5 + 195*e^4 + 2959*e^3 + 4763/9*e^2 - 21211/9*e - 1469/3, -16/3*e^8 - 20*e^7 + 38*e^6 + 536/3*e^5 - 56*e^4 - 472*e^3 - 136/3*e^2 + 1112/3*e + 103, 164/9*e^8 + 193/3*e^7 - 404/3*e^6 - 5068/9*e^5 + 230*e^4 + 1435*e^3 + 656/9*e^2 - 9559/9*e - 629/3, 74/9*e^8 + 106/3*e^7 - 143/3*e^6 - 2872/9*e^5 - e^4 + 871*e^3 + 2231/9*e^2 - 6553/9*e - 506/3, -62/3*e^8 - 75*e^7 + 147*e^6 + 1972/3*e^5 - 218*e^4 - 1694*e^3 - 488/3*e^2 + 3892/3*e + 274, -2*e^8 - 7*e^7 + 15*e^6 + 60*e^5 - 28*e^4 - 146*e^3 - 5*e^2 + 97*e + 21, 50/9*e^8 + 52/3*e^7 - 137/3*e^6 - 1375/9*e^5 + 100*e^4 + 393*e^3 - 97/9*e^2 - 2608/9*e - 272/3, 281/9*e^8 + 340/3*e^7 - 668/3*e^6 - 8956/9*e^5 + 335*e^4 + 2566*e^3 + 1979/9*e^2 - 17578/9*e - 1142/3, -34*e^8 - 124*e^7 + 244*e^6 + 1097*e^5 - 374*e^4 - 2870*e^3 - 237*e^2 + 2241*e + 441, -49/9*e^8 - 50/3*e^7 + 130/3*e^6 + 1262/9*e^5 - 89*e^4 - 326*e^3 + 56/9*e^2 + 1739/9*e + 214/3, -56/3*e^8 - 74*e^7 + 119*e^6 + 1966/3*e^5 - 94*e^4 - 1728*e^3 - 977/3*e^2 + 4090/3*e + 299, 22/3*e^8 + 25*e^7 - 58*e^6 - 677/3*e^5 + 114*e^4 + 603*e^3 + 55/3*e^2 - 1442/3*e - 112, -56/9*e^8 - 73/3*e^7 + 116/3*e^6 + 1864/9*e^5 - 34*e^4 - 516*e^3 - 593/9*e^2 + 3367/9*e + 161/3, 53/3*e^8 + 64*e^7 - 123*e^6 - 1669/3*e^5 + 168*e^4 + 1424*e^3 + 494/3*e^2 - 3292/3*e - 228, -205/9*e^8 - 257/3*e^7 + 469/3*e^6 + 6767/9*e^5 - 204*e^4 - 1942*e^3 - 2017/9*e^2 + 13328/9*e + 916/3, 158/9*e^8 + 202/3*e^7 - 359/3*e^6 - 5434/9*e^5 + 138*e^4 + 1617*e^3 + 2234/9*e^2 - 11698/9*e - 842/3, 176/9*e^8 + 229/3*e^7 - 377/3*e^6 - 6055/9*e^5 + 101*e^4 + 1757*e^3 + 3170/9*e^2 - 12391/9*e - 932/3, -413/9*e^8 - 505/3*e^7 + 980/3*e^6 + 13414/9*e^5 - 481*e^4 - 3903*e^3 - 3389/9*e^2 + 27364/9*e + 1859/3, -67/9*e^8 - 68/3*e^7 + 193/3*e^6 + 1757/9*e^5 - 167*e^4 - 471*e^3 + 1046/9*e^2 + 2783/9*e + 97/3, -65/9*e^8 - 85/3*e^7 + 134/3*e^6 + 2170/9*e^5 - 27*e^4 - 590*e^3 - 1313/9*e^2 + 3844/9*e + 347/3, -40/9*e^8 - 41/3*e^7 + 127/3*e^6 + 1136/9*e^5 - 131*e^4 - 341*e^3 + 1154/9*e^2 + 2459/9*e + 13/3, 31/3*e^8 + 40*e^7 - 67*e^6 - 1049/3*e^5 + 64*e^4 + 903*e^3 + 451/3*e^2 - 2081/3*e - 176, -76/9*e^8 - 104/3*e^7 + 160/3*e^6 + 2810/9*e^5 - 39*e^4 - 847*e^3 - 1339/9*e^2 + 6392/9*e + 382/3, -194/9*e^8 - 241/3*e^7 + 449/3*e^6 + 6370/9*e^5 - 199*e^4 - 1840*e^3 - 2018/9*e^2 + 12904/9*e + 947/3, -149/9*e^8 - 199/3*e^7 + 311/3*e^6 + 5272/9*e^5 - 72*e^4 - 1548*e^3 - 2783/9*e^2 + 11338/9*e + 785/3, 188/9*e^8 + 226/3*e^7 - 455/3*e^6 - 5980/9*e^5 + 247*e^4 + 1725*e^3 + 869/9*e^2 - 11992/9*e - 773/3, 21*e^8 + 84*e^7 - 133*e^6 - 746*e^5 + 96*e^4 + 1973*e^3 + 388*e^2 - 1562*e - 322, -94/9*e^8 - 122/3*e^7 + 208/3*e^6 + 3224/9*e^5 - 70*e^4 - 926*e^3 - 1537/9*e^2 + 6464/9*e + 547/3, 179/9*e^8 + 217/3*e^7 - 416/3*e^6 - 5701/9*e^5 + 186*e^4 + 1638*e^3 + 1868/9*e^2 - 11362/9*e - 797/3, 39*e^8 + 146*e^7 - 269*e^6 - 1295*e^5 + 341*e^4 + 3412*e^3 + 441*e^2 - 2690*e - 556, 82/9*e^8 + 101/3*e^7 - 199/3*e^6 - 2732/9*e^5 + 101*e^4 + 812*e^3 + 760/9*e^2 - 5810/9*e - 376/3, -410/9*e^8 - 526/3*e^7 + 908/3*e^6 + 13948/9*e^5 - 317*e^4 - 4064*e^3 - 5852/9*e^2 + 28807/9*e + 1982/3, -238/9*e^8 - 308/3*e^7 + 529/3*e^6 + 8246/9*e^5 - 186*e^4 - 2444*e^3 - 3526/9*e^2 + 17714/9*e + 1231/3, 101/9*e^8 + 127/3*e^7 - 233/3*e^6 - 3376/9*e^5 + 107*e^4 + 993*e^3 + 818/9*e^2 - 7174/9*e - 413/3, 74/3*e^8 + 96*e^7 - 158*e^6 - 2533/3*e^5 + 130*e^4 + 2207*e^3 + 1232/3*e^2 - 5206/3*e - 371, -73/9*e^8 - 74/3*e^7 + 214/3*e^6 + 2003/9*e^5 - 185*e^4 - 592*e^3 + 950/9*e^2 + 4136/9*e + 217/3, 19/9*e^8 + 32/3*e^7 - 22/3*e^6 - 851/9*e^5 - 37*e^4 + 252*e^3 + 1039/9*e^2 - 1805/9*e - 61/3, -58/3*e^8 - 71*e^7 + 137*e^6 + 1871/3*e^5 - 201*e^4 - 1610*e^3 - 490/3*e^2 + 3686/3*e + 274, 40/3*e^8 + 51*e^7 - 91*e^6 - 1370/3*e^5 + 108*e^4 + 1231*e^3 + 541/3*e^2 - 3068/3*e - 217, -14/3*e^8 - 17*e^7 + 31*e^6 + 430/3*e^5 - 41*e^4 - 356*e^3 - 62/3*e^2 + 826/3*e + 16, -40/3*e^8 - 50*e^7 + 93*e^6 + 1337/3*e^5 - 124*e^4 - 1174*e^3 - 391/3*e^2 + 2714/3*e + 156, 137/9*e^8 + 190/3*e^7 - 275/3*e^6 - 5140/9*e^5 + 25*e^4 + 1555*e^3 + 3563/9*e^2 - 11494/9*e - 857/3, 208/9*e^8 + 251/3*e^7 - 490/3*e^6 - 6557/9*e^5 + 240*e^4 + 1860*e^3 + 1570/9*e^2 - 12731/9*e - 895/3, -83/9*e^8 - 112/3*e^7 + 176/3*e^6 + 3097/9*e^5 - 33*e^4 - 969*e^3 - 1988/9*e^2 + 7579/9*e + 557/3, -28/9*e^8 - 44/3*e^7 + 55/3*e^6 + 1301/9*e^5 + 7*e^4 - 443*e^3 - 1219/9*e^2 + 3812/9*e + 319/3, 112/3*e^8 + 144*e^7 - 244*e^6 - 3812/3*e^5 + 229*e^4 + 3336*e^3 + 1744/3*e^2 - 7916/3*e - 559, -22/3*e^8 - 24*e^7 + 54*e^6 + 590/3*e^5 - 97*e^4 - 443*e^3 + 20/3*e^2 + 806/3*e + 60, -268/9*e^8 - 329/3*e^7 + 628/3*e^6 + 8765/9*e^5 - 288*e^4 - 2574*e^3 - 2692/9*e^2 + 18386/9*e + 1240/3, 22/3*e^8 + 36*e^7 - 31*e^6 - 965/3*e^5 - 84*e^4 + 866*e^3 + 1066/3*e^2 - 2120/3*e - 181, 10/3*e^8 + 14*e^7 - 15*e^6 - 353/3*e^5 - 31*e^4 + 290*e^3 + 400/3*e^2 - 638/3*e - 43, -115/3*e^8 - 138*e^7 + 278*e^6 + 3650/3*e^5 - 446*e^4 - 3165*e^3 - 643/3*e^2 + 7370/3*e + 478, 85/3*e^8 + 107*e^7 - 190*e^6 - 2822/3*e^5 + 211*e^4 + 2449*e^3 + 1099/3*e^2 - 5732/3*e - 370, 124/9*e^8 + 152/3*e^7 - 280/3*e^6 - 3905/9*e^5 + 115*e^4 + 1069*e^3 + 1288/9*e^2 - 6812/9*e - 535/3, -26*e^8 - 101*e^7 + 169*e^6 + 886*e^5 - 164*e^4 - 2302*e^3 - 361*e^2 + 1786*e + 353, -33*e^8 - 120*e^7 + 233*e^6 + 1056*e^5 - 333*e^4 - 2744*e^3 - 282*e^2 + 2104*e + 440, 95/3*e^8 + 123*e^7 - 206*e^6 - 3292/3*e^5 + 174*e^4 + 2931*e^3 + 1682/3*e^2 - 7081/3*e - 495];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, 4*w + 19}>] := -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;