/*
  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![-39, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {12,6,-2*w + 14}>;

primesArray := [
[3, 3, w + 6],
[3, 3, -w + 7],
[4, 2, 2],
[11, 11, -3*w - 17],
[11, 11, 3*w - 20],
[13, 13, 2*w - 13],
[13, 13, 2*w + 11],
[17, 17, w + 7],
[17, 17, -w + 8],
[19, 19, -w - 4],
[19, 19, -w + 5],
[25, 5, 5],
[31, 31, -6*w - 35],
[31, 31, -6*w + 41],
[37, 37, -w - 1],
[37, 37, w - 2],
[47, 47, 3*w + 16],
[47, 47, -3*w + 19],
[49, 7, -7],
[67, 67, 3*w - 22],
[67, 67, -3*w - 19],
[71, 71, -w - 10],
[71, 71, w - 11],
[89, 89, -8*w - 47],
[89, 89, 8*w - 55],
[101, 101, 4*w - 29],
[101, 101, -4*w - 25],
[109, 109, -3*w - 20],
[109, 109, 3*w - 23],
[113, 113, 3*w - 17],
[113, 113, -3*w - 14],
[127, 127, -9*w - 53],
[127, 127, 9*w - 62],
[157, 157, 2*w - 1],
[167, 167, 2*w - 19],
[167, 167, -2*w - 17],
[173, 173, 13*w - 89],
[173, 173, -13*w - 76],
[193, 193, 11*w - 73],
[193, 193, -11*w - 62],
[197, 197, 3*w - 14],
[197, 197, -3*w - 11],
[199, 199, 3*w - 25],
[199, 199, -3*w - 22],
[233, 233, -w - 16],
[233, 233, w - 17],
[239, 239, 7*w - 50],
[239, 239, -7*w - 43],
[257, 257, 6*w + 31],
[257, 257, -6*w + 37],
[263, 263, 3*w - 11],
[263, 263, -3*w - 8],
[277, 277, -12*w - 71],
[277, 277, 12*w - 83],
[281, 281, -3*w - 7],
[281, 281, 3*w - 10],
[283, 283, -7*w + 44],
[283, 283, 7*w + 37],
[311, 311, 3*w - 8],
[311, 311, -3*w - 5],
[313, 313, -13*w - 73],
[313, 313, 13*w - 86],
[317, 317, -9*w + 58],
[317, 317, 9*w + 49],
[331, 331, -5*w - 23],
[331, 331, 5*w - 28],
[347, 347, -3*w - 1],
[347, 347, 3*w - 4],
[349, 349, 3*w - 28],
[349, 349, -3*w - 25],
[353, 353, 3*w - 2],
[353, 353, 3*w - 1],
[389, 389, 6*w - 35],
[389, 389, 6*w + 29],
[419, 419, 2*w - 25],
[419, 419, -2*w - 23],
[431, 431, 10*w - 71],
[431, 431, -10*w - 61],
[457, 457, 27*w + 157],
[457, 457, 27*w - 184],
[461, 461, -4*w - 31],
[461, 461, 4*w - 35],
[467, 467, -w - 22],
[467, 467, w - 23],
[487, 487, 8*w + 41],
[487, 487, -8*w + 49],
[523, 523, -6*w - 41],
[523, 523, 6*w - 47],
[529, 23, -23],
[547, 547, 4*w - 11],
[547, 547, -4*w - 7],
[557, 557, 28*w - 191],
[557, 557, -28*w - 163],
[571, 571, 20*w - 133],
[571, 571, -20*w - 113],
[577, 577, -3*w - 29],
[577, 577, 3*w - 32],
[593, 593, 19*w - 131],
[593, 593, -19*w - 112],
[601, 601, 5*w - 22],
[601, 601, -5*w - 17],
[617, 617, -18*w - 101],
[617, 617, 18*w - 119],
[619, 619, -4*w - 1],
[619, 619, 4*w - 5],
[631, 631, 17*w - 112],
[631, 631, -17*w - 95],
[641, 641, 15*w + 83],
[641, 641, 15*w - 98],
[647, 647, -17*w - 101],
[647, 647, 17*w - 118],
[653, 653, -11*w - 68],
[653, 653, 11*w - 79],
[659, 659, 5*w - 43],
[659, 659, -5*w - 38],
[661, 661, -25*w - 142],
[661, 661, 25*w - 167],
[677, 677, 13*w - 92],
[677, 677, -13*w - 79],
[709, 709, 5*w - 19],
[709, 709, -5*w - 14],
[727, 727, -9*w - 58],
[727, 727, 9*w - 67],
[733, 733, 7*w - 38],
[733, 733, -7*w - 31],
[739, 739, -18*w - 107],
[739, 739, 18*w - 125],
[743, 743, 2*w - 31],
[743, 743, -2*w - 29],
[769, 769, -3*w - 32],
[769, 769, 3*w - 35],
[773, 773, -w - 28],
[773, 773, w - 29],
[797, 797, 21*w - 139],
[797, 797, -21*w - 118],
[821, 821, -15*w + 97],
[821, 821, 15*w + 82],
[827, 827, 9*w - 53],
[827, 827, -9*w - 44],
[829, 829, -19*w - 106],
[829, 829, 19*w - 125],
[841, 29, -29],
[853, 853, 9*w - 68],
[853, 853, -9*w - 59],
[907, 907, 3*w - 37],
[907, 907, -3*w - 34],
[911, 911, 5*w - 46],
[911, 911, -5*w - 41],
[929, 929, -6*w - 19],
[929, 929, 6*w - 25],
[941, 941, -20*w - 119],
[941, 941, 20*w - 139],
[953, 953, -w - 31],
[953, 953, w - 32],
[967, 967, 11*w + 56],
[967, 967, -11*w + 67],
[977, 977, 16*w - 113],
[977, 977, -16*w - 97],
[991, 991, -8*w - 35],
[991, 991, 8*w - 43]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + x^5 - 10*x^4 - 3*x^3 + 16*x^2 - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1, 1, -3/5*e^5 - 4/5*e^4 + 27/5*e^3 + 13/5*e^2 - 27/5*e + 6/5, -2/5*e^5 - 1/5*e^4 + 23/5*e^3 - 3/5*e^2 - 48/5*e + 14/5, 2/5*e^5 + 1/5*e^4 - 23/5*e^3 - 2/5*e^2 + 43/5*e + 1/5, -8/5*e^5 - 9/5*e^4 + 77/5*e^3 + 33/5*e^2 - 107/5*e - 9/5, -11/5*e^5 - 13/5*e^4 + 104/5*e^3 + 46/5*e^2 - 144/5*e - 3/5, 4/5*e^5 + 2/5*e^4 - 41/5*e^3 + 6/5*e^2 + 61/5*e + 2/5, e^5 + e^4 - 10*e^3 - 3*e^2 + 18*e + 1, e^3 + 2*e^2 - 7*e - 5, 2/5*e^5 + 1/5*e^4 - 18/5*e^3 + 3/5*e^2 + 13/5*e - 4/5, 7/5*e^5 + 6/5*e^4 - 73/5*e^3 - 17/5*e^2 + 128/5*e + 16/5, 4/5*e^5 + 7/5*e^4 - 31/5*e^3 - 34/5*e^2 + 21/5*e + 27/5, 1/5*e^5 + 3/5*e^4 - 9/5*e^3 - 21/5*e^2 + 14/5*e + 28/5, e^5 + e^4 - 10*e^3 - 4*e^2 + 16*e, 9/5*e^5 + 12/5*e^4 - 86/5*e^3 - 49/5*e^2 + 141/5*e - 3/5, -e^5 + 11*e^3 - 5*e^2 - 18*e + 4, 3/5*e^5 + 4/5*e^4 - 32/5*e^3 - 23/5*e^2 + 57/5*e + 19/5, 3/5*e^5 + 14/5*e^4 - 17/5*e^3 - 108/5*e^2 + 7/5*e + 84/5, -7/5*e^5 - 11/5*e^4 + 63/5*e^3 + 52/5*e^2 - 93/5*e - 16/5, -1/5*e^5 + 2/5*e^4 + 14/5*e^3 - 24/5*e^2 - 54/5*e + 17/5, 14/5*e^5 + 17/5*e^4 - 131/5*e^3 - 69/5*e^2 + 166/5*e + 42/5, 8/5*e^5 + 9/5*e^4 - 67/5*e^3 - 23/5*e^2 + 42/5*e - 11/5, 16/5*e^5 + 13/5*e^4 - 164/5*e^3 - 26/5*e^2 + 249/5*e + 28/5, -6/5*e^5 - 3/5*e^4 + 59/5*e^3 - 9/5*e^2 - 64/5*e + 47/5, 6/5*e^5 + 13/5*e^4 - 54/5*e^3 - 76/5*e^2 + 79/5*e + 38/5, -9/5*e^5 - 17/5*e^4 + 81/5*e^3 + 104/5*e^2 - 121/5*e - 72/5, -3*e^5 - 2*e^4 + 31*e^3 + e^2 - 51*e + 6, 2/5*e^5 - 9/5*e^4 - 33/5*e^3 + 93/5*e^2 + 68/5*e - 69/5, -1/5*e^5 - 3/5*e^4 + 4/5*e^3 + 26/5*e^2 + 26/5*e - 48/5, 16/5*e^5 + 18/5*e^4 - 159/5*e^3 - 71/5*e^2 + 239/5*e + 23/5, -e^5 - 2*e^4 + 10*e^3 + 13*e^2 - 21*e - 3, -21/5*e^5 - 23/5*e^4 + 209/5*e^3 + 76/5*e^2 - 344/5*e + 2/5, 16/5*e^5 + 18/5*e^4 - 149/5*e^3 - 56/5*e^2 + 179/5*e + 13/5, e^5 + e^4 - 11*e^3 - 4*e^2 + 21*e + 2, 13/5*e^5 + 14/5*e^4 - 127/5*e^3 - 43/5*e^2 + 182/5*e - 26/5, 4/5*e^5 + 7/5*e^4 - 36/5*e^3 - 19/5*e^2 + 71/5*e - 28/5, -18/5*e^5 - 9/5*e^4 + 197/5*e^3 - 17/5*e^2 - 342/5*e + 31/5, 2*e^4 + 4*e^3 - 16*e^2 - 18*e + 12, 3*e^5 + 4*e^4 - 27*e^3 - 19*e^2 + 26*e + 8, 3/5*e^5 + 4/5*e^4 - 22/5*e^3 - 23/5*e^2 - 33/5*e + 49/5, 1/5*e^5 - 12/5*e^4 - 34/5*e^3 + 99/5*e^2 + 109/5*e - 107/5, -6/5*e^5 - 13/5*e^4 + 49/5*e^3 + 86/5*e^2 - 34/5*e - 53/5, 9/5*e^5 + 12/5*e^4 - 81/5*e^3 - 44/5*e^2 + 96/5*e + 62/5, 21/5*e^5 + 28/5*e^4 - 194/5*e^3 - 111/5*e^2 + 249/5*e + 13/5, -18/5*e^5 - 29/5*e^4 + 167/5*e^3 + 143/5*e^2 - 232/5*e + 1/5, 12/5*e^5 + 16/5*e^4 - 103/5*e^3 - 52/5*e^2 + 103/5*e + 41/5, 29/5*e^5 + 37/5*e^4 - 281/5*e^3 - 144/5*e^2 + 461/5*e + 2/5, -e^2 + e + 12, 6/5*e^5 + 18/5*e^4 - 44/5*e^3 - 106/5*e^2 + 29/5*e + 38/5, -17/5*e^5 - 26/5*e^4 + 153/5*e^3 + 127/5*e^2 - 158/5*e - 41/5, -14/5*e^5 - 12/5*e^4 + 136/5*e^3 + 24/5*e^2 - 176/5*e + 8/5, -32/5*e^5 - 41/5*e^4 + 308/5*e^3 + 177/5*e^2 - 453/5*e + 14/5, -2*e^5 - 2*e^4 + 20*e^3 + 6*e^2 - 40*e - 8, -3/5*e^5 + 1/5*e^4 + 32/5*e^3 - 27/5*e^2 + 8/5*e + 41/5, 17/5*e^5 + 16/5*e^4 - 183/5*e^3 - 57/5*e^2 + 348/5*e + 6/5, -e^5 + 9*e^3 - 8*e^2 - 4*e + 16, 2/5*e^5 + 11/5*e^4 - 13/5*e^3 - 107/5*e^2 - 2/5*e + 151/5, -12/5*e^5 - 6/5*e^4 + 133/5*e^3 + 2/5*e^2 - 253/5*e + 69/5, -1/5*e^5 + 7/5*e^4 + 34/5*e^3 - 54/5*e^2 - 174/5*e + 57/5, -6/5*e^5 - 13/5*e^4 + 44/5*e^3 + 71/5*e^2 - 29/5*e - 18/5, 28/5*e^5 + 29/5*e^4 - 287/5*e^3 - 88/5*e^2 + 497/5*e + 9/5, 3*e^5 + e^4 - 31*e^3 + 8*e^2 + 39*e - 20, 12/5*e^5 + 6/5*e^4 - 128/5*e^3 + 18/5*e^2 + 213/5*e - 124/5, -e^5 + 12*e^3 - 7*e^2 - 33*e + 15, -3/5*e^5 - 14/5*e^4 + 22/5*e^3 + 103/5*e^2 - 27/5*e - 99/5, -22/5*e^5 - 26/5*e^4 + 203/5*e^3 + 72/5*e^2 - 233/5*e + 39/5, -12/5*e^5 - 16/5*e^4 + 118/5*e^3 + 87/5*e^2 - 208/5*e - 126/5, -24/5*e^5 - 32/5*e^4 + 226/5*e^3 + 119/5*e^2 - 316/5*e - 22/5, -3/5*e^5 - 29/5*e^4 - 3/5*e^3 + 228/5*e^2 + 53/5*e - 174/5, -8/5*e^5 - 14/5*e^4 + 72/5*e^3 + 63/5*e^2 - 107/5*e + 16/5, 8/5*e^5 + 9/5*e^4 - 87/5*e^3 - 43/5*e^2 + 162/5*e + 29/5, -7/5*e^5 - 21/5*e^4 + 48/5*e^3 + 137/5*e^2 - 48/5*e - 81/5, 7*e^5 + 8*e^4 - 68*e^3 - 27*e^2 + 107*e - 11, -29/5*e^5 - 22/5*e^4 + 296/5*e^3 + 39/5*e^2 - 441/5*e - 27/5, 13/5*e^5 + 19/5*e^4 - 107/5*e^3 - 73/5*e^2 + 62/5*e + 94/5, 16/5*e^5 + 28/5*e^4 - 149/5*e^3 - 156/5*e^2 + 189/5*e + 118/5, -2/5*e^5 - 6/5*e^4 + 18/5*e^3 + 22/5*e^2 - 58/5*e + 44/5, -21/5*e^5 - 23/5*e^4 + 199/5*e^3 + 76/5*e^2 - 239/5*e + 2/5, -27/5*e^5 - 41/5*e^4 + 248/5*e^3 + 202/5*e^2 - 338/5*e - 131/5, -23/5*e^5 - 24/5*e^4 + 222/5*e^3 + 78/5*e^2 - 257/5*e - 9/5, 18/5*e^5 + 14/5*e^4 - 187/5*e^3 - 28/5*e^2 + 317/5*e + 29/5, -2*e^5 - e^4 + 23*e^3 + e^2 - 56*e + 3, 23/5*e^5 + 44/5*e^4 - 202/5*e^3 - 243/5*e^2 + 267/5*e + 99/5, 4*e^5 + e^4 - 42*e^3 + 15*e^2 + 60*e - 22, 6/5*e^5 - 12/5*e^4 - 74/5*e^3 + 134/5*e^2 + 104/5*e - 82/5, 7/5*e^5 + 26/5*e^4 - 38/5*e^3 - 172/5*e^2 - 22/5*e + 131/5, 3*e^5 + 3*e^4 - 32*e^3 - 12*e^2 + 56*e - 4, -6/5*e^5 - 28/5*e^4 + 29/5*e^3 + 196/5*e^2 - 29/5*e - 133/5, 6/5*e^5 + 8/5*e^4 - 69/5*e^3 - 51/5*e^2 + 179/5*e + 53/5, -22/5*e^5 - 26/5*e^4 + 218/5*e^3 + 92/5*e^2 - 388/5*e - 1/5, -34/5*e^5 - 52/5*e^4 + 316/5*e^3 + 254/5*e^2 - 426/5*e - 132/5, 9/5*e^5 + 7/5*e^4 - 91/5*e^3 - 14/5*e^2 + 131/5*e + 42/5, -8/5*e^5 + 6/5*e^4 + 107/5*e^3 - 67/5*e^2 - 227/5*e - 39/5, 9/5*e^5 + 17/5*e^4 - 86/5*e^3 - 84/5*e^2 + 146/5*e - 28/5, 22/5*e^5 + 26/5*e^4 - 193/5*e^3 - 82/5*e^2 + 188/5*e - 9/5, 9/5*e^5 + 2/5*e^4 - 101/5*e^3 + 41/5*e^2 + 236/5*e - 88/5, 32/5*e^5 + 31/5*e^4 - 308/5*e^3 - 57/5*e^2 + 413/5*e - 14/5, -18/5*e^5 - 24/5*e^4 + 182/5*e^3 + 103/5*e^2 - 352/5*e + 46/5, 31/5*e^5 + 23/5*e^4 - 324/5*e^3 - 41/5*e^2 + 499/5*e - 47/5, -36/5*e^5 - 33/5*e^4 + 354/5*e^3 + 81/5*e^2 - 519/5*e - 48/5, 27/5*e^5 + 41/5*e^4 - 258/5*e^3 - 207/5*e^2 + 363/5*e + 131/5, -27/5*e^5 - 16/5*e^4 + 278/5*e^3 - 33/5*e^2 - 473/5*e + 79/5, 32/5*e^5 + 26/5*e^4 - 313/5*e^3 - 2/5*e^2 + 433/5*e - 104/5, 31/5*e^5 + 18/5*e^4 - 319/5*e^3 + 19/5*e^2 + 524/5*e + 3/5, 18/5*e^5 + 14/5*e^4 - 192/5*e^3 - 18/5*e^2 + 332/5*e - 126/5, -26/5*e^5 - 33/5*e^4 + 234/5*e^3 + 101/5*e^2 - 329/5*e + 87/5, -38/5*e^5 - 24/5*e^4 + 392/5*e^3 + 18/5*e^2 - 552/5*e - 14/5, -7/5*e^5 - 11/5*e^4 + 43/5*e^3 + 52/5*e^2 + 97/5*e - 36/5, -19/5*e^5 - 12/5*e^4 + 181/5*e^3 - 31/5*e^2 - 166/5*e + 208/5, -44/5*e^5 - 57/5*e^4 + 411/5*e^3 + 229/5*e^2 - 526/5*e - 77/5, 42/5*e^5 + 46/5*e^4 - 408/5*e^3 - 157/5*e^2 + 593/5*e - 64/5, 48/5*e^5 + 49/5*e^4 - 482/5*e^3 - 163/5*e^2 + 757/5*e - 61/5, -4/5*e^5 - 2/5*e^4 + 41/5*e^3 - 1/5*e^2 - 41/5*e + 3/5, 4/5*e^5 + 2/5*e^4 - 31/5*e^3 + 6/5*e^2 - 49/5*e + 12/5, 16/5*e^5 + 28/5*e^4 - 129/5*e^3 - 156/5*e^2 + 104/5*e + 113/5, e^5 + 2*e^4 - 6*e^3 - 11*e^2 - 5*e + 23, 26/5*e^5 + 8/5*e^4 - 279/5*e^3 + 54/5*e^2 + 434/5*e - 7/5, -12/5*e^5 - 16/5*e^4 + 118/5*e^3 + 52/5*e^2 - 188/5*e + 4/5, 49/5*e^5 + 57/5*e^4 - 461/5*e^3 - 164/5*e^2 + 611/5*e - 58/5, -41/5*e^5 - 33/5*e^4 + 409/5*e^3 + 66/5*e^2 - 589/5*e + 42/5, 1/5*e^5 + 13/5*e^4 + 11/5*e^3 - 51/5*e^2 - 26/5*e - 152/5, 1/5*e^5 + 8/5*e^4 + 6/5*e^3 - 21/5*e^2 - 11/5*e - 137/5, 2/5*e^5 - 19/5*e^4 - 48/5*e^3 + 133/5*e^2 + 83/5*e + 46/5, -37/5*e^5 - 41/5*e^4 + 343/5*e^3 + 97/5*e^2 - 478/5*e + 134/5, -22/5*e^5 - 26/5*e^4 + 193/5*e^3 + 82/5*e^2 - 163/5*e + 49/5, 34/5*e^5 + 22/5*e^4 - 371/5*e^3 - 24/5*e^2 + 691/5*e - 13/5, -7/5*e^5 - 16/5*e^4 + 68/5*e^3 + 117/5*e^2 - 143/5*e - 171/5, -47/5*e^5 - 51/5*e^4 + 448/5*e^3 + 192/5*e^2 - 588/5*e - 91/5, -9*e^5 - 8*e^4 + 90*e^3 + 15*e^2 - 137*e + 3, -12/5*e^5 - 21/5*e^4 + 128/5*e^3 + 127/5*e^2 - 333/5*e - 86/5, -14/5*e^5 - 17/5*e^4 + 126/5*e^3 + 19/5*e^2 - 176/5*e + 188/5, 43/5*e^5 + 49/5*e^4 - 427/5*e^3 - 198/5*e^2 + 637/5*e + 44/5, -21/5*e^5 - 28/5*e^4 + 184/5*e^3 + 121/5*e^2 - 199/5*e - 163/5, -32/5*e^5 - 21/5*e^4 + 328/5*e^3 + 27/5*e^2 - 533/5*e - 86/5, 6/5*e^5 + 13/5*e^4 - 79/5*e^3 - 96/5*e^2 + 229/5*e + 13/5, -29/5*e^5 - 7/5*e^4 + 321/5*e^3 - 101/5*e^2 - 606/5*e + 118/5, -16/5*e^5 + 7/5*e^4 + 189/5*e^3 - 184/5*e^2 - 369/5*e + 177/5, -56/5*e^5 - 83/5*e^4 + 514/5*e^3 + 371/5*e^2 - 689/5*e - 118/5, 21/5*e^5 + 3/5*e^4 - 234/5*e^3 + 64/5*e^2 + 404/5*e - 67/5, 14/5*e^5 + 32/5*e^4 - 136/5*e^3 - 214/5*e^2 + 276/5*e + 102/5, 26/5*e^5 + 13/5*e^4 - 284/5*e^3 - 16/5*e^2 + 509/5*e + 88/5, -48/5*e^5 - 39/5*e^4 + 492/5*e^3 + 53/5*e^2 - 782/5*e + 56/5, 23/5*e^5 + 39/5*e^4 - 217/5*e^3 - 223/5*e^2 + 357/5*e + 104/5, 7*e^5 + 7*e^4 - 75*e^3 - 30*e^2 + 141*e + 18, 13/5*e^5 + 24/5*e^4 - 102/5*e^3 - 83/5*e^2 + 77/5*e - 81/5, 13/5*e^5 - 1/5*e^4 - 147/5*e^3 + 92/5*e^2 + 247/5*e - 86/5, -39/5*e^5 - 42/5*e^4 + 406/5*e^3 + 169/5*e^2 - 701/5*e - 87/5, 42/5*e^5 + 51/5*e^4 - 423/5*e^3 - 227/5*e^2 + 728/5*e + 71/5, -24/5*e^5 - 37/5*e^4 + 206/5*e^3 + 169/5*e^2 - 186/5*e - 122/5, -14/5*e^5 - 17/5*e^4 + 176/5*e^3 + 119/5*e^2 - 441/5*e - 132/5, 51/5*e^5 + 43/5*e^4 - 524/5*e^3 - 96/5*e^2 + 844/5*e + 23/5, -37/5*e^5 - 61/5*e^4 + 338/5*e^3 + 327/5*e^2 - 503/5*e - 161/5, -32/5*e^5 - 46/5*e^4 + 308/5*e^3 + 242/5*e^2 - 458/5*e - 146/5, 52/5*e^5 + 51/5*e^4 - 513/5*e^3 - 117/5*e^2 + 788/5*e + 16/5, 34/5*e^5 + 47/5*e^4 - 331/5*e^3 - 219/5*e^2 + 561/5*e + 137/5, -28/5*e^5 - 39/5*e^4 + 262/5*e^3 + 163/5*e^2 - 347/5*e + 16/5, 57/5*e^5 + 61/5*e^4 - 563/5*e^3 - 217/5*e^2 + 853/5*e + 6/5, 64/5*e^5 + 72/5*e^4 - 626/5*e^3 - 264/5*e^2 + 886/5*e + 92/5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3,3,-w + 7}>] := -1;
ALEigenvalues[ideal<ZF | {4,2,2}>] := -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;