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

NN := ideal<ZF | {19, 19, -w^2 + 2*w + 5}>;

primesArray := [
[3, 3, w + 1],
[7, 7, w - 1],
[8, 2, 2],
[9, 3, -w^2 + 2*w + 4],
[11, 11, -w^2 + 2*w + 2],
[13, 13, -w^2 + w + 4],
[19, 19, w + 3],
[19, 19, -w^2 + 2*w + 5],
[19, 19, -w^2 + 3*w + 2],
[23, 23, w^2 - w - 3],
[23, 23, -w^2 + 2],
[23, 23, -w + 4],
[31, 31, w^2 - 5],
[43, 43, w^2 - 3*w - 3],
[49, 7, w^2 - 6],
[53, 53, 2*w - 5],
[61, 61, 2*w^2 - 2*w - 9],
[71, 71, 2*w - 3],
[73, 73, 2*w^2 - 5*w - 5],
[83, 83, w^2 - w - 9],
[97, 97, 2*w^2 - 3*w - 7],
[103, 103, w^2 - 3*w - 6],
[109, 109, w^2 - 4*w - 3],
[121, 11, 2*w^2 - w - 7],
[125, 5, -5],
[127, 127, w^2 + w - 5],
[131, 131, w^2 - 11],
[137, 137, 3*w^2 - 7*w - 8],
[137, 137, 2*w^2 - 5*w - 6],
[137, 137, -w^2 + w - 2],
[139, 139, 3*w^2 - 2*w - 16],
[139, 139, 3*w - 2],
[139, 139, -w^2 + 3*w - 3],
[151, 151, w^2 + w - 11],
[157, 157, 2*w^2 - 3*w - 3],
[163, 163, w^2 - w - 10],
[169, 13, 2*w^2 - 3*w - 4],
[173, 173, -3*w^2 + 6*w + 8],
[191, 191, -4*w^2 + 7*w + 15],
[193, 193, 2*w^2 - 3],
[197, 197, w^2 - 3*w - 11],
[199, 199, 3*w^2 - 5*w - 16],
[211, 211, w^2 - 4*w - 9],
[223, 223, 2*w^2 - w - 4],
[227, 227, 3*w - 4],
[229, 229, w^2 + w - 8],
[241, 241, w^2 - 4*w - 6],
[251, 251, w - 7],
[257, 257, w^2 - 4*w - 7],
[263, 263, 3*w^2 - 5*w - 10],
[263, 263, 2*w^2 - 5*w - 15],
[263, 263, -w^2 + 6*w - 1],
[269, 269, -w^2 - w + 14],
[277, 277, 2*w^2 - 5*w - 8],
[277, 277, w^2 + 3*w - 3],
[277, 277, 3*w^2 - 4*w - 12],
[281, 281, -2*w - 7],
[293, 293, 3*w^2 - 2*w - 12],
[307, 307, 2*w^2 + w - 8],
[313, 313, w^2 + 4*w - 2],
[317, 317, 3*w^2 - 6*w - 5],
[331, 331, 4*w^2 - 4*w - 19],
[337, 337, -5*w^2 + 8*w + 21],
[347, 347, 3*w^2 - 6*w - 13],
[347, 347, 3*w^2 - 5*w - 22],
[347, 347, 3*w^2 - 5*w - 9],
[349, 349, w^2 - 5*w - 11],
[353, 353, -w^2 + 6*w - 7],
[359, 359, -4*w^2 + 8*w + 11],
[367, 367, 3*w^2 - 4*w - 11],
[373, 373, w^2 + 2*w - 6],
[379, 379, 5*w^2 - 9*w - 18],
[409, 409, -w^2 - 4],
[439, 439, 3*w^2 - 6*w - 14],
[443, 443, -w^2 - 2*w + 13],
[443, 443, 2*w^2 - 15],
[443, 443, 3*w^2 - 13],
[449, 449, w^2 + 2*w - 7],
[449, 449, 2*w^2 - 6*w - 7],
[449, 449, 3*w^2 - 3*w - 11],
[461, 461, -w^2 - 2*w - 5],
[467, 467, 3*w^2 - 5*w - 7],
[479, 479, -w^2 + 6*w - 4],
[491, 491, 3*w^2 - 5*w - 6],
[499, 499, w^2 - 5*w - 8],
[509, 509, 3*w^2 - 6*w - 19],
[521, 521, 3*w^2 - 2*w - 10],
[523, 523, 4*w - 9],
[541, 541, 3*w^2 - 3*w - 10],
[547, 547, 2*w^2 + w - 20],
[547, 547, w^2 - 6*w - 5],
[547, 547, 6*w^2 - 11*w - 24],
[557, 557, -5*w^2 + 11*w + 11],
[563, 563, 3*w^2 - 6*w - 16],
[569, 569, 4*w^2 - 5*w - 17],
[587, 587, 5*w^2 - 8*w - 20],
[593, 593, 5*w - 3],
[593, 593, 3*w^2 - w - 17],
[593, 593, w - 9],
[613, 613, 2*w^2 - w - 19],
[613, 613, 3*w^2 - 4*w - 5],
[613, 613, 3*w^2 - w - 8],
[617, 617, 3*w^2 - 2*w - 9],
[617, 617, -w^2 + w - 5],
[617, 617, 4*w^2 - 6*w - 15],
[619, 619, -w^2 + 3*w - 6],
[631, 631, 4*w^2 - 7*w - 21],
[641, 641, -w^2 - 3*w + 20],
[643, 643, 3*w^2 - 14],
[643, 643, w^2 - w - 13],
[643, 643, 2*w^2 - 9*w + 2],
[647, 647, 2*w^2 - 6*w - 9],
[653, 653, 3*w^2 - 5],
[659, 659, 6*w^2 - 11*w - 21],
[661, 661, w^2 - 6*w - 6],
[661, 661, w^2 + 3*w - 6],
[661, 661, 5*w^2 - 7*w - 22],
[673, 673, w^2 - 6*w - 12],
[677, 677, 4*w^2 - 7*w - 22],
[677, 677, 3*w^2 - 7*w - 12],
[677, 677, -2*w^2 - 3],
[683, 683, -2*w - 9],
[683, 683, 4*w^2 - 7*w - 12],
[683, 683, 3*w^2 - 3*w - 8],
[691, 691, 2*w^2 + w - 11],
[701, 701, 3*w^2 - 3*w - 5],
[709, 709, -w^2 - 2*w - 6],
[727, 727, 3*w^2 - w - 6],
[727, 727, 3*w^2 - w - 27],
[727, 727, 3*w^2 - w - 5],
[733, 733, -2*w^2 + 9*w - 5],
[739, 739, 3*w^2 - 4*w - 23],
[743, 743, 2*w^2 - 7*w - 7],
[743, 743, 3*w^2 - w - 20],
[743, 743, 3*w^2 - 2*w - 7],
[757, 757, -w - 9],
[757, 757, 4*w^2 - 2*w - 29],
[757, 757, 2*w^2 - 3*w - 18],
[761, 761, 3*w^2 - 2*w - 6],
[761, 761, 3*w^2 - 2*w - 25],
[773, 773, -5*w - 12],
[787, 787, -3*w - 10],
[809, 809, 4*w^2 - w - 19],
[821, 821, 5*w^2 - 10*w - 19],
[823, 823, -4*w - 11],
[827, 827, -7*w - 5],
[827, 827, 2*w^2 - 6*w - 13],
[827, 827, -w^2 - 2*w + 18],
[839, 839, 6*w^2 - 9*w - 26],
[839, 839, 5*w^2 - 12*w - 13],
[839, 839, w - 10],
[857, 857, -w^2 + 4*w - 8],
[859, 859, 3*w^2 - 8*w - 10],
[859, 859, 4*w^2 - 9*w - 14],
[859, 859, 3*w^2 - 26],
[863, 863, 2*w^2 + 3*w - 7],
[877, 877, -5*w^2 + 11*w + 9],
[877, 877, -5*w^2 + 10*w + 13],
[877, 877, -w^2 + w - 6],
[881, 881, w^2 - 3*w - 14],
[883, 883, -w^2 + 3*w - 7],
[887, 887, -3*w^2 + 6*w - 2],
[907, 907, w^2 + 3*w - 8],
[911, 911, 4*w^2 - 6*w - 29],
[937, 937, -6*w^2 + 11*w + 25],
[953, 953, 6*w^2 - 6*w - 29],
[961, 31, -w^2 + 7*w - 5],
[967, 967, 2*w^2 + w - 14],
[971, 971, 5*w^2 - 3*w - 27],
[971, 971, 4*w - 15],
[971, 971, w^2 - 2*w - 14],
[977, 977, 3*w^2 - 5*w - 24],
[983, 983, w^2 - 4*w - 15],
[991, 991, 6*w^2 - 10*w - 23],
[991, 991, 5*w^2 - 6*w - 22],
[991, 991, 3*w^2 + w - 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 2*x^7 - 16*x^6 - 32*x^5 + 72*x^4 + 156*x^3 - 64*x^2 - 240*x - 104;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^7 + 3/4*e^6 - 17*e^5 - 11*e^4 + 86*e^3 + 51*e^2 - 126*e - 88, 3/2*e^7 + e^6 - 51/2*e^5 - 29/2*e^4 + 259/2*e^3 + 68*e^2 - 191*e - 125, 3/4*e^7 + 1/2*e^6 - 25/2*e^5 - 7*e^4 + 123/2*e^3 + 31*e^2 - 87*e - 56, 1/4*e^7 + 1/4*e^6 - 4*e^5 - 3*e^4 + 37/2*e^3 + 9*e^2 - 23*e - 10, -1/2*e^7 - 1/2*e^6 + 17/2*e^5 + 15/2*e^4 - 44*e^3 - 35*e^2 + 69*e + 56, -7/2*e^7 - 5/2*e^6 + 59*e^5 + 71/2*e^4 - 295*e^3 - 159*e^2 + 422*e + 276, -1, 1/2*e^7 + 1/2*e^6 - 8*e^5 - 7*e^4 + 73/2*e^3 + 28*e^2 - 45*e - 38, -11/4*e^7 - 2*e^6 + 93/2*e^5 + 57/2*e^4 - 467/2*e^3 - 128*e^2 + 336*e + 220, -1/2*e^7 - 1/4*e^6 + 8*e^5 + 3*e^4 - 73/2*e^3 - 11*e^2 + 44*e + 18, -7/4*e^7 - 5/4*e^6 + 59/2*e^5 + 35/2*e^4 - 147*e^3 - 77*e^2 + 207*e + 132, -9/4*e^7 - 7/4*e^6 + 75/2*e^5 + 25*e^4 - 185*e^3 - 110*e^2 + 263*e + 178, e^7 + e^6 - 33/2*e^5 - 14*e^4 + 161/2*e^3 + 58*e^2 - 113*e - 86, -1/4*e^7 + 5*e^5 + 1/2*e^4 - 31*e^3 - 7*e^2 + 58*e + 26, -7/4*e^7 - 5/4*e^6 + 59/2*e^5 + 18*e^4 - 148*e^3 - 84*e^2 + 217*e + 152, -e^7 - 3/4*e^6 + 33/2*e^5 + 10*e^4 - 161/2*e^3 - 41*e^2 + 113*e + 68, -15/4*e^7 - 3*e^6 + 63*e^5 + 87/2*e^4 - 314*e^3 - 195*e^2 + 452*e + 318, 1/2*e^7 - 9*e^5 + 1/2*e^4 + 97/2*e^3 - e^2 - 74*e - 22, -5/4*e^7 - 3/4*e^6 + 21*e^5 + 11*e^4 - 104*e^3 - 53*e^2 + 145*e + 96, e^7 + 5/4*e^6 - 17*e^5 - 18*e^4 + 175/2*e^3 + 79*e^2 - 136*e - 122, -11/2*e^7 - 15/4*e^6 + 93*e^5 + 53*e^4 - 935/2*e^3 - 239*e^2 + 677*e + 430, 11/4*e^7 + 2*e^6 - 93/2*e^5 - 29*e^4 + 234*e^3 + 134*e^2 - 339*e - 242, -2*e^7 - e^6 + 69/2*e^5 + 16*e^4 - 178*e^3 - 88*e^2 + 266*e + 178, 17/2*e^7 + 23/4*e^6 - 143*e^5 - 82*e^4 + 713*e^3 + 371*e^2 - 1018*e - 648, 21/4*e^7 + 13/4*e^6 - 89*e^5 - 91/2*e^4 + 448*e^3 + 204*e^2 - 643*e - 380, -27/4*e^7 - 19/4*e^6 + 227/2*e^5 + 66*e^4 - 565*e^3 - 286*e^2 + 801*e + 494, 3/4*e^7 + 1/2*e^6 - 13*e^5 - 7*e^4 + 69*e^3 + 33*e^2 - 110*e - 70, -13/4*e^7 - 5/2*e^6 + 109/2*e^5 + 35*e^4 - 271*e^3 - 152*e^2 + 390*e + 250, 5/4*e^7 + 5/4*e^6 - 43/2*e^5 - 18*e^4 + 112*e^3 + 82*e^2 - 175*e - 144, -17/4*e^7 - 5/2*e^6 + 72*e^5 + 71/2*e^4 - 362*e^3 - 166*e^2 + 524*e + 322, -9/4*e^7 - 7/4*e^6 + 77/2*e^5 + 26*e^4 - 197*e^3 - 122*e^2 + 293*e + 208, 35/4*e^7 + 21/4*e^6 - 297/2*e^5 - 151/2*e^4 + 749*e^3 + 355*e^2 - 1087*e - 672, 11/2*e^7 + 9/2*e^6 - 185/2*e^5 - 64*e^4 + 462*e^3 + 281*e^2 - 662*e - 460, 5*e^7 + 13/4*e^6 - 169/2*e^5 - 47*e^4 + 847/2*e^3 + 219*e^2 - 609*e - 392, -5/4*e^7 - 1/2*e^6 + 21*e^5 + 13/2*e^4 - 105*e^3 - 30*e^2 + 150*e + 72, 13/2*e^7 + 5*e^6 - 110*e^5 - 73*e^4 + 554*e^3 + 334*e^2 - 803*e - 556, -17/4*e^7 - 3*e^6 + 143/2*e^5 + 41*e^4 - 713/2*e^3 - 173*e^2 + 509*e + 300, 8*e^7 + 11/2*e^6 - 135*e^5 - 159/2*e^4 + 677*e^3 + 367*e^2 - 979*e - 656, 13/4*e^7 + 2*e^6 - 109/2*e^5 - 29*e^4 + 270*e^3 + 134*e^2 - 384*e - 242, -1/4*e^7 + 1/4*e^6 + 6*e^5 - 3/2*e^4 - 43*e^3 - 12*e^2 + 89*e + 62, -37/4*e^7 - 13/2*e^6 + 311/2*e^5 + 92*e^4 - 774*e^3 - 408*e^2 + 1098*e + 706, 9/2*e^7 + 4*e^6 - 75*e^5 - 57*e^4 + 741/2*e^3 + 244*e^2 - 529*e - 374, 15/4*e^7 + 9/4*e^6 - 129/2*e^5 - 34*e^4 + 665/2*e^3 + 173*e^2 - 497*e - 340, 5/2*e^7 + 2*e^6 - 43*e^5 - 30*e^4 + 221*e^3 + 141*e^2 - 326*e - 242, 23/4*e^7 + 13/4*e^6 - 193/2*e^5 - 46*e^4 + 478*e^3 + 212*e^2 - 675*e - 404, -13/4*e^7 - 3*e^6 + 54*e^5 + 85/2*e^4 - 265*e^3 - 178*e^2 + 373*e + 254, -10*e^7 - 7*e^6 + 170*e^5 + 101*e^4 - 862*e^3 - 468*e^2 + 1260*e + 848, -e^7 + 35/2*e^5 - 91*e^3 - 8*e^2 + 136*e + 56, 43/4*e^7 + 7*e^6 - 363/2*e^5 - 99*e^4 + 1821/2*e^3 + 451*e^2 - 1315*e - 824, 19/4*e^7 + 13/4*e^6 - 159/2*e^5 - 46*e^4 + 787/2*e^3 + 207*e^2 - 552*e - 360, 15/4*e^7 + 9/4*e^6 - 127/2*e^5 - 33*e^4 + 319*e^3 + 162*e^2 - 461*e - 318, 5/4*e^7 + 5/4*e^6 - 19*e^5 - 15*e^4 + 79*e^3 + 43*e^2 - 77*e - 36, 35/4*e^7 + 6*e^6 - 148*e^5 - 175/2*e^4 + 745*e^3 + 410*e^2 - 1083*e - 738, -11/4*e^7 - 9/4*e^6 + 95/2*e^5 + 69/2*e^4 - 248*e^3 - 171*e^2 + 385*e + 300, -1/4*e^7 + 1/4*e^6 + 4*e^5 - 4*e^4 - 37/2*e^3 + 11*e^2 + 23*e + 14, -11/4*e^7 - 7/4*e^6 + 47*e^5 + 25*e^4 - 239*e^3 - 115*e^2 + 347*e + 214, 21/4*e^7 + 4*e^6 - 175/2*e^5 - 113/2*e^4 + 863/2*e^3 + 250*e^2 - 608*e - 424, -1/2*e^7 + 8*e^5 - e^4 - 35*e^3 + 6*e^2 + 38*e, -23/4*e^7 - 13/4*e^6 + 99*e^5 + 47*e^4 - 509*e^3 - 227*e^2 + 749*e + 454, 3/4*e^7 + 3/4*e^6 - 25/2*e^5 - 10*e^4 + 119/2*e^3 + 33*e^2 - 77*e - 20, -4*e^7 - 2*e^6 + 70*e^5 + 61/2*e^4 - 369*e^3 - 157*e^2 + 567*e + 320, -9*e^7 - 7*e^6 + 301/2*e^5 + 100*e^4 - 744*e^3 - 442*e^2 + 1054*e + 722, -13/4*e^7 - 5/2*e^6 + 54*e^5 + 35*e^4 - 265*e^3 - 146*e^2 + 371*e + 220, 45/4*e^7 + 29/4*e^6 - 190*e^5 - 205/2*e^4 + 952*e^3 + 462*e^2 - 1368*e - 834, -9/4*e^7 - 5/4*e^6 + 73/2*e^5 + 16*e^4 - 341/2*e^3 - 61*e^2 + 218*e + 104, 27/2*e^7 + 17/2*e^6 - 461/2*e^5 - 122*e^4 + 1174*e^3 + 574*e^2 - 1722*e - 1088, 10*e^7 + 7*e^6 - 170*e^5 - 102*e^4 + 861*e^3 + 476*e^2 - 1256*e - 852, -7/2*e^7 - 5/2*e^6 + 57*e^5 + 67/2*e^4 - 272*e^3 - 137*e^2 + 367*e + 220, e^7 - 1/4*e^6 - 37/2*e^5 + 5*e^4 + 201/2*e^3 - 19*e^2 - 147*e - 36, 9*e^7 + 6*e^6 - 151*e^5 - 86*e^4 + 749*e^3 + 391*e^2 - 1060*e - 684, -7/2*e^7 - 7/2*e^6 + 115/2*e^5 + 48*e^4 - 279*e^3 - 194*e^2 + 386*e + 286, 55/4*e^7 + 37/4*e^6 - 234*e^5 - 267/2*e^4 + 1189*e^3 + 623*e^2 - 1744*e - 1146, -23/2*e^7 - 15/2*e^6 + 196*e^5 + 215/2*e^4 - 1985/2*e^3 - 495*e^2 + 1432*e + 906, -3/2*e^7 - 1/2*e^6 + 26*e^5 + 9*e^4 - 134*e^3 - 59*e^2 + 198*e + 136, -17/4*e^7 - 13/4*e^6 + 71*e^5 + 45*e^4 - 352*e^3 - 191*e^2 + 499*e + 312, -51/4*e^7 - 37/4*e^6 + 216*e^5 + 135*e^4 - 1088*e^3 - 621*e^2 + 1577*e + 1072, 15/4*e^7 + 13/4*e^6 - 125/2*e^5 - 44*e^4 + 306*e^3 + 178*e^2 - 419*e - 288, -71/4*e^7 - 13*e^6 + 601/2*e^5 + 375/2*e^4 - 3023/2*e^3 - 852*e^2 + 2182*e + 1468, 5/4*e^7 - 1/2*e^6 - 43/2*e^5 + 13/2*e^4 + 219/2*e^3 - 8*e^2 - 162*e - 64, -7/4*e^7 - 5/4*e^6 + 29*e^5 + 16*e^4 - 143*e^3 - 63*e^2 + 211*e + 100, -25/4*e^7 - 4*e^6 + 211/2*e^5 + 111/2*e^4 - 528*e^3 - 244*e^2 + 749*e + 458, -5/2*e^7 - 2*e^6 + 41*e^5 + 28*e^4 - 195*e^3 - 121*e^2 + 260*e + 194, -4*e^7 - 2*e^6 + 137/2*e^5 + 57/2*e^4 - 699/2*e^3 - 147*e^2 + 510*e + 326, 69/4*e^7 + 25/2*e^6 - 291*e^5 - 179*e^4 + 2915/2*e^3 + 803*e^2 - 2095*e - 1384, 7/4*e^7 + 3/4*e^6 - 31*e^5 - 13*e^4 + 331/2*e^3 + 77*e^2 - 260*e - 166, 31/2*e^7 + 21/2*e^6 - 264*e^5 - 151*e^4 + 1340*e^3 + 698*e^2 - 1957*e - 1264, 15/2*e^7 + 11/2*e^6 - 126*e^5 - 75*e^4 + 623*e^3 + 310*e^2 - 866*e - 510, -43/4*e^7 - 29/4*e^6 + 365/2*e^5 + 104*e^4 - 1845/2*e^3 - 479*e^2 + 1346*e + 860, 25/4*e^7 + 19/4*e^6 - 103*e^5 - 135/2*e^4 + 499*e^3 + 297*e^2 - 688*e - 488, 15/4*e^7 + 3*e^6 - 127/2*e^5 - 43*e^4 + 318*e^3 + 195*e^2 - 452*e - 338, -63/4*e^7 - 39/4*e^6 + 269*e^5 + 143*e^4 - 2741/2*e^3 - 685*e^2 + 2020*e + 1286, 11/2*e^7 + 3*e^6 - 187/2*e^5 - 42*e^4 + 472*e^3 + 198*e^2 - 687*e - 414, -27/4*e^7 - 7/2*e^6 + 229/2*e^5 + 50*e^4 - 580*e^3 - 246*e^2 + 854*e + 510, -9/4*e^7 - 9/4*e^6 + 38*e^5 + 33*e^4 - 379/2*e^3 - 149*e^2 + 270*e + 242, -9/4*e^7 - e^6 + 41*e^5 + 17*e^4 - 228*e^3 - 107*e^2 + 376*e + 250, 27/4*e^7 + 9/2*e^6 - 115*e^5 - 131/2*e^4 + 584*e^3 + 305*e^2 - 856*e - 556, -11/4*e^7 - 9/4*e^6 + 93/2*e^5 + 63/2*e^4 - 232*e^3 - 137*e^2 + 333*e + 242, 33/4*e^7 + 23/4*e^6 - 140*e^5 - 82*e^4 + 711*e^3 + 377*e^2 - 1057*e - 684, -15/4*e^7 - 5/4*e^6 + 65*e^5 + 35/2*e^4 - 339*e^3 - 98*e^2 + 511*e + 260, 75/4*e^7 + 29/2*e^6 - 316*e^5 - 208*e^4 + 1584*e^3 + 930*e^2 - 2289*e - 1572, 9/4*e^7 + e^6 - 40*e^5 - 17*e^4 + 427/2*e^3 + 101*e^2 - 329*e - 232, 5/4*e^7 - 45/2*e^5 - 1/2*e^4 + 245/2*e^3 + 16*e^2 - 192*e - 104, -5/4*e^7 - 1/2*e^6 + 19*e^5 + 9/2*e^4 - 80*e^3 - 9*e^2 + 84*e + 20, 55/4*e^7 + 15/2*e^6 - 467/2*e^5 - 108*e^4 + 1178*e^3 + 517*e^2 - 1706*e - 1018, 33/2*e^7 + 21/2*e^6 - 557/2*e^5 - 299/2*e^4 + 1396*e^3 + 684*e^2 - 2010*e - 1254, 15*e^7 + 43/4*e^6 - 253*e^5 - 154*e^4 + 2537/2*e^3 + 701*e^2 - 1843*e - 1226, -13/4*e^7 - 3*e^6 + 52*e^5 + 83/2*e^4 - 243*e^3 - 170*e^2 + 324*e + 250, 51/4*e^7 + 27/4*e^6 - 217*e^5 - 193/2*e^4 + 1098*e^3 + 467*e^2 - 1600*e - 942, 1/4*e^7 - 3/4*e^6 - 7*e^5 + 9*e^4 + 56*e^3 - 11*e^2 - 127*e - 58, -27/4*e^7 - 21/4*e^6 + 116*e^5 + 76*e^4 - 598*e^3 - 349*e^2 + 899*e + 602, 1/2*e^7 - 19/2*e^5 - 1/2*e^4 + 52*e^3 + 9*e^2 - 67*e - 40, 3/4*e^7 - 16*e^5 - 3*e^4 + 105*e^3 + 33*e^2 - 200*e - 90, 3/4*e^7 + 2*e^6 - 23/2*e^5 - 53/2*e^4 + 99/2*e^3 + 82*e^2 - 56*e - 44, 27/2*e^7 + 8*e^6 - 461/2*e^5 - 114*e^4 + 1173*e^3 + 536*e^2 - 1718*e - 1050, -19/4*e^7 - 17/4*e^6 + 79*e^5 + 60*e^4 - 389*e^3 - 253*e^2 + 555*e + 384, -15/4*e^7 - 3/2*e^6 + 127/2*e^5 + 21*e^4 - 321*e^3 - 108*e^2 + 474*e + 270, -5*e^7 - 5/2*e^6 + 86*e^5 + 73/2*e^4 - 883/2*e^3 - 183*e^2 + 650*e + 378, -59/4*e^7 - 39/4*e^6 + 248*e^5 + 139*e^4 - 2473/2*e^3 - 633*e^2 + 1772*e + 1142, 6*e^7 + 11/2*e^6 - 100*e^5 - 155/2*e^4 + 492*e^3 + 325*e^2 - 683*e - 496, -41/2*e^7 - 27/2*e^6 + 346*e^5 + 190*e^4 - 1735*e^3 - 853*e^2 + 2506*e + 1548, -16*e^7 - 21/2*e^6 + 271*e^5 + 150*e^4 - 1364*e^3 - 688*e^2 + 1982*e + 1252, -7/2*e^7 - 3*e^6 + 58*e^5 + 83/2*e^4 - 281*e^3 - 169*e^2 + 381*e + 252, 9/2*e^7 + 13/4*e^6 - 153/2*e^5 - 49*e^4 + 783/2*e^3 + 235*e^2 - 597*e - 412, 45/4*e^7 + 31/4*e^6 - 188*e^5 - 217/2*e^4 + 930*e^3 + 478*e^2 - 1321*e - 820, -6*e^7 - 11/2*e^6 + 197/2*e^5 + 78*e^4 - 475*e^3 - 328*e^2 + 660*e + 476, 33/4*e^7 + 23/4*e^6 - 139*e^5 - 83*e^4 + 693*e^3 + 383*e^2 - 979*e - 674, -67/4*e^7 - 47/4*e^6 + 565/2*e^5 + 167*e^4 - 2825/2*e^3 - 749*e^2 + 2025*e + 1320, -49/4*e^7 - 31/4*e^6 + 413/2*e^5 + 110*e^4 - 1031*e^3 - 508*e^2 + 1473*e + 930, -11/4*e^7 - 2*e^6 + 97/2*e^5 + 67/2*e^4 - 525/2*e^3 - 182*e^2 + 434*e + 344, e^7 + 3/2*e^6 - 19*e^5 - 25*e^4 + 113*e^3 + 126*e^2 - 212*e - 204, -10*e^7 - 27/4*e^6 + 168*e^5 + 96*e^4 - 838*e^3 - 437*e^2 + 1208*e + 796, -9/4*e^7 - 2*e^6 + 75/2*e^5 + 28*e^4 - 188*e^3 - 118*e^2 + 275*e + 182, 25/4*e^7 + 11/2*e^6 - 105*e^5 - 157/2*e^4 + 524*e^3 + 340*e^2 - 750*e - 522, -23/4*e^7 - 4*e^6 + 199/2*e^5 + 121/2*e^4 - 520*e^3 - 300*e^2 + 793*e + 554, 35/4*e^7 + 13/2*e^6 - 148*e^5 - 191/2*e^4 + 745*e^3 + 445*e^2 - 1084*e - 746, -45/4*e^7 - 33/4*e^6 + 188*e^5 + 116*e^4 - 929*e^3 - 507*e^2 + 1315*e + 862, 77/4*e^7 + 53/4*e^6 - 327*e^5 - 193*e^4 + 1655*e^3 + 903*e^2 - 2419*e - 1616, 35/2*e^7 + 12*e^6 - 295*e^5 - 169*e^4 + 1476*e^3 + 750*e^2 - 2118*e - 1328, 9*e^7 + 25/4*e^6 - 152*e^5 - 90*e^4 + 764*e^3 + 408*e^2 - 1116*e - 706, 21/4*e^7 + 13/4*e^6 - 87*e^5 - 89/2*e^4 + 423*e^3 + 195*e^2 - 582*e - 354, 5/2*e^7 + 5/2*e^6 - 41*e^5 - 34*e^4 + 196*e^3 + 135*e^2 - 252*e - 196, -47/4*e^7 - 23/4*e^6 + 201*e^5 + 167/2*e^4 - 1023*e^3 - 412*e^2 + 1493*e + 860, -10*e^7 - 8*e^6 + 168*e^5 + 223/2*e^4 - 1677/2*e^3 - 475*e^2 + 1204*e + 766, -15/4*e^7 - 1/4*e^6 + 129/2*e^5 + 9/2*e^4 - 329*e^3 - 51*e^2 + 479*e + 210, 25*e^7 + 33/2*e^6 - 847/2*e^5 - 238*e^4 + 4269/2*e^3 + 1104*e^2 - 3095*e - 1994, -61/4*e^7 - 10*e^6 + 517/2*e^5 + 289/2*e^4 - 1301*e^3 - 670*e^2 + 1873*e + 1206, -9/2*e^7 - 5/2*e^6 + 155/2*e^5 + 36*e^4 - 398*e^3 - 179*e^2 + 592*e + 384, 19*e^7 + 13*e^6 - 641/2*e^5 - 185*e^4 + 3213/2*e^3 + 840*e^2 - 2313*e - 1490, 87/4*e^7 + 16*e^6 - 733/2*e^5 - 459/2*e^4 + 3667/2*e^3 + 1036*e^2 - 2646*e - 1784, -19/4*e^7 - 13/4*e^6 + 155/2*e^5 + 89/2*e^4 - 367*e^3 - 187*e^2 + 481*e + 304, 12*e^7 + 15/2*e^6 - 409/2*e^5 - 217/2*e^4 + 1041*e^3 + 518*e^2 - 1534*e - 996, -63/4*e^7 - 45/4*e^6 + 264*e^5 + 157*e^4 - 2627/2*e^3 - 687*e^2 + 1882*e + 1182, -63/4*e^7 - 19/2*e^6 + 270*e^5 + 141*e^4 - 1383*e^3 - 692*e^2 + 2054*e + 1324, 37/4*e^7 + 6*e^6 - 157*e^5 - 177/2*e^4 + 797*e^3 + 429*e^2 - 1180*e - 792, 55/4*e^7 + 45/4*e^6 - 231*e^5 - 315/2*e^4 + 1151*e^3 + 675*e^2 - 1644*e - 1092, -17/4*e^7 - 5*e^6 + 70*e^5 + 69*e^4 - 677/2*e^3 - 273*e^2 + 455*e + 360, -19/4*e^7 - 15/4*e^6 + 163/2*e^5 + 54*e^4 - 835/2*e^3 - 243*e^2 + 618*e + 420, -13/4*e^7 - 2*e^6 + 57*e^5 + 67/2*e^4 - 301*e^3 - 187*e^2 + 462*e + 370, 31/4*e^7 + 9/2*e^6 - 133*e^5 - 131/2*e^4 + 680*e^3 + 313*e^2 - 998*e - 588, 41/4*e^7 + 7*e^6 - 174*e^5 - 201/2*e^4 + 879*e^3 + 455*e^2 - 1282*e - 792, 11*e^7 + 7*e^6 - 189*e^5 - 99*e^4 + 972*e^3 + 454*e^2 - 1448*e - 860, 8*e^7 + 21/4*e^6 - 137*e^5 - 77*e^4 + 701*e^3 + 367*e^2 - 1040*e - 678, -31/2*e^7 - 11*e^6 + 262*e^5 + 158*e^4 - 1315*e^3 - 715*e^2 + 1892*e + 1234, 81/4*e^7 + 13*e^6 - 681/2*e^5 - 182*e^4 + 1698*e^3 + 819*e^2 - 2436*e - 1486, 39/2*e^7 + 57/4*e^6 - 330*e^5 - 207*e^4 + 3325/2*e^3 + 951*e^2 - 2426*e - 1662, 65/4*e^7 + 25/2*e^6 - 275*e^5 - 357/2*e^4 + 1382*e^3 + 793*e^2 - 1994*e - 1348, 25/4*e^7 + 17/4*e^6 - 106*e^5 - 129/2*e^4 + 535*e^3 + 319*e^2 - 778*e - 568, -19*e^7 - 14*e^6 + 321*e^5 + 201*e^4 - 1611*e^3 - 911*e^2 + 2320*e + 1570, -29/4*e^7 - 13/4*e^6 + 245/2*e^5 + 87/2*e^4 - 609*e^3 - 203*e^2 + 847*e + 444, -47/4*e^7 - 39/4*e^6 + 198*e^5 + 283/2*e^4 - 995*e^3 - 634*e^2 + 1450*e + 1028, -21/2*e^7 - 8*e^6 + 177*e^5 + 231/2*e^4 - 884*e^3 - 511*e^2 + 1262*e + 822, 21/4*e^7 + 9/4*e^6 - 90*e^5 - 33*e^4 + 462*e^3 + 175*e^2 - 687*e - 388, -20*e^7 - 15*e^6 + 339*e^5 + 216*e^4 - 1709*e^3 - 978*e^2 + 2475*e + 1680, 29/4*e^7 + 9/2*e^6 - 123*e^5 - 133/2*e^4 + 1247/2*e^3 + 324*e^2 - 918*e - 600, 1/4*e^7 + 1/2*e^6 - 4*e^5 - 15/2*e^4 + 23*e^3 + 34*e^2 - 58*e - 60];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^2 + 2*w + 5}>] := 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;