/*
  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![-87, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {1, 1, 1}>;

primesArray := [
[2, 2, w + 1],
[3, 3, w],
[13, 13, -w + 10],
[13, 13, w + 10],
[17, 17, w + 6],
[17, 17, w + 11],
[19, 19, w + 7],
[19, 19, w + 12],
[23, 23, -w - 8],
[23, 23, w - 8],
[25, 5, -5],
[29, 29, w],
[31, 31, w + 5],
[31, 31, w + 26],
[41, 41, w + 13],
[41, 41, w + 28],
[43, 43, w + 1],
[43, 43, w + 42],
[49, 7, -7],
[59, 59, -2*w + 17],
[59, 59, 2*w + 17],
[71, 71, -w - 4],
[71, 71, w - 4],
[79, 79, w + 18],
[79, 79, w + 61],
[83, 83, -w - 2],
[83, 83, w - 2],
[89, 89, w + 40],
[89, 89, w + 49],
[101, 101, w + 17],
[101, 101, w + 84],
[107, 107, -3*w + 26],
[107, 107, 3*w + 26],
[109, 109, -w - 14],
[109, 109, w - 14],
[113, 113, w + 55],
[113, 113, w + 58],
[121, 11, -11],
[127, 127, w + 50],
[127, 127, w + 77],
[137, 137, w + 19],
[137, 137, w + 118],
[163, 163, w + 24],
[163, 163, w + 139],
[167, 167, 4*w - 35],
[167, 167, -4*w - 35],
[179, 179, 2*w - 13],
[179, 179, -2*w - 13],
[181, 181, 2*w - 23],
[181, 181, -2*w - 23],
[211, 211, w + 64],
[211, 211, w + 147],
[227, 227, 2*w - 11],
[227, 227, -2*w - 11],
[239, 239, -5*w + 44],
[239, 239, -8*w + 73],
[241, 241, 12*w - 113],
[241, 241, -3*w + 32],
[269, 269, w + 25],
[269, 269, w + 244],
[271, 271, w + 30],
[271, 271, w + 241],
[277, 277, 2*w - 25],
[277, 277, -2*w - 25],
[293, 293, w + 110],
[293, 293, w + 183],
[307, 307, w + 98],
[307, 307, w + 209],
[313, 313, -w - 20],
[313, 313, w - 20],
[317, 317, w + 74],
[317, 317, w + 243],
[331, 331, w + 114],
[331, 331, w + 217],
[347, 347, 2*w - 1],
[347, 347, -2*w - 1],
[349, 349, -9*w + 86],
[349, 349, 6*w - 59],
[367, 367, w + 120],
[367, 367, w + 247],
[373, 373, 3*w - 34],
[373, 373, -3*w - 34],
[379, 379, w + 128],
[379, 379, w + 251],
[383, 383, 3*w - 20],
[383, 383, -3*w - 20],
[389, 389, w + 101],
[389, 389, w + 288],
[397, 397, -w - 22],
[397, 397, w - 22],
[419, 419, -7*w + 62],
[419, 419, -10*w + 91],
[431, 431, 4*w - 31],
[431, 431, -4*w - 31],
[449, 449, w + 136],
[449, 449, w + 313],
[457, 457, -4*w - 43],
[457, 457, 4*w - 43],
[461, 461, w + 116],
[461, 461, w + 345],
[569, 569, w + 35],
[569, 569, w + 534],
[587, 587, 3*w - 14],
[587, 587, -3*w - 14],
[607, 607, w + 249],
[607, 607, w + 358],
[613, 613, 2*w - 31],
[613, 613, -2*w - 31],
[617, 617, w + 127],
[617, 617, w + 490],
[619, 619, w + 169],
[619, 619, w + 450],
[641, 641, w + 37],
[641, 641, w + 604],
[647, 647, -9*w + 80],
[647, 647, -12*w + 109],
[653, 653, w + 273],
[653, 653, w + 380],
[661, 661, -3*w - 38],
[661, 661, 3*w - 38],
[673, 673, -13*w + 124],
[673, 673, 8*w - 79],
[677, 677, w + 133],
[677, 677, w + 544],
[683, 683, -3*w - 10],
[683, 683, 3*w - 10],
[709, 709, 10*w - 97],
[709, 709, -11*w + 106],
[719, 719, 3*w - 8],
[719, 719, -3*w - 8],
[727, 727, w + 304],
[727, 727, w + 423],
[739, 739, w + 48],
[739, 739, w + 691],
[751, 751, w + 350],
[751, 751, w + 401],
[773, 773, w + 297],
[773, 773, w + 476],
[797, 797, w + 41],
[797, 797, w + 756],
[809, 809, w + 286],
[809, 809, w + 523],
[823, 823, w + 160],
[823, 823, w + 663],
[859, 859, w + 199],
[859, 859, w + 660],
[863, 863, 4*w - 23],
[863, 863, -4*w - 23],
[877, 877, 2*w - 35],
[877, 877, -2*w - 35],
[881, 881, w + 43],
[881, 881, w + 838],
[907, 907, w + 359],
[907, 907, w + 548],
[937, 937, -w - 32],
[937, 937, w - 32],
[967, 967, w + 455],
[967, 967, w + 512]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - 72*x^10 + 1800*x^8 - 17280*x^6 + 42128*x^4 - 15744*x^2 + 256;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/1024*e^11 + 9/128*e^9 - 451/256*e^7 + 273/16*e^5 - 697/16*e^3 + 20*e, 1/1024*e^11 - 9/128*e^9 + 451/256*e^7 - 273/16*e^5 + 697/16*e^3 - 19*e, 1, 1, -1/1024*e^11 + 37/512*e^9 - 471/256*e^7 + 2251/128*e^5 - 599/16*e^3 + 25/8*e, -1/1024*e^11 + 37/512*e^9 - 471/256*e^7 + 2251/128*e^5 - 599/16*e^3 + 25/8*e, -3/1024*e^11 + 107/512*e^9 - 1325/256*e^7 + 6293/128*e^5 - 1879/16*e^3 + 327/8*e, -3/1024*e^11 + 107/512*e^9 - 1325/256*e^7 + 6293/128*e^5 - 1879/16*e^3 + 327/8*e, 1/512*e^10 - 15/128*e^8 + 303/128*e^6 - 567/32*e^4 + 34*e^2 - 3, 1/512*e^10 - 15/128*e^8 + 303/128*e^6 - 567/32*e^4 + 34*e^2 - 3, 1/64*e^8 - 3/4*e^6 + 159/16*e^4 - 45/2*e^2 + 7, 3/512*e^11 - 109/256*e^9 + 1373/128*e^7 - 6619/64*e^5 + 1993/8*e^3 - 345/4*e, -3/1024*e^11 + 53/256*e^9 - 1301/256*e^7 + 3069/64*e^5 - 921/8*e^3 + 35*e, -3/1024*e^11 + 53/256*e^9 - 1301/256*e^7 + 3069/64*e^5 - 921/8*e^3 + 35*e, 5/1024*e^11 - 183/512*e^9 + 2319/256*e^7 - 11209/128*e^5 + 1659/8*e^3 - 481/8*e, 5/1024*e^11 - 183/512*e^9 + 2319/256*e^7 - 11209/128*e^5 + 1659/8*e^3 - 481/8*e, 1/256*e^11 - 141/512*e^9 + 431/64*e^7 - 8063/128*e^5 + 2327/16*e^3 - 303/8*e, 1/256*e^11 - 141/512*e^9 + 431/64*e^7 - 8063/128*e^5 + 2327/16*e^3 - 303/8*e, 1/128*e^8 - 3/8*e^6 + 163/32*e^4 - 57/4*e^2 + 25/2, -1/512*e^10 + 15/128*e^8 - 307/128*e^6 + 603/32*e^4 - 351/8*e^2 + 27/2, -1/512*e^10 + 15/128*e^8 - 307/128*e^6 + 603/32*e^4 - 351/8*e^2 + 27/2, 1/32*e^6 - 9/8*e^4 + 79/8*e^2 - 21/2, 1/32*e^6 - 9/8*e^4 + 79/8*e^2 - 21/2, 1/512*e^11 - 71/512*e^9 + 437/128*e^7 - 4109/128*e^5 + 591/8*e^3 - 175/8*e, 1/512*e^11 - 71/512*e^9 + 437/128*e^7 - 4109/128*e^5 + 591/8*e^3 - 175/8*e, -1/512*e^10 + 15/128*e^8 - 307/128*e^6 + 603/32*e^4 - 343/8*e^2 + 3/2, -1/512*e^10 + 15/128*e^8 - 307/128*e^6 + 603/32*e^4 - 343/8*e^2 + 3/2, -5/1024*e^11 + 183/512*e^9 - 2323/256*e^7 + 11297/128*e^5 - 3445/16*e^3 + 567/8*e, -5/1024*e^11 + 183/512*e^9 - 2323/256*e^7 + 11297/128*e^5 - 3445/16*e^3 + 567/8*e, 1/64*e^7 - 11/16*e^5 + 127/16*e^3 - 43/4*e, 1/64*e^7 - 11/16*e^5 + 127/16*e^3 - 43/4*e, -1/512*e^10 + 15/128*e^8 - 303/128*e^6 + 567/32*e^4 - 35*e^2 + 15, -1/512*e^10 + 15/128*e^8 - 303/128*e^6 + 567/32*e^4 - 35*e^2 + 15, 5/256*e^8 - 15/16*e^6 + 823/64*e^4 - 309/8*e^2 + 31/4, 5/256*e^8 - 15/16*e^6 + 823/64*e^4 - 309/8*e^2 + 31/4, -1/256*e^11 + 9/32*e^9 - 225/32*e^7 + 1081/16*e^5 - 2661/16*e^3 + 277/4*e, -1/256*e^11 + 9/32*e^9 - 225/32*e^7 + 1081/16*e^5 - 2661/16*e^3 + 277/4*e, 3/256*e^8 - 9/16*e^6 + 481/64*e^4 - 147/8*e^2 + 89/4, 3/1024*e^11 - 109/512*e^9 + 1377/256*e^7 - 6707/128*e^5 + 133*e^3 - 503/8*e, 3/1024*e^11 - 109/512*e^9 + 1377/256*e^7 - 6707/128*e^5 + 133*e^3 - 503/8*e, 1/256*e^9 - 3/16*e^7 + 155/64*e^5 - 29/8*e^3 - 49/4*e, 1/256*e^9 - 3/16*e^7 + 155/64*e^5 - 29/8*e^3 - 49/4*e, 5/1024*e^11 - 89/256*e^9 + 2199/256*e^7 - 5201/64*e^5 + 3061/16*e^3 - 251/4*e, 5/1024*e^11 - 89/256*e^9 + 2199/256*e^7 - 5201/64*e^5 + 3061/16*e^3 - 251/4*e, 1/256*e^10 - 15/64*e^8 + 307/64*e^6 - 603/16*e^4 + 347/4*e^2 - 15, 1/256*e^10 - 15/64*e^8 + 307/64*e^6 - 603/16*e^4 + 347/4*e^2 - 15, -1/256*e^10 + 15/64*e^8 - 301/64*e^6 + 549/16*e^4 - 465/8*e^2 - 9/2, -1/256*e^10 + 15/64*e^8 - 301/64*e^6 + 549/16*e^4 - 465/8*e^2 - 9/2, 15/256*e^8 - 45/16*e^6 + 2405/64*e^4 - 735/8*e^2 + 85/4, 15/256*e^8 - 45/16*e^6 + 2405/64*e^4 - 735/8*e^2 + 85/4, 11/1024*e^11 - 99/128*e^9 + 4953/256*e^7 - 2977/16*e^5 + 7317/16*e^3 - 377/2*e, 11/1024*e^11 - 99/128*e^9 + 4953/256*e^7 - 2977/16*e^5 + 7317/16*e^3 - 377/2*e, 1/256*e^10 - 15/64*e^8 + 303/64*e^6 - 567/16*e^4 + 69*e^2 - 18, 1/256*e^10 - 15/64*e^8 + 303/64*e^6 - 567/16*e^4 + 69*e^2 - 18, -3/512*e^10 + 45/128*e^8 - 909/128*e^6 + 1701/32*e^4 - 102*e^2 + 9, -3/512*e^10 + 45/128*e^8 - 909/128*e^6 + 1701/32*e^4 - 102*e^2 + 9, -3/128*e^8 + 9/8*e^6 - 465/32*e^4 + 99/4*e^2 + 9/2, -3/128*e^8 + 9/8*e^6 - 465/32*e^4 + 99/4*e^2 + 9/2, -9/1024*e^11 + 331/512*e^9 - 4215/256*e^7 + 20477/128*e^5 - 6095/16*e^3 + 839/8*e, -9/1024*e^11 + 331/512*e^9 - 4215/256*e^7 + 20477/128*e^5 - 6095/16*e^3 + 839/8*e, -9/1024*e^11 + 159/256*e^9 - 3899/256*e^7 + 9155/64*e^5 - 5351/16*e^3 + 379/4*e, -9/1024*e^11 + 159/256*e^9 - 3899/256*e^7 + 9155/64*e^5 - 5351/16*e^3 + 379/4*e, -7/128*e^8 + 21/8*e^6 - 1141/32*e^4 + 399/4*e^2 - 63/2, -7/128*e^8 + 21/8*e^6 - 1141/32*e^4 + 399/4*e^2 - 63/2, 5/512*e^11 - 183/256*e^9 + 2317/128*e^7 - 11165/64*e^5 + 6509/16*e^3 - 219/2*e, 5/512*e^11 - 183/256*e^9 + 2317/128*e^7 - 11165/64*e^5 + 6509/16*e^3 - 219/2*e, -3/256*e^11 + 429/512*e^9 - 333/16*e^7 + 25431/128*e^5 - 483*e^3 + 1437/8*e, -3/256*e^11 + 429/512*e^9 - 333/16*e^7 + 25431/128*e^5 - 483*e^3 + 1437/8*e, -19/256*e^8 + 57/16*e^6 - 3057/64*e^4 + 963/8*e^2 - 89/4, -19/256*e^8 + 57/16*e^6 - 3057/64*e^4 + 963/8*e^2 - 89/4, 9/1024*e^11 - 327/512*e^9 + 4119/256*e^7 - 19857/128*e^5 + 5979/16*e^3 - 1035/8*e, 9/1024*e^11 - 327/512*e^9 + 4119/256*e^7 - 19857/128*e^5 + 5979/16*e^3 - 1035/8*e, -7/1024*e^11 + 127/256*e^9 - 3201/256*e^7 + 7747/64*e^5 - 2389/8*e^3 + 269/2*e, -7/1024*e^11 + 127/256*e^9 - 3201/256*e^7 + 7747/64*e^5 - 2389/8*e^3 + 269/2*e, -1/32*e^6 + 9/8*e^4 - 95/8*e^2 + 69/2, -1/32*e^6 + 9/8*e^4 - 95/8*e^2 + 69/2, -7/128*e^8 + 21/8*e^6 - 1141/32*e^4 + 399/4*e^2 - 49/2, -7/128*e^8 + 21/8*e^6 - 1141/32*e^4 + 399/4*e^2 - 49/2, -3/1024*e^11 + 111/512*e^9 - 1425/256*e^7 + 7017/128*e^5 - 1101/8*e^3 + 597/8*e, -3/1024*e^11 + 111/512*e^9 - 1425/256*e^7 + 7017/128*e^5 - 1101/8*e^3 + 597/8*e, 11/256*e^8 - 33/16*e^6 + 1769/64*e^4 - 555/8*e^2 + 45/4, 11/256*e^8 - 33/16*e^6 + 1769/64*e^4 - 555/8*e^2 + 45/4, 23/1024*e^11 - 815/512*e^9 + 10025/256*e^7 - 47281/128*e^5 + 13975/16*e^3 - 2147/8*e, 23/1024*e^11 - 815/512*e^9 + 10025/256*e^7 - 47281/128*e^5 + 13975/16*e^3 - 2147/8*e, 1/512*e^10 - 15/128*e^8 + 303/128*e^6 - 567/32*e^4 + 36*e^2 - 27, 1/512*e^10 - 15/128*e^8 + 303/128*e^6 - 567/32*e^4 + 36*e^2 - 27, 3/512*e^11 - 53/128*e^9 + 1303/128*e^7 - 3099/32*e^5 + 3939/16*e^3 - 535/4*e, 3/512*e^11 - 53/128*e^9 + 1303/128*e^7 - 3099/32*e^5 + 3939/16*e^3 - 535/4*e, -3/256*e^8 + 9/16*e^6 - 529/64*e^4 + 291/8*e^2 - 93/4, -3/256*e^8 + 9/16*e^6 - 529/64*e^4 + 291/8*e^2 - 93/4, -1/32*e^6 + 9/8*e^4 - 71/8*e^2 - 3/2, -1/32*e^6 + 9/8*e^4 - 71/8*e^2 - 3/2, 1/512*e^10 - 15/128*e^8 + 307/128*e^6 - 603/32*e^4 + 359/8*e^2 - 51/2, 1/512*e^10 - 15/128*e^8 + 307/128*e^6 - 603/32*e^4 + 359/8*e^2 - 51/2, -3/256*e^11 + 217/256*e^9 - 85/4*e^7 + 13039/64*e^5 - 7787/16*e^3 + 174*e, -3/256*e^11 + 217/256*e^9 - 85/4*e^7 + 13039/64*e^5 - 7787/16*e^3 + 174*e, -3/4*e^4 + 18*e^2 - 11, -3/4*e^4 + 18*e^2 - 11, -3/256*e^11 + 215/256*e^9 - 1337/64*e^7 + 12773/64*e^5 - 3899/8*e^3 + 837/4*e, -3/256*e^11 + 215/256*e^9 - 1337/64*e^7 + 12773/64*e^5 - 3899/8*e^3 + 837/4*e, 3/256*e^11 - 219/256*e^9 + 693/32*e^7 - 13437/64*e^5 + 8157/16*e^3 - 171*e, 3/256*e^11 - 219/256*e^9 + 693/32*e^7 - 13437/64*e^5 + 8157/16*e^3 - 171*e, 3/512*e^10 - 45/128*e^8 + 913/128*e^6 - 1737/32*e^4 + 887/8*e^2 - 15/2, 3/512*e^10 - 45/128*e^8 + 913/128*e^6 - 1737/32*e^4 + 887/8*e^2 - 15/2, 1/1024*e^11 - 7/128*e^9 + 243/256*e^7 - 33/8*e^5 - 299/16*e^3 + 69*e, 1/1024*e^11 - 7/128*e^9 + 243/256*e^7 - 33/8*e^5 - 299/16*e^3 + 69*e, -23/256*e^8 + 69/16*e^6 - 3677/64*e^4 + 1095/8*e^2 - 77/4, -23/256*e^8 + 69/16*e^6 - 3677/64*e^4 + 1095/8*e^2 - 77/4, -11/512*e^11 + 201/128*e^9 - 5089/128*e^7 + 12301/32*e^5 - 7333/8*e^3 + 561/2*e, -11/512*e^11 + 201/128*e^9 - 5089/128*e^7 + 12301/32*e^5 - 7333/8*e^3 + 561/2*e, 25/1024*e^11 - 445/256*e^9 + 11007/256*e^7 - 26161/64*e^5 + 7915/8*e^3 - 689/2*e, 25/1024*e^11 - 445/256*e^9 + 11007/256*e^7 - 26161/64*e^5 + 7915/8*e^3 - 689/2*e, -9/1024*e^11 + 333/512*e^9 - 4263/256*e^7 + 20787/128*e^5 - 6153/16*e^3 + 741/8*e, -9/1024*e^11 + 333/512*e^9 - 4263/256*e^7 + 20787/128*e^5 - 6153/16*e^3 + 741/8*e, -1/512*e^10 + 15/128*e^8 - 303/128*e^6 + 567/32*e^4 - 32*e^2 - 21, -1/512*e^10 + 15/128*e^8 - 303/128*e^6 + 567/32*e^4 - 32*e^2 - 21, 3/1024*e^11 - 101/512*e^9 + 1189/256*e^7 - 5555/128*e^5 + 2015/16*e^3 - 909/8*e, 3/1024*e^11 - 101/512*e^9 + 1189/256*e^7 - 5555/128*e^5 + 2015/16*e^3 - 909/8*e, 1/4*e^4 - 6*e^2 + 21, 1/4*e^4 - 6*e^2 + 21, 3/64*e^8 - 9/4*e^6 + 489/16*e^4 - 171/2*e^2 + 20, 3/64*e^8 - 9/4*e^6 + 489/16*e^4 - 171/2*e^2 + 20, -1/64*e^7 + 11/16*e^5 - 127/16*e^3 + 43/4*e, -1/64*e^7 + 11/16*e^5 - 127/16*e^3 + 43/4*e, -1/256*e^10 + 15/64*e^8 - 303/64*e^6 + 567/16*e^4 - 70*e^2 + 30, -1/256*e^10 + 15/64*e^8 - 303/64*e^6 + 567/16*e^4 - 70*e^2 + 30, -7/128*e^8 + 21/8*e^6 - 1125/32*e^4 + 351/4*e^2 - 73/2, -7/128*e^8 + 21/8*e^6 - 1125/32*e^4 + 351/4*e^2 - 73/2, -3/256*e^10 + 45/64*e^8 - 909/64*e^6 + 1701/16*e^4 - 207*e^2 + 54, -3/256*e^10 + 45/64*e^8 - 909/64*e^6 + 1701/16*e^4 - 207*e^2 + 54, -23/1024*e^11 + 819/512*e^9 - 10129/256*e^7 + 48109/128*e^5 - 14473/16*e^3 + 2499/8*e, -23/1024*e^11 + 819/512*e^9 - 10129/256*e^7 + 48109/128*e^5 - 14473/16*e^3 + 2499/8*e, 19/1024*e^11 - 21/16*e^9 + 8245/256*e^7 - 9675/32*e^5 + 1403/2*e^3 - 793/4*e, 19/1024*e^11 - 21/16*e^9 + 8245/256*e^7 - 9675/32*e^5 + 1403/2*e^3 - 793/4*e, 11/1024*e^11 - 387/512*e^9 + 4721/256*e^7 - 22005/128*e^5 + 1571/4*e^3 - 757/8*e, 11/1024*e^11 - 387/512*e^9 + 4721/256*e^7 - 22005/128*e^5 + 1571/4*e^3 - 757/8*e, -5/1024*e^11 + 177/512*e^9 - 2191/256*e^7 + 10631/128*e^5 - 913/4*e^3 + 1119/8*e, -5/1024*e^11 + 177/512*e^9 - 2191/256*e^7 + 10631/128*e^5 - 913/4*e^3 + 1119/8*e, 7/1024*e^11 - 251/512*e^9 + 3141/256*e^7 - 15309/128*e^5 + 319*e^3 - 1341/8*e, 7/1024*e^11 - 251/512*e^9 + 3141/256*e^7 - 15309/128*e^5 + 319*e^3 - 1341/8*e, -7/512*e^11 + 127/128*e^9 - 3203/128*e^7 + 7777/32*e^5 - 9747/16*e^3 + 979/4*e, -7/512*e^11 + 127/128*e^9 - 3203/128*e^7 + 7777/32*e^5 - 9747/16*e^3 + 979/4*e, 23/1024*e^11 - 821/512*e^9 + 10181/256*e^7 - 48523/128*e^5 + 7361/8*e^3 - 2675/8*e, 23/1024*e^11 - 821/512*e^9 + 10181/256*e^7 - 48523/128*e^5 + 7361/8*e^3 - 2675/8*e, -3/256*e^11 + 429/512*e^9 - 1333/64*e^7 + 25535/128*e^5 - 7903/16*e^3 + 1519/8*e, -3/256*e^11 + 429/512*e^9 - 1333/64*e^7 + 25535/128*e^5 - 7903/16*e^3 + 1519/8*e, -1/256*e^10 + 15/64*e^8 - 313/64*e^6 + 657/16*e^4 - 931/8*e^2 + 93/2, -1/256*e^10 + 15/64*e^8 - 313/64*e^6 + 657/16*e^4 - 931/8*e^2 + 93/2, -11/128*e^8 + 33/8*e^6 - 1729/32*e^4 + 435/4*e^2 - 5/2, -11/128*e^8 + 33/8*e^6 - 1729/32*e^4 + 435/4*e^2 - 5/2, -5/256*e^11 + 45/32*e^9 - 1125/32*e^7 + 5405/16*e^5 - 13305/16*e^3 + 1385/4*e, -5/256*e^11 + 45/32*e^9 - 1125/32*e^7 + 5405/16*e^5 - 13305/16*e^3 + 1385/4*e, -5/1024*e^11 + 169/512*e^9 - 1967/256*e^7 + 8591/128*e^5 - 507/4*e^3 - 249/8*e, -5/1024*e^11 + 169/512*e^9 - 1967/256*e^7 + 8591/128*e^5 - 507/4*e^3 - 249/8*e, 1/128*e^8 - 3/8*e^6 + 147/32*e^4 - 9/4*e^2 + 49/2, 1/128*e^8 - 3/8*e^6 + 147/32*e^4 - 9/4*e^2 + 49/2, -9/512*e^11 + 639/512*e^9 - 3941/128*e^7 + 37397/128*e^5 - 5669/8*e^3 + 1903/8*e, -9/512*e^11 + 639/512*e^9 - 3941/128*e^7 + 37397/128*e^5 - 5669/8*e^3 + 1903/8*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;