/*
  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![2, 0, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {26,26,w^3 - 4*w - 3}>;

primesArray := [
[2, 2, w],
[2, 2, w + 1],
[13, 13, -w^2 + w + 3],
[13, 13, w^2 + w - 3],
[19, 19, -w^3 + 3*w + 1],
[19, 19, -w^3 + 3*w - 1],
[43, 43, -w^2 + w - 1],
[43, 43, w^2 + w + 1],
[49, 7, w^3 + w^2 - 6*w - 3],
[49, 7, w^3 - w^2 - 6*w + 3],
[53, 53, 2*w^3 - w^2 - 9*w + 3],
[53, 53, 2*w^3 + w^2 - 9*w - 3],
[59, 59, w^3 - w^2 - 4*w + 1],
[59, 59, -w^3 - w^2 + 4*w + 1],
[67, 67, 3*w^3 - 13*w + 1],
[67, 67, -w^3 + w^2 + 6*w - 5],
[81, 3, -3],
[83, 83, -2*w^3 - w^2 + 9*w + 7],
[83, 83, 4*w^3 - 18*w - 1],
[89, 89, -2*w^3 + 10*w + 1],
[89, 89, -w^2 + w + 5],
[89, 89, w^2 + w - 5],
[89, 89, 2*w^3 - 10*w + 1],
[101, 101, -2*w^3 + 8*w + 1],
[101, 101, 2*w^3 - 8*w + 1],
[103, 103, w^3 + 2*w^2 - 5*w - 7],
[103, 103, w^3 + w^2 - 6*w - 7],
[103, 103, -w^2 - 3*w + 1],
[103, 103, -w^3 + 2*w^2 + 5*w - 7],
[127, 127, -w^3 + 2*w^2 + 5*w - 5],
[127, 127, w^3 + 3*w^2 - 4*w - 13],
[127, 127, -w^3 + 3*w^2 + 4*w - 13],
[127, 127, w^3 + 2*w^2 - 5*w - 5],
[149, 149, -2*w^3 - w^2 + 7*w - 1],
[149, 149, 2*w^3 - w^2 - 7*w - 1],
[151, 151, -w^3 + w^2 + 2*w - 3],
[151, 151, 2*w^3 - 10*w + 3],
[151, 151, -2*w^3 + 10*w + 3],
[151, 151, w^3 + w^2 - 2*w - 3],
[157, 157, w^3 + 2*w^2 - 3*w - 1],
[157, 157, -w^3 + 2*w^2 + 3*w - 1],
[169, 13, 2*w^2 - 3],
[179, 179, 2*w^3 + w^2 - 7*w - 3],
[179, 179, -2*w^3 + w^2 + 7*w - 3],
[229, 229, -w^3 + 2*w^2 + 3*w - 7],
[229, 229, w^3 + 2*w^2 - 3*w - 7],
[251, 251, -2*w^3 + 2*w^2 + 8*w - 9],
[251, 251, -2*w^3 - 2*w^2 + 8*w + 9],
[263, 263, 2*w^3 - w^2 - 3*w + 1],
[263, 263, -2*w^3 + 2*w^2 + 10*w - 5],
[263, 263, -2*w^3 - 2*w^2 + 10*w + 5],
[263, 263, -2*w^3 - w^2 + 3*w + 1],
[289, 17, 2*w^2 - 5],
[293, 293, 4*w^3 - w^2 - 19*w + 7],
[293, 293, -4*w^3 - w^2 + 19*w + 7],
[307, 307, -w^3 + 3*w - 5],
[307, 307, w^3 - 3*w - 5],
[331, 331, 4*w^3 - 2*w^2 - 18*w + 7],
[331, 331, 5*w^3 - 3*w^2 - 22*w + 11],
[349, 349, -3*w^3 + 13*w + 3],
[349, 349, -3*w^3 + 13*w - 3],
[361, 19, 2*w^2 - 11],
[373, 373, w^3 + 2*w^2 + w - 3],
[373, 373, -w^3 + 2*w^2 - w - 3],
[383, 383, -w^3 + 5*w + 5],
[383, 383, -3*w^3 + 2*w^2 + 11*w - 5],
[383, 383, 3*w^3 + 2*w^2 - 11*w - 5],
[383, 383, w^3 - 5*w + 5],
[389, 389, w^3 - 4*w^2 + w + 7],
[389, 389, -2*w^3 - w^2 + 7*w + 7],
[421, 421, -w^3 - w^2 - 2*w + 1],
[421, 421, w^3 - w^2 + 2*w + 1],
[433, 433, 4*w^3 - 2*w^2 - 16*w + 5],
[433, 433, -4*w + 1],
[433, 433, 4*w + 1],
[433, 433, -5*w^3 + 3*w^2 + 24*w - 13],
[443, 443, -w^3 + w^2 + 3],
[443, 443, -4*w^2 + 6*w + 7],
[457, 457, 3*w^3 + w^2 - 12*w - 3],
[457, 457, -2*w^3 + 6*w - 1],
[457, 457, 2*w^3 - 6*w - 1],
[457, 457, 3*w^3 - w^2 - 12*w + 3],
[461, 461, w^3 - 3*w^2 - 2*w + 7],
[461, 461, w^3 + 3*w^2 - 2*w - 7],
[463, 463, -2*w^3 + 3*w^2 + 9*w - 11],
[463, 463, 2*w^3 - w^2 - 5*w + 1],
[463, 463, -2*w^3 - w^2 + 5*w + 1],
[463, 463, -2*w^3 - 3*w^2 + 9*w + 11],
[467, 467, w^3 - 7*w + 1],
[467, 467, -w^3 + 7*w + 1],
[491, 491, 2*w^2 - 2*w - 7],
[491, 491, 2*w^2 + 2*w - 7],
[509, 509, w^3 + 3*w^2 - 4*w - 11],
[509, 509, -w^3 + 3*w^2 + 4*w - 11],
[523, 523, 2*w^3 - w^2 - 11*w + 3],
[523, 523, 2*w^3 + w^2 - 11*w - 3],
[529, 23, 3*w^2 - w - 15],
[529, 23, -3*w^2 - w + 15],
[557, 557, 3*w^2 + w - 13],
[557, 557, -3*w^2 + w + 13],
[563, 563, -3*w^3 + 3*w^2 + 12*w - 7],
[563, 563, 2*w^3 - 2*w^2 - 8*w + 3],
[569, 569, w^3 + 3*w^2 - 6*w - 9],
[569, 569, 4*w^3 - w^2 - 17*w + 1],
[569, 569, -4*w^3 - w^2 + 17*w + 1],
[569, 569, -w^3 + 3*w^2 + 6*w - 9],
[587, 587, -w^3 + w^2 + 2*w - 5],
[587, 587, w^3 + w^2 - 2*w - 5],
[593, 593, 3*w^3 - 2*w^2 - 15*w + 5],
[593, 593, -w^3 - 2*w^2 + 5*w + 1],
[593, 593, w^3 - 2*w^2 - 5*w + 1],
[593, 593, -3*w^3 - 2*w^2 + 15*w + 5],
[599, 599, w^3 + w^2 - 3],
[599, 599, -w^2 - w - 3],
[599, 599, -w^2 + w - 3],
[599, 599, -w^3 + w^2 - 3],
[613, 613, -w^3 + w^2 + 6*w - 9],
[613, 613, w^3 + w^2 - 6*w - 9],
[625, 5, -5],
[659, 659, -2*w^3 + 4*w^2 + 4*w - 1],
[659, 659, 2*w^3 + 4*w^2 - 4*w - 1],
[661, 661, w^3 + 3*w^2 - 6*w - 7],
[661, 661, -2*w^3 + 6*w^2 - 7],
[701, 701, -w^3 - 2*w^2 + 3*w + 9],
[701, 701, -w^3 + 2*w^2 + 3*w - 9],
[733, 733, w^3 + 2*w^2 - 7*w - 1],
[733, 733, -4*w^3 + w^2 + 19*w - 11],
[739, 739, -w^3 - w^2 + 2*w + 9],
[739, 739, -w^3 + w^2 + 2*w - 9],
[757, 757, 3*w^3 - w^2 - 10*w + 5],
[757, 757, -3*w^3 - w^2 + 10*w + 5],
[773, 773, -4*w^3 + 5*w^2 + 19*w - 23],
[773, 773, -7*w^3 + 6*w^2 + 33*w - 29],
[797, 797, 3*w^3 - 13*w + 5],
[797, 797, 3*w^3 - 13*w - 5],
[829, 829, -2*w^3 - w^2 + 9*w - 1],
[829, 829, 2*w^3 - w^2 - 9*w - 1],
[859, 859, 3*w^3 - w^2 - 14*w + 1],
[859, 859, -3*w^3 - w^2 + 14*w + 1],
[863, 863, -5*w^3 + 21*w + 3],
[863, 863, -3*w^3 - w^2 + 12*w - 1],
[863, 863, 3*w^3 - w^2 - 12*w - 1],
[863, 863, -5*w^3 + 21*w - 3],
[883, 883, -2*w^3 + 4*w^2 + 8*w - 17],
[883, 883, 2*w^3 + 4*w^2 - 8*w - 17],
[953, 953, -w^3 - w^2 + 8*w + 3],
[953, 953, 4*w^3 - 2*w^2 - 16*w + 7],
[953, 953, -3*w^3 + w^2 + 16*w - 7],
[953, 953, w^3 - w^2 - 8*w + 3],
[961, 31, -4*w^3 + 2*w^2 + 12*w + 3],
[961, 31, 3*w^3 - w^2 - 10*w - 1],
[971, 971, 3*w^3 - 4*w^2 - 13*w + 15],
[971, 971, -4*w^3 + 3*w^2 + 19*w - 17],
[977, 977, -2*w^3 + w^2 + 9*w + 3],
[977, 977, -4*w^3 + w^2 + 19*w - 9],
[977, 977, 3*w^3 - 15*w + 5],
[977, 977, 2*w^3 + w^2 - 9*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 3*x^7 - 67*x^6 - 255*x^5 + 1318*x^4 + 6568*x^3 - 4288*x^2 - 52336*x - 60832;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/64*e^7 - 1/64*e^6 - 63/64*e^5 - 7/64*e^4 + 673/32*e^3 + 343/16*e^2 - 1195/8*e - 254, -1, -1, e, 11/64*e^7 - 13/64*e^6 - 679/64*e^5 + 33/64*e^4 + 887/4*e^3 + 1603/8*e^2 - 6169/4*e - 10067/4, -1/8*e^7 + 3/16*e^6 + 121/16*e^5 - 37/16*e^4 - 2495/16*e^3 - 909/8*e^2 + 4303/4*e + 3299/2, -3/32*e^7 + 3/32*e^6 + 189/32*e^5 + 13/32*e^4 - 2003/16*e^3 - 963/8*e^2 + 3517/4*e + 1451, 11/64*e^7 - 13/64*e^6 - 679/64*e^5 + 33/64*e^4 + 887/4*e^3 + 1595/8*e^2 - 6165/4*e - 9995/4, 3/16*e^7 - 3/16*e^6 - 189/16*e^5 - 13/16*e^4 + 2003/8*e^3 + 967/4*e^2 - 3515/2*e - 2918, -1/4*e^7 + 3/8*e^6 + 121/8*e^5 - 35/8*e^4 - 2499/8*e^3 - 236*e^2 + 4321/2*e + 3370, -e^2 + 2*e + 22, -1/32*e^7 + 3/32*e^6 + 53/32*e^5 - 87/32*e^4 - 123/4*e^3 + 31/4*e^2 + 393/2*e + 369/2, -1/16*e^6 + 5/16*e^5 + 43/16*e^4 - 173/16*e^3 - 327/8*e^2 + 369/4*e + 473/2, 3/32*e^7 - 3/32*e^6 - 189/32*e^5 - 13/32*e^4 + 2003/16*e^3 + 963/8*e^2 - 3509/4*e - 1447, -1/16*e^6 + 5/16*e^5 + 43/16*e^4 - 173/16*e^3 - 327/8*e^2 + 369/4*e + 473/2, 3/32*e^7 - 3/32*e^6 - 189/32*e^5 - 13/32*e^4 + 2003/16*e^3 + 971/8*e^2 - 3513/4*e - 1465, 3/16*e^7 - 3/16*e^6 - 189/16*e^5 - 13/16*e^4 + 2003/8*e^3 + 967/4*e^2 - 3515/2*e - 2914, -1/8*e^7 + 3/16*e^6 + 121/16*e^5 - 37/16*e^4 - 2495/16*e^3 - 917/8*e^2 + 4315/4*e + 3343/2, -17/64*e^7 + 15/64*e^6 + 1077/64*e^5 + 165/64*e^4 - 1429/4*e^3 - 2897/8*e^2 + 10011/4*e + 16801/4, 7/32*e^7 - 9/32*e^6 - 431/32*e^5 + 61/32*e^4 + 2249/8*e^3 + 234*e^2 - 1955*e - 6205/2, -7/16*e^7 + 7/16*e^6 + 439/16*e^5 + 33/16*e^4 - 579*e^3 - 1125/2*e^2 + 4045*e + 6715, -1/32*e^7 + 3/32*e^6 + 53/32*e^5 - 87/32*e^4 - 123/4*e^3 + 23/4*e^2 + 399/2*e + 437/2, 5/32*e^7 - 7/32*e^6 - 305/32*e^5 + 67/32*e^4 + 395/2*e^3 + 158*e^2 - 2731/2*e - 4339/2, -1/32*e^7 + 3/32*e^6 + 53/32*e^5 - 87/32*e^4 - 123/4*e^3 + 23/4*e^2 + 401/2*e + 441/2, -1/8*e^7 + 1/4*e^6 + 29/4*e^5 - 19/4*e^4 - 1169/8*e^3 - 81*e^2 + 2005/2*e + 1486, 9/64*e^7 - 11/64*e^6 - 553/64*e^5 + 31/64*e^4 + 2891/16*e^3 + 659/4*e^2 - 2523/2*e - 8255/4, 5/64*e^7 - 7/64*e^6 - 301/64*e^5 + 59/64*e^4 + 1545/16*e^3 + 79*e^2 - 662*e - 4191/4, -7/32*e^7 + 9/32*e^6 + 431/32*e^5 - 61/32*e^4 - 2249/8*e^3 - 234*e^2 + 1955*e + 6201/2, 7/32*e^7 - 9/32*e^6 - 431/32*e^5 + 61/32*e^4 + 2249/8*e^3 + 235*e^2 - 1956*e - 6237/2, 3/32*e^7 - 1/32*e^6 - 199/32*e^5 - 99/32*e^4 + 271/2*e^3 + 655/4*e^2 - 1925/2*e - 3439/2, -e^2 + e + 18, 3/64*e^7 - 9/64*e^6 - 163/64*e^5 + 261/64*e^4 + 777/16*e^3 - 29/4*e^2 - 641/2*e - 1357/4, 5/64*e^7 + 1/64*e^6 - 341/64*e^5 - 285/64*e^4 + 1883/16*e^3 + 653/4*e^2 - 1683/2*e - 6211/4, 5/32*e^7 - 7/32*e^6 - 305/32*e^5 + 59/32*e^4 + 199*e^3 + 659/4*e^2 - 2789/2*e - 4453/2, -3/16*e^7 + 5/16*e^6 + 179/16*e^5 - 73/16*e^4 - 919/4*e^3 - 157*e^2 + 1592*e + 2415, 7/32*e^7 - 9/32*e^6 - 431/32*e^5 + 61/32*e^4 + 2249/8*e^3 + 233*e^2 - 1954*e - 6165/2, 15/64*e^7 - 29/64*e^6 - 879/64*e^5 + 553/64*e^4 + 4453/16*e^3 + 603/4*e^2 - 3825/2*e - 11105/4, 3/64*e^7 - 1/64*e^6 - 195/64*e^5 - 115/64*e^4 + 1053/16*e^3 + 347/4*e^2 - 933/2*e - 3469/4, -27/64*e^7 + 33/64*e^6 + 1667/64*e^5 - 157/64*e^4 - 8703/16*e^3 - 470*e^2 + 3778*e + 24281/4, 1/16*e^7 - 3/16*e^6 - 53/16*e^5 + 83/16*e^4 + 62*e^3 - 19/4*e^2 - 403*e - 462, -1/8*e^7 + 1/4*e^6 + 29/4*e^5 - 19/4*e^4 - 1169/8*e^3 - 81*e^2 + 2005/2*e + 1490, -7/16*e^7 + 7/16*e^6 + 439/16*e^5 + 33/16*e^4 - 579*e^3 - 1125/2*e^2 + 4045*e + 6719, 9/32*e^7 - 13/32*e^6 - 547/32*e^5 + 149/32*e^4 + 5647/16*e^3 + 2103/8*e^2 - 9737/4*e - 3748, -7/32*e^7 + 7/32*e^6 + 441/32*e^5 + 25/32*e^4 - 4663/16*e^3 - 2211/8*e^2 + 8157/4*e + 3347, 11/32*e^7 - 13/32*e^6 - 679/32*e^5 + 33/32*e^4 + 887/2*e^3 + 1595/4*e^2 - 6165/2*e - 9991/2, -1/32*e^7 + 3/32*e^6 + 53/32*e^5 - 87/32*e^4 - 123/4*e^3 + 31/4*e^2 + 397/2*e + 369/2, -21/32*e^7 + 21/32*e^6 + 1319/32*e^5 + 107/32*e^4 - 13951/16*e^3 - 6839/8*e^2 + 24433/4*e + 10196, -1/8*e^7 + 3/16*e^6 + 121/16*e^5 - 37/16*e^4 - 2495/16*e^3 - 901/8*e^2 + 4291/4*e + 3255/2, -15/32*e^7 + 21/32*e^6 + 915/32*e^5 - 193/32*e^4 - 594*e^3 - 1915/4*e^2 + 8245/2*e + 13083/2, 25/64*e^7 - 43/64*e^6 - 1481/64*e^5 + 671/64*e^4 + 7535/16*e^3 + 1241/4*e^2 - 6457/2*e - 19559/4, 1/2*e^7 - 5/8*e^6 - 247/8*e^5 + 27/8*e^4 + 5171/8*e^3 + 2219/4*e^2 - 9017/2*e - 7231, 1/64*e^7 - 11/64*e^6 - 9/64*e^5 + 431/64*e^4 - 139/16*e^3 - 88*e^2 + 108*e + 1693/4, -21/32*e^7 + 27/32*e^6 + 1293/32*e^5 - 183/32*e^4 - 6747/8*e^3 - 702*e^2 + 5863*e + 18599/2, -5/16*e^7 + 7/16*e^6 + 307/16*e^5 - 71/16*e^4 - 3203/8*e^3 - 312*e^2 + 5581/2*e + 4377, 13/16*e^7 - 15/16*e^6 - 809/16*e^5 + 35/16*e^4 + 1063*e^3 + 1901/2*e^2 - 7423*e - 12011, -37/64*e^7 + 43/64*e^6 + 2297/64*e^5 - 103/64*e^4 - 3013/4*e^3 - 5401/8*e^2 + 21007/4*e + 34045/4, 5/64*e^7 - 3/64*e^6 - 329/64*e^5 - 113/64*e^4 + 451/4*e^3 + 1009/8*e^2 - 3255/4*e - 5589/4, 37/64*e^7 - 43/64*e^6 - 2297/64*e^5 + 103/64*e^4 + 3013/4*e^3 + 5401/8*e^2 - 21007/4*e - 34045/4, -11/32*e^7 + 15/32*e^6 + 673/32*e^5 - 135/32*e^4 - 6993/16*e^3 - 2789/8*e^2 + 12143/4*e + 4768, -15/32*e^7 + 17/32*e^6 + 931/32*e^5 - 21/32*e^4 - 2439/4*e^3 - 2219/4*e^2 + 8483/2*e + 13803/2, -5/32*e^7 + 11/32*e^6 + 285/32*e^5 - 247/32*e^4 - 1411/8*e^3 - 133/2*e^2 + 1192*e + 3279/2, 17/32*e^7 - 23/32*e^6 - 1037/32*e^5 + 195/32*e^4 + 2685/4*e^3 + 2165/4*e^2 - 9279/2*e - 14593/2, -29/32*e^7 + 31/32*e^6 + 1809/32*e^5 + 45/32*e^4 - 4755/4*e^3 - 4477/4*e^2 + 16577/2*e + 27317/2, -3/32*e^7 + 5/32*e^6 + 179/32*e^5 - 73/32*e^4 - 919/8*e^3 - 76*e^2 + 791*e + 2317/2, 19/32*e^7 - 21/32*e^6 - 1187/32*e^5 + 9/32*e^4 + 6255/8*e^3 + 1431/2*e^2 - 5468*e - 17809/2, 1/64*e^7 + 5/64*e^6 - 89/64*e^5 - 289/64*e^4 + 561/16*e^3 + 185/2*e^2 - 267*e - 2587/4, -11/64*e^7 + 1/64*e^6 + 739/64*e^5 + 483/64*e^4 - 4059/16*e^3 - 649/2*e^2 + 1811*e + 12945/4, -9/64*e^7 + 19/64*e^6 + 521/64*e^5 - 439/64*e^4 - 2591/16*e^3 - 215/4*e^2 + 2169/2*e + 5591/4, 13/32*e^7 - 19/32*e^6 - 789/32*e^5 + 223/32*e^4 + 4071/8*e^3 + 753/2*e^2 - 3512*e - 10799/2, -1/2*e^7 + 5/8*e^6 + 123/4*e^5 - 25/8*e^4 - 641*e^3 - 2231/4*e^2 + 4450*e + 7191, -13/16*e^7 + 13/16*e^6 + 817/16*e^5 + 59/16*e^4 - 4319/4*e^3 - 1045*e^2 + 7558*e + 12535, -9/32*e^7 + 7/32*e^6 + 573/32*e^5 + 125/32*e^4 - 380*e^3 - 1587/4*e^2 + 5303/2*e + 8929/2, 27/32*e^7 - 29/32*e^6 - 1687/32*e^5 - 31/32*e^4 + 2221/2*e^3 + 4145/4*e^2 - 15511/2*e - 25399/2, -19/32*e^7 + 17/32*e^6 + 1203/32*e^5 + 179/32*e^4 - 6389/8*e^3 - 808*e^2 + 5599*e + 18757/2, -7/32*e^7 + 9/32*e^6 + 431/32*e^5 - 61/32*e^4 - 2249/8*e^3 - 234*e^2 + 1955*e + 6181/2, 5/8*e^7 - e^6 - 299/8*e^5 + 55/4*e^4 + 1529/2*e^3 + 1079/2*e^2 - 5250*e - 8064, 3/8*e^7 - 7/16*e^6 - 373/16*e^5 + 25/16*e^4 + 7815/16*e^3 + 3397/8*e^2 - 13579/4*e - 10879/2, 39/64*e^7 - 57/64*e^6 - 2371/64*e^5 + 653/64*e^4 + 3069/4*e^3 + 4599/8*e^2 - 21257/4*e - 32847/4, -1/4*e^7 + 1/2*e^6 + 29/2*e^5 - 19/2*e^4 - 1169/4*e^3 - 160*e^2 + 2003*e + 2938, -11/16*e^7 + 15/16*e^6 + 671/16*e^5 - 127/16*e^4 - 3481/4*e^3 - 707*e^2 + 6036*e + 9545, 9/16*e^7 - 13/16*e^6 - 545/16*e^5 + 133/16*e^4 + 703*e^3 + 553*e^2 - 4855*e - 7663, -9/32*e^7 + 7/32*e^6 + 573/32*e^5 + 133/32*e^4 - 763/2*e^3 - 811/2*e^2 + 5355/2*e + 9127/2, -5/32*e^7 + 11/32*e^6 + 289/32*e^5 - 263/32*e^4 - 721/4*e^3 - 215/4*e^2 + 2437/2*e + 3141/2, 1/4*e^7 - 3/8*e^6 - 15*e^5 + 33/8*e^4 + 308*e^3 + 239*e^2 - 2123*e - 3346, -3/32*e^7 + 5/32*e^6 + 179/32*e^5 - 57/32*e^4 - 927/8*e^3 - 191/2*e^2 + 813*e + 2653/2, 1/16*e^7 - 1/16*e^6 - 63/16*e^5 - 7/16*e^4 + 673/8*e^3 + 343/4*e^2 - 1195/2*e - 1020, 3/16*e^7 - 5/16*e^6 - 181/16*e^5 + 81/16*e^4 + 1869/8*e^3 + 593/4*e^2 - 3251/2*e - 2406, 43/64*e^7 - 49/64*e^6 - 2667/64*e^5 + 45/64*e^4 + 13993/16*e^3 + 3233/4*e^2 - 12209/2*e - 40125/4, 7/16*e^7 - 3/8*e^6 - 223/8*e^5 - 17/4*e^4 + 9499/16*e^3 + 4741/8*e^2 - 16679/4*e - 13825/2, 19/64*e^7 - 21/64*e^6 - 1183/64*e^5 + 9/64*e^4 + 388*e^3 + 2859/8*e^2 - 10805/4*e - 17739/4, 11/32*e^7 - 19/32*e^6 - 649/32*e^5 + 307/32*e^4 + 6561/16*e^3 + 2069/8*e^2 - 11159/4*e - 4146, 5/64*e^7 - 3/64*e^6 - 321/64*e^5 - 145/64*e^4 + 871/8*e^3 + 1103/8*e^2 - 3145/4*e - 5857/4, -1/16*e^7 + 1/16*e^6 + 61/16*e^5 + 7/16*e^4 - 313/4*e^3 - 82*e^2 + 534*e + 943, 13/16*e^7 - 15/16*e^6 - 807/16*e^5 + 39/16*e^4 + 8453/8*e^3 + 936*e^2 - 14699/2*e - 11825, -1/4*e^7 + 7/16*e^6 + 237/16*e^5 - 109/16*e^4 - 4849/16*e^3 - 1639/8*e^2 + 8397/4*e + 6497/2, 39/64*e^7 - 49/64*e^6 - 2403/64*e^5 + 277/64*e^4 + 784*e^3 + 5331/8*e^2 - 21801/4*e - 34759/4, 33/32*e^7 - 39/32*e^6 - 2041/32*e^5 + 123/32*e^4 + 10671/8*e^3 + 4715/4*e^2 - 9271*e - 29841/2, -5/16*e^7 + 7/16*e^6 + 307/16*e^5 - 75/16*e^4 - 3199/8*e^3 - 1209/4*e^2 + 5561/2*e + 4286, 5/16*e^7 - 9/16*e^6 - 293/16*e^5 + 145/16*e^4 + 741/2*e^3 + 242*e^2 - 2537*e - 3875, 15/32*e^7 - 17/32*e^6 - 931/32*e^5 + 29/32*e^4 + 2437/4*e^3 + 546*e^2 - 8473/2*e - 13657/2, -41/64*e^7 + 47/64*e^6 + 2549/64*e^5 - 75/64*e^4 - 6691/8*e^3 - 6095/8*e^2 + 23309/4*e + 38093/4, -27/64*e^7 + 45/64*e^6 + 1615/64*e^5 - 673/64*e^4 - 4143/8*e^3 - 2829/8*e^2 + 14311/4*e + 21951/4, 7/32*e^7 - 5/32*e^6 - 447/32*e^5 - 111/32*e^4 + 2375/8*e^3 + 311*e^2 - 2074*e - 6945/2, 5/32*e^7 - 3/32*e^6 - 325/32*e^5 - 113/32*e^4 + 1757/8*e^3 + 491/2*e^2 - 1552*e - 5331/2, -3/16*e^7 + 3/16*e^6 + 189/16*e^5 + 17/16*e^4 - 2007/8*e^3 - 501/2*e^2 + 3531/2*e + 2993, 19/32*e^7 - 25/32*e^6 - 1167/32*e^5 + 197/32*e^4 + 3037/4*e^3 + 2465/4*e^2 - 10533/2*e - 16563/2, -35/64*e^7 + 45/64*e^6 + 2143/64*e^5 - 305/64*e^4 - 694*e^3 - 4623/8*e^2 + 19141/4*e + 30459/4, 15/16*e^7 - e^6 - 469/8*e^5 - 15/8*e^4 + 19803/16*e^3 + 9401/8*e^2 - 34691/4*e - 28661/2, 23/32*e^7 - 29/32*e^6 - 1419/32*e^5 + 177/32*e^4 + 927*e^3 + 780*e^2 - 12905/2*e - 20505/2, 17/32*e^7 - 19/32*e^6 - 1053/32*e^5 - 9/32*e^4 + 690*e^3 + 2613/4*e^2 - 9625/2*e - 15949/2, -5/8*e^7 + 7/8*e^6 + 38*e^5 - 69/8*e^4 - 6277/8*e^3 - 1227/2*e^2 + 10811/2*e + 8468, -1/16*e^7 - 1/16*e^6 + 71/16*e^5 + 93/16*e^4 - 791/8*e^3 - 651/4*e^2 + 1399/2*e + 1358, 23/64*e^7 - 21/64*e^6 - 1447/64*e^5 - 223/64*e^4 + 7649/16*e^3 + 1983/4*e^2 - 6689/2*e - 22801/4, -9/8*e^7 + 11/8*e^6 + 139/2*e^5 - 51/8*e^4 - 11625/8*e^3 - 5031/4*e^2 + 20205/2*e + 16201, -43/64*e^7 + 65/64*e^6 + 2603/64*e^5 - 797/64*e^4 - 13449/16*e^3 - 2475/4*e^2 + 11645/2*e + 35909/4, 5/32*e^7 - 3/32*e^6 - 325/32*e^5 - 113/32*e^4 + 1757/8*e^3 + 499/2*e^2 - 1560*e - 5495/2, 3/32*e^7 - 5/32*e^6 - 175/32*e^5 + 57/32*e^4 + 111*e^3 + 363/4*e^2 - 1533/2*e - 2527/2, 17/32*e^7 - 23/32*e^6 - 1037/32*e^5 + 187/32*e^4 + 2687/4*e^3 + 548*e^2 - 9291/2*e - 14663/2, -1/8*e^7 + 1/8*e^6 + 31/4*e^5 + 5/8*e^4 - 1287/8*e^3 - 159*e^2 + 2209/2*e + 1850, 1/8*e^7 - 1/16*e^6 - 131/16*e^5 - 57/16*e^4 + 2857/16*e^3 + 1695/8*e^2 - 5101/4*e - 4493/2, -51/64*e^7 + 45/64*e^6 + 3231/64*e^5 + 495/64*e^4 - 4291/4*e^3 - 8699/8*e^2 + 30129/4*e + 50611/4, -1/8*e^7 - 1/8*e^6 + 71/8*e^5 + 99/8*e^4 - 805/4*e^3 - 1399/4*e^2 + 1455*e + 2975, 1/2*e^7 - 1/2*e^6 - 63/2*e^5 - 5/2*e^4 + 669*e^3 + 653*e^2 - 4710*e - 7842, -27/32*e^7 + 29/32*e^6 + 1687/32*e^5 + 31/32*e^4 - 2221/2*e^3 - 4149/4*e^2 + 15519/2*e + 25423/2, 23/16*e^7 - 27/16*e^6 - 1425/16*e^5 + 79/16*e^4 + 14923/8*e^3 + 3313/2*e^2 - 25965/2*e - 20967, 3/4*e^7 - 7/8*e^6 - 93/2*e^5 + 13/8*e^4 + 977*e^3 + 892*e^2 - 6831*e - 11166, -7/16*e^7 + 5/16*e^6 + 449/16*e^5 + 119/16*e^4 - 4805/8*e^3 - 2577/4*e^2 + 8451/2*e + 7196, -1/8*e^7 + 5/16*e^6 + 111/16*e^5 - 123/16*e^4 - 2157/16*e^3 - 267/8*e^2 + 3633/4*e + 2417/2, 5/8*e^7 - 9/16*e^6 - 635/16*e^5 - 81/16*e^4 + 13505/16*e^3 + 6659/8*e^2 - 23693/4*e - 19593/2, -5/16*e^7 + 7/16*e^6 + 305/16*e^5 - 67/16*e^4 - 396*e^3 - 312*e^2 + 2744*e + 4279, -1/8*e^7 - 1/8*e^6 + 9*e^5 + 95/8*e^4 - 1649/8*e^3 - 339*e^2 + 3013/2*e + 2956, -19/32*e^7 + 21/32*e^6 + 1183/32*e^5 + 23/32*e^4 - 779*e^3 - 2975/4*e^2 + 10905/2*e + 18143/2, 1/16*e^7 - 1/16*e^6 - 65/16*e^5 + 13/16*e^4 + 171/2*e^3 + 211/4*e^2 - 585*e - 818, 21/32*e^7 - 31/32*e^6 - 1269/32*e^5 + 355/32*e^4 + 6535/8*e^3 + 1227/2*e^2 - 5630*e - 17419/2, 5/32*e^7 - 3/32*e^6 - 329/32*e^5 - 81/32*e^4 + 445/2*e^3 + 885/4*e^2 - 3147/2*e - 5113/2, -13/32*e^7 + 11/32*e^6 + 825/32*e^5 + 153/32*e^4 - 2197/4*e^3 - 2273/4*e^2 + 7733/2*e + 13081/2, 3/32*e^7 - 13/32*e^6 - 143/32*e^5 + 449/32*e^4 + 149/2*e^3 - 455/4*e^2 - 881/2*e - 131/2, 13/16*e^7 - 7/8*e^6 - 407/8*e^5 - 1/2*e^4 + 17173/16*e^3 + 7935/8*e^2 - 30037/4*e - 24479/2, 63/64*e^7 - 73/64*e^6 - 3907/64*e^5 + 141/64*e^4 + 10247/8*e^3 + 9293/8*e^2 - 35735/4*e - 58227/4, -3/64*e^7 + 9/64*e^6 + 155/64*e^5 - 229/64*e^4 - 723/16*e^3 - 3*e^2 + 301*e + 1553/4, 1/64*e^7 - 19/64*e^6 + 23/64*e^5 + 807/64*e^4 - 399/16*e^3 - 183*e^2 + 230*e + 3725/4, -29/64*e^7 + 23/64*e^6 + 1853/64*e^5 + 389/64*e^4 - 9887/16*e^3 - 2581/4*e^2 + 8691/2*e + 29395/4, -7/8*e^7 + e^6 + 109/2*e^5 - 7/4*e^4 - 9177/8*e^3 - 4149/4*e^2 + 16053/2*e + 13045, 13/32*e^7 - 13/32*e^6 - 819/32*e^5 - 51/32*e^4 + 8653/16*e^3 + 4145/8*e^2 - 15095/4*e - 6245, -45/64*e^7 + 51/64*e^6 + 2793/64*e^5 - 47/64*e^4 - 7317/8*e^3 - 6727/8*e^2 + 25429/4*e + 41697/4, 19/32*e^7 - 21/32*e^6 - 1183/32*e^5 - 23/32*e^4 + 779*e^3 + 2983/4*e^2 - 10915/2*e - 18171/2, 5/32*e^7 + 1/32*e^6 - 341/32*e^5 - 301/32*e^4 + 1891/8*e^3 + 340*e^2 - 1687*e - 6355/2, -5/8*e^7 + 7/8*e^6 + 38*e^5 - 63/8*e^4 - 6305/8*e^3 - 2547/4*e^2 + 10939/2*e + 8645, 21/16*e^7 - 29/16*e^6 - 1283/16*e^5 + 261/16*e^4 + 13337/8*e^3 + 5359/4*e^2 - 23167/2*e - 18294, -5/16*e^7 + 7/16*e^6 + 307/16*e^5 - 75/16*e^4 - 3199/8*e^3 - 1209/4*e^2 + 5549/2*e + 4282, 3/8*e^7 - 3/8*e^6 - 189/8*e^5 - 9/8*e^4 + 1991/4*e^3 + 468*e^2 - 3455*e - 5684, -11/64*e^7 + 13/64*e^6 + 679/64*e^5 - 1/64*e^4 - 895/4*e^3 - 1719/8*e^2 + 6297/4*e + 10427/4, -15/64*e^7 + 25/64*e^6 + 891/64*e^5 - 381/64*e^4 - 1129/4*e^3 - 1503/8*e^2 + 7653/4*e + 11655/4, -1/2*e^7 + 5/8*e^6 + 31*e^5 - 31/8*e^4 - 2605/4*e^3 - 541*e^2 + 4551*e + 7176, 11/32*e^7 - 17/32*e^6 - 659/32*e^5 + 205/32*e^4 + 3375/8*e^3 + 319*e^2 - 2896*e - 9133/2, -19/32*e^7 + 25/32*e^6 + 1163/32*e^5 - 181/32*e^4 - 6035/8*e^3 - 631*e^2 + 5218*e + 16717/2, -19/16*e^7 + 21/16*e^6 + 1183/16*e^5 + 3/16*e^4 - 3109/2*e^3 - 5783/4*e^2 + 10844*e + 17800];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,-w + 1}>] := 1;
ALEigenvalues[ideal<ZF | {13,13,w^2 - w - 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;