/*
  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![3, -4, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[3, 3, w],
[3, 3, w^3 - 2*w^2 - 6*w + 1],
[9, 3, w^3 - 2*w^2 - 5*w + 1],
[16, 2, 2],
[17, 17, w^3 - w^2 - 8*w - 5],
[17, 17, -2*w^3 + 4*w^2 + 11*w - 2],
[17, 17, -w^3 + 3*w^2 + 2*w - 2],
[17, 17, w^3 - 2*w^2 - 4*w + 1],
[25, 5, w^3 - 3*w^2 - 3*w + 1],
[25, 5, w^2 - 2*w - 7],
[29, 29, w^3 - 2*w^2 - 6*w - 1],
[29, 29, w - 2],
[53, 53, w^3 - 3*w^2 - 4*w + 4],
[53, 53, -w^3 + 3*w^2 + 4*w - 5],
[61, 61, -w^3 + 2*w^2 + 7*w - 4],
[61, 61, -4*w^3 + 11*w^2 + 14*w - 8],
[101, 101, -w^3 + 2*w^2 + 7*w - 1],
[103, 103, -w^3 + 3*w^2 + 2*w - 5],
[103, 103, -w^3 + w^2 + 8*w + 2],
[113, 113, w^3 - 3*w^2 - 3*w + 7],
[113, 113, w^2 - 2*w - 1],
[127, 127, w^3 - 2*w^2 - 4*w - 2],
[127, 127, -2*w^3 + 4*w^2 + 11*w + 1],
[131, 131, -3*w^3 + 7*w^2 + 14*w - 8],
[131, 131, -w^2 + w + 8],
[131, 131, 2*w^3 - 5*w^2 - 9*w + 1],
[131, 131, -3*w^3 + 6*w^2 + 16*w + 1],
[139, 139, -2*w^3 + 5*w^2 + 9*w - 2],
[139, 139, -w^2 + w + 7],
[157, 157, -2*w^3 + 5*w^2 + 7*w - 5],
[157, 157, -2*w^3 + 3*w^2 + 13*w + 2],
[157, 157, -2*w^3 + 5*w^2 + 6*w - 2],
[157, 157, -3*w^3 + 5*w^2 + 19*w + 4],
[169, 13, -2*w^3 + 4*w^2 + 10*w - 1],
[173, 173, 3*w^3 - 8*w^2 - 10*w + 7],
[173, 173, 2*w^3 - 2*w^2 - 15*w - 8],
[179, 179, -3*w^2 + 6*w + 17],
[179, 179, -2*w^3 + 3*w^2 + 12*w + 1],
[179, 179, -3*w^3 + 7*w^2 + 13*w - 7],
[179, 179, -2*w^3 + 3*w^2 + 14*w + 4],
[191, 191, w^3 - 3*w^2 - w + 4],
[191, 191, -2*w^3 + 3*w^2 + 14*w + 2],
[199, 199, -w^3 + 8*w + 10],
[199, 199, 5*w^3 - 15*w^2 - 14*w + 14],
[211, 211, -w - 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 5],
[211, 211, 2*w^3 - 3*w^2 - 12*w - 4],
[211, 211, -3*w^3 + 7*w^2 + 13*w - 4],
[257, 257, -3*w^3 + 8*w^2 + 10*w - 2],
[257, 257, 2*w^3 - 2*w^2 - 16*w - 7],
[263, 263, 3*w^3 - 6*w^2 - 17*w + 1],
[263, 263, w^3 - 2*w^2 - 3*w - 1],
[269, 269, 5*w^3 - 13*w^2 - 19*w + 7],
[269, 269, -3*w^3 + 6*w^2 + 18*w - 1],
[277, 277, w^3 - 4*w^2 - w + 5],
[277, 277, -w^3 + 4*w^2 + w - 11],
[283, 283, -w^3 + 5*w^2 - w - 14],
[283, 283, 2*w^3 - 7*w^2 - 4*w + 10],
[283, 283, -w^3 + w^2 + 9*w + 1],
[283, 283, w^2 - 4*w - 5],
[311, 311, -w^3 + w^2 + 9*w + 5],
[311, 311, w^2 - 4*w - 1],
[313, 313, -4*w^3 + 9*w^2 + 19*w - 10],
[313, 313, -2*w^2 + 5*w + 11],
[337, 337, -3*w^3 + 6*w^2 + 14*w + 2],
[337, 337, -4*w^3 + 8*w^2 + 21*w + 1],
[347, 347, -2*w^3 + 6*w^2 + 7*w - 10],
[347, 347, -w^3 + 4*w^2 + 2*w - 7],
[361, 19, -4*w^3 + 11*w^2 + 13*w - 10],
[361, 19, -2*w^3 + 5*w^2 + 8*w - 10],
[373, 373, 2*w^3 - 5*w^2 - 10*w + 2],
[373, 373, -w^3 + 2*w^2 + 8*w - 2],
[373, 373, -2*w^3 + 4*w^2 + 13*w - 1],
[373, 373, -w^3 + 3*w^2 + 5*w - 8],
[389, 389, -w^3 + 4*w^2 + 2*w - 10],
[389, 389, -2*w^3 + 6*w^2 + 7*w - 7],
[419, 419, -2*w^3 + 5*w^2 + 6*w - 1],
[419, 419, -3*w^3 + 5*w^2 + 19*w + 5],
[433, 433, w^2 - 4*w - 2],
[433, 433, w^3 - w^2 - 9*w - 4],
[439, 439, -w^3 + 4*w^2 + w - 2],
[439, 439, w^3 - 4*w^2 - w + 14],
[443, 443, -2*w^3 + 5*w^2 + 9*w + 1],
[443, 443, -w^2 + w + 10],
[467, 467, w^3 - w^2 - 8*w + 4],
[467, 467, w^3 - 4*w^2 + 2],
[491, 491, -4*w^3 + 12*w^2 + 10*w - 11],
[491, 491, 2*w^3 - w^2 - 18*w - 11],
[491, 491, 3*w^3 - 9*w^2 - 7*w + 11],
[491, 491, 2*w^3 - 20*w - 19],
[503, 503, w^3 - 3*w^2 - 4*w + 10],
[503, 503, -w^3 + w^2 + 10*w - 2],
[523, 523, -4*w^3 + 9*w^2 + 22*w - 7],
[523, 523, w^3 - 3*w^2 - 7*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w - 7],
[547, 547, 2*w^3 - w^2 - 18*w - 14],
[547, 547, 3*w^3 - 8*w^2 - 14*w + 8],
[571, 571, 5*w^3 - 11*w^2 - 25*w + 5],
[571, 571, -4*w^3 + 9*w^2 + 19*w - 11],
[599, 599, -5*w^3 + 12*w^2 + 20*w - 4],
[599, 599, 4*w^3 - 6*w^2 - 25*w - 11],
[601, 601, -4*w^3 + 9*w^2 + 20*w - 5],
[601, 601, w^3 - 9*w - 14],
[601, 601, 3*w^3 - 8*w^2 - 11*w + 2],
[601, 601, -2*w^3 + 3*w^2 + 14*w - 1],
[641, 641, -2*w^3 + 3*w^2 + 15*w - 5],
[641, 641, -4*w^3 + 8*w^2 + 23*w - 2],
[647, 647, -3*w^3 + 8*w^2 + 13*w - 11],
[647, 647, -w^3 + 4*w^2 + 3*w - 7],
[659, 659, 3*w^3 - 7*w^2 - 14*w - 2],
[659, 659, w^3 - w^2 - 6*w - 11],
[673, 673, -3*w^3 + 7*w^2 + 11*w - 1],
[673, 673, -4*w^3 + 7*w^2 + 24*w + 5],
[677, 677, 5*w^3 - 12*w^2 - 21*w + 11],
[677, 677, -2*w^3 + 6*w^2 + 5*w - 1],
[677, 677, -2*w^3 + 6*w^2 + 8*w - 7],
[677, 677, 2*w^3 - 6*w^2 - 8*w + 11],
[719, 719, -2*w^3 + 6*w^2 + 10*w - 7],
[719, 719, -4*w^3 + 10*w^2 + 20*w - 13],
[727, 727, 3*w^3 - 10*w^2 - 8*w + 10],
[727, 727, 3*w^3 - 8*w^2 - 13*w + 10],
[727, 727, 2*w^3 - 8*w^2 - 3*w + 23],
[727, 727, -w^3 + 4*w^2 + 3*w - 8],
[751, 751, -7*w^3 + 18*w^2 + 25*w - 7],
[751, 751, w^3 - 7*w^2 + 5*w + 26],
[757, 757, -4*w^3 + 9*w^2 + 18*w - 7],
[757, 757, -3*w^3 + 5*w^2 + 17*w + 1],
[797, 797, 2*w^2 - 4*w - 5],
[797, 797, 2*w^3 - 6*w^2 - 6*w + 11],
[823, 823, -2*w^3 + 2*w^2 + 15*w + 14],
[823, 823, -3*w^3 + 8*w^2 + 10*w - 1],
[829, 829, -6*w^3 + 16*w^2 + 22*w - 7],
[829, 829, 5*w^3 - 13*w^2 - 20*w + 8],
[829, 829, -w^3 + 7*w + 4],
[829, 829, -5*w^3 + 12*w^2 + 23*w - 14],
[841, 29, -3*w^3 + 6*w^2 + 15*w - 2],
[859, 859, -4*w^3 + 7*w^2 + 24*w - 1],
[859, 859, -5*w^3 + 15*w^2 + 14*w - 13],
[881, 881, 3*w^3 - 7*w^2 - 16*w + 4],
[881, 881, -w^3 + 3*w^2 + 6*w - 7],
[883, 883, -2*w^3 + 3*w^2 + 12*w + 10],
[883, 883, -3*w^3 + 7*w^2 + 13*w + 2],
[887, 887, 4*w^3 - 11*w^2 - 11*w + 11],
[887, 887, 2*w^3 - 7*w^2 - 3*w + 7],
[887, 887, 4*w^3 - 5*w^2 - 29*w - 10],
[887, 887, 3*w^2 - 7*w - 16],
[907, 907, -w^3 + 3*w^2 + 8*w - 5],
[907, 907, -5*w^3 + 11*w^2 + 28*w - 8],
[911, 911, -4*w^3 + 10*w^2 + 18*w - 17],
[911, 911, 2*w^2 - 2*w - 1],
[919, 919, 4*w^3 - 7*w^2 - 25*w - 4],
[919, 919, 2*w^3 - 5*w^2 - 5*w + 1],
[937, 937, -w^3 + 3*w^2 + w - 8],
[937, 937, -2*w^3 + 3*w^2 + 14*w - 2],
[953, 953, -2*w^3 + w^2 + 17*w + 19],
[953, 953, 4*w^3 - 11*w^2 - 13*w + 4],
[961, 31, 2*w^3 - 3*w^2 - 11*w - 8],
[961, 31, w^3 - 4*w^2 + 2*w + 11],
[971, 971, 2*w^2 - 5*w - 4],
[971, 971, -w^3 + 4*w^2 - 11],
[991, 991, -2*w^3 + 6*w^2 + 9*w - 8],
[991, 991, -2*w^3 + 3*w^2 + 13*w - 1],
[991, 991, 3*w^3 - 8*w^2 - 14*w + 11],
[991, 991, -2*w^3 + 5*w^2 + 7*w - 8],
[997, 997, -2*w^3 + 5*w^2 + 6*w + 1],
[997, 997, -3*w^3 + 5*w^2 + 19*w + 7],
[997, 997, -6*w^3 + 12*w^2 + 33*w - 4],
[997, 997, 4*w^3 - 11*w^2 - 15*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^13 + 3*x^12 - 23*x^11 - 68*x^10 + 187*x^9 + 554*x^8 - 631*x^7 - 1957*x^6 + 796*x^5 + 2828*x^4 - 244*x^3 - 1052*x^2 + 324*x - 20;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, 11449/85624*e^12 + 27611/85624*e^11 - 272411/85624*e^10 - 302153/42812*e^9 + 2319209/85624*e^8 + 1170663/21406*e^7 - 762905/7784*e^6 - 15307547/85624*e^5 + 1489800/10703*e^4 + 9512369/42812*e^3 - 1852831/42812*e^2 - 66093/1529*e + 207895/21406, -1, 2697/85624*e^12 + 12605/85624*e^11 - 50109/85624*e^10 - 75001/21406*e^9 + 226081/85624*e^8 + 1301577/42812*e^7 + 49303/7784*e^6 - 10029941/85624*e^5 - 2539487/42812*e^4 + 8252807/42812*e^3 + 3581525/42812*e^2 - 284219/3058*e + 244469/21406, -3925/85624*e^12 - 8185/85624*e^11 + 96807/85624*e^10 + 90623/42812*e^9 - 850995/85624*e^8 - 706429/42812*e^7 + 283411/7784*e^6 + 4553089/85624*e^5 - 502451/10703*e^4 - 631625/10703*e^3 - 56263/42812*e^2 - 8669/3058*e + 39267/10703, 2697/85624*e^12 + 12605/85624*e^11 - 50109/85624*e^10 - 75001/21406*e^9 + 226081/85624*e^8 + 1301577/42812*e^7 + 49303/7784*e^6 - 10029941/85624*e^5 - 2539487/42812*e^4 + 8252807/42812*e^3 + 3581525/42812*e^2 - 284219/3058*e + 244469/21406, -3925/85624*e^12 - 8185/85624*e^11 + 96807/85624*e^10 + 90623/42812*e^9 - 850995/85624*e^8 - 706429/42812*e^7 + 283411/7784*e^6 + 4553089/85624*e^5 - 502451/10703*e^4 - 631625/10703*e^3 - 56263/42812*e^2 - 8669/3058*e + 39267/10703, 7687/85624*e^12 + 7195/85624*e^11 - 206015/85624*e^10 - 16988/10703*e^9 + 2045331/85624*e^8 + 437523/42812*e^7 - 835491/7784*e^6 - 2256267/85624*e^5 + 9218269/42812*e^4 + 662583/42812*e^3 - 6495953/42812*e^2 + 72029/3058*e + 123547/21406, 7687/85624*e^12 + 7195/85624*e^11 - 206015/85624*e^10 - 16988/10703*e^9 + 2045331/85624*e^8 + 437523/42812*e^7 - 835491/7784*e^6 - 2256267/85624*e^5 + 9218269/42812*e^4 + 662583/42812*e^3 - 6495953/42812*e^2 + 72029/3058*e + 123547/21406, -1553/85624*e^12 - 3893/85624*e^11 + 39307/85624*e^10 + 42543/42812*e^9 - 380015/85624*e^8 - 322313/42812*e^7 + 160007/7784*e^6 + 1925141/85624*e^5 - 499099/10703*e^4 - 216101/10703*e^3 + 1894429/42812*e^2 - 13419/3058*e - 66316/10703, -1553/85624*e^12 - 3893/85624*e^11 + 39307/85624*e^10 + 42543/42812*e^9 - 380015/85624*e^8 - 322313/42812*e^7 + 160007/7784*e^6 + 1925141/85624*e^5 - 499099/10703*e^4 - 216101/10703*e^3 + 1894429/42812*e^2 - 13419/3058*e - 66316/10703, 14925/85624*e^12 + 45849/85624*e^11 - 330755/85624*e^10 - 512165/42812*e^9 + 2497783/85624*e^8 + 4085937/42812*e^7 - 636103/7784*e^6 - 27980889/85624*e^5 + 888225/21406*e^4 + 9628849/21406*e^3 + 3727799/42812*e^2 - 444707/3058*e + 205725/10703, 14925/85624*e^12 + 45849/85624*e^11 - 330755/85624*e^10 - 512165/42812*e^9 + 2497783/85624*e^8 + 4085937/42812*e^7 - 636103/7784*e^6 - 27980889/85624*e^5 + 888225/21406*e^4 + 9628849/21406*e^3 + 3727799/42812*e^2 - 444707/3058*e + 205725/10703, 565/12232*e^12 + 1373/12232*e^11 - 12603/12232*e^10 - 15137/6116*e^9 + 93595/12232*e^8 + 122379/6116*e^7 - 20683/1112*e^6 - 883637/12232*e^5 - 27351/3058*e^4 + 159347/1529*e^3 + 374015/6116*e^2 - 72031/3058*e - 4252/1529, 565/12232*e^12 + 1373/12232*e^11 - 12603/12232*e^10 - 15137/6116*e^9 + 93595/12232*e^8 + 122379/6116*e^7 - 20683/1112*e^6 - 883637/12232*e^5 - 27351/3058*e^4 + 159347/1529*e^3 + 374015/6116*e^2 - 72031/3058*e - 4252/1529, -689/1946*e^12 - 1355/1946*e^11 + 16479/1946*e^10 + 14002/973*e^9 - 141053/1946*e^8 - 100365/973*e^7 + 515293/1946*e^6 + 587451/1946*e^5 - 380910/973*e^4 - 298894/973*e^3 + 146159/973*e^2 - 32/139*e + 3210/973, 5455/21406*e^12 + 13698/10703*e^11 - 206191/42812*e^10 - 1275515/42812*e^9 + 1089057/42812*e^8 + 5337011/21406*e^7 + 18575/3892*e^6 - 9636325/10703*e^5 - 13221099/42812*e^4 + 56646553/42812*e^3 + 10745447/21406*e^2 - 766303/1529*e + 1176319/21406, 5455/21406*e^12 + 13698/10703*e^11 - 206191/42812*e^10 - 1275515/42812*e^9 + 1089057/42812*e^8 + 5337011/21406*e^7 + 18575/3892*e^6 - 9636325/10703*e^5 - 13221099/42812*e^4 + 56646553/42812*e^3 + 10745447/21406*e^2 - 766303/1529*e + 1176319/21406, -3617/85624*e^12 + 747/85624*e^11 + 107895/85624*e^10 - 11171/42812*e^9 - 1163307/85624*e^8 + 113775/42812*e^7 + 496267/7784*e^6 - 1103331/85624*e^5 - 2593797/21406*e^4 + 915465/21406*e^3 + 2316261/42812*e^2 - 201125/3058*e + 230304/10703, -3617/85624*e^12 + 747/85624*e^11 + 107895/85624*e^10 - 11171/42812*e^9 - 1163307/85624*e^8 + 113775/42812*e^7 + 496267/7784*e^6 - 1103331/85624*e^5 - 2593797/21406*e^4 + 915465/21406*e^3 + 2316261/42812*e^2 - 201125/3058*e + 230304/10703, 8875/42812*e^12 + 8009/42812*e^11 - 58013/10703*e^10 - 136795/42812*e^9 + 1110693/21406*e^8 + 165318/10703*e^7 - 431813/1946*e^6 - 304053/42812*e^5 + 18014611/42812*e^4 - 4189013/42812*e^3 - 2888612/10703*e^2 + 226348/1529*e - 531651/21406, 8875/42812*e^12 + 8009/42812*e^11 - 58013/10703*e^10 - 136795/42812*e^9 + 1110693/21406*e^8 + 165318/10703*e^7 - 431813/1946*e^6 - 304053/42812*e^5 + 18014611/42812*e^4 - 4189013/42812*e^3 - 2888612/10703*e^2 + 226348/1529*e - 531651/21406, -22277/42812*e^12 - 51391/42812*e^11 + 134080/10703*e^10 + 1128117/42812*e^9 - 2317831/21406*e^8 - 2197163/10703*e^7 + 778605/1946*e^6 + 28976847/42812*e^5 - 25097733/42812*e^4 - 36232541/42812*e^3 + 2093101/10703*e^2 + 242144/1529*e - 556335/21406, -327/3892*e^12 - 529/973*e^11 + 1366/973*e^10 + 49625/3892*e^9 - 19675/3892*e^8 - 420685/3892*e^7 - 86575/3892*e^6 + 387636/973*e^5 + 609961/3892*e^4 - 589829/973*e^3 - 435271/1946*e^2 + 69159/278*e - 21517/973, -327/3892*e^12 - 529/973*e^11 + 1366/973*e^10 + 49625/3892*e^9 - 19675/3892*e^8 - 420685/3892*e^7 - 86575/3892*e^6 + 387636/973*e^5 + 609961/3892*e^4 - 589829/973*e^3 - 435271/1946*e^2 + 69159/278*e - 21517/973, -22277/42812*e^12 - 51391/42812*e^11 + 134080/10703*e^10 + 1128117/42812*e^9 - 2317831/21406*e^8 - 2197163/10703*e^7 + 778605/1946*e^6 + 28976847/42812*e^5 - 25097733/42812*e^4 - 36232541/42812*e^3 + 2093101/10703*e^2 + 242144/1529*e - 556335/21406, -3499/21406*e^12 - 1460/10703*e^11 + 182725/42812*e^10 + 97197/42812*e^9 - 1730111/42812*e^8 - 209695/21406*e^7 + 651807/3892*e^6 - 81945/10703*e^5 - 12514899/42812*e^4 + 5214733/42812*e^3 + 3025053/21406*e^2 - 286351/1529*e + 745075/21406, -3499/21406*e^12 - 1460/10703*e^11 + 182725/42812*e^10 + 97197/42812*e^9 - 1730111/42812*e^8 - 209695/21406*e^7 + 651807/3892*e^6 - 81945/10703*e^5 - 12514899/42812*e^4 + 5214733/42812*e^3 + 3025053/21406*e^2 - 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711451/10703*e^8 - 17794173/42812*e^7 + 105484/973*e^6 + 32129589/21406*e^5 + 6016817/21406*e^4 - 93086093/42812*e^3 - 7646587/10703*e^2 + 2221555/3058*e - 1595261/21406, -21269/42812*e^12 - 45621/21406*e^11 + 437361/42812*e^10 + 1062343/21406*e^9 - 711451/10703*e^8 - 17794173/42812*e^7 + 105484/973*e^6 + 32129589/21406*e^5 + 6016817/21406*e^4 - 93086093/42812*e^3 - 7646587/10703*e^2 + 2221555/3058*e - 1595261/21406, 18313/85624*e^12 + 15665/85624*e^11 - 465659/85624*e^10 - 133479/42812*e^9 + 4242603/85624*e^8 + 714107/42812*e^7 - 1505587/7784*e^6 - 2822217/85624*e^5 + 3317945/10703*e^4 + 490335/21406*e^3 - 6093705/42812*e^2 - 20525/3058*e - 79637/10703, 18313/85624*e^12 + 15665/85624*e^11 - 465659/85624*e^10 - 133479/42812*e^9 + 4242603/85624*e^8 + 714107/42812*e^7 - 1505587/7784*e^6 - 2822217/85624*e^5 + 3317945/10703*e^4 + 490335/21406*e^3 - 6093705/42812*e^2 - 20525/3058*e - 79637/10703, 22039/85624*e^12 + 17245/85624*e^11 - 602295/85624*e^10 - 69749/21406*e^9 + 6164273/85624*e^8 + 264113/21406*e^7 - 2669657/7784*e^6 + 1840163/85624*e^5 + 33164017/42812*e^4 - 3179305/21406*e^3 - 29390003/42812*e^2 + 166929/1529*e + 402800/10703, 22039/85624*e^12 + 17245/85624*e^11 - 602295/85624*e^10 - 69749/21406*e^9 + 6164273/85624*e^8 + 264113/21406*e^7 - 2669657/7784*e^6 + 1840163/85624*e^5 + 33164017/42812*e^4 - 3179305/21406*e^3 - 29390003/42812*e^2 + 166929/1529*e + 402800/10703, 49131/42812*e^12 + 57779/21406*e^11 - 1162655/42812*e^10 - 1254499/21406*e^9 + 2452138/10703*e^8 + 19290039/42812*e^7 - 795092/973*e^6 - 31429973/21406*e^5 + 24465181/21406*e^4 + 79365791/42812*e^3 - 3961312/10703*e^2 - 1319961/3058*e + 1825255/21406, 49131/42812*e^12 + 57779/21406*e^11 - 1162655/42812*e^10 - 1254499/21406*e^9 + 2452138/10703*e^8 + 19290039/42812*e^7 - 795092/973*e^6 - 31429973/21406*e^5 + 24465181/21406*e^4 + 79365791/42812*e^3 - 3961312/10703*e^2 - 1319961/3058*e + 1825255/21406, 62199/85624*e^12 + 156621/85624*e^11 - 1438359/85624*e^10 - 851787/21406*e^9 + 11702345/85624*e^8 + 6591967/21406*e^7 - 3524881/7784*e^6 - 87140917/85624*e^5 + 21818141/42812*e^4 + 28229407/21406*e^3 + 1203537/42812*e^2 - 516751/1529*e + 326929/10703, 62199/85624*e^12 + 156621/85624*e^11 - 1438359/85624*e^10 - 851787/21406*e^9 + 11702345/85624*e^8 + 6591967/21406*e^7 - 3524881/7784*e^6 - 87140917/85624*e^5 + 21818141/42812*e^4 + 28229407/21406*e^3 + 1203537/42812*e^2 - 516751/1529*e + 326929/10703, -10319/85624*e^12 - 30981/85624*e^11 + 231915/85624*e^10 + 83642/10703*e^9 - 1853741/85624*e^8 - 1266931/21406*e^7 + 584485/7784*e^6 + 16298589/85624*e^5 - 5112979/42812*e^4 - 2909140/10703*e^3 + 3527435/42812*e^2 + 264839/1529*e - 143796/10703, -10319/85624*e^12 - 30981/85624*e^11 + 231915/85624*e^10 + 83642/10703*e^9 - 1853741/85624*e^8 - 1266931/21406*e^7 + 584485/7784*e^6 + 16298589/85624*e^5 - 5112979/42812*e^4 - 2909140/10703*e^3 + 3527435/42812*e^2 + 264839/1529*e - 143796/10703, 1416/10703*e^12 + 27341/42812*e^11 - 97865/42812*e^10 - 608481/42812*e^9 + 89840/10703*e^8 + 4752955/42812*e^7 + 37858/973*e^6 - 15303861/42812*e^5 - 11453063/42812*e^4 + 9080895/21406*e^3 + 4139239/10703*e^2 - 227171/3058*e - 3594/10703, 13049/42812*e^12 + 16105/10703*e^11 - 126657/21406*e^10 - 1523045/42812*e^9 + 1414883/42812*e^8 + 13025359/42812*e^7 - 22679/3892*e^6 - 12139041/10703*e^5 - 15848265/42812*e^4 + 37396175/21406*e^3 + 13656459/21406*e^2 - 2219693/3058*e + 1096680/10703, 13049/42812*e^12 + 16105/10703*e^11 - 126657/21406*e^10 - 1523045/42812*e^9 + 1414883/42812*e^8 + 13025359/42812*e^7 - 22679/3892*e^6 - 12139041/10703*e^5 - 15848265/42812*e^4 + 37396175/21406*e^3 + 13656459/21406*e^2 - 2219693/3058*e + 1096680/10703, -41191/85624*e^12 - 191237/85624*e^11 + 802145/85624*e^10 + 2225981/42812*e^9 - 4509179/85624*e^8 - 9303787/21406*e^7 + 113355/7784*e^6 + 133763557/85624*e^5 + 11827173/21406*e^4 - 96194281/42812*e^3 - 42514347/42812*e^2 + 1159091/1529*e - 910999/21406, -41191/85624*e^12 - 191237/85624*e^11 + 802145/85624*e^10 + 2225981/42812*e^9 - 4509179/85624*e^8 - 9303787/21406*e^7 + 113355/7784*e^6 + 133763557/85624*e^5 + 11827173/21406*e^4 - 96194281/42812*e^3 - 42514347/42812*e^2 + 1159091/1529*e - 910999/21406, 28337/42812*e^12 + 46637/21406*e^11 - 310729/21406*e^10 - 2100797/42812*e^9 + 4646993/42812*e^8 + 16980933/42812*e^7 - 1174501/3892*e^6 - 29649581/21406*e^5 + 6704055/42812*e^4 + 20961177/10703*e^3 + 6294069/21406*e^2 - 2046859/3058*e + 1083079/10703, 28337/42812*e^12 + 46637/21406*e^11 - 310729/21406*e^10 - 2100797/42812*e^9 + 4646993/42812*e^8 + 16980933/42812*e^7 - 1174501/3892*e^6 - 29649581/21406*e^5 + 6704055/42812*e^4 + 20961177/10703*e^3 + 6294069/21406*e^2 - 2046859/3058*e + 1083079/10703, 41659/42812*e^12 + 115015/42812*e^11 - 474311/21406*e^10 - 2517989/42812*e^9 + 3772119/21406*e^8 + 9736465/21406*e^7 - 1090179/1946*e^6 - 63051371/42812*e^5 + 23622853/42812*e^4 + 75960163/42812*e^3 + 2091178/10703*e^2 - 404427/1529*e - 211531/21406, 3777/3892*e^12 + 11399/3892*e^11 - 85455/3892*e^10 - 127055/1946*e^9 + 676085/3892*e^8 + 504417/973*e^7 - 2142327/3892*e^6 - 6813215/3892*e^5 + 525876/973*e^4 + 4480489/1946*e^3 + 327347/1946*e^2 - 81935/139*e + 74391/973, 41659/42812*e^12 + 115015/42812*e^11 - 474311/21406*e^10 - 2517989/42812*e^9 + 3772119/21406*e^8 + 9736465/21406*e^7 - 1090179/1946*e^6 - 63051371/42812*e^5 + 23622853/42812*e^4 + 75960163/42812*e^3 + 2091178/10703*e^2 - 404427/1529*e - 211531/21406, 3777/3892*e^12 + 11399/3892*e^11 - 85455/3892*e^10 - 127055/1946*e^9 + 676085/3892*e^8 + 504417/973*e^7 - 2142327/3892*e^6 - 6813215/3892*e^5 + 525876/973*e^4 + 4480489/1946*e^3 + 327347/1946*e^2 - 81935/139*e + 74391/973, 24099/42812*e^12 + 58081/42812*e^11 - 139644/10703*e^10 - 1239919/42812*e^9 + 2275615/21406*e^8 + 2303663/10703*e^7 - 689529/1946*e^6 - 27864685/42812*e^5 + 17764955/42812*e^4 + 28771079/42812*e^3 + 61973/10703*e^2 + 41515/1529*e - 527207/21406, 24099/42812*e^12 + 58081/42812*e^11 - 139644/10703*e^10 - 1239919/42812*e^9 + 2275615/21406*e^8 + 2303663/10703*e^7 - 689529/1946*e^6 - 27864685/42812*e^5 + 17764955/42812*e^4 + 28771079/42812*e^3 + 61973/10703*e^2 + 41515/1529*e - 527207/21406, 17289/21406*e^12 + 55469/21406*e^11 - 766105/42812*e^10 - 2479353/42812*e^9 + 5824341/42812*e^8 + 4927326/10703*e^7 - 1510245/3892*e^6 - 33120431/21406*e^5 + 8383779/42812*e^4 + 84780893/42812*e^3 + 10744871/21406*e^2 - 614292/1529*e - 341525/21406, 17289/21406*e^12 + 55469/21406*e^11 - 766105/42812*e^10 - 2479353/42812*e^9 + 5824341/42812*e^8 + 4927326/10703*e^7 - 1510245/3892*e^6 - 33120431/21406*e^5 + 8383779/42812*e^4 + 84780893/42812*e^3 + 10744871/21406*e^2 - 614292/1529*e - 341525/21406, -3375/6116*e^12 - 13641/6116*e^11 + 70737/6116*e^10 + 158075/3058*e^9 - 479063/6116*e^8 - 656787/1529*e^7 + 85827/556*e^6 + 9372321/6116*e^5 + 303669/1529*e^4 - 6674633/3058*e^3 - 2005947/3058*e^2 + 1082426/1529*e - 104605/1529, -3375/6116*e^12 - 13641/6116*e^11 + 70737/6116*e^10 + 158075/3058*e^9 - 479063/6116*e^8 - 656787/1529*e^7 + 85827/556*e^6 + 9372321/6116*e^5 + 303669/1529*e^4 - 6674633/3058*e^3 - 2005947/3058*e^2 + 1082426/1529*e - 104605/1529, -43873/85624*e^12 - 117505/85624*e^11 + 1049201/85624*e^10 + 332531/10703*e^9 - 9007345/85624*e^8 - 10757895/42812*e^7 + 2999825/7784*e^6 + 74392489/85624*e^5 - 23455007/42812*e^4 - 50488927/42812*e^3 + 5981487/42812*e^2 + 1014703/3058*e - 739251/21406, -43873/85624*e^12 - 117505/85624*e^11 + 1049201/85624*e^10 + 332531/10703*e^9 - 9007345/85624*e^8 - 10757895/42812*e^7 + 2999825/7784*e^6 + 74392489/85624*e^5 - 23455007/42812*e^4 - 50488927/42812*e^3 + 5981487/42812*e^2 + 1014703/3058*e - 739251/21406, 54521/85624*e^12 + 154135/85624*e^11 - 1268299/85624*e^10 - 1762237/42812*e^9 + 10326069/85624*e^8 + 3627595/10703*e^7 - 3064245/7784*e^6 - 103633679/85624*e^5 + 4307854/10703*e^4 + 74383523/42812*e^3 + 4152593/42812*e^2 - 857674/1529*e + 1514267/21406, 54521/85624*e^12 + 154135/85624*e^11 - 1268299/85624*e^10 - 1762237/42812*e^9 + 10326069/85624*e^8 + 3627595/10703*e^7 - 3064245/7784*e^6 - 103633679/85624*e^5 + 4307854/10703*e^4 + 74383523/42812*e^3 + 4152593/42812*e^2 - 857674/1529*e + 1514267/21406, 37963/85624*e^12 + 221891/85624*e^11 - 642855/85624*e^10 - 1284363/21406*e^9 + 2290759/85624*e^8 + 21342829/42812*e^7 + 1005097/7784*e^6 - 152636867/85624*e^5 - 37311301/42812*e^4 + 111071215/42812*e^3 + 50815019/42812*e^2 - 2975523/3058*e + 2089697/21406, 37963/85624*e^12 + 221891/85624*e^11 - 642855/85624*e^10 - 1284363/21406*e^9 + 2290759/85624*e^8 + 21342829/42812*e^7 + 1005097/7784*e^6 - 152636867/85624*e^5 - 37311301/42812*e^4 + 111071215/42812*e^3 + 50815019/42812*e^2 - 2975523/3058*e + 2089697/21406, -261/1529*e^12 - 391/6116*e^11 + 28719/6116*e^10 + 4119/6116*e^9 - 72977/1529*e^8 + 9803/6116*e^7 + 31158/139*e^6 - 165593/6116*e^5 - 3121915/6116*e^4 + 141791/3058*e^3 + 751460/1529*e^2 + 72197/3058*e - 87728/1529, -261/1529*e^12 - 391/6116*e^11 + 28719/6116*e^10 + 4119/6116*e^9 - 72977/1529*e^8 + 9803/6116*e^7 + 31158/139*e^6 - 165593/6116*e^5 - 3121915/6116*e^4 + 141791/3058*e^3 + 751460/1529*e^2 + 72197/3058*e - 87728/1529, 18995/21406*e^12 + 75473/42812*e^11 - 932671/42812*e^10 - 1626645/42812*e^9 + 2059157/10703*e^8 + 12330497/42812*e^7 - 709994/973*e^6 - 38931017/42812*e^5 + 47577825/42812*e^4 + 11037508/10703*e^3 - 4219349/10703*e^2 - 189779/3058*e - 374153/10703, 5515/10703*e^12 + 46071/42812*e^11 - 134087/10703*e^10 - 499151/21406*e^9 + 4689317/42812*e^8 + 7643613/42812*e^7 - 1612265/3892*e^6 - 24829151/42812*e^5 + 7028034/10703*e^4 + 31800141/42812*e^3 - 6723723/21406*e^2 - 651645/3058*e + 441213/21406, 18995/21406*e^12 + 75473/42812*e^11 - 932671/42812*e^10 - 1626645/42812*e^9 + 2059157/10703*e^8 + 12330497/42812*e^7 - 709994/973*e^6 - 38931017/42812*e^5 + 47577825/42812*e^4 + 11037508/10703*e^3 - 4219349/10703*e^2 - 189779/3058*e - 374153/10703, 5515/10703*e^12 + 46071/42812*e^11 - 134087/10703*e^10 - 499151/21406*e^9 + 4689317/42812*e^8 + 7643613/42812*e^7 - 1612265/3892*e^6 - 24829151/42812*e^5 + 7028034/10703*e^4 + 31800141/42812*e^3 - 6723723/21406*e^2 - 651645/3058*e + 441213/21406, -107365/85624*e^12 - 279931/85624*e^11 + 2494249/85624*e^10 + 770617/10703*e^9 - 20467915/85624*e^8 - 6052721/10703*e^7 + 6257371/7784*e^6 + 162340459/85624*e^5 - 39411585/42812*e^4 - 52664389/21406*e^3 - 2755923/42812*e^2 + 832825/1529*e - 650186/10703, -107365/85624*e^12 - 279931/85624*e^11 + 2494249/85624*e^10 + 770617/10703*e^9 - 20467915/85624*e^8 - 6052721/10703*e^7 + 6257371/7784*e^6 + 162340459/85624*e^5 - 39411585/42812*e^4 - 52664389/21406*e^3 - 2755923/42812*e^2 + 832825/1529*e - 650186/10703, 91739/85624*e^12 + 283531/85624*e^11 - 2081561/85624*e^10 - 3200909/42812*e^9 + 16480993/85624*e^8 + 25800741/42812*e^7 - 4746745/7784*e^6 - 177916091/85624*e^5 + 6513322/10703*e^4 + 61035089/21406*e^3 + 6190029/42812*e^2 - 2735749/3058*e + 1296632/10703, 91739/85624*e^12 + 283531/85624*e^11 - 2081561/85624*e^10 - 3200909/42812*e^9 + 16480993/85624*e^8 + 25800741/42812*e^7 - 4746745/7784*e^6 - 177916091/85624*e^5 + 6513322/10703*e^4 + 61035089/21406*e^3 + 6190029/42812*e^2 - 2735749/3058*e + 1296632/10703];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 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;