/*
  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, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -w^3 + 4*w + 1}>;

primesArray := [
[3, 3, w],
[3, 3, w - 2],
[3, 3, w - 1],
[16, 2, 2],
[23, 23, -w^3 + 4*w + 1],
[25, 5, w^3 - 2*w^2 - 2*w + 2],
[25, 5, w^3 - w^2 - 3*w + 1],
[29, 29, w^3 - 2*w^2 - 4*w + 4],
[29, 29, w^3 - w^2 - 5*w + 1],
[43, 43, -w^3 + 2*w^2 + 3*w - 2],
[43, 43, w^3 - w^2 - 4*w + 2],
[61, 61, -w^2 + 2*w + 4],
[61, 61, w^2 - 5],
[79, 79, 3*w^3 - 7*w^2 - 8*w + 16],
[79, 79, -w^3 + w^2 + 6*w - 4],
[101, 101, -w^3 + 3*w^2 + w - 7],
[101, 101, w^3 - 4*w - 4],
[103, 103, 2*w^3 - w^2 - 8*w - 1],
[103, 103, 2*w^3 - 5*w^2 - 4*w + 8],
[107, 107, 2*w^2 - 3*w - 4],
[107, 107, 3*w - 5],
[107, 107, w^3 - 4*w^2 + 10],
[107, 107, 2*w^2 - w - 5],
[113, 113, -w^3 + 4*w^2 + w - 11],
[113, 113, -2*w^3 + 4*w^2 + 5*w - 5],
[113, 113, 2*w^3 - 2*w^2 - 7*w + 2],
[113, 113, w^3 + w^2 - 6*w - 7],
[121, 11, w^3 - 3*w^2 - 4*w + 8],
[121, 11, w^3 - 7*w - 2],
[127, 127, w^2 - 4*w + 2],
[127, 127, w^3 - w^2 - 2*w - 2],
[127, 127, -w^3 + 2*w^2 + w - 4],
[127, 127, -2*w^3 + 5*w^2 + 3*w - 8],
[131, 131, 2*w^3 - 2*w^2 - 9*w + 2],
[131, 131, -w^3 - w^2 + 7*w + 5],
[157, 157, w^3 - w^2 - 7*w + 8],
[157, 157, w^3 - w^2 - 5*w + 4],
[169, 13, 2*w^2 - 2*w - 5],
[173, 173, -w^3 + 7*w - 1],
[173, 173, -w^3 - w^2 + 6*w + 4],
[179, 179, -2*w^3 + 4*w^2 + 5*w - 8],
[179, 179, -2*w^3 + 2*w^2 + 7*w + 1],
[181, 181, -w^3 + 4*w^2 - 7],
[181, 181, w^2 - 4*w + 5],
[191, 191, 3*w^3 - 7*w^2 - 9*w + 20],
[191, 191, -w^3 + 4*w + 5],
[191, 191, -w^3 + 3*w^2 + w - 8],
[191, 191, 3*w^3 - 3*w^2 - 12*w + 2],
[199, 199, 2*w^2 - 7],
[199, 199, 3*w^3 - 6*w^2 - 9*w + 13],
[233, 233, w^3 - 3*w^2 - 4*w + 11],
[233, 233, 3*w - 7],
[251, 251, 3*w^3 - 7*w^2 - 7*w + 13],
[251, 251, 4*w^3 - w^2 - 20*w - 11],
[251, 251, -w^3 + w^2 + 5*w - 7],
[251, 251, -2*w^3 + 4*w^2 + 8*w - 11],
[257, 257, w^2 - 2*w - 7],
[257, 257, w^2 - 8],
[269, 269, 2*w^3 - 3*w^2 - 8*w + 2],
[269, 269, 2*w^3 - 3*w^2 - 8*w + 7],
[283, 283, -2*w^3 + 4*w^2 + 4*w - 7],
[283, 283, 2*w^3 - 2*w^2 - 6*w - 1],
[289, 17, -w^2 + w - 2],
[289, 17, w^2 - w - 7],
[311, 311, -w^3 + 2*w^2 + 3*w - 8],
[311, 311, -w^3 + w^2 + 4*w + 4],
[313, 313, -w^3 + 2*w^2 + 5*w - 5],
[313, 313, w^3 - w^2 - 6*w + 1],
[337, 337, 3*w - 1],
[337, 337, 3*w - 2],
[347, 347, -2*w^3 - w^2 + 11*w + 11],
[347, 347, -2*w^3 + 7*w^2 + 3*w - 19],
[367, 367, 4*w^3 - 8*w^2 - 13*w + 19],
[367, 367, 4*w^3 - 4*w^2 - 17*w - 2],
[373, 373, w^2 - 3*w - 5],
[373, 373, w^2 + w - 7],
[433, 433, 2*w^3 - 3*w^2 - 5*w + 1],
[433, 433, -2*w^3 + 3*w^2 + 5*w - 5],
[443, 443, w^3 - 7*w + 2],
[443, 443, 3*w^3 - 6*w^2 - 12*w + 20],
[491, 491, -2*w^3 + 3*w^2 + 8*w - 8],
[491, 491, 2*w^3 - 3*w^2 - 11*w + 14],
[521, 521, w^3 + w^2 - 8*w - 7],
[521, 521, w^3 - 5*w^2 + 2*w + 10],
[521, 521, w^3 + 2*w^2 - 5*w - 8],
[521, 521, w^3 - 4*w^2 - 3*w + 13],
[523, 523, 3*w^3 - 5*w^2 - 11*w + 11],
[523, 523, w^3 + w^2 - 6*w - 1],
[529, 23, 3*w^2 - 3*w - 10],
[547, 547, w^3 - 3*w^2 - 4*w + 14],
[547, 547, -2*w^3 + 2*w^2 + 7*w - 5],
[547, 547, w^3 - 4*w + 4],
[547, 547, 3*w^3 - 9*w^2 - 6*w + 22],
[563, 563, w^3 - 2*w^2 - 6*w + 8],
[563, 563, -3*w^3 + 4*w^2 + 11*w - 7],
[563, 563, 3*w^3 - 5*w^2 - 10*w + 5],
[563, 563, 3*w^3 - 8*w^2 - 3*w + 10],
[571, 571, 2*w^3 - 3*w^2 - 5*w + 4],
[571, 571, -2*w^3 + 3*w^2 + 5*w - 2],
[599, 599, 2*w^3 - 2*w^2 - 5*w + 1],
[599, 599, -2*w^3 + 4*w^2 + 3*w - 4],
[641, 641, -2*w^2 + 7*w - 7],
[641, 641, 2*w^2 + w - 8],
[641, 641, 2*w^2 - 5*w - 5],
[641, 641, w^3 - 4*w^2 - 2*w + 16],
[647, 647, -w^3 + 3*w^2 - 2*w - 4],
[647, 647, w^3 - w - 4],
[653, 653, -2*w^3 + 4*w^2 + 9*w - 13],
[653, 653, 3*w^3 - 7*w^2 - 6*w + 11],
[673, 673, w^3 - 2*w^2 - 5*w + 11],
[673, 673, -2*w^3 + 2*w^2 + 10*w - 5],
[673, 673, 2*w^3 - 4*w^2 - 8*w + 5],
[673, 673, 2*w^3 - 7*w^2 - w + 13],
[677, 677, -w^3 + 2*w^2 + 6*w - 2],
[677, 677, 3*w^3 - 7*w^2 - 8*w + 13],
[677, 677, 3*w^3 - 2*w^2 - 13*w - 1],
[677, 677, -w^3 + w^2 + 7*w - 5],
[701, 701, w^2 - 5*w - 1],
[701, 701, -w^3 + 4*w^2 + 3*w - 7],
[701, 701, -w^3 - w^2 + 8*w + 1],
[701, 701, w^2 + 3*w - 5],
[719, 719, -2*w^3 + w^2 + 7*w + 5],
[719, 719, 2*w^3 - 5*w^2 - 3*w + 11],
[727, 727, 2*w^2 - 4*w - 11],
[727, 727, -3*w^3 + w^2 + 10*w + 5],
[751, 751, 4*w^3 - 12*w^2 - 7*w + 28],
[751, 751, 2*w^3 - 6*w^2 - 5*w + 20],
[757, 757, -w^3 + 2*w^2 - 5],
[757, 757, w^3 - w^2 - w - 4],
[797, 797, -4*w^3 + 7*w^2 + 13*w - 11],
[797, 797, 4*w^3 - 5*w^2 - 17*w + 2],
[797, 797, -4*w^3 + 7*w^2 + 15*w - 16],
[797, 797, 4*w^3 - 5*w^2 - 15*w + 5],
[809, 809, -w^3 + 4*w^2 + 3*w - 16],
[809, 809, 3*w^3 - 8*w^2 - 10*w + 26],
[841, 29, 3*w^2 - 3*w - 8],
[857, 857, 4*w^3 - 2*w^2 - 21*w - 7],
[857, 857, -w^3 + 2*w^2 - w + 4],
[881, 881, -w^3 + 2*w^2 + 7*w - 10],
[881, 881, w^3 - 5*w^2 - w + 13],
[881, 881, w^3 + 2*w^2 - 8*w - 8],
[881, 881, -w^3 + w^2 + 8*w + 2],
[887, 887, w^3 - w^2 - 5*w - 5],
[887, 887, -w^3 + 2*w^2 + 4*w - 10],
[907, 907, -3*w^3 + 3*w^2 + 9*w - 5],
[907, 907, 3*w^3 - 6*w^2 - 6*w + 4],
[919, 919, 4*w^3 - 3*w^2 - 16*w - 2],
[919, 919, -4*w^3 + 9*w^2 + 10*w - 17],
[937, 937, -3*w^3 + w^2 + 12*w + 7],
[937, 937, -3*w^3 + 8*w^2 + 5*w - 17],
[953, 953, -2*w^3 + 6*w^2 + 2*w - 13],
[953, 953, -w^3 + 3*w^2 + 4*w - 2],
[953, 953, w^3 - 7*w + 4],
[953, 953, 2*w^3 - 8*w - 7],
[961, 31, 3*w^3 - 4*w^2 - 13*w + 1],
[961, 31, 3*w^3 - 5*w^2 - 12*w + 13],
[991, 991, 3*w^3 - w^2 - 15*w - 4],
[991, 991, -w^3 - 3*w^2 + 7*w + 5],
[991, 991, w^3 - 6*w^2 + 2*w + 8],
[991, 991, 3*w^3 - 8*w^2 - 8*w + 17]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - 20*x^10 + 146*x^8 - 467*x^6 + 589*x^4 - 128*x^2 + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 21/146*e^11 - 198/73*e^9 + 1338/73*e^7 - 7729/146*e^5 + 8479/146*e^3 - 639/73*e, -7/146*e^11 + 66/73*e^9 - 446/73*e^7 + 2625/146*e^5 - 3313/146*e^3 + 724/73*e, -11/73*e^10 + 197/73*e^8 - 1214/73*e^6 + 2957/73*e^4 - 2380/73*e^2 + 75/73, 1, -99/584*e^11 + 923/292*e^9 - 6193/292*e^7 + 35957/584*e^5 - 41349/584*e^3 + 4827/292*e, -209/584*e^11 + 1981/292*e^9 - 13431/292*e^7 + 77207/584*e^5 - 81063/584*e^3 + 2209/292*e, -55/584*e^11 + 675/292*e^9 - 6101/292*e^7 + 48073/584*e^5 - 71833/584*e^3 + 6867/292*e, -17/584*e^11 + 129/292*e^9 - 447/292*e^7 - 2385/584*e^5 + 14897/584*e^3 - 7179/292*e, 15/292*e^10 - 131/146*e^8 + 841/146*e^6 - 5041/292*e^4 + 6849/292*e^2 - 1395/146, -15/292*e^10 + 131/146*e^8 - 841/146*e^6 + 5041/292*e^4 - 6849/292*e^2 + 227/146, 13/584*e^11 - 133/292*e^9 + 943/292*e^7 - 5459/584*e^5 + 7571/584*e^3 - 5005/292*e, 7/8*e^11 - 71/4*e^9 + 525/4*e^7 - 3377/8*e^5 + 4153/8*e^3 - 303/4*e, 295/292*e^11 - 3063/146*e^9 + 23353/146*e^7 - 156177/292*e^5 + 201565/292*e^3 - 16193/146*e, -89/292*e^11 + 933/146*e^9 - 7287/146*e^7 + 51187/292*e^5 - 72699/292*e^3 + 9007/146*e, 49/584*e^11 - 389/292*e^9 + 2027/292*e^7 - 7863/584*e^5 + 5087/584*e^3 - 177/292*e, -293/584*e^11 + 3065/292*e^9 - 23455/292*e^7 + 156595/584*e^5 - 200827/584*e^3 + 16445/292*e, 1/292*e^10 + 147/146*e^8 - 2387/146*e^6 + 24445/292*e^4 - 41533/292*e^2 + 2535/146, -105/292*e^10 + 771/146*e^8 - 3113/146*e^6 + 831/292*e^4 + 23597/292*e^2 - 2207/146, -109/292*e^10 + 913/146*e^8 - 5099/146*e^6 + 20727/292*e^4 - 9707/292*e^2 + 501/146, -3/146*e^11 + 70/73*e^9 - 869/73*e^7 + 8279/146*e^5 - 14101/146*e^3 + 1593/73*e, 101/146*e^11 - 994/73*e^9 + 7113/73*e^7 - 44299/146*e^5 + 53183/146*e^3 - 3991/73*e, -47/292*e^10 + 537/146*e^8 - 4465/146*e^6 + 31933/292*e^4 - 42309/292*e^2 + 2619/146, -29/292*e^10 + 263/146*e^8 - 1587/146*e^6 + 7079/292*e^4 - 5299/292*e^2 + 945/146, -415/584*e^11 + 4111/292*e^9 - 29497/292*e^7 + 183073/584*e^5 - 218105/584*e^3 + 16987/292*e, 47/584*e^11 - 391/292*e^9 + 2129/292*e^7 - 8281/584*e^5 + 4057/584*e^3 + 885/292*e, -123/292*e^10 + 1045/146*e^8 - 5845/146*e^6 + 23057/292*e^4 - 9325/292*e^2 - 241/146, -749/584*e^11 + 7281/292*e^9 - 51007/292*e^7 + 307155/584*e^5 - 352739/584*e^3 + 27025/292*e, -251/584*e^11 + 2523/292*e^9 - 18589/292*e^7 + 120405/584*e^5 - 154085/584*e^3 + 19255/292*e, -209/146*e^11 + 2054/73*e^9 - 14672/73*e^7 + 91223/146*e^5 - 109825/146*e^3 + 8925/73*e, -239/292*e^11 + 2389/146*e^9 - 17449/146*e^7 + 111525/292*e^5 - 137685/292*e^3 + 9817/146*e, 177/292*e^11 - 1867/146*e^9 + 14479/146*e^7 - 98787/292*e^5 + 131451/292*e^3 - 12519/146*e, -33/146*e^11 + 259/73*e^9 - 1237/73*e^7 + 2593/146*e^5 + 5635/146*e^3 - 2771/73*e, -273/292*e^10 + 2355/146*e^8 - 13963/146*e^6 + 65583/292*e^4 - 50075/292*e^2 + 1445/146, -7/292*e^10 + 285/146*e^8 - 3877/146*e^6 + 36497/292*e^4 - 56165/292*e^2 + 2403/146, -39/292*e^10 + 399/146*e^8 - 2975/146*e^6 + 19881/292*e^4 - 27093/292*e^2 + 999/146, 147/292*e^10 - 1459/146*e^8 + 10607/146*e^6 - 67681/292*e^4 + 82421/292*e^2 - 4327/146, -13/73*e^10 + 266/73*e^8 - 2032/73*e^6 + 6846/73*e^4 - 8520/73*e^2 + 958/73, 9/146*e^10 - 137/73*e^8 + 1366/73*e^6 - 10821/146*e^4 + 15293/146*e^2 - 1859/73, 171/146*e^10 - 1508/73*e^8 + 9164/73*e^6 - 44269/146*e^4 + 35067/146*e^2 - 1449/73, 177/146*e^11 - 1721/73*e^9 + 12070/73*e^7 - 72945/146*e^5 + 85169/146*e^3 - 8431/73*e, 207/146*e^11 - 2056/73*e^9 + 14847/73*e^7 - 93247/146*e^5 + 113467/146*e^3 - 10637/73*e, -22/73*e^10 + 321/73*e^8 - 1406/73*e^6 + 1607/73*e^4 + 277/73*e^2 + 1464/73, 3/73*e^10 + 6/73*e^8 - 452/73*e^6 + 2160/73*e^4 - 2105/73*e^2 - 704/73, 35/292*e^10 - 549/146*e^8 + 5661/146*e^6 - 47873/292*e^4 + 74381/292*e^2 - 5007/146, -207/292*e^11 + 2129/146*e^9 - 16015/146*e^7 + 105073/292*e^5 - 131133/292*e^3 + 7425/146*e, 241/292*e^11 - 2387/146*e^9 + 17201/146*e^7 - 108187/292*e^5 + 132875/292*e^3 - 12047/146*e, 117/292*e^10 - 759/146*e^8 + 1771/146*e^6 + 17737/292*e^4 - 59757/292*e^2 + 4303/146, 137/292*e^11 - 1323/146*e^9 + 9219/146*e^7 - 55463/292*e^5 + 65007/292*e^3 - 6755/146*e, 201/292*e^11 - 1989/146*e^9 + 14277/146*e^7 - 88515/292*e^5 + 103223/292*e^3 - 5699/146*e, 1173/584*e^11 - 11529/292*e^9 + 82235/292*e^7 - 508787/584*e^5 + 603803/584*e^3 - 45725/292*e, 527/584*e^11 - 4875/292*e^9 + 31961/292*e^7 - 176017/584*e^5 + 181761/584*e^3 - 17183/292*e, 114/73*e^11 - 2327/73*e^9 + 17353/73*e^7 - 56401/73*e^5 + 69879/73*e^3 - 8575/73*e, 301/292*e^10 - 2765/146*e^8 + 17937/146*e^6 - 96815/292*e^4 + 94863/292*e^2 - 6093/146, 259/292*e^10 - 2369/146*e^8 + 15115/146*e^6 - 77853/292*e^4 + 67393/292*e^2 - 3355/146, -153/146*e^11 + 1526/73*e^9 - 11104/73*e^7 + 70807/146*e^5 - 88577/146*e^3 + 8170/73*e, -81/146*e^10 + 795/73*e^8 - 5578/73*e^6 + 32857/146*e^4 - 34269/146*e^2 + 963/73, -67/146*e^10 + 590/73*e^8 - 3664/73*e^6 + 18993/146*e^4 - 18299/146*e^2 + 1705/73, 187/292*e^10 - 1711/146*e^8 + 11049/146*e^6 - 59029/292*e^4 + 54841/292*e^2 - 2499/146, 49/292*e^10 - 535/146*e^8 + 4363/146*e^6 - 31223/292*e^4 + 41879/292*e^2 - 3973/146, 289/146*e^11 - 2850/73*e^9 + 20447/73*e^7 - 127939/146*e^5 + 155843/146*e^3 - 13956/73*e, -8/73*e^11 + 276/73*e^9 - 3053/73*e^7 + 13585/73*e^5 - 21348/73*e^3 + 2145/73*e, 52/73*e^10 - 918/73*e^8 + 5500/73*e^6 - 12638/73*e^4 + 8968/73*e^2 - 1788/73, 30/73*e^10 - 597/73*e^8 + 4386/73*e^6 - 14097/73*e^4 + 16764/73*e^2 - 1638/73, -46/73*e^10 + 857/73*e^8 - 5674/73*e^6 + 16009/73*e^4 - 17266/73*e^2 + 1256/73, 44/73*e^10 - 861/73*e^8 + 6170/73*e^6 - 19493/73*e^4 + 23682/73*e^2 - 2344/73, -521/584*e^11 + 4881/292*e^9 - 32559/292*e^7 + 184279/584*e^5 - 203783/584*e^3 + 29765/292*e, -1719/584*e^11 + 17115/292*e^9 - 123885/292*e^7 + 778361/584*e^5 - 934633/584*e^3 + 67595/292*e, 853/584*e^11 - 8929/292*e^9 + 68771/292*e^7 - 464123/584*e^5 + 594931/584*e^3 - 33777/292*e, -285/584*e^11 + 3073/292*e^9 - 24739/292*e^7 + 179875/584*e^5 - 269707/584*e^3 + 45193/292*e, -11/73*e^10 + 197/73*e^8 - 1214/73*e^6 + 2957/73*e^4 - 2088/73*e^2 - 728/73, 33/73*e^10 - 591/73*e^8 + 3642/73*e^6 - 8871/73*e^4 + 6848/73*e^2 - 152/73, 251/292*e^10 - 2377/146*e^8 + 16253/146*e^6 - 96461/292*e^4 + 110869/292*e^2 - 6115/146, -7/292*e^10 + 139/146*e^8 - 1833/146*e^6 + 20145/292*e^4 - 39813/292*e^2 + 3571/146, 869/584*e^11 - 8621/292*e^9 + 62115/292*e^7 - 388947/584*e^5 + 471187/584*e^3 - 41689/292*e, -53/584*e^11 + 677/292*e^9 - 6203/292*e^7 + 47323/584*e^5 - 56787/584*e^3 - 13175/292*e, -731/584*e^11 + 7299/292*e^9 - 53093/292*e^7 + 336029/584*e^5 - 408293/584*e^3 + 30899/292*e, -609/584*e^11 + 5961/292*e^9 - 42087/292*e^7 + 254071/584*e^5 - 281807/584*e^3 + 9041/292*e, -37/146*e^10 + 182/73*e^8 + 208/73*e^6 - 11967/146*e^4 + 28395/146*e^2 - 3129/73, 103/146*e^10 - 700/73*e^8 + 2120/73*e^6 + 8971/146*e^4 - 39227/146*e^2 + 1809/73, -101/292*e^10 + 921/146*e^8 - 5799/146*e^6 + 29407/292*e^4 - 26027/292*e^2 + 49/146, 49/292*e^10 - 535/146*e^8 + 4363/146*e^6 - 31515/292*e^4 + 42171/292*e^2 - 1929/146, -643/584*e^11 + 6511/292*e^9 - 48821/292*e^7 + 328141/584*e^5 - 451157/584*e^3 + 62135/292*e, -31/146*e^10 + 188/73*e^8 - 244/73*e^6 - 7501/146*e^4 + 22725/146*e^2 - 2665/73, 5/146*e^10 + 78/73*e^8 - 1788/73*e^6 + 21193/146*e^4 - 39765/146*e^2 + 2893/73, 427/584*e^11 - 4975/292*e^9 + 42901/292*e^7 - 326565/584*e^5 + 487085/584*e^3 - 59567/292*e, -17/292*e^10 + 129/146*e^8 - 885/146*e^6 + 8127/292*e^4 - 17515/292*e^2 - 755/146, 253/292*e^10 - 2229/146*e^8 + 13669/146*e^6 - 68887/292*e^4 + 64011/292*e^2 - 6885/146, 91/73*e^10 - 1643/73*e^8 + 10282/73*e^6 - 25803/73*e^4 + 21972/73*e^2 - 2910/73, 97/73*e^10 - 1777/73*e^8 + 11276/73*e^6 - 28783/73*e^4 + 25573/73*e^2 - 3442/73, -135/146*e^10 + 1252/73*e^8 - 8372/73*e^6 + 47997/146*e^4 - 50399/146*e^2 + 1897/73, -17/146*e^10 + 275/73*e^8 - 3002/73*e^6 + 26377/146*e^4 - 39561/146*e^2 + 2019/73, 86/73*e^10 - 1434/73*e^8 + 7726/73*e^6 - 13708/73*e^4 + 2607/73*e^2 - 1250/73, -175/292*e^11 + 1723/146*e^9 - 12245/146*e^7 + 75261/292*e^5 - 93629/292*e^3 + 17735/146*e, -233/292*e^10 + 1811/146*e^8 - 8557/146*e^6 + 19339/292*e^4 + 23085/292*e^2 - 5779/146, 117/292*e^10 - 759/146*e^8 + 1917/146*e^6 + 14525/292*e^4 - 52749/292*e^2 + 2843/146, -977/292*e^11 + 9827/146*e^9 - 71937/146*e^7 + 457479/292*e^5 - 556883/292*e^3 + 42097/146*e, -85/146*e^11 + 864/73*e^9 - 6323/73*e^7 + 39321/146*e^5 - 44359/146*e^3 + 1481/73*e, -79/146*e^11 + 870/73*e^9 - 7067/73*e^7 + 50649/146*e^5 - 71929/146*e^3 + 9683/73*e, -53/292*e^11 + 823/146*e^9 - 8831/146*e^7 + 79151/292*e^5 - 130955/292*e^3 + 15149/146*e, 65/292*e^11 - 811/146*e^9 + 7635/146*e^7 - 64671/292*e^5 + 110855/292*e^3 - 21521/146*e, 347/292*e^10 - 3011/146*e^8 + 18073/146*e^6 - 86909/292*e^4 + 67453/292*e^2 + 1017/146, -899/584*e^11 + 8299/292*e^9 - 54161/292*e^7 + 295661/584*e^5 - 299173/584*e^3 + 23455/292*e, -1149/584*e^11 + 11261/292*e^9 - 80247/292*e^7 + 498035/584*e^5 - 599035/584*e^3 + 52545/292*e, -195/292*e^10 + 1411/146*e^8 - 5385/146*e^6 - 2211/292*e^4 + 47619/292*e^2 - 2889/146, 228/73*e^11 - 4508/73*e^9 + 32370/73*e^7 - 101049/73*e^5 + 122384/73*e^3 - 21238/73*e, 15/73*e^11 - 262/73*e^9 + 1536/73*e^7 - 3070/73*e^5 - 451/73*e^3 + 3196/73*e, -395/584*e^11 + 3839/292*e^9 - 27305/292*e^7 + 172653/584*e^5 - 225909/584*e^3 + 32647/292*e, 275/584*e^11 - 3083/292*e^9 + 25541/292*e^7 - 187221/584*e^5 + 272733/584*e^3 - 35503/292*e, -39/73*e^10 + 579/73*e^8 - 2373/73*e^6 + 609/73*e^4 + 7071/73*e^2 + 976/73, 85/146*e^10 - 718/73*e^8 + 4133/73*e^6 - 18589/146*e^4 + 12531/146*e^2 - 167/73, -133/146*e^10 + 1108/73*e^8 - 6065/73*e^6 + 23449/146*e^4 - 9219/146*e^2 - 917/73, -54/73*e^10 + 987/73*e^8 - 6099/73*e^6 + 14337/73*e^4 - 10071/73*e^2 + 700/73, -645/584*e^11 + 6217/292*e^9 - 42879/292*e^7 + 252387/584*e^5 - 288083/584*e^3 + 33997/292*e, 32/73*e^10 - 593/73*e^8 + 4109/73*e^6 - 12876/73*e^4 + 15604/73*e^2 - 1572/73, -3/73*e^10 - 79/73*e^8 + 1839/73*e^6 - 10336/73*e^4 + 16924/73*e^2 - 756/73, -599/584*e^11 + 6263/292*e^9 - 48145/292*e^7 + 325657/584*e^5 - 432001/584*e^3 + 49575/292*e, -867/584*e^11 + 8915/292*e^9 - 67181/292*e^7 + 443093/584*e^5 - 560677/584*e^3 + 37999/292*e, 29/292*e^10 + 175/146*e^8 - 5567/146*e^6 + 68257/292*e^4 - 127561/292*e^2 + 9859/146, -493/292*e^10 + 4179/146*e^8 - 23621/146*e^6 + 96691/292*e^4 - 41319/292*e^2 - 4667/146, 723/584*e^11 - 7307/292*e^9 + 54085/292*e^7 - 352301/584*e^5 + 452645/584*e^3 - 45631/292*e, 267/292*e^11 - 2361/146*e^9 + 14561/146*e^7 - 74429/292*e^5 + 75017/292*e^3 - 14757/146*e, 507/292*e^11 - 4749/146*e^9 + 31813/146*e^7 - 182533/292*e^5 + 204457/292*e^3 - 23061/146*e, 415/292*e^11 - 4257/146*e^9 + 31833/146*e^7 - 207017/292*e^5 + 259277/292*e^3 - 25017/146*e, 527/292*e^11 - 5313/146*e^9 + 38969/146*e^7 - 249017/292*e^5 + 312285/292*e^3 - 35433/146*e, -75/146*e^10 + 655/73*e^8 - 3767/73*e^6 + 15423/146*e^4 - 7965/146*e^2 + 1135/73, -103/146*e^10 + 846/73*e^8 - 4383/73*e^6 + 13659/146*e^4 + 1413/146*e^2 + 381/73, -379/584*e^11 + 4147/292*e^9 - 33377/292*e^7 + 233813/584*e^5 - 302933/584*e^3 + 11595/292*e, 135/584*e^11 - 1471/292*e^9 + 11949/292*e^7 - 85665/584*e^5 + 113033/584*e^3 - 2335/292*e, 993/584*e^11 - 10249/292*e^9 + 77691/292*e^7 - 515455/584*e^5 + 650679/584*e^3 - 39205/292*e, -207/292*e^10 + 1837/146*e^8 - 11489/146*e^6 + 60689/292*e^4 - 62221/292*e^2 + 6111/146, 31/292*e^10 - 261/146*e^8 + 1777/146*e^6 - 13377/292*e^4 + 24141/292*e^2 - 1423/146, 335/584*e^11 - 3023/292*e^9 + 18685/292*e^7 - 87665/584*e^5 + 44921/584*e^3 + 23157/292*e, 1137/584*e^11 - 11273/292*e^9 + 81735/292*e^7 - 521567/584*e^5 + 661767/584*e^3 - 75373/292*e, 375/584*e^11 - 2983/292*e^9 + 14601/292*e^7 - 36673/584*e^5 - 33759/584*e^3 + 8341/292*e, -99/73*e^10 + 1700/73*e^8 - 9612/73*e^6 + 19240/73*e^4 - 8864/73*e^2 + 2500/73, -251/584*e^11 + 2523/292*e^9 - 18297/292*e^7 + 112813/584*e^5 - 119629/584*e^3 - 11989/292*e, 1407/584*e^11 - 13923/292*e^9 + 99793/292*e^7 - 617561/584*e^5 + 723145/584*e^3 - 43543/292*e, -1721/584*e^11 + 17113/292*e^9 - 124075/292*e^7 + 785535/584*e^5 - 960775/584*e^3 + 73329/292*e, -46/73*e^10 + 711/73*e^8 - 3411/73*e^6 + 4913/73*e^4 + 108/73*e^2 + 234/73, 118/73*e^10 - 2100/73*e^8 + 13149/73*e^6 - 34322/73*e^4 + 32592/73*e^2 - 2530/73, 901/584*e^11 - 9757/292*e^9 + 77711/292*e^7 - 541691/584*e^5 + 717179/584*e^3 - 45541/292*e, -31/292*e^10 + 553/146*e^8 - 6303/146*e^6 + 57469/292*e^4 - 90425/292*e^2 + 2299/146, 63/292*e^10 - 667/146*e^8 + 4963/146*e^6 - 31217/292*e^4 + 37993/292*e^2 - 3523/146, -843/292*e^11 + 8501/146*e^9 - 62565/146*e^7 + 402557/292*e^5 - 504809/292*e^3 + 47447/146*e, 215/292*e^11 - 2267/146*e^9 + 17651/146*e^7 - 120921/292*e^5 + 156277/292*e^3 - 4227/146*e, 739/292*e^11 - 7291/146*e^9 + 52393/146*e^7 - 329101/292*e^5 + 403653/292*e^3 - 34563/146*e, -519/292*e^11 + 5467/146*e^9 - 42297/146*e^7 + 287481/292*e^5 - 380289/292*e^3 + 34543/146*e, -2483/584*e^11 + 24819/292*e^9 - 180405/292*e^7 + 1138445/584*e^5 - 1374813/584*e^3 + 101563/292*e, -665/584*e^11 + 6489/292*e^9 - 45655/292*e^7 + 275655/584*e^5 - 318823/584*e^3 + 34105/292*e, 325/584*e^11 - 3325/292*e^9 + 25327/292*e^7 - 175019/584*e^5 + 253515/584*e^3 - 37525/292*e, -4/73*e^10 + 284/73*e^8 - 3826/73*e^6 + 18290/73*e^4 - 29362/73*e^2 + 2642/73, -116/73*e^10 + 1958/73*e^8 - 11090/73*e^6 + 23352/73*e^4 - 12582/73*e^2 - 1054/73, -1529/584*e^11 + 15553/292*e^9 - 115763/292*e^7 + 755583/584*e^5 - 965847/584*e^3 + 94017/292*e, 57/73*e^10 - 1200/73*e^8 + 9224/73*e^6 - 31011/73*e^4 + 40378/73*e^2 - 5930/73, 50/73*e^10 - 703/73*e^8 + 2346/73*e^6 + 3588/73*e^4 - 19510/73*e^2 + 2234/73, -9/292*e^10 - 301/146*e^8 + 5423/146*e^6 - 55609/292*e^4 + 90265/292*e^2 - 5587/146, 213/292*e^10 - 1539/146*e^8 + 6073/146*e^6 - 1911/292*e^4 - 39225/292*e^2 - 1121/146, -81/292*e^10 + 503/146*e^8 - 1417/146*e^6 - 4081/292*e^4 + 17561/292*e^2 - 2103/146, 637/292*e^10 - 5495/146*e^8 + 32337/146*e^6 - 148063/292*e^4 + 104967/292*e^2 - 2301/146];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -w^3 + 4*w + 1}>] := -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;