/*
  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 | {16, 2, 2}>;

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 - 34*x^10 + 431*x^8 - 2532*x^6 + 6946*x^4 - 7344*x^2 + 800;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-259/43520*e^11 + 3913/21760*e^9 - 81429/43520*e^7 + 10479/1360*e^5 - 223347/21760*e^3 - 5123/5440*e, e, -259/43520*e^11 + 3913/21760*e^9 - 81429/43520*e^7 + 10479/1360*e^5 - 223347/21760*e^3 - 5123/5440*e, -1, -7/136*e^10 + 103/68*e^8 - 2085/136*e^6 + 1052/17*e^4 - 5939/68*e^2 + 321/17, 473/43520*e^11 - 6411/21760*e^9 + 110063/43520*e^7 - 8873/1360*e^5 - 68311/21760*e^3 + 77881/5440*e, 473/43520*e^11 - 6411/21760*e^9 + 110063/43520*e^7 - 8873/1360*e^5 - 68311/21760*e^3 + 77881/5440*e, -11/4352*e^11 + 225/2176*e^9 - 6861/4352*e^7 + 1443/136*e^5 - 61819/2176*e^3 + 11093/544*e, -11/4352*e^11 + 225/2176*e^9 - 6861/4352*e^7 + 1443/136*e^5 - 61819/2176*e^3 + 11093/544*e, -63/4352*e^10 + 893/2176*e^8 - 17337/4352*e^6 + 2163/136*e^4 - 59503/2176*e^2 + 8193/544, -63/4352*e^10 + 893/2176*e^8 - 17337/4352*e^6 + 2163/136*e^4 - 59503/2176*e^2 + 8193/544, 39/4352*e^11 - 501/2176*e^9 + 7313/4352*e^7 - 115/136*e^5 - 51241/2176*e^3 + 24071/544*e, 39/4352*e^11 - 501/2176*e^9 + 7313/4352*e^7 - 115/136*e^5 - 51241/2176*e^3 + 24071/544*e, 601/21760*e^11 - 9227/10880*e^9 + 199791/21760*e^7 - 28681/680*e^5 + 879913/10880*e^3 - 133543/2720*e, 601/21760*e^11 - 9227/10880*e^9 + 199791/21760*e^7 - 28681/680*e^5 + 879913/10880*e^3 - 133543/2720*e, -37/1280*e^11 + 559/640*e^9 - 11907/1280*e^7 + 1677/40*e^5 - 50101/640*e^3 + 8731/160*e, -37/1280*e^11 + 559/640*e^9 - 11907/1280*e^7 + 1677/40*e^5 - 50101/640*e^3 + 8731/160*e, -137/1088*e^10 + 2011/544*e^8 - 40447/1088*e^6 + 5021/34*e^4 - 107385/544*e^2 + 1911/136, -137/1088*e^10 + 2011/544*e^8 - 40447/1088*e^6 + 5021/34*e^4 - 107385/544*e^2 + 1911/136, 263/4352*e^10 - 3797/2176*e^8 + 74033/4352*e^6 - 8667/136*e^4 + 173623/2176*e^2 - 10681/544, 407/8704*e^11 - 6149/4352*e^9 + 129825/8704*e^7 - 17691/272*e^5 + 470343/4352*e^3 - 43241/1088*e, 407/8704*e^11 - 6149/4352*e^9 + 129825/8704*e^7 - 17691/272*e^5 + 470343/4352*e^3 - 43241/1088*e, 263/4352*e^10 - 3797/2176*e^8 + 74033/4352*e^6 - 8667/136*e^4 + 173623/2176*e^2 - 10681/544, 235/4352*e^10 - 3521/2176*e^8 + 73581/4352*e^6 - 9859/136*e^4 + 247515/2176*e^2 - 12661/544, -235/8704*e^11 + 3521/4352*e^9 - 73581/8704*e^7 + 9859/272*e^5 - 251867/4352*e^3 + 21365/1088*e, -235/8704*e^11 + 3521/4352*e^9 - 73581/8704*e^7 + 9859/272*e^5 - 251867/4352*e^3 + 21365/1088*e, 235/4352*e^10 - 3521/2176*e^8 + 73581/4352*e^6 - 9859/136*e^4 + 247515/2176*e^2 - 12661/544, 1351/43520*e^11 - 20117/21760*e^9 + 412401/43520*e^7 - 53411/1360*e^5 + 1386743/21760*e^3 - 216313/5440*e, 1351/43520*e^11 - 20117/21760*e^9 + 412401/43520*e^7 - 53411/1360*e^5 + 1386743/21760*e^3 - 216313/5440*e, 83/21760*e^11 - 1401/10880*e^9 + 36933/21760*e^7 - 7723/680*e^5 + 422339/10880*e^3 - 124749/2720*e, 319/10880*e^11 - 4893/5440*e^9 + 106489/10880*e^7 - 15599/340*e^5 + 505647/5440*e^3 - 94577/1360*e, 319/10880*e^11 - 4893/5440*e^9 + 106489/10880*e^7 - 15599/340*e^5 + 505647/5440*e^3 - 94577/1360*e, 83/21760*e^11 - 1401/10880*e^9 + 36933/21760*e^7 - 7723/680*e^5 + 422339/10880*e^3 - 124749/2720*e, 215/4352*e^10 - 3013/2176*e^8 + 56161/4352*e^6 - 5999/136*e^4 + 84871/2176*e^2 + 4343/544, 215/4352*e^10 - 3013/2176*e^8 + 56161/4352*e^6 - 5999/136*e^4 + 84871/2176*e^2 + 4343/544, 15/544*e^10 - 245/272*e^8 + 5721/544*e^6 - 872/17*e^4 + 23791/272*e^2 - 513/68, 15/544*e^10 - 245/272*e^8 + 5721/544*e^6 - 872/17*e^4 + 23791/272*e^2 - 513/68, -59/4352*e^10 + 1009/2176*e^8 - 24733/4352*e^6 + 3975/136*e^4 - 111403/2176*e^2 + 5445/544, 1/1088*e^10 + 29/544*e^8 - 1849/1088*e^6 + 487/34*e^4 - 20591/544*e^2 + 1489/136, 1/1088*e^10 + 29/544*e^8 - 1849/1088*e^6 + 487/34*e^4 - 20591/544*e^2 + 1489/136, -973/43520*e^11 + 13671/21760*e^9 - 251803/43520*e^7 + 24453/1360*e^5 - 37469/21760*e^3 - 207661/5440*e, -973/43520*e^11 + 13671/21760*e^9 - 251803/43520*e^7 + 24453/1360*e^5 - 37469/21760*e^3 - 207661/5440*e, 3/544*e^10 - 49/272*e^8 + 1253/544*e^6 - 222/17*e^4 + 6227/272*e^2 + 523/68, 3/544*e^10 - 49/272*e^8 + 1253/544*e^6 - 222/17*e^4 + 6227/272*e^2 + 523/68, 65/544*e^10 - 971/272*e^8 + 19895/544*e^6 - 2515/17*e^4 + 54225/272*e^2 - 455/68, -81/5440*e^11 + 1187/2720*e^9 - 24311/5440*e^7 + 1688/85*e^5 - 119713/2720*e^3 + 31983/680*e, -81/5440*e^11 + 1187/2720*e^9 - 24311/5440*e^7 + 1688/85*e^5 - 119713/2720*e^3 + 31983/680*e, 65/544*e^10 - 971/272*e^8 + 19895/544*e^6 - 2515/17*e^4 + 54225/272*e^2 - 455/68, -787/10880*e^11 + 11449/5440*e^9 - 225797/10880*e^7 + 26567/340*e^5 - 447811/5440*e^3 - 61539/1360*e, -787/10880*e^11 + 11449/5440*e^9 - 225797/10880*e^7 + 26567/340*e^5 - 447811/5440*e^3 - 61539/1360*e, 2657/43520*e^11 - 38819/21760*e^9 + 772007/43520*e^7 - 92017/1360*e^5 + 1562161/21760*e^3 + 222529/5440*e, 2657/43520*e^11 - 38819/21760*e^9 + 772007/43520*e^7 - 92017/1360*e^5 + 1562161/21760*e^3 + 222529/5440*e, -479/5440*e^11 + 7053/2720*e^9 - 142489/5440*e^7 + 8917/85*e^5 - 393487/2720*e^3 + 14777/680*e, -83/4352*e^10 + 1401/2176*e^8 - 32581/4352*e^6 + 4595/136*e^4 - 95939/2176*e^2 - 371/544, -83/4352*e^10 + 1401/2176*e^8 - 32581/4352*e^6 + 4595/136*e^4 - 95939/2176*e^2 - 371/544, -479/5440*e^11 + 7053/2720*e^9 - 142489/5440*e^7 + 8917/85*e^5 - 393487/2720*e^3 + 14777/680*e, -607/4352*e^10 + 9053/2176*e^8 - 186521/4352*e^6 + 24195/136*e^4 - 571407/2176*e^2 + 23969/544, -607/4352*e^10 + 9053/2176*e^8 - 186521/4352*e^6 + 24195/136*e^4 - 571407/2176*e^2 + 23969/544, 163/2176*e^10 - 2345/1088*e^8 + 45685/2176*e^6 - 5347/68*e^4 + 99155/1088*e^2 + 355/272, 163/2176*e^10 - 2345/1088*e^8 + 45685/2176*e^6 - 5347/68*e^4 + 99155/1088*e^2 + 355/272, 2859/43520*e^11 - 42753/21760*e^9 + 888109/43520*e^7 - 116859/1360*e^5 + 2784027/21760*e^3 - 113717/5440*e, 2859/43520*e^11 - 42753/21760*e^9 + 888109/43520*e^7 - 116859/1360*e^5 + 2784027/21760*e^3 - 113717/5440*e, -e^2 + 34, 11/136*e^10 - 157/68*e^8 + 3053/136*e^6 - 1454/17*e^4 + 7419/68*e^2 + 25/17, -111/2176*e^10 + 1677/1088*e^8 - 35209/2176*e^6 + 4627/68*e^4 - 103647/1088*e^2 - 175/272, -111/2176*e^10 + 1677/1088*e^8 - 35209/2176*e^6 + 4627/68*e^4 - 103647/1088*e^2 - 175/272, 683/8704*e^11 - 10113/4352*e^9 + 207021/8704*e^7 - 26691/272*e^5 + 618907/4352*e^3 - 13621/1088*e, 683/8704*e^11 - 10113/4352*e^9 + 207021/8704*e^7 - 26691/272*e^5 + 618907/4352*e^3 - 13621/1088*e, -953/43520*e^11 + 14251/21760*e^9 - 288783/43520*e^7 + 34193/1360*e^5 - 449289/21760*e^3 - 205081/5440*e, -953/43520*e^11 + 14251/21760*e^9 - 288783/43520*e^7 + 34193/1360*e^5 - 449289/21760*e^3 - 205081/5440*e, -581/4352*e^10 + 8719/2176*e^8 - 180195/4352*e^6 + 23121/136*e^4 - 519253/2176*e^2 + 19707/544, -581/4352*e^10 + 8719/2176*e^8 - 180195/4352*e^6 + 23121/136*e^4 - 519253/2176*e^2 + 19707/544, 55/1088*e^10 - 853/544*e^8 + 18529/1088*e^6 - 2591/34*e^4 + 67015/544*e^2 - 1337/136, 55/1088*e^10 - 853/544*e^8 + 18529/1088*e^6 - 2591/34*e^4 + 67015/544*e^2 - 1337/136, 861/21760*e^11 - 12567/10880*e^9 + 252171/21760*e^7 - 31601/680*e^5 + 705133/10880*e^3 - 2083/2720*e, 861/21760*e^11 - 12567/10880*e^9 + 252171/21760*e^7 - 31601/680*e^5 + 705133/10880*e^3 - 2083/2720*e, 903/8704*e^11 - 13525/4352*e^9 + 283313/8704*e^7 - 38619/272*e^5 + 1080631/4352*e^3 - 150073/1088*e, 903/8704*e^11 - 13525/4352*e^9 + 283313/8704*e^7 - 38619/272*e^5 + 1080631/4352*e^3 - 150073/1088*e, -193/1088*e^10 + 2835/544*e^8 - 57127/1088*e^6 + 7125/34*e^4 - 155441/544*e^2 + 3391/136, -193/1088*e^10 + 2835/544*e^8 - 57127/1088*e^6 + 7125/34*e^4 - 155441/544*e^2 + 3391/136, -87/1088*e^10 + 1285/544*e^8 - 26273/1088*e^6 + 3395/34*e^4 - 81031/544*e^2 + 5097/136, -87/1088*e^10 + 1285/544*e^8 - 26273/1088*e^6 + 3395/34*e^4 - 81031/544*e^2 + 5097/136, -5257/43520*e^11 + 77659/21760*e^9 - 1578687/43520*e^7 + 201117/1360*e^5 - 4775641/21760*e^3 + 402231/5440*e, -1033/4352*e^10 + 15195/2176*e^8 - 304063/4352*e^6 + 36985/136*e^4 - 758233/2176*e^2 + 13079/544, -1033/4352*e^10 + 15195/2176*e^8 - 304063/4352*e^6 + 36985/136*e^4 - 758233/2176*e^2 + 13079/544, -5257/43520*e^11 + 77659/21760*e^9 - 1578687/43520*e^7 + 201117/1360*e^5 - 4775641/21760*e^3 + 402231/5440*e, -217/4352*e^10 + 2955/2176*e^8 - 52463/4352*e^6 + 5161/136*e^4 - 74153/2176*e^2 + 9543/544, -217/4352*e^10 + 2955/2176*e^8 - 52463/4352*e^6 + 5161/136*e^4 - 74153/2176*e^2 + 9543/544, -65/4352*e^10 + 835/2176*e^8 - 15815/4352*e^6 + 2345/136*e^4 - 94481/2176*e^2 + 24799/544, 565/4352*e^10 - 8095/2176*e^8 + 155379/4352*e^6 - 17585/136*e^4 + 319941/2176*e^2 - 17419/544, -5/4352*e^10 + 399/2176*e^8 - 17955/4352*e^6 + 4229/136*e^4 - 159253/2176*e^2 + 17307/544, -5/4352*e^10 + 399/2176*e^8 - 17955/4352*e^6 + 4229/136*e^4 - 159253/2176*e^2 + 17307/544, 565/4352*e^10 - 8095/2176*e^8 + 155379/4352*e^6 - 17585/136*e^4 + 319941/2176*e^2 - 17419/544, -2633/43520*e^11 + 41691/21760*e^9 - 946943/43520*e^7 + 146273/1360*e^5 - 4893849/21760*e^3 + 774999/5440*e, 205/4352*e^10 - 3303/2176*e^8 + 76827/4352*e^6 - 12025/136*e^4 + 373469/2176*e^2 - 31219/544, 205/4352*e^10 - 3303/2176*e^8 + 76827/4352*e^6 - 12025/136*e^4 + 373469/2176*e^2 - 31219/544, -2633/43520*e^11 + 41691/21760*e^9 - 946943/43520*e^7 + 146273/1360*e^5 - 4893849/21760*e^3 + 774999/5440*e, 445/8704*e^11 - 6135/4352*e^9 + 109611/8704*e^7 - 10201/272*e^5 + 38221/4352*e^3 + 44349/1088*e, 445/8704*e^11 - 6135/4352*e^9 + 109611/8704*e^7 - 10201/272*e^5 + 38221/4352*e^3 + 44349/1088*e, -2203/21760*e^11 + 32401/10880*e^9 - 654013/21760*e^7 + 82323/680*e^5 - 1932299/10880*e^3 + 155909/2720*e, -2203/21760*e^11 + 32401/10880*e^9 - 654013/21760*e^7 + 82323/680*e^5 - 1932299/10880*e^3 + 155909/2720*e, 219/4352*e^10 - 2897/2176*e^8 + 50941/4352*e^6 - 5343/136*e^4 + 111307/2176*e^2 - 23973/544, -1/16*e^11 + 15/8*e^9 - 311/16*e^7 + 163/2*e^5 - 1009/8*e^3 + 67/2*e, -1/16*e^11 + 15/8*e^9 - 311/16*e^7 + 163/2*e^5 - 1009/8*e^3 + 67/2*e, 219/4352*e^10 - 2897/2176*e^8 + 50941/4352*e^6 - 5343/136*e^4 + 111307/2176*e^2 - 23973/544, 801/10880*e^11 - 11587/5440*e^9 + 227111/10880*e^7 - 26821/340*e^5 + 509873/5440*e^3 + 20097/1360*e, 801/10880*e^11 - 11587/5440*e^9 + 227111/10880*e^7 - 26821/340*e^5 + 509873/5440*e^3 + 20097/1360*e, -33/1360*e^11 + 471/680*e^9 - 9023/1360*e^7 + 2011/85*e^5 - 13689/680*e^3 - 4121/170*e, -33/1360*e^11 + 471/680*e^9 - 9023/1360*e^7 + 2011/85*e^5 - 13689/680*e^3 - 4121/170*e, 11/4352*e^10 - 225/2176*e^8 + 6861/4352*e^6 - 1307/136*e^4 + 40059/2176*e^2 - 2389/544, -911/4352*e^10 + 13293/2176*e^8 - 266345/4352*e^6 + 33023/136*e^4 - 704831/2176*e^2 + 21201/544, -911/4352*e^10 + 13293/2176*e^8 - 266345/4352*e^6 + 33023/136*e^4 - 704831/2176*e^2 + 21201/544, 11/4352*e^10 - 225/2176*e^8 + 6861/4352*e^6 - 1307/136*e^4 + 40059/2176*e^2 - 2389/544, 37/640*e^11 - 559/320*e^9 + 11907/640*e^7 - 1677/20*e^5 + 49141/320*e^3 - 5611/80*e, 139/544*e^10 - 2089/272*e^8 + 43549/544*e^6 - 5747/17*e^4 + 138283/272*e^2 - 6549/68, 139/544*e^10 - 2089/272*e^8 + 43549/544*e^6 - 5747/17*e^4 + 138283/272*e^2 - 6549/68, 37/640*e^11 - 559/320*e^9 + 11907/640*e^7 - 1677/20*e^5 + 49141/320*e^3 - 5611/80*e, -103/2720*e^11 + 1501/1360*e^9 - 29873/2720*e^7 + 7441/170*e^5 - 92119/1360*e^3 + 11669/340*e, 183/2176*e^10 - 2853/1088*e^8 + 63105/2176*e^6 - 9207/68*e^4 + 261799/1088*e^2 - 18825/272, 183/2176*e^10 - 2853/1088*e^8 + 63105/2176*e^6 - 9207/68*e^4 + 261799/1088*e^2 - 18825/272, -103/2720*e^11 + 1501/1360*e^9 - 29873/2720*e^7 + 7441/170*e^5 - 92119/1360*e^3 + 11669/340*e, 19/640*e^11 - 313/320*e^9 + 7429/640*e^7 - 1199/20*e^5 + 42307/320*e^3 - 9117/80*e, 19/640*e^11 - 313/320*e^9 + 7429/640*e^7 - 1199/20*e^5 + 42307/320*e^3 - 9117/80*e, -2923/21760*e^11 + 44161/10880*e^9 - 932973/21760*e^7 + 127443/680*e^5 - 3426779/10880*e^3 + 348629/2720*e, -2923/21760*e^11 + 44161/10880*e^9 - 932973/21760*e^7 + 127443/680*e^5 - 3426779/10880*e^3 + 348629/2720*e, 163/2176*e^10 - 2345/1088*e^8 + 45685/2176*e^6 - 5347/68*e^4 + 105683/1088*e^2 - 5629/272, 163/2176*e^10 - 2345/1088*e^8 + 45685/2176*e^6 - 5347/68*e^4 + 105683/1088*e^2 - 5629/272, 215/4352*e^11 - 3013/2176*e^9 + 56161/4352*e^7 - 6135/136*e^5 + 117511/2176*e^3 - 15785/544*e, 215/4352*e^11 - 3013/2176*e^9 + 56161/4352*e^7 - 6135/136*e^5 + 117511/2176*e^3 - 15785/544*e, 161/2720*e^11 - 2267/1360*e^9 + 42311/2720*e^7 - 8987/170*e^5 + 66353/1360*e^3 - 923/340*e, -49/1088*e^10 + 755/544*e^8 - 16023/1088*e^6 + 2113/34*e^4 - 46401/544*e^2 - 337/136, -49/1088*e^10 + 755/544*e^8 - 16023/1088*e^6 + 2113/34*e^4 - 46401/544*e^2 - 337/136, 161/2720*e^11 - 2267/1360*e^9 + 42311/2720*e^7 - 8987/170*e^5 + 66353/1360*e^3 - 923/340*e, -347/5440*e^11 + 5169/2720*e^9 - 106397/5440*e^7 + 6991/85*e^5 - 379531/2720*e^3 + 64581/680*e, -347/5440*e^11 + 5169/2720*e^9 - 106397/5440*e^7 + 6991/85*e^5 - 379531/2720*e^3 + 64581/680*e, -29/4352*e^10 + 791/2176*e^8 - 25803/4352*e^6 + 4985/136*e^4 - 161197/2176*e^2 + 20195/544, 5443/43520*e^11 - 79881/21760*e^9 + 1615573/43520*e^7 - 206823/1360*e^5 + 5252019/21760*e^3 - 676349/5440*e, 5443/43520*e^11 - 79881/21760*e^9 + 1615573/43520*e^7 - 206823/1360*e^5 + 5252019/21760*e^3 - 676349/5440*e, -689/8704*e^11 + 9939/4352*e^9 - 193751/8704*e^7 + 22477/272*e^5 - 397441/4352*e^3 - 15985/1088*e, 47/256*e^10 - 717/128*e^8 + 15177/256*e^6 - 2019/8*e^4 + 47071/128*e^2 - 1105/32, 47/256*e^10 - 717/128*e^8 + 15177/256*e^6 - 2019/8*e^4 + 47071/128*e^2 - 1105/32, -689/8704*e^11 + 9939/4352*e^9 - 193751/8704*e^7 + 22477/272*e^5 - 397441/4352*e^3 - 15985/1088*e, -25/128*e^10 + 363/64*e^8 - 7151/128*e^6 + 849/4*e^4 - 16297/64*e^2 - 217/16, -25/128*e^10 + 363/64*e^8 - 7151/128*e^6 + 849/4*e^4 - 16297/64*e^2 - 217/16, -975/8704*e^11 + 14701/4352*e^9 - 311209/8704*e^7 + 43335/272*e^5 - 1277951/4352*e^3 + 189201/1088*e, -975/8704*e^11 + 14701/4352*e^9 - 311209/8704*e^7 + 43335/272*e^5 - 1277951/4352*e^3 + 189201/1088*e, -1631/21760*e^11 + 22877/10880*e^9 - 427801/21760*e^7 + 46591/680*e^5 - 789263/10880*e^3 + 19713/2720*e, -1631/21760*e^11 + 22877/10880*e^9 - 427801/21760*e^7 + 46591/680*e^5 - 789263/10880*e^3 + 19713/2720*e, 3/160*e^11 - 41/80*e^9 + 693/160*e^7 - 48/5*e^5 - 1261/80*e^3 + 771/20*e, 3/160*e^11 - 41/80*e^9 + 693/160*e^7 - 48/5*e^5 - 1261/80*e^3 + 771/20*e, -241/5440*e^11 + 3347/2720*e^9 - 60311/5440*e^7 + 5611/170*e^5 - 7553/2720*e^3 - 33537/680*e, -261/4352*e^10 + 3855/2176*e^8 - 79907/4352*e^6 + 10525/136*e^4 - 234389/2176*e^2 + 8219/544, -261/4352*e^10 + 3855/2176*e^8 - 79907/4352*e^6 + 10525/136*e^4 - 234389/2176*e^2 + 8219/544, -241/5440*e^11 + 3347/2720*e^9 - 60311/5440*e^7 + 5611/170*e^5 - 7553/2720*e^3 - 33537/680*e, -695/4352*e^10 + 9765/2176*e^8 - 184833/4352*e^6 + 21119/136*e^4 - 428391/2176*e^2 + 31657/544, -695/4352*e^10 + 9765/2176*e^8 - 184833/4352*e^6 + 21119/136*e^4 - 428391/2176*e^2 + 31657/544, -401/2176*e^10 + 5779/1088*e^8 - 111543/2176*e^6 + 12657/68*e^4 - 222881/1088*e^2 - 4401/272, -19/1088*e^10 + 265/544*e^8 - 4581/1088*e^6 + 301/34*e^4 + 12061/544*e^2 - 7347/136, -19/1088*e^10 + 265/544*e^8 - 4581/1088*e^6 + 301/34*e^4 + 12061/544*e^2 - 7347/136, -401/2176*e^10 + 5779/1088*e^8 - 111543/2176*e^6 + 12657/68*e^4 - 222881/1088*e^2 - 4401/272];
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;