/*
  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![1, 7, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^2 + 4}>;

primesArray := [
[2, 2, -w + 1],
[4, 2, -w^2 - w + 3],
[5, 5, w^3 - 5*w + 1],
[11, 11, -w^3 + 6*w - 2],
[13, 13, -w^2 + 4],
[17, 17, 2*w^3 + w^2 - 10*w - 2],
[37, 37, w^3 + w^2 - 6*w - 3],
[37, 37, -w^3 + 6*w - 4],
[41, 41, -w^3 + 5*w - 5],
[43, 43, w^3 - 4*w + 4],
[43, 43, -2*w^3 + 11*w - 2],
[47, 47, -2*w^3 - w^2 + 12*w],
[47, 47, 2*w - 1],
[61, 61, -w^3 + w^2 + 6*w - 5],
[71, 71, w^3 + w^2 - 4*w - 3],
[71, 71, 2*w^3 + 2*w^2 - 9*w - 2],
[79, 79, w^3 + w^2 - 6*w - 5],
[81, 3, -3],
[83, 83, -3*w^3 - w^2 + 16*w + 3],
[97, 97, w^3 + w^2 - 6*w + 1],
[103, 103, -2*w^3 - 2*w^2 + 7*w - 2],
[103, 103, -3*w^3 + 14*w - 12],
[107, 107, -w^3 + 3*w + 3],
[125, 5, -2*w^3 - w^2 + 8*w - 2],
[127, 127, w^3 - w^2 - 4*w + 1],
[131, 131, 2*w^3 - 10*w + 3],
[137, 137, -5*w^3 - 2*w^2 + 26*w + 2],
[137, 137, w^3 - 7*w + 1],
[139, 139, -2*w^3 - w^2 + 8*w],
[149, 149, w^3 - 2*w^2 - 7*w + 9],
[149, 149, -2*w^3 + 11*w],
[151, 151, 2*w^2 + 2*w - 9],
[151, 151, -5*w^3 - 2*w^2 + 27*w + 1],
[151, 151, w^3 - 7*w + 3],
[151, 151, 2*w^3 + 2*w^2 - 11*w - 8],
[157, 157, w^3 - 7*w - 3],
[169, 13, 6*w^3 + 4*w^2 - 32*w - 11],
[173, 173, w^3 + w^2 - 4*w - 7],
[191, 191, -3*w^3 + 14*w - 10],
[193, 193, 2*w^3 - 12*w + 3],
[197, 197, w + 4],
[197, 197, w^3 + 2*w^2 - 6*w - 6],
[197, 197, 2*w^3 + w^2 - 10*w + 2],
[197, 197, 2*w^2 - 3],
[199, 199, w^3 + 2*w^2 - 4*w - 4],
[223, 223, -w^3 - 2*w^2 + 5*w + 7],
[229, 229, 5*w^3 + 3*w^2 - 26*w - 5],
[233, 233, 2*w + 3],
[241, 241, 2*w^3 + 2*w^2 - 10*w - 3],
[251, 251, w^3 - 2*w^2 - 5*w + 7],
[263, 263, 2*w^3 + w^2 - 12*w + 2],
[269, 269, 2*w^3 + 3*w^2 - 14*w - 4],
[271, 271, 3*w^3 + 2*w^2 - 15*w - 7],
[281, 281, 2*w^2 + w - 4],
[281, 281, w^2 - 8],
[283, 283, -3*w^3 + 18*w - 2],
[283, 283, -w^3 - w^2 + 4*w - 3],
[283, 283, -w^3 + w^2 + 6*w - 3],
[283, 283, w^3 + 2*w^2 - 5*w - 3],
[293, 293, -4*w^3 + 19*w - 16],
[307, 307, -w^3 + 2*w^2 + 6*w - 8],
[311, 311, 3*w^3 + 2*w^2 - 17*w - 9],
[311, 311, w^3 + 2*w^2 - 6*w - 8],
[313, 313, -4*w^3 - 2*w^2 + 17*w - 6],
[317, 317, 2*w^3 - 9*w + 2],
[317, 317, 3*w^2 + 2*w - 10],
[331, 331, 5*w^3 + 4*w^2 - 26*w - 12],
[337, 337, 2*w^3 + w^2 - 8*w + 6],
[347, 347, 2*w^3 - 9*w + 4],
[353, 353, w^3 - 5*w - 3],
[359, 359, -w^3 + w + 3],
[367, 367, -2*w^3 + 12*w - 7],
[367, 367, 2*w^3 + 4*w^2 - 6*w - 7],
[379, 379, -w^3 - 2*w^2 + w + 3],
[379, 379, -3*w^2 - 2*w + 12],
[397, 397, -2*w^3 - w^2 + 6*w + 2],
[397, 397, -w^3 + 3*w - 7],
[397, 397, -2*w^3 - 2*w^2 + 9*w + 6],
[397, 397, w^3 + 2*w^2 - 5*w - 5],
[401, 401, 3*w^3 + 3*w^2 - 16*w - 13],
[401, 401, w^3 + 2*w^2 - 6*w - 4],
[431, 431, 3*w^3 + 2*w^2 - 14*w - 4],
[431, 431, 2*w^3 + w^2 - 10*w + 4],
[433, 433, -2*w^3 + 2*w^2 + 13*w - 12],
[433, 433, -w^3 + 4*w^2 + 8*w - 18],
[439, 439, 2*w^3 + 2*w^2 - 8*w + 3],
[439, 439, -w^3 + 3*w^2 + 6*w - 11],
[443, 443, -7*w^3 - 2*w^2 + 39*w + 1],
[457, 457, w^3 + 2*w^2 - 3*w - 7],
[461, 461, w^2 - 2*w - 4],
[461, 461, 4*w^3 + 2*w^2 - 20*w - 5],
[463, 463, 3*w^3 + 2*w^2 - 13*w - 3],
[467, 467, -3*w^3 + 15*w - 11],
[467, 467, -4*w^3 + 18*w + 1],
[479, 479, 2*w^2 + 2*w - 11],
[487, 487, -3*w^3 - 2*w^2 + 19*w + 1],
[487, 487, 4*w^3 + 2*w^2 - 19*w],
[491, 491, 3*w^3 + 2*w^2 - 18*w],
[499, 499, -w^3 + 3*w - 5],
[499, 499, -6*w^3 - 2*w^2 + 31*w - 2],
[523, 523, 2*w^2 + 3*w - 10],
[541, 541, 2*w^3 + 3*w^2 - 8*w - 6],
[547, 547, -3*w^3 - w^2 + 16*w - 3],
[563, 563, -w^3 + 5*w - 7],
[563, 563, 2*w^2 + w - 2],
[571, 571, 3*w^3 + 2*w^2 - 12*w + 6],
[577, 577, -2*w^3 + w^2 + 8*w - 6],
[577, 577, -2*w^3 - 2*w^2 + 12*w + 11],
[587, 587, -5*w^3 - w^2 + 26*w - 5],
[593, 593, -w^3 + 8*w - 4],
[599, 599, -w^3 + w^2 + 4*w - 9],
[601, 601, 3*w^3 + w^2 - 14*w + 1],
[607, 607, 2*w^2 + 2*w - 3],
[607, 607, 4*w^3 - 24*w + 1],
[613, 613, -3*w^3 - 3*w^2 + 16*w + 9],
[617, 617, -2*w^3 + 13*w - 6],
[619, 619, -4*w^3 + 20*w - 1],
[631, 631, -5*w^3 - 2*w^2 + 29*w + 5],
[643, 643, -w^3 - 2*w^2 + 6*w + 10],
[647, 647, -5*w^3 - 4*w^2 + 20*w - 4],
[647, 647, -3*w^3 - 3*w^2 + 18*w + 7],
[653, 653, 4*w^3 - 20*w + 3],
[653, 653, w^3 - 8*w + 2],
[659, 659, -9*w^3 - 4*w^2 + 48*w + 4],
[659, 659, -5*w^3 - 2*w^2 + 25*w - 1],
[661, 661, 4*w^3 + 2*w^2 - 21*w],
[673, 673, -3*w^3 + w^2 + 18*w - 9],
[683, 683, 2*w^3 + 2*w^2 - 7*w - 4],
[691, 691, -5*w^3 - 3*w^2 + 28*w + 7],
[691, 691, 3*w^3 - 15*w + 5],
[701, 701, -2*w^3 + 2*w^2 + 11*w - 10],
[701, 701, -w^3 + 8*w - 10],
[709, 709, -4*w^3 - w^2 + 20*w - 4],
[709, 709, w^3 + 4*w^2 - w - 13],
[719, 719, -2*w^3 + 12*w - 9],
[739, 739, -6*w^3 - 4*w^2 + 29*w + 6],
[739, 739, -w^3 + w^2 + 8*w - 7],
[769, 769, -w^3 + w^2 + 8*w - 5],
[769, 769, 3*w^3 - 2*w^2 - 15*w + 19],
[773, 773, -4*w^3 + 22*w - 9],
[797, 797, -2*w^3 + 13*w - 2],
[827, 827, -w^3 + 6*w - 8],
[827, 827, -4*w^2 - 3*w + 14],
[839, 839, w^2 + 4*w - 6],
[857, 857, 7*w^3 + 2*w^2 - 37*w + 1],
[863, 863, 3*w^3 - 14*w + 2],
[881, 881, -3*w^3 + 2*w^2 + 15*w - 13],
[883, 883, 2*w^3 - w^2 - 10*w + 14],
[887, 887, -5*w^3 - 4*w^2 + 25*w + 11],
[887, 887, 3*w^3 - 14*w + 8],
[907, 907, 4*w^3 - 20*w + 13],
[907, 907, 2*w^2 + 4*w - 5],
[907, 907, -3*w^3 + 18*w - 4],
[907, 907, w - 6],
[947, 947, 3*w^3 + 3*w^2 - 12*w - 5],
[947, 947, 2*w^3 + 3*w^2 - 10*w - 10],
[961, 31, 2*w^3 - w^2 - 12*w + 4],
[961, 31, -w^3 + 2*w^2 + 5*w - 5],
[971, 971, -4*w^3 - w^2 + 20*w - 2],
[991, 991, -2*w^3 + 3*w^2 + 14*w - 18],
[997, 997, 7*w^3 + 2*w^2 - 36*w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^17 + 3*x^16 - 24*x^15 - 76*x^14 + 213*x^13 + 748*x^12 - 808*x^11 - 3581*x^10 + 833*x^9 + 8474*x^8 + 2163*x^7 - 8726*x^6 - 4788*x^5 + 2572*x^4 + 1996*x^3 + 32*x^2 - 128*x - 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -16605/29144*e^16 - 42915/29144*e^15 + 103871/7286*e^14 + 135987/3643*e^13 - 3964245/29144*e^12 - 2682995/7286*e^11 + 2201076/3643*e^10 + 51698949/29144*e^9 - 33917721/29144*e^8 - 62163177/14572*e^7 + 12143953/29144*e^6 + 67074477/14572*e^5 + 6984749/7286*e^4 - 12046155/7286*e^3 - 1857938/3643*e^2 + 456056/3643*e + 112470/3643, 47957/29144*e^16 + 130821/29144*e^15 - 296159/7286*e^14 - 1659327/14572*e^13 + 11067813/29144*e^12 + 16385853/14572*e^11 - 23577595/14572*e^10 - 158085065/29144*e^9 + 80378219/29144*e^8 + 190406441/14572*e^7 + 6742417/29144*e^6 - 51479560/3643*e^5 - 62714039/14572*e^4 + 37142305/7286*e^3 + 14675143/7286*e^2 - 1407090/3643*e - 431295/3643, -31787/14572*e^16 - 93157/14572*e^15 + 773221/14572*e^14 + 2369209/14572*e^13 - 7036899/14572*e^12 - 5865410/3643*e^11 + 28360497/14572*e^10 + 113516925/14572*e^9 - 39643017/14572*e^8 - 274319653/14572*e^7 - 9100246/3643*e^6 + 148884725/7286*e^5 + 116457201/14572*e^4 - 54169387/7286*e^3 - 12516747/3643*e^2 + 2157864/3643*e + 713747/3643, -1, -31303/29144*e^16 - 87549/29144*e^15 + 384753/14572*e^14 + 1111153/14572*e^13 - 7135599/29144*e^12 - 5490851/7286*e^11 + 14983657/14572*e^10 + 106054891/29144*e^9 - 48871747/29144*e^8 - 63942437/7286*e^7 - 12165391/29144*e^6 + 138495463/14572*e^5 + 44451191/14572*e^4 - 25060379/7286*e^3 - 9937897/7286*e^2 + 972295/3643*e + 274889/3643, 33637/7286*e^16 + 46292/3643*e^15 - 414462/3643*e^14 - 2348847/7286*e^13 + 7712855/7286*e^12 + 23194369/7286*e^11 - 32582847/7286*e^10 - 111860159/7286*e^9 + 27017354/3643*e^8 + 134639708/3643*e^7 + 5113989/3643*e^6 - 290702971/7286*e^5 - 93551033/7286*e^4 + 52173669/3643*e^3 + 21378510/3643*e^2 - 3929472/3643*e - 1232176/3643, -37981/14572*e^16 - 99773/14572*e^15 + 473043/7286*e^14 + 631994/3643*e^13 - 8965413/14572*e^12 - 12464039/7286*e^11 + 9830639/3643*e^10 + 120039109/14572*e^9 - 73221143/14572*e^8 - 72124182/3643*e^7 + 17855649/14572*e^6 + 77698723/3643*e^5 + 18475780/3643*e^4 - 27733522/3643*e^3 - 9306768/3643*e^2 + 2013638/3643*e + 556660/3643, 25601/14572*e^16 + 74109/14572*e^15 - 156171/3643*e^14 - 942199/7286*e^13 + 5716369/14572*e^12 + 9329687/7286*e^11 - 11660163/7286*e^10 - 90304401/14572*e^9 + 34210331/14572*e^8 + 109229985/7286*e^7 + 23929397/14572*e^6 - 59487692/3643*e^5 - 44109373/7286*e^4 + 21912564/3643*e^3 + 9655103/3643*e^2 - 1827208/3643*e - 558500/3643, -5273/3643*e^16 - 64677/14572*e^15 + 126212/3643*e^14 + 1644839/14572*e^13 - 1114926/3643*e^12 - 16286073/14572*e^11 + 16737187/14572*e^10 + 39384345/7286*e^9 - 16220221/14572*e^8 - 95092833/7286*e^7 - 48447131/14572*e^6 + 206095329/14572*e^5 + 103159773/14572*e^4 - 37438749/7286*e^3 - 10449711/3643*e^2 + 1529907/3643*e + 563437/3643, 1955/7286*e^16 + 3739/3643*e^15 - 43495/7286*e^14 - 95154/3643*e^13 + 331491/7286*e^12 + 1886395/7286*e^11 - 379616/3643*e^10 - 9135785/7286*e^9 - 1125881/3643*e^8 + 22085193/7286*e^7 + 13437611/7286*e^6 - 23973047/7286*e^5 - 9628336/3643*e^4 + 4407413/3643*e^3 + 3724148/3643*e^2 - 399504/3643*e - 219788/3643, 19533/3643*e^16 + 53839/3643*e^15 - 481614/3643*e^14 - 1366411/3643*e^13 + 4485612/3643*e^12 + 13498989/3643*e^11 - 18990177/3643*e^10 - 65136082/3643*e^9 + 31724721/3643*e^8 + 156902348/3643*e^7 + 5088840/3643*e^6 - 169533453/3643*e^5 - 53444290/3643*e^4 + 60933065/3643*e^3 + 24599597/3643*e^2 - 4572999/3643*e - 1414878/3643, -41873/14572*e^16 - 113557/14572*e^15 + 518313/7286*e^14 + 720201/3643*e^13 - 9724917/14572*e^12 - 14225267/7286*e^11 + 10452976/3643*e^10 + 137259073/14572*e^9 - 73427047/14572*e^8 - 82673271/3643*e^7 + 2281477/14572*e^6 + 89416187/3643*e^5 + 25114203/3643*e^4 - 32242735/3643*e^3 - 12031725/3643*e^2 + 2412406/3643*e + 712348/3643, -42821/7286*e^16 - 119423/7286*e^15 + 1052453/7286*e^14 + 3031611/7286*e^13 - 9748911/7286*e^12 - 14978881/3643*e^11 + 40802717/7286*e^10 + 144597343/7286*e^9 - 65597163/7286*e^8 - 348432497/7286*e^7 - 10266177/3643*e^6 + 188338998/3643*e^5 + 127071661/7286*e^4 - 67800230/3643*e^3 - 28649407/3643*e^2 + 5130632/3643*e + 1645878/3643, -5567/3643*e^16 - 27515/7286*e^15 + 280963/7286*e^14 + 348078/3643*e^13 - 1359900/3643*e^12 - 6856707/7286*e^11 + 6212744/3643*e^10 + 16500998/3643*e^9 - 25738127/7286*e^8 - 79409955/7286*e^7 + 7639967/3643*e^6 + 86027905/7286*e^5 + 5660530/3643*e^4 - 15678047/3643*e^3 - 3822227/3643*e^2 + 1241622/3643*e + 257056/3643, 193947/29144*e^16 + 549131/29144*e^15 - 594066/3643*e^14 - 6975223/14572*e^13 + 43791323/29144*e^12 + 68984847/14572*e^11 - 90605879/14572*e^10 - 666553047/29144*e^9 + 279821533/29144*e^8 + 803949645/14572*e^7 + 134451099/29144*e^6 - 217619210/3643*e^5 - 310116899/14572*e^4 + 157371651/7286*e^3 + 68909047/7286*e^2 - 6104794/3643*e - 1983087/3643, 18103/7286*e^16 + 89443/14572*e^15 - 457225/7286*e^14 - 2262177/14572*e^13 + 4431803/7286*e^12 + 22259037/14572*e^11 - 40595873/14572*e^10 - 26726418/3643*e^9 + 84692119/14572*e^8 + 64014253/3643*e^7 - 52574669/14572*e^6 - 274269025/14572*e^5 - 32563055/14572*e^4 + 48091023/7286*e^3 + 5644213/3643*e^2 - 1584451/3643*e - 384667/3643, 25983/14572*e^16 + 16751/3643*e^15 - 324865/7286*e^14 - 1696543/14572*e^13 + 6193771/14572*e^12 + 16707471/14572*e^11 - 27470195/14572*e^10 - 80261453/14572*e^9 + 26359379/7286*e^8 + 96006511/7286*e^7 - 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6251486/3643*e^12 - 18663133/3643*e^11 + 26682652/3643*e^10 + 90172713/3643*e^9 - 45782647/3643*e^8 - 217703743/3643*e^7 - 2608072/3643*e^6 + 236353422/3643*e^5 + 69340544/3643*e^4 - 86102503/3643*e^3 - 32634009/3643*e^2 + 6716692/3643*e + 1960432/3643, -54861/7286*e^16 - 141569/7286*e^15 + 685442/3643*e^14 + 1791506/3643*e^13 - 13060365/7286*e^12 - 17644137/3643*e^11 + 28943488/3643*e^10 + 169687217/7286*e^9 - 110951931/7286*e^8 - 203546126/3643*e^7 + 38009137/7286*e^6 + 218651249/3643*e^5 + 47047443/3643*e^4 - 77444147/3643*e^3 - 24898620/3643*e^2 + 5496736/3643*e + 1582116/3643, -51300/3643*e^16 - 138003/3643*e^15 + 2544679/7286*e^14 + 6999449/7286*e^13 - 11969771/3643*e^12 - 34542420/3643*e^11 + 103478199/7286*e^10 + 166491601/3643*e^9 - 92305004/3643*e^8 - 800972673/7286*e^7 + 16535595/7286*e^6 + 431739021/3643*e^5 + 234059135/7286*e^4 - 154122124/3643*e^3 - 56464506/3643*e^2 + 11100792/3643*e + 3356762/3643, -9831/7286*e^16 - 13665/14572*e^15 + 141062/3643*e^14 + 322081/14572*e^13 - 3246667/7286*e^12 - 2905635/14572*e^11 + 38254489/14572*e^10 + 3150184/3643*e^9 - 122231693/14572*e^8 - 13526351/7286*e^7 + 207135657/14572*e^6 + 25919915/14572*e^5 - 168679585/14572*e^4 - 3363715/7286*e^3 + 12498369/3643*e^2 - 399447/3643*e - 574273/3643, 142927/14572*e^16 + 387667/14572*e^15 - 884829/3643*e^14 - 4921249/7286*e^13 + 33199283/14572*e^12 + 48637387/7286*e^11 - 71302545/7286*e^10 - 469617331/14572*e^9 + 249529017/14572*e^8 + 566091127/7286*e^7 - 4639105/14572*e^6 - 306387854/3643*e^5 - 172537341/7286*e^4 + 110759262/3643*e^3 + 40853295/3643*e^2 - 8500278/3643*e - 2495876/3643, -108627/14572*e^16 - 290773/14572*e^15 + 673554/3643*e^14 + 1843094/3643*e^13 - 25340367/14572*e^12 - 18184591/3643*e^11 + 27361053/3643*e^10 + 350372247/14572*e^9 - 194686835/14572*e^8 - 420930701/7286*e^7 + 15297047/14572*e^6 + 452678205/7286*e^5 + 62608633/3643*e^4 - 80167630/3643*e^3 - 30220986/3643*e^2 + 5602540/3643*e + 1809004/3643, 58871/7286*e^16 + 161417/7286*e^15 - 727981/3643*e^14 - 2049693/3643*e^13 + 13629131/7286*e^12 + 20262739/3643*e^11 - 29155190/3643*e^10 - 195687307/7286*e^9 + 100829819/7286*e^8 + 235906495/3643*e^7 + 2504007/7286*e^6 - 255307869/3643*e^5 - 73316819/3643*e^4 + 92200544/3643*e^3 + 34451233/3643*e^2 - 7050332/3643*e - 2067650/3643, -17552/3643*e^16 - 46692/3643*e^15 + 435556/3643*e^14 + 1182919/3643*e^13 - 4101677/3643*e^12 - 11667112/3643*e^11 + 17764474/3643*e^10 + 56229141/3643*e^9 - 31870236/3643*e^8 - 135447272/3643*e^7 + 3504656/3643*e^6 + 146883810/3643*e^5 + 39246179/3643*e^4 - 53784340/3643*e^3 - 18938232/3643*e^2 + 4421026/3643*e + 1123062/3643, -188751/14572*e^16 - 543541/14572*e^15 + 1150455/3643*e^14 + 3452500/3643*e^13 - 42029131/14572*e^12 - 34149241/3643*e^11 + 42645014/3643*e^10 + 659976523/14572*e^9 - 244646155/14572*e^8 - 795986747/7286*e^7 - 196974329/14572*e^6 + 861562921/7286*e^5 + 168479066/3643*e^4 - 155567766/3643*e^3 - 72955381/3643*e^2 + 11970414/3643*e + 4112372/3643, 349527/14572*e^16 + 942593/14572*e^15 - 4330097/7286*e^14 - 11956611/7286*e^13 + 81318547/14572*e^12 + 59037547/3643*e^11 - 175016991/7286*e^10 - 1139141483/14572*e^9 + 616103639/14572*e^8 + 686004337/3643*e^7 - 23180733/14572*e^6 - 1483779461/7286*e^5 - 418249439/7286*e^4 + 267714506/3643*e^3 + 100226485/3643*e^2 - 20365866/3643*e - 6021338/3643, 142553/29144*e^16 + 378697/29144*e^15 - 886471/7286*e^14 - 4799331/14572*e^13 + 33533873/29144*e^12 + 47338865/14572*e^11 - 73286919/14572*e^10 - 455909805/29144*e^9 + 270572263/29144*e^8 + 547427721/14572*e^7 - 58282611/29144*e^6 - 147002511/3643*e^5 - 142684023/14572*e^4 + 103861253/7286*e^3 + 35477525/7286*e^2 - 3630064/3643*e - 1019575/3643, 30815/29144*e^16 + 110015/29144*e^15 - 176567/7286*e^14 - 1401545/14572*e^13 + 5749975/29144*e^12 + 13915367/14572*e^11 - 8588905/14572*e^10 - 135120275/29144*e^9 - 10019791/29144*e^8 + 164005195/14572*e^7 + 148700547/29144*e^6 - 44848248/3643*e^5 - 115594373/14572*e^4 + 33443463/7286*e^3 + 22006921/7286*e^2 - 1554508/3643*e - 660789/3643, 44373/7286*e^16 + 61462/3643*e^15 - 1094473/7286*e^14 - 1559958/3643*e^13 + 10203771/7286*e^12 + 30820327/7286*e^11 - 21658554/3643*e^10 - 148664515/7286*e^9 + 36563386/3643*e^8 + 357714297/7286*e^7 + 8537913/7286*e^6 - 385192823/7286*e^5 - 58845969/3643*e^4 + 68295641/3643*e^3 + 27199435/3643*e^2 - 4869774/3643*e - 1517176/3643, 49597/3643*e^16 + 136004/3643*e^15 - 1224682/3643*e^14 - 3452546/3643*e^13 + 11434517/3643*e^12 + 34120621/3643*e^11 - 48657112/3643*e^10 - 164744941/3643*e^9 + 82646821/3643*e^8 + 397358116/3643*e^7 + 7953286/3643*e^6 - 430760474/3643*e^5 - 130575494/3643*e^4 + 156633880/3643*e^3 + 61032683/3643*e^2 - 12346650/3643*e - 3614244/3643, 41428/3643*e^16 + 120203/3643*e^15 - 4041375/14572*e^14 - 12224403/14572*e^13 + 9235614/3643*e^12 + 30251225/3643*e^11 - 150327899/14572*e^10 - 292572509/7286*e^9 + 109174703/7286*e^8 + 1412912821/14572*e^7 + 161950037/14572*e^6 - 382908572/3643*e^5 - 576518183/14572*e^4 + 277754669/7286*e^3 + 62327920/3643*e^2 - 11022708/3643*e - 3527437/3643, -516115/29144*e^16 - 1439155/29144*e^15 + 1585985/3643*e^14 + 18267147/14572*e^13 - 117570899/29144*e^12 - 180517199/14572*e^11 + 246226475/14572*e^10 + 1742704375/29144*e^9 - 793919709/29144*e^8 - 2099962381/14572*e^7 - 238477723/29144*e^6 + 567790524/3643*e^5 + 759168275/14572*e^4 - 409645495/7286*e^3 - 170564679/7286*e^2 + 15747412/3643*e + 4905491/3643, -73201/7286*e^16 - 102986/3643*e^15 + 899184/3643*e^14 + 5233589/7286*e^13 - 16643883/7286*e^12 - 51765473/7286*e^11 + 69530947/7286*e^10 + 250080355/7286*e^9 - 55563632/3643*e^8 - 301545221/3643*e^7 - 18560218/3643*e^6 + 652405515/7286*e^5 + 218406619/7286*e^4 - 117595269/3643*e^3 - 48764974/3643*e^2 + 9015572/3643*e + 2759786/3643, 11797/7286*e^16 + 17734/3643*e^15 - 561491/14572*e^14 - 1798797/14572*e^13 + 2450559/7286*e^12 + 8887275/7286*e^11 - 17762929/14572*e^10 - 21488547/3643*e^9 + 6407097/7286*e^8 + 208396433/14572*e^7 + 69502593/14572*e^6 - 114751267/7286*e^5 - 136226937/14572*e^4 + 44380731/7286*e^3 + 14334966/3643*e^2 - 2277022/3643*e - 868571/3643, 356505/14572*e^16 + 991351/14572*e^15 - 4386375/7286*e^14 - 12582665/7286*e^13 + 81437657/14572*e^12 + 62166648/3643*e^11 - 171228683/7286*e^10 - 1200121909/14572*e^9 + 559720909/14572*e^8 + 722822936/3643*e^7 + 137439293/14572*e^6 - 1562069221/7286*e^5 - 510524079/7286*e^4 + 280817315/3643*e^3 + 116043317/3643*e^2 - 21246212/3643*e - 6705538/3643, -134457/14572*e^16 - 391531/14572*e^15 + 1634237/7286*e^14 + 4975561/7286*e^13 - 29695185/14572*e^12 - 24620037/3643*e^11 + 59546985/7286*e^10 + 476238045/14572*e^9 - 162517261/14572*e^8 - 287684656/3643*e^7 - 169077997/14572*e^6 + 625383247/7286*e^5 + 256070423/7286*e^4 - 114675426/3643*e^3 - 55035103/3643*e^2 + 9543956/3643*e + 3147378/3643, 731053/29144*e^16 + 1991337/29144*e^15 - 2258954/3643*e^14 - 25260455/14572*e^13 + 169071485/29144*e^12 + 249466511/14572*e^11 - 361300091/14572*e^10 - 2406833529/29144*e^9 + 1245067871/29144*e^8 + 2898624139/14572*e^7 + 50392897/29144*e^6 - 1566634845/7286*e^5 - 924436503/14572*e^4 + 564060133/7286*e^3 + 217290865/7286*e^2 - 21198170/3643*e - 6418307/3643, 2867/3643*e^16 + 18286/3643*e^15 - 100057/7286*e^14 - 946307/7286*e^13 + 136870/3643*e^12 + 4776386/3643*e^11 + 4656475/7286*e^10 - 23609491/3643*e^9 - 19810398/3643*e^8 + 116944507/7286*e^7 + 118100459/7286*e^6 - 65679079/3643*e^5 - 139646117/7286*e^4 + 25967341/3643*e^3 + 24736028/3643*e^2 - 2818244/3643*e - 1148742/3643, 28557/14572*e^16 + 99809/14572*e^15 - 329693/7286*e^14 - 637044/3643*e^13 + 5433065/14572*e^12 + 12669691/7286*e^11 - 4194077/3643*e^10 - 123136181/14572*e^9 - 5896693/14572*e^8 + 74720394/3643*e^7 + 131130035/14572*e^6 - 81585732/3643*e^5 - 51996037/3643*e^4 + 30385682/3643*e^3 + 19675523/3643*e^2 - 2923590/3643*e - 965700/3643, 51925/7286*e^16 + 128505/7286*e^15 - 2615497/14572*e^14 - 6506981/14572*e^13 + 12604969/7286*e^12 + 16028722/3643*e^11 - 114081473/14572*e^10 - 77151418/3643*e^9 + 57583587/3643*e^8 + 742288127/14572*e^7 - 117650863/14572*e^6 - 200823661/3643*e^5 - 132020573/14572*e^4 + 145894205/7286*e^3 + 19749368/3643*e^2 - 5527028/3643*e - 1235197/3643, 102382/3643*e^16 + 279622/3643*e^15 - 5056747/7286*e^14 - 14183535/7286*e^13 + 23616115/3643*e^12 + 70007696/3643*e^11 - 201201861/7286*e^10 - 337520167/3643*e^9 + 171543760/3643*e^8 + 1624448353/7286*e^7 + 27382061/7286*e^6 - 876385407/3643*e^5 - 531358309/7286*e^4 + 313978162/3643*e^3 + 123604314/3643*e^2 - 23173016/3643*e - 7176756/3643, 367535/29144*e^16 + 1001975/29144*e^15 - 2272705/7286*e^14 - 12718509/14572*e^13 + 85120727/29144*e^12 + 125686707/14572*e^11 - 182191565/14572*e^10 - 1213398043/29144*e^9 + 631194529/29144*e^8 + 1462284951/14572*e^7 + 11437003/29144*e^6 - 395465715/3643*e^5 - 455066509/14572*e^4 + 285275063/7286*e^3 + 106482663/7286*e^2 - 10859942/3643*e - 3112041/3643, -4025/14572*e^16 - 181/14572*e^15 + 63525/7286*e^14 + 1306/3643*e^13 - 1617661/14572*e^12 - 27695/7286*e^11 + 2663156/3643*e^10 + 314101/14572*e^9 - 38603215/14572*e^8 - 299034/3643*e^7 + 75339941/14572*e^6 + 739167/3643*e^5 - 17707775/3643*e^4 - 910222/3643*e^3 + 5883485/3643*e^2 + 488400/3643*e - 307340/3643, -41857/29144*e^16 - 196709/29144*e^15 + 216027/7286*e^14 + 2527833/14572*e^13 - 5375969/29144*e^12 - 25327971/14572*e^11 - 422183/14572*e^10 + 248323085/29144*e^9 + 130696773/29144*e^8 - 304650621/14572*e^7 - 484698969/29144*e^6 + 169170017/7286*e^5 + 309746557/14572*e^4 - 65799441/7286*e^3 - 56714247/7286*e^2 + 3535372/3643*e + 1510041/3643];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^2 + 4}>] := 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;