/*
  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 | {25, 5, w^3 - 2*w^2 - 2*w + 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^16 + 6*x^15 - 12*x^14 - 128*x^13 - 31*x^12 + 1031*x^11 + 1051*x^10 - 3828*x^9 - 5696*x^8 + 6123*x^7 + 12576*x^6 - 1815*x^5 - 10952*x^4 - 3792*x^3 + 1950*x^2 + 1362*x + 213;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [13535/163*e^15 + 101017/163*e^14 - 14366/163*e^13 - 1751859/163*e^12 - 2983969/163*e^11 + 9558016/163*e^10 + 28170895/163*e^9 - 10404915/163*e^8 - 91896197/163*e^7 - 51963831/163*e^6 + 92684374/163*e^5 + 110801250/163*e^4 + 15710370/163*e^3 - 27186224/163*e^2 - 13663365/163*e - 1921675/163, e, 8940/163*e^15 + 66982/163*e^14 - 7678/163*e^13 - 1158250/163*e^12 - 2003896/163*e^11 + 6271605/163*e^10 + 18808939/163*e^9 - 6429077/163*e^8 - 61101535/163*e^7 - 35886196/163*e^6 + 60947037/163*e^5 + 75069847/163*e^4 + 11623314/163*e^3 - 18222741/163*e^2 - 9418544/163*e - 1349900/163, -132230/163*e^15 - 987868/163*e^14 + 133361/163*e^13 + 17118647/163*e^12 + 29279246/163*e^11 - 93210925/163*e^10 - 275998917/163*e^9 + 99904571/163*e^8 + 899374775/163*e^7 + 513811095/163*e^6 - 904570355/163*e^5 - 1089942912/163*e^4 - 158094776/163*e^3 + 266754446/163*e^2 + 134947239/163*e + 19056532/163, -221477/163*e^15 - 1654954/163*e^14 + 220990/163*e^13 + 28674035/163*e^12 + 49084344/163*e^11 - 156066431/163*e^10 - 462548331/163*e^9 + 166741356/163*e^8 + 1506942118/163*e^7 + 862693790/163*e^6 - 1514791881/163*e^5 - 1828137997/163*e^4 - 266372983/163*e^3 + 447208973/163*e^2 + 226530092/163*e + 32011986/163, 1, 56868/163*e^15 + 424561/163*e^14 - 59371/163*e^13 - 7360824/163*e^12 - 12555140/163*e^11 + 40131617/163*e^10 + 118468863/163*e^9 - 43449121/163*e^8 - 386308597/163*e^7 - 219256063/163*e^6 + 389220527/163*e^5 + 466650170/163*e^4 + 66735511/163*e^3 - 114381763/163*e^2 - 57632367/163*e - 8120336/163, 152586/163*e^15 + 1139730/163*e^14 - 155405/163*e^13 - 19753123/163*e^12 - 33759203/163*e^11 + 107596366/163*e^10 + 318321515/163*e^9 - 115662763/163*e^8 - 1037512612/163*e^7 - 591587559/163*e^6 + 1044120196/163*e^5 + 1256165989/163*e^4 + 181303419/163*e^3 - 307636086/163*e^2 - 155369314/163*e - 21911316/163, 47210/163*e^15 + 352347/163*e^14 - 49984/163*e^13 - 6109890/163*e^12 - 10409900/163*e^11 + 33326760/163*e^10 + 98265403/163*e^9 - 36210068/163*e^8 - 320502092/163*e^7 - 181498366/163*e^6 + 323085085/163*e^5 + 386744281/163*e^4 + 55091004/163*e^3 - 94839349/163*e^2 - 47738258/163*e - 6721929/163, -238740/163*e^15 - 1784239/163*e^14 + 236094/163*e^13 + 30909661/163*e^12 + 52947787/163*e^11 - 168172549/163*e^10 - 498817552/163*e^9 + 179161616/163*e^8 + 1624725368/163*e^7 + 931899226/163*e^6 - 1632080409/163*e^5 - 1972876997/163*e^4 - 289164361/163*e^3 + 482212795/163*e^2 + 244801526/163*e + 34660240/163, 67448/163*e^15 + 505109/163*e^14 - 59479/163*e^13 - 8736951/163*e^12 - 15090133/163*e^11 + 47345290/163*e^10 + 141730684/163*e^9 - 48843918/163*e^8 - 460632950/163*e^7 - 269528886/163*e^6 + 460048676/163*e^5 + 564923575/163*e^4 + 86609189/163*e^3 - 137326030/163*e^2 - 70717304/163*e - 10107299/163, 303011/163*e^15 + 2263461/163*e^14 - 307559/163*e^13 - 39226830/163*e^12 - 67059087/163*e^11 + 213640655/163*e^10 + 632245610/163*e^9 - 229407042/163*e^8 - 2060520433/163*e^7 - 1175757222/163*e^6 + 2073153839/163*e^5 + 2495657259/163*e^4 + 360937968/163*e^3 - 611019249/163*e^2 - 308814694/163*e - 43577330/163, 442770/163*e^15 + 3308510/163*e^14 - 441903/163*e^13 - 57323766/163*e^12 - 98125256/163*e^11 + 311999777/163*e^10 + 924686388/163*e^9 - 333337329/163*e^8 - 3012506605/163*e^7 - 1724645395/163*e^6 + 3028020874/163*e^5 + 3654625791/163*e^4 + 532845684/163*e^3 - 893878208/163*e^2 - 452956412/163*e - 64040686/163, -487040/163*e^15 - 3640362/163*e^14 + 478825/163*e^13 + 63060288/163*e^12 + 108068722/163*e^11 - 343035061/163*e^10 - 1017956236/163*e^9 + 364928656/163*e^8 + 3315373181/163*e^7 + 1903240159/163*e^6 - 3329825039/163*e^5 - 4027478027/163*e^4 - 590959908/163*e^3 + 984327535/163*e^2 + 499783610/163*e + 70755538/163, 314278/163*e^15 + 2347969/163*e^14 - 316543/163*e^13 - 40686713/163*e^12 - 69597083/163*e^11 + 221525147/163*e^10 + 656025613/163*e^9 - 237317505/163*e^8 - 2137667219/163*e^7 - 1221647921/163*e^6 + 2149822786/163*e^5 + 2591064457/163*e^4 + 376104052/163*e^3 - 634098392/163*e^2 - 320847096/163*e - 45313072/163, 533133/163*e^15 + 3985528/163*e^14 - 519443/163*e^13 - 69030227/163*e^12 - 118381349/163*e^11 + 375379345/163*e^10 + 1114810761/163*e^9 - 398239172/163*e^8 - 3630132033/163*e^7 - 2087637975/163*e^6 + 3644100979/163*e^5 + 4413791675/163*e^4 + 650288187/163*e^3 - 1078245543/163*e^2 - 548147022/163*e - 77662581/163, -281831/163*e^15 - 2105225/163*e^14 + 286297/163*e^13 + 36485116/163*e^12 + 62367501/163*e^11 - 198716955/163*e^10 - 588026855/163*e^9 + 213453861/163*e^8 + 1916443361/163*e^7 + 1093296095/163*e^6 - 1928283240/163*e^5 - 2320863902/163*e^4 - 335547251/163*e^3 + 568221385/163*e^2 + 287177619/163*e + 40527822/163, -9654/163*e^15 - 71808/163*e^14 + 11926/163*e^13 + 1248315/163*e^12 + 2097529/163*e^11 - 6853362/163*e^10 - 19900978/163*e^9 + 7816906/163*e^8 + 65135722/163*e^7 + 35655974/163*e^6 - 66248478/163*e^5 - 77317637/163*e^4 - 10181730/163*e^3 + 19131855/163*e^2 + 9411811/163*e + 1301431/163, -399099/163*e^15 - 2983911/163*e^14 + 386422/163*e^13 + 51678256/163*e^12 + 88664906/163*e^11 - 280967613/163*e^10 - 834836482/163*e^9 + 297626334/163*e^8 + 2718227099/163*e^7 + 1564629317/163*e^6 - 2728202458/163*e^5 - 3306490106/163*e^4 - 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3313430553/163*e^5 - 4024387736/163*e^4 - 597156533/163*e^3 + 982384771/163*e^2 + 500386319/163*e + 70965211/163, 302146/163*e^15 + 2259038/163*e^14 - 292760/163*e^13 - 39125388/163*e^12 - 67122768/163*e^11 + 212736039/163*e^10 + 632026787/163*e^9 - 225486954/163*e^8 - 2057986775/163*e^7 - 1184060022/163*e^6 + 2065915564/163*e^5 + 2502773744/163*e^4 + 368570500/163*e^3 - 611513940/163*e^2 - 310730752/163*e - 43992848/163, -420799/163*e^15 - 3144397/163*e^14 + 419588/163*e^13 + 54479591/163*e^12 + 93262721/163*e^11 - 296509740/163*e^10 - 878836216/163*e^9 + 316709546/163*e^8 + 2863041959/163*e^7 + 1639351171/163*e^6 - 2877494691/163*e^5 - 3473540321/163*e^4 - 506946986/163*e^3 + 849436424/163*e^2 + 430637394/163*e + 60913373/163, -1176834/163*e^15 - 8791537/163*e^14 + 1189207/163*e^13 + 152350079/163*e^12 + 260537067/163*e^11 - 829582754/163*e^10 - 2456039963/163*e^9 + 889479544/163*e^8 + 8003357455/163*e^7 + 4571444285/163*e^6 - 8049445719/163*e^5 - 9698355826/163*e^4 - 1407255105/163*e^3 + 2373410321/163*e^2 + 1200996756/163*e + 169644183/163, 574782/163*e^15 + 4293974/163*e^14 - 580224/163*e^13 - 74409295/163*e^12 - 127259440/163*e^11 + 405152581/163*e^10 + 1199603047/163*e^9 - 434211427/163*e^8 - 3908883369/163*e^7 - 2233442148/163*e^6 + 3930738285/163*e^5 + 4737462858/163*e^4 + 688516729/163*e^3 - 1159053788/163*e^2 - 586928235/163*e - 82970487/163, -855788/163*e^15 - 6397369/163*e^14 + 835623/163*e^13 + 110808075/163*e^12 + 189994789/163*e^11 - 602622821/163*e^10 - 1789324891/163*e^9 + 639819328/163*e^8 + 5826891598/163*e^7 + 3349211458/163*e^6 - 5850378235/163*e^5 - 7082910727/163*e^4 - 1041909101/163*e^3 + 1730667787/163*e^2 + 879304379/163*e + 124516185/163, 637758/163*e^15 + 4768649/163*e^14 - 614721/163*e^13 - 82582356/163*e^12 - 141735589/163*e^11 + 448908079/163*e^10 + 1334353502/163*e^9 - 474849837/163*e^8 - 4344183570/163*e^7 - 2502851252/163*e^6 + 4358771990/163*e^5 + 5286744541/163*e^4 + 781845775/163*e^3 - 1290930506/163*e^2 - 656998807/163*e - 93149625/163, -5322*e^15 - 39743*e^14 + 5484*e^13 + 688915*e^12 + 1176306*e^11 - 3754166*e^10 - 11095305*e^9 + 4049103*e^8 + 36171192*e^7 + 20580242*e^6 - 36421736*e^5 - 43747220*e^4 - 6286372*e^3 + 10718257*e^2 + 5406977*e + 762131, -1139391/163*e^15 - 8518148/163*e^14 + 1107499/163*e^13 + 147532784/163*e^12 + 253050728/163*e^11 - 802218758/163*e^10 - 2382879073/163*e^9 + 850648250/163*e^8 + 7759165401/163*e^7 + 4463430513/163*e^6 - 7788902232/163*e^5 - 9435493460/163*e^4 - 1390029971/163*e^3 + 2305170332/163*e^2 + 1171641822/163*e + 165941573/163, 404405/163*e^15 + 3021408/163*e^14 - 406638/163*e^13 - 52354872/163*e^12 - 89567009/163*e^11 + 285033489/163*e^10 + 844209903/163*e^9 - 305185543/163*e^8 - 2750678320/163*e^7 - 1572599795/163*e^6 + 2765695697/163*e^5 + 3334723507/163*e^4 + 485101292/163*e^3 - 815833726/163*e^2 - 413180459/163*e - 58394501/163, 1361020/163*e^15 + 10176305/163*e^14 - 1313996/163*e^13 - 176234132/163*e^12 - 302432842/163*e^11 + 958034998/163*e^10 + 2847328192/163*e^9 - 1013796565/163*e^8 - 9270047858/163*e^7 - 5339622697/163*e^6 + 9301405927/163*e^5 + 11280129204/163*e^4 + 1668072694/163*e^3 - 2754387814/163*e^2 - 1401892482/163*e - 198782887/163, 942379/163*e^15 + 7041327/163*e^14 - 943288/163*e^13 - 122003753/163*e^12 - 208795731/163*e^11 + 664103572/163*e^10 + 1967746536/163*e^9 - 710076558/163*e^8 - 6410955189/163*e^7 - 3668451613/163*e^6 + 6444647875/163*e^5 + 7775564890/163*e^4 + 1132832192/163*e^3 - 1901905935/163*e^2 - 963623039/163*e - 136240282/163, 1150789/163*e^15 + 8597778/163*e^14 - 1157333/163*e^13 - 148982636/163*e^12 - 254873538/163*e^11 + 811107267/163*e^10 + 2402331957/163*e^9 - 868503324/163*e^8 - 7827678099/163*e^7 - 4474906851/163*e^6 + 7871133192/163*e^5 + 9489448002/163*e^4 + 1379104036/163*e^3 - 2321880391/163*e^2 - 1175405060/163*e - 166075097/163];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {25, 5, w^3 - 2*w^2 - 2*w + 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;