/*
  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, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w^2 - w + 4}>;

primesArray := [
[3, 3, w],
[3, 3, w + 1],
[3, 3, w + 2],
[8, 2, 2],
[11, 11, -w^2 + 5],
[13, 13, w^2 - w - 7],
[17, 17, -w^2 + w + 4],
[19, 19, -w^2 - w + 4],
[29, 29, 2*w^2 - 2*w - 11],
[31, 31, w^2 - 2*w - 4],
[37, 37, -2*w^2 + 3*w + 7],
[41, 41, w^2 - 2],
[59, 59, 2*w^2 - 13],
[61, 61, w^2 + w - 10],
[67, 67, 2*w^2 - w - 11],
[73, 73, -w^2 - 1],
[83, 83, w^2 - w - 10],
[89, 89, -w^2 + 4*w - 2],
[97, 97, w^2 - 2*w - 7],
[97, 97, -w^2 - 4*w - 5],
[97, 97, 2*w^2 + w - 8],
[101, 101, -4*w^2 + 2*w + 25],
[103, 103, w^2 + w - 7],
[107, 107, -3*w - 4],
[109, 109, 5*w^2 - 3*w - 31],
[121, 11, 2*w^2 - 3*w - 4],
[125, 5, -5],
[137, 137, -5*w^2 + 3*w + 34],
[139, 139, w^2 + 2*w - 4],
[151, 151, 3*w^2 - 23],
[151, 151, w^2 - 3*w - 5],
[151, 151, 3*w^2 - 3*w - 17],
[157, 157, 4*w^2 - 9*w - 8],
[157, 157, w^2 + 3*w - 2],
[157, 157, 2*w^2 - w - 2],
[167, 167, 2*w^2 + w - 11],
[167, 167, -w^2 + 11],
[167, 167, -3*w^2 + 19],
[169, 13, w^2 - 4*w - 4],
[173, 173, 2*w^2 - 3*w - 10],
[179, 179, -3*w + 7],
[191, 191, -w^2 + 5*w - 5],
[193, 193, -w^2 + w - 2],
[199, 199, 3*w - 2],
[211, 211, w^2 - 3*w - 11],
[223, 223, 3*w^2 - 20],
[223, 223, w^2 - 5*w - 4],
[223, 223, 2*w^2 - w - 8],
[229, 229, 4*w^2 - 3*w - 26],
[229, 229, 2*w^2 + w - 20],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, -6*w^2 + 3*w + 38],
[241, 241, -w^2 + 4*w - 5],
[257, 257, -2*w^2 + 5*w - 1],
[263, 263, -2*w^2 - w + 17],
[269, 269, -w^2 - w + 13],
[269, 269, 3*w - 4],
[269, 269, -4*w^2 + w + 31],
[271, 271, 2*w^2 - 7],
[271, 271, 3*w - 5],
[271, 271, w^2 - 3*w - 8],
[289, 17, 2*w^2 + w - 5],
[307, 307, 2*w^2 - w - 5],
[311, 311, 3*w^2 - 3*w - 19],
[311, 311, -7*w^2 + 13*w + 19],
[311, 311, w^2 - 2*w - 13],
[313, 313, w^2 + 2*w - 7],
[317, 317, -5*w^2 + w + 35],
[331, 331, -w^2 + 3*w - 4],
[337, 337, 3*w^2 - 3*w - 13],
[343, 7, -7],
[347, 347, 2*w^2 - 3*w - 13],
[349, 349, 3*w^2 - 3*w - 20],
[359, 359, -w^2 + 5*w - 2],
[361, 19, -2*w^2 + 7*w - 1],
[389, 389, -2*w^2 + w - 1],
[397, 397, w^2 - 6*w + 10],
[401, 401, 6*w^2 - 3*w - 44],
[419, 419, -3*w^2 + 6*w + 10],
[419, 419, 4*w^2 - 2*w - 31],
[419, 419, -4*w^2 + 4*w + 19],
[421, 421, 6*w^2 - 12*w - 13],
[431, 431, 4*w^2 - 7*w - 10],
[433, 433, w^2 - 4*w - 7],
[443, 443, 2*w^2 + 4*w - 5],
[443, 443, 5*w^2 - 11*w - 11],
[443, 443, 3*w^2 - 14],
[449, 449, -2*w^2 + 2*w - 1],
[457, 457, w^2 - 6*w - 5],
[457, 457, w^2 + 3*w - 5],
[457, 457, 6*w^2 - 3*w - 37],
[463, 463, w^2 + 5*w - 1],
[467, 467, w^2 - w - 13],
[479, 479, 5*w^2 - 9*w - 16],
[487, 487, -6*w - 5],
[499, 499, 3*w^2 - 3*w - 4],
[499, 499, 3*w^2 + 3*w - 1],
[499, 499, w^2 - 5*w + 8],
[503, 503, 4*w^2 - 5*w - 16],
[509, 509, 2*w^2 - 5*w - 8],
[521, 521, 4*w^2 - w - 22],
[523, 523, w^2 - 5*w - 16],
[547, 547, 5*w^2 - 2*w - 38],
[547, 547, 8*w^2 - 3*w - 58],
[547, 547, 4*w^2 - 29],
[557, 557, 7*w^2 - 4*w - 43],
[557, 557, 3*w - 11],
[557, 557, 2*w^2 - 4*w - 11],
[569, 569, 5*w^2 - 4*w - 32],
[569, 569, 3*w^2 - 3*w - 11],
[569, 569, w^2 + 2*w - 16],
[571, 571, w^2 - 4*w - 10],
[593, 593, 2*w^2 - 6*w - 7],
[601, 601, -w^2 - 3*w + 14],
[607, 607, 3*w^2 + 3*w - 7],
[619, 619, 3*w^2 - 6*w - 11],
[631, 631, -w^2 - w - 5],
[641, 641, 3*w^2 - 3*w - 5],
[647, 647, -w^2 + 6*w - 1],
[653, 653, -5*w^2 + 4*w + 35],
[661, 661, 3*w^2 - 3*w - 10],
[673, 673, w^2 + 4*w - 4],
[677, 677, 5*w^2 - 6*w - 25],
[677, 677, w^2 - 14],
[677, 677, 4*w^2 - 3*w - 20],
[691, 691, 5*w^2 - 2*w - 29],
[701, 701, 5*w^2 - 10*w - 14],
[719, 719, -3*w - 11],
[727, 727, -2*w^2 + 9*w - 8],
[733, 733, -w^2 - 4*w - 8],
[739, 739, w^2 + 3*w - 8],
[743, 743, 2*w^2 - 4*w - 17],
[757, 757, -w^2 + w - 5],
[769, 769, 3*w^2 - 3*w - 7],
[769, 769, -w^2 - 3*w - 7],
[769, 769, 5*w^2 - 3*w - 28],
[773, 773, 7*w^2 - 6*w - 38],
[787, 787, 2*w^2 - w - 20],
[797, 797, -5*w^2 + 14*w - 1],
[821, 821, -w^2 + 6*w - 4],
[823, 823, 7*w^2 - 6*w - 41],
[839, 839, 4*w^2 - 6*w - 11],
[841, 29, 5*w^2 - 4*w - 26],
[857, 857, 3*w^2 - 11],
[863, 863, 2*w^2 + 2*w - 17],
[881, 881, 5*w^2 - 5*w - 29],
[883, 883, -4*w^2 + 8*w + 13],
[907, 907, 4*w^2 - 9*w - 11],
[911, 911, 2*w^2 + 3*w - 10],
[947, 947, 2*w^2 + 5*w - 5],
[953, 953, 2*w^2 - 5*w - 11],
[961, 31, w^2 - 5*w - 13],
[967, 967, 2*w^2 - 3*w - 22],
[967, 967, 5*w^2 - 9*w - 10],
[967, 967, 3*w^2 - 10],
[977, 977, 5*w^2 - 37],
[983, 983, 7*w^2 - 5*w - 40],
[997, 997, -7*w^2 + 5*w + 49]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^16 - 38*x^14 + 570*x^12 - 4330*x^10 + 17802*x^8 - 39556*x^6 + 46044*x^4 - 26272*x^2 + 5776;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-2383/83596*e^14 + 88451/83596*e^12 - 638999/41798*e^10 + 2279976/20899*e^8 - 8393043/20899*e^6 + 30300539/41798*e^4 - 11818090/20899*e^2 + 3177061/20899, e, 1469529/12706592*e^15 - 717255/167192*e^13 + 20739811/334384*e^11 - 2822284403/6353296*e^9 + 10497720375/6353296*e^7 - 4856516049/1588324*e^5 + 7978362867/3176648*e^3 - 1153443205/1588324*e, -2383/83596*e^14 + 88451/83596*e^12 - 638999/41798*e^10 + 2279976/20899*e^8 - 8393043/20899*e^6 + 30300539/41798*e^4 - 11818090/20899*e^2 + 3156162/20899, 56599/12706592*e^15 - 29191/167192*e^13 + 908053/334384*e^11 - 136320613/6353296*e^9 + 580760993/6353296*e^7 - 326035241/1588324*e^5 + 697044285/3176648*e^3 - 131840835/1588324*e, 2077463/12706592*e^15 - 1012613/167192*e^13 + 29228309/334384*e^11 - 3967938869/6353296*e^9 + 14709940665/6353296*e^7 - 6771366171/1588324*e^5 + 11040937269/3176648*e^3 - 1580258035/1588324*e, -255633/3176648*e^15 + 62364/20899*e^13 - 3605457/83596*e^11 + 490586225/1588324*e^9 - 1825942167/1588324*e^7 + 847142287/397081*e^5 - 1404114533/794162*e^3 + 206647228/397081*e, 1, 2362103/12706592*e^15 - 1149607/167192*e^13 + 33111549/334384*e^11 - 4481013709/6353296*e^9 + 16529895441/6353296*e^7 - 7542582645/1588324*e^5 + 12088041725/3176648*e^3 - 1681021739/1588324*e, 689689/6353296*e^15 - 336335/83596*e^13 + 9714711/167192*e^11 - 1320222079/3176648*e^9 + 4903248391/3176648*e^7 - 2265670901/794162*e^5 + 3729105827/1588324*e^3 - 546340169/794162*e, 35345/3176648*e^15 - 8829/20899*e^13 + 528511/83596*e^11 - 75786853/1588324*e^9 + 305806907/1588324*e^7 - 160640415/397081*e^5 + 315649835/794162*e^3 - 54309623/397081*e, 307687/12706592*e^15 - 150791/167192*e^13 + 4388957/334384*e^11 - 603875821/6353296*e^9 + 2290253169/6353296*e^7 - 1098112709/1588324*e^5 + 1917865165/3176648*e^3 - 290975915/1588324*e, -11533/83596*e^14 + 213411/41798*e^12 - 3075671/41798*e^10 + 21928215/41798*e^8 - 80984315/41798*e^6 + 73984492/20899*e^4 - 59278825/20899*e^2 + 16394892/20899, -127283/334384*e^14 + 1179425/83596*e^12 - 34059207/167192*e^10 + 243387753/167192*e^8 - 901513477/167192*e^6 + 413535145/41798*e^4 - 667435953/83596*e^2 + 93835003/41798, -3427159/6353296*e^15 + 417776/20899*e^13 - 48259645/167192*e^11 + 6556211577/3176648*e^9 - 24328731965/3176648*e^7 + 5605936520/397081*e^5 - 18286799729/1588324*e^3 + 2615630243/794162*e, -6767/83596*e^14 + 62480/20899*e^12 - 1797673/41798*e^10 + 12808311/41798*e^8 - 47412143/41798*e^6 + 43683953/20899*e^4 - 35684443/20899*e^2 + 10416952/20899, -132927/334384*e^14 + 1231561/83596*e^12 - 35563975/167192*e^10 + 254218365/167192*e^8 - 942734297/167192*e^6 + 433906537/41798*e^4 - 706290329/83596*e^2 + 100889251/41798, -9841/41798*e^14 + 363607/41798*e^12 - 2615096/20899*e^10 + 18606314/20899*e^8 - 68607519/20899*e^6 + 125453971/20899*e^4 - 101405260/20899*e^2 + 28870464/20899, 302079/1588324*e^15 - 587859/83596*e^13 + 4229323/41798*e^11 - 285640662/397081*e^9 + 1049445459/397081*e^7 - 3797945209/794162*e^5 + 1495815780/397081*e^3 - 406684372/397081*e, 104013/334384*e^14 - 962367/83596*e^12 + 27748729/167192*e^10 - 198058143/167192*e^8 + 733647515/167192*e^6 - 337718525/41798*e^4 + 551954875/83596*e^2 - 79566617/41798, -609087/12706592*e^15 + 295963/167192*e^13 - 8506181/334384*e^11 + 1147925029/6353296*e^9 - 4220279561/6353296*e^7 + 1919761253/1588324*e^5 - 3077660325/3176648*e^3 + 428656651/1588324*e, -91051/668768*e^15 + 839847/167192*e^13 - 24099651/334384*e^11 + 170646721/334384*e^9 - 623121973/334384*e^7 + 278732903/83596*e^5 - 428233897/167192*e^3 + 55568399/83596*e, -25729/167192*e^14 + 475117/83596*e^12 - 6830097/83596*e^10 + 48551337/83596*e^8 - 178752165/83596*e^6 + 163053981/41798*e^4 - 131725651/41798*e^2 + 18917276/20899, 198943/1588324*e^15 - 193933/41798*e^13 + 2800449/41798*e^11 - 190466643/397081*e^9 + 1419596175/794162*e^7 - 1323908109/397081*e^5 + 1112666181/397081*e^3 - 336882044/397081*e, 195195/1588324*e^15 - 378785/83596*e^13 + 2713169/41798*e^11 - 363863083/794162*e^9 + 660152718/397081*e^7 - 2333891725/794162*e^5 + 877552482/397081*e^3 - 220520279/397081*e, -514639/3176648*e^15 + 125622/20899*e^13 - 7273157/83596*e^11 + 992826337/1588324*e^9 - 3719675113/1588324*e^7 + 1747481218/397081*e^5 - 2951824025/794162*e^3 + 437397351/397081*e, -13725/41798*e^14 + 1015113/83596*e^12 - 3655008/20899*e^10 + 52104789/41798*e^8 - 192594687/41798*e^6 + 353005335/41798*e^4 - 142382205/20899*e^2 + 40029675/20899, -1200963/3176648*e^15 + 1171101/83596*e^13 - 16909217/83596*e^11 + 2297287253/1588324*e^9 - 8527712567/1588324*e^7 + 7869048383/794162*e^5 - 6435972111/794162*e^3 + 920627050/397081*e, 2547/167192*e^14 - 44967/83596*e^12 + 601379/83596*e^10 - 3768703/83596*e^8 + 10753439/83596*e^6 - 4690155/41798*e^4 - 3722025/41798*e^2 + 2086544/20899, 67723/83596*e^14 - 2508727/83596*e^12 + 9052884/20899*e^10 - 64715445/20899*e^8 + 240195958/20899*e^6 - 886684503/41798*e^4 + 362866804/20899*e^2 - 104595525/20899, -1122987/3176648*e^15 + 273609/20899*e^13 - 15790299/83596*e^11 + 2143132577/1588324*e^9 - 7944974853/1588324*e^7 + 3659061840/397081*e^5 - 5972422423/794162*e^3 + 856884710/397081*e, 4252517/12706592*e^15 - 2076035/167192*e^13 + 60056423/334384*e^11 - 8179815159/6353296*e^9 + 30479390971/6353296*e^7 - 14149281885/1588324*e^5 + 23378358327/3176648*e^3 - 3392299177/1588324*e, 9616297/12706592*e^15 - 4691233/167192*e^13 + 135580835/334384*e^11 - 18442665763/6353296*e^9 + 68598045855/6353296*e^7 - 31764068283/1588324*e^5 + 52326845107/3176648*e^3 - 7607764717/1588324*e, -21195/334384*e^14 + 196661/83596*e^12 - 5706851/167192*e^10 + 41283841/167192*e^8 - 157156589/167192*e^6 + 76353583/41798*e^4 - 137229945/83596*e^2 + 22115807/41798, -1378477/6353296*e^15 + 334857/41798*e^13 - 19229271/167192*e^11 + 2587416439/3176648*e^9 - 9440186647/3176648*e^7 + 2106331449/397081*e^5 - 6452012243/1588324*e^3 + 839911867/794162*e, 69975/334384*e^14 - 647055/83596*e^12 + 18645855/167192*e^10 - 133022701/167192*e^8 + 492700913/167192*e^6 - 226931571/41798*e^4 + 371133365/83596*e^2 - 53776415/41798, -81499/334384*e^14 + 757595/83596*e^12 - 21988651/167192*e^10 + 158453505/167192*e^8 - 595582909/167192*e^6 + 280719545/41798*e^4 - 475849577/83596*e^2 + 71444567/41798, 12584/20899*e^14 - 465508/20899*e^12 + 6705958/20899*e^10 - 47802134/20899*e^8 + 176683125/20899*e^6 - 324094886/20899*e^4 + 263005379/20899*e^2 - 75026820/20899, -3591491/6353296*e^15 + 875929/41798*e^13 - 50614473/167192*e^11 + 6880008113/3176648*e^9 - 25549243201/3176648*e^7 + 5894006889/397081*e^5 - 19271832781/1588324*e^3 + 2763955925/794162*e, 2884529/12706592*e^15 - 1408399/167192*e^13 + 40756587/334384*e^11 - 5555331923/6353296*e^9 + 20734208847/6353296*e^7 - 9659724325/1588324*e^5 + 16075064571/3176648*e^3 - 2345287421/1588324*e, -13099/41798*e^14 + 970659/83596*e^12 - 7007453/41798*e^10 + 50117535/41798*e^8 - 186158155/41798*e^6 + 344000895/41798*e^4 - 140871405/20899*e^2 + 40392941/20899, 145369/334384*e^14 - 1348063/83596*e^12 + 38995913/167192*e^10 - 279674395/167192*e^8 + 1043731647/167192*e^6 - 486283749/41798*e^4 + 808949003/83596*e^2 - 118538969/41798, -41465/334384*e^14 + 382215/83596*e^12 - 10970373/167192*e^10 + 77891667/167192*e^8 - 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329788243/83596*e^8 + 1224371771/83596*e^6 - 1131977583/41798*e^4 + 931036963/41798*e^2 - 135455734/20899, 431532/397081*e^15 - 3366715/83596*e^13 + 24307609/41798*e^11 - 1651273557/397081*e^9 + 12257425855/794162*e^7 - 22603975597/794162*e^5 + 9221132791/397081*e^3 - 2635424354/397081*e, -421189/334384*e^15 + 3905141/83596*e^13 - 112904493/167192*e^11 + 808717935/167192*e^9 - 3010112027/167192*e^7 + 1395320057/41798*e^5 - 2303184383/83596*e^3 + 336550245/41798*e, 27165663/12706592*e^15 - 13238077/167192*e^13 + 381951109/334384*e^11 - 51815812765/6353296*e^9 + 191846026161/6353296*e^7 - 88091630083/1588324*e^5 + 142890304437/3176648*e^3 - 20271831819/1588324*e, -42685/41798*e^14 + 1583729/41798*e^12 - 11454095/20899*e^10 + 82117604/20899*e^8 - 306069940/20899*e^6 + 568821845/20899*e^4 - 471462067/20899*e^2 + 137921254/20899, 451651/334384*e^14 - 4183569/83596*e^12 + 120752039/167192*e^10 - 862317745/167192*e^8 + 3191377989/167192*e^6 - 1462671003/41798*e^4 + 2359871993/83596*e^2 - 332031939/41798, -104361/83596*e^14 + 967359/20899*e^12 - 13979979/20899*e^10 + 200209785/41798*e^8 - 744924729/41798*e^6 + 690132265/20899*e^4 - 568058007/20899*e^2 + 164466670/20899, 189363/334384*e^14 - 1749981/83596*e^12 + 50349551/167192*e^10 - 357932241/167192*e^8 + 1315631813/167192*e^6 - 596056171/41798*e^4 + 941603937/83596*e^2 - 128855039/41798, -830121/1588324*e^15 + 1616471/83596*e^13 - 11641405/41798*e^11 + 787606795/397081*e^9 - 2903003276/397081*e^7 + 10573361517/794162*e^5 - 4215995391/397081*e^3 + 1165371964/397081*e, 93795/167192*e^14 - 435641/20899*e^12 + 25258343/83596*e^10 - 181645517/83596*e^8 + 680280817/83596*e^6 - 318450327/20899*e^4 + 533247253/41798*e^2 - 79146743/20899, -1604915/3176648*e^15 + 782603/41798*e^13 - 22608149/83596*e^11 + 3074207159/1588324*e^9 - 11431654017/1588324*e^7 + 5290918656/397081*e^5 - 8691952997/794162*e^3 + 1250538150/397081*e, -191199/167192*e^14 + 3544709/83596*e^12 - 51238401/83596*e^10 + 367136455/83596*e^8 - 1368248023/83596*e^6 + 1272594073/41798*e^4 - 1057072939/41798*e^2 + 155380800/20899, 4057137/3176648*e^15 - 1977757/41798*e^13 + 57095719/83596*e^11 - 7753490217/1588324*e^9 + 28762453683/1588324*e^7 - 13260003601/397081*e^5 + 21696067119/794162*e^3 - 3132866897/397081*e, -2445863/3176648*e^15 + 2383399/83596*e^13 - 34363509/83596*e^11 + 4654833605/1588324*e^9 - 17174469563/1588324*e^7 + 15647640811/794162*e^5 - 12475329835/794162*e^3 + 1727890896/397081*e, 40413/167192*e^14 - 743617/83596*e^12 + 10627539/83596*e^10 - 74791653/83596*e^8 + 270376637/83596*e^6 - 237793911/41798*e^4 + 177595589/41798*e^2 - 22232430/20899, 16121/334384*e^14 - 145495/83596*e^12 + 4059097/167192*e^10 - 27699843/167192*e^8 + 96044431/167192*e^6 - 39376247/41798*e^4 + 49618763/83596*e^2 - 2971857/41798, 102659/167192*e^14 - 474715/20899*e^12 + 27345699/83596*e^10 - 194707329/83596*e^8 + 717491041/83596*e^6 - 326526135/20899*e^4 + 520132285/41798*e^2 - 71741337/20899, -141823/6353296*e^15 + 34535/41798*e^13 - 1999613/167192*e^11 + 274843125/3176648*e^9 - 1053422725/3176648*e^7 + 261279594/397081*e^5 - 970619957/1588324*e^3 + 151250315/794162*e, -18272937/12706592*e^15 + 8906939/167192*e^13 - 257148027/334384*e^11 + 34931998531/6353296*e^9 - 129697142823/6353296*e^7 + 59896185093/1588324*e^5 - 98234840011/3176648*e^3 + 14202125613/1588324*e, -2865393/6353296*e^15 + 1398503/83596*e^13 - 40463615/167192*e^11 + 5518032399/3176648*e^9 - 20636499439/3176648*e^7 + 9667281039/794162*e^5 - 16291604255/1588324*e^3 + 2420758831/794162*e, -1/76*e^15 + 1/2*e^13 - 15/2*e^11 + 2165/38*e^9 - 8901/38*e^7 + 9870/19*e^5 - 11131/19*e^3 + 4592/19*e, -131779/167192*e^14 + 1223741/41798*e^12 - 35451133/83596*e^10 + 254553257/83596*e^8 - 950124193/83596*e^6 + 441566598/20899*e^4 - 729071749/41798*e^2 + 106039707/20899, -1920979/6353296*e^15 + 936267/83596*e^13 - 26994857/167192*e^11 + 3652424185/3176648*e^9 - 13428959221/3176648*e^7 + 6065294441/794162*e^5 - 9488213265/1588324*e^3 + 1261849499/794162*e, -7681355/12706592*e^15 + 3744993/167192*e^13 - 108174481/334384*e^11 + 14710860393/6353296*e^9 - 54743984357/6353296*e^7 + 25402251263/1588324*e^5 - 42048144113/3176648*e^3 + 6142096607/1588324*e, 376703/794162*e^15 - 1473473/83596*e^13 + 10673945/41798*e^11 - 1456033233/794162*e^9 + 5430173049/794162*e^7 - 10076667899/794162*e^5 + 4150261279/397081*e^3 - 1197080965/397081*e, 39310/20899*e^14 - 5830541/83596*e^12 + 42129129/41798*e^10 - 301573697/41798*e^8 + 1121082577/41798*e^6 - 2073691697/41798*e^4 + 852296083/20899*e^2 - 247368267/20899, -210535/167192*e^14 + 975318/20899*e^12 - 56325719/83596*e^10 + 402529057/83596*e^8 - 1491688249/83596*e^6 + 685363004/20899*e^4 - 1111067707/41798*e^2 + 157402645/20899, -3548833/6353296*e^15 + 1730689/83596*e^13 - 49995847/167192*e^11 + 6796640999/3176648*e^9 - 25260720935/3176648*e^7 + 11689607701/794162*e^5 - 19284864111/1588324*e^3 + 2816436423/794162*e, -6547713/6353296*e^15 + 3198075/83596*e^13 - 92586223/167192*e^11 + 12626884711/3176648*e^9 - 47164511039/3176648*e^7 + 22006313639/794162*e^5 - 36775456471/1588324*e^3 + 5451072195/794162*e, 48457/83596*e^14 - 1792411/83596*e^12 + 12902995/41798*e^10 - 45904855/20899*e^8 + 168854149/20899*e^6 - 611823431/41798*e^4 + 240864559/20899*e^2 - 65652429/20899, 142527/6353296*e^15 - 68805/83596*e^13 + 1976841/167192*e^11 - 270775737/3176648*e^9 + 1043583033/3176648*e^7 - 528510971/794162*e^5 + 1021549093/1588324*e^3 - 168423443/794162*e, -5505237/12706592*e^15 + 2682209/167192*e^13 - 77355135/334384*e^11 + 10484573439/6353296*e^9 - 38743522019/6353296*e^7 + 17714041495/1588324*e^5 - 28453120503/3176648*e^3 + 3932324513/1588324*e, -282727/334384*e^14 + 2617963/83596*e^12 - 75558939/167192*e^10 + 539944373/167192*e^8 - 2002934217/167192*e^6 + 923440209/41798*e^4 - 1510186905/83596*e^2 + 218394007/41798, -7429203/6353296*e^15 + 3624031/83596*e^13 - 104702345/167192*e^11 + 14228894521/3176648*e^9 - 52803480149/3176648*e^7 + 24320673903/794162*e^5 - 39598310513/1588324*e^3 + 5631291971/794162*e, 11341147/6353296*e^15 - 2766073/41798*e^13 + 159855449/167192*e^11 - 21737033705/3176648*e^9 + 80791115777/3176648*e^7 - 18673165046/397081*e^5 + 61285050061/1588324*e^3 - 8857147665/794162*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^2 - w + 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;