/*
  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![-11, -12, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {11,11,-w^2 + 2*w + 8}>;

primesArray := [
[8, 2, 2],
[11, 11, w],
[11, 11, -w^2 + 3*w + 7],
[11, 11, w^2 - 2*w - 8],
[23, 23, w^2 - 2*w - 7],
[23, 23, w^2 - 3*w - 8],
[23, 23, w - 1],
[27, 3, 3],
[29, 29, w^2 - 2*w - 5],
[29, 29, w^2 - 3*w - 10],
[29, 29, w - 3],
[31, 31, w^2 - 2*w - 6],
[31, 31, -w^2 + 3*w + 9],
[31, 31, w - 2],
[37, 37, -w^2 + 4*w + 7],
[43, 43, 2*w^2 - 6*w - 13],
[43, 43, 2*w^2 - 4*w - 17],
[43, 43, w^2 - 2*w - 12],
[47, 47, w^2 - 4*w - 4],
[47, 47, w^2 - w - 5],
[47, 47, 2*w^2 - 5*w - 18],
[73, 73, -2*w^2 + 5*w + 14],
[73, 73, w^2 - w - 9],
[73, 73, -w^2 + 4*w + 8],
[97, 97, -2*w^2 + 5*w + 10],
[97, 97, 2*w^2 - 3*w - 15],
[97, 97, w^2 - 4*w - 12],
[101, 101, -w - 5],
[101, 101, w^2 - 3*w - 2],
[101, 101, w^2 - 2*w - 13],
[103, 103, w^2 - w - 10],
[103, 103, w^2 - 4*w - 9],
[103, 103, -2*w^2 + 5*w + 13],
[125, 5, -5],
[137, 137, 2*w - 1],
[137, 137, 2*w^2 - 6*w - 15],
[137, 137, -2*w^2 + 4*w + 15],
[149, 149, 3*w^2 - 7*w - 24],
[149, 149, -2*w^2 + 3*w + 18],
[149, 149, 3*w^2 - 8*w - 20],
[179, 179, 2*w^2 - 6*w - 21],
[179, 179, -w^2 + 4*w + 15],
[179, 179, -2*w^2 + 5*w + 7],
[191, 191, -w - 6],
[191, 191, w^2 - 3*w - 1],
[191, 191, w^2 - 2*w - 14],
[193, 193, -3*w^2 + 9*w + 20],
[193, 193, 2*w^2 - 4*w - 23],
[193, 193, -3*w^2 + 6*w + 25],
[199, 199, w^2 - 3*w - 14],
[199, 199, w^2 - 2*w - 1],
[199, 199, w - 7],
[211, 211, 2*w^2 - 3*w - 19],
[211, 211, w^2 - 5*w - 10],
[211, 211, 3*w^2 - 8*w - 19],
[223, 223, 2*w^2 - 6*w - 17],
[223, 223, 2*w^2 - 4*w - 13],
[223, 223, 2*w - 3],
[233, 233, -3*w^2 + 7*w + 23],
[233, 233, w^2 - 8],
[233, 233, -2*w^2 + 7*w + 14],
[251, 251, w^2 - 5*w - 4],
[251, 251, 3*w^2 - 8*w - 25],
[251, 251, 2*w^2 - 3*w - 13],
[269, 269, w^2 - 5*w - 12],
[269, 269, 2*w^2 - 3*w - 21],
[269, 269, 3*w^2 - 8*w - 17],
[307, 307, w^2 - 5*w - 3],
[307, 307, 2*w^2 - 3*w - 12],
[307, 307, -3*w^2 + 8*w + 26],
[343, 7, -7],
[347, 347, 2*w^2 - 3*w - 10],
[347, 347, 3*w^2 - 8*w - 28],
[347, 347, w^2 - 5*w - 1],
[359, 359, 3*w^2 - 5*w - 23],
[359, 359, 2*w^2 - 7*w - 23],
[359, 359, -3*w^2 + 7*w + 14],
[397, 397, 2*w^2 - 8*w - 7],
[397, 397, 4*w^2 - 10*w - 37],
[397, 397, 2*w^2 - 2*w - 9],
[401, 401, 2*w^2 - 7*w - 16],
[401, 401, 3*w^2 - 7*w - 21],
[401, 401, w^2 - 10],
[421, 421, 3*w^2 - 5*w - 29],
[421, 421, w^2 - 6*w - 12],
[421, 421, -4*w^2 + 11*w + 24],
[433, 433, w^2 - 6*w - 15],
[433, 433, 3*w^2 - 5*w - 32],
[433, 433, -4*w^2 + 11*w + 21],
[443, 443, 5*w^2 - 13*w - 36],
[443, 443, -2*w^2 + 9*w + 15],
[443, 443, 5*w^2 - 12*w - 40],
[467, 467, w^2 - 6*w - 13],
[467, 467, 3*w^2 - 5*w - 30],
[467, 467, -4*w^2 + 11*w + 23],
[487, 487, -3*w^2 + 10*w + 20],
[487, 487, -4*w^2 + 9*w + 32],
[487, 487, w^2 + w - 7],
[491, 491, -w - 8],
[491, 491, -w^2 + 3*w - 1],
[491, 491, w^2 - 2*w - 16],
[541, 541, 3*w^2 - 7*w - 16],
[541, 541, 2*w^2 - 7*w - 21],
[541, 541, w^2 - 15],
[547, 547, 2*w^2 - 4*w - 25],
[547, 547, 2*w^2 - 6*w - 5],
[547, 547, -2*w - 9],
[563, 563, 3*w^2 - 7*w - 18],
[563, 563, 2*w^2 - 7*w - 19],
[563, 563, w^2 - 13],
[569, 569, w^2 - 14],
[569, 569, 3*w^2 - 7*w - 17],
[569, 569, 2*w^2 - 7*w - 20],
[593, 593, w^2 - w - 18],
[593, 593, w^2 - 4*w - 17],
[593, 593, 2*w^2 - 5*w - 5],
[619, 619, -5*w^2 + 14*w + 30],
[619, 619, 4*w^2 - 7*w - 38],
[619, 619, -5*w^2 + 11*w + 42],
[643, 643, w^2 + w - 8],
[643, 643, -3*w^2 + 10*w + 21],
[643, 643, -4*w^2 + 9*w + 31],
[677, 677, -2*w^2 + 7*w + 27],
[677, 677, -3*w^2 + 7*w + 10],
[677, 677, 3*w^2 - 9*w - 31],
[709, 709, w^2 - 7*w - 14],
[709, 709, 4*w^2 - 7*w - 39],
[709, 709, 5*w^2 - 14*w - 29],
[739, 739, -5*w^2 + 14*w + 27],
[739, 739, w^2 - 7*w - 16],
[739, 739, 4*w^2 - 7*w - 41],
[751, 751, 4*w^2 - 9*w - 43],
[751, 751, 4*w^2 - 7*w - 40],
[751, 751, -3*w^2 + 10*w + 9],
[769, 769, w^2 - 3*w - 17],
[769, 769, -w^2 + 2*w - 2],
[769, 769, w - 10],
[787, 787, 3*w^2 - 6*w - 19],
[787, 787, 3*w - 5],
[787, 787, 3*w^2 - 9*w - 26],
[857, 857, 3*w^2 - 9*w - 28],
[857, 857, 3*w^2 - 6*w - 17],
[857, 857, 3*w - 7],
[859, 859, 4*w^2 - 11*w - 40],
[859, 859, 3*w^2 - 4*w - 19],
[859, 859, 5*w^2 - 13*w - 42],
[877, 877, 6*w^2 - 14*w - 49],
[877, 877, -3*w^2 + 13*w + 21],
[877, 877, -4*w^2 + 14*w + 25],
[887, 887, 2*w^2 - 5*w - 4],
[887, 887, w^2 - w - 19],
[887, 887, w^2 - 4*w - 18],
[911, 911, 2*w^2 - 2*w - 21],
[911, 911, 4*w^2 - 10*w - 25],
[911, 911, 2*w^2 - 8*w - 19],
[919, 919, 4*w^2 - 11*w - 34],
[919, 919, 3*w^2 - 5*w - 19],
[919, 919, w^2 - 6*w - 2],
[991, 991, -w - 10],
[991, 991, -w^2 + 3*w - 3],
[991, 991, w^2 - 2*w - 18]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 + 6*x^6 - 10*x^5 - 104*x^4 - 112*x^3 + 125*x^2 + 129*x - 51;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -7/2732*e^6 - 231/2732*e^5 - 703/2732*e^4 + 3603/2732*e^3 + 13373/2732*e^2 + 576/683*e - 12459/2732, -279/5464*e^6 - 1011/5464*e^5 + 5545/5464*e^4 + 17543/5464*e^3 - 16903/5464*e^2 - 6021/1366*e + 34989/5464, -1, 485/2732*e^6 + 2345/2732*e^5 - 7103/2732*e^4 - 40053/2732*e^3 - 15631/2732*e^2 + 10048/683*e + 309/2732, -399/5464*e^6 - 2239/5464*e^5 + 3641/5464*e^4 + 35987/5464*e^3 + 49209/5464*e^2 - 659/683*e - 29895/5464, 579/5464*e^6 + 2715/5464*e^5 - 8981/5464*e^4 - 45895/5464*e^3 - 9045/5464*e^2 + 4474/683*e - 24405/5464, 203/2732*e^6 + 1235/2732*e^5 - 1469/2732*e^4 - 19795/2732*e^3 - 32657/2732*e^2 + 371/683*e + 28007/2732, 171/1366*e^6 + 431/683*e^5 - 2341/1366*e^4 - 7370/683*e^3 - 8015/1366*e^2 + 17383/1366*e + 1869/683, -663/5464*e^6 - 2755/5464*e^5 + 11473/5464*e^4 + 48151/5464*e^3 - 5327/5464*e^2 - 20477/1366*e - 18555/5464, -571/5464*e^6 - 2451/5464*e^5 + 10565/5464*e^4 + 44119/5464*e^3 - 17947/5464*e^2 - 10072/683*e - 3507/5464, 143/1366*e^6 + 621/1366*e^5 - 2421/1366*e^4 - 10573/1366*e^3 + 1765/1366*e^2 + 6128/683*e - 12631/1366, -289/2732*e^6 - 1341/2732*e^5 + 4931/2732*e^4 + 23861/2732*e^3 - 3653/2732*e^2 - 10467/683*e + 9775/2732, 275/2732*e^6 + 879/2732*e^5 - 6337/2732*e^4 - 16655/2732*e^3 + 30399/2732*e^2 + 10936/683*e - 31961/2732, -677/5464*e^6 - 3217/5464*e^5 + 10067/5464*e^4 + 55357/5464*e^3 + 21419/5464*e^2 - 16593/1366*e - 32545/5464, -68/683*e^6 - 195/683*e^5 + 1562/683*e^4 + 3485/683*e^3 - 7569/683*e^2 - 7280/683*e + 11179/683, -475/2732*e^6 - 2015/2732*e^5 + 7717/2732*e^4 + 33735/2732*e^3 + 2381/2732*e^2 - 8334/683*e + 3049/2732, -671/5464*e^6 - 3019/5464*e^5 + 9889/5464*e^4 + 49927/5464*e^3 + 21665/5464*e^2 - 9281/1366*e - 12499/5464, -7/5464*e^6 - 231/5464*e^5 - 703/5464*e^4 + 3603/5464*e^3 + 16105/5464*e^2 + 971/683*e - 53439/5464, 265/5464*e^6 + 549/5464*e^5 - 6951/5464*e^4 - 10337/5464*e^3 + 43649/5464*e^2 + 8539/1366*e - 43515/5464, 241/1366*e^6 + 1123/1366*e^5 - 3507/1366*e^4 - 18669/1366*e^3 - 7877/1366*e^2 + 6392/683*e - 759/1366, -1861/5464*e^6 - 9505/5464*e^5 + 26979/5464*e^4 + 165213/5464*e^3 + 60299/5464*e^2 - 52059/1366*e - 32353/5464, 481/2732*e^6 + 2213/2732*e^5 - 7895/2732*e^4 - 39165/2732*e^3 + 597/2732*e^2 + 14963/683*e - 21251/2732, 259/5464*e^6 + 351/5464*e^5 - 6773/5464*e^4 - 4907/5464*e^3 + 37939/5464*e^2 - 139/1366*e - 36241/5464, 125/2732*e^6 + 27/2732*e^5 - 4619/2732*e^4 - 2479/2732*e^3 + 40641/2732*e^2 + 18945/1366*e - 20861/2732, -523/2732*e^6 - 2233/2732*e^5 + 9141/2732*e^4 + 40293/2732*e^3 - 7783/2732*e^2 - 38723/1366*e + 10699/2732, 827/2732*e^6 + 4069/2732*e^5 - 11785/2732*e^4 - 69533/2732*e^3 - 31661/2732*e^2 + 40211/1366*e + 13249/2732, -25/1366*e^6 - 71/683*e^5 - 169/1366*e^4 + 726/683*e^3 + 9903/1366*e^2 + 12229/1366*e - 4539/683, 331/2732*e^6 + 1361/2732*e^5 - 6177/2732*e^4 - 24989/2732*e^3 + 10839/2732*e^2 + 29731/1366*e - 15615/2732, -219/2732*e^6 - 397/2732*e^5 + 6497/2732*e^4 + 8321/2732*e^3 - 49959/2732*e^2 - 20843/1366*e + 51039/2732, 1919/5464*e^6 + 8687/5464*e^5 - 30521/5464*e^4 - 149403/5464*e^3 - 25137/5464*e^2 + 22326/683*e - 625/5464, -493/2732*e^6 - 2609/2732*e^5 + 5519/2732*e^4 + 41829/2732*e^3 + 45355/2732*e^2 + 465/683*e - 32501/2732, 41/5464*e^6 + 1353/5464*e^5 + 3337/5464*e^4 - 23445/5464*e^3 - 63887/5464*e^2 + 1533/683*e + 35897/5464, 679/5464*e^6 + 3283/5464*e^5 - 8305/5464*e^4 - 51703/5464*e^3 - 48657/5464*e^2 - 1915/1366*e + 11907/5464, -937/5464*e^6 - 3601/5464*e^5 + 15959/5464*e^4 + 58437/5464*e^3 - 5633/5464*e^2 - 5649/683*e - 13305/5464, 17/1366*e^6 - 61/683*e^5 - 1415/1366*e^4 + 845/683*e^3 + 19821/1366*e^2 + 5235/1366*e - 8142/683, 303/1366*e^6 + 560/683*e^5 - 6257/1366*e^4 - 10411/683*e^3 + 20619/1366*e^2 + 47505/1366*e - 11211/683, -621/5464*e^6 - 4101/5464*e^5 + 4763/5464*e^4 + 67513/5464*e^3 + 89283/5464*e^2 - 2063/683*e - 17565/5464, 57/5464*e^6 + 1881/5464*e^5 + 6505/5464*e^4 - 26997/5464*e^3 - 123335/5464*e^2 - 8297/683*e + 67497/5464, -603/2732*e^6 - 2141/2732*e^5 + 12425/2732*e^4 + 38929/2732*e^3 - 41115/2732*e^2 - 42925/1366*e + 46671/2732, -279/5464*e^6 - 1011/5464*e^5 + 5545/5464*e^4 + 17543/5464*e^3 - 22367/5464*e^2 - 7387/1366*e + 67773/5464, 294/683*e^6 + 1506/683*e^5 - 3941/683*e^4 - 25654/683*e^3 - 14583/683*e^2 + 25489/683*e + 4881/683, -1989/5464*e^6 - 10997/5464*e^5 + 23491/5464*e^4 + 185433/5464*e^3 + 153403/5464*e^2 - 18080/683*e - 96645/5464, -499/1366*e^6 - 2807/1366*e^5 + 5697/1366*e^4 + 47259/1366*e^3 + 42377/1366*e^2 - 15743/683*e - 21129/1366, 541/5464*e^6 + 1461/5464*e^5 - 15139/5464*e^4 - 27897/5464*e^3 + 109605/5464*e^2 + 11599/683*e - 113115/5464, -199/1366*e^6 - 1103/1366*e^5 + 2261/1366*e^4 + 18907/1366*e^3 + 17795/1366*e^2 - 9206/683*e - 18741/1366, -167/2732*e^6 - 1413/2732*e^5 + 401/2732*e^4 + 26829/2732*e^3 + 42329/2732*e^2 - 31157/1366*e - 59357/2732, 1579/5464*e^6 + 8395/5464*e^5 - 18613/5464*e^4 - 142223/5464*e^3 - 126501/5464*e^2 + 14361/683*e + 21803/5464, 527/2732*e^6 + 2365/2732*e^5 - 8349/2732*e^4 - 38449/2732*e^3 - 5713/2732*e^2 + 7037/1366*e - 21923/2732, -25/5464*e^6 + 1907/5464*e^5 + 10759/5464*e^4 - 26551/5464*e^3 - 175873/5464*e^2 - 16579/1366*e + 135035/5464, -237/1366*e^6 - 991/1366*e^5 + 4299/1366*e^4 + 16415/1366*e^3 - 8351/1366*e^2 - 4611/683*e + 30515/1366, 61/5464*e^6 + 2013/5464*e^5 + 4565/5464*e^4 - 36081/5464*e^3 - 84923/5464*e^2 + 6662/683*e + 31685/5464, 283/1366*e^6 + 1143/1366*e^5 - 4753/1366*e^4 - 18431/1366*e^3 + 2041/1366*e^2 + 6993/683*e - 21625/1366, 246/683*e^6 + 1288/683*e^5 - 3200/683*e^4 - 21828/683*e^3 - 13819/683*e^2 + 20993/683*e + 7750/683, 269/2732*e^6 + 681/2732*e^5 - 6159/2732*e^4 - 11225/2732*e^3 + 24689/2732*e^2 + 2941/683*e - 11027/2732, -51/1366*e^6 - 317/1366*e^5 + 147/1366*e^4 + 5175/1366*e^3 + 12935/1366*e^2 - 681/683*e - 20131/1366, -1527/5464*e^6 - 6679/5464*e^5 + 26177/5464*e^4 + 117019/5464*e^3 - 18895/5464*e^2 - 21379/683*e + 48113/5464, -2693/5464*e^6 - 15105/5464*e^5 + 31627/5464*e^4 + 257029/5464*e^3 + 206499/5464*e^2 - 56971/1366*e - 96457/5464, 69/683*e^6 + 1139/1366*e^5 + 2/683*e^4 - 19025/1366*e^3 - 19027/683*e^2 + 9511/1366*e + 24621/1366, 1299/5464*e^6 + 7351/5464*e^5 - 13949/5464*e^4 - 126507/5464*e^3 - 121589/5464*e^2 + 34687/1366*e + 45255/5464, -101/5464*e^6 - 601/5464*e^5 - 1557/5464*e^4 + 3981/5464*e^3 + 61427/5464*e^2 + 22849/1366*e - 53313/5464, -1265/5464*e^6 - 6229/5464*e^5 + 16583/5464*e^4 + 101201/5464*e^3 + 73807/5464*e^2 - 9189/1366*e - 57333/5464, -2105/5464*e^6 - 9361/5464*e^5 + 36039/5464*e^4 + 164741/5464*e^3 - 26201/5464*e^2 - 32529/683*e + 94983/5464, -459/1366*e^6 - 2853/1366*e^5 + 4055/1366*e^4 + 47941/1366*e^3 + 57677/1366*e^2 - 16374/683*e - 29553/1366, -465/5464*e^6 - 1685/5464*e^5 + 11063/5464*e^4 + 38345/5464*e^3 - 57313/5464*e^2 - 44185/1366*e + 20067/5464, 205/2732*e^6 + 1301/2732*e^5 - 2439/2732*e^4 - 24337/2732*e^3 - 18915/2732*e^2 + 10549/683*e + 42885/2732, 165/683*e^6 + 664/683*e^5 - 2846/683*e^4 - 11359/683*e^3 + 618/683*e^2 + 14089/683*e + 6231/683, 161/5464*e^6 - 151/5464*e^5 - 5687/5464*e^4 + 4555/5464*e^3 + 55777/5464*e^2 + 206/683*e - 104119/5464, 1297/2732*e^6 + 7285/2732*e^5 - 15711/2732*e^4 - 124697/2732*e^3 - 94351/2732*e^2 + 29973/683*e + 49501/2732, -431/2732*e^6 - 563/2732*e^5 + 13697/2732*e^4 + 13039/2732*e^3 - 113291/2732*e^2 - 21419/683*e + 103609/2732, -141/5464*e^6 - 1921/5464*e^5 - 4013/5464*e^4 + 29253/5464*e^3 + 108963/5464*e^2 + 7797/1366*e - 99529/5464, -24/683*e^6 - 109/683*e^5 + 712/683*e^4 + 2596/683*e^3 - 5765/683*e^2 - 10444/683*e + 18168/683, 15/5464*e^6 + 495/5464*e^5 + 2287/5464*e^4 - 5379/5464*e^3 - 37633/5464*e^2 - 1105/683*e - 72825/5464, 5/1366*e^6 + 165/1366*e^5 + 307/1366*e^4 - 3159/1366*e^3 - 6625/1366*e^2 + 3080/683*e + 7143/1366, 363/5464*e^6 + 3783/5464*e^5 + 2891/5464*e^4 - 66243/5464*e^3 - 143573/5464*e^2 + 16867/1366*e + 114519/5464, -9/2732*e^6 - 297/2732*e^5 + 267/2732*e^4 + 8145/2732*e^3 - 8565/2732*e^2 - 12334/683*e + 35499/2732, -1599/2732*e^6 - 9055/2732*e^5 + 17385/2732*e^4 + 152127/2732*e^3 + 147537/2732*e^2 - 28735/683*e - 72231/2732, 407/1366*e^6 + 910/683*e^5 - 6155/1366*e^4 - 15125/683*e^3 - 13365/1366*e^2 + 22641/1366*e + 15676/683, -1557/5464*e^6 - 7669/5464*e^5 + 21603/5464*e^4 + 127777/5464*e^3 + 61835/5464*e^2 - 9607/683*e + 62627/5464, 575/2732*e^6 + 1217/2732*e^5 - 15237/2732*e^4 - 24517/2732*e^3 + 97339/2732*e^2 + 52997/1366*e - 82847/2732, 599/5464*e^6 + 6107/5464*e^5 + 3175/5464*e^4 - 104975/5464*e^3 - 221321/5464*e^2 + 18523/1366*e + 209067/5464, 537/5464*e^6 + 1329/5464*e^5 - 13199/5464*e^4 - 24277/5464*e^3 + 71193/5464*e^2 + 7568/683*e - 807/5464, 1621/5464*e^6 + 9781/5464*e^5 - 14395/5464*e^4 - 163841/5464*e^3 - 212203/5464*e^2 + 13316/683*e + 134805/5464, -1069/5464*e^6 - 7957/5464*e^5 + 3483/5464*e^4 + 128721/5464*e^3 + 229371/5464*e^2 - 1389/683*e - 148333/5464, 2021/5464*e^6 + 9321/5464*e^5 - 33547/5464*e^4 - 162485/5464*e^3 + 17293/5464*e^2 + 53529/1366*e - 143407/5464, -1513/2732*e^6 - 7583/2732*e^5 + 22119/2732*e^4 + 130303/2732*e^3 + 44515/2732*e^2 - 72111/1366*e - 10295/2732, -277/1366*e^6 - 814/683*e^5 + 3209/1366*e^4 + 14355/683*e^3 + 22793/1366*e^2 - 33231/1366*e - 18437/683, -1097/5464*e^6 - 6149/5464*e^5 + 11599/5464*e^4 + 96689/5464*e^3 + 113479/5464*e^2 + 9771/1366*e - 58837/5464, 38/683*e^6 + 459/1366*e^5 - 672/683*e^4 - 9359/1366*e^3 + 875/683*e^2 + 31479/1366*e + 1889/1366, -123/5464*e^6 + 1405/5464*e^5 + 11845/5464*e^4 - 22553/5464*e^3 - 168963/5464*e^2 - 3916/683*e + 105405/5464, -1983/5464*e^6 - 10799/5464*e^5 + 23313/5464*e^4 + 185467/5464*e^3 + 164577/5464*e^2 - 26035/683*e - 158559/5464, 399/2732*e^6 + 2239/2732*e^5 - 3641/2732*e^4 - 35987/2732*e^3 - 46477/2732*e^2 + 635/683*e + 29895/2732, -1901/5464*e^6 - 10825/5464*e^5 + 24523/5464*e^4 + 190485/5464*e^3 + 113299/5464*e^2 - 68477/1366*e - 149601/5464, -299/5464*e^6 - 4403/5464*e^5 - 6611/5464*e^4 + 76623/5464*e^3 + 178981/5464*e^2 - 7798/683*e - 50955/5464, 235/1366*e^6 + 925/1366*e^5 - 4695/1366*e^4 - 17337/1366*e^3 + 16465/1366*e^2 + 23186/683*e - 37197/1366, 1541/2732*e^6 + 7141/2732*e^5 - 24771/2732*e^4 - 124225/2732*e^3 - 7851/2732*e^2 + 45021/683*e - 31391/2732, 283/1366*e^6 + 1143/1366*e^5 - 4753/1366*e^4 - 19797/1366*e^3 + 675/1366*e^2 + 17238/683*e + 15257/1366, 731/1366*e^6 + 3633/1366*e^5 - 10303/1366*e^4 - 61881/1366*e^3 - 30133/1366*e^2 + 31617/683*e - 2869/1366, 905/5464*e^6 + 8009/5464*e^5 - 439/5464*e^4 - 138757/5464*e^3 - 241559/5464*e^2 + 13698/683*e + 190521/5464, 151/683*e^6 + 202/683*e^5 - 4935/683*e^4 - 5519/683*e^3 + 41024/683*e^2 + 37821/683*e - 37128/683, 63/1366*e^6 + 713/1366*e^5 + 863/1366*e^4 - 11937/1366*e^3 - 28835/1366*e^2 + 3975/683*e - 10809/1366, -37/683*e^6 - 393/1366*e^5 + 187/683*e^4 + 6183/1366*e^3 + 7362/683*e^2 - 329/1366*e - 3403/1366, 164/683*e^6 + 1945/1366*e^5 - 1678/683*e^4 - 33885/1366*e^3 - 16498/683*e^2 + 38691/1366*e + 20123/1366, -177/683*e^6 - 1437/1366*e^5 + 3202/683*e^4 + 25997/1366*e^3 - 3842/683*e^2 - 46135/1366*e + 2291/1366, 381/5464*e^6 + 1645/5464*e^5 - 3107/5464*e^4 - 25161/5464*e^3 - 66339/5464*e^2 - 2796/683*e + 46253/5464, -389/5464*e^6 + 823/5464*e^5 + 17915/5464*e^4 - 8579/5464*e^3 - 196261/5464*e^2 - 31705/1366*e + 159239/5464, -1487/5464*e^6 - 8091/5464*e^5 + 17705/5464*e^4 + 138191/5464*e^3 + 119345/5464*e^2 - 43415/1366*e - 132427/5464, -721/2732*e^6 - 1937/2732*e^5 + 17747/2732*e^4 + 37805/2732*e^3 - 95129/2732*e^2 - 31511/683*e + 69063/2732, 1399/5464*e^6 + 5187/5464*e^5 - 29665/5464*e^4 - 102263/5464*e^3 + 106535/5464*e^2 + 76415/1366*e - 74157/5464, 609/2732*e^6 + 3705/2732*e^5 - 4407/2732*e^4 - 59385/2732*e^3 - 97971/2732*e^2 + 2479/683*e + 67629/2732, -1479/5464*e^6 - 7827/5464*e^5 + 19289/5464*e^4 + 125487/5464*e^3 + 81425/5464*e^2 + 1395/1366*e + 9045/5464, -23/1366*e^6 - 38/683*e^5 + 227/1366*e^4 + 504/683*e^3 - 943/1366*e^2 - 333/1366*e + 18609/683, 273/5464*e^6 + 813/5464*e^5 - 10831/5464*e^4 - 28505/5464*e^3 + 104081/5464*e^2 + 57447/1366*e - 71427/5464, -571/5464*e^6 - 2451/5464*e^5 + 10565/5464*e^4 + 38655/5464*e^3 - 23411/5464*e^2 + 3588/683*e + 127629/5464, 239/5464*e^6 - 309/5464*e^5 - 13465/5464*e^4 - 8663/5464*e^3 + 157327/5464*e^2 + 53805/1366*e - 141309/5464, -863/2732*e^6 - 2525/2732*e^5 + 21049/2732*e^4 + 50205/2732*e^3 - 117343/2732*e^2 - 80145/1366*e + 150603/2732, 539/5464*e^6 - 1337/5464*e^5 - 22365/5464*e^4 + 20357/5464*e^3 + 202411/5464*e^2 + 17801/1366*e - 13249/5464, -1657/5464*e^6 - 5505/5464*e^5 + 31855/5464*e^4 + 87141/5464*e^3 - 73401/5464*e^2 - 5883/683*e + 67295/5464, -397/1366*e^6 - 2173/1366*e^5 + 5403/1366*e^4 + 38275/1366*e^3 + 16507/1366*e^2 - 25309/683*e + 23231/1366, -1113/5464*e^6 - 6677/5464*e^5 + 8431/5464*e^4 + 111169/5464*e^3 + 178391/5464*e^2 - 11549/1366*e - 248893/5464, -907/2732*e^6 - 5343/2732*e^5 + 9605/2732*e^4 + 91391/2732*e^3 + 88485/2732*e^2 - 21182/683*e - 14159/2732, 257/5464*e^6 + 3017/5464*e^5 + 2393/5464*e^4 - 49541/5464*e^3 - 104207/5464*e^2 + 2013/683*e + 36305/5464, -4689/5464*e^6 - 20869/5464*e^5 + 76271/5464*e^4 + 358641/5464*e^3 + 20847/5464*e^2 - 108947/1366*e + 168723/5464, -1205/5464*e^6 - 9713/5464*e^5 - 1589/5464*e^4 + 156181/5464*e^3 + 354931/5464*e^2 + 2461/1366*e - 233889/5464, 259/1366*e^6 + 1200/683*e^5 + 1423/1366*e^4 - 19187/683*e^3 - 95929/1366*e^2 - 8069/1366*e + 38910/683, 3/2732*e^6 + 2831/2732*e^5 + 10839/2732*e^4 - 46427/2732*e^3 - 180189/2732*e^2 - 3174/683*e + 163015/2732, 1157/2732*e^6 + 5397/2732*e^5 - 18843/2732*e^4 - 93617/2732*e^3 - 1739/2732*e^2 + 30565/683*e - 52151/2732, 2565/5464*e^6 + 13613/5464*e^5 - 35115/5464*e^4 - 231345/5464*e^3 - 102467/5464*e^2 + 30971/683*e - 82579/5464, 19/5464*e^6 - 2105/5464*e^5 - 5117/5464*e^4 + 48373/5464*e^3 + 66347/5464*e^2 - 48105/1366*e - 51265/5464, -227/1366*e^6 - 661/1366*e^5 + 4913/1366*e^4 + 10097/1366*e^3 - 20235/1366*e^2 - 1866/683*e + 44801/1366, 25/683*e^6 + 142/683*e^5 - 514/683*e^4 - 3501/683*e^3 + 1025/683*e^2 + 17823/683*e + 14542/683, 2665/2732*e^6 + 11449/2732*e^5 - 45367/2732*e^4 - 198905/2732*e^3 + 24573/2732*e^2 + 74984/683*e - 43535/2732, -359/1366*e^6 - 919/1366*e^5 + 8829/1366*e^4 + 18911/1366*e^3 - 46137/1366*e^2 - 38783/683*e + 29981/1366, 629/5464*e^6 + 4365/5464*e^5 - 3179/5464*e^4 - 80217/5464*e^3 - 121739/5464*e^2 + 16272/683*e + 66149/5464, 2487/5464*e^6 + 11039/5464*e^5 - 43729/5464*e^4 - 204467/5464*e^3 + 41863/5464*e^2 + 55158/683*e + 1055/5464, 2227/5464*e^6 + 13387/5464*e^5 - 21445/5464*e^4 - 225975/5464*e^3 - 258389/5464*e^2 + 20582/683*e + 203339/5464, -499/2732*e^6 - 2807/2732*e^5 + 2965/2732*e^4 + 39063/2732*e^3 + 91553/2732*e^2 + 22522/683*e - 68939/2732, 2695/5464*e^6 + 12439/5464*e^5 - 40793/5464*e^4 - 212395/5464*e^3 - 64353/5464*e^2 + 30501/683*e + 78551/5464, 2455/5464*e^6 + 15447/5464*e^5 - 22745/5464*e^4 - 262931/5464*e^3 - 287289/5464*e^2 + 27008/683*e + 145487/5464, 3549/5464*e^6 + 18765/5464*e^5 - 47915/5464*e^4 - 324121/5464*e^3 - 165939/5464*e^2 + 45907/683*e + 106877/5464, -1033/2732*e^6 - 5403/2732*e^5 + 13343/2732*e^4 + 94775/2732*e^3 + 64195/2732*e^2 - 68755/1366*e - 43083/2732, 247/2732*e^6 + 1321/2732*e^5 - 953/2732*e^4 - 20001/2732*e^3 - 66369/2732*e^2 - 8353/1366*e + 170913/2732, -1803/2732*e^6 - 8957/2732*e^5 + 26169/2732*e^4 + 155069/2732*e^3 + 59945/2732*e^2 - 89563/1366*e - 20253/2732, -407/1366*e^6 - 2503/1366*e^5 + 4789/1366*e^4 + 45959/1366*e^3 + 32489/1366*e^2 - 38299/683*e - 36133/1366, -97/683*e^6 - 469/683*e^5 + 1967/683*e^4 + 9923/683*e^3 - 6026/683*e^2 - 36861/683*e + 7861/683, 417/2732*e^6 + 2833/2732*e^5 - 1443/2732*e^4 - 44081/2732*e^3 - 97647/2732*e^2 - 13628/683*e + 144673/2732, -477/5464*e^6 + 651/5464*e^5 + 16883/5464*e^4 - 8167/5464*e^3 - 145229/5464*e^2 - 14720/683*e + 15491/5464, -1015/1366*e^6 - 2063/683*e^5 + 18273/1366*e^4 + 36169/683*e^3 - 23857/1366*e^2 - 118749/1366*e + 17065/683, -3159/5464*e^6 - 16823/5464*e^5 + 41809/5464*e^4 + 288083/5464*e^3 + 165537/5464*e^2 - 35023/683*e - 60607/5464, -36/683*e^6 + 178/683*e^5 + 1751/683*e^4 - 2936/683*e^3 - 17185/683*e^2 - 4738/683*e + 10860/683, -1447/2732*e^6 - 6771/2732*e^5 + 22893/2732*e^4 + 118383/2732*e^3 + 17169/2732*e^2 - 44755/683*e - 94407/2732, -513/5464*e^6 - 3269/5464*e^5 + 1559/5464*e^4 + 49001/5464*e^3 + 131959/5464*e^2 + 9331/1366*e - 189477/5464, 727/1366*e^6 + 3501/1366*e^5 - 9729/1366*e^4 - 55529/1366*e^3 - 38493/1366*e^2 + 6614/683*e + 24747/1366, -2717/5464*e^6 - 13165/5464*e^5 + 43267/5464*e^4 + 232305/5464*e^3 + 36131/5464*e^2 - 40036/683*e - 31845/5464, -419/5464*e^6 - 167/5464*e^5 + 13341/5464*e^4 + 7643/5464*e^3 - 110067/5464*e^2 - 34115/1366*e + 130041/5464, -4635/5464*e^6 - 24551/5464*e^5 + 58277/5464*e^4 + 413587/5464*e^3 + 301725/5464*e^2 - 85485/1366*e - 82519/5464, 1/5464*e^6 + 2765/5464*e^5 + 11809/5464*e^4 - 44617/5464*e^3 - 196663/5464*e^2 - 11303/1366*e + 216437/5464, 1317/5464*e^6 + 7945/5464*e^5 - 14483/5464*e^4 - 137333/5464*e^3 - 115387/5464*e^2 + 33401/1366*e - 96775/5464, -663/2732*e^6 - 2755/2732*e^5 + 14205/2732*e^4 + 56347/2732*e^3 - 51771/2732*e^2 - 48480/683*e + 19693/2732, 338/683*e^6 + 1592/683*e^5 - 4791/683*e^4 - 26543/683*e^3 - 15511/683*e^2 + 21642/683*e + 16651/683, -245/1366*e^6 - 1255/1366*e^5 + 4081/1366*e^4 + 20923/1366*e^3 - 1849/1366*e^2 - 4758/683*e + 21545/1366];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11,11,-w^2 + 2*w + 8}>] := 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;