/*
  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![5, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {5, 5, -w + 2}>;

primesArray := [
[2, 2, -w + 1],
[5, 5, w],
[5, 5, -w^2 + 6],
[5, 5, -w + 2],
[7, 7, w + 2],
[11, 11, 2*w - 1],
[13, 13, w^2 + w - 3],
[27, 3, 3],
[41, 41, w^2 - w - 1],
[41, 41, 3*w^2 - w - 23],
[41, 41, w^2 - 2],
[43, 43, w^2 - w - 3],
[47, 47, -w - 4],
[49, 7, 3*w^2 - 2*w - 24],
[53, 53, 2*w^2 - w - 12],
[59, 59, w^2 - 2*w - 4],
[61, 61, -w^2 + 3*w - 3],
[71, 71, w^2 + w - 7],
[79, 79, 2*w + 3],
[89, 89, 2*w - 7],
[97, 97, 2*w^2 - 11],
[101, 101, 3*w - 4],
[103, 103, -2*w^2 - 2*w + 9],
[107, 107, 2*w^2 - 2*w - 17],
[113, 113, -3*w^2 - 3*w + 11],
[121, 11, 4*w^2 - 2*w - 29],
[127, 127, -w^2 - w - 1],
[131, 131, -w^2 + w + 11],
[137, 137, w^2 - 2*w - 6],
[149, 149, w^2 + 2*w - 6],
[163, 163, -2*w^2 - w + 8],
[167, 167, -2*w^2 + 2*w + 13],
[169, 13, 2*w^2 - 3*w - 6],
[173, 173, w^2 + 6*w - 4],
[179, 179, 3*w^2 - 2*w - 22],
[191, 191, -w^2 + 6*w - 6],
[191, 191, w^2 + 2*w - 12],
[191, 191, 9*w^2 - 2*w - 64],
[193, 193, 6*w^2 - 2*w - 41],
[197, 197, 4*w^2 - w - 26],
[199, 199, 6*w^2 - w - 44],
[211, 211, -w^2 - 2],
[223, 223, 3*w^2 - 3*w - 17],
[223, 223, w^2 + w - 13],
[223, 223, 2*w^2 - 9],
[227, 227, -w^2 + 4*w - 6],
[227, 227, -2*w^2 - 3*w + 8],
[227, 227, 2*w^2 + w - 14],
[229, 229, 2*w^2 - 2*w - 1],
[241, 241, -2*w^2 + 4*w - 3],
[241, 241, -3*w - 2],
[241, 241, 7*w^2 - 3*w - 51],
[251, 251, w^2 - 12],
[263, 263, 2*w^2 - w - 2],
[263, 263, -2*w^2 + 5*w - 4],
[263, 263, -5*w + 2],
[277, 277, w^2 - 6*w + 2],
[283, 283, 7*w^2 - 2*w - 48],
[283, 283, 2*w^2 - 2*w - 7],
[283, 283, 4*w - 9],
[307, 307, -w^2 + 2*w - 4],
[307, 307, 2*w^2 - 3*w - 4],
[307, 307, 3*w^2 - 8*w + 2],
[311, 311, 2*w^2 - 4*w - 7],
[313, 313, 3*w^2 - 2*w - 26],
[313, 313, -2*w^2 + 19],
[313, 313, 7*w^2 - w - 49],
[317, 317, 4*w - 7],
[331, 331, w^2 - 4*w - 2],
[347, 347, 3*w^2 - w - 17],
[349, 349, 11*w^2 - 3*w - 77],
[349, 349, w^2 + 3*w - 7],
[359, 359, w^2 + 4*w - 6],
[367, 367, 3*w^2 + 2*w - 12],
[373, 373, 10*w^2 - 4*w - 73],
[379, 379, 7*w - 6],
[383, 383, 2*w^2 - 2*w - 3],
[389, 389, 5*w - 6],
[397, 397, 3*w^2 - 16],
[397, 397, 2*w^2 - w - 4],
[397, 397, w - 8],
[401, 401, -w^2 - w - 3],
[421, 421, -4*w^2 + 3*w + 26],
[431, 431, -w^2 + 2*w + 14],
[433, 433, 7*w^2 - 46],
[439, 439, -w^2 + 3*w + 11],
[461, 461, w^2 + 2*w - 16],
[463, 463, 13*w^2 - 4*w - 96],
[467, 467, 5*w - 12],
[467, 467, -2*w^2 - 1],
[467, 467, 5*w^2 + w - 31],
[487, 487, 3*w^2 + w - 21],
[491, 491, 2*w^2 - 3*w - 18],
[491, 491, 2*w^2 + 2*w - 13],
[491, 491, 3*w^2 - 7*w + 1],
[499, 499, w^2 + 3*w - 9],
[503, 503, 5*w^2 - 38],
[509, 509, 3*w^2 - 3*w - 19],
[521, 521, 5*w^2 - 3*w - 37],
[523, 523, -3*w^2 - 4*w + 12],
[557, 557, 7*w^2 - 4*w - 54],
[569, 569, 5*w^2 - 36],
[571, 571, 8*w^2 - 3*w - 54],
[577, 577, w^2 - 4*w - 4],
[577, 577, w^2 + 3*w - 11],
[577, 577, 3*w^2 + w - 13],
[593, 593, -4*w - 3],
[599, 599, -2*w - 9],
[601, 601, 4*w^2 + w - 26],
[613, 613, 8*w^2 - w - 56],
[613, 613, -3*w^2 + 9*w - 7],
[613, 613, 2*w^2 + 4*w - 9],
[619, 619, w^2 + 5*w - 7],
[631, 631, 6*w^2 - 2*w - 47],
[631, 631, 4*w^2 - 3*w - 28],
[631, 631, 6*w^2 - 37],
[641, 641, -4*w^2 + 2*w + 23],
[641, 641, w^2 + 7*w - 7],
[641, 641, 2*w^2 - 3*w - 16],
[643, 643, 3*w^2 - 11*w + 3],
[659, 659, 4*w^2 - 4*w - 19],
[661, 661, 3*w^2 - 2*w - 14],
[661, 661, -w^2 + 4*w - 8],
[661, 661, 9*w^2 - 4*w - 64],
[673, 673, -3*w^2 + 3*w + 23],
[673, 673, -2*w^2 - 2*w - 1],
[673, 673, 2*w^2 - 5*w - 6],
[677, 677, -2*w^2 + 7*w - 8],
[683, 683, 3*w^2 - 14],
[683, 683, -2*w^2 + 8*w - 1],
[683, 683, 3*w^2 + 2*w - 18],
[691, 691, 4*w^2 + 2*w - 23],
[709, 709, 2*w^2 + w - 22],
[719, 719, 3*w^2 - 8*w - 4],
[727, 727, w^2 - 4*w - 14],
[727, 727, -2*w^2 + 2*w - 3],
[727, 727, 3*w^2 - 4*w - 16],
[733, 733, 6*w^2 - w - 48],
[751, 751, 4*w^2 - 10*w + 3],
[757, 757, 3*w^2 - w - 27],
[761, 761, -w^2 + 10*w - 8],
[769, 769, -2*w^2 + 12*w - 9],
[787, 787, 2*w^2 - 21],
[797, 797, 8*w^2 + w - 48],
[811, 811, 6*w^2 - 4*w - 47],
[821, 821, 2*w - 11],
[821, 821, 3*w^2 - 3*w - 29],
[821, 821, -2*w^2 + 3*w - 4],
[823, 823, -6*w + 1],
[823, 823, 3*w^2 + w - 11],
[823, 823, 4*w^2 + w - 32],
[827, 827, 8*w^2 - 4*w - 63],
[829, 829, -3*w^2 + 7*w + 7],
[839, 839, 8*w^2 - w - 58],
[857, 857, 2*w^2 + 3*w - 22],
[859, 859, 9*w^2 - 3*w - 61],
[863, 863, 6*w^2 - w - 38],
[877, 877, -2*w^2 + 5*w - 6],
[881, 881, w^2 - 5*w - 3],
[887, 887, 6*w^2 + w - 38],
[907, 907, 3*w^2 - 12*w + 4],
[911, 911, 3*w^2 - w - 13],
[919, 919, w^2 - 7*w + 1],
[947, 947, -2*w^2 - w - 2],
[947, 947, -5*w - 2],
[947, 947, 7*w^2 - 4*w - 48],
[967, 967, w^2 + 6*w - 8],
[967, 967, 8*w^2 - w - 52],
[967, 967, 2*w^2 - 7*w - 2],
[971, 971, 4*w^2 - 6*w - 13],
[971, 971, 3*w^2 - 2*w - 12],
[971, 971, 6*w^2 - 43],
[977, 977, -2*w^2 - 3*w - 2],
[977, 977, 4*w^2 - w - 22],
[977, 977, 3*w^2 - 4*w - 8],
[997, 997, w^2 - 4*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 32*x^8 + 314*x^6 - 936*x^4 + 584*x^2 - 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/1620*e^8 + 1/180*e^6 + 149/810*e^4 - 355/162*e^2 + 1109/405, e, -53/3240*e^9 + 47/90*e^7 - 8303/1620*e^5 + 1270/81*e^3 - 5239/405*e, 1, 1/1620*e^9 - 1/180*e^7 - 149/810*e^5 + 355/162*e^3 - 1514/405*e, 71/1620*e^9 - 251/180*e^7 + 5443/405*e^5 - 6061/162*e^3 + 6716/405*e, 2/45*e^9 - 7/5*e^7 + 1193/90*e^5 - 317/9*e^3 + 533/45*e, 1/810*e^8 - 1/90*e^6 - 149/405*e^4 + 274/81*e^2 + 212/405, 1/216*e^9 - 1/6*e^7 + 229/108*e^5 - 292/27*e^3 + 377/27*e, -1/45*e^8 + 7/10*e^6 - 287/45*e^4 + 127/9*e^2 - 154/45, 49/3240*e^9 - 23/45*e^7 + 8899/1620*e^5 - 1625/81*e^3 + 8267/405*e, -7/90*e^9 + 49/20*e^7 - 2099/90*e^5 + 1141/18*e^3 - 944/45*e, 23/1620*e^8 - 17/45*e^6 + 2243/810*e^4 - 332/81*e^2 + 8/405, -2/45*e^9 + 7/5*e^7 - 1193/90*e^5 + 317/9*e^3 - 533/45*e, 299/3240*e^9 - 133/45*e^7 + 46979/1620*e^5 - 6937/81*e^3 + 18277/405*e, 44/405*e^9 - 313/90*e^7 + 27641/810*e^5 - 8207/81*e^3 + 24326/405*e, 8/405*e^8 - 61/90*e^6 + 2881/405*e^4 - 1772/81*e^2 + 2582/405, -19/1620*e^8 + 16/45*e^6 - 2839/810*e^4 + 1042/81*e^2 - 2824/405, -23/540*e^9 + 83/60*e^7 - 1864/135*e^5 + 2257/54*e^3 - 2978/135*e, 17/810*e^8 - 31/45*e^6 + 2732/405*e^4 - 1498/81*e^2 + 2794/405, -151/3240*e^9 + 139/90*e^7 - 26101/1620*e^5 + 4439/81*e^3 - 16913/405*e, 1/810*e^8 - 1/90*e^6 - 149/405*e^4 + 274/81*e^2 + 4262/405, 91/1620*e^8 - 79/45*e^6 + 13171/810*e^4 - 3166/81*e^2 + 3976/405, -49/405*e^9 + 691/180*e^7 - 14963/405*e^5 + 16847/162*e^3 - 22396/405*e, -19/1620*e^9 + 16/45*e^7 - 1217/405*e^5 + 313/81*e^3 + 6491/405*e, -197/3240*e^9 + 173/90*e^7 - 29777/1620*e^5 + 4042/81*e^3 - 3961/405*e, 8/405*e^8 - 61/90*e^6 + 2881/405*e^4 - 1772/81*e^2 + 3392/405, -1/15*e^8 + 21/10*e^6 - 302/15*e^4 + 172/3*e^2 - 244/15, 53/1620*e^9 - 47/45*e^7 + 8303/810*e^5 - 2459/81*e^3 + 6023/405*e, 19/270*e^8 - 32/15*e^6 + 2569/135*e^4 - 1166/27*e^2 + 1058/135, -53/810*e^9 + 94/45*e^7 - 8303/405*e^5 + 5080/81*e^3 - 20146/405*e, -37/810*e^9 + 127/90*e^7 - 10439/810*e^5 + 2498/81*e^3 - 1364/405*e, -659/3240*e^9 + 581/90*e^7 - 101069/1620*e^5 + 14353/81*e^3 - 30877/405*e, -557/3240*e^9 + 244/45*e^7 - 83867/1620*e^5 + 11620/81*e^3 - 27091/405*e, -61/405*e^9 + 437/90*e^7 - 19487/405*e^5 + 11689/81*e^3 - 31534/405*e, 17/540*e^8 - 31/30*e^6 + 2867/270*e^4 - 938/27*e^2 + 3152/135, 5/54*e^8 - 17/6*e^6 + 686/27*e^4 - 1574/27*e^2 + 88/27, -49/1620*e^8 + 46/45*e^6 - 8089/810*e^4 + 1954/81*e^2 - 4384/405, -71/270*e^9 + 251/30*e^7 - 21907/270*e^5 + 6304/27*e^3 - 16267/135*e, -47/270*e^9 + 167/30*e^7 - 14749/270*e^5 + 4402/27*e^3 - 13609/135*e, 53/810*e^8 - 94/45*e^6 + 7898/405*e^4 - 3622/81*e^2 + 3136/405, -1/54*e^8 + 2/3*e^6 - 202/27*e^4 + 736/27*e^2 - 644/27, -23/162*e^9 + 163/36*e^7 - 7159/162*e^5 + 21139/162*e^3 - 6334/81*e, 11/162*e^8 - 19/9*e^6 + 1601/81*e^4 - 4370/81*e^2 + 2008/81, -8/405*e^9 + 61/90*e^7 - 6167/810*e^5 + 2501/81*e^3 - 14732/405*e, -23/810*e^8 + 34/45*e^6 - 2243/405*e^4 + 664/81*e^2 + 1604/405, -41/540*e^8 + 73/30*e^6 - 6311/270*e^4 + 1646/27*e^2 - 836/135, -29/270*e^8 + 52/15*e^6 - 4589/135*e^4 + 2584/27*e^2 - 3988/135, 773/3240*e^9 - 677/90*e^7 + 116483/1620*e^5 - 16264/81*e^3 + 40969/405*e, 1/45*e^8 - 7/10*e^6 + 287/45*e^4 - 136/9*e^2 + 154/45, 41/540*e^9 - 73/30*e^7 + 3223/135*e^5 - 1916/27*e^3 + 6101/135*e, 803/3240*e^9 - 707/90*e^7 + 122543/1620*e^5 - 17368/81*e^3 + 44179/405*e, -37/810*e^8 + 127/90*e^6 - 5017/405*e^4 + 1850/81*e^2 + 1876/405, 49/324*e^9 - 175/36*e^7 + 7765/162*e^5 - 23023/162*e^3 + 5842/81*e, 91/810*e^8 - 158/45*e^6 + 13171/405*e^4 - 6494/81*e^2 + 7952/405, 139/810*e^9 - 499/90*e^7 + 44843/810*e^5 - 13916/81*e^3 + 45668/405*e, -103/810*e^9 + 373/90*e^7 - 17053/405*e^5 + 11063/81*e^3 - 44111/405*e, 157/810*e^9 - 281/45*e^7 + 50009/810*e^5 - 14978/81*e^3 + 42194/405*e, 1/324*e^9 - 1/36*e^7 - 149/162*e^5 + 1775/162*e^3 - 1676/81*e, -1/405*e^8 + 1/45*e^6 - 107/405*e^4 + 748/81*e^2 - 11764/405, 26/405*e^8 - 187/90*e^6 + 8047/405*e^4 - 4058/81*e^2 + 9404/405, 19/810*e^9 - 32/45*e^7 + 2434/405*e^5 - 626/81*e^3 - 11362/405*e, -5/27*e^9 + 71/12*e^7 - 1561/27*e^5 + 9131/54*e^3 - 2444/27*e, 34/135*e^9 - 481/60*e^7 + 10523/135*e^5 - 12281/54*e^3 + 17926/135*e, 31/270*e^8 - 53/15*e^6 + 4291/135*e^4 - 1928/27*e^2 + 902/135, -23/270*e^8 + 83/30*e^6 - 3728/135*e^4 + 2149/27*e^2 - 2446/135, 13/135*e^8 - 43/15*e^6 + 3416/135*e^4 - 1624/27*e^2 + 1462/135, -23/270*e^8 + 83/30*e^6 - 3728/135*e^4 + 2203/27*e^2 - 2986/135, -89/405*e^9 + 314/45*e^7 - 54281/810*e^5 + 15056/81*e^3 - 36926/405*e, -28/405*e^9 + 191/90*e^7 - 15307/810*e^5 + 3124/81*e^3 + 8378/405*e, 31/216*e^9 - 14/3*e^7 + 5155/108*e^5 - 4327/27*e^3 + 3479/27*e, 1/54*e^8 - 2/3*e^6 + 202/27*e^4 - 736/27*e^2 + 266/27, 23/270*e^8 - 83/30*e^6 + 3728/135*e^4 - 2230/27*e^2 + 4336/135, -79/405*e^9 + 563/90*e^7 - 49711/810*e^5 + 14623/81*e^3 - 41596/405*e, -5/36*e^9 + 9/2*e^7 - 406/9*e^5 + 1264/9*e^3 - 863/9*e, -37/405*e^8 + 127/45*e^6 - 10844/405*e^4 + 6130/81*e^2 - 4348/405, -2/45*e^9 + 7/5*e^7 - 1193/90*e^5 + 299/9*e^3 + 412/45*e, -7/81*e^8 + 25/9*e^6 - 2207/81*e^4 + 6416/81*e^2 - 1834/81, 53/135*e^9 - 188/15*e^7 + 16471/135*e^5 - 9485/27*e^3 + 22607/135*e, -35/162*e^9 + 125/18*e^7 - 5558/81*e^5 + 16688/81*e^3 - 9283/81*e, -11/162*e^8 + 19/9*e^6 - 1520/81*e^4 + 3074/81*e^2 + 422/81, -19/810*e^8 + 32/45*e^6 - 2839/405*e^4 + 2084/81*e^2 - 4838/405, -25/216*e^9 + 11/3*e^7 - 3835/108*e^5 + 2818/27*e^3 - 2081/27*e, -43/270*e^8 + 74/15*e^6 - 6148/135*e^4 + 3122/27*e^2 - 5336/135, -5/54*e^8 + 17/6*e^6 - 686/27*e^4 + 1574/27*e^2 - 466/27, 43/405*e^9 - 311/90*e^7 + 28237/810*e^5 - 8917/81*e^3 + 27952/405*e, 4/81*e^8 - 13/9*e^6 + 995/81*e^4 - 2324/81*e^2 + 2830/81, -19/108*e^8 + 16/3*e^6 - 2569/54*e^4 + 2996/27*e^2 - 664/27, -1/54*e^8 + 2/3*e^6 - 202/27*e^4 + 682/27*e^2 - 428/27, 5/54*e^8 - 17/6*e^6 + 686/27*e^4 - 1574/27*e^2 + 628/27, -41/324*e^8 + 35/9*e^6 - 5717/162*e^4 + 6826/81*e^2 - 620/81, -7/81*e^8 + 25/9*e^6 - 2126/81*e^4 + 5120/81*e^2 - 376/81, -49/540*e^9 + 169/60*e^7 - 7009/270*e^5 + 3341/54*e^3 + 1556/135*e, -7/405*e^8 + 59/90*e^6 - 3179/405*e^4 + 2320/81*e^2 - 7828/405, 1/1620*e^9 - 1/180*e^7 - 277/405*e^5 + 1975/162*e^3 - 18524/405*e, 11/135*e^8 - 41/15*e^6 + 3742/135*e^4 - 2126/27*e^2 + 3044/135, -32/405*e^8 + 122/45*e^6 - 11524/405*e^4 + 7250/81*e^2 - 20048/405, -8/405*e^9 + 61/90*e^7 - 2881/405*e^5 + 1853/81*e^3 - 9467/405*e, -191/810*e^9 + 343/45*e^7 - 61747/810*e^5 + 19189/81*e^3 - 63577/405*e, -53/810*e^8 + 94/45*e^6 - 7898/405*e^4 + 3622/81*e^2 + 4964/405, 71/810*e^9 - 251/90*e^7 + 22177/810*e^5 - 6628/81*e^3 + 12217/405*e, 4/45*e^8 - 14/5*e^6 + 1193/45*e^4 - 652/9*e^2 + 1426/45, -47/135*e^9 + 167/15*e^7 - 29363/270*e^5 + 8588/27*e^3 - 25058/135*e, -47/360*e^9 + 43/10*e^7 - 8027/180*e^5 + 1366/9*e^3 - 5821/45*e, -17/810*e^8 + 31/45*e^6 - 3137/405*e^4 + 2794/81*e^2 - 10894/405, 91/1080*e^9 - 79/30*e^7 + 13171/540*e^5 - 1610/27*e^3 + 233/135*e, -61/324*e^9 + 107/18*e^7 - 9217/162*e^5 + 12890/81*e^3 - 6061/81*e, -41/540*e^8 + 73/30*e^6 - 6311/270*e^4 + 1646/27*e^2 - 2456/135, -2/27*e^8 + 13/6*e^6 - 511/27*e^4 + 1297/27*e^2 - 470/27, -58/405*e^8 + 193/45*e^6 - 15116/405*e^4 + 6448/81*e^2 - 2722/405, -34/405*e^8 + 124/45*e^6 - 11333/405*e^4 + 6802/81*e^2 - 15226/405, 64/405*e^8 - 443/90*e^6 + 18593/405*e^4 - 9478/81*e^2 + 6886/405, -17/405*e^8 + 62/45*e^6 - 5059/405*e^4 + 1700/81*e^2 + 892/405, 35/324*e^8 - 29/9*e^6 + 4667/162*e^4 - 5914/81*e^2 + 1928/81, 29/270*e^9 - 52/15*e^7 + 4724/135*e^5 - 3070/27*e^3 + 10468/135*e, 1/90*e^9 - 7/20*e^7 + 166/45*e^5 - 271/18*e^3 + 842/45*e, -5/324*e^9 + 7/18*e^7 - 227/162*e^5 - 1400/81*e^3 + 3763/81*e, -2/405*e^8 + 2/45*e^6 + 1001/405*e^4 - 2392/81*e^2 + 12922/405, 65/648*e^9 - 59/18*e^7 + 10889/324*e^5 - 9206/81*e^3 + 8143/81*e, -107/810*e^9 + 377/90*e^7 - 16457/405*e^5 + 9805/81*e^3 - 36454/405*e, -53/810*e^9 + 94/45*e^7 - 8303/405*e^5 + 4918/81*e^3 - 13666/405*e, 41/108*e^9 - 73/6*e^7 + 6419/54*e^5 - 9256/27*e^3 + 4427/27*e, 2/81*e^8 - 13/18*e^6 + 619/81*e^4 - 3025/81*e^2 + 2954/81, 7/540*e^9 - 11/30*e^7 + 577/270*e^5 + 257/27*e^3 - 6413/135*e, -1/20*e^9 + 17/10*e^7 - 181/10*e^5 + 61*e^3 - 181/5*e, -181/810*e^8 + 631/90*e^6 - 26896/405*e^4 + 14396/81*e^2 - 22982/405, 41/180*e^9 - 73/10*e^7 + 3223/45*e^5 - 1916/9*e^3 + 6191/45*e, -1/45*e^8 + 7/10*e^6 - 332/45*e^4 + 235/9*e^2 + 1106/45, -47/810*e^9 + 319/180*e^7 - 6362/405*e^5 + 5591/162*e^3 - 3484/405*e, 167/540*e^9 - 587/60*e^7 + 25337/270*e^5 - 14155/54*e^3 + 17972/135*e, -103/1620*e^8 + 82/45*e^6 - 12193/810*e^4 + 2332/81*e^2 + 9332/405, -1/405*e^8 + 1/45*e^6 + 703/405*e^4 - 1844/81*e^2 + 10916/405, -71/810*e^8 + 251/90*e^6 - 10886/405*e^4 + 6061/81*e^2 - 9382/405, -463/1620*e^9 + 1633/180*e^7 - 35369/405*e^5 + 39431/162*e^3 - 39358/405*e, 37/270*e^9 - 269/60*e^7 + 6232/135*e^5 - 8425/54*e^3 + 18374/135*e, 7/36*e^8 - 6*e^6 + 991/18*e^4 - 1262/9*e^2 + 472/9, -47/405*e^9 + 683/180*e^7 - 31523/810*e^5 + 20983/162*e^3 - 43418/405*e, 8/45*e^8 - 28/5*e^6 + 2386/45*e^4 - 1268/9*e^2 + 1142/45, 13/270*e^9 - 43/30*e^7 + 1573/135*e^5 - 353/27*e^3 - 4804/135*e, -113/810*e^8 + 383/90*e^6 - 15158/405*e^4 + 6379/81*e^2 - 5326/405, 2/81*e^8 - 13/18*e^6 + 619/81*e^4 - 2944/81*e^2 + 2954/81, 4/81*e^8 - 13/9*e^6 + 914/81*e^4 - 866/81*e^2 - 2678/81, 2/15*e^8 - 21/5*e^6 + 604/15*e^4 - 338/3*e^2 + 788/15, 113/648*e^9 - 49/9*e^7 + 16373/324*e^5 - 10181/81*e^3 + 2101/81*e, -122/405*e^9 + 437/45*e^7 - 78353/810*e^5 + 24431/81*e^3 - 91418/405*e, -23/135*e^8 + 83/15*e^6 - 7456/135*e^4 + 4406/27*e^2 - 6782/135, -7/270*e^8 + 11/15*e^6 - 847/135*e^4 + 431/27*e^2 - 674/135, -119/810*e^8 + 217/45*e^6 - 19529/405*e^4 + 11458/81*e^2 - 14698/405, -323/810*e^9 + 1133/90*e^7 - 48668/405*e^5 + 26761/81*e^3 - 57136/405*e, 59/270*e^9 - 209/30*e^7 + 9164/135*e^5 - 5299/27*e^3 + 10888/135*e, 1/810*e^8 - 1/90*e^6 - 149/405*e^4 + 274/81*e^2 + 11552/405, -1/81*e^8 + 1/9*e^6 + 298/81*e^4 - 3226/81*e^2 + 3140/81, -799/3240*e^9 + 353/45*e^7 - 123139/1620*e^5 + 17723/81*e^3 - 53687/405*e, -11/162*e^8 + 19/9*e^6 - 1520/81*e^4 + 3074/81*e^2 - 64/81, -5/54*e^8 + 17/6*e^6 - 686/27*e^4 + 1655/27*e^2 - 574/27, 46/135*e^9 - 649/60*e^7 + 14102/135*e^5 - 16193/54*e^3 + 23824/135*e, 179/324*e^9 - 629/36*e^7 + 27113/162*e^5 - 75125/162*e^3 + 16706/81*e, 101/810*e^8 - 371/90*e^6 + 16946/405*e^4 - 10396/81*e^2 + 25462/405, -1129/3240*e^9 + 991/90*e^7 - 171169/1620*e^5 + 24116/81*e^3 - 59837/405*e, 4/405*e^8 - 4/45*e^6 - 382/405*e^4 - 76/81*e^2 + 17896/405, 103/270*e^9 - 731/60*e^7 + 32081/270*e^5 - 18589/54*e^3 + 22376/135*e, 409/810*e^9 - 722/45*e^7 + 125573/810*e^5 - 35516/81*e^3 + 78608/405*e, 349/810*e^9 - 2453/180*e^7 + 52894/405*e^5 - 58957/162*e^3 + 71558/405*e, -28/135*e^8 + 191/30*e^6 - 7991/135*e^4 + 4312/27*e^2 - 7012/135, 31/90*e^9 - 111/10*e^7 + 9977/90*e^5 - 3125/9*e^3 + 10802/45*e, -14/405*e^9 + 191/180*e^7 - 7451/810*e^5 + 2233/162*e^3 + 15124/405*e, -23/810*e^8 + 34/45*e^6 - 2243/405*e^4 + 664/81*e^2 + 16184/405, -1/810*e^9 + 1/90*e^7 + 149/405*e^5 - 355/81*e^3 + 4648/405*e, -2/5*e^9 + 63/5*e^7 - 1203/10*e^5 + 335*e^3 - 808/5*e, -121/810*e^9 + 218/45*e^7 - 19636/405*e^5 + 12206/81*e^3 - 45902/405*e, -181/810*e^9 + 631/90*e^7 - 53387/810*e^5 + 13910/81*e^3 - 22982/405*e, 4/27*e^8 - 29/6*e^6 + 1265/27*e^4 - 3188/27*e^2 + 292/27, -1/5*e^8 + 63/10*e^6 - 297/5*e^4 + 154*e^2 - 254/5, 41/180*e^9 - 73/10*e^7 + 3223/45*e^5 - 1916/9*e^3 + 6551/45*e, -67/270*e^9 + 116/15*e^7 - 9727/135*e^5 + 4970/27*e^3 - 5699/135*e, 1897/3240*e^9 - 839/45*e^7 + 293257/1620*e^5 - 42221/81*e^3 + 103781/405*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -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;