/*
  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![-25, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17,17,-w + 7}>;

primesArray := [
[4, 2, 2],
[5, 5, -w + 5],
[5, 5, -w - 4],
[9, 3, 3],
[13, 13, w + 3],
[13, 13, w - 4],
[17, 17, w + 6],
[17, 17, -w + 7],
[19, 19, w + 2],
[19, 19, w - 3],
[23, 23, w + 1],
[23, 23, -w + 2],
[31, 31, -w - 7],
[31, 31, w - 8],
[37, 37, 2*w - 9],
[37, 37, 2*w + 7],
[43, 43, 4*w + 17],
[43, 43, 4*w - 21],
[47, 47, -w - 8],
[47, 47, w - 9],
[49, 7, -7],
[71, 71, 3*w - 14],
[71, 71, -5*w + 29],
[79, 79, 5*w + 21],
[79, 79, 5*w - 26],
[97, 97, 2*w - 3],
[97, 97, -2*w - 1],
[101, 101, 2*w - 1],
[107, 107, -w - 11],
[107, 107, w - 12],
[121, 11, -11],
[131, 131, -w - 12],
[131, 131, w - 13],
[137, 137, 3*w - 11],
[137, 137, -3*w - 8],
[157, 157, -w - 13],
[157, 157, w - 14],
[179, 179, -4*w - 13],
[179, 179, 4*w - 17],
[181, 181, 7*w + 29],
[181, 181, 7*w - 36],
[193, 193, 3*w - 22],
[193, 193, -3*w - 19],
[197, 197, -3*w - 4],
[197, 197, 3*w - 7],
[211, 211, -9*w + 52],
[211, 211, 5*w - 23],
[223, 223, 2*w - 19],
[223, 223, -2*w - 17],
[227, 227, 3*w - 2],
[227, 227, 3*w - 1],
[233, 233, 6*w - 29],
[233, 233, 6*w + 23],
[239, 239, 9*w - 47],
[239, 239, 9*w + 38],
[251, 251, -5*w + 22],
[251, 251, 5*w + 17],
[281, 281, -w - 17],
[281, 281, w - 18],
[283, 283, -4*w - 9],
[283, 283, 4*w - 13],
[307, 307, 7*w - 34],
[307, 307, 7*w + 27],
[317, 317, -w - 18],
[317, 317, w - 19],
[359, 359, 5*w - 19],
[359, 359, -5*w - 14],
[367, 367, -11*w + 64],
[367, 367, 7*w - 33],
[373, 373, -3*w - 23],
[373, 373, 3*w - 26],
[379, 379, 4*w - 7],
[379, 379, -4*w - 3],
[383, 383, 2*w - 23],
[383, 383, -2*w - 21],
[409, 409, 10*w + 41],
[409, 409, 10*w - 51],
[421, 421, 5*w - 17],
[421, 421, -5*w - 12],
[449, 449, -5*w - 11],
[449, 449, 5*w - 16],
[491, 491, 5*w - 36],
[491, 491, -5*w - 31],
[499, 499, 5*w - 14],
[499, 499, -5*w - 9],
[509, 509, 6*w - 23],
[509, 509, -6*w - 17],
[521, 521, -5*w - 8],
[521, 521, 5*w - 13],
[541, 541, 5*w - 12],
[541, 541, -5*w - 7],
[557, 557, 4*w - 33],
[557, 557, -4*w - 29],
[563, 563, 9*w + 34],
[563, 563, 9*w - 43],
[569, 569, 15*w + 64],
[569, 569, 15*w - 79],
[587, 587, -7*w - 22],
[587, 587, 7*w - 29],
[593, 593, 14*w + 59],
[593, 593, 14*w - 73],
[601, 601, -5*w - 3],
[601, 601, 5*w - 8],
[607, 607, 13*w - 67],
[607, 607, 13*w + 54],
[619, 619, -5*w - 1],
[619, 619, 5*w - 6],
[631, 631, 5*w - 2],
[631, 631, 5*w - 3],
[643, 643, 3*w - 31],
[643, 643, -3*w - 28],
[653, 653, -6*w - 13],
[653, 653, 6*w - 19],
[677, 677, -w - 26],
[677, 677, w - 27],
[683, 683, 2*w - 29],
[683, 683, -2*w - 27],
[691, 691, 6*w - 43],
[691, 691, -6*w - 37],
[701, 701, -5*w - 34],
[701, 701, 5*w - 39],
[727, 727, 13*w + 53],
[727, 727, 13*w - 66],
[743, 743, 7*w + 41],
[743, 743, 19*w + 82],
[761, 761, -16*w + 93],
[761, 761, 10*w - 47],
[787, 787, -w - 28],
[787, 787, w - 29],
[809, 809, -6*w - 7],
[809, 809, 6*w - 13],
[821, 821, 4*w - 37],
[821, 821, -4*w - 33],
[827, 827, 12*w - 59],
[827, 827, 12*w + 47],
[829, 829, 3*w - 34],
[829, 829, -3*w - 31],
[839, 839, 16*w - 83],
[839, 839, 16*w + 67],
[841, 29, -29],
[853, 853, 14*w + 57],
[853, 853, 14*w - 71],
[857, 857, 7*w - 23],
[857, 857, -7*w - 16],
[887, 887, 8*w - 31],
[887, 887, -8*w - 23],
[929, 929, -5*w - 37],
[929, 929, 5*w - 42],
[967, 967, -w - 31],
[967, 967, w - 32],
[977, 977, 17*w + 71],
[977, 977, 17*w - 88],
[991, 991, 8*w - 29],
[991, 991, -8*w - 21],
[997, 997, -7*w - 12],
[997, 997, 7*w - 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 + 2*x^8 - 18*x^7 - 17*x^6 + 117*x^5 + 4*x^4 - 258*x^3 + 118*x^2 + 91*x - 29;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 321/254*e^8 + 1269/254*e^7 - 1636/127*e^6 - 5838/127*e^5 + 14715/254*e^4 + 28519/254*e^3 - 13786/127*e^2 - 11967/254*e + 2746/127, 13/127*e^8 + 87/127*e^7 + 57/127*e^6 - 569/127*e^5 - 807/127*e^4 + 808/127*e^3 + 1385/127*e^2 - 17/127*e - 157/127, 57/127*e^8 + 421/254*e^7 - 1327/254*e^6 - 2065/127*e^5 + 3134/127*e^4 + 10749/254*e^3 - 11037/254*e^2 - 2722/127*e + 1427/254, -226/127*e^8 - 897/127*e^7 + 2272/127*e^6 + 8192/127*e^5 - 10042/127*e^4 - 19625/127*e^3 + 18438/127*e^2 + 7515/127*e - 3435/127, -48/127*e^8 - 204/127*e^7 + 405/127*e^6 + 1759/127*e^5 - 1563/127*e^4 - 4058/127*e^3 + 2682/127*e^2 + 1704/127*e - 153/127, -96/127*e^8 - 408/127*e^7 + 810/127*e^6 + 3518/127*e^5 - 3126/127*e^4 - 7989/127*e^3 + 5745/127*e^2 + 3281/127*e - 1195/127, 1, -137/254*e^8 - 307/127*e^7 + 1021/254*e^6 + 2583/127*e^5 - 3707/254*e^4 - 5762/127*e^3 + 7893/254*e^2 + 4165/254*e - 2683/254, -57/127*e^8 - 421/254*e^7 + 1327/254*e^6 + 2065/127*e^5 - 3134/127*e^4 - 10495/254*e^3 + 11545/254*e^2 + 1960/127*e - 2697/254, e^8 + 4*e^7 - 10*e^6 - 37*e^5 + 43*e^4 + 91*e^3 - 75*e^2 - 40*e + 10, -51/127*e^8 - 185/127*e^7 + 597/127*e^6 + 1734/127*e^5 - 3018/127*e^4 - 4137/127*e^3 + 5977/127*e^2 + 1239/127*e - 1377/127, 271/254*e^8 + 993/254*e^7 - 1560/127*e^6 - 4734/127*e^5 + 14849/254*e^4 + 23477/254*e^3 - 13167/127*e^2 - 9049/254*e + 2071/127, -28/127*e^8 - 119/127*e^7 + 268/127*e^6 + 1206/127*e^5 - 880/127*e^4 - 3362/127*e^3 + 866/127*e^2 + 2137/127*e + 6/127, -222/127*e^8 - 880/127*e^7 + 2270/127*e^6 + 8183/127*e^5 - 10134/127*e^4 - 20451/127*e^3 + 18532/127*e^2 + 9405/127*e - 3200/127, -207/254*e^8 - 424/127*e^7 + 1945/254*e^6 + 3773/127*e^5 - 8447/254*e^4 - 8885/127*e^3 + 16789/254*e^2 + 6269/254*e - 5335/254, 213/127*e^8 + 810/127*e^7 - 2329/127*e^6 - 7623/127*e^5 + 10849/127*e^4 + 18817/127*e^3 - 19950/127*e^2 - 7879/127*e + 3465/127, -63/127*e^8 - 236/127*e^7 + 730/127*e^6 + 2396/127*e^5 - 3250/127*e^4 - 6485/127*e^3 + 5314/127*e^2 + 3443/127*e - 685/127, -340/127*e^8 - 1445/127*e^7 + 2964/127*e^6 + 12830/127*e^5 - 11738/127*e^4 - 30755/127*e^3 + 21728/127*e^2 + 13975/127*e - 4100/127, -162/127*e^8 - 625/127*e^7 + 1732/127*e^6 + 5889/127*e^5 - 7831/127*e^4 - 14807/127*e^3 + 13592/127*e^2 + 7021/127*e - 1834/127, 206/127*e^8 + 812/127*e^7 - 2135/127*e^6 - 7639/127*e^5 + 9486/127*e^4 + 19437/127*e^3 - 16876/127*e^2 - 9599/127*e + 2133/127, 3*e^8 + 12*e^7 - 30*e^6 - 110*e^5 + 134*e^4 + 270*e^3 - 254*e^2 - 121*e + 51, 105/127*e^8 + 829/254*e^7 - 2137/254*e^6 - 3824/127*e^5 + 4697/127*e^4 + 18357/254*e^3 - 17417/254*e^2 - 3156/127*e + 3003/254, 268/127*e^8 + 1139/127*e^7 - 2293/127*e^6 - 9874/127*e^5 + 9076/127*e^4 + 22255/127*e^3 - 17197/127*e^2 - 7228/127*e + 2918/127, -60/127*e^8 - 383/254*e^7 + 1711/254*e^6 + 2040/127*e^5 - 4589/127*e^4 - 10653/254*e^3 + 18135/254*e^2 + 1241/127*e - 4637/254, -648/127*e^8 - 2627/127*e^7 + 6293/127*e^6 + 24064/127*e^5 - 26752/127*e^4 - 59101/127*e^3 + 48145/127*e^2 + 27068/127*e - 8860/127, 363/127*e^8 + 3149/254*e^7 - 5951/254*e^6 - 13739/127*e^5 + 11082/127*e^4 + 64965/254*e^3 - 40597/254*e^2 - 15490/127*e + 8807/254, 285/127*e^8 + 1116/127*e^7 - 3000/127*e^6 - 10579/127*e^5 + 13384/127*e^4 + 26809/127*e^3 - 24100/127*e^2 - 12340/127*e + 4901/127, -114/127*e^8 - 421/127*e^7 + 1327/127*e^6 + 4130/127*e^5 - 6141/127*e^4 - 10368/127*e^3 + 10148/127*e^2 + 3666/127*e - 792/127, 154/127*e^8 + 1055/254*e^7 - 4091/254*e^6 - 5617/127*e^5 + 10301/127*e^4 + 32283/254*e^3 - 35815/254*e^2 - 9912/127*e + 4633/254, 417/127*e^8 + 3227/254*e^7 - 8799/254*e^6 - 14940/127*e^5 + 20000/127*e^4 + 71619/254*e^3 - 73111/254*e^2 - 12581/127*e + 12993/254, 907/127*e^8 + 3696/127*e^7 - 8772/127*e^6 - 34013/127*e^5 + 37051/127*e^4 + 84421/127*e^3 - 66506/127*e^2 - 40009/127*e + 12170/127, 423/254*e^8 + 1010/127*e^7 - 2307/254*e^6 - 7826/127*e^5 + 4749/254*e^4 + 14777/127*e^3 - 8919/254*e^2 - 4539/254*e - 263/254, -350/127*e^8 - 1424/127*e^7 + 3350/127*e^6 + 12916/127*e^5 - 14048/127*e^4 - 31103/127*e^3 + 24795/127*e^2 + 13441/127*e - 3481/127, 284/127*e^8 + 1207/127*e^7 - 2428/127*e^6 - 10545/127*e^5 + 9343/127*e^4 + 24285/127*e^3 - 16948/127*e^2 - 9193/127*e + 4239/127, -1295/254*e^8 - 2482/127*e^7 + 13919/254*e^6 + 23158/127*e^5 - 64195/254*e^4 - 56569/127*e^3 + 117459/254*e^2 + 45401/254*e - 22265/254, -268/127*e^8 - 1266/127*e^7 + 1658/127*e^6 + 10382/127*e^5 - 4504/127*e^4 - 22763/127*e^3 + 9450/127*e^2 + 9514/127*e - 3045/127, 314/127*e^8 + 1398/127*e^7 - 2443/127*e^6 - 12200/127*e^5 + 8526/127*e^4 + 28885/127*e^3 - 15354/127*e^2 - 13306/127*e + 2382/127, -168/127*e^8 - 841/127*e^7 + 719/127*e^6 + 6347/127*e^5 - 454/127*e^4 - 11663/127*e^3 + 624/127*e^2 + 1646/127*e + 1306/127, -98/127*e^8 - 480/127*e^7 + 557/127*e^6 + 4094/127*e^5 - 1048/127*e^4 - 10243/127*e^3 + 1380/127*e^2 + 7416/127*e - 360/127, 103/127*e^8 + 939/254*e^7 - 1627/254*e^6 - 4391/127*e^5 + 2711/127*e^4 + 24263/254*e^3 - 9891/254*e^2 - 8673/127*e + 2387/254, -424/127*e^8 - 1802/127*e^7 + 3641/127*e^6 + 15686/127*e^5 - 14505/127*e^4 - 35888/127*e^3 + 27501/127*e^2 + 12766/127*e - 5479/127, 1367/254*e^8 + 2762/127*e^7 - 13447/254*e^6 - 25398/127*e^5 + 58475/254*e^4 + 62851/127*e^3 - 108401/254*e^2 - 59387/254*e + 21923/254, 1859/254*e^8 + 7615/254*e^7 - 8688/127*e^6 - 33889/127*e^5 + 73321/254*e^4 + 158597/254*e^3 - 67787/127*e^2 - 60089/254*e + 12206/127, -231/127*e^8 - 950/127*e^7 + 2211/127*e^6 + 8870/127*e^5 - 8784/127*e^4 - 22212/127*e^3 + 14320/127*e^2 + 11185/127*e - 2427/127, 1413/254*e^8 + 2955/127*e^7 - 12835/254*e^6 - 26561/127*e^5 + 52591/254*e^4 + 64642/127*e^3 - 98049/254*e^2 - 60893/254*e + 19609/254, -517/127*e^8 - 2102/127*e^7 + 4894/127*e^6 + 18594/127*e^5 - 21256/127*e^4 - 42655/127*e^3 + 40619/127*e^2 + 12956/127*e - 7482/127, -675/254*e^8 - 2837/254*e^7 + 3058/127*e^6 + 12872/127*e^5 - 24099/254*e^4 - 61717/254*e^3 + 21501/127*e^2 + 24407/254*e - 3969/127, -909/254*e^8 - 1757/127*e^7 + 9789/254*e^6 + 16723/127*e^5 - 44879/254*e^4 - 43020/127*e^3 + 82207/254*e^2 + 44271/254*e - 16161/254, 345/127*e^8 + 3123/254*e^7 - 4917/254*e^6 - 13127/127*e^5 + 7813/127*e^4 + 58683/254*e^3 - 28235/254*e^2 - 11549/127*e + 3517/254, -125/127*e^8 - 309/127*e^7 + 2158/127*e^6 + 3615/127*e^5 - 12365/127*e^4 - 9684/127*e^3 + 22780/127*e^2 + 2088/127*e - 4137/127, -188/127*e^8 - 672/127*e^7 + 2380/127*e^6 + 7154/127*e^5 - 11297/127*e^4 - 20614/127*e^3 + 18442/127*e^2 + 12135/127*e - 1012/127, -1184/127*e^8 - 4778/127*e^7 + 11641/127*e^6 + 43939/127*e^5 - 49984/127*e^4 - 108056/127*e^3 + 89651/127*e^2 + 48890/127*e - 15712/127, 411/254*e^8 + 794/127*e^7 - 4333/254*e^6 - 7241/127*e^5 + 20519/254*e^4 + 17032/127*e^3 - 41713/254*e^2 - 11225/254*e + 11097/254, 117/127*e^8 + 931/254*e^7 - 2149/254*e^6 - 3724/127*e^5 + 4929/127*e^4 + 12639/254*e^3 - 21425/254*e^2 + 2641/127*e + 5937/254, -139/127*e^8 - 991/254*e^7 + 3441/254*e^6 + 5107/127*e^5 - 8233/127*e^4 - 29207/254*e^3 + 27503/254*e^2 + 10459/127*e - 3061/254, 345/127*e^8 + 1371/127*e^7 - 3411/127*e^6 - 12365/127*e^5 + 14925/127*e^4 + 29532/127*e^3 - 27008/127*e^2 - 12692/127*e + 4235/127, 1557/254*e^8 + 3007/127*e^7 - 16463/254*e^6 - 27739/127*e^5 + 75949/254*e^4 + 66665/127*e^3 - 141655/254*e^2 - 48733/254*e + 26545/254, 677/127*e^8 + 5437/254*e^7 - 13377/254*e^6 - 24796/127*e^5 + 29387/127*e^4 + 120195/254*e^3 - 108897/254*e^2 - 24478/127*e + 18905/254, -1173/127*e^8 - 4636/127*e^7 + 11953/127*e^6 + 42676/127*e^5 - 53539/127*e^4 - 104295/127*e^3 + 99244/127*e^2 + 44118/127*e - 17574/127, 633/254*e^8 + 1234/127*e^7 - 6603/254*e^6 - 11396/127*e^5 + 30145/254*e^4 + 27575/127*e^3 - 56943/254*e^2 - 20757/254*e + 11503/254, -2153/254*e^8 - 9055/254*e^7 + 9460/127*e^6 + 39776/127*e^5 - 75449/254*e^4 - 185135/254*e^3 + 68841/127*e^2 + 74209/254*e - 11603/127, -503/254*e^8 - 1725/254*e^7 + 3269/127*e^6 + 8805/127*e^5 - 33643/254*e^4 - 47705/254*e^3 + 32031/127*e^2 + 26937/254*e - 9267/127, -87/127*e^8 - 211/127*e^7 + 1631/127*e^6 + 2958/127*e^5 - 9175/127*e^4 - 9149/127*e^3 + 16053/127*e^2 + 3406/127*e - 1079/127, -27/127*e^8 + 44/127*e^7 + 1093/127*e^6 + 664/127*e^5 - 7126/127*e^4 - 4521/127*e^3 + 11875/127*e^2 + 5086/127*e - 1618/127, 128/127*e^8 + 707/254*e^7 - 4319/254*e^6 - 4606/127*e^5 + 11407/127*e^4 + 29813/254*e^3 - 38307/254*e^2 - 10513/127*e + 9833/254, 963/254*e^8 + 3553/254*e^7 - 5543/127*e^6 - 17006/127*e^5 + 53289/254*e^4 + 84541/254*e^3 - 49359/127*e^2 - 31837/254*e + 10016/127, 75/127*e^8 + 414/127*e^7 - 101/127*e^6 - 2804/127*e^5 - 963/127*e^4 + 4261/127*e^3 - 587/127*e^2 - 313/127*e + 2533/127, 418/127*e^8 + 3299/254*e^7 - 8419/254*e^6 - 14847/127*e^5 + 19088/127*e^4 + 70571/254*e^3 - 70905/254*e^2 - 15728/127*e + 11015/254, 1330/127*e^8 + 5462/127*e^7 - 12476/127*e^6 - 49157/127*e^5 + 51960/127*e^4 + 117150/127*e^3 - 93459/127*e^2 - 47850/127*e + 14828/127, -626/127*e^8 - 5067/254*e^7 + 11929/254*e^6 + 22554/127*e^5 - 25480/127*e^4 - 105063/254*e^3 + 93387/254*e^2 + 20826/127*e - 16913/254, -803/254*e^8 - 1500/127*e^7 + 9101/254*e^6 + 14159/127*e^5 - 45031/254*e^4 - 35486/127*e^3 + 87873/254*e^2 + 29713/254*e - 18125/254, -517/127*e^8 - 2229/127*e^7 + 4259/127*e^6 + 19229/127*e^5 - 16303/127*e^4 - 44306/127*e^3 + 30586/127*e^2 + 18544/127*e - 5704/127, -275/127*e^8 - 1264/127*e^7 + 1725/127*e^6 + 9858/127*e^5 - 4851/127*e^4 - 18206/127*e^3 + 8968/127*e^2 + 428/127*e - 313/127, 208/127*e^8 + 1011/127*e^7 - 1120/127*e^6 - 8088/127*e^5 + 2328/127*e^4 + 17246/127*e^3 - 5780/127*e^2 - 7257/127*e + 2949/127, 819/254*e^8 + 2941/254*e^7 - 4999/127*e^6 - 14812/127*e^5 + 48219/254*e^4 + 78971/254*e^3 - 42796/127*e^2 - 39425/254*e + 7310/127, 933/127*e^8 + 3616/127*e^7 - 9928/127*e^6 - 34008/127*e^5 + 45216/127*e^4 + 85021/127*e^3 - 81897/127*e^2 - 37376/127*e + 12999/127, 1361/254*e^8 + 2527/127*e^7 - 15349/254*e^6 - 23645/127*e^5 + 74615/254*e^4 + 57692/127*e^3 - 141943/254*e^2 - 42283/254*e + 29381/254, -996/127*e^8 - 8339/254*e^7 + 17633/254*e^6 + 36404/127*e^5 - 35893/127*e^4 - 165867/254*e^3 + 134671/254*e^2 + 28119/127*e - 22161/254, 536/127*e^8 + 2278/127*e^7 - 4713/127*e^6 - 20510/127*e^5 + 18279/127*e^4 + 50225/127*e^3 - 32616/127*e^2 - 25251/127*e + 7741/127, -145/254*e^8 - 775/254*e^7 + 195/127*e^6 + 2973/127*e^5 + 2065/254*e^4 - 12285/254*e^3 - 3704/127*e^2 + 5719/254*e + 519/127, 1048/127*e^8 + 4454/127*e^7 - 9033/127*e^6 - 39061/127*e^5 + 35459/127*e^4 + 90547/127*e^3 - 65161/127*e^2 - 34283/127*e + 9373/127, -229/127*e^8 - 878/127*e^7 + 2464/127*e^6 + 8167/127*e^5 - 11370/127*e^4 - 19196/127*e^3 + 21352/127*e^2 + 5018/127*e - 5167/127, -1725/254*e^8 - 6855/254*e^7 + 8845/127*e^6 + 32119/127*e^5 - 79197/254*e^4 - 163535/254*e^3 + 72981/127*e^2 + 81367/254*e - 13445/127, 1189/127*e^8 + 4704/127*e^7 - 12215/127*e^6 - 43982/127*e^5 + 54187/127*e^4 + 110389/127*e^3 - 98741/127*e^2 - 52179/127*e + 18387/127, 918/127*e^8 + 7803/254*e^7 - 16031/254*e^6 - 34768/127*e^5 + 31337/127*e^4 + 167347/254*e^3 - 111413/254*e^2 - 38558/127*e + 14901/254, -443/254*e^8 - 1597/254*e^7 + 2746/127*e^6 + 8293/127*e^5 - 26133/254*e^4 - 45109/254*e^3 + 23211/127*e^2 + 22267/254*e - 6806/127, -3513/254*e^8 - 13819/254*e^7 + 18182/127*e^6 + 64547/127*e^5 - 162279/254*e^4 - 322633/254*e^3 + 147222/127*e^2 + 148651/254*e - 27804/127, -1098/127*e^8 - 4349/127*e^7 + 10963/127*e^6 + 39110/127*e^5 - 49041/127*e^4 - 91271/127*e^3 + 92688/127*e^2 + 31486/127*e - 18851/127, -441/127*e^8 - 1779/127*e^7 + 4348/127*e^6 + 16391/127*e^5 - 18940/127*e^4 - 41077/127*e^3 + 35039/127*e^2 + 20672/127*e - 7589/127, 446/127*e^8 + 3283/254*e^7 - 10479/254*e^6 - 16180/127*e^5 + 25175/127*e^4 + 85423/254*e^3 - 91687/254*e^2 - 21675/127*e + 15575/254, 3863/254*e^8 + 7431/127*e^7 - 41365/254*e^6 - 69354/127*e^5 + 191059/254*e^4 + 170391/127*e^3 - 352513/254*e^2 - 139613/254*e + 67217/254, 1765/254*e^8 + 7025/254*e^7 - 8982/127*e^6 - 32926/127*e^5 + 78023/254*e^4 + 165181/254*e^3 - 68828/127*e^2 - 75167/254*e + 9921/127, 566/127*e^8 + 2215/127*e^7 - 5998/127*e^6 - 21022/127*e^5 + 27368/127*e^4 + 53555/127*e^3 - 51215/127*e^2 - 23776/127*e + 12869/127, -1353/127*e^8 - 5401/127*e^7 + 13440/127*e^6 + 49050/127*e^5 - 59559/127*e^4 - 118179/127*e^3 + 111524/127*e^2 + 50889/127*e - 23450/127, 983/254*e^8 + 1946/127*e^7 - 9953/254*e^6 - 17981/127*e^5 + 43685/254*e^4 + 44968/127*e^3 - 78563/254*e^2 - 47533/254*e + 17651/254, -1829/254*e^8 - 7297/254*e^7 + 9125/127*e^6 + 33379/127*e^5 - 80107/254*e^4 - 162379/254*e^3 + 72902/127*e^2 + 71089/254*e - 12436/127, 1042/127*e^8 + 4238/127*e^7 - 9919/127*e^6 - 37968/127*e^5 + 42201/127*e^4 + 88611/127*e^3 - 78256/127*e^2 - 31911/127*e + 18101/127, 377/127*e^8 + 1507/127*e^7 - 3808/127*e^6 - 14215/127*e^5 + 15967/127*e^4 + 35878/127*e^3 - 26383/127*e^2 - 17765/127*e + 3321/127, 126/127*e^8 + 726/127*e^7 - 63/127*e^6 - 5300/127*e^5 - 3660/127*e^4 + 9795/127*e^3 + 7152/127*e^2 - 3584/127*e + 100/127, 759/254*e^8 + 1851/127*e^7 - 3745/254*e^6 - 14173/127*e^5 + 5911/254*e^4 + 26948/127*e^3 - 11691/254*e^2 - 12149/254*e + 935/254, -519/127*e^8 - 1920/127*e^7 + 5911/127*e^6 + 18281/127*e^5 - 28195/127*e^4 - 45925/127*e^3 + 51494/127*e^2 + 20139/127*e - 11346/127, 539/127*e^8 + 2513/127*e^7 - 3381/127*e^6 - 20358/127*e^5 + 9193/127*e^4 + 43192/127*e^3 - 16988/127*e^2 - 15261/127*e + 1853/127, -479/127*e^8 - 2004/127*e^7 + 4367/127*e^6 + 18064/127*e^5 - 18066/127*e^4 - 44660/127*e^3 + 34019/127*e^2 + 24434/127*e - 7726/127, -525/254*e^8 - 1068/127*e^7 + 5025/254*e^6 + 9560/127*e^5 - 22723/254*e^4 - 23486/127*e^3 + 47289/254*e^2 + 23273/254*e - 16969/254, 2189/254*e^8 + 4477/127*e^7 - 20843/254*e^6 - 40642/127*e^5 + 87829/254*e^4 + 98312/127*e^3 - 158807/254*e^2 - 85647/254*e + 25829/254, 1/127*e^8 - 218/127*e^7 - 1207/127*e^6 + 347/127*e^5 + 8359/127*e^4 + 4175/127*e^3 - 11851/127*e^2 - 8100/127*e - 608/127, -59/254*e^8 - 473/254*e^7 - 271/127*e^6 + 1765/127*e^5 + 6691/254*e^4 - 7565/254*e^3 - 7837/127*e^2 + 5079/254*e + 3585/127, 537/254*e^8 + 1157/127*e^7 - 4015/254*e^6 - 9002/127*e^5 + 14827/254*e^4 + 15389/127*e^3 - 30243/254*e^2 + 7543/254*e + 7895/254, -697/254*e^8 - 3629/254*e^7 + 1349/127*e^6 + 14516/127*e^5 + 1807/254*e^4 - 64413/254*e^3 - 3205/127*e^2 + 36491/254*e + 687/127, -556/127*e^8 - 2490/127*e^7 + 4088/127*e^6 + 20936/127*e^5 - 14009/127*e^4 - 47111/127*e^3 + 27447/127*e^2 + 19738/127*e - 8789/127, -1254/127*e^8 - 10405/254*e^7 + 23225/254*e^6 + 47081/127*e^5 - 47866/127*e^4 - 228985/254*e^3 + 176393/254*e^2 + 50105/127*e - 38887/254, -650/127*e^8 - 2445/127*e^7 + 7183/127*e^6 + 22862/127*e^5 - 34199/127*e^4 - 56275/127*e^3 + 63846/127*e^2 + 22186/127*e - 10565/127, 714/127*e^8 + 2590/127*e^7 - 8612/127*e^6 - 25800/127*e^5 + 42125/127*e^4 + 68205/127*e^3 - 77201/127*e^2 - 31824/127*e + 14706/127, -3925/254*e^8 - 7658/127*e^7 + 41269/254*e^6 + 71678/127*e^5 - 187347/254*e^4 - 178404/127*e^3 + 343309/254*e^2 + 156165/254*e - 61017/254, 160/127*e^8 + 1061/127*e^7 + 428/127*e^6 - 7980/127*e^5 - 8379/127*e^4 + 17125/127*e^3 + 15317/127*e^2 - 9363/127*e - 3046/127, -530/127*e^8 - 2189/127*e^7 + 4583/127*e^6 + 17893/127*e^5 - 19687/127*e^4 - 35462/127*e^3 + 39869/127*e^2 + 3956/127*e - 8595/127, 555/127*e^8 + 2200/127*e^7 - 5548/127*e^6 - 19759/127*e^5 + 25208/127*e^4 + 46365/127*e^3 - 48743/127*e^2 - 16083/127*e + 9651/127, -961/254*e^8 - 3481/254*e^7 + 5733/127*e^6 + 16845/127*e^5 - 57653/254*e^4 - 86605/254*e^3 + 55161/127*e^2 + 40529/254*e - 12148/127, 325/127*e^8 + 1032/127*e^7 - 4544/127*e^6 - 10669/127*e^5 + 24021/127*e^4 + 27820/127*e^3 - 43226/127*e^2 - 9823/127*e + 6235/127, 57/127*e^8 + 274/127*e^7 - 346/127*e^6 - 2319/127*e^5 + 848/127*e^4 + 5311/127*e^3 - 2407/127*e^2 - 2087/127*e + 1793/127, -1757/254*e^8 - 7245/254*e^7 + 8091/127*e^6 + 32155/127*e^5 - 67285/254*e^4 - 150069/254*e^3 + 62953/127*e^2 + 60405/254*e - 16544/127, 1137/127*e^8 + 8839/254*e^7 - 24251/254*e^6 - 41960/127*e^5 + 55002/127*e^4 + 213933/254*e^3 - 199545/254*e^2 - 51476/127*e + 38665/254, 874/127*e^8 + 3778/127*e^7 - 7168/127*e^6 - 32637/127*e^5 + 26761/127*e^4 + 73773/127*e^3 - 48549/127*e^2 - 25439/127*e + 5945/127, -1612/127*e^8 - 6597/127*e^7 + 15284/127*e^6 + 59634/127*e^5 - 64397/127*e^4 - 143372/127*e^3 + 116423/127*e^2 + 60401/127*e - 18124/127, -1259/127*e^8 - 5192/127*e^7 + 11615/127*e^6 + 46108/127*e^5 - 48894/127*e^4 - 109015/127*e^3 + 91762/127*e^2 + 46663/127*e - 16975/127, 523/127*e^8 + 2064/127*e^7 - 5151/127*e^6 - 18036/127*e^5 + 23404/127*e^4 + 40019/127*e^3 - 45431/127*e^2 - 11264/127*e + 9803/127, -307/127*e^8 - 1273/127*e^7 + 2757/127*e^6 + 11073/127*e^5 - 11481/127*e^4 - 25441/127*e^3 + 21551/127*e^2 + 11343/127*e - 3209/127, 1994/127*e^8 + 8030/127*e^7 - 19666/127*e^6 - 73765/127*e^5 + 85075/127*e^4 + 181329/127*e^3 - 153928/127*e^2 - 82090/127*e + 25898/127, -849/254*e^8 - 1566/127*e^7 + 9759/254*e^6 + 14814/127*e^5 - 47529/254*e^4 - 35372/127*e^3 + 90475/254*e^2 + 18773/254*e - 15557/254, -283/127*e^8 - 1044/127*e^7 + 3126/127*e^6 + 9241/127*e^5 - 15843/127*e^4 - 20618/127*e^3 + 32402/127*e^2 + 3887/127*e - 8784/127, -287/127*e^8 - 1188/127*e^7 + 2747/127*e^6 + 11028/127*e^5 - 12068/127*e^4 - 28936/127*e^3 + 24688/127*e^2 + 17999/127*e - 6352/127, -504/127*e^8 - 3649/254*e^7 + 11807/254*e^6 + 17517/127*e^5 - 28159/127*e^4 - 86361/254*e^3 + 100645/254*e^2 + 17130/127*e - 19977/254, -275/127*e^8 - 2401/254*e^7 + 4593/254*e^6 + 10874/127*e^5 - 7899/127*e^4 - 52287/254*e^3 + 24159/254*e^2 + 10461/127*e + 2041/254, 703/127*e^8 + 2448/127*e^7 - 9051/127*e^6 - 25299/127*e^5 + 45680/127*e^4 + 69778/127*e^3 - 84254/127*e^2 - 36196/127*e + 18092/127, -138/127*e^8 - 523/127*e^7 + 1593/127*e^6 + 5327/127*e^5 - 7240/127*e^4 - 14683/127*e^3 + 12759/127*e^2 + 8963/127*e - 2075/127, 593/127*e^8 + 2552/127*e^7 - 5059/127*e^6 - 22956/127*e^5 + 18365/127*e^4 + 55790/127*e^3 - 29689/127*e^2 - 26195/127*e + 2676/127, 826/127*e^8 + 3701/127*e^7 - 6128/127*e^6 - 31386/127*e^5 + 20753/127*e^4 + 70477/127*e^3 - 38628/127*e^2 - 26148/127*e + 3633/127, 555/127*e^8 + 2073/127*e^7 - 6056/127*e^6 - 18616/127*e^5 + 29272/127*e^4 + 41793/127*e^3 - 55982/127*e^2 - 10622/127*e + 11556/127, -781/254*e^8 - 1485/127*e^7 + 8455/254*e^6 + 13658/127*e^5 - 40711/254*e^4 - 33249/127*e^3 + 77087/254*e^2 + 24741/254*e - 10419/254, -466/127*e^8 - 1917/127*e^7 + 4297/127*e^6 + 16733/127*e^5 - 19000/127*e^4 - 38645/127*e^3 + 39214/127*e^2 + 15146/127*e - 11185/127, 148/127*e^8 + 502/127*e^7 - 1979/127*e^6 - 5159/127*e^5 + 10947/127*e^4 + 14904/127*e^3 - 22684/127*e^2 - 9826/127*e + 6282/127, 1175/254*e^8 + 2608/127*e^7 - 9033/254*e^6 - 22515/127*e^5 + 30125/254*e^4 + 50544/127*e^3 - 52715/254*e^2 - 31997/254*e + 5563/254, 789/127*e^8 + 3131/127*e^7 - 7824/127*e^6 - 27969/127*e^5 + 35193/127*e^4 + 64973/127*e^3 - 66866/127*e^2 - 22104/127*e + 13048/127, -855/254*e^8 - 1801/127*e^7 + 7603/254*e^6 + 16059/127*e^5 - 30881/254*e^4 - 39134/127*e^3 + 57187/254*e^2 + 37147/254*e - 7337/254, 737/127*e^8 + 3291/127*e^7 - 5639/127*e^6 - 28868/127*e^5 + 17847/127*e^4 + 68345/127*e^3 - 27702/127*e^2 - 33085/127*e + 3516/127, 1475/127*e^8 + 5856/127*e^7 - 14771/127*e^6 - 53071/127*e^5 + 66405/127*e^4 + 127149/127*e^3 - 126183/127*e^2 - 51156/127*e + 25220/127, 2931/254*e^8 + 11663/254*e^7 - 14798/127*e^6 - 53764/127*e^5 + 131215/254*e^4 + 264635/254*e^3 - 120723/127*e^2 - 119481/254*e + 21598/127, -388/127*e^8 - 1649/127*e^7 + 3242/127*e^6 + 14081/127*e^5 - 12158/127*e^4 - 30368/127*e^3 + 21362/127*e^2 + 7424/127*e - 3999/127, 603/127*e^8 + 2277/127*e^7 - 6715/127*e^6 - 21518/127*e^5 + 32359/127*e^4 + 53979/127*e^3 - 62855/127*e^2 - 23248/127*e + 17424/127, -900/127*e^8 - 3571/127*e^7 + 9213/127*e^6 + 33521/127*e^5 - 40260/127*e^4 - 85041/127*e^3 + 70417/127*e^2 + 42491/127*e - 10076/127, 163/127*e^8 + 534/127*e^7 - 2431/127*e^6 - 6558/127*e^5 + 12126/127*e^4 + 20887/127*e^3 - 19982/127*e^2 - 13089/127*e - 1568/127, -129/127*e^8 - 707/127*e^7 + 382/127*e^6 + 5910/127*e^5 + 2078/127*e^4 - 13303/127*e^3 - 6143/127*e^2 + 4643/127*e + 2486/127, 67/254*e^8 + 1015/254*e^7 + 1539/127*e^6 - 2917/127*e^5 - 25671/254*e^4 + 10739/254*e^3 + 24314/127*e^2 - 17555/254*e - 5255/127, -389/127*e^8 - 1431/127*e^7 + 4576/127*e^6 + 14242/127*e^5 - 21533/127*e^4 - 37718/127*e^3 + 37277/127*e^2 + 17810/127*e - 2629/127, -583/254*e^8 - 1096/127*e^7 + 6451/254*e^6 + 10292/127*e^5 - 29771/254*e^4 - 24546/127*e^3 + 51895/254*e^2 + 15045/254*e - 4819/254];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17,17,-w + 7}>] := -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;