/*
  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![-55, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {2, 2, w + 1}>;

primesArray := [
[2, 2, w + 1],
[3, 3, w + 1],
[3, 3, w + 2],
[5, 5, -2*w + 15],
[11, 11, 3*w - 22],
[13, 13, w + 4],
[13, 13, w + 9],
[17, 17, w + 2],
[17, 17, w + 15],
[19, 19, -w - 6],
[19, 19, w - 6],
[23, 23, w + 3],
[23, 23, w + 20],
[47, 47, w + 14],
[47, 47, w + 33],
[49, 7, -7],
[67, 67, w + 16],
[67, 67, w + 51],
[73, 73, w + 36],
[73, 73, w + 37],
[79, 79, 5*w + 36],
[79, 79, 5*w - 36],
[89, 89, -w - 12],
[89, 89, w - 12],
[103, 103, w + 40],
[103, 103, w + 63],
[131, 131, -6*w - 43],
[131, 131, -6*w + 43],
[139, 139, 2*w - 9],
[139, 139, -2*w - 9],
[151, 151, 4*w - 27],
[151, 151, 4*w + 27],
[163, 163, w + 50],
[163, 163, w + 113],
[173, 173, w + 48],
[173, 173, w + 125],
[181, 181, -3*w - 26],
[181, 181, 3*w - 26],
[193, 193, w + 21],
[193, 193, w + 172],
[197, 197, w + 45],
[197, 197, w + 152],
[211, 211, 2*w - 3],
[211, 211, -2*w - 3],
[223, 223, w + 72],
[223, 223, w + 151],
[229, 229, 6*w + 47],
[229, 229, 6*w - 47],
[233, 233, w + 88],
[233, 233, w + 145],
[239, 239, -3*w - 16],
[239, 239, 3*w - 16],
[269, 269, -w - 18],
[269, 269, w - 18],
[271, 271, -8*w - 57],
[271, 271, -8*w + 57],
[277, 277, w + 71],
[277, 277, w + 206],
[293, 293, w + 73],
[293, 293, w + 220],
[337, 337, w + 27],
[337, 337, w + 310],
[359, 359, 9*w + 64],
[359, 359, 9*w - 64],
[367, 367, w + 34],
[367, 367, w + 333],
[373, 373, w + 146],
[373, 373, w + 227],
[383, 383, w + 173],
[383, 383, w + 210],
[389, 389, -5*w - 42],
[389, 389, 5*w - 42],
[401, 401, 29*w - 216],
[401, 401, 11*w - 84],
[421, 421, -6*w + 49],
[421, 421, -6*w - 49],
[431, 431, 3*w - 8],
[431, 431, -3*w - 8],
[439, 439, -4*w - 21],
[439, 439, 4*w - 21],
[443, 443, w + 99],
[443, 443, w + 344],
[449, 449, -8*w - 63],
[449, 449, -8*w + 63],
[457, 457, w + 91],
[457, 457, w + 366],
[463, 463, w + 38],
[463, 463, w + 425],
[467, 467, w + 191],
[467, 467, w + 276],
[479, 479, -3*w - 4],
[479, 479, 3*w - 4],
[487, 487, w + 223],
[487, 487, w + 264],
[491, 491, 3*w - 2],
[491, 491, -3*w - 2],
[509, 509, 2*w - 27],
[509, 509, -2*w - 27],
[521, 521, -w - 24],
[521, 521, w - 24],
[557, 557, w + 75],
[557, 557, w + 482],
[571, 571, 11*w + 78],
[571, 571, 11*w - 78],
[587, 587, w + 246],
[587, 587, w + 341],
[593, 593, w + 184],
[593, 593, w + 409],
[613, 613, w + 235],
[613, 613, w + 378],
[641, 641, -4*w - 39],
[641, 641, 4*w - 39],
[643, 643, w + 174],
[643, 643, w + 469],
[647, 647, w + 257],
[647, 647, w + 390],
[659, 659, -18*w + 131],
[659, 659, 30*w - 221],
[661, 661, 3*w - 34],
[661, 661, -3*w - 34],
[673, 673, w + 196],
[673, 673, w + 477],
[677, 677, w + 240],
[677, 677, w + 437],
[683, 683, w + 136],
[683, 683, w + 547],
[709, 709, 21*w - 158],
[709, 709, 27*w - 202],
[727, 727, w + 339],
[727, 727, w + 388],
[733, 733, w + 39],
[733, 733, w + 694],
[739, 739, 10*w - 69],
[739, 739, 10*w + 69],
[811, 811, 35*w - 258],
[811, 811, -19*w + 138],
[823, 823, w + 177],
[823, 823, w + 646],
[829, 829, -6*w - 53],
[829, 829, 6*w - 53],
[841, 29, -29],
[853, 853, w + 316],
[853, 853, w + 537],
[857, 857, w + 270],
[857, 857, w + 587],
[863, 863, w + 405],
[863, 863, w + 458],
[877, 877, w + 377],
[877, 877, w + 500],
[881, 881, -19*w + 144],
[881, 881, -37*w + 276],
[883, 883, w + 52],
[883, 883, w + 831],
[907, 907, w + 350],
[907, 907, w + 557],
[919, 919, -8*w - 51],
[919, 919, 8*w - 51],
[929, 929, 5*w - 48],
[929, 929, -5*w - 48],
[937, 937, w + 176],
[937, 937, w + 761],
[947, 947, w + 252],
[947, 947, w + 695],
[953, 953, w + 334],
[953, 953, w + 619],
[961, 31, -31],
[983, 983, w + 277],
[983, 983, w + 706],
[997, 997, w + 371],
[997, 997, w + 626]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 16*x^6 + 138*x^4 - 272*x^2 + 1369;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/504*e^6 - 5/168*e^4 - 55/168*e^2 + 101/504, 4/3367*e^7 + 219/13468*e^5 + 478/3367*e^3 + 3011/13468*e, 113/40404*e^7 + 363/6734*e^5 + 7307/13468*e^3 + 15157/20202*e, 2, -4/3367*e^7 - 219/13468*e^5 - 478/3367*e^3 + 10457/13468*e, -5/546*e^6 - 57/364*e^4 - 275/182*e^2 + 905/1092, -1/364*e^6 - 2/91*e^4 - 165/364*e^2 + 34/91, 9/728*e^6 + 163/728*e^4 + 1485/728*e^2 - 769/728, -1/2184*e^6 - 33/728*e^4 - 55/728*e^2 - 319/2184, -29/3848*e^7 - 427/3848*e^5 - 3225/3848*e^3 + 17841/3848*e, -3/3848*e^7 - 11/3848*e^5 - 599/3848*e^3 + 3073/3848*e, -5/1554*e^7 - 39/518*e^5 - 415/518*e^3 - 1637/1554*e, 5/1554*e^7 + 39/518*e^5 + 415/518*e^3 + 1637/1554*e, 23/5772*e^7 + 135/1924*e^5 + 1317/1924*e^3 + 5621/5772*e, 23/5772*e^7 + 135/1924*e^5 + 1317/1924*e^3 + 5621/5772*e, 3, 23/5772*e^7 + 135/1924*e^5 + 1317/1924*e^3 + 5621/5772*e, 23/5772*e^7 + 135/1924*e^5 + 1317/1924*e^3 + 5621/5772*e, -1/182*e^6 - 4/91*e^4 - 165/182*e^2 + 68/91, -5/273*e^6 - 57/182*e^4 - 275/91*e^2 + 905/546, -75/13468*e^7 - 1237/13468*e^5 - 7279/13468*e^3 + 41231/13468*e, 107/13468*e^7 + 1675/13468*e^5 + 11103/13468*e^3 - 62145/13468*e, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 + 68/91, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 - 796/91, 97/13468*e^7 + 1959/13468*e^5 + 20009/13468*e^3 + 27303/13468*e, 31/40404*e^7 - 69/13468*e^5 - 1571/13468*e^3 - 3215/40404*e, 535/26936*e^7 + 8375/26936*e^5 + 55515/26936*e^3 - 310725/26936*e, -375/26936*e^7 - 6185/26936*e^5 - 36395/26936*e^3 + 206155/26936*e, 44/3367*e^7 + 2409/13468*e^5 + 5258/3367*e^3 - 115027/13468*e, 44/3367*e^7 + 2409/13468*e^5 + 5258/3367*e^3 - 115027/13468*e, 27/13468*e^7 + 145/3367*e^5 + 1543/13468*e^3 - 2465/3367*e, -155/13468*e^7 - 583/3367*e^5 - 16839/13468*e^3 + 23379/3367*e, 17/40404*e^7 + 72/3367*e^5 + 3483/13468*e^3 + 3062/10101*e, -89/20202*e^7 - 1233/13468*e^5 - 6351/6734*e^3 - 51595/40404*e, -2/91*e^6 - 155/364*e^4 - 330/91*e^2 + 633/364, 11/1092*e^6 + 45/182*e^4 + 605/364*e^2 - 293/546, 3/182*e^6 + 12/91*e^4 - 51/182*e^2 - 477/91, -3/182*e^6 - 12/91*e^4 + 51/182*e^2 + 2115/91, -23/2184*e^6 - 31/728*e^4 - 1265/728*e^2 + 3583/2184, -107/2184*e^6 - 619/728*e^4 - 5885/728*e^2 + 9547/2184, 5/156*e^6 + 35/52*e^4 + 275/52*e^2 - 355/156, -5/156*e^6 - 35/52*e^4 - 275/52*e^2 + 355/156, 641/26936*e^7 + 9405/26936*e^5 + 71549/26936*e^3 - 395575/26936*e, 95/26936*e^7 + 669/26936*e^5 + 16403/26936*e^3 - 85447/26936*e, -257/20202*e^7 - 1383/6734*e^5 - 13043/6734*e^3 - 57413/20202*e, -129/6734*e^7 - 2397/6734*e^5 - 23833/6734*e^3 - 33325/6734*e, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 + 1115/91, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 + 1979/91, -71/2184*e^6 - 523/728*e^4 - 3905/728*e^2 + 4651/2184, 97/2184*e^6 + 653/728*e^4 + 5335/728*e^2 - 7277/2184, 51/6734*e^7 + 1817/13468*e^5 + 4411/6734*e^3 - 51091/13468*e, -131/6734*e^7 - 4007/13468*e^5 - 13971/6734*e^3 + 155661/13468*e, 3/182*e^6 + 12/91*e^4 - 51/182*e^2 - 1933/91, -3/182*e^6 - 12/91*e^4 + 51/182*e^2 + 659/91, 51/6734*e^7 + 1817/13468*e^5 + 4411/6734*e^3 - 51091/13468*e, -131/6734*e^7 - 4007/13468*e^5 - 13971/6734*e^3 + 155661/13468*e, -19/546*e^6 - 81/182*e^4 - 1045/182*e^2 + 2129/546, -11/182*e^6 - 179/182*e^4 - 1815/182*e^2 + 1041/182, 4/91*e^6 + 155/182*e^4 + 660/91*e^2 - 633/182, -11/546*e^6 - 45/91*e^4 - 605/182*e^2 + 293/273, -89/2184*e^6 - 571/728*e^4 - 4895/728*e^2 + 7099/2184, 37/2184*e^6 + 311/728*e^4 + 2035/728*e^2 - 1847/2184, 139/6734*e^7 + 2113/6734*e^5 + 14927/6734*e^3 - 83059/6734*e, -43/6734*e^7 - 799/6734*e^5 - 3455/6734*e^3 + 20317/6734*e, 935/40404*e^7 + 5739/13468*e^5 + 56885/13468*e^3 + 239297/40404*e, 225/13468*e^7 + 3711/13468*e^5 + 35305/13468*e^3 + 51391/13468*e, -1/39*e^6 - 7/13*e^4 - 55/13*e^2 + 71/39, 1/39*e^6 + 7/13*e^4 + 55/13*e^2 - 71/39, 1/13468*e^7 + 645/13468*e^5 + 8537/13468*e^3 + 9237/13468*e, -647/40404*e^7 - 4425/13468*e^5 - 45413/13468*e^3 - 185099/40404*e, -20, -20, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 - 2252/91, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 - 1388/91, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 + 705/91, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 - 159/91, 59/13468*e^7 + 509/6734*e^5 + 5367/13468*e^3 - 15387/6734*e, -123/13468*e^7 - 947/6734*e^5 - 13015/13468*e^3 + 36301/6734*e, 219/13468*e^7 + 802/3367*e^5 + 24487/13468*e^3 - 33836/3367*e, 1/364*e^7 + 2/91*e^5 + 165/364*e^3 - 216/91*e, -291/13468*e^7 - 5877/13468*e^5 - 60027/13468*e^3 - 81909/13468*e, -31/13468*e^7 + 207/13468*e^5 + 4713/13468*e^3 + 3215/13468*e, -2/91*e^6 - 16/91*e^4 + 34/91*e^2 + 818/91, 2/91*e^6 + 16/91*e^4 - 34/91*e^2 - 2638/91, 37/2184*e^6 + 311/728*e^4 + 2035/728*e^2 - 1847/2184, -89/2184*e^6 - 571/728*e^4 - 4895/728*e^2 + 7099/2184, -55/3108*e^7 - 429/1036*e^5 - 4565/1036*e^3 - 18007/3108*e, 55/3108*e^7 + 429/1036*e^5 + 4565/1036*e^3 + 18007/3108*e, 305/40404*e^7 + 801/6734*e^5 + 14955/13468*e^3 + 33223/20202*e, 125/10101*e^7 + 3123/13468*e^5 + 7785/3367*e^3 + 130289/40404*e, -59/6734*e^7 - 509/3367*e^5 - 5367/6734*e^3 + 15387/3367*e, 123/6734*e^7 + 947/3367*e^5 + 13015/6734*e^3 - 36301/3367*e, -503/20202*e^7 - 1884/3367*e^5 - 39677/6734*e^3 - 79000/10101*e, 57/3367*e^7 + 2823/6734*e^5 + 15229/3367*e^3 + 39551/6734*e, 619/26936*e^7 + 8683/26936*e^5 + 72287/26936*e^3 - 396769/26936*e, 437/26936*e^7 + 5771/26936*e^5 + 53905/26936*e^3 - 293393/26936*e, -3/182*e^6 - 12/91*e^4 + 51/182*e^2 + 1205/91, 3/182*e^6 + 12/91*e^4 - 51/182*e^2 - 1387/91, -2/91*e^6 - 16/91*e^4 + 34/91*e^2 + 3184/91, 2/91*e^6 + 16/91*e^4 - 34/91*e^2 - 272/91, 19/546*e^6 + 81/182*e^4 + 1045/182*e^2 - 2129/546, 11/182*e^6 + 179/182*e^4 + 1815/182*e^2 - 1041/182, 81/3848*e^7 + 1259/3848*e^5 + 8477/3848*e^3 - 47377/3848*e, -49/3848*e^7 - 821/3848*e^5 - 4653/3848*e^3 + 26463/3848*e, -29/10101*e^7 - 1371/13468*e^5 - 3961/3367*e^3 - 58025/40404*e, 599/40404*e^7 + 2103/6734*e^5 + 43501/13468*e^3 + 88033/20202*e, 55/1092*e^6 + 359/364*e^4 + 3025/364*e^2 - 4295/1092, -29/1092*e^6 - 229/364*e^4 - 1595/364*e^2 + 1669/1092, -3/91*e^6 - 187/364*e^4 - 495/91*e^2 + 1177/364, -29/1092*e^6 - 69/182*e^4 - 1595/364*e^2 + 1517/546, 3/182*e^6 + 12/91*e^4 - 51/182*e^2 - 386/91, -3/182*e^6 - 12/91*e^4 + 51/182*e^2 + 2206/91, 137/20202*e^7 + 1671/13468*e^5 + 8263/6734*e^3 + 69661/40404*e, 209/40404*e^7 + 291/3367*e^5 + 11131/13468*e^3 + 12095/10101*e, -31/40404*e^7 + 69/13468*e^5 + 1571/13468*e^3 + 3215/40404*e, -97/13468*e^7 - 1959/13468*e^5 - 20009/13468*e^3 - 27303/13468*e, -95/26936*e^7 - 669/26936*e^5 - 16403/26936*e^3 + 85447/26936*e, -641/26936*e^7 - 9405/26936*e^5 - 71549/26936*e^3 + 395575/26936*e, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 + 2161/91, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 + 1297/91, -17/2184*e^6 - 197/728*e^4 - 935/728*e^2 + 37/2184, 95/2184*e^6 + 587/728*e^4 + 5225/728*e^2 - 7915/2184, 5/364*e^6 + 131/364*e^4 + 825/364*e^2 - 225/364, -41/1092*e^6 - 261/364*e^4 - 2255/364*e^2 + 3301/1092, 344/10101*e^7 + 9645/13468*e^5 + 24926/3367*e^3 + 403727/40404*e, -83/13468*e^7 - 1515/6734*e^5 - 35171/13468*e^3 - 21383/6734*e, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 + 2207/91, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 + 3071/91, -79/20202*e^7 - 75/3367*e^5 - 341/6734*e^3 - 2909/10101*e, -202/10101*e^7 - 2685/6734*e^5 - 13658/3367*e^3 - 112223/20202*e, 1/546*e^6 + 33/182*e^4 + 55/182*e^2 + 319/546, -9/182*e^6 - 163/182*e^4 - 1485/182*e^2 + 769/182, -99/6734*e^7 - 3131/13468*e^5 - 10147/6734*e^3 + 113833/13468*e, 83/6734*e^7 + 2693/13468*e^5 + 8235/6734*e^3 - 92919/13468*e, -277/26936*e^7 - 3581/26936*e^5 - 34785/26936*e^3 + 188823/26936*e, -459/26936*e^7 - 6493/26936*e^5 - 53167/26936*e^3 + 292199/26936*e, -5/273*e^7 - 57/182*e^5 - 275/91*e^3 - 2371/546*e, -145/6734*e^7 - 1308/3367*e^5 - 25745/6734*e^3 - 18168/3367*e, -5/182*e^6 - 20/91*e^4 + 85/182*e^2 + 2251/91, 5/182*e^6 + 20/91*e^4 - 85/182*e^2 - 2069/91, -23, -4/273*e^6 - 41/91*e^4 - 220/91*e^2 + 89/273, 17/273*e^6 + 106/91*e^4 + 935/91*e^2 - 1402/273, -3/104*e^6 - 63/104*e^4 - 495/104*e^2 + 213/104, 3/104*e^6 + 63/104*e^4 + 495/104*e^2 - 213/104, -202/10101*e^7 - 2685/6734*e^5 - 13658/3367*e^3 - 112223/20202*e, -79/20202*e^7 - 75/3367*e^5 - 341/6734*e^3 - 2909/10101*e, -101/1092*e^6 - 347/182*e^4 - 5555/364*e^2 + 3683/546, 22/273*e^6 + 629/364*e^4 + 1210/91*e^2 - 6053/1092, -1/91*e^6 - 8/91*e^4 + 17/91*e^2 + 2138/91, 1/91*e^6 + 8/91*e^4 - 17/91*e^2 + 410/91, -47/13468*e^7 - 3/3367*e^5 + 2801/13468*e^3 + 51/3367*e, -493/20202*e^7 - 6603/13468*e^5 - 33667/6734*e^3 - 276041/40404*e, 113/40404*e^7 + 363/6734*e^5 + 7307/13468*e^3 + 15157/20202*e, 4/3367*e^7 + 219/13468*e^5 + 478/3367*e^3 + 3011/13468*e, -545/13468*e^7 - 8091/13468*e^5 - 60077/13468*e^3 + 332833/13468*e, 1/13468*e^7 + 645/13468*e^5 - 4931/13468*e^3 + 22705/13468*e, -1/182*e^6 - 4/91*e^4 + 17/182*e^2 - 1570/91, 1/182*e^6 + 4/91*e^4 - 17/182*e^2 - 2434/91, 20/273*e^6 + 114/91*e^4 + 1100/91*e^2 - 1810/273, 2/91*e^6 + 16/91*e^4 + 330/91*e^2 - 272/91, -863/20202*e^7 - 10821/13468*e^5 - 54017/6734*e^3 - 451495/40404*e, -337/13468*e^7 - 1311/3367*e^5 - 48689/13468*e^3 - 18117/3367*e, 95/2184*e^6 + 405/728*e^4 + 5225/728*e^2 - 10645/2184, 55/728*e^6 + 895/728*e^4 + 9075/728*e^2 - 5205/728, -7, -1675/40404*e^7 - 9957/13468*e^5 - 97585/13468*e^3 - 414751/40404*e, -515/13468*e^7 - 8943/13468*e^5 - 86795/13468*e^3 - 124063/13468*e, 5/182*e^6 + 20/91*e^4 + 825/182*e^2 - 340/91, 25/273*e^6 + 285/182*e^4 + 1375/91*e^2 - 4525/546];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, w + 1}>] := 1/504*e^6 + 5/168*e^4 + 55/168*e^2 - 101/504;

// 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;