/*
  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![-57, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[3, 3, w],
[3, 3, w + 2],
[4, 2, 2],
[5, 5, w + 1],
[5, 5, w + 3],
[11, 11, w + 1],
[11, 11, w + 9],
[17, 17, w + 2],
[17, 17, w + 14],
[19, 19, w],
[19, 19, w + 18],
[37, 37, -w - 4],
[37, 37, w - 5],
[43, 43, w + 16],
[43, 43, w + 26],
[49, 7, -7],
[53, 53, -w - 10],
[53, 53, w - 11],
[61, 61, w + 15],
[61, 61, w + 45],
[71, 71, w + 33],
[71, 71, w + 37],
[83, 83, w + 17],
[83, 83, w + 65],
[97, 97, w + 30],
[97, 97, w + 66],
[103, 103, w + 34],
[103, 103, w + 68],
[149, 149, w + 61],
[149, 149, w + 87],
[151, 151, w + 28],
[151, 151, w + 122],
[167, 167, w + 39],
[167, 167, w + 127],
[169, 13, -13],
[173, 173, 3*w - 20],
[173, 173, -3*w - 17],
[181, 181, w + 24],
[181, 181, w + 156],
[193, 193, 2*w - 7],
[193, 193, -2*w - 5],
[229, 229, 2*w - 1],
[233, 233, w + 27],
[233, 233, w + 205],
[241, 241, -3*w - 26],
[241, 241, 3*w - 29],
[271, 271, w + 29],
[271, 271, w + 241],
[277, 277, w + 50],
[277, 277, w + 226],
[293, 293, w + 112],
[293, 293, w + 180],
[307, 307, w + 132],
[307, 307, w + 174],
[311, 311, w + 88],
[311, 311, w + 222],
[337, 337, w + 71],
[337, 337, w + 265],
[347, 347, 2*w - 25],
[347, 347, -2*w - 23],
[359, 359, 3*w - 14],
[359, 359, -3*w - 11],
[367, 367, w + 43],
[367, 367, w + 323],
[373, 373, w + 64],
[373, 373, w + 308],
[383, 383, -3*w - 10],
[383, 383, 3*w - 13],
[397, 397, w + 82],
[397, 397, w + 314],
[401, 401, w + 35],
[401, 401, w + 365],
[409, 409, w + 67],
[409, 409, w + 341],
[421, 421, w + 46],
[421, 421, w + 374],
[431, 431, w + 119],
[431, 431, w + 311],
[433, 433, w + 175],
[433, 433, w + 257],
[439, 439, 5*w - 34],
[439, 439, -5*w - 29],
[443, 443, -3*w - 7],
[443, 443, 3*w - 10],
[449, 449, -w - 22],
[449, 449, w - 23],
[457, 457, w + 208],
[457, 457, w + 248],
[461, 461, 5*w - 46],
[461, 461, -5*w - 41],
[463, 463, w + 186],
[463, 463, w + 276],
[467, 467, w + 94],
[467, 467, w + 372],
[491, 491, w + 145],
[491, 491, w + 345],
[503, 503, 3*w - 5],
[503, 503, -3*w - 2],
[509, 509, -3*w - 1],
[509, 509, 3*w - 4],
[529, 23, -23],
[541, 541, 3*w - 34],
[541, 541, -3*w - 31],
[557, 557, w + 158],
[557, 557, w + 398],
[569, 569, w + 262],
[569, 569, w + 306],
[587, 587, w + 223],
[587, 587, w + 363],
[593, 593, -w - 25],
[593, 593, w - 26],
[607, 607, -3*w - 32],
[607, 607, 3*w - 35],
[617, 617, 6*w - 41],
[617, 617, -6*w - 35],
[619, 619, -5*w - 26],
[619, 619, 5*w - 31],
[631, 631, w + 144],
[631, 631, w + 486],
[641, 641, w + 44],
[641, 641, w + 596],
[643, 643, 7*w + 43],
[643, 643, 7*w - 50],
[661, 661, w + 77],
[661, 661, w + 583],
[673, 673, w + 58],
[673, 673, w + 614],
[683, 683, w + 101],
[683, 683, w + 581],
[691, 691, -4*w - 13],
[691, 691, 4*w - 17],
[701, 701, w + 196],
[701, 701, w + 504],
[733, 733, w + 47],
[733, 733, w + 685],
[743, 743, w + 136],
[743, 743, w + 606],
[751, 751, w + 318],
[751, 751, w + 432],
[757, 757, w + 353],
[757, 757, w + 403],
[769, 769, w + 209],
[769, 769, w + 559],
[787, 787, w + 84],
[787, 787, w + 702],
[821, 821, w + 354],
[821, 821, w + 466],
[841, 29, -29],
[859, 859, w + 97],
[859, 859, w + 761],
[863, 863, w + 264],
[863, 863, w + 598],
[883, 883, w + 250],
[883, 883, w + 632],
[907, 907, -4*w - 1],
[907, 907, 4*w - 5],
[911, 911, w + 261],
[911, 911, w + 649],
[919, 919, w + 203],
[919, 919, w + 715],
[941, 941, w + 356],
[941, 941, w + 584],
[953, 953, w + 177],
[953, 953, w + 775],
[961, 31, -31],
[967, 967, 3*w - 40],
[967, 967, -3*w - 37],
[971, 971, w + 54],
[971, 971, w + 916],
[977, 977, -7*w - 58],
[977, 977, 7*w - 65],
[991, 991, w + 459],
[991, 991, w + 531],
[997, 997, w + 385],
[997, 997, w + 611]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 + 2*x^11 - 22*x^10 - 52*x^9 + 150*x^8 + 439*x^7 - 270*x^6 - 1363*x^5 - 343*x^4 + 1278*x^3 + 720*x^2 - 135*x - 81;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-120332/566811*e^11 - 35563/566811*e^10 + 388940/80973*e^9 + 1627772/566811*e^8 - 7053910/188937*e^7 - 17079890/566811*e^6 + 7143865/62979*e^5 + 57323240/566811*e^4 - 65096494/566811*e^3 - 5820838/62979*e^2 + 1354495/62979*e + 203243/20993, e, -95302/566811*e^11 - 66884/566811*e^10 + 313561/80973*e^9 + 2082484/566811*e^8 - 5732845/188937*e^7 - 19186990/566811*e^6 + 17263868/188937*e^5 + 61337623/566811*e^4 - 49248446/566811*e^3 - 18683264/188937*e^2 + 626693/62979*e + 204053/20993, 14825/80973*e^11 + 1339/80973*e^10 - 335864/80973*e^9 - 128222/80973*e^8 + 291938/8997*e^7 + 1523660/80973*e^6 - 2710709/26991*e^5 - 5277788/80973*e^4 + 8617261/80973*e^3 + 1582082/26991*e^2 - 60479/2999*e - 15197/2999, -1, -17056/188937*e^11 + 16745/188937*e^10 + 17776/8997*e^9 - 75974/62979*e^8 - 2923204/188937*e^7 + 915638/188937*e^6 + 9616009/188937*e^5 - 1264826/188937*e^4 - 1351092/20993*e^3 + 748943/188937*e^2 + 1262273/62979*e + 9130/20993, -100852/188937*e^11 - 44339/188937*e^10 + 326677/26991*e^9 + 1668802/188937*e^8 - 5903150/62979*e^7 - 16492990/188937*e^6 + 5876736/20993*e^5 + 54045757/188937*e^4 - 50182571/188937*e^3 - 5473079/20993*e^2 + 668496/20993*e + 564149/20993, 34598/566811*e^11 - 29450/566811*e^10 - 108440/80973*e^9 + 379297/566811*e^8 + 217514/20993*e^7 - 1222924/566811*e^6 - 6242165/188937*e^5 + 94195/566811*e^4 + 22817902/566811*e^3 + 883748/188937*e^2 - 912293/62979*e - 72922/20993, -138217/566811*e^11 - 39791/566811*e^10 + 444694/80973*e^9 + 1850197/566811*e^8 - 7999792/188937*e^7 - 19491229/566811*e^6 + 23888357/188937*e^5 + 65632156/566811*e^4 - 69212837/566811*e^3 - 20329196/188937*e^2 + 1263068/62979*e + 301775/20993, -3091/20993*e^11 + 1186/62979*e^10 + 29300/8997*e^9 + 50765/62979*e^8 - 1576993/62979*e^7 - 281180/20993*e^6 + 4821913/62979*e^5 + 1128292/20993*e^4 - 5052109/62979*e^3 - 3624799/62979*e^2 + 336253/20993*e + 121454/20993, 599626/566811*e^11 + 251099/566811*e^10 - 1931791/80973*e^9 - 9828382/566811*e^8 + 3852304/20993*e^7 + 98861713/566811*e^6 - 102259894/188937*e^5 - 329307061/566811*e^4 + 279722054/566811*e^3 + 102697921/188937*e^2 - 2185651/62979*e - 1159726/20993, -159673/188937*e^11 - 70421/188937*e^10 + 517012/26991*e^9 + 2677012/188937*e^8 - 3118421/20993*e^7 - 26627086/188937*e^6 + 28048336/62979*e^5 + 88259713/188937*e^4 - 80296487/188937*e^3 - 27744922/62979*e^2 + 976538/20993*e + 1146276/20993, -76322/566811*e^11 - 3274/566811*e^10 + 247049/80973*e^9 + 599714/566811*e^8 - 1516085/62979*e^7 - 7370105/566811*e^6 + 14443763/188937*e^5 + 24715412/566811*e^4 - 49801441/566811*e^3 - 5914457/188937*e^2 + 1617848/62979*e - 44892/20993, -316210/566811*e^11 - 150971/566811*e^10 + 1029868/80973*e^9 + 5439256/566811*e^8 - 6242351/62979*e^7 - 52807033/566811*e^6 + 56372815/188937*e^5 + 170653300/566811*e^4 - 161631683/566811*e^3 - 50516029/188937*e^2 + 657596/20993*e + 582184/20993, -184237/566811*e^11 - 75830/566811*e^10 + 590557/80973*e^9 + 2987500/566811*e^8 - 10520600/188937*e^7 - 29798122/566811*e^6 + 10176098/62979*e^5 + 96805279/566811*e^4 - 78803615/566811*e^3 - 3107712/20993*e^2 + 214624/62979*e + 195933/20993, -276901/566811*e^11 - 97373/566811*e^10 + 880723/80973*e^9 + 4279897/566811*e^8 - 5200483/62979*e^7 - 44604310/566811*e^6 + 45281533/188937*e^5 + 151529338/566811*e^4 - 121191575/566811*e^3 - 48750337/188937*e^2 + 1060117/62979*e + 627975/20993, -15241/62979*e^11 - 9635/62979*e^10 + 50368/8997*e^9 + 307138/62979*e^8 - 929960/20993*e^7 - 2852194/62979*e^6 + 2875225/20993*e^5 + 9112357/62979*e^4 - 9093197/62979*e^3 - 2763914/20993*e^2 + 766961/20993*e + 271372/20993, -265061/566811*e^11 - 103483/566811*e^10 + 859340/80973*e^9 + 4087577/566811*e^8 - 15546029/188937*e^7 - 41219555/566811*e^6 + 46508338/188937*e^5 + 137220677/566811*e^4 - 132151456/566811*e^3 - 43219516/188937*e^2 + 1484566/62979*e + 685441/20993, 401309/566811*e^11 + 170620/566811*e^10 - 1283465/80973*e^9 - 6799412/566811*e^8 + 22843942/188937*e^7 + 68868281/566811*e^6 - 22143964/62979*e^5 - 231520190/566811*e^4 + 174904831/566811*e^3 + 8247234/20993*e^2 - 832082/62979*e - 1020216/20993, -104336/566811*e^11 + 75119/566811*e^10 + 320966/80973*e^9 - 723979/566811*e^8 - 5669102/188937*e^7 + 156526/566811*e^6 + 17384071/188937*e^5 + 8481299/566811*e^4 - 56858638/566811*e^3 - 2885893/188937*e^2 + 1255822/62979*e - 28043/20993, 43130/188937*e^11 + 22814/188937*e^10 - 46031/8997*e^9 - 284966/62979*e^8 + 7363898/188937*e^7 + 8666297/188937*e^6 - 21297608/188937*e^5 - 30247745/188937*e^4 + 2042459/20993*e^3 + 32047679/188937*e^2 - 301891/62979*e - 559487/20993, 608626/566811*e^11 + 270710/566811*e^10 - 1963273/80973*e^9 - 10362658/566811*e^8 + 11773792/62979*e^7 + 103607998/566811*e^6 - 104745031/188937*e^5 - 345792325/566811*e^4 + 291536471/566811*e^3 + 109797637/188937*e^2 - 2774293/62979*e - 1507677/20993, 368029/566811*e^11 + 10772/566811*e^10 - 1174729/80973*e^9 - 2859673/566811*e^8 + 7010474/62979*e^7 + 36441934/566811*e^6 - 62720461/188937*e^5 - 128977573/566811*e^4 + 180330446/566811*e^3 + 37994101/188937*e^2 - 2199313/62979*e - 165000/20993, -58258/566811*e^11 - 127490/566811*e^10 + 190243/80973*e^9 + 3278224/566811*e^8 - 3279016/188937*e^7 - 27685753/566811*e^6 + 8052029/188937*e^5 + 86772298/566811*e^4 - 5675504/566811*e^3 - 26786219/188937*e^2 - 1722598/62979*e + 195464/20993, -171397/566811*e^11 - 100754/566811*e^10 + 559678/80973*e^9 + 3449578/566811*e^8 - 3398984/62979*e^7 - 33569440/566811*e^6 + 30716227/188937*e^5 + 113854855/566811*e^4 - 87485105/566811*e^3 - 39129901/188937*e^2 + 958540/62979*e + 706907/20993, -268103/566811*e^11 - 132658/566811*e^10 + 860732/80973*e^9 + 4817843/566811*e^8 - 5099687/62979*e^7 - 46774472/566811*e^6 + 43837403/188937*e^5 + 150626780/566811*e^4 - 105878659/566811*e^3 - 44215712/188937*e^2 - 1330480/62979*e + 501463/20993, 90086/62979*e^11 + 85606/188937*e^10 - 868840/26991*e^9 - 3834403/188937*e^8 + 46872590/188937*e^7 + 4442257/20993*e^6 - 139753448/188937*e^5 - 44686855/62979*e^4 + 132821060/188937*e^3 + 123269906/188937*e^2 - 4877095/62979*e - 1312886/20993, -57083/62979*e^11 - 71965/188937*e^10 + 554050/26991*e^9 + 2796391/188937*e^8 - 30027215/188937*e^7 - 3129123/20993*e^6 + 89881520/188937*e^5 + 31499368/62979*e^4 - 86242601/188937*e^3 - 90088469/188937*e^2 + 3663292/62979*e + 1040162/20993, -962201/566811*e^11 - 480112/566811*e^10 + 3120155/80973*e^9 + 17179910/566811*e^8 - 18771005/62979*e^7 - 167348624/566811*e^6 + 167103086/188937*e^5 + 548815094/566811*e^4 - 462688591/566811*e^3 - 170059679/188937*e^2 + 4354982/62979*e + 2063633/20993, 77005/188937*e^11 + 6946/20993*e^10 - 252839/26991*e^9 - 1847693/188937*e^8 + 13682051/188937*e^7 + 16572232/188937*e^6 - 39521102/188937*e^5 - 51800950/188937*e^4 + 32182135/188937*e^3 + 45656999/188937*e^2 + 257207/62979*e - 540875/20993, 699404/566811*e^11 + 184396/566811*e^10 - 2234162/80973*e^9 - 9127388/566811*e^8 + 13269784/62979*e^7 + 97166159/566811*e^6 - 116601494/188937*e^5 - 325080839/566811*e^4 + 315752734/566811*e^3 + 96710786/188937*e^2 - 2332217/62979*e - 917925/20993, 402014/566811*e^11 + 190735/566811*e^10 - 1295198/80973*e^9 - 7044056/566811*e^8 + 2575135/20993*e^7 + 69292031/566811*e^6 - 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5344995/20993*e - 4091828/20993, 492175/188937*e^11 + 32771/188937*e^10 - 1584124/26991*e^9 - 4174927/188937*e^8 + 28765244/62979*e^7 + 51056017/188937*e^6 - 29431226/20993*e^5 - 179007295/188937*e^4 + 273350132/188937*e^3 + 17773880/20993*e^2 - 5028049/20993*e - 1381260/20993, -3151/20993*e^11 + 12130/62979*e^10 + 28850/8997*e^9 - 182908/62979*e^8 - 1578244/62979*e^7 + 344259/20993*e^6 + 5400538/62979*e^5 - 1140733/20993*e^4 - 7967524/62979*e^3 + 6481826/62979*e^2 + 1491758/20993*e - 808044/20993, 863215/188937*e^11 + 334916/188937*e^10 - 2797810/26991*e^9 - 13376302/188937*e^8 + 50701792/62979*e^7 + 135672358/188937*e^6 - 152728430/62979*e^5 - 454632013/188937*e^4 + 446224490/188937*e^3 + 143337728/62979*e^2 - 6798858/20993*e - 5478863/20993, 385793/188937*e^11 + 219691/188937*e^10 - 1259276/26991*e^9 - 7393343/188937*e^8 + 7622259/20993*e^7 + 70476401/188937*e^6 - 68363879/62979*e^5 - 230066195/188937*e^4 + 191382766/188937*e^3 + 72945389/62979*e^2 - 1443297/20993*e - 3301649/20993, -277615/80973*e^11 - 104345/80973*e^10 + 6269527/80973*e^9 + 4246996/80973*e^8 - 5373077/8997*e^7 - 43180279/80973*e^6 + 47872495/26991*e^5 + 143881987/80973*e^4 - 134038127/80973*e^3 - 44491597/26991*e^2 + 1487485/8997*e + 565947/2999, 1317497/566811*e^11 + 114634/566811*e^10 - 4220252/80973*e^9 - 11912723/566811*e^8 + 76013023/188937*e^7 + 142936037/566811*e^6 - 76591889/62979*e^5 - 499981691/566811*e^4 + 693503968/566811*e^3 + 51500497/62979*e^2 - 13382081/62979*e - 2189181/20993, -924568/566811*e^11 - 169265/566811*e^10 + 2972818/80973*e^9 + 10245196/566811*e^8 - 5941711/20993*e^7 - 114996001/566811*e^6 + 159358834/188937*e^5 + 398929168/566811*e^4 - 447001034/566811*e^3 - 127098961/188937*e^2 + 3537598/62979*e + 1971913/20993, -321991/566811*e^11 - 189956/566811*e^10 + 1064899/80973*e^9 + 5921245/566811*e^8 - 6505379/62979*e^7 - 53825602/566811*e^6 + 59052739/188937*e^5 + 162944164/566811*e^4 - 173880482/566811*e^3 - 39917200/188937*e^2 + 923962/20993*e - 401336/20993, 133982/566811*e^11 + 150328/566811*e^10 - 471776/80973*e^9 - 3798437/566811*e^8 + 3121091/62979*e^7 + 30277172/566811*e^6 - 32345687/188937*e^5 - 81532355/566811*e^4 + 126876751/566811*e^3 + 16365173/188937*e^2 - 4495619/62979*e + 289641/20993, -490205/566811*e^11 - 5596/566811*e^10 + 1601045/80973*e^9 + 3125051/566811*e^8 - 9874964/62979*e^7 - 39811361/566811*e^6 + 93928019/188937*e^5 + 134795033/566811*e^4 - 314529664/566811*e^3 - 34055657/188937*e^2 + 8356469/62979*e + 276470/20993, 796465/566811*e^11 + 362660/566811*e^10 - 2595808/80973*e^9 - 13507390/566811*e^8 + 47282743/188937*e^7 + 134998981/566811*e^6 - 143150795/188937*e^5 - 458387284/566811*e^4 + 419832230/566811*e^3 + 153303620/188937*e^2 - 6495242/62979*e - 2133938/20993, -273365/566811*e^11 - 26857/566811*e^10 + 892562/80973*e^9 + 2687135/566811*e^8 - 5599885/62979*e^7 - 32169152/566811*e^6 + 55714202/188937*e^5 + 113352863/566811*e^4 - 205937512/566811*e^3 - 34977008/188937*e^2 + 6380291/62979*e + 409051/20993, -1201157/566811*e^11 - 638290/566811*e^10 + 3863318/80973*e^9 + 22731125/566811*e^8 - 7653199/20993*e^7 - 220885643/566811*e^6 + 199316714/188937*e^5 + 724218431/566811*e^4 - 511559422/566811*e^3 - 226704671/188937*e^2 + 869777/62979*e + 3201717/20993, 672802/566811*e^11 + 97166/566811*e^10 - 2171710/80973*e^9 - 6891022/566811*e^8 + 4394099/20993*e^7 + 79539205/566811*e^6 - 122281480/188937*e^5 - 279317098/566811*e^4 + 393284768/566811*e^3 + 89761579/188937*e^2 - 9658471/62979*e - 1303487/20993, -1621555/566811*e^11 - 643367/566811*e^10 + 5211523/80973*e^9 + 26121439/566811*e^8 - 31159645/62979*e^7 - 265924552/566811*e^6 + 276572461/188937*e^5 + 892246528/566811*e^4 - 773186441/566811*e^3 - 281831572/188937*e^2 + 10497898/62979*e + 3955485/20993, -2222119/566811*e^11 - 644300/566811*e^10 + 7146967/80973*e^9 + 30080776/566811*e^8 - 42860267/62979*e^7 - 317415004/566811*e^6 + 383666338/188937*e^5 + 1069545523/566811*e^4 - 1097165855/566811*e^3 - 327746158/188937*e^2 + 15079852/62979*e + 3659652/20993, -3914285/566811*e^11 - 1478623/566811*e^10 + 12591326/80973*e^9 + 60704075/566811*e^8 - 75323173/62979*e^7 - 619047881/566811*e^6 + 668962517/188937*e^5 + 2070821477/566811*e^4 - 1868637682/566811*e^3 - 648473831/188937*e^2 + 21510689/62979*e + 8310410/20993, -272176/188937*e^11 - 216709/188937*e^10 + 99004/2999*e^9 + 727317/20993*e^8 - 48403714/188937*e^7 - 59297905/188937*e^6 + 142661605/188937*e^5 + 187362940/188937*e^4 - 42302419/62979*e^3 - 170524036/188937*e^2 + 1383080/62979*e + 2176147/20993, -3698575/566811*e^11 - 1389194/566811*e^10 + 11957992/80973*e^9 + 56255560/566811*e^8 - 215851498/188937*e^7 - 570967627/566811*e^6 + 644801411/188937*e^5 + 1899851113/566811*e^4 - 1835501300/566811*e^3 - 586821113/188937*e^2 + 21282680/62979*e + 6718811/20993, -3598244/566811*e^11 - 1409746/566811*e^10 + 11586527/80973*e^9 + 57054806/566811*e^8 - 208255070/188937*e^7 - 579995471/566811*e^6 + 617984635/188937*e^5 + 1943665205/566811*e^4 - 1732462924/566811*e^3 - 614189419/188937*e^2 + 19400170/62979*e + 8095879/20993, -373258/188937*e^11 - 205078/188937*e^10 + 404713/8997*e^9 + 2378611/62979*e^8 - 66104266/188937*e^7 - 68981350/188937*e^6 + 197911543/188937*e^5 + 227815900/188937*e^4 - 20499928/20993*e^3 - 220075822/188937*e^2 + 4136123/62979*e + 3217552/20993, -1167884/566811*e^11 - 545194/566811*e^10 + 3744068/80973*e^9 + 20709806/566811*e^8 - 22231687/62979*e^7 - 205999805/566811*e^6 + 193573547/188937*e^5 + 681029465/566811*e^4 - 505400065/566811*e^3 - 207963362/188937*e^2 + 3001709/62979*e + 2164718/20993, 1342643/566811*e^11 + 298786/566811*e^10 - 4337327/80973*e^9 - 15664169/566811*e^8 + 8726906/20993*e^7 + 169768502/566811*e^6 - 237711143/188937*e^5 - 575370587/566811*e^4 + 705877624/566811*e^3 + 175431245/188937*e^2 - 9861713/62979*e - 1722118/20993, 465805/188937*e^11 + 145564/188937*e^10 - 500033/8997*e^9 - 2155724/62979*e^8 + 80895700/188937*e^7 + 67376173/188937*e^6 - 240015997/188937*e^5 - 226237057/188937*e^4 + 74078987/62979*e^3 + 210027994/188937*e^2 - 4856129/62979*e - 2364553/20993, -81196/62979*e^11 - 30227/62979*e^10 + 258856/8997*e^9 + 1276333/62979*e^8 - 4593835/20993*e^7 - 13077466/62979*e^6 + 13362658/20993*e^5 + 43519717/62979*e^4 - 36448676/62979*e^3 - 13270105/20993*e^2 + 1845321/20993*e + 1184443/20993, -216355/80973*e^11 - 68207/80973*e^10 + 4840411/80973*e^9 + 3058621/80973*e^8 - 1363061/2999*e^7 - 31978162/80973*e^6 + 35376982/26991*e^5 + 107451142/80973*e^4 - 90288962/80973*e^3 - 32921221/26991*e^2 + 132271/8997*e + 427397/2999, -62278/62979*e^11 - 17006/20993*e^10 + 204185/8997*e^9 + 1523372/62979*e^8 - 11082581/62979*e^7 - 13702030/62979*e^6 + 32355287/62979*e^5 + 42903559/62979*e^4 - 27100006/62979*e^3 - 37849943/62979*e^2 - 473476/20993*e + 1076693/20993, 544142/566811*e^11 + 278242/566811*e^10 - 1749608/80973*e^9 - 10225871/566811*e^8 + 3480830/20993*e^7 + 101439797/566811*e^6 - 92297078/188937*e^5 - 341153810/566811*e^4 + 254832223/566811*e^3 + 110765828/188937*e^2 - 2423873/62979*e - 1670000/20993, 3723452/566811*e^11 + 1455346/566811*e^10 - 12011261/80973*e^9 - 58584758/566811*e^8 + 24024464/20993*e^7 + 594793631/566811*e^6 - 642753962/188937*e^5 - 1990029515/566811*e^4 + 1807887661/566811*e^3 + 624073283/188937*e^2 - 6957703/20993*e - 7722248/20993, 1561033/566811*e^11 + 708110/566811*e^10 - 5047933/80973*e^9 - 26241349/566811*e^8 + 10086631/20993*e^7 + 258132733/566811*e^6 - 267790261/188937*e^5 - 844942495/566811*e^4 + 731282348/566811*e^3 + 256645243/188937*e^2 - 6082921/62979*e - 2645813/20993, 1056185/188937*e^11 + 274448/188937*e^10 - 1128676/8997*e^9 - 4578571/62979*e^8 + 182196431/188937*e^7 + 147381161/188937*e^6 - 541373039/188937*e^5 - 499920509/188937*e^4 + 169908280/62979*e^3 + 462130670/188937*e^2 - 17028478/62979*e - 5014319/20993, 232438/566811*e^11 + 369734/566811*e^10 - 807442/80973*e^9 - 9154294/566811*e^8 + 1696741/20993*e^7 + 76067785/566811*e^6 - 46676092/188937*e^5 - 242681344/566811*e^4 + 129245912/566811*e^3 + 88147705/188937*e^2 - 103076/20993*e - 1860226/20993, 974363/566811*e^11 + 143728/566811*e^10 - 3170723/80973*e^9 - 9639554/566811*e^8 + 58249454/188937*e^7 + 109431278/566811*e^6 - 181622011/188937*e^5 - 375637610/566811*e^4 + 581988868/566811*e^3 + 112013980/188937*e^2 - 13264396/62979*e - 928720/20993, -1077385/566811*e^11 - 413012/566811*e^10 + 3521965/80973*e^9 + 16094263/566811*e^8 - 64476025/188937*e^7 - 161017480/566811*e^6 + 197146745/188937*e^5 + 528688420/566811*e^4 - 591775076/566811*e^3 - 158501822/188937*e^2 + 9780632/62979*e + 2067932/20993, 162458/566811*e^11 - 56921/566811*e^10 - 499553/80973*e^9 + 142444/566811*e^8 + 8467763/188937*e^7 + 4291310/566811*e^6 - 22866847/188937*e^5 - 12694964/566811*e^4 + 49498996/566811*e^3 - 6209036/188937*e^2 + 1006598/62979*e + 808292/20993];
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;