/*
  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![-34, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, w - 6],
[3, 3, w + 1],
[3, 3, w + 2],
[5, 5, w + 2],
[5, 5, w + 3],
[11, 11, w + 1],
[11, 11, w + 10],
[17, 17, -3*w + 17],
[29, 29, w + 11],
[29, 29, w + 18],
[37, 37, w + 16],
[37, 37, w + 21],
[47, 47, -w - 9],
[47, 47, w - 9],
[49, 7, -7],
[61, 61, w + 20],
[61, 61, w + 41],
[89, 89, 2*w - 15],
[89, 89, -2*w - 15],
[103, 103, -14*w + 81],
[103, 103, 4*w - 21],
[107, 107, w + 26],
[107, 107, w + 81],
[109, 109, w + 19],
[109, 109, w + 90],
[127, 127, 2*w - 3],
[127, 127, -2*w - 3],
[131, 131, w + 54],
[131, 131, w + 77],
[137, 137, -3*w - 13],
[137, 137, 3*w - 13],
[139, 139, w + 27],
[139, 139, w + 112],
[151, 151, 8*w - 45],
[151, 151, -10*w + 57],
[163, 163, w + 69],
[163, 163, w + 94],
[169, 13, -13],
[173, 173, w + 42],
[173, 173, w + 131],
[181, 181, w + 45],
[181, 181, w + 136],
[191, 191, -w - 15],
[191, 191, w - 15],
[197, 197, w + 25],
[197, 197, w + 172],
[211, 211, w + 33],
[211, 211, w + 178],
[223, 223, -3*w - 23],
[223, 223, 3*w - 23],
[227, 227, w + 48],
[227, 227, w + 179],
[239, 239, -5*w + 33],
[239, 239, -23*w + 135],
[257, 257, -3*w - 7],
[257, 257, 3*w - 7],
[263, 263, -24*w + 139],
[263, 263, 6*w - 31],
[269, 269, w + 29],
[269, 269, w + 240],
[271, 271, -9*w + 55],
[271, 271, -15*w + 89],
[277, 277, w + 119],
[277, 277, w + 158],
[281, 281, 3*w - 5],
[281, 281, -3*w - 5],
[283, 283, w + 113],
[283, 283, w + 170],
[317, 317, w + 44],
[317, 317, w + 273],
[347, 347, w + 46],
[347, 347, w + 301],
[353, 353, 9*w - 49],
[353, 353, -21*w + 121],
[359, 359, -25*w + 147],
[359, 359, -7*w + 45],
[361, 19, -19],
[379, 379, w + 105],
[379, 379, w + 274],
[383, 383, 6*w - 29],
[383, 383, -36*w + 209],
[397, 397, w + 35],
[397, 397, w + 362],
[409, 409, -5*w - 21],
[409, 409, 5*w - 21],
[419, 419, w + 156],
[419, 419, w + 263],
[433, 433, -12*w + 73],
[433, 433, -18*w + 107],
[457, 457, 6*w + 41],
[457, 457, -6*w + 41],
[463, 463, -4*w - 9],
[463, 463, 4*w - 9],
[499, 499, w + 212],
[499, 499, w + 287],
[529, 23, -23],
[541, 541, w + 121],
[541, 541, w + 420],
[547, 547, w + 157],
[547, 547, w + 390],
[569, 569, -28*w + 165],
[569, 569, -10*w + 63],
[571, 571, w + 264],
[571, 571, w + 307],
[577, 577, -47*w + 273],
[577, 577, 7*w - 33],
[593, 593, 2*w - 27],
[593, 593, -2*w - 27],
[599, 599, -6*w - 25],
[599, 599, 6*w - 25],
[619, 619, w + 167],
[619, 619, w + 452],
[631, 631, -15*w + 91],
[631, 631, -21*w + 125],
[643, 643, w + 57],
[643, 643, w + 586],
[647, 647, -11*w + 69],
[647, 647, -29*w + 171],
[653, 653, w + 120],
[653, 653, w + 533],
[677, 677, w + 64],
[677, 677, w + 613],
[683, 683, w + 248],
[683, 683, w + 435],
[691, 691, w + 322],
[691, 691, w + 369],
[709, 709, w + 265],
[709, 709, w + 444],
[727, 727, -9*w + 59],
[727, 727, -39*w + 229],
[761, 761, -27*w + 155],
[761, 761, 15*w - 83],
[769, 769, 5*w - 9],
[769, 769, -5*w - 9],
[787, 787, w + 63],
[787, 787, w + 724],
[811, 811, w + 70],
[811, 811, w + 741],
[821, 821, w + 149],
[821, 821, w + 672],
[827, 827, w + 193],
[827, 827, w + 634],
[853, 853, w + 253],
[853, 853, w + 600],
[863, 863, -6*w - 19],
[863, 863, 6*w - 19],
[877, 877, w + 339],
[877, 877, w + 538],
[907, 907, w + 74],
[907, 907, w + 833],
[919, 919, -3*w - 35],
[919, 919, 3*w - 35],
[937, 937, 7*w - 27],
[937, 937, -7*w - 27],
[941, 941, w + 144],
[941, 941, w + 797],
[947, 947, w + 292],
[947, 947, w + 655],
[953, 953, 2*w - 33],
[953, 953, -2*w - 33],
[961, 31, -31],
[967, 967, 9*w - 61],
[967, 967, -9*w - 61],
[977, 977, -4*w - 39],
[977, 977, 4*w - 39],
[997, 997, w + 55],
[997, 997, w + 942]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 + 22*x^8 + 159*x^6 + 439*x^4 + 351*x^2 + 25;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-77/1339*e^8 - 1372/1339*e^6 - 6749/1339*e^4 - 9110/1339*e^2 - 373/1339, e, -31/6695*e^9 + 178/6695*e^7 + 9786/6695*e^5 + 52466/6695*e^3 + 64174/6695*e, -13/515*e^9 - 241/515*e^7 - 1312/515*e^5 - 2552/515*e^3 - 2028/515*e, -3/1339*e^9 + 190/1339*e^7 + 3841/1339*e^5 + 16826/1339*e^3 + 12819/1339*e, -4/103*e^9 - 90/103*e^7 - 681/103*e^5 - 2045/103*e^3 - 1757/103*e, -249/6695*e^9 - 6993/6695*e^7 - 64151/6695*e^5 - 204886/6695*e^3 - 139784/6695*e, -55/1339*e^8 - 980/1339*e^6 - 5012/1339*e^4 - 8420/1339*e^2 - 649/1339, -492/6695*e^9 - 10349/6695*e^7 - 70373/6695*e^5 - 183658/6695*e^3 - 129797/6695*e, 461/6695*e^9 + 10527/6695*e^7 + 80159/6695*e^5 + 236124/6695*e^3 + 200666/6695*e, 2218/6695*e^9 + 43416/6695*e^7 + 260452/6695*e^5 + 559882/6695*e^3 + 327983/6695*e, -876/6695*e^9 - 15487/6695*e^7 - 75494/6695*e^5 - 113414/6695*e^3 - 81036/6695*e, 271/1339*e^8 + 4707/1339*e^6 + 21701/1339*e^4 + 27489/1339*e^2 + 14981/1339, -333/1339*e^8 - 5690/1339*e^6 - 24892/1339*e^4 - 22982/1339*e^2 + 891/1339, 19/1339*e^8 + 582/1339*e^6 + 5578/1339*e^4 + 14838/1339*e^2 - 2186/1339, 356/6695*e^9 + 10482/6695*e^7 + 100779/6695*e^5 + 336299/6695*e^3 + 254326/6695*e, 1052/6695*e^9 + 23979/6695*e^7 + 179103/6695*e^5 + 495193/6695*e^3 + 368052/6695*e, 189/1339*e^8 + 4098/1339*e^6 + 28495/1339*e^4 + 67400/1339*e^2 + 25261/1339, 216/1339*e^8 + 3727/1339*e^6 + 16689/1339*e^4 + 16391/1339*e^2 + 4959/1339, 672/1339*e^8 + 12339/1339*e^6 + 64865/1339*e^4 + 106042/1339*e^2 + 30157/1339, 43/103*e^8 + 710/103*e^6 + 2866/103*e^4 + 1976/103*e^2 + 734/103, 1461/6695*e^9 + 31997/6695*e^7 + 230324/6695*e^5 + 635104/6695*e^3 + 466876/6695*e, 972/6695*e^9 + 13424/6695*e^7 + 18193/6695*e^5 - 178642/6695*e^3 - 267578/6695*e, 98/6695*e^9 + 1381/6695*e^7 - 53/6695*e^5 - 47078/6695*e^3 - 79987/6695*e, -2978/6695*e^9 - 60001/6695*e^7 - 376452/6695*e^5 - 852127/6695*e^3 - 515038/6695*e, 3/103*e^8 + 119/103*e^6 + 1309/103*e^4 + 4289/103*e^2 + 1189/103, 49/1339*e^8 + 1360/1339*e^6 + 11355/1339*e^4 + 26004/1339*e^2 + 6202/1339, 197/6695*e^9 + 4484/6695*e^7 + 28518/6695*e^5 + 28333/6695*e^3 - 82568/6695*e, 3017/6695*e^9 + 53514/6695*e^7 + 260908/6695*e^5 + 373623/6695*e^3 + 166287/6695*e, -101/1339*e^8 - 1191/1339*e^6 + 1216/1339*e^4 + 27751/1339*e^2 + 23178/1339, -539/1339*e^8 - 9604/1339*e^6 - 45904/1339*e^4 - 54397/1339*e^2 - 9306/1339, -1126/6695*e^9 - 24202/6695*e^7 - 168269/6695*e^5 - 440789/6695*e^3 - 365176/6695*e, 173/1339*e^9 + 3326/1339*e^7 + 20415/1339*e^5 + 54482/1339*e^3 + 64171/1339*e, 332/1339*e^8 + 5307/1339*e^6 + 19031/1339*e^4 + 918/1339*e^2 - 10008/1339, -396/1339*e^8 - 7056/1339*e^6 - 33944/1339*e^4 - 36522/1339*e^2 + 13002/1339, -779/6695*e^9 - 15828/6695*e^7 - 104171/6695*e^5 - 282981/6695*e^3 - 291989/6695*e, -1519/6695*e^9 - 38143/6695*e^7 - 317191/6695*e^5 - 944041/6695*e^3 - 604674/6695*e, -454/1339*e^8 - 7846/1339*e^6 - 36454/1339*e^4 - 45523/1339*e^2 - 8303/1339, -3624/6695*e^9 - 67738/6695*e^7 - 375966/6695*e^5 - 751391/6695*e^3 - 581309/6695*e, 2209/6695*e^9 + 41308/6695*e^7 + 223771/6695*e^5 + 381391/6695*e^3 + 113369/6695*e, 42/6695*e^9 - 1321/6695*e^7 - 29672/6695*e^5 - 131122/6695*e^3 - 150008/6695*e, 223/1339*e^9 + 3730/1339*e^7 + 14868/1339*e^5 + 2125/1339*e^3 - 23613/1339*e, -467/1339*e^8 - 8808/1339*e^6 - 48375/1339*e^4 - 79284/1339*e^2 - 10331/1339, -82/1339*e^8 - 609/1339*e^6 + 6794/1339*e^4 + 41250/1339*e^2 + 14297/1339, -452/6695*e^9 - 8419/6695*e^7 - 43478/6695*e^5 - 44243/6695*e^3 + 121068/6695*e, 24/65*e^9 + 508/65*e^7 + 3436/65*e^5 + 8366/65*e^3 + 4594/65*e, 1866/6695*e^9 + 39822/6695*e^7 + 274169/6695*e^5 + 693454/6695*e^3 + 396671/6695*e, 68/1339*e^9 + 603/1339*e^7 - 4491/1339*e^5 - 38159/1339*e^3 - 45527/1339*e, 185/1339*e^8 + 2566/1339*e^6 + 3712/1339*e^4 - 27551/1339*e^2 - 19241/1339, -66/103*e^8 - 1176/103*e^6 - 5726/103*e^4 - 7426/103*e^2 - 1335/103, 454/1339*e^9 + 7846/1339*e^7 + 35115/1339*e^5 + 30794/1339*e^3 - 19816/1339*e, -24/1339*e^9 + 181/1339*e^7 + 9304/1339*e^5 + 56946/1339*e^3 + 83806/1339*e, 1447/1339*e^8 + 25296/1339*e^6 + 118812/1339*e^4 + 146782/1339*e^2 + 24573/1339, 2/1339*e^8 - 573/1339*e^6 - 8363/1339*e^4 - 21483/1339*e^2 + 12878/1339, 46/1339*e^8 + 211/1339*e^6 - 4889/1339*e^4 - 24120/1339*e^2 - 22488/1339, 178/1339*e^8 + 3902/1339*e^6 + 26957/1339*e^4 + 60360/1339*e^2 + 28077/1339, -841/1339*e^8 - 14133/1339*e^6 - 59158/1339*e^4 - 45488/1339*e^2 + 3734/1339, 229/1339*e^8 + 3350/1339*e^6 + 7186/1339*e^4 - 27510/1339*e^2 - 18454/1339, -1004/1339*e^9 - 18985/1339*e^7 - 106659/1339*e^5 - 206046/1339*e^3 - 113879/1339*e, -1301/6695*e^9 - 24277/6695*e^7 - 136134/6695*e^5 - 298379/6695*e^3 - 320376/6695*e, -945/1339*e^8 - 16473/1339*e^6 - 76864/1339*e^4 - 94641/1339*e^2 - 25880/1339, 246/1339*e^8 + 4505/1339*e^6 + 22466/1339*e^4 + 27557/1339*e^2 + 13347/1339, 1826/6695*e^9 + 37892/6695*e^7 + 247274/6695*e^5 + 580819/6695*e^3 + 386826/6695*e, 1067/6695*e^9 + 23029/6695*e^7 + 153203/6695*e^5 + 317333/6695*e^3 + 42852/6695*e, 269/1339*e^8 + 5280/1339*e^6 + 31403/1339*e^4 + 61023/1339*e^2 + 10137/1339, -420/1339*e^8 - 8214/1339*e^6 - 50081/1339*e^4 - 110798/1339*e^2 - 46465/1339, -86/6695*e^9 - 802/6695*e^7 + 8791/6695*e^5 + 97606/6695*e^3 + 158594/6695*e, -3317/6695*e^9 - 61294/6695*e^7 - 325373/6695*e^5 - 525453/6695*e^3 - 82792/6695*e, -3301/6695*e^9 - 73912/6695*e^7 - 536889/6695*e^5 - 1404309/6695*e^3 - 819321/6695*e, -19/6695*e^9 + 757/6695*e^7 + 18524/6695*e^5 + 116384/6695*e^3 + 269986/6695*e, -4333/6695*e^9 - 83536/6695*e^7 - 484957/6695*e^5 - 969487/6695*e^3 - 422568/6695*e, 743/6695*e^9 + 14091/6695*e^7 + 77957/6695*e^5 + 118007/6695*e^3 - 104512/6695*e, 365/1339*e^8 + 5895/1339*e^6 + 23645/1339*e^4 + 22038/1339*e^2 + 21714/1339, -1281/1339*e^8 - 21973/1339*e^6 - 96576/1339*e^4 - 87407/1339*e^2 + 7915/1339, 1660/1339*e^8 + 29213/1339*e^6 + 138003/1339*e^4 + 163931/1339*e^2 + 8876/1339, 26/103*e^8 + 482/103*e^6 + 2521/103*e^4 + 3559/103*e^2 + 657/103, 864/1339*e^8 + 14908/1339*e^6 + 66756/1339*e^4 + 61547/1339*e^2 - 31046/1339, 755/1339*e^9 + 13331/1339*e^7 + 63932/1339*e^5 + 81500/1339*e^3 + 6231/1339*e, -1563/6695*e^9 - 26876/6695*e^7 - 117137/6695*e^5 - 73732/6695*e^3 + 147057/6695*e, -382/1339*e^8 - 7050/1339*e^6 - 37586/1339*e^4 - 65054/1339*e^2 + 5401/1339, 161/1339*e^8 + 2747/1339*e^6 + 11677/1339*e^4 - 63/1339*e^2 - 29165/1339, 688/6695*e^9 + 19806/6695*e^7 + 184082/6695*e^5 + 578237/6695*e^3 + 378218/6695*e, 958/6695*e^9 + 2706/6695*e^7 - 154913/6695*e^5 - 869153/6695*e^3 - 735322/6695*e, 327/1339*e^8 + 6070/1339*e^6 + 31235/1339*e^4 + 36549/1339*e^2 - 31491/1339, -1408/1339*e^8 - 25088/1339*e^6 - 121880/1339*e^4 - 160653/1339*e^2 - 34557/1339, 1331/6695*e^9 + 35767/6695*e^7 + 318659/6695*e^5 + 1013859/6695*e^3 + 767956/6695*e, 3466/6695*e^9 + 63462/6695*e^7 + 332329/6695*e^5 + 539134/6695*e^3 + 107581/6695*e, 625/1339*e^8 + 11745/1339*e^6 + 63893/1339*e^4 + 112115/1339*e^2 + 59596/1339, 1345/1339*e^8 + 23722/1339*e^6 + 111489/1339*e^4 + 125689/1339*e^2 - 4214/1339, 44/1339*e^8 - 555/1339*e^6 - 17950/1339*e^4 - 78960/1339*e^2 - 17959/1339, 629/1339*e^8 + 9260/1339*e^6 + 24404/1339*e^4 - 32615/1339*e^2 - 33819/1339, -279/1339*e^8 - 3754/1339*e^6 - 4317/1339*e^4 + 46392/1339*e^2 + 32593/1339, -55/103*e^8 - 980/103*e^6 - 4806/103*e^4 - 6875/103*e^2 - 3842/103, 603/1339*e^9 + 11353/1339*e^7 + 63495/1339*e^5 + 123476/1339*e^3 + 75940/1339*e, 268/1339*e^9 + 6236/1339*e^7 + 46966/1339*e^5 + 123316/1339*e^3 + 54580/1339*e, 422/1339*e^8 + 6302/1339*e^6 + 18955/1339*e^4 - 398/1339*e^2 + 15156/1339, -518/6695*e^9 - 8256/6695*e^7 - 31282/6695*e^5 - 14177/6695*e^3 + 40217/6695*e, -678/1339*e^9 - 13298/1339*e^7 - 79946/1339*e^5 - 170137/1339*e^3 - 95571/1339*e, 1428/6695*e^9 + 28731/6695*e^7 + 182862/6695*e^5 + 455982/6695*e^3 + 429798/6695*e, -3994/6695*e^9 - 75548/6695*e^7 - 422221/6695*e^5 - 800731/6695*e^3 - 439724/6695*e, 997/1339*e^8 + 17643/1339*e^6 + 83039/1339*e^4 + 82395/1339*e^2 - 46348/1339, 1511/1339*e^8 + 25706/1339*e^6 + 112301/1339*e^4 + 108741/1339*e^2 + 26935/1339, 342/1339*e^9 + 9137/1339*e^7 + 80319/1339*e^5 + 246999/1339*e^3 + 164180/1339*e, 2673/6695*e^9 + 57001/6695*e^7 + 389802/6695*e^5 + 971592/6695*e^3 + 613203/6695*e, 1351/1339*e^8 + 22003/1339*e^6 + 83722/1339*e^4 + 27765/1339*e^2 - 20479/1339, -964/1339*e^8 - 14377/1339*e^6 - 38255/1339*e^4 + 63252/1339*e^2 + 49951/1339, -774/1339*e^8 - 13913/1339*e^6 - 68171/1339*e^4 - 88304/1339*e^2 - 21452/1339, 656/1339*e^8 + 12906/1339*e^6 + 74192/1339*e^4 + 121243/1339*e^2 - 1900/1339, -1477/1339*e^8 - 24735/1339*e^6 - 104504/1339*e^4 - 90998/1339*e^2 - 18232/1339, 44/103*e^8 + 784/103*e^6 + 3783/103*e^4 + 4161/103*e^2 - 3024/103, 646/6695*e^9 + 21127/6695*e^7 + 220449/6695*e^5 + 769614/6695*e^3 + 481361/6695*e, -3854/6695*e^9 - 88878/6695*e^7 - 672881/6695*e^5 - 1855976/6695*e^3 - 1151759/6695*e, -1605/1339*e^8 - 28233/1339*e^6 - 135669/1339*e^4 - 182291/1339*e^2 - 20278/1339, -609/1339*e^8 - 10973/1339*e^6 - 54474/1339*e^4 - 76434/1339*e^2 - 30217/1339, -1282/1339*e^9 - 25034/1339*e^7 - 147963/1339*e^5 - 298270/1339*e^3 - 137780/1339*e, -1967/6695*e^9 - 32979/6695*e^7 - 145748/6695*e^5 - 210443/6695*e^3 - 301187/6695*e, -135/103*e^8 - 2368/103*e^6 - 11010/103*e^4 - 11931/103*e^2 + 1291/103, 135/1339*e^8 + 3501/1339*e^6 + 30683/1339*e^4 + 103807/1339*e^2 + 59170/1339, -6114/6695*e^9 - 117583/6695*e^7 - 682726/6695*e^5 - 1394301/6695*e^3 - 787439/6695*e, 592/6695*e^9 + 15174/6695*e^7 + 134263/6695*e^5 + 475288/6695*e^3 + 472422/6695*e, 4259/6695*e^9 + 69923/6695*e^7 + 281551/6695*e^5 + 247271/6695*e^3 + 304934/6695*e, -1881/6695*e^9 - 25482/6695*e^7 - 27334/6695*e^5 + 368146/6695*e^3 + 477519/6695*e, 6561/6695*e^9 + 130782/6695*e^7 + 810714/6695*e^5 + 1850434/6695*e^3 + 1283591/6695*e, -7872/6695*e^9 - 152194/6695*e^7 - 891643/6695*e^5 - 1833853/6695*e^3 - 1005552/6695*e, 2482/6695*e^9 + 42764/6695*e^7 + 191583/6695*e^5 + 165123/6695*e^3 - 185488/6695*e, -5587/6695*e^9 - 113914/6695*e^7 - 728578/6695*e^5 - 1676978/6695*e^3 - 947792/6695*e, 329/1339*e^9 + 6836/1339*e^7 + 46974/1339*e^5 + 134237/1339*e^3 + 134033/1339*e, -4827/6695*e^9 - 90634/6695*e^7 - 498763/6695*e^5 - 902693/6695*e^3 - 392512/6695*e, 407/1339*e^8 + 7252/1339*e^6 + 35482/1339*e^4 + 48918/1339*e^2 - 43937/1339, 318/1339*e^8 + 5301/1339*e^6 + 24012/1339*e^4 + 40162/1339*e^2 + 57848/1339, 914/1339*e^8 + 15312/1339*e^6 + 62548/1339*e^4 + 26597/1339*e^2 - 50541/1339, 955/1339*e^8 + 14947/1339*e^6 + 51117/1339*e^4 - 2062/1339*e^2 - 30240/1339, -1841/1339*e^8 - 30247/1339*e^6 - 122288/1339*e^4 - 97667/1339*e^2 - 61626/1339, 461/1339*e^8 + 6510/1339*e^6 + 11870/1339*e^4 - 59795/1339*e^2 - 71151/1339, -4678/6695*e^9 - 88466/6695*e^7 - 491807/6695*e^5 - 889012/6695*e^3 - 361028/6695*e, 993/6695*e^9 + 16111/6695*e^7 + 63612/6695*e^5 + 77157/6695*e^3 + 213103/6695*e, -1290/1339*e^9 - 25420/1339*e^7 - 154681/1339*e^5 - 342221/1339*e^3 - 208038/1339*e, -3157/6695*e^9 - 60269/6695*e^7 - 344998/6695*e^5 - 690853/6695*e^3 - 384857/6695*e, -1636/6695*e^9 - 38767/6695*e^7 - 298614/6695*e^5 - 787274/6695*e^3 - 375211/6695*e, 489/1339*e^9 + 11878/1339*e^7 + 95638/1339*e^5 + 276807/1339*e^3 + 153328/1339*e, -2482/6695*e^9 - 42764/6695*e^7 - 191583/6695*e^5 - 165123/6695*e^3 + 212268/6695*e, 1621/6695*e^9 + 39717/6695*e^7 + 317819/6695*e^5 + 891489/6695*e^3 + 546426/6695*e, -581/6695*e^9 - 29707/6695*e^7 - 368389/6695*e^5 - 1330564/6695*e^3 - 839446/6695*e, -428/1339*e^9 - 7261/1339*e^7 - 30019/1339*e^5 - 8798/1339*e^3 + 49313/1339*e, -9/13*e^8 - 171/13*e^6 - 944/13*e^4 - 1561/13*e^2 - 322/13, -1267/1339*e^8 - 21967/1339*e^6 - 101557/1339*e^4 - 121295/1339*e^2 - 37178/1339, 174/515*e^9 + 2988/515*e^7 + 13401/515*e^5 + 13201/515*e^3 - 8391/515*e, -2308/6695*e^9 - 44411/6695*e^7 - 259037/6695*e^5 - 563922/6695*e^3 - 519183/6695*e, -7357/6695*e^9 - 142409/6695*e^7 - 839113/6695*e^5 - 1751968/6695*e^3 - 943237/6695*e, -7002/6695*e^9 - 133649/6695*e^7 - 766958/6695*e^5 - 1578328/6695*e^3 - 1094372/6695*e, 646/1339*e^8 + 10415/1339*e^6 + 38345/1339*e^4 + 6384/1339*e^2 - 56917/1339, 5/1339*e^8 + 1915/1339*e^6 + 30644/1339*e^4 + 115676/1339*e^2 + 49602/1339, -1884/1339*e^8 - 31987/1339*e^6 - 137308/1339*e^4 - 107780/1339*e^2 + 45790/1339, 517/1339*e^8 + 7873/1339*e^6 + 24082/1339*e^4 - 11904/1339*e^2 + 16277/1339, -49/1339*e^9 - 2699/1339*e^7 - 34118/1339*e^5 - 122412/1339*e^3 - 75830/1339*e, -2226/6695*e^9 - 43802/6695*e^7 - 261814/6695*e^5 - 548934/6695*e^3 - 351376/6695*e, -2746/6695*e^9 - 55502/6695*e^7 - 350344/6695*e^5 - 788004/6695*e^3 - 472666/6695*e, 6654/6695*e^9 + 130248/6695*e^7 + 774661/6695*e^5 + 1579221/6695*e^3 + 729539/6695*e, -1333/1339*e^8 - 23143/1339*e^6 - 105429/1339*e^4 - 109975/1339*e^2 + 17210/1339, -474/1339*e^8 - 8811/1339*e^6 - 46554/1339*e^4 - 77069/1339*e^2 - 44692/1339, -633/1339*e^8 - 10792/1339*e^6 - 46509/1339*e^4 - 30200/1339*e^2 + 8063/1339, 1626/1339*e^8 + 28242/1339*e^6 + 127528/1339*e^4 + 118069/1339*e^2 - 57404/1339, 670/1339*e^8 + 10234/1339*e^6 + 30380/1339*e^4 - 26460/1339*e^2 - 25569/1339, -1268/1339*e^8 - 19672/1339*e^6 - 63231/1339*e^4 + 21338/1339*e^2 + 7265/1339, 758/1339*e^8 + 13141/1339*e^6 + 57413/1339*e^4 + 40572/1339*e^2 - 9266/1339, -399/6695*e^9 - 17578/6695*e^7 - 206851/6695*e^5 - 736061/6695*e^3 - 476304/6695*e, 2976/6695*e^9 + 63252/6695*e^7 + 426324/6695*e^5 + 1035629/6695*e^3 + 668196/6695*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}>] := 13/515*e^9 + 241/515*e^7 + 1312/515*e^5 + 2552/515*e^3 + 2028/515*e;

// 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;