/*
  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![-1, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, w^2 - w - 3}>;

primesArray := [
[3, 3, w + 1],
[7, 7, w - 1],
[8, 2, 2],
[9, 3, -w^2 + 2*w + 4],
[11, 11, -w^2 + 2*w + 2],
[13, 13, -w^2 + w + 4],
[19, 19, w + 3],
[19, 19, -w^2 + 2*w + 5],
[19, 19, -w^2 + 3*w + 2],
[23, 23, w^2 - w - 3],
[23, 23, -w^2 + 2],
[23, 23, -w + 4],
[31, 31, w^2 - 5],
[43, 43, w^2 - 3*w - 3],
[49, 7, w^2 - 6],
[53, 53, 2*w - 5],
[61, 61, 2*w^2 - 2*w - 9],
[71, 71, 2*w - 3],
[73, 73, 2*w^2 - 5*w - 5],
[83, 83, w^2 - w - 9],
[97, 97, 2*w^2 - 3*w - 7],
[103, 103, w^2 - 3*w - 6],
[109, 109, w^2 - 4*w - 3],
[121, 11, 2*w^2 - w - 7],
[125, 5, -5],
[127, 127, w^2 + w - 5],
[131, 131, w^2 - 11],
[137, 137, 3*w^2 - 7*w - 8],
[137, 137, 2*w^2 - 5*w - 6],
[137, 137, -w^2 + w - 2],
[139, 139, 3*w^2 - 2*w - 16],
[139, 139, 3*w - 2],
[139, 139, -w^2 + 3*w - 3],
[151, 151, w^2 + w - 11],
[157, 157, 2*w^2 - 3*w - 3],
[163, 163, w^2 - w - 10],
[169, 13, 2*w^2 - 3*w - 4],
[173, 173, -3*w^2 + 6*w + 8],
[191, 191, -4*w^2 + 7*w + 15],
[193, 193, 2*w^2 - 3],
[197, 197, w^2 - 3*w - 11],
[199, 199, 3*w^2 - 5*w - 16],
[211, 211, w^2 - 4*w - 9],
[223, 223, 2*w^2 - w - 4],
[227, 227, 3*w - 4],
[229, 229, w^2 + w - 8],
[241, 241, w^2 - 4*w - 6],
[251, 251, w - 7],
[257, 257, w^2 - 4*w - 7],
[263, 263, 3*w^2 - 5*w - 10],
[263, 263, 2*w^2 - 5*w - 15],
[263, 263, -w^2 + 6*w - 1],
[269, 269, -w^2 - w + 14],
[277, 277, 2*w^2 - 5*w - 8],
[277, 277, w^2 + 3*w - 3],
[277, 277, 3*w^2 - 4*w - 12],
[281, 281, -2*w - 7],
[293, 293, 3*w^2 - 2*w - 12],
[307, 307, 2*w^2 + w - 8],
[313, 313, w^2 + 4*w - 2],
[317, 317, 3*w^2 - 6*w - 5],
[331, 331, 4*w^2 - 4*w - 19],
[337, 337, -5*w^2 + 8*w + 21],
[347, 347, 3*w^2 - 6*w - 13],
[347, 347, 3*w^2 - 5*w - 22],
[347, 347, 3*w^2 - 5*w - 9],
[349, 349, w^2 - 5*w - 11],
[353, 353, -w^2 + 6*w - 7],
[359, 359, -4*w^2 + 8*w + 11],
[367, 367, 3*w^2 - 4*w - 11],
[373, 373, w^2 + 2*w - 6],
[379, 379, 5*w^2 - 9*w - 18],
[409, 409, -w^2 - 4],
[439, 439, 3*w^2 - 6*w - 14],
[443, 443, -w^2 - 2*w + 13],
[443, 443, 2*w^2 - 15],
[443, 443, 3*w^2 - 13],
[449, 449, w^2 + 2*w - 7],
[449, 449, 2*w^2 - 6*w - 7],
[449, 449, 3*w^2 - 3*w - 11],
[461, 461, -w^2 - 2*w - 5],
[467, 467, 3*w^2 - 5*w - 7],
[479, 479, -w^2 + 6*w - 4],
[491, 491, 3*w^2 - 5*w - 6],
[499, 499, w^2 - 5*w - 8],
[509, 509, 3*w^2 - 6*w - 19],
[521, 521, 3*w^2 - 2*w - 10],
[523, 523, 4*w - 9],
[541, 541, 3*w^2 - 3*w - 10],
[547, 547, 2*w^2 + w - 20],
[547, 547, w^2 - 6*w - 5],
[547, 547, 6*w^2 - 11*w - 24],
[557, 557, -5*w^2 + 11*w + 11],
[563, 563, 3*w^2 - 6*w - 16],
[569, 569, 4*w^2 - 5*w - 17],
[587, 587, 5*w^2 - 8*w - 20],
[593, 593, 5*w - 3],
[593, 593, 3*w^2 - w - 17],
[593, 593, w - 9],
[613, 613, 2*w^2 - w - 19],
[613, 613, 3*w^2 - 4*w - 5],
[613, 613, 3*w^2 - w - 8],
[617, 617, 3*w^2 - 2*w - 9],
[617, 617, -w^2 + w - 5],
[617, 617, 4*w^2 - 6*w - 15],
[619, 619, -w^2 + 3*w - 6],
[631, 631, 4*w^2 - 7*w - 21],
[641, 641, -w^2 - 3*w + 20],
[643, 643, 3*w^2 - 14],
[643, 643, w^2 - w - 13],
[643, 643, 2*w^2 - 9*w + 2],
[647, 647, 2*w^2 - 6*w - 9],
[653, 653, 3*w^2 - 5],
[659, 659, 6*w^2 - 11*w - 21],
[661, 661, w^2 - 6*w - 6],
[661, 661, w^2 + 3*w - 6],
[661, 661, 5*w^2 - 7*w - 22],
[673, 673, w^2 - 6*w - 12],
[677, 677, 4*w^2 - 7*w - 22],
[677, 677, 3*w^2 - 7*w - 12],
[677, 677, -2*w^2 - 3],
[683, 683, -2*w - 9],
[683, 683, 4*w^2 - 7*w - 12],
[683, 683, 3*w^2 - 3*w - 8],
[691, 691, 2*w^2 + w - 11],
[701, 701, 3*w^2 - 3*w - 5],
[709, 709, -w^2 - 2*w - 6],
[727, 727, 3*w^2 - w - 6],
[727, 727, 3*w^2 - w - 27],
[727, 727, 3*w^2 - w - 5],
[733, 733, -2*w^2 + 9*w - 5],
[739, 739, 3*w^2 - 4*w - 23],
[743, 743, 2*w^2 - 7*w - 7],
[743, 743, 3*w^2 - w - 20],
[743, 743, 3*w^2 - 2*w - 7],
[757, 757, -w - 9],
[757, 757, 4*w^2 - 2*w - 29],
[757, 757, 2*w^2 - 3*w - 18],
[761, 761, 3*w^2 - 2*w - 6],
[761, 761, 3*w^2 - 2*w - 25],
[773, 773, -5*w - 12],
[787, 787, -3*w - 10],
[809, 809, 4*w^2 - w - 19],
[821, 821, 5*w^2 - 10*w - 19],
[823, 823, -4*w - 11],
[827, 827, -7*w - 5],
[827, 827, 2*w^2 - 6*w - 13],
[827, 827, -w^2 - 2*w + 18],
[839, 839, 6*w^2 - 9*w - 26],
[839, 839, 5*w^2 - 12*w - 13],
[839, 839, w - 10],
[857, 857, -w^2 + 4*w - 8],
[859, 859, 3*w^2 - 8*w - 10],
[859, 859, 4*w^2 - 9*w - 14],
[859, 859, 3*w^2 - 26],
[863, 863, 2*w^2 + 3*w - 7],
[877, 877, -5*w^2 + 11*w + 9],
[877, 877, -5*w^2 + 10*w + 13],
[877, 877, -w^2 + w - 6],
[881, 881, w^2 - 3*w - 14],
[883, 883, -w^2 + 3*w - 7],
[887, 887, -3*w^2 + 6*w - 2],
[907, 907, w^2 + 3*w - 8],
[911, 911, 4*w^2 - 6*w - 29],
[937, 937, -6*w^2 + 11*w + 25],
[953, 953, 6*w^2 - 6*w - 29],
[961, 31, -w^2 + 7*w - 5],
[967, 967, 2*w^2 + w - 14],
[971, 971, 5*w^2 - 3*w - 27],
[971, 971, 4*w - 15],
[971, 971, w^2 - 2*w - 14],
[977, 977, 3*w^2 - 5*w - 24],
[983, 983, w^2 - 4*w - 15],
[991, 991, 6*w^2 - 10*w - 23],
[991, 991, 5*w^2 - 6*w - 22],
[991, 991, 3*w^2 + w - 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 + 4*x^9 - 11*x^8 - 54*x^7 + 25*x^6 + 223*x^5 + 22*x^4 - 339*x^3 - 80*x^2 + 165*x + 29;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -221/751*e^9 - 678/751*e^8 + 3029/751*e^7 + 8713/751*e^6 - 13684/751*e^5 - 31981/751*e^4 + 27847/751*e^3 + 35947/751*e^2 - 22508/751*e - 6646/751, 364/751*e^9 + 940/751*e^8 - 5254/751*e^7 - 12142/751*e^6 + 25454/751*e^5 + 44767/751*e^4 - 54215/751*e^3 - 48737/751*e^2 + 43080/751*e + 6352/751, -170/751*e^9 - 406/751*e^8 + 2330/751*e^7 + 5027/751*e^6 - 10064/751*e^5 - 17033/751*e^4 + 17608/751*e^3 + 15520/751*e^2 - 11768/751*e - 1184/751, -230/751*e^9 - 726/751*e^8 + 3064/751*e^7 + 9496/751*e^6 - 12865/751*e^5 - 35900/751*e^4 + 22674/751*e^3 + 41363/751*e^2 - 15568/751*e - 8140/751, 582/751*e^9 + 1602/751*e^8 - 8021/751*e^7 - 20594/751*e^6 + 35656/751*e^5 + 75692/751*e^4 - 67014/751*e^3 - 84631/751*e^2 + 45872/751*e + 14002/751, -155/751*e^9 - 326/751*e^8 + 2522/751*e^7 + 4473/751*e^6 - 14433/751*e^5 - 18512/751*e^4 + 35492/751*e^3 + 24267/751*e^2 - 28842/751*e - 3951/751, -1116/751*e^9 - 2948/751*e^8 + 16356/751*e^7 + 38514/751*e^6 - 80937/751*e^5 - 145002/751*e^4 + 173383/751*e^3 + 165410/751*e^2 - 131511/751*e - 29799/751, -300/751*e^9 - 849/751*e^8 + 4421/751*e^7 + 11080/751*e^6 - 22266/751*e^5 - 41014/751*e^4 + 49362/751*e^3 + 42850/751*e^2 - 38526/751*e - 3238/751, -1, 269/751*e^9 + 934/751*e^8 - 3466/751*e^7 - 12138/751*e^6 + 13822/751*e^5 + 45122/751*e^4 - 23789/751*e^3 - 50313/751*e^2 + 17287/751*e + 6353/751, 991/751*e^9 + 2782/751*e^8 - 14201/751*e^7 - 35900/751*e^6 + 68280/751*e^5 + 131793/751*e^4 - 143428/751*e^3 - 143175/751*e^2 + 107573/751*e + 20314/751, 24/751*e^9 + 128/751*e^8 - 594/751*e^7 - 2088/751*e^6 + 5326/751*e^5 + 10701/751*e^4 - 18999/751*e^3 - 18448/751*e^2 + 19544/751*e + 6988/751, 595/751*e^9 + 1421/751*e^8 - 8906/751*e^7 - 18721/751*e^6 + 44987/751*e^5 + 72007/751*e^4 - 96174/751*e^3 - 85862/751*e^2 + 71228/751*e + 13156/751, -1594/751*e^9 - 4496/751*e^8 + 22554/751*e^7 + 58321/751*e^6 - 105780/751*e^5 - 216379/751*e^4 + 215144/751*e^3 + 238691/751*e^2 - 159281/751*e - 32545/751, 1358/751*e^9 + 3738/751*e^8 - 19717/751*e^7 - 49054/751*e^6 + 95714/751*e^5 + 185877/751*e^4 - 198422/751*e^3 - 213744/751*e^2 + 145586/751*e + 37678/751, 1431/751*e^9 + 3877/751*e^8 - 21336/751*e^7 - 50899/751*e^6 + 108597/751*e^5 + 192798/751*e^4 - 239407/751*e^3 - 222043/751*e^2 + 183754/751*e + 38531/751, -103/751*e^9 - 299/751*e^8 + 1986/751*e^7 + 4455/751*e^6 - 13908/751*e^5 - 19734/751*e^4 + 41265/751*e^3 + 25351/751*e^2 - 41570/751*e - 4331/751, 1855/751*e^9 + 5137/751*e^8 - 26573/751*e^7 - 66759/751*e^6 + 127089/751*e^5 + 248171/751*e^4 - 262640/751*e^3 - 271840/751*e^2 + 190831/751*e + 33815/751, -1105/751*e^9 - 2639/751*e^8 + 16647/751*e^7 + 34553/751*e^6 - 84942/751*e^5 - 131367/751*e^4 + 183544/751*e^3 + 154952/751*e^2 - 137323/751*e - 30226/751, 2505/751*e^9 + 6601/751*e^8 - 37028/751*e^7 - 86510/751*e^6 + 185095/751*e^5 + 326020/751*e^4 - 397378/751*e^3 - 363430/751*e^2 + 299087/751*e + 47840/751, 1406/751*e^9 + 3994/751*e^8 - 19403/751*e^7 - 50977/751*e^6 + 87591/751*e^5 + 183247/751*e^4 - 172585/751*e^3 - 188307/751*e^2 + 127598/751*e + 12602/751, 1693/751*e^9 + 4273/751*e^8 - 25192/751*e^7 - 56420/751*e^6 + 126811/751*e^5 + 216681/751*e^4 - 273144/751*e^3 - 253958/751*e^2 + 205354/751*e + 43722/751, -1412/751*e^9 - 4026/751*e^8 + 19927/751*e^7 + 52250/751*e^6 - 92302/751*e^5 - 193620/751*e^4 + 180151/751*e^3 + 210943/751*e^2 - 123472/751*e - 27116/751, -589/751*e^9 - 1389/751*e^8 + 8382/751*e^7 + 17448/751*e^6 - 38774/751*e^5 - 60883/751*e^4 + 74339/751*e^3 + 58720/751*e^2 - 52073/751*e - 4650/751, -2011/751*e^9 - 5969/751*e^8 + 28181/751*e^7 + 78078/751*e^6 - 130166/751*e^5 - 294071/751*e^4 + 260341/751*e^3 + 333925/751*e^2 - 191699/751*e - 55956/751, -840/751*e^9 - 2227/751*e^8 + 11778/751*e^7 + 28020/751*e^6 - 54985/751*e^5 - 98918/751*e^4 + 114482/751*e^3 + 102707/751*e^2 - 90750/751*e - 15525/751, -3128/751*e^9 - 8672/751*e^8 + 44374/751*e^7 + 112173/751*e^6 - 208759/751*e^5 - 414642/751*e^4 + 424471/751*e^3 + 455294/751*e^2 - 312509/751*e - 62640/751, -91/751*e^9 - 235/751*e^8 + 1689/751*e^7 + 3411/751*e^6 - 11245/751*e^5 - 15510/751*e^4 + 30639/751*e^3 + 27392/751*e^2 - 23537/751*e - 18861/751, -640/751*e^9 - 1661/751*e^8 + 9081/751*e^7 + 21885/751*e^6 - 41643/751*e^5 - 84092/751*e^4 + 77819/751*e^3 + 100175/751*e^2 - 51548/751*e - 22128/751, 302/751*e^9 + 1110/751*e^8 - 4095/751*e^7 - 15009/751*e^6 + 17578/751*e^5 + 59742/751*e^4 - 30856/751*e^3 - 73426/751*e^2 + 22381/751*e + 9578/751, 358/751*e^9 + 908/751*e^8 - 5481/751*e^7 - 12371/751*e^6 + 28253/751*e^5 + 49414/751*e^4 - 59416/751*e^3 - 57643/751*e^2 + 39696/751*e + 6107/751, 1520/751*e^9 + 3851/751*e^8 - 23351/751*e^7 - 51132/751*e^6 + 123028/751*e^5 + 197090/751*e^4 - 275034/751*e^3 - 229373/751*e^2 + 209167/751*e + 37534/751, 1203/751*e^9 + 3412/751*e^8 - 17195/751*e^7 - 44581/751*e^6 + 82032/751*e^5 + 167365/751*e^4 - 171191/751*e^3 - 186473/751*e^2 + 134017/751*e + 23213/751, 1322/751*e^9 + 3546/751*e^8 - 18826/751*e^7 - 45171/751*e^6 + 89227/751*e^5 + 161940/751*e^4 - 183817/751*e^3 - 165795/751*e^2 + 140302/751*e + 10674/751, -570/751*e^9 - 1538/751*e^8 + 8475/751*e^7 + 20301/751*e^6 - 42756/751*e^5 - 76725/751*e^4 + 91685/751*e^3 + 80664/751*e^2 - 67642/751*e - 745/751, 916/751*e^9 + 2382/751*e^8 - 13659/751*e^7 - 31628/751*e^6 + 68346/751*e^5 + 121164/751*e^4 - 142728/751*e^3 - 135091/751*e^2 + 100570/751*e + 10868/751, -2278/751*e^9 - 5891/751*e^8 + 33475/751*e^7 + 76524/751*e^6 - 165198/751*e^5 - 283666/751*e^4 + 346945/751*e^3 + 307100/751*e^2 - 254420/751*e - 37945/751, 599/751*e^9 + 1192/751*e^8 - 9756/751*e^7 - 16065/751*e^6 + 54386/751*e^5 + 64403/751*e^4 - 122246/751*e^3 - 83179/751*e^2 + 85500/751*e + 20579/751, 364/751*e^9 + 940/751*e^8 - 5254/751*e^7 - 12142/751*e^6 + 24703/751*e^5 + 44767/751*e^4 - 48207/751*e^3 - 50239/751*e^2 + 37823/751*e + 9356/751, -3949/751*e^9 - 11048/751*e^8 + 56245/751*e^7 + 143797/751*e^6 - 266224/751*e^5 - 537663/751*e^4 + 543532/751*e^3 + 606230/751*e^2 - 394045/751*e - 98292/751, 419/751*e^9 + 983/751*e^8 - 6803/751*e^7 - 13172/751*e^6 + 39224/751*e^5 + 51360/751*e^4 - 98036/751*e^3 - 57469/751*e^2 + 79357/751*e - 3293/751, -1498/751*e^9 - 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57757/751*e^7 - 129859/751*e^6 + 307295/751*e^5 + 511854/751*e^4 - 696760/751*e^3 - 626767/751*e^2 + 535825/751*e + 129188/751, 2831/751*e^9 + 7088/751*e^8 - 42468/751*e^7 - 93093/751*e^6 + 217762/751*e^5 + 354407/751*e^4 - 484032/751*e^3 - 416252/751*e^2 + 367297/751*e + 88438/751, 6534/751*e^9 + 17575/751*e^8 - 95253/751*e^7 - 228255/751*e^6 + 467320/751*e^5 + 846783/751*e^4 - 991473/751*e^3 - 925763/751*e^2 + 754023/751*e + 121862/751, -2352/751*e^9 - 5785/751*e^8 + 36433/751*e^7 + 77705/751*e^6 - 195263/751*e^5 - 306710/751*e^4 + 452275/751*e^3 + 375747/751*e^2 - 358489/751*e - 75763/751, 1576/751*e^9 + 3649/751*e^8 - 23986/751*e^7 - 46992/751*e^6 + 126193/751*e^5 + 172493/751*e^4 - 289325/751*e^3 - 185052/751*e^2 + 218221/751*e + 24300/751, -3279/751*e^9 - 9978/751*e^8 + 45295/751*e^7 + 128314/751*e^6 - 206283/751*e^5 - 466292/751*e^4 + 415116/751*e^3 + 485248/751*e^2 - 312059/751*e - 36638/751, 3894/751*e^9 + 10254/751*e^8 - 56198/751*e^7 - 133755/751*e^6 + 269727/751*e^5 + 502532/751*e^4 - 554534/751*e^3 - 571964/751*e^2 + 414093/751*e + 105684/751, 544/751*e^9 + 1900/751*e^8 - 6705/751*e^7 - 24047/751*e^6 + 25596/751*e^5 + 86348/751*e^4 - 46883/751*e^3 - 99981/751*e^2 + 33452/751*e + 42991/751, 4279/751*e^9 + 12057/751*e^8 - 61033/751*e^7 - 155985/751*e^6 + 291017/751*e^5 + 573466/751*e^4 - 608194/751*e^3 - 613562/751*e^2 + 469768/751*e + 68960/751, -4042/751*e^9 - 11544/751*e^8 + 58359/751*e^7 + 149635/751*e^6 - 285548/751*e^5 - 553877/751*e^4 + 617247/751*e^3 + 610126/751*e^2 - 475035/751*e - 85192/751, -4158/751*e^9 - 10911/751*e^8 + 62732/751*e^7 + 143205/751*e^6 - 326060/751*e^5 - 542139/751*e^4 + 740242/751*e^3 + 617933/751*e^2 - 587021/751*e - 91681/751, -1116/751*e^9 - 2948/751*e^8 + 15605/751*e^7 + 37012/751*e^6 - 71925/751*e^5 - 126978/751*e^4 + 144845/751*e^3 + 107583/751*e^2 - 112736/751*e + 14510/751, -4471/751*e^9 - 12330/751*e^8 + 63532/751*e^7 + 159171/751*e^6 - 300581/751*e^5 - 587729/751*e^4 + 621251/751*e^3 + 659761/751*e^2 - 463153/751*e - 131623/751, -5669/751*e^9 - 15465/751*e^8 + 81542/751*e^7 + 201815/751*e^6 - 390578/751*e^5 - 760093/751*e^4 + 801157/751*e^3 + 874183/751*e^2 - 583836/751*e - 154006/751, 302/751*e^9 + 359/751*e^8 - 5597/751*e^7 - 6748/751*e^6 + 34851/751*e^5 + 40216/751*e^4 - 80422/751*e^3 - 71924/751*e^2 + 49417/751*e + 19341/751, -3795/751*e^9 - 10477/751*e^8 + 54311/751*e^7 + 137158/751*e^6 - 258459/751*e^5 - 518752/751*e^4 + 529578/751*e^3 + 597251/751*e^2 - 392803/751*e - 90752/751, 5508/751*e^9 + 14356/751*e^8 - 81500/751*e^7 - 187057/751*e^6 + 409735/751*e^5 + 700417/751*e^4 - 891026/751*e^3 - 785224/751*e^2 + 671770/751*e + 111509/751, 1648/751*e^9 + 4784/751*e^8 - 22764/751*e^7 - 60766/751*e^6 + 103119/751*e^5 + 215861/751*e^4 - 202881/751*e^3 - 212609/751*e^2 + 141673/751*e + 955/751, -438/751*e^9 - 1585/751*e^8 + 5208/751*e^7 + 20082/751*e^6 - 17218/751*e^5 - 73068/751*e^4 + 22112/751*e^3 + 84340/751*e^2 - 23985/751*e - 8873/751, 1896/751*e^9 + 4855/751*e^8 - 29653/751*e^7 - 65069/751*e^6 + 160908/751*e^5 + 253591/751*e^4 - 373670/751*e^3 - 290338/751*e^2 + 290557/751*e + 27103/751, 2499/751*e^9 + 6569/751*e^8 - 37255/751*e^7 - 86739/751*e^6 + 187894/751*e^5 + 331418/751*e^4 - 400326/751*e^3 - 381348/751*e^2 + 285189/751*e + 64868/751, 5244/751*e^9 + 14450/751*e^8 - 76468/751*e^7 - 188872/751*e^6 + 375932/751*e^5 + 707372/751*e^4 - 797691/751*e^3 - 776805/751*e^2 + 588962/751*e + 84958/751, 1544/751*e^9 + 4730/751*e^8 - 20190/751*e^7 - 60730/751*e^6 + 81792/751*e^5 + 219056/751*e^4 - 140829/751*e^3 - 218532/751*e^2 + 104796/751*e + 20490/751, -1243/751*e^9 - 2624/751*e^8 + 20438/751*e^7 + 34543/751*e^6 - 115942/751*e^5 - 130377/751*e^4 + 268944/751*e^3 + 144623/751*e^2 - 194878/751*e - 13331/751, 657/751*e^9 + 1251/751*e^8 - 9314/751*e^7 - 15854/751*e^6 + 41598/751*e^5 + 57783/751*e^4 - 72971/751*e^3 - 65679/751*e^2 + 51373/751*e + 15938/751, 2839/751*e^9 + 8132/751*e^8 - 38911/751*e^7 - 104303/751*e^6 + 173476/751*e^5 + 380504/751*e^4 - 337912/751*e^3 - 410886/751*e^2 + 242637/751*e + 46208/751, -2202/751*e^9 - 5736/751*e^8 + 32345/751*e^7 + 74418/751*e^6 - 161600/751*e^5 - 275689/751*e^4 + 354747/751*e^3 + 294242/751*e^2 - 278395/751*e - 27582/751, 351/751*e^9 + 370/751*e^8 - 6622/751*e^7 - 5754/751*e^6 + 43159/751*e^5 + 28926/751*e^4 - 108416/751*e^3 - 50510/751*e^2 + 65037/751*e + 21467/751, -3042/751*e^9 - 7212/751*e^8 + 47878/751*e^7 + 96430/751*e^6 - 259392/751*e^5 - 377611/751*e^4 + 590891/751*e^3 + 457780/751*e^2 - 452506/751*e - 85914/751, 3022/751*e^9 + 7606/751*e^8 - 45881/751*e^7 - 99947/751*e^6 + 237180/751*e^5 + 379583/751*e^4 - 516105/751*e^3 - 436649/751*e^2 + 374387/751*e + 69076/751, 958/751*e^9 + 2606/751*e^8 - 14323/751*e^7 - 33780/751*e^6 + 74287/751*e^5 + 124683/751*e^4 - 170907/751*e^3 - 131327/751*e^2 + 136274/751*e - 8445/751, -6285/751*e^9 - 16998/751*e^8 + 92282/751*e^7 + 223114/751*e^6 - 459939/751*e^5 - 846251/751*e^4 + 998912/751*e^3 + 972432/751*e^2 - 760783/751*e - 162387/751, -2418/751*e^9 - 6137/751*e^8 + 37691/751*e^7 + 81194/751*e^6 - 202775/751*e^5 - 308163/751*e^4 + 463405/751*e^3 + 338612/751*e^2 - 361167/751*e - 41659/751, -1492/751*e^9 - 4703/751*e^8 + 18903/751*e^7 + 58459/751*e^6 - 70002/751*e^5 - 196997/751*e^4 + 90277/751*e^3 + 167797/751*e^2 - 30408/751*e + 158/751, -2642/751*e^9 - 6831/751*e^8 + 39480/751*e^7 + 88666/751*e^6 - 201917/751*e^5 - 329184/751*e^4 + 447722/751*e^3 + 363347/751*e^2 - 335050/751*e - 49554/751, -6498/751*e^9 - 17383/751*e^8 + 94362/751*e^7 + 226625/751*e^6 - 459331/751*e^5 - 852886/751*e^4 + 958844/751*e^3 + 985958/751*e^2 - 695418/751*e - 179721/751, 1608/751*e^9 + 4070/751*e^8 - 24778/751*e^7 - 52029/751*e^6 + 133795/751*e^5 + 187512/751*e^4 - 316910/751*e^3 - 197383/751*e^2 + 254293/751*e + 28110/751, 4798/751*e^9 + 14074/751*e^8 - 65054/751*e^7 - 180110/751*e^6 + 282840/751*e^5 + 654018/751*e^4 - 525922/751*e^3 - 707594/751*e^2 + 374968/751*e + 100291/751, 1596/751*e^9 + 4006/751*e^8 - 24481/751*e^7 - 52487/751*e^6 + 131132/751*e^5 + 200561/751*e^4 - 310039/751*e^3 - 241480/751*e^2 + 251280/751*e + 50150/751, 5483/751*e^9 + 14473/751*e^8 - 78816/751*e^7 - 187135/751*e^6 + 376713/751*e^5 + 693870/751*e^4 - 768630/751*e^3 - 780026/751*e^2 + 545020/751*e + 120126/751, -4263/751*e^9 - 11471/751*e^8 + 62890/751*e^7 + 150838/751*e^6 - 312750/751*e^5 - 576095/751*e^4 + 660865/751*e^3 + 685125/751*e^2 - 472009/751*e - 133143/751];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, w^2 - w - 3}>] := 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;