/*
  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 | {1, 1, 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^12 - 50*x^10 + 857*x^8 - 5988*x^6 + 16760*x^4 - 19952*x^2 + 8464;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-4893/1292048*e^11 + 58529/323012*e^9 - 3687761/1292048*e^7 + 10639339/646024*e^5 - 2173977/80753*e^3 + 930274/80753*e, 4893/1292048*e^11 - 58529/323012*e^9 + 3687761/1292048*e^7 - 10639339/646024*e^5 + 2173977/80753*e^3 - 849521/80753*e, 4893/1292048*e^11 - 58529/323012*e^9 + 3687761/1292048*e^7 - 10639339/646024*e^5 + 2173977/80753*e^3 - 849521/80753*e, 531/28088*e^10 - 12895/14044*e^8 + 418287/28088*e^6 - 646745/7022*e^4 + 1317987/7022*e^2 - 410638/3511, -4553/112352*e^10 + 55247/28088*e^8 - 3577113/112352*e^6 + 10987823/56176*e^4 - 10915793/28088*e^2 + 3293019/14044, 69317/5168192*e^11 - 838529/1292048*e^9 + 53984581/5168192*e^7 - 163743903/2584096*e^5 + 156719381/1292048*e^3 - 43714187/646024*e, 69317/5168192*e^11 - 838529/1292048*e^9 + 53984581/5168192*e^7 - 163743903/2584096*e^5 + 156719381/1292048*e^3 - 43714187/646024*e, -63195/5168192*e^11 + 765431/1292048*e^9 - 49426923/5168192*e^7 + 151211601/2584096*e^5 - 149492719/1292048*e^3 + 45901585/646024*e, -63195/5168192*e^11 + 765431/1292048*e^9 - 49426923/5168192*e^7 + 151211601/2584096*e^5 - 149492719/1292048*e^3 + 45901585/646024*e, 445/112352*e^10 - 2819/14044*e^8 + 396165/112352*e^6 - 1440249/56176*e^4 + 2054521/28088*e^2 - 868117/14044, 445/112352*e^10 - 2819/14044*e^8 + 396165/112352*e^6 - 1440249/56176*e^4 + 2054521/28088*e^2 - 868117/14044, -19197/2584096*e^11 + 117365/323012*e^9 - 15434677/2584096*e^7 + 49153705/1292048*e^5 - 53758673/646024*e^3 + 17308961/323012*e, -19197/2584096*e^11 + 117365/323012*e^9 - 15434677/2584096*e^7 + 49153705/1292048*e^5 - 53758673/646024*e^3 + 17308961/323012*e, 66553/2584096*e^11 - 404623/323012*e^9 + 52626913/2584096*e^7 - 163489093/1292048*e^5 + 168426733/646024*e^3 - 54114333/323012*e, 66553/2584096*e^11 - 404623/323012*e^9 + 52626913/2584096*e^7 - 163489093/1292048*e^5 + 168426733/646024*e^3 - 54114333/323012*e, 1135/56176*e^10 - 27695/28088*e^8 + 906139/56176*e^6 - 1426941/14044*e^4 + 3047583/14044*e^2 - 468308/3511, 19197/2584096*e^11 - 117365/323012*e^9 + 15434677/2584096*e^7 - 49153705/1292048*e^5 + 53758673/646024*e^3 - 17308961/323012*e, 19197/2584096*e^11 - 117365/323012*e^9 + 15434677/2584096*e^7 - 49153705/1292048*e^5 + 53758673/646024*e^3 - 17308961/323012*e, -256053/5168192*e^11 + 3109985/1292048*e^9 - 201803413/5168192*e^7 + 623425487/2584096*e^5 - 630009053/1292048*e^3 + 193176963/646024*e, -256053/5168192*e^11 + 3109985/1292048*e^9 - 201803413/5168192*e^7 + 623425487/2584096*e^5 - 630009053/1292048*e^3 + 193176963/646024*e, 4553/112352*e^10 - 55247/28088*e^8 + 3577113/112352*e^6 - 10987823/56176*e^4 + 10915793/28088*e^2 - 3293019/14044, 4553/112352*e^10 - 55247/28088*e^8 + 3577113/112352*e^6 - 10987823/56176*e^4 + 10915793/28088*e^2 - 3293019/14044, -2197/56176*e^10 + 53485/28088*e^8 - 1742713/56176*e^6 + 2720431/14044*e^4 - 5683557/14044*e^2 + 907034/3511, -2197/56176*e^10 + 53485/28088*e^8 - 1742713/56176*e^6 + 2720431/14044*e^4 - 5683557/14044*e^2 + 907034/3511, 47731/2584096*e^11 - 286951/323012*e^9 + 36508603/2584096*e^7 - 107739039/1292048*e^5 + 95693019/646024*e^3 - 26292579/323012*e, 47731/2584096*e^11 - 286951/323012*e^9 + 36508603/2584096*e^7 - 107739039/1292048*e^5 + 95693019/646024*e^3 - 26292579/323012*e, -375/56176*e^10 + 9305/28088*e^8 - 313491/56176*e^6 + 260019/7022*e^4 - 1185121/14044*e^2 + 162832/3511, -375/56176*e^10 + 9305/28088*e^8 - 313491/56176*e^6 + 260019/7022*e^4 - 1185121/14044*e^2 + 162832/3511, -3491/56176*e^10 + 10588/3511*e^8 - 2740539/56176*e^6 + 8400843/28088*e^4 - 8265775/14044*e^2 + 2415567/7022, -3491/56176*e^10 + 10588/3511*e^8 - 2740539/56176*e^6 + 8400843/28088*e^4 - 8265775/14044*e^2 + 2415567/7022, 1437/28088*e^10 - 35095/14044*e^8 + 1150065/28088*e^6 - 906764/3511*e^4 + 3835139/7022*e^2 - 1203116/3511, 1437/28088*e^10 - 35095/14044*e^8 + 1150065/28088*e^6 - 906764/3511*e^4 + 3835139/7022*e^2 - 1203116/3511, -18991/646024*e^11 + 463459/323012*e^9 - 15171517/646024*e^7 + 23920515/161506*e^5 - 102396795/323012*e^3 + 34012605/161506*e, -18991/646024*e^11 + 463459/323012*e^9 - 15171517/646024*e^7 + 23920515/161506*e^5 - 102396795/323012*e^3 + 34012605/161506*e, 6027/224704*e^11 - 74143/56176*e^9 + 4925963/224704*e^7 - 16003537/112352*e^5 + 18317707/56176*e^3 - 6457557/28088*e, 6027/224704*e^11 - 74143/56176*e^9 + 4925963/224704*e^7 - 16003537/112352*e^5 + 18317707/56176*e^3 - 6457557/28088*e, -677/28088*e^10 + 16705/14044*e^8 - 557417/28088*e^6 + 913647/7022*e^4 - 2141205/7022*e^2 + 788780/3511, -677/28088*e^10 + 16705/14044*e^8 - 557417/28088*e^6 + 913647/7022*e^4 - 2141205/7022*e^2 + 788780/3511, 180627/5168192*e^11 - 2170127/1292048*e^9 + 137933187/5168192*e^7 - 406555737/2584096*e^5 + 358194511/1292048*e^3 - 90554737/646024*e, 180627/5168192*e^11 - 2170127/1292048*e^9 + 137933187/5168192*e^7 - 406555737/2584096*e^5 + 358194511/1292048*e^3 - 90554737/646024*e, -102763/5168192*e^11 + 1257087/1292048*e^9 - 82562795/5168192*e^7 + 260952177/2584096*e^5 - 275909675/1292048*e^3 + 82114613/646024*e, -102763/5168192*e^11 + 1257087/1292048*e^9 - 82562795/5168192*e^7 + 260952177/2584096*e^5 - 275909675/1292048*e^3 + 82114613/646024*e, -687/56176*e^10 + 16485/28088*e^8 - 523083/56176*e^6 + 193363/3511*e^4 - 1464897/14044*e^2 + 275894/3511, -687/56176*e^10 + 16485/28088*e^8 - 523083/56176*e^6 + 193363/3511*e^4 - 1464897/14044*e^2 + 275894/3511, 229841/2584096*e^11 - 1395457/323012*e^9 + 181059945/2584096*e^7 - 559554013/1292048*e^5 + 567696749/646024*e^3 - 173747885/323012*e, 229841/2584096*e^11 - 1395457/323012*e^9 + 181059945/2584096*e^7 - 559554013/1292048*e^5 + 567696749/646024*e^3 - 173747885/323012*e, 3425/28088*e^10 - 82645/14044*e^8 + 2649749/28088*e^6 - 3993411/7022*e^4 + 7579473/7022*e^2 - 2132226/3511, 3425/28088*e^10 - 82645/14044*e^8 + 2649749/28088*e^6 - 3993411/7022*e^4 + 7579473/7022*e^2 - 2132226/3511, -397523/5168192*e^11 + 4840111/1292048*e^9 - 315562499/5168192*e^7 + 985460057/2584096*e^5 - 1026504799/1292048*e^3 + 329942753/646024*e, -397523/5168192*e^11 + 4840111/1292048*e^9 - 315562499/5168192*e^7 + 985460057/2584096*e^5 - 1026504799/1292048*e^3 + 329942753/646024*e, 13659/112352*e^10 - 165741/28088*e^8 + 10731339/112352*e^6 - 32963469/56176*e^4 + 32747379/28088*e^2 - 9879057/14044, 13659/112352*e^10 - 165741/28088*e^8 + 10731339/112352*e^6 - 32963469/56176*e^4 + 32747379/28088*e^2 - 9879057/14044, 2051/28088*e^10 - 49675/14044*e^8 + 1603583/28088*e^6 - 2453529/7022*e^4 + 4860339/7022*e^2 - 1485080/3511, 2051/28088*e^10 - 49675/14044*e^8 + 1603583/28088*e^6 - 2453529/7022*e^4 + 4860339/7022*e^2 - 1485080/3511, -1523/28088*e^10 + 18357/7022*e^8 - 1172187/28088*e^6 + 3480297/14044*e^4 - 3105627/7022*e^2 + 773725/3511, -1523/28088*e^10 + 18357/7022*e^8 - 1172187/28088*e^6 + 3480297/14044*e^4 - 3105627/7022*e^2 + 773725/3511, -620567/5168192*e^11 + 7526731/1292048*e^9 - 487093495/5168192*e^7 + 1495709589/2584096*e^5 - 1486304751/1292048*e^3 + 444952497/646024*e, -620567/5168192*e^11 + 7526731/1292048*e^9 - 487093495/5168192*e^7 + 1495709589/2584096*e^5 - 1486304751/1292048*e^3 + 444952497/646024*e, 147579/5168192*e^11 - 1831455/1292048*e^9 + 123644731/5168192*e^7 - 415160417/2584096*e^5 + 511590907/1292048*e^3 - 195673829/646024*e, 147579/5168192*e^11 - 1831455/1292048*e^9 + 123644731/5168192*e^7 - 415160417/2584096*e^5 + 511590907/1292048*e^3 - 195673829/646024*e, 292361/5168192*e^11 - 3525149/1292048*e^9 + 225515577/5168192*e^7 - 673993435/2584096*e^5 + 616519333/1292048*e^3 - 158723931/646024*e, 292361/5168192*e^11 - 3525149/1292048*e^9 + 225515577/5168192*e^7 - 673993435/2584096*e^5 + 616519333/1292048*e^3 - 158723931/646024*e, 10645/112352*e^10 - 128675/28088*e^8 + 8265861/112352*e^6 - 24909011/56176*e^4 + 23338301/28088*e^2 - 6387919/14044, 10645/112352*e^10 - 128675/28088*e^8 + 8265861/112352*e^6 - 24909011/56176*e^4 + 23338301/28088*e^2 - 6387919/14044, -14111/112352*e^11 + 85651/14044*e^9 - 11106279/112352*e^7 + 34270643/56176*e^5 - 34653603/28088*e^3 + 10754723/14044*e, -14111/112352*e^11 + 85651/14044*e^9 - 11106279/112352*e^7 + 34270643/56176*e^5 - 34653603/28088*e^3 + 10754723/14044*e, -261751/5168192*e^11 + 3159659/1292048*e^9 - 202727287/5168192*e^7 + 611331925/2584096*e^5 - 580386023/1292048*e^3 + 169660921/646024*e, -261751/5168192*e^11 + 3159659/1292048*e^9 - 202727287/5168192*e^7 + 611331925/2584096*e^5 - 580386023/1292048*e^3 + 169660921/646024*e, 171125/2584096*e^11 - 1044283/323012*e^9 + 136806813/2584096*e^7 - 431881945/1292048*e^5 + 463345853/646024*e^3 - 153359381/323012*e, 171125/2584096*e^11 - 1044283/323012*e^9 + 136806813/2584096*e^7 - 431881945/1292048*e^5 + 463345853/646024*e^3 - 153359381/323012*e, 911/7022*e^10 - 22090/3511*e^8 + 714611/7022*e^6 - 2196882/3511*e^4 + 4407150/3511*e^2 - 2681884/3511, 911/7022*e^10 - 22090/3511*e^8 + 714611/7022*e^6 - 2196882/3511*e^4 + 4407150/3511*e^2 - 2681884/3511, -2383/56176*e^10 + 56415/28088*e^8 - 1744507/56176*e^6 + 2412509/14044*e^4 - 3380223/14044*e^2 + 211334/3511, -2383/56176*e^10 + 56415/28088*e^8 - 1744507/56176*e^6 + 2412509/14044*e^4 - 3380223/14044*e^2 + 211334/3511, -989/28088*e^10 + 23885/14044*e^8 - 767009/28088*e^6 + 1160039/7022*e^4 - 2224365/7022*e^2 + 628694/3511, -989/28088*e^10 + 23885/14044*e^8 - 767009/28088*e^6 + 1160039/7022*e^4 - 2224365/7022*e^2 + 628694/3511, -16673/112352*e^10 + 202807/28088*e^8 - 13196817/112352*e^6 + 41017927/56176*e^4 - 42156457/28088*e^2 + 13370195/14044, -16673/112352*e^10 + 202807/28088*e^8 - 13196817/112352*e^6 + 41017927/56176*e^4 - 42156457/28088*e^2 + 13370195/14044, 124/3511*e^10 - 23819/14044*e^8 + 188475/7022*e^6 - 2186807/14044*e^4 + 890309/3511*e^2 - 334999/3511, 124/3511*e^10 - 23819/14044*e^8 + 188475/7022*e^6 - 2186807/14044*e^4 + 890309/3511*e^2 - 334999/3511, -363697/2584096*e^11 + 2204089/323012*e^9 - 284946505/2584096*e^7 + 873339501/1292048*e^5 - 866600133/646024*e^3 + 260950965/323012*e, -363697/2584096*e^11 + 2204089/323012*e^9 - 284946505/2584096*e^7 + 873339501/1292048*e^5 - 866600133/646024*e^3 + 260950965/323012*e, 687/14044*e^10 - 16485/7022*e^8 + 523083/14044*e^6 - 769941/3511*e^4 + 1359567/3511*e^2 - 703322/3511, 687/14044*e^10 - 16485/7022*e^8 + 523083/14044*e^6 - 769941/3511*e^4 + 1359567/3511*e^2 - 703322/3511, 192845/5168192*e^11 - 2372785/1292048*e^9 + 157704477/5168192*e^7 - 512807431/2584096*e^5 + 588384833/1292048*e^3 - 208370815/646024*e, 192845/5168192*e^11 - 2372785/1292048*e^9 + 157704477/5168192*e^7 - 512807431/2584096*e^5 + 588384833/1292048*e^3 - 208370815/646024*e, -228193/2584096*e^11 + 1383455/323012*e^9 - 178954665/2584096*e^7 + 549052613/1292048*e^5 - 547214545/646024*e^3 + 171326617/323012*e, -228193/2584096*e^11 + 1383455/323012*e^9 - 178954665/2584096*e^7 + 549052613/1292048*e^5 - 547214545/646024*e^3 + 171326617/323012*e, -33501/1292048*e^11 + 410931/323012*e^9 - 27181593/1292048*e^7 + 87668071/646024*e^5 - 49410719/161506*e^3 + 16459440/80753*e, -33501/1292048*e^11 + 410931/323012*e^9 - 27181593/1292048*e^7 + 87668071/646024*e^5 - 49410719/161506*e^3 + 16459440/80753*e, 13315/112352*e^10 - 162503/28088*e^8 + 10642851/112352*e^6 - 33550505/56176*e^4 + 35665427/28088*e^2 - 11596621/14044, 13315/112352*e^10 - 162503/28088*e^8 + 10642851/112352*e^6 - 33550505/56176*e^4 + 35665427/28088*e^2 - 11596621/14044, 323803/2584096*e^11 - 1970587/323012*e^9 + 256811683/2584096*e^7 - 801417487/1292048*e^5 + 834982951/646024*e^3 - 268384215/323012*e, 323803/2584096*e^11 - 1970587/323012*e^9 + 256811683/2584096*e^7 - 801417487/1292048*e^5 + 834982951/646024*e^3 - 268384215/323012*e, 9895/112352*e^10 - 59685/14044*e^8 + 7638879/112352*e^6 - 22828859/56176*e^4 + 20968059/28088*e^2 - 5736591/14044, 9895/112352*e^10 - 59685/14044*e^8 + 7638879/112352*e^6 - 22828859/56176*e^4 + 20968059/28088*e^2 - 5736591/14044, -4487/28088*e^10 + 108435/14044*e^8 - 3486323/28088*e^6 + 5286901/7022*e^4 - 10215447/7022*e^2 + 3009678/3511, -4487/28088*e^10 + 108435/14044*e^8 - 3486323/28088*e^6 + 5286901/7022*e^4 - 10215447/7022*e^2 + 3009678/3511, 1593/28088*e^10 - 38685/14044*e^8 + 1254861/28088*e^6 - 1940235/7022*e^4 + 3953961/7022*e^2 - 1238936/3511, 1593/28088*e^10 - 38685/14044*e^8 + 1254861/28088*e^6 - 1940235/7022*e^4 + 3953961/7022*e^2 - 1238936/3511, -153277/5168192*e^11 + 1881129/1292048*e^9 - 124568605/5168192*e^7 + 403066855/2584096*e^5 - 461967877/1292048*e^3 + 172157787/646024*e, -153277/5168192*e^11 + 1881129/1292048*e^9 - 124568605/5168192*e^7 + 403066855/2584096*e^5 - 461967877/1292048*e^3 + 172157787/646024*e, 11675/112352*e^10 - 70961/14044*e^8 + 9223539/112352*e^6 - 28589855/56176*e^4 + 29186143/28088*e^2 - 9209059/14044, 11675/112352*e^10 - 70961/14044*e^8 + 9223539/112352*e^6 - 28589855/56176*e^4 + 29186143/28088*e^2 - 9209059/14044, -47393/1292048*e^11 + 581131/323012*e^9 - 38402189/1292048*e^7 + 123557087/646024*e^5 - 34690913/80753*e^3 + 23961741/80753*e, -47393/1292048*e^11 + 581131/323012*e^9 - 38402189/1292048*e^7 + 123557087/646024*e^5 - 34690913/80753*e^3 + 23961741/80753*e, 229153/5168192*e^11 - 2787949/1292048*e^9 + 181416641/5168192*e^7 - 563375379/2584096*e^5 + 574895113/1292048*e^3 - 173917783/646024*e, 229153/5168192*e^11 - 2787949/1292048*e^9 + 181416641/5168192*e^7 - 563375379/2584096*e^5 + 574895113/1292048*e^3 - 173917783/646024*e, 36719/5168192*e^11 - 466819/1292048*e^9 + 32673935/5168192*e^7 - 115787357/2584096*e^5 + 151228471/1292048*e^3 - 47971049/646024*e, 36719/5168192*e^11 - 466819/1292048*e^9 + 32673935/5168192*e^7 - 115787357/2584096*e^5 + 151228471/1292048*e^3 - 47971049/646024*e, 11795/56176*e^10 - 284715/28088*e^8 + 9134543/56176*e^6 - 13787017/14044*e^4 + 26280771/14044*e^2 - 3721516/3511, 11795/56176*e^10 - 284715/28088*e^8 + 9134543/56176*e^6 - 13787017/14044*e^4 + 26280771/14044*e^2 - 3721516/3511, 78973/646024*e^11 - 1910523/323012*e^9 + 61504499/646024*e^7 - 93394633/161506*e^5 + 361357965/323012*e^3 - 106211083/161506*e, 78973/646024*e^11 - 1910523/323012*e^9 + 61504499/646024*e^7 - 93394633/161506*e^5 + 361357965/323012*e^3 - 106211083/161506*e, 1125/2584096*e^11 + 921/323012*e^9 - 2050899/2584096*e^7 + 19789047/1292048*e^5 - 56925123/646024*e^3 + 31538379/323012*e, 1125/2584096*e^11 + 921/323012*e^9 - 2050899/2584096*e^7 + 19789047/1292048*e^5 - 56925123/646024*e^3 + 31538379/323012*e, 203/56176*e^10 - 6067/28088*e^8 + 269247/56176*e^6 - 666797/14044*e^4 + 2644145/14044*e^2 - 592223/3511, 203/56176*e^10 - 6067/28088*e^8 + 269247/56176*e^6 - 666797/14044*e^4 + 2644145/14044*e^2 - 592223/3511, 385/28088*e^10 - 9085/14044*e^8 + 279157/28088*e^6 - 379843/7022*e^4 + 494769/7022*e^2 - 4408/3511, 385/28088*e^10 - 9085/14044*e^8 + 279157/28088*e^6 - 379843/7022*e^4 + 494769/7022*e^2 - 4408/3511, 174055/5168192*e^11 - 2130067/1292048*e^9 + 140397719/5168192*e^7 - 450584661/2584096*e^5 + 509855155/1292048*e^3 - 193604365/646024*e, 174055/5168192*e^11 - 2130067/1292048*e^9 + 140397719/5168192*e^7 - 450584661/2584096*e^5 + 509855155/1292048*e^3 - 193604365/646024*e, -804017/5168192*e^11 + 9778157/1292048*e^9 - 636144593/5168192*e^7 + 1977405635/2584096*e^5 - 2035424745/1292048*e^3 + 645940087/646024*e, -804017/5168192*e^11 + 9778157/1292048*e^9 - 636144593/5168192*e^7 + 1977405635/2584096*e^5 - 2035424745/1292048*e^3 + 645940087/646024*e, 114527/646024*e^11 - 2785527/323012*e^9 + 90608629/646024*e^7 - 140881253/161506*e^5 + 581551239/323012*e^3 - 183655361/161506*e, 114527/646024*e^11 - 2785527/323012*e^9 + 90608629/646024*e^7 - 140881253/161506*e^5 + 581551239/323012*e^3 - 183655361/161506*e, -911/7022*e^10 + 22090/3511*e^8 - 714611/7022*e^6 + 2196882/3511*e^4 - 4407150/3511*e^2 + 2667840/3511, -911/7022*e^10 + 22090/3511*e^8 - 714611/7022*e^6 + 2196882/3511*e^4 - 4407150/3511*e^2 + 2667840/3511, -554019/2584096*e^11 + 3365737/323012*e^9 - 437187995/2584096*e^7 + 1354375151/1292048*e^5 - 1383704659/646024*e^3 + 431619307/323012*e, -554019/2584096*e^11 + 3365737/323012*e^9 - 437187995/2584096*e^7 + 1354375151/1292048*e^5 - 1383704659/646024*e^3 + 431619307/323012*e, -915777/5168192*e^11 + 11076717/1292048*e^9 - 713071009/5168192*e^7 + 2163656243/2584096*e^5 - 2078012569/1292048*e^3 + 591918407/646024*e, -915777/5168192*e^11 + 11076717/1292048*e^9 - 713071009/5168192*e^7 + 2163656243/2584096*e^5 - 2078012569/1292048*e^3 + 591918407/646024*e, -2569/56176*e^10 + 7857/3511*e^8 - 2069313/56176*e^6 + 6614209/28088*e^4 - 7354557/14044*e^2 + 2623021/7022, -2569/56176*e^10 + 7857/3511*e^8 - 2069313/56176*e^6 + 6614209/28088*e^4 - 7354557/14044*e^2 + 2623021/7022, 5647/112352*e^10 - 33895/14044*e^8 + 4292583/112352*e^6 - 12480939/56176*e^4 + 10367987/28088*e^2 - 2226783/14044, 5647/112352*e^10 - 33895/14044*e^8 + 4292583/112352*e^6 - 12480939/56176*e^4 + 10367987/28088*e^2 - 2226783/14044, -211019/2584096*e^11 + 1277785/323012*e^9 - 164941635/2584096*e^7 + 503803959/1292048*e^5 - 494963035/646024*e^3 + 145926131/323012*e, -211019/2584096*e^11 + 1277785/323012*e^9 - 164941635/2584096*e^7 + 503803959/1292048*e^5 - 494963035/646024*e^3 + 145926131/323012*e, 1749/14044*e^10 - 42275/7022*e^8 + 1359657/14044*e^6 - 2063431/3511*e^4 + 3995541/3511*e^2 - 2388006/3511, 1749/14044*e^10 - 42275/7022*e^8 + 1359657/14044*e^6 - 2063431/3511*e^4 + 3995541/3511*e^2 - 2388006/3511, -843/56176*e^10 + 20075/28088*e^8 - 627879/56176*e^6 + 893137/14044*e^4 - 1401147/14044*e^2 + 167408/3511, -114981/5168192*e^11 + 1459745/1292048*e^9 - 102334085/5168192*e^7 + 367203871/2584096*e^5 - 506099997/1292048*e^3 + 199930691/646024*e, -114981/5168192*e^11 + 1459745/1292048*e^9 - 102334085/5168192*e^7 + 367203871/2584096*e^5 - 506099997/1292048*e^3 + 199930691/646024*e, 29091/224704*e^11 - 354111/56176*e^9 + 23068563/224704*e^7 - 71849097/112352*e^5 + 74043711/56176*e^3 - 22970241/28088*e, 29091/224704*e^11 - 354111/56176*e^9 + 23068563/224704*e^7 - 71849097/112352*e^5 + 74043711/56176*e^3 - 22970241/28088*e, -241951/2584096*e^11 + 1458759/323012*e^9 - 186753575/2584096*e^7 + 559697995/1292048*e^5 - 519429503/646024*e^3 + 136305431/323012*e, -241951/2584096*e^11 + 1458759/323012*e^9 - 186753575/2584096*e^7 + 559697995/1292048*e^5 - 519429503/646024*e^3 + 136305431/323012*e, 279667/5168192*e^11 - 3411991/1292048*e^9 + 223422451/5168192*e^7 - 705490057/2584096*e^5 + 760953315/1292048*e^3 - 263960957/646024*e, 279667/5168192*e^11 - 3411991/1292048*e^9 + 223422451/5168192*e^7 - 705490057/2584096*e^5 + 760953315/1292048*e^3 - 263960957/646024*e, 823/28088*e^10 - 20515/14044*e^8 + 696547/28088*e^6 - 1180549/7022*e^4 + 2964423/7022*e^2 - 1131812/3511, 823/28088*e^10 - 20515/14044*e^8 + 696547/28088*e^6 - 1180549/7022*e^4 + 2964423/7022*e^2 - 1131812/3511, -2601/161506*e^11 + 252119/323012*e^9 - 4082501/323012*e^7 + 25216703/323012*e^5 - 50145285/323012*e^3 + 8612119/161506*e, -2601/161506*e^11 + 252119/323012*e^9 - 4082501/323012*e^7 + 25216703/323012*e^5 - 50145285/323012*e^3 + 8612119/161506*e, -581/646024*e^11 + 4773/323012*e^9 + 420473/646024*e^7 - 2641837/161506*e^5 + 32829531/323012*e^3 - 20420269/161506*e, -581/646024*e^11 + 4773/323012*e^9 + 420473/646024*e^7 - 2641837/161506*e^5 + 32829531/323012*e^3 - 20420269/161506*e, -205/7022*e^10 + 20581/14044*e^8 - 88707/3511*e^6 + 2480325/14044*e^4 - 1619821/3511*e^2 + 1193781/3511, -205/7022*e^10 + 20581/14044*e^8 - 88707/3511*e^6 + 2480325/14044*e^4 - 1619821/3511*e^2 + 1193781/3511, 4945/56176*e^10 - 119425/28088*e^8 + 3835045/56176*e^6 - 5800195/14044*e^4 + 11121825/14044*e^2 - 1547158/3511, 4945/56176*e^10 - 119425/28088*e^8 + 3835045/56176*e^6 - 5800195/14044*e^4 + 11121825/14044*e^2 - 1547158/3511, 298457/5168192*e^11 - 3654709/1292048*e^9 + 240729209/5168192*e^7 - 767712827/2584096*e^5 + 839482993/1292048*e^3 - 278727407/646024*e, 298457/5168192*e^11 - 3654709/1292048*e^9 + 240729209/5168192*e^7 - 767712827/2584096*e^5 + 839482993/1292048*e^3 - 278727407/646024*e, 143/3511*e^11 - 27497/14044*e^9 + 436917/14044*e^7 - 2573443/14044*e^5 + 4538649/14044*e^3 - 1217027/7022*e, 143/3511*e^11 - 27497/14044*e^9 + 436917/14044*e^7 - 2573443/14044*e^5 + 4538649/14044*e^3 - 1217027/7022*e, -399961/5168192*e^11 + 4813293/1292048*e^9 - 307062665/5168192*e^7 + 914193867/2584096*e^5 - 836975093/1292048*e^3 + 235760651/646024*e, -399961/5168192*e^11 + 4813293/1292048*e^9 - 307062665/5168192*e^7 + 914193867/2584096*e^5 - 836975093/1292048*e^3 + 235760651/646024*e, -14649/56176*e^10 + 355345/28088*e^8 - 11503341/56176*e^6 + 17708507/14044*e^4 - 35668809/14044*e^2 + 5554856/3511, 481129/2584096*e^11 - 2906437/323012*e^9 + 373452769/2584096*e^7 - 1128683637/1292048*e^5 + 1075301925/646024*e^3 - 301727973/323012*e, 481129/2584096*e^11 - 2906437/323012*e^9 + 373452769/2584096*e^7 - 1128683637/1292048*e^5 + 1075301925/646024*e^3 - 301727973/323012*e, -262599/5168192*e^11 + 3206507/1292048*e^9 - 209994855/5168192*e^7 + 660583653/2584096*e^5 - 694085407/1292048*e^3 + 212318209/646024*e, -262599/5168192*e^11 + 3206507/1292048*e^9 - 209994855/5168192*e^7 + 660583653/2584096*e^5 - 694085407/1292048*e^3 + 212318209/646024*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;