/*
  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![-27, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {7, 7, w - 5}>;

primesArray := [
[3, 3, -w + 6],
[3, 3, w + 5],
[4, 2, 2],
[5, 5, -3*w + 17],
[5, 5, -3*w - 14],
[7, 7, w - 5],
[7, 7, w + 4],
[29, 29, -w - 7],
[29, 29, -w + 8],
[31, 31, -5*w + 28],
[31, 31, -5*w - 23],
[43, 43, 6*w + 29],
[43, 43, -6*w + 35],
[61, 61, 3*w - 19],
[61, 61, -3*w - 16],
[71, 71, -7*w - 34],
[71, 71, 7*w - 41],
[73, 73, 2*w - 7],
[73, 73, -2*w - 5],
[83, 83, -w - 10],
[83, 83, w - 11],
[89, 89, 3*w - 14],
[89, 89, 3*w + 11],
[97, 97, 3*w + 17],
[97, 97, 3*w - 20],
[109, 109, 2*w - 1],
[113, 113, -3*w - 10],
[113, 113, 3*w - 13],
[121, 11, -11],
[131, 131, 5*w - 31],
[131, 131, -5*w - 26],
[137, 137, -9*w - 41],
[137, 137, 9*w - 50],
[157, 157, -14*w + 79],
[157, 157, 14*w + 65],
[169, 13, -13],
[173, 173, -3*w - 7],
[173, 173, 3*w - 10],
[191, 191, -10*w - 49],
[191, 191, 10*w - 59],
[193, 193, 22*w + 103],
[193, 193, 22*w - 125],
[197, 197, 6*w + 25],
[197, 197, -6*w + 31],
[211, 211, -4*w - 13],
[211, 211, 4*w - 17],
[223, 223, 8*w + 35],
[223, 223, -8*w + 43],
[227, 227, 9*w - 49],
[227, 227, 9*w + 40],
[233, 233, 3*w - 5],
[233, 233, -3*w - 2],
[239, 239, -3*w - 1],
[239, 239, 3*w - 4],
[263, 263, 33*w - 188],
[263, 263, 33*w + 155],
[281, 281, 8*w - 49],
[281, 281, -8*w - 41],
[289, 17, -17],
[293, 293, 4*w - 29],
[293, 293, 4*w + 25],
[307, 307, 3*w - 25],
[307, 307, -3*w - 22],
[311, 311, -5*w - 29],
[311, 311, 5*w - 34],
[331, 331, 19*w + 88],
[331, 331, -19*w + 107],
[347, 347, 23*w - 133],
[347, 347, 36*w - 205],
[349, 349, 15*w - 88],
[349, 349, -15*w - 73],
[353, 353, -w - 19],
[353, 353, w - 20],
[361, 19, -19],
[373, 373, 23*w - 130],
[373, 373, 23*w + 107],
[401, 401, -9*w + 47],
[401, 401, 9*w + 38],
[409, 409, 5*w - 19],
[409, 409, -5*w - 14],
[421, 421, -10*w + 53],
[421, 421, 10*w + 43],
[431, 431, -15*w - 68],
[431, 431, 15*w - 83],
[433, 433, 43*w - 245],
[433, 433, 43*w + 202],
[439, 439, 35*w - 199],
[439, 439, 35*w + 164],
[443, 443, 12*w + 53],
[443, 443, 12*w - 65],
[457, 457, 3*w - 28],
[457, 457, -3*w - 25],
[461, 461, 21*w - 118],
[461, 461, 21*w + 97],
[463, 463, -6*w - 35],
[463, 463, 6*w - 41],
[467, 467, 2*w - 25],
[467, 467, -2*w - 23],
[479, 479, -w - 22],
[479, 479, w - 23],
[499, 499, -5*w - 11],
[499, 499, 5*w - 16],
[509, 509, -5*w - 32],
[509, 509, 5*w - 37],
[523, 523, -7*w - 25],
[523, 523, 7*w - 32],
[529, 23, -23],
[541, 541, 11*w + 47],
[541, 541, -11*w + 58],
[557, 557, 7*w - 47],
[557, 557, -7*w - 40],
[571, 571, -5*w - 8],
[571, 571, 5*w - 13],
[593, 593, -16*w - 79],
[593, 593, 16*w - 95],
[619, 619, 6*w - 43],
[619, 619, 6*w + 37],
[647, 647, -9*w + 44],
[647, 647, -9*w - 35],
[653, 653, -4*w - 31],
[653, 653, 4*w - 35],
[659, 659, 45*w + 211],
[659, 659, 45*w - 256],
[661, 661, 5*w - 7],
[661, 661, -5*w - 2],
[683, 683, 27*w + 125],
[683, 683, -27*w + 152],
[727, 727, -27*w - 130],
[727, 727, 27*w - 157],
[743, 743, 14*w - 85],
[743, 743, -14*w - 71],
[751, 751, 40*w + 187],
[751, 751, 40*w - 227],
[797, 797, -13*w - 67],
[797, 797, 13*w - 80],
[809, 809, -30*w + 169],
[809, 809, 30*w + 139],
[811, 811, 3*w - 34],
[811, 811, -3*w - 31],
[823, 823, -13*w + 68],
[823, 823, 13*w + 55],
[827, 827, 7*w - 50],
[827, 827, 7*w + 43],
[829, 829, -7*w - 19],
[829, 829, 7*w - 26],
[857, 857, -8*w - 47],
[857, 857, 8*w - 55],
[863, 863, -11*w + 70],
[863, 863, -11*w - 59],
[877, 877, -3*w - 32],
[877, 877, 3*w - 35],
[881, 881, -6*w - 7],
[881, 881, 6*w - 13],
[887, 887, -21*w - 95],
[887, 887, 21*w - 116],
[907, 907, 19*w - 104],
[907, 907, 19*w + 85],
[947, 947, 9*w - 40],
[947, 947, -9*w - 31],
[953, 953, 29*w + 140],
[953, 953, -29*w + 169],
[977, 977, 6*w - 5],
[977, 977, 6*w - 1],
[997, 997, -26*w - 119],
[997, 997, 26*w - 145]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 2*x^5 - 17*x^4 + 30*x^3 + 81*x^2 - 87*x - 143;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/5*e^5 + 17/5*e^3 + 4/5*e^2 - 68/5*e - 44/5, e, 8/5*e^5 + e^4 - 121/5*e^3 - 77/5*e^2 + 414/5*e + 402/5, 9/5*e^5 + e^4 - 133/5*e^3 - 76/5*e^2 + 442/5*e + 406/5, 3/5*e^5 - 46/5*e^3 - 7/5*e^2 + 154/5*e + 112/5, -1, 7/5*e^5 + e^4 - 104/5*e^3 - 73/5*e^2 + 346/5*e + 353/5, 3*e^5 + 2*e^4 - 45*e^3 - 31*e^2 + 152*e + 159, 7/5*e^5 + 2*e^4 - 104/5*e^3 - 138/5*e^2 + 361/5*e + 503/5, -3*e^5 - 3*e^4 + 44*e^3 + 41*e^2 - 148*e - 166, -1/5*e^5 + 17/5*e^3 + 9/5*e^2 - 63/5*e - 84/5, e^4 - 2*e^3 - 15*e^2 + 20*e + 48, -9/5*e^5 - e^4 + 143/5*e^3 + 101/5*e^2 - 527/5*e - 566/5, -17/5*e^5 - e^4 + 254/5*e^3 + 93/5*e^2 - 836/5*e - 683/5, -27/5*e^5 - 4*e^4 + 409/5*e^3 + 308/5*e^2 - 1421/5*e - 1503/5, -23/5*e^5 - e^4 + 346/5*e^3 + 112/5*e^2 - 1149/5*e - 917/5, 7*e^5 + 4*e^4 - 104*e^3 - 62*e^2 + 349*e + 333, 17/5*e^5 + 2*e^4 - 259/5*e^3 - 163/5*e^2 + 896/5*e + 868/5, 7/5*e^5 + 2*e^4 - 104/5*e^3 - 138/5*e^2 + 351/5*e + 488/5, 19/5*e^5 + 2*e^4 - 293/5*e^3 - 181/5*e^2 + 1027/5*e + 1036/5, 4/5*e^5 - 53/5*e^3 + 4/5*e^2 + 132/5*e + 81/5, -21/5*e^5 - 2*e^4 + 327/5*e^3 + 184/5*e^2 - 1158/5*e - 1074/5, 7/5*e^5 - 109/5*e^3 - 18/5*e^2 + 371/5*e + 243/5, 1/5*e^5 - 27/5*e^3 - 19/5*e^2 + 138/5*e + 164/5, -3/5*e^5 + 46/5*e^3 + 2/5*e^2 - 159/5*e - 37/5, -2*e^5 + 26*e^3 - 3*e^2 - 64*e - 16, 16/5*e^5 + 2*e^4 - 242/5*e^3 - 154/5*e^2 + 833/5*e + 814/5, -3/5*e^5 + 61/5*e^3 + 37/5*e^2 - 274/5*e - 297/5, -24/5*e^5 - 4*e^4 + 368/5*e^3 + 296/5*e^2 - 1312/5*e - 1346/5, -12/5*e^5 - 2*e^4 + 169/5*e^3 + 138/5*e^2 - 526/5*e - 613/5, 22/5*e^5 + 3*e^4 - 329/5*e^3 - 238/5*e^2 + 1116/5*e + 1183/5, 2*e^5 + e^4 - 31*e^3 - 19*e^2 + 110*e + 115, 12/5*e^5 - 184/5*e^3 - 33/5*e^2 + 611/5*e + 443/5, -33/5*e^5 - 3*e^4 + 501/5*e^3 + 252/5*e^2 - 1699/5*e - 1542/5, -9/5*e^5 - 2*e^4 + 128/5*e^3 + 126/5*e^2 - 412/5*e - 421/5, 4/5*e^5 + e^4 - 63/5*e^3 - 66/5*e^2 + 232/5*e + 206/5, 39/5*e^5 + 3*e^4 - 593/5*e^3 - 281/5*e^2 + 2022/5*e + 1846/5, 3/5*e^5 - 31/5*e^3 + 8/5*e^2 + 24/5*e - 38/5, -11/5*e^5 - 2*e^4 + 177/5*e^3 + 174/5*e^2 - 678/5*e - 814/5, 11/5*e^5 - 177/5*e^3 - 39/5*e^2 + 613/5*e + 489/5, -39/5*e^5 - 4*e^4 + 568/5*e^3 + 281/5*e^2 - 1822/5*e - 1566/5, 32/5*e^5 + 4*e^4 - 489/5*e^3 - 303/5*e^2 + 1711/5*e + 1568/5, 17/5*e^5 + 2*e^4 - 259/5*e^3 - 158/5*e^2 + 916/5*e + 883/5, 21/5*e^5 + 5*e^4 - 302/5*e^3 - 319/5*e^2 + 983/5*e + 1174/5, -10*e^5 - 8*e^4 + 151*e^3 + 122*e^2 - 521*e - 574, 38/5*e^5 + 5*e^4 - 566/5*e^3 - 392/5*e^2 + 1919/5*e + 1987/5, -7/5*e^5 + 109/5*e^3 + 8/5*e^2 - 361/5*e - 233/5, -34/5*e^5 - 2*e^4 + 508/5*e^3 + 201/5*e^2 - 1682/5*e - 1421/5, 31/5*e^5 + 6*e^4 - 472/5*e^3 - 444/5*e^2 + 1668/5*e + 1904/5, 23/5*e^5 + 5*e^4 - 351/5*e^3 - 367/5*e^2 + 1234/5*e + 1522/5, -58/5*e^5 - 6*e^4 + 866/5*e^3 + 482/5*e^2 - 2894/5*e - 2722/5, -43/5*e^5 - 7*e^4 + 671/5*e^3 + 572/5*e^2 - 2439/5*e - 2682/5, 9*e^5 + 4*e^4 - 138*e^3 - 71*e^2 + 474*e + 427, -17/5*e^5 - 3*e^4 + 264/5*e^3 + 228/5*e^2 - 936/5*e - 1063/5, 37/5*e^5 + 4*e^4 - 579/5*e^3 - 373/5*e^2 + 2061/5*e + 2143/5, -16/5*e^5 + 242/5*e^3 + 49/5*e^2 - 813/5*e - 634/5, 44/5*e^5 + 5*e^4 - 658/5*e^3 - 386/5*e^2 + 2242/5*e + 2076/5, 27/5*e^5 + 4*e^4 - 419/5*e^3 - 333/5*e^2 + 1516/5*e + 1613/5, 14*e^5 + 10*e^4 - 212*e^3 - 154*e^2 + 729*e + 763, -7/5*e^5 - e^4 + 69/5*e^3 + 23/5*e^2 - 61/5*e - 13/5, -23/5*e^5 - 6*e^4 + 361/5*e^3 + 432/5*e^2 - 1339/5*e - 1712/5, -7/5*e^5 - 2*e^4 + 104/5*e^3 + 128/5*e^2 - 351/5*e - 458/5, -39/5*e^5 - 6*e^4 + 573/5*e^3 + 421/5*e^2 - 1902/5*e - 1926/5, e^5 + e^4 - 13*e^3 - 10*e^2 + 36*e + 46, -9*e^5 - 8*e^4 + 138*e^3 + 119*e^2 - 485*e - 537, 21/5*e^5 + 4*e^4 - 327/5*e^3 - 294/5*e^2 + 1213/5*e + 1259/5, 27/5*e^5 + 2*e^4 - 384/5*e^3 - 133/5*e^2 + 1141/5*e + 878/5, 22/5*e^5 + e^4 - 309/5*e^3 - 63/5*e^2 + 911/5*e + 608/5, 43/5*e^5 + 4*e^4 - 641/5*e^3 - 302/5*e^2 + 2154/5*e + 1817/5, 18/5*e^5 + 4*e^4 - 241/5*e^3 - 237/5*e^2 + 709/5*e + 837/5, -54/5*e^5 - 7*e^4 + 798/5*e^3 + 511/5*e^2 - 2652/5*e - 2576/5, -66/5*e^5 - 10*e^4 + 992/5*e^3 + 754/5*e^2 - 3428/5*e - 3539/5, -3*e^5 - 3*e^4 + 51*e^3 + 56*e^2 - 207*e - 271, -61/5*e^5 - 8*e^4 + 922/5*e^3 + 634/5*e^2 - 3148/5*e - 3239/5, -22/5*e^5 - 3*e^4 + 319/5*e^3 + 213/5*e^2 - 1036/5*e - 1073/5, 32/5*e^5 + 2*e^4 - 464/5*e^3 - 148/5*e^2 + 1486/5*e + 1093/5, -e^4 + 3*e^3 + 19*e^2 - 28*e - 81, 3/5*e^5 + 5*e^4 - 36/5*e^3 - 297/5*e^2 + 144/5*e + 662/5, -39/5*e^5 - 6*e^4 + 613/5*e^3 + 481/5*e^2 - 2242/5*e - 2296/5, 14*e^5 + 7*e^4 - 211*e^3 - 113*e^2 + 714*e + 657, 5*e^5 + 5*e^4 - 74*e^3 - 68*e^2 + 254*e + 279, -48/5*e^5 - 5*e^4 + 696/5*e^3 + 342/5*e^2 - 2209/5*e - 1902/5, -5*e^5 - 3*e^4 + 70*e^3 + 35*e^2 - 214*e - 165, 29/5*e^5 + 4*e^4 - 428/5*e^3 - 296/5*e^2 + 1457/5*e + 1446/5, 5*e^5 + 8*e^4 - 75*e^3 - 108*e^2 + 271*e + 381, -7/5*e^5 - 2*e^4 + 94/5*e^3 + 133/5*e^2 - 276/5*e - 443/5, -23/5*e^5 - 3*e^4 + 356/5*e^3 + 247/5*e^2 - 1289/5*e - 1207/5, -48/5*e^5 - 4*e^4 + 751/5*e^3 + 382/5*e^2 - 2654/5*e - 2397/5, 39/5*e^5 + 3*e^4 - 568/5*e^3 - 236/5*e^2 + 1812/5*e + 1486/5, -e^5 - 2*e^4 + 12*e^3 + 17*e^2 - 31*e - 27, -39/5*e^5 - 3*e^4 + 608/5*e^3 + 281/5*e^2 - 2137/5*e - 1806/5, 58/5*e^5 + 10*e^4 - 881/5*e^3 - 747/5*e^2 + 3069/5*e + 3352/5, -81/5*e^5 - 11*e^4 + 1217/5*e^3 + 834/5*e^2 - 4153/5*e - 4189/5, -34/5*e^5 - 7*e^4 + 508/5*e^3 + 486/5*e^2 - 1772/5*e - 1896/5, -24/5*e^5 - e^4 + 393/5*e^3 + 156/5*e^2 - 1467/5*e - 1206/5, -16/5*e^5 - 3*e^4 + 222/5*e^3 + 189/5*e^2 - 678/5*e - 659/5, -6*e^5 - 6*e^4 + 87*e^3 + 78*e^2 - 287*e - 310, 52/5*e^5 + 6*e^4 - 789/5*e^3 - 463/5*e^2 + 2711/5*e + 2518/5, -51/5*e^5 - 11*e^4 + 767/5*e^3 + 774/5*e^2 - 2673/5*e - 3209/5, -56/5*e^5 - 13*e^4 + 832/5*e^3 + 889/5*e^2 - 2873/5*e - 3419/5, 5*e^5 + 5*e^4 - 76*e^3 - 75*e^2 + 269*e + 322, -48/5*e^5 - 8*e^4 + 721/5*e^3 + 567/5*e^2 - 2504/5*e - 2502/5, 39/5*e^5 + 6*e^4 - 573/5*e^3 - 436/5*e^2 + 1867/5*e + 2076/5, 5*e^5 + 5*e^4 - 78*e^3 - 81*e^2 + 288*e + 353, 54/5*e^5 + 8*e^4 - 823/5*e^3 - 621/5*e^2 + 2907/5*e + 2891/5, -23/5*e^5 - 3*e^4 + 311/5*e^3 + 197/5*e^2 - 889/5*e - 907/5, -42/5*e^5 - 8*e^4 + 629/5*e^3 + 588/5*e^2 - 2171/5*e - 2558/5, 13/5*e^5 + 4*e^4 - 191/5*e^3 - 277/5*e^2 + 704/5*e + 1022/5, -4/5*e^5 + 2*e^4 + 88/5*e^3 - 74/5*e^2 - 402/5*e - 96/5, -31/5*e^5 - 2*e^4 + 487/5*e^3 + 244/5*e^2 - 1743/5*e - 1624/5, -1/5*e^5 + 4*e^4 + 27/5*e^3 - 226/5*e^2 - 58/5*e + 326/5, -4*e^5 - 3*e^4 + 62*e^3 + 46*e^2 - 222*e - 210, 20*e^5 + 15*e^4 - 299*e^3 - 219*e^2 + 1016*e + 1041, -94/5*e^5 - 14*e^4 + 1408/5*e^3 + 1041/5*e^2 - 4797/5*e - 5011/5, -63/5*e^5 - 9*e^4 + 961/5*e^3 + 687/5*e^2 - 3339/5*e - 3407/5, 44/5*e^5 + 5*e^4 - 653/5*e^3 - 386/5*e^2 + 2197/5*e + 2171/5, -2*e^5 - e^4 + 32*e^3 + 19*e^2 - 116*e - 110, 32/5*e^5 + 5*e^4 - 489/5*e^3 - 373/5*e^2 + 1741/5*e + 1738/5, 6/5*e^5 - 2*e^4 - 102/5*e^3 + 71/5*e^2 + 368/5*e + 184/5, 79/5*e^5 + 14*e^4 - 1203/5*e^3 - 1051/5*e^2 + 4227/5*e + 4781/5, -7/5*e^5 - e^4 + 114/5*e^3 + 73/5*e^2 - 431/5*e - 403/5, e^5 + 7*e^4 - 13*e^3 - 85*e^2 + 52*e + 191, 39/5*e^5 + 5*e^4 - 593/5*e^3 - 411/5*e^2 + 2057/5*e + 2131/5, 10*e^5 + 4*e^4 - 150*e^3 - 69*e^2 + 505*e + 451, 34/5*e^5 + 3*e^4 - 523/5*e^3 - 296/5*e^2 + 1787/5*e + 1766/5, -17/5*e^5 + 229/5*e^3 - 7/5*e^2 - 601/5*e - 233/5, 48/5*e^5 + 6*e^4 - 736/5*e^3 - 502/5*e^2 + 2569/5*e + 2617/5, -66/5*e^5 - 9*e^4 + 982/5*e^3 + 644/5*e^2 - 3268/5*e - 3139/5, -4*e^4 - 3*e^3 + 41*e^2 + 14*e - 55, 18/5*e^5 + 6*e^4 - 296/5*e^3 - 427/5*e^2 + 1169/5*e + 1522/5, 61/5*e^5 + 7*e^4 - 927/5*e^3 - 554/5*e^2 + 3178/5*e + 2989/5, 19/5*e^5 + 2*e^4 - 283/5*e^3 - 196/5*e^2 + 957/5*e + 1096/5, 14/5*e^5 + 2*e^4 - 208/5*e^3 - 156/5*e^2 + 692/5*e + 801/5, 48/5*e^5 + 7*e^4 - 736/5*e^3 - 602/5*e^2 + 2589/5*e + 3007/5, -12*e^5 - 10*e^4 + 180*e^3 + 143*e^2 - 620*e - 623, 19*e^5 + 13*e^4 - 284*e^3 - 199*e^2 + 965*e + 993, -7/5*e^5 - e^4 + 104/5*e^3 + 83/5*e^2 - 321/5*e - 403/5, 64/5*e^5 + 10*e^4 - 993/5*e^3 - 831/5*e^2 + 3552/5*e + 4036/5, -38/5*e^5 - 7*e^4 + 586/5*e^3 + 522/5*e^2 - 2079/5*e - 2237/5, 24/5*e^5 + 2*e^4 - 393/5*e^3 - 251/5*e^2 + 1447/5*e + 1696/5, -49/5*e^5 - 3*e^4 + 758/5*e^3 + 321/5*e^2 - 2642/5*e - 2246/5, -20*e^5 - 15*e^4 + 303*e^3 + 227*e^2 - 1059*e - 1086, -11*e^5 - 6*e^4 + 170*e^3 + 102*e^2 - 594*e - 564, 86/5*e^5 + 16*e^4 - 1327/5*e^3 - 1189/5*e^2 + 4743/5*e + 5244/5, 28/5*e^5 + 2*e^4 - 446/5*e^3 - 197/5*e^2 + 1599/5*e + 1282/5, 79/5*e^5 + 11*e^4 - 1178/5*e^3 - 806/5*e^2 + 3982/5*e + 3966/5, 29*e^5 + 17*e^4 - 435*e^3 - 264*e^2 + 1479*e + 1395, -62/5*e^5 - 12*e^4 + 954/5*e^3 + 883/5*e^2 - 3406/5*e - 3878/5, -19/5*e^5 - 2*e^4 + 308/5*e^3 + 196/5*e^2 - 1097/5*e - 1146/5, -7/5*e^5 + 2*e^4 + 84/5*e^3 - 122/5*e^2 - 111/5*e + 117/5, -2/5*e^5 - 2*e^4 + 59/5*e^3 + 168/5*e^2 - 346/5*e - 613/5, 11/5*e^5 + e^4 - 167/5*e^3 - 79/5*e^2 + 593/5*e + 594/5, -3/5*e^5 + e^4 + 31/5*e^3 - 78/5*e^2 + 6/5*e + 148/5, -107/5*e^5 - 15*e^4 + 1614/5*e^3 + 1163/5*e^2 - 5546/5*e - 5763/5, -17*e^5 - 12*e^4 + 250*e^3 + 164*e^2 - 825*e - 765, 71/5*e^5 + 11*e^4 - 1082/5*e^3 - 809/5*e^2 + 3758/5*e + 3804/5, 5*e^5 + 7*e^4 - 77*e^3 - 107*e^2 + 283*e + 408, 16/5*e^5 + 5*e^4 - 297/5*e^3 - 414/5*e^2 + 1303/5*e + 1689/5, 2/5*e^5 - 2*e^4 - 29/5*e^3 + 92/5*e^2 + 81/5*e + 18/5, -39/5*e^5 - 10*e^4 + 638/5*e^3 + 811/5*e^2 - 2512/5*e - 3391/5, -46/5*e^5 - 5*e^4 + 722/5*e^3 + 419/5*e^2 - 2583/5*e - 2369/5, -43/5*e^5 - 6*e^4 + 626/5*e^3 + 407/5*e^2 - 2024/5*e - 1952/5, -29/5*e^5 - 5*e^4 + 428/5*e^3 + 336/5*e^2 - 1407/5*e - 1466/5, 11/5*e^5 + 3*e^4 - 172/5*e^3 - 184/5*e^2 + 628/5*e + 694/5, 16*e^5 + 11*e^4 - 236*e^3 - 157*e^2 + 781*e + 748];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, w - 5}>] := 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;