/*
  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![3, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -w^3 + 4*w + 1}>;

primesArray := [
[3, 3, w],
[3, 3, w - 2],
[3, 3, w - 1],
[16, 2, 2],
[23, 23, -w^3 + 4*w + 1],
[25, 5, w^3 - 2*w^2 - 2*w + 2],
[25, 5, w^3 - w^2 - 3*w + 1],
[29, 29, w^3 - 2*w^2 - 4*w + 4],
[29, 29, w^3 - w^2 - 5*w + 1],
[43, 43, -w^3 + 2*w^2 + 3*w - 2],
[43, 43, w^3 - w^2 - 4*w + 2],
[61, 61, -w^2 + 2*w + 4],
[61, 61, w^2 - 5],
[79, 79, 3*w^3 - 7*w^2 - 8*w + 16],
[79, 79, -w^3 + w^2 + 6*w - 4],
[101, 101, -w^3 + 3*w^2 + w - 7],
[101, 101, w^3 - 4*w - 4],
[103, 103, 2*w^3 - w^2 - 8*w - 1],
[103, 103, 2*w^3 - 5*w^2 - 4*w + 8],
[107, 107, 2*w^2 - 3*w - 4],
[107, 107, 3*w - 5],
[107, 107, w^3 - 4*w^2 + 10],
[107, 107, 2*w^2 - w - 5],
[113, 113, -w^3 + 4*w^2 + w - 11],
[113, 113, -2*w^3 + 4*w^2 + 5*w - 5],
[113, 113, 2*w^3 - 2*w^2 - 7*w + 2],
[113, 113, w^3 + w^2 - 6*w - 7],
[121, 11, w^3 - 3*w^2 - 4*w + 8],
[121, 11, w^3 - 7*w - 2],
[127, 127, w^2 - 4*w + 2],
[127, 127, w^3 - w^2 - 2*w - 2],
[127, 127, -w^3 + 2*w^2 + w - 4],
[127, 127, -2*w^3 + 5*w^2 + 3*w - 8],
[131, 131, 2*w^3 - 2*w^2 - 9*w + 2],
[131, 131, -w^3 - w^2 + 7*w + 5],
[157, 157, w^3 - w^2 - 7*w + 8],
[157, 157, w^3 - w^2 - 5*w + 4],
[169, 13, 2*w^2 - 2*w - 5],
[173, 173, -w^3 + 7*w - 1],
[173, 173, -w^3 - w^2 + 6*w + 4],
[179, 179, -2*w^3 + 4*w^2 + 5*w - 8],
[179, 179, -2*w^3 + 2*w^2 + 7*w + 1],
[181, 181, -w^3 + 4*w^2 - 7],
[181, 181, w^2 - 4*w + 5],
[191, 191, 3*w^3 - 7*w^2 - 9*w + 20],
[191, 191, -w^3 + 4*w + 5],
[191, 191, -w^3 + 3*w^2 + w - 8],
[191, 191, 3*w^3 - 3*w^2 - 12*w + 2],
[199, 199, 2*w^2 - 7],
[199, 199, 3*w^3 - 6*w^2 - 9*w + 13],
[233, 233, w^3 - 3*w^2 - 4*w + 11],
[233, 233, 3*w - 7],
[251, 251, 3*w^3 - 7*w^2 - 7*w + 13],
[251, 251, 4*w^3 - w^2 - 20*w - 11],
[251, 251, -w^3 + w^2 + 5*w - 7],
[251, 251, -2*w^3 + 4*w^2 + 8*w - 11],
[257, 257, w^2 - 2*w - 7],
[257, 257, w^2 - 8],
[269, 269, 2*w^3 - 3*w^2 - 8*w + 2],
[269, 269, 2*w^3 - 3*w^2 - 8*w + 7],
[283, 283, -2*w^3 + 4*w^2 + 4*w - 7],
[283, 283, 2*w^3 - 2*w^2 - 6*w - 1],
[289, 17, -w^2 + w - 2],
[289, 17, w^2 - w - 7],
[311, 311, -w^3 + 2*w^2 + 3*w - 8],
[311, 311, -w^3 + w^2 + 4*w + 4],
[313, 313, -w^3 + 2*w^2 + 5*w - 5],
[313, 313, w^3 - w^2 - 6*w + 1],
[337, 337, 3*w - 1],
[337, 337, 3*w - 2],
[347, 347, -2*w^3 - w^2 + 11*w + 11],
[347, 347, -2*w^3 + 7*w^2 + 3*w - 19],
[367, 367, 4*w^3 - 8*w^2 - 13*w + 19],
[367, 367, 4*w^3 - 4*w^2 - 17*w - 2],
[373, 373, w^2 - 3*w - 5],
[373, 373, w^2 + w - 7],
[433, 433, 2*w^3 - 3*w^2 - 5*w + 1],
[433, 433, -2*w^3 + 3*w^2 + 5*w - 5],
[443, 443, w^3 - 7*w + 2],
[443, 443, 3*w^3 - 6*w^2 - 12*w + 20],
[491, 491, -2*w^3 + 3*w^2 + 8*w - 8],
[491, 491, 2*w^3 - 3*w^2 - 11*w + 14],
[521, 521, w^3 + w^2 - 8*w - 7],
[521, 521, w^3 - 5*w^2 + 2*w + 10],
[521, 521, w^3 + 2*w^2 - 5*w - 8],
[521, 521, w^3 - 4*w^2 - 3*w + 13],
[523, 523, 3*w^3 - 5*w^2 - 11*w + 11],
[523, 523, w^3 + w^2 - 6*w - 1],
[529, 23, 3*w^2 - 3*w - 10],
[547, 547, w^3 - 3*w^2 - 4*w + 14],
[547, 547, -2*w^3 + 2*w^2 + 7*w - 5],
[547, 547, w^3 - 4*w + 4],
[547, 547, 3*w^3 - 9*w^2 - 6*w + 22],
[563, 563, w^3 - 2*w^2 - 6*w + 8],
[563, 563, -3*w^3 + 4*w^2 + 11*w - 7],
[563, 563, 3*w^3 - 5*w^2 - 10*w + 5],
[563, 563, 3*w^3 - 8*w^2 - 3*w + 10],
[571, 571, 2*w^3 - 3*w^2 - 5*w + 4],
[571, 571, -2*w^3 + 3*w^2 + 5*w - 2],
[599, 599, 2*w^3 - 2*w^2 - 5*w + 1],
[599, 599, -2*w^3 + 4*w^2 + 3*w - 4],
[641, 641, -2*w^2 + 7*w - 7],
[641, 641, 2*w^2 + w - 8],
[641, 641, 2*w^2 - 5*w - 5],
[641, 641, w^3 - 4*w^2 - 2*w + 16],
[647, 647, -w^3 + 3*w^2 - 2*w - 4],
[647, 647, w^3 - w - 4],
[653, 653, -2*w^3 + 4*w^2 + 9*w - 13],
[653, 653, 3*w^3 - 7*w^2 - 6*w + 11],
[673, 673, w^3 - 2*w^2 - 5*w + 11],
[673, 673, -2*w^3 + 2*w^2 + 10*w - 5],
[673, 673, 2*w^3 - 4*w^2 - 8*w + 5],
[673, 673, 2*w^3 - 7*w^2 - w + 13],
[677, 677, -w^3 + 2*w^2 + 6*w - 2],
[677, 677, 3*w^3 - 7*w^2 - 8*w + 13],
[677, 677, 3*w^3 - 2*w^2 - 13*w - 1],
[677, 677, -w^3 + w^2 + 7*w - 5],
[701, 701, w^2 - 5*w - 1],
[701, 701, -w^3 + 4*w^2 + 3*w - 7],
[701, 701, -w^3 - w^2 + 8*w + 1],
[701, 701, w^2 + 3*w - 5],
[719, 719, -2*w^3 + w^2 + 7*w + 5],
[719, 719, 2*w^3 - 5*w^2 - 3*w + 11],
[727, 727, 2*w^2 - 4*w - 11],
[727, 727, -3*w^3 + w^2 + 10*w + 5],
[751, 751, 4*w^3 - 12*w^2 - 7*w + 28],
[751, 751, 2*w^3 - 6*w^2 - 5*w + 20],
[757, 757, -w^3 + 2*w^2 - 5],
[757, 757, w^3 - w^2 - w - 4],
[797, 797, -4*w^3 + 7*w^2 + 13*w - 11],
[797, 797, 4*w^3 - 5*w^2 - 17*w + 2],
[797, 797, -4*w^3 + 7*w^2 + 15*w - 16],
[797, 797, 4*w^3 - 5*w^2 - 15*w + 5],
[809, 809, -w^3 + 4*w^2 + 3*w - 16],
[809, 809, 3*w^3 - 8*w^2 - 10*w + 26],
[841, 29, 3*w^2 - 3*w - 8],
[857, 857, 4*w^3 - 2*w^2 - 21*w - 7],
[857, 857, -w^3 + 2*w^2 - w + 4],
[881, 881, -w^3 + 2*w^2 + 7*w - 10],
[881, 881, w^3 - 5*w^2 - w + 13],
[881, 881, w^3 + 2*w^2 - 8*w - 8],
[881, 881, -w^3 + w^2 + 8*w + 2],
[887, 887, w^3 - w^2 - 5*w - 5],
[887, 887, -w^3 + 2*w^2 + 4*w - 10],
[907, 907, -3*w^3 + 3*w^2 + 9*w - 5],
[907, 907, 3*w^3 - 6*w^2 - 6*w + 4],
[919, 919, 4*w^3 - 3*w^2 - 16*w - 2],
[919, 919, -4*w^3 + 9*w^2 + 10*w - 17],
[937, 937, -3*w^3 + w^2 + 12*w + 7],
[937, 937, -3*w^3 + 8*w^2 + 5*w - 17],
[953, 953, -2*w^3 + 6*w^2 + 2*w - 13],
[953, 953, -w^3 + 3*w^2 + 4*w - 2],
[953, 953, w^3 - 7*w + 4],
[953, 953, 2*w^3 - 8*w - 7],
[961, 31, 3*w^3 - 4*w^2 - 13*w + 1],
[961, 31, 3*w^3 - 5*w^2 - 12*w + 13],
[991, 991, 3*w^3 - w^2 - 15*w - 4],
[991, 991, -w^3 - 3*w^2 + 7*w + 5],
[991, 991, w^3 - 6*w^2 + 2*w + 8],
[991, 991, 3*w^3 - 8*w^2 - 8*w + 17]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 14*x^6 + 32*x^4 - 11*x^2 + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/3*e^7 + 13/3*e^5 - 19/3*e^3 - 11/3*e, 19/3*e^7 - 262/3*e^5 + 553/3*e^3 - 94/3*e, -16/3*e^6 + 220/3*e^4 - 460/3*e^2 + 88/3, -1, -13/3*e^7 + 178/3*e^5 - 361/3*e^3 + 28/3*e, -8/3*e^7 + 110/3*e^5 - 230/3*e^3 + 41/3*e, 5/3*e^7 - 71/3*e^5 + 173/3*e^3 - 74/3*e, 10/3*e^7 - 139/3*e^5 + 304/3*e^3 - 61/3*e, -4/3*e^6 + 58/3*e^4 - 148/3*e^2 + 52/3, 12*e^6 - 166*e^4 + 356*e^2 - 62, -59/3*e^7 + 815/3*e^5 - 1736/3*e^3 + 326/3*e, 5/3*e^7 - 68/3*e^5 + 134/3*e^3 - 20/3*e, 52/3*e^7 - 718/3*e^5 + 1525/3*e^3 - 277/3*e, -12*e^7 + 165*e^5 - 343*e^3 + 51*e, 91/3*e^7 - 1258/3*e^5 + 2689/3*e^3 - 517/3*e, -23/3*e^7 + 314/3*e^5 - 629/3*e^3 + 65/3*e, -11/3*e^6 + 155/3*e^4 - 365/3*e^2 + 110/3, -1/3*e^6 + 10/3*e^4 + 20/3*e^2 - 35/3, 14/3*e^6 - 191/3*e^4 + 380/3*e^2 - 38/3, -88/3*e^7 + 1213/3*e^5 - 2557/3*e^3 + 445/3*e, 28/3*e^7 - 388/3*e^5 + 841/3*e^3 - 205/3*e, 2/3*e^6 - 29/3*e^4 + 80/3*e^2 - 35/3, 12*e^6 - 165*e^4 + 345*e^2 - 55, -46/3*e^7 + 637/3*e^5 - 1375/3*e^3 + 313/3*e, 36*e^7 - 496*e^5 + 1043*e^3 - 177*e, 12*e^6 - 165*e^4 + 345*e^2 - 55, 113/3*e^7 - 1559/3*e^5 + 3299/3*e^3 - 572/3*e, 6*e^7 - 83*e^5 + 178*e^3 - 29*e, -11/3*e^7 + 152/3*e^5 - 320/3*e^3 + 11/3*e, 116/3*e^7 - 1601/3*e^5 + 3392/3*e^3 - 566/3*e, 106/3*e^7 - 1465/3*e^5 + 3130/3*e^3 - 592/3*e, -98/3*e^7 + 1355/3*e^5 - 2903/3*e^3 + 590/3*e, 16/3*e^6 - 220/3*e^4 + 454/3*e^2 - 67/3, -16/3*e^6 + 220/3*e^4 - 454/3*e^2 + 55/3, -47/3*e^6 + 650/3*e^4 - 1391/3*e^2 + 260/3, 11/3*e^6 - 155/3*e^4 + 356/3*e^2 - 68/3, 4/3*e^6 - 55/3*e^4 + 115/3*e^2 - 34/3, -1/3*e^6 + 13/3*e^4 - 19/3*e^2 + 28/3, -e^6 + 14*e^4 - 32*e^2 + 19, 9*e^7 - 123*e^5 + 247*e^3 - 20*e, -26*e^7 + 359*e^5 - 762*e^3 + 133*e, 8*e^6 - 111*e^4 + 240*e^2 - 43, -32/3*e^6 + 443/3*e^4 - 950/3*e^2 + 170/3, -64/3*e^6 + 883/3*e^4 - 1867/3*e^2 + 337/3, -31/3*e^7 + 430/3*e^5 - 940/3*e^3 + 262/3*e, 89/3*e^7 - 1229/3*e^5 + 2618/3*e^3 - 524/3*e, 8/3*e^6 - 113/3*e^4 + 257/3*e^2 - 23/3, -8/3*e^7 + 110/3*e^5 - 233/3*e^3 + 80/3*e, 79/3*e^7 - 1093/3*e^5 + 2350/3*e^3 - 499/3*e, 62/3*e^7 - 860/3*e^5 + 1877/3*e^3 - 467/3*e, 28/3*e^7 - 385/3*e^5 + 793/3*e^3 - 43/3*e, -97/3*e^7 + 1339/3*e^5 - 2842/3*e^3 + 532/3*e, 77/3*e^6 - 1067/3*e^4 + 2303/3*e^2 - 416/3, -41/3*e^6 + 572/3*e^4 - 1268/3*e^2 + 269/3, 119/3*e^7 - 1640/3*e^5 + 3452/3*e^3 - 590/3*e, -17/3*e^6 + 239/3*e^4 - 551/3*e^2 + 170/3, 29/3*e^6 - 404/3*e^4 + 896/3*e^2 - 155/3, 62/3*e^6 - 860/3*e^4 + 1859/3*e^2 - 320/3, -70/3*e^6 + 970/3*e^4 - 2089/3*e^2 + 397/3, -14/3*e^7 + 200/3*e^5 - 497/3*e^3 + 212/3*e, -7/3*e^7 + 103/3*e^5 - 286/3*e^3 + 157/3*e, -14, 20*e^6 - 275*e^4 + 575*e^2 - 115, 16/3*e^6 - 223/3*e^4 + 481/3*e^2 - 49/3, -88/3*e^6 + 1213/3*e^4 - 2551/3*e^2 + 433/3, -64/3*e^7 + 877/3*e^5 - 1792/3*e^3 + 229/3*e, 145/3*e^7 - 2005/3*e^5 + 4291/3*e^3 - 838/3*e, 8/3*e^7 - 113/3*e^5 + 272/3*e^3 - 122/3*e, 22*e^7 - 303*e^5 + 634*e^3 - 88*e, 14*e^6 - 192*e^4 + 396*e^2 - 61, 46/3*e^6 - 634/3*e^4 + 1342/3*e^2 - 241/3, 32/3*e^6 - 440/3*e^4 + 905/3*e^2 - 113/3, -16*e^6 + 220*e^4 - 455*e^2 + 64, -134/3*e^7 + 1850/3*e^5 - 3935/3*e^3 + 755/3*e, -38/3*e^7 + 521/3*e^5 - 1061/3*e^3 + 53/3*e, -262/3*e^7 + 3616/3*e^5 - 7666/3*e^3 + 1378/3*e, 118/3*e^7 - 1624/3*e^5 + 3394/3*e^3 - 562/3*e, -41/3*e^6 + 557/3*e^4 - 1100/3*e^2 + 143/3, -107/3*e^6 + 1478/3*e^4 - 3155/3*e^2 + 587/3, -26/3*e^6 + 353/3*e^4 - 698/3*e^2 + 104/3, -86/3*e^6 + 1187/3*e^4 - 2522/3*e^2 + 461/3, -328/3*e^7 + 4531/3*e^5 - 9646/3*e^3 + 1744/3*e, 17/3*e^6 - 239/3*e^4 + 548/3*e^2 - 59/3, -15*e^6 + 208*e^4 - 451*e^2 + 109, -22*e^7 + 307*e^5 - 688*e^3 + 174*e, -7/3*e^6 + 100/3*e^4 - 247/3*e^2 + 52/3, 19/3*e^6 - 265/3*e^4 + 592/3*e^2 - 154/3, 80/3*e^6 - 1100/3*e^4 + 2300/3*e^2 - 377/3, 8/3*e^6 - 110/3*e^4 + 239/3*e^2 - 65/3, 37/3*e^6 - 508/3*e^4 + 1060/3*e^2 - 157/3, 71/3*e^6 - 977/3*e^4 + 2045/3*e^2 - 308/3, 56/3*e^6 - 770/3*e^4 + 1601/3*e^2 - 248/3, 70/3*e^7 - 967/3*e^5 + 2047/3*e^3 - 274/3*e, -1/3*e^6 + 19/3*e^4 - 88/3*e^2 + 31/3, 61/3*e^6 - 844/3*e^4 + 1813/3*e^2 - 355/3, 9*e^7 - 128*e^5 + 316*e^3 - 153*e, 48*e^7 - 661*e^5 + 1385*e^3 - 227*e, -78*e^7 + 1076*e^5 - 2275*e^3 + 397*e, -38/3*e^7 + 521/3*e^5 - 1067/3*e^3 + 107/3*e, -34*e^7 + 468*e^5 - 979*e^3 + 151*e, 2/3*e^6 - 26/3*e^4 + 50/3*e^2 - 32/3, -295/3*e^7 + 4075/3*e^5 - 8677/3*e^3 + 1597/3*e, -29/3*e^7 + 407/3*e^5 - 935/3*e^3 + 239/3*e, 70/3*e^6 - 964/3*e^4 + 2020/3*e^2 - 334/3, -64/3*e^7 + 883/3*e^5 - 1870/3*e^3 + 322/3*e, -88/3*e^7 + 1213/3*e^5 - 2554/3*e^3 + 406/3*e, 15*e^7 - 208*e^5 + 449*e^3 - 70*e, 148/3*e^7 - 2050/3*e^5 + 4432/3*e^3 - 967/3*e, 59/3*e^6 - 812/3*e^4 + 1691/3*e^2 - 335/3, -62/3*e^6 + 857/3*e^4 - 1823/3*e^2 + 347/3, 46/3*e^6 - 637/3*e^4 + 1363/3*e^2 - 193/3, -23/3*e^6 + 317/3*e^4 - 656/3*e^2 - 1/3, -23/3*e^7 + 320/3*e^5 - 704/3*e^3 + 137/3*e, 11*e^6 - 152*e^4 + 321*e^2 - 47, -53/3*e^6 + 731/3*e^4 - 1538/3*e^2 + 251/3, -40*e^7 + 553*e^5 - 1183*e^3 + 230*e, -8/3*e^7 + 110/3*e^5 - 227/3*e^3 + 53/3*e, -e^6 + 13*e^4 - 17*e^2 - 32, 23/3*e^6 - 314/3*e^4 + 626/3*e^2 - 131/3, 218/3*e^7 - 3005/3*e^5 + 6329/3*e^3 - 1043/3*e, -54*e^7 + 744*e^5 - 1564*e^3 + 266*e, 52*e^7 - 718*e^5 + 1526*e^3 - 288*e, -94/3*e^7 + 1294/3*e^5 - 2716/3*e^3 + 514/3*e, 52*e^7 - 720*e^5 + 1554*e^3 - 344*e, -80/3*e^6 + 1097/3*e^4 - 2264/3*e^2 + 350/3, -104/3*e^6 + 1433/3*e^4 - 3026/3*e^2 + 527/3, -209/3*e^7 + 2891/3*e^5 - 6206/3*e^3 + 1286/3*e, 53/3*e^7 - 728/3*e^5 + 1514/3*e^3 - 302/3*e, -125/3*e^7 + 1727/3*e^5 - 3683/3*e^3 + 743/3*e, 26/3*e^6 - 356/3*e^4 + 710/3*e^2 + 13/3, -22*e^6 + 302*e^4 - 620*e^2 + 107, 245/3*e^7 - 3377/3*e^5 + 7115/3*e^3 - 1223/3*e, 56*e^7 - 775*e^5 + 1669*e^3 - 369*e, -160/3*e^7 + 2206/3*e^5 - 4663/3*e^3 + 895/3*e, 20/3*e^6 - 275/3*e^4 + 575/3*e^2 - 92/3, -5/3*e^7 + 80/3*e^5 - 299/3*e^3 + 314/3*e, -188/3*e^7 + 2594/3*e^5 - 5480/3*e^3 + 863/3*e, -23*e^7 + 316*e^5 - 652*e^3 + 64*e, 49/3*e^6 - 685/3*e^4 + 1525/3*e^2 - 319/3, -47*e^6 + 650*e^4 - 1390*e^2 + 242, -103/3*e^7 + 1423/3*e^5 - 3040/3*e^3 + 616/3*e, 83/3*e^6 - 1151/3*e^4 + 2489/3*e^2 - 437/3, -71/3*e^6 + 986/3*e^4 - 2144/3*e^2 + 428/3, -226/3*e^7 + 3130/3*e^5 - 6748/3*e^3 + 1351/3*e, -59/3*e^7 + 830/3*e^5 - 1931/3*e^3 + 614/3*e, 68*e^7 - 940*e^5 + 2008*e^3 - 364*e, 232/3*e^7 - 3208/3*e^5 + 6868/3*e^3 - 1312/3*e, 244/3*e^7 - 3361/3*e^5 + 7054/3*e^3 - 1126/3*e, -8/3*e^7 + 113/3*e^5 - 266/3*e^3 + 122/3*e, -65*e^7 + 901*e^5 - 1952*e^3 + 398*e, -142/3*e^6 + 1954/3*e^4 - 4078/3*e^2 + 730/3, -10/3*e^6 + 136/3*e^4 - 292/3*e^2 + 184/3, -359/3*e^7 + 4964/3*e^5 - 10622/3*e^3 + 2012/3*e, 74/3*e^6 - 1022/3*e^4 + 2165/3*e^2 - 314/3, -50/3*e^6 + 692/3*e^4 - 1475/3*e^2 + 287/3, 25/3*e^6 - 334/3*e^4 + 595/3*e^2 + 35/3, 7*e^6 - 95*e^4 + 192*e^2 - 59, 143/3*e^6 - 1970/3*e^4 + 4139/3*e^2 - 749/3, 67/3*e^6 - 931/3*e^4 + 2050/3*e^2 - 403/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -w^3 + 4*w + 1}>] := 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;