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

NN := ideal<ZF | {71,71,w^5 - 3*w^4 - 2*w^3 + 7*w^2 + w}>;

primesArray := [
[13, 13, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 4],
[13, 13, -w^2 + w + 2],
[27, 3, 2*w^5 - 4*w^4 - 7*w^3 + 9*w^2 + 4*w - 2],
[27, 3, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - w + 5],
[29, 29, w^3 - 2*w^2 - 2*w + 3],
[29, 29, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 4*w - 2],
[43, 43, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w + 1],
[43, 43, -w^4 + w^3 + 5*w^2 - 4],
[49, 7, w^5 - 4*w^4 + 11*w^2 - 3*w - 4],
[64, 2, -2],
[71, 71, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 - w - 6],
[71, 71, 2*w^4 - 4*w^3 - 6*w^2 + 7*w + 2],
[71, 71, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w],
[71, 71, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 3*w + 5],
[83, 83, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 5*w + 5],
[83, 83, 3*w^5 - 6*w^4 - 10*w^3 + 12*w^2 + 5*w - 2],
[83, 83, -2*w^5 + 5*w^4 + 5*w^3 - 11*w^2 - 3*w + 3],
[83, 83, 3*w^5 - 7*w^4 - 8*w^3 + 15*w^2 + 2*w - 4],
[97, 97, -3*w^5 + 6*w^4 + 10*w^3 - 12*w^2 - 5*w + 3],
[97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 3],
[97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 1],
[97, 97, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 - 2*w + 3],
[113, 113, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 7*w + 3],
[113, 113, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 - 6],
[113, 113, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 1],
[113, 113, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 2*w + 5],
[125, 5, 3*w^5 - 6*w^4 - 10*w^3 + 11*w^2 + 6*w - 1],
[125, 5, -2*w^5 + 4*w^4 + 6*w^3 - 6*w^2 - 3*w - 1],
[127, 127, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 2*w + 5],
[127, 127, 2*w^5 - 3*w^4 - 9*w^3 + 6*w^2 + 7*w - 2],
[127, 127, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 4*w + 4],
[127, 127, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - w + 3],
[139, 139, 3*w^5 - 7*w^4 - 9*w^3 + 16*w^2 + 6*w - 4],
[139, 139, -w^5 + w^4 + 5*w^3 - 6*w - 1],
[167, 167, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 4],
[167, 167, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 4],
[169, 13, -w^5 + 4*w^4 - 11*w^2 + 3*w + 5],
[169, 13, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 3*w - 3],
[181, 181, 2*w^5 - 4*w^4 - 6*w^3 + 6*w^2 + 2*w + 1],
[197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 1],
[197, 197, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 6*w + 4],
[211, 211, 2*w^5 - 5*w^4 - 6*w^3 + 12*w^2 + 6*w - 1],
[211, 211, -2*w^5 + 4*w^4 + 8*w^3 - 9*w^2 - 8*w + 2],
[211, 211, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w + 3],
[211, 211, -2*w^4 + 5*w^3 + 5*w^2 - 9*w - 3],
[223, 223, -w^5 + 8*w^3 + w^2 - 11*w + 1],
[223, 223, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 + w - 1],
[239, 239, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 2*w + 3],
[239, 239, 2*w^5 - 5*w^4 - 4*w^3 + 9*w^2 + w],
[239, 239, -3*w^5 + 7*w^4 + 10*w^3 - 18*w^2 - 7*w + 5],
[239, 239, w^5 - w^4 - 6*w^3 + 2*w^2 + 5*w - 2],
[251, 251, w^5 - w^4 - 6*w^3 + 3*w^2 + 6*w - 2],
[251, 251, -w^5 + 4*w^4 - 12*w^2 + 3*w + 6],
[281, 281, -5*w^5 + 11*w^4 + 17*w^3 - 27*w^2 - 10*w + 9],
[281, 281, 2*w^5 - 5*w^4 - 5*w^3 + 10*w^2 + 3*w + 1],
[293, 293, -2*w^5 + 4*w^4 + 7*w^3 - 7*w^2 - 7*w + 2],
[293, 293, -w^4 + 2*w^3 + 2*w^2 - 3*w + 3],
[293, 293, -2*w^5 + 7*w^4 + w^3 - 19*w^2 + 7*w + 9],
[293, 293, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 7*w + 2],
[307, 307, 2*w^2 - 3*w - 4],
[307, 307, -w^5 + 4*w^4 - 13*w^2 + 6*w + 6],
[337, 337, 4*w^5 - 9*w^4 - 12*w^3 + 20*w^2 + 6*w - 5],
[337, 337, -2*w^5 + 3*w^4 + 10*w^3 - 6*w^2 - 12*w + 2],
[349, 349, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 4],
[349, 349, 5*w^5 - 11*w^4 - 15*w^3 + 24*w^2 + 5*w - 7],
[379, 379, -2*w^5 + 5*w^4 + 4*w^3 - 10*w^2 + w + 5],
[379, 379, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 5],
[379, 379, -4*w^5 + 8*w^4 + 14*w^3 - 17*w^2 - 10*w + 4],
[379, 379, -5*w^5 + 12*w^4 + 13*w^3 - 27*w^2 - 2*w + 7],
[421, 421, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + w - 3],
[421, 421, 3*w^5 - 9*w^4 - 6*w^3 + 25*w^2 - w - 9],
[433, 433, -5*w^5 + 10*w^4 + 18*w^3 - 21*w^2 - 13*w + 4],
[433, 433, 5*w^5 - 12*w^4 - 15*w^3 + 30*w^2 + 7*w - 9],
[433, 433, -3*w^5 + 8*w^4 + 8*w^3 - 22*w^2 - w + 8],
[433, 433, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 9*w - 3],
[449, 449, 3*w^5 - 9*w^4 - 6*w^3 + 24*w^2 + w - 9],
[449, 449, 3*w^5 - 6*w^4 - 11*w^3 + 15*w^2 + 6*w - 5],
[449, 449, 2*w^5 - 3*w^4 - 8*w^3 + 3*w^2 + 7*w],
[449, 449, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 8*w - 2],
[449, 449, -3*w^4 + 5*w^3 + 12*w^2 - 9*w - 6],
[449, 449, -3*w^5 + 8*w^4 + 7*w^3 - 20*w^2 - 2*w + 6],
[461, 461, 2*w^4 - 4*w^3 - 7*w^2 + 7*w + 2],
[461, 461, -4*w^5 + 10*w^4 + 11*w^3 - 24*w^2 - 6*w + 8],
[463, 463, 4*w^5 - 8*w^4 - 14*w^3 + 18*w^2 + 6*w - 5],
[463, 463, w^5 - w^4 - 5*w^3 + w^2 + 4*w - 3],
[491, 491, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 + 5],
[491, 491, -2*w^5 + 6*w^4 + 3*w^3 - 14*w^2 + 3],
[547, 547, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 5],
[547, 547, w^5 - 4*w^4 + 12*w^2 - 2*w - 5],
[547, 547, -3*w^5 + 5*w^4 + 13*w^3 - 11*w^2 - 11*w + 3],
[547, 547, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + w + 6],
[547, 547, -2*w^5 + 6*w^4 + 4*w^3 - 15*w^2 - w + 4],
[547, 547, 4*w^5 - 8*w^4 - 15*w^3 + 18*w^2 + 12*w - 6],
[587, 587, -5*w^5 + 12*w^4 + 14*w^3 - 28*w^2 - 6*w + 7],
[587, 587, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 6*w - 6],
[601, 601, -3*w^5 + 9*w^4 + 5*w^3 - 24*w^2 + 5*w + 10],
[601, 601, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - w + 9],
[601, 601, -w^5 + 8*w^3 + 2*w^2 - 11*w - 1],
[601, 601, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 4],
[617, 617, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3],
[617, 617, -5*w^5 + 11*w^4 + 16*w^3 - 26*w^2 - 7*w + 8],
[617, 617, 3*w^5 - 7*w^4 - 9*w^3 + 15*w^2 + 7*w - 2],
[617, 617, -2*w^5 + 2*w^4 + 11*w^3 - w^2 - 13*w - 2],
[631, 631, 2*w^4 - 4*w^3 - 7*w^2 + 8*w + 2],
[631, 631, w^5 - w^4 - 5*w^3 - w^2 + 4*w + 5],
[631, 631, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 6],
[631, 631, -3*w^5 + 7*w^4 + 9*w^3 - 15*w^2 - 7*w + 3],
[643, 643, -2*w^5 + 3*w^4 + 9*w^3 - 6*w^2 - 8*w + 4],
[643, 643, 3*w^5 - 8*w^4 - 7*w^3 + 20*w^2 + 2*w - 5],
[659, 659, 3*w^5 - 9*w^4 - 5*w^3 + 23*w^2 - 3*w - 10],
[659, 659, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 5],
[673, 673, 3*w^5 - 6*w^4 - 11*w^3 + 14*w^2 + 8*w - 2],
[673, 673, 3*w^5 - 8*w^4 - 6*w^3 + 19*w^2 - 3*w - 8],
[673, 673, 3*w^5 - 6*w^4 - 10*w^3 + 13*w^2 + 3*w - 2],
[673, 673, 2*w^5 - 3*w^4 - 10*w^3 + 6*w^2 + 11*w - 2],
[701, 701, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 4],
[701, 701, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 2*w - 6],
[701, 701, -w^5 + w^4 + 5*w^3 - 5*w - 5],
[701, 701, -w^5 + w^4 + 6*w^3 - 2*w^2 - 9*w + 2],
[727, 727, -w^5 + 5*w^4 - w^3 - 16*w^2 + 4*w + 8],
[727, 727, w^5 - w^4 - 4*w^3 - 2*w^2 + 4*w + 4],
[727, 727, -3*w^5 + 9*w^4 + 5*w^3 - 22*w^2 + 2*w + 7],
[727, 727, -w^5 + 2*w^4 + 3*w^3 - 5*w^2 + 5],
[727, 727, -5*w^5 + 11*w^4 + 16*w^3 - 24*w^2 - 10*w + 6],
[727, 727, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 7*w - 4],
[743, 743, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 6*w - 8],
[743, 743, -4*w^5 + 7*w^4 + 16*w^3 - 14*w^2 - 12*w + 1],
[743, 743, -4*w^5 + 9*w^4 + 14*w^3 - 23*w^2 - 10*w + 7],
[743, 743, w^5 - 3*w^4 + 6*w^2 - 6*w - 2],
[757, 757, -w^5 + 7*w^3 + 3*w^2 - 7*w - 6],
[757, 757, 5*w^5 - 11*w^4 - 16*w^3 + 24*w^2 + 11*w - 5],
[769, 769, -3*w^5 + 4*w^4 + 14*w^3 - 6*w^2 - 12*w + 1],
[769, 769, 4*w^5 - 7*w^4 - 17*w^3 + 16*w^2 + 14*w - 5],
[797, 797, -5*w^5 + 12*w^4 + 14*w^3 - 27*w^2 - 7*w + 7],
[797, 797, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 4*w - 7],
[811, 811, 5*w^5 - 13*w^4 - 13*w^3 + 33*w^2 + 3*w - 10],
[811, 811, -4*w^5 + 10*w^4 + 9*w^3 - 22*w^2 + w + 6],
[841, 29, 3*w^5 - 6*w^4 - 9*w^3 + 10*w^2 + 4*w - 1],
[841, 29, -w^5 + w^4 + 4*w^3 + 3*w^2 - 3*w - 7],
[853, 853, -4*w^5 + 9*w^4 + 12*w^3 - 21*w^2 - 4*w + 5],
[853, 853, -3*w^5 + 6*w^4 + 10*w^3 - 13*w^2 - 4*w + 2],
[853, 853, -w^5 + w^4 + 5*w^3 + w^2 - 4*w - 6],
[853, 853, -3*w^5 + 7*w^4 + 8*w^3 - 16*w^2 - w + 7],
[853, 853, -4*w^5 + 10*w^4 + 10*w^3 - 24*w^2 - w + 10],
[853, 853, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2],
[883, 883, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 4*w + 6],
[883, 883, -w^5 + 5*w^4 - 3*w^3 - 12*w^2 + 9*w + 3],
[883, 883, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 7*w - 6],
[883, 883, 2*w^5 - 6*w^4 - 3*w^3 + 16*w^2 - 3*w - 9],
[937, 937, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 8*w - 7],
[937, 937, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w],
[937, 937, 2*w^5 - 6*w^4 - 4*w^3 + 18*w^2 - 3*w - 10],
[937, 937, 2*w^5 - 7*w^4 - 3*w^3 + 22*w^2 - 2*w - 10],
[953, 953, 4*w^5 - 7*w^4 - 16*w^3 + 14*w^2 + 13*w - 3],
[953, 953, 5*w^5 - 11*w^4 - 15*w^3 + 23*w^2 + 7*w - 3],
[953, 953, w^5 + w^4 - 9*w^3 - 6*w^2 + 12*w + 3],
[953, 953, -3*w^5 + 7*w^4 + 10*w^3 - 17*w^2 - 9*w + 5],
[953, 953, w^5 - 4*w^4 + w^3 + 8*w^2 - 5*w + 1],
[953, 953, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 4*w + 7],
[967, 967, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + w - 9],
[967, 967, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 3*x^4 - 41*x^3 + 35*x^2 + 448*x + 506;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-25/161*e^4 + 5/7*e^3 + 841/161*e^2 - 2285/161*e - 300/7, e, -45/161*e^4 + 9/7*e^3 + 1385/161*e^2 - 3791/161*e - 470/7, 40/161*e^4 - 8/7*e^3 - 1249/161*e^2 + 3334/161*e + 438/7, 53/161*e^4 - 12/7*e^3 - 1667/161*e^2 + 5456/161*e + 650/7, -68/161*e^4 + 15/7*e^3 + 2075/161*e^2 - 6505/161*e - 739/7, -50/161*e^4 + 10/7*e^3 + 1682/161*e^2 - 4570/161*e - 593/7, -13/161*e^4 + 4/7*e^3 + 418/161*e^2 - 2122/161*e - 191/7, 9/23*e^4 - 2*e^3 - 277/23*e^2 + 887/23*e + 98, -4/23*e^4 + e^3 + 118/23*e^2 - 453/23*e - 45, 18/161*e^4 - 5/7*e^3 - 393/161*e^2 + 1935/161*e + 97/7, 45/161*e^4 - 9/7*e^3 - 1385/161*e^2 + 3791/161*e + 442/7, -1, 10/161*e^4 - 2/7*e^3 - 272/161*e^2 + 592/161*e + 92/7, 15/161*e^4 - 3/7*e^3 - 569/161*e^2 + 1693/161*e + 215/7, -15/161*e^4 + 3/7*e^3 + 569/161*e^2 - 1371/161*e - 222/7, 93/161*e^4 - 20/7*e^3 - 2916/161*e^2 + 8951/161*e + 1095/7, 38/161*e^4 - 9/7*e^3 - 1098/161*e^2 + 3763/161*e + 428/7, 30/161*e^4 - 6/7*e^3 - 1138/161*e^2 + 3064/161*e + 486/7, 12/161*e^4 - 1/7*e^3 - 584/161*e^2 + 807/161*e + 249/7, -88/161*e^4 + 19/7*e^3 + 2619/161*e^2 - 8011/161*e - 909/7, 14/23*e^4 - 3*e^3 - 436/23*e^2 + 1367/23*e + 154, -18/23*e^4 + 4*e^3 + 554/23*e^2 - 1820/23*e - 208, 68/161*e^4 - 15/7*e^3 - 2075/161*e^2 + 6827/161*e + 795/7, -18/161*e^4 + 5/7*e^3 + 393/161*e^2 - 2257/161*e - 146/7, 73/161*e^4 - 16/7*e^3 - 2050/161*e^2 + 6640/161*e + 645/7, 1/7*e^4 - 6/7*e^3 - 30/7*e^2 + 132/7*e + 241/7, -86/161*e^4 + 20/7*e^3 + 2468/161*e^2 - 8762/161*e - 892/7, -30/161*e^4 + 6/7*e^3 + 977/161*e^2 - 2903/161*e - 297/7, -15/161*e^4 + 3/7*e^3 + 247/161*e^2 - 727/161*e + 16/7, -131/161*e^4 + 29/7*e^3 + 4175/161*e^2 - 13519/161*e - 1502/7, -90/161*e^4 + 18/7*e^3 + 2931/161*e^2 - 8387/161*e - 1031/7, 183/161*e^4 - 38/7*e^3 - 5686/161*e^2 + 16694/161*e + 1979/7, -20/161*e^4 + 4/7*e^3 + 705/161*e^2 - 2472/161*e - 254/7, 165/161*e^4 - 33/7*e^3 - 5293/161*e^2 + 14759/161*e + 1805/7, 3/161*e^4 - 2/7*e^3 + 15/161*e^2 + 725/161*e + 36/7, -73/161*e^4 + 16/7*e^3 + 2211/161*e^2 - 7123/161*e - 708/7, -37/161*e^4 + 6/7*e^3 + 1264/161*e^2 - 2609/161*e - 451/7, 121/161*e^4 - 27/7*e^3 - 3903/161*e^2 + 12283/161*e + 1508/7, -61/161*e^4 + 15/7*e^3 + 1788/161*e^2 - 6960/161*e - 613/7, 81/161*e^4 - 19/7*e^3 - 2493/161*e^2 + 8627/161*e + 909/7, 5/161*e^4 - 1/7*e^3 - 297/161*e^2 + 457/161*e + 144/7, -30/161*e^4 + 6/7*e^3 + 977/161*e^2 - 2742/161*e - 290/7, -38/161*e^4 + 9/7*e^3 + 1259/161*e^2 - 4407/161*e - 568/7, -32/23*e^4 + 7*e^3 + 990/23*e^2 - 3118/23*e - 362, 2/161*e^4 + 1/7*e^3 - 151/161*e^2 - 751/161*e + 59/7, 3/161*e^4 - 2/7*e^3 - 146/161*e^2 + 1369/161*e + 169/7, -113/161*e^4 + 24/7*e^3 + 3460/161*e^2 - 10296/161*e - 1195/7, -78/161*e^4 + 17/7*e^3 + 2347/161*e^2 - 7419/161*e - 831/7, -121/161*e^4 + 27/7*e^3 + 3742/161*e^2 - 12283/161*e - 1333/7, 19/23*e^4 - 4*e^3 - 595/23*e^2 + 1824/23*e + 203, 118/161*e^4 - 25/7*e^3 - 3596/161*e^2 + 10914/161*e + 1262/7, -40/161*e^4 + 8/7*e^3 + 1088/161*e^2 - 3334/161*e - 291/7, -166/161*e^4 + 36/7*e^3 + 5288/161*e^2 - 16396/161*e - 1866/7, -86/161*e^4 + 20/7*e^3 + 2468/161*e^2 - 8762/161*e - 892/7, -83/161*e^4 + 18/7*e^3 + 2644/161*e^2 - 7715/161*e - 1031/7, -44/161*e^4 + 6/7*e^3 + 1551/161*e^2 - 2798/161*e - 416/7, 83/161*e^4 - 18/7*e^3 - 2322/161*e^2 + 7232/161*e + 639/7, 51/161*e^4 - 13/7*e^3 - 1677/161*e^2 + 6529/161*e + 724/7, -7/23*e^4 + 2*e^3 + 172/23*e^2 - 856/23*e - 59, 164/161*e^4 - 37/7*e^3 - 5137/161*e^2 + 16825/161*e + 1891/7, 10/23*e^4 - 2*e^3 - 341/23*e^2 + 868/23*e + 123, 14/23*e^4 - 3*e^3 - 413/23*e^2 + 1321/23*e + 135, 95/161*e^4 - 19/7*e^3 - 3228/161*e^2 + 9005/161*e + 1210/7, 68/161*e^4 - 15/7*e^3 - 2075/161*e^2 + 6827/161*e + 795/7, -163/161*e^4 + 34/7*e^3 + 5142/161*e^2 - 14544/161*e - 1858/7, -2/7*e^4 + 12/7*e^3 + 60/7*e^2 - 229/7*e - 538/7, 5/23*e^4 - e^3 - 159/23*e^2 + 365/23*e + 70, 61/161*e^4 - 15/7*e^3 - 1949/161*e^2 + 7604/161*e + 802/7, -86/161*e^4 + 20/7*e^3 + 2468/161*e^2 - 8118/161*e - 850/7, -80/161*e^4 + 16/7*e^3 + 2498/161*e^2 - 7151/161*e - 848/7, 66/161*e^4 - 16/7*e^3 - 1924/161*e^2 + 6934/161*e + 750/7, 18/161*e^4 - 5/7*e^3 - 393/161*e^2 + 1291/161*e + 160/7, -24/161*e^4 + 2/7*e^3 + 1007/161*e^2 - 648/161*e - 267/7, -31/161*e^4 + 9/7*e^3 + 811/161*e^2 - 4540/161*e - 218/7, 73/161*e^4 - 16/7*e^3 - 2050/161*e^2 + 5996/161*e + 687/7, 76/161*e^4 - 18/7*e^3 - 2196/161*e^2 + 8170/161*e + 772/7, -40/161*e^4 + 8/7*e^3 + 1410/161*e^2 - 3656/161*e - 578/7, 73/161*e^4 - 16/7*e^3 - 2211/161*e^2 + 6640/161*e + 778/7, -159/161*e^4 + 36/7*e^3 + 4840/161*e^2 - 16368/161*e - 1719/7, 28/23*e^4 - 6*e^3 - 826/23*e^2 + 2596/23*e + 280, -272/161*e^4 + 60/7*e^3 + 8300/161*e^2 - 26342/161*e - 2998/7, -39/161*e^4 + 12/7*e^3 + 932/161*e^2 - 5400/161*e - 279/7, 148/161*e^4 - 31/7*e^3 - 4734/161*e^2 + 13817/161*e + 1706/7, -18/23*e^4 + 4*e^3 + 531/23*e^2 - 1682/23*e - 192, -272/161*e^4 + 60/7*e^3 + 8622/161*e^2 - 27308/161*e - 3110/7, -26/161*e^4 + 8/7*e^3 + 514/161*e^2 - 3278/161*e - 130/7, -15/23*e^4 + 3*e^3 + 477/23*e^2 - 1371/23*e - 166, 8/23*e^4 - 2*e^3 - 282/23*e^2 + 1090/23*e + 116, -121/161*e^4 + 27/7*e^3 + 3581/161*e^2 - 11639/161*e - 1354/7, 102/161*e^4 - 26/7*e^3 - 3032/161*e^2 + 12092/161*e + 1070/7, 36/23*e^4 - 8*e^3 - 1108/23*e^2 + 3548/23*e + 394, -198/161*e^4 + 41/7*e^3 + 6094/161*e^2 - 18226/161*e - 2096/7, 78/161*e^4 - 17/7*e^3 - 2347/161*e^2 + 8385/161*e + 733/7, -88/161*e^4 + 19/7*e^3 + 2780/161*e^2 - 7689/161*e - 972/7, 174/161*e^4 - 39/7*e^3 - 5409/161*e^2 + 17739/161*e + 2053/7, 311/161*e^4 - 65/7*e^3 - 9715/161*e^2 + 28361/161*e + 3424/7, 129/161*e^4 - 30/7*e^3 - 4024/161*e^2 + 14270/161*e + 1401/7, 58/161*e^4 - 13/7*e^3 - 1803/161*e^2 + 6235/161*e + 745/7, 29/161*e^4 - 10/7*e^3 - 499/161*e^2 + 3842/161*e + 138/7, -199/161*e^4 + 44/7*e^3 + 6411/161*e^2 - 20668/161*e - 2528/7, -65/161*e^4 + 20/7*e^3 + 1607/161*e^2 - 8678/161*e - 703/7, 36/161*e^4 - 10/7*e^3 - 786/161*e^2 + 3548/161*e + 292/7, -394/161*e^4 + 83/7*e^3 + 12520/161*e^2 - 37364/161*e - 4406/7, 51/161*e^4 - 13/7*e^3 - 1677/161*e^2 + 6529/161*e + 808/7, 5/161*e^4 - 1/7*e^3 - 297/161*e^2 + 1423/161*e + 179/7, 272/161*e^4 - 60/7*e^3 - 8300/161*e^2 + 25698/161*e + 2956/7, 61/161*e^4 - 15/7*e^3 - 2110/161*e^2 + 7926/161*e + 977/7, -171/161*e^4 + 37/7*e^3 + 5585/161*e^2 - 17497/161*e - 1982/7, -80/161*e^4 + 16/7*e^3 + 2498/161*e^2 - 7634/161*e - 862/7, 41/23*e^4 - 9*e^3 - 1313/23*e^2 + 4143/23*e + 496, 312/161*e^4 - 68/7*e^3 - 9549/161*e^2 + 29676/161*e + 3296/7, 275/161*e^4 - 55/7*e^3 - 8607/161*e^2 + 24169/161*e + 2852/7, 19/23*e^4 - 4*e^3 - 572/23*e^2 + 1732/23*e + 211, -150/161*e^4 + 30/7*e^3 + 4724/161*e^2 - 12744/161*e - 1534/7, -4/161*e^4 - 2/7*e^3 + 302/161*e^2 + 1180/161*e + 22/7, -57/161*e^4 + 10/7*e^3 + 1486/161*e^2 - 3632/161*e - 236/7, -242/161*e^4 + 54/7*e^3 + 7484/161*e^2 - 24244/161*e - 2806/7, -5/161*e^4 + 1/7*e^3 + 297/161*e^2 - 457/161*e - 186/7, 130/161*e^4 - 26/7*e^3 - 4180/161*e^2 + 12204/161*e + 1448/7, 29/161*e^4 - 10/7*e^3 - 338/161*e^2 + 3520/161*e + 61/7, 270/161*e^4 - 54/7*e^3 - 8471/161*e^2 + 23712/161*e + 2820/7, -16/23*e^4 + 4*e^3 + 472/23*e^2 - 1766/23*e - 182, -146/161*e^4 + 32/7*e^3 + 4583/161*e^2 - 13602/161*e - 1696/7, 72/161*e^4 - 13/7*e^3 - 2538/161*e^2 + 5164/161*e + 990/7, -139/161*e^4 + 25/7*e^3 + 4457/161*e^2 - 10032/161*e - 1570/7, -13/7*e^4 + 64/7*e^3 + 404/7*e^2 - 1240/7*e - 3350/7, 27/23*e^4 - 6*e^3 - 854/23*e^2 + 2730/23*e + 333, 268/161*e^4 - 62/7*e^3 - 7998/161*e^2 + 27522/161*e + 2964/7, 288/161*e^4 - 66/7*e^3 - 8864/161*e^2 + 29672/161*e + 3456/7, -8/161*e^4 + 3/7*e^3 + 443/161*e^2 - 1665/161*e - 313/7, -416/161*e^4 + 93/7*e^3 + 12571/161*e^2 - 41339/161*e - 4628/7, -27/161*e^4 + 4/7*e^3 + 1314/161*e^2 - 2822/161*e - 618/7, -421/161*e^4 + 94/7*e^3 + 12868/161*e^2 - 41796/161*e - 4695/7, 20/161*e^4 - 4/7*e^3 - 544/161*e^2 + 1184/161*e + 198/7, 1/23*e^4 - 41/23*e^2 - 42/23*e - 4, 64/23*e^4 - 14*e^3 - 2026/23*e^2 + 6282/23*e + 734, -14/23*e^4 + 3*e^3 + 413/23*e^2 - 1229/23*e - 131, 14/23*e^4 - 3*e^3 - 436/23*e^2 + 1367/23*e + 146, -321/161*e^4 + 67/7*e^3 + 9987/161*e^2 - 29919/161*e - 3516/7, -139/161*e^4 + 25/7*e^3 + 4457/161*e^2 - 10515/161*e - 1374/7, -142/161*e^4 + 27/7*e^3 + 4442/161*e^2 - 11079/161*e - 1529/7, -162/161*e^4 + 38/7*e^3 + 4986/161*e^2 - 17898/161*e - 1888/7, -44/23*e^4 + 10*e^3 + 1344/23*e^2 - 4408/23*e - 510, 159/161*e^4 - 36/7*e^3 - 5001/161*e^2 + 16368/161*e + 1866/7, 117/161*e^4 - 22/7*e^3 - 3440/161*e^2 + 8472/161*e + 1159/7, -360/161*e^4 + 79/7*e^3 + 11080/161*e^2 - 34836/161*e - 3942/7, 347/161*e^4 - 75/7*e^3 - 10662/161*e^2 + 32392/161*e + 3737/7, -129/161*e^4 + 30/7*e^3 + 4024/161*e^2 - 14109/161*e - 1653/7, 159/161*e^4 - 36/7*e^3 - 5484/161*e^2 + 17656/161*e + 2293/7, 86/161*e^4 - 20/7*e^3 - 2629/161*e^2 + 10050/161*e + 955/7, -250/161*e^4 + 57/7*e^3 + 7766/161*e^2 - 26070/161*e - 2860/7, 432/161*e^4 - 92/7*e^3 - 13618/161*e^2 + 40322/161*e + 4792/7, 382/161*e^4 - 82/7*e^3 - 11936/161*e^2 + 36718/161*e + 4304/7, 439/161*e^4 - 92/7*e^3 - 13744/161*e^2 + 40672/161*e + 4848/7, 65/161*e^4 - 13/7*e^3 - 1929/161*e^2 + 4975/161*e + 836/7, 40/161*e^4 - 8/7*e^3 - 1410/161*e^2 + 3656/161*e + 424/7, -185/161*e^4 + 37/7*e^3 + 5515/161*e^2 - 14977/161*e - 1814/7, 26/161*e^4 - 1/7*e^3 - 836/161*e^2 - 586/161*e + 256/7, -346/161*e^4 + 72/7*e^3 + 10989/161*e^2 - 31560/161*e - 3851/7, 27/23*e^4 - 6*e^3 - 785/23*e^2 + 2500/23*e + 258];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {71,71,w^5 - 3*w^4 - 2*w^3 + 7*w^2 + w}>] := 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;