/* 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 := PolynomialRing(Rationals()); g := P![1, 2, -3, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, w^3 - 2*w^2 - 2*w + 1], [7, 7, -w^3 + 3*w^2 + w - 3], [7, 7, -w^2 + w + 2], [17, 17, -w^3 + 3*w^2 - 3], [17, 17, -w^3 + w^2 + 4*w], [25, 5, -w^3 + 3*w^2 + 2*w - 2], [25, 5, -2*w^3 + 4*w^2 + 5*w - 1], [41, 41, -w^3 + 2*w^2 + 4*w - 2], [47, 47, -2*w^3 + 5*w^2 + 4*w - 4], [47, 47, 2*w^3 - 4*w^2 - 5*w], [49, 7, w^2 - 4*w - 1], [71, 71, 2*w - 3], [71, 71, -w^3 + w^2 + 6*w - 2], [73, 73, -w^3 + 3*w^2 + 3*w - 5], [73, 73, 2*w^3 - 5*w^2 - 5*w + 4], [73, 73, -w^3 + 3*w^2 - 5], [73, 73, w^3 - w^2 - 4*w + 2], [79, 79, 2*w^3 - 3*w^2 - 6*w + 2], [79, 79, 2*w^3 - 3*w^2 - 5*w], [81, 3, -3], [89, 89, -w^3 + w^2 + 5*w - 3], [89, 89, w - 4], [97, 97, -3*w^3 + 7*w^2 + 6*w - 5], [97, 97, -2*w^3 + 3*w^2 + 7*w - 2], [103, 103, -w - 3], [103, 103, -w^3 + 2*w^2 + 3*w - 5], [103, 103, -w^3 + 4*w^2 - w - 5], [103, 103, -3*w^3 + 6*w^2 + 7*w - 3], [113, 113, w^3 - 4*w^2 + w + 4], [113, 113, 2*w^2 - 3*w - 4], [113, 113, 3*w^3 - 6*w^2 - 7*w + 4], [113, 113, -2*w^3 + 6*w^2 + 2*w - 7], [137, 137, -w^3 + 4*w^2 - 4], [137, 137, w^3 - 4*w^2 + 6], [151, 151, -3*w^3 + 8*w^2 + 3*w - 4], [151, 151, -2*w^3 + 2*w^2 + 7*w + 4], [167, 167, -2*w^3 + 3*w^2 + 6*w - 3], [167, 167, -2*w^3 + 5*w^2 + 2*w - 6], [191, 191, w^3 - 2*w^2 - w + 5], [191, 191, -w^3 + 5*w^2 - 3*w - 3], [193, 193, -w^3 + 2*w^2 + 5*w - 2], [193, 193, -2*w^3 + 4*w^2 + 7*w - 4], [199, 199, -2*w^3 + 5*w^2 + 4*w - 2], [199, 199, w^2 - 5], [223, 223, -w^3 + 8*w - 2], [223, 223, -2*w^3 + 2*w^2 + 9*w], [223, 223, -3*w^3 + 5*w^2 + 11*w - 1], [223, 223, -w^3 + 4*w^2 + w - 8], [233, 233, -3*w^3 + 7*w^2 + 6*w - 4], [233, 233, w^3 - w^2 - 2*w - 3], [239, 239, -w^3 + 6*w], [239, 239, 2*w^3 - 5*w^2 - 4*w + 1], [257, 257, -3*w^3 + 7*w^2 + 7*w - 5], [257, 257, 3*w^3 - 7*w^2 - 5*w + 1], [263, 263, w^3 - w^2 - 4*w - 5], [263, 263, -2*w^3 + 3*w^2 + 9*w - 4], [281, 281, -w^3 + 3*w^2 + 4*w - 6], [281, 281, 3*w^3 - 7*w^2 - 8*w + 5], [289, 17, -3*w^3 + 6*w^2 + 6*w - 4], [311, 311, -3*w^3 + 5*w^2 + 11*w - 3], [311, 311, w - 5], [313, 313, -2*w^3 + 6*w^2 + w - 9], [313, 313, -w^3 + 3*w^2 + 3*w - 7], [359, 359, 3*w^3 - 6*w^2 - 8*w], [359, 359, -w^2 + 3*w - 4], [359, 359, w^3 - w^2 - 5*w - 5], [359, 359, w^3 - 2*w^2 - 4], [383, 383, -4*w^3 + 8*w^2 + 9*w - 3], [383, 383, -w^3 + 5*w^2 - 3*w - 7], [401, 401, -3*w^3 + 8*w^2 + 3*w - 9], [401, 401, -3*w^3 + 7*w^2 + 6*w - 3], [401, 401, w^3 - w^2 - 2*w - 4], [401, 401, -2*w^3 + 2*w^2 + 7*w - 1], [439, 439, -4*w^3 + 9*w^2 + 7*w - 6], [439, 439, 3*w^3 - 5*w^2 - 7*w + 1], [449, 449, -4*w^3 + 9*w^2 + 9*w - 6], [449, 449, 2*w^3 - 2*w^2 - 11*w - 2], [449, 449, -w^3 + 5*w^2 - 3*w - 11], [449, 449, -3*w^3 + 5*w^2 + 10*w - 5], [457, 457, -2*w^3 + 2*w^2 + 9*w - 1], [457, 457, w^3 - 4*w^2 + 3*w + 5], [463, 463, 3*w^2 - 5*w - 6], [463, 463, -w^3 + 5*w^2 - 3*w - 5], [479, 479, -2*w^3 + 3*w^2 + 9*w - 2], [479, 479, -w^3 + w^2 + 7*w - 1], [487, 487, -2*w^2 + 5*w + 6], [487, 487, -w^3 + 7*w - 2], [487, 487, -w^3 + 6*w - 2], [487, 487, -w^3 + 4*w^2 - 2*w - 8], [503, 503, w^3 - 4*w - 6], [503, 503, -3*w^3 + 8*w^2 + 4*w - 4], [521, 521, 3*w^2 - 4*w - 7], [521, 521, 2*w^3 - 7*w^2 + 6], [529, 23, -w^3 + 2*w^2 + 2*w - 6], [529, 23, w^3 - 2*w^2 - 2*w - 4], [569, 569, 2*w^3 - 7*w^2 - 2*w + 7], [569, 569, 4*w^3 - 7*w^2 - 10*w + 4], [569, 569, 3*w^3 - 4*w^2 - 13*w], [569, 569, 4*w^3 - 8*w^2 - 11*w + 3], [577, 577, 4*w^2 - 7*w - 10], [577, 577, w^3 + w^2 - 6*w - 9], [593, 593, w^3 - 4*w^2 + 2*w - 2], [593, 593, 3*w^3 - 10*w^2 + w + 11], [601, 601, -2*w^3 + 4*w^2 + 8*w - 3], [601, 601, -2*w^3 + 4*w^2 + 8*w - 5], [617, 617, 5*w^3 - 12*w^2 - 11*w + 11], [617, 617, 3*w^2 - 4*w - 5], [617, 617, -2*w^3 + 7*w^2 - 8], [617, 617, 5*w^3 - 10*w^2 - 13*w + 11], [631, 631, -3*w^2 + 8*w + 5], [631, 631, 2*w^3 - 7*w^2 - w + 10], [631, 631, 5*w^3 - 11*w^2 - 10*w + 9], [631, 631, 2*w^3 - w^2 - 12*w], [641, 641, -3*w^3 + 8*w^2 + 5*w - 5], [641, 641, 2*w^2 - w - 7], [647, 647, -2*w^3 + 2*w^2 + 11*w - 1], [647, 647, 2*w^3 - 2*w^2 - 11*w - 1], [647, 647, -3*w^3 + 9*w^2 + 4*w - 10], [647, 647, w^3 - 5*w^2 + 6], [673, 673, -5*w^3 + 11*w^2 + 12*w - 8], [673, 673, 3*w^3 - 5*w^2 - 10*w + 6], [719, 719, -3*w^3 + 4*w^2 + 10*w - 2], [719, 719, -3*w^3 + 8*w^2 + 2*w - 8], [727, 727, -w^3 + 5*w^2 - 2*w - 6], [727, 727, -w^3 + 5*w^2 - 2*w - 7], [743, 743, -5*w^3 + 10*w^2 + 13*w - 4], [743, 743, -4*w^3 + 9*w^2 + 10*w - 6], [743, 743, 2*w^3 - 2*w^2 - 10*w + 3], [743, 743, w^2 + 2*w - 5], [751, 751, -w^3 + 4*w^2 - 3*w - 6], [751, 751, 2*w^3 - 2*w^2 - 9*w + 2], [761, 761, 4*w^3 - 9*w^2 - 6*w + 4], [761, 761, 3*w^3 - 3*w^2 - 12*w - 1], [761, 761, 3*w^3 - 9*w^2 + 8], [761, 761, 4*w^3 - 7*w^2 - 10*w + 1], [769, 769, 4*w^3 - 7*w^2 - 10*w + 3], [769, 769, -4*w^3 + 7*w^2 + 14*w - 5], [769, 769, 4*w^3 - 9*w^2 - 6*w + 6], [769, 769, -w^2 + 6*w], [809, 809, -2*w^3 + 2*w^2 + 7*w - 3], [809, 809, -3*w^3 + 8*w^2 + 3*w - 11], [823, 823, 3*w^3 - 8*w^2 - 7*w + 6], [823, 823, -2*w^3 + 6*w^2 + 5*w - 10], [839, 839, -3*w^3 + 5*w^2 + 12*w - 1], [839, 839, -w^3 + w^2 + 8*w - 4], [887, 887, -4*w^3 + 9*w^2 + 5*w - 6], [887, 887, -5*w^3 + 9*w^2 + 13*w - 5], [919, 919, -4*w^3 + 6*w^2 + 14*w - 3], [919, 919, -5*w^3 + 12*w^2 + 9*w - 8], [929, 929, -2*w^3 + 3*w^2 + 10*w - 5], [929, 929, -2*w^3 + 3*w^2 + 10*w], [937, 937, -3*w^3 + 7*w^2 + 8*w - 3], [937, 937, -w^3 + 3*w^2 + 4*w - 8], [961, 31, -4*w^3 + 8*w^2 + 8*w - 5], [961, 31, -4*w^3 + 8*w^2 + 8*w - 3], [967, 967, 4*w^3 - 8*w^2 - 7*w + 1], [967, 967, -5*w^3 + 10*w^2 + 11*w - 3], [977, 977, 3*w^3 - 8*w^2 - w + 7], [977, 977, -4*w^3 + 6*w^2 + 13*w - 3], [991, 991, w^2 - w - 8], [991, 991, -3*w^3 + 5*w^2 + 12*w - 5]]; primes := [ideal : I in primesArray]; heckePol := x^5 - 4*x^4 - 8*x^3 + 38*x^2 - 9*x - 14; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 1/4*e^4 - 3/4*e^3 - 9/4*e^2 + 25/4*e + 1/2, 2, -3/4*e^4 + 3/4*e^3 + 31/4*e^2 - 21/4*e - 5/2, 3/4*e^4 - 1/4*e^3 - 31/4*e^2 - 5/4*e + 11/2, 3/2*e^4 - 3/2*e^3 - 31/2*e^2 + 21/2*e + 7, -e^4 + 11*e^2 + 2*e - 6, -2*e^4 + 2*e^3 + 20*e^2 - 14*e - 6, -1/2*e^4 - 1/2*e^3 + 13/2*e^2 + 11/2*e - 7, 1/4*e^4 + 3/4*e^3 - 13/4*e^2 - 41/4*e + 11/2, -2*e^4 + 3*e^3 + 20*e^2 - 23*e - 6, -1, -9/4*e^4 + 5/4*e^3 + 93/4*e^2 - 35/4*e - 3/2, 5/4*e^4 - 1/4*e^3 - 49/4*e^2 - 5/4*e + 11/2, -9/4*e^4 + 11/4*e^3 + 85/4*e^2 - 73/4*e - 5/2, -7/4*e^4 + 7/4*e^3 + 75/4*e^2 - 53/4*e - 29/2, 11/4*e^4 - 5/4*e^3 - 115/4*e^2 + 15/4*e + 31/2, 1/4*e^4 + 5/4*e^3 - 17/4*e^2 - 47/4*e + 9/2, 1/4*e^4 - 5/4*e^3 - 13/4*e^2 + 51/4*e + 15/2, 2*e^4 - e^3 - 22*e^2 + 5*e + 16, -2*e^4 + 3*e^3 + 20*e^2 - 25*e - 4, -3/2*e^4 - 1/2*e^3 + 31/2*e^2 + 19/2*e - 5, 9/4*e^4 - 15/4*e^3 - 89/4*e^2 + 125/4*e + 9/2, e^4 - e^3 - 9*e^2 + 5*e - 4, -2*e^2 + 16, 2*e^4 - 2*e^3 - 20*e^2 + 12*e + 6, 3/2*e^4 - 3/2*e^3 - 35/2*e^2 + 25/2*e + 17, -1/4*e^4 + 5/4*e^3 + 5/4*e^2 - 35/4*e + 17/2, e^3 - 9*e - 2, e^4 - 2*e^3 - 9*e^2 + 18*e - 6, 3/4*e^4 + 1/4*e^3 - 39/4*e^2 - 3/4*e + 29/2, 13/4*e^4 - 5/4*e^3 - 137/4*e^2 + 19/4*e + 27/2, 3/4*e^4 - 3/4*e^3 - 31/4*e^2 + 29/4*e - 3/2, 11/4*e^4 - 15/4*e^3 - 111/4*e^2 + 109/4*e + 13/2, -e^4 + 2*e^3 + 11*e^2 - 16*e - 4, 5/2*e^4 - 1/2*e^3 - 49/2*e^2 - 9/2*e + 1, 2*e^4 - e^3 - 20*e^2 + 3*e + 8, -5/4*e^4 + 5/4*e^3 + 49/4*e^2 - 23/4*e - 19/2, 1/4*e^4 + 1/4*e^3 - 9/4*e^2 - 11/4*e + 5/2, 7/4*e^4 - 9/4*e^3 - 59/4*e^2 + 67/4*e - 29/2, -3/2*e^4 + 3/2*e^3 + 27/2*e^2 - 21/2*e + 5, -3/4*e^4 + 3/4*e^3 + 31/4*e^2 - 37/4*e + 3/2, -e^3 + 9*e - 8, -e^4 + 4*e^3 + 9*e^2 - 34*e, 2*e^4 - 2*e^3 - 22*e^2 + 18*e + 20, -9/4*e^4 + 9/4*e^3 + 101/4*e^2 - 59/4*e - 55/2, -19/4*e^4 + 11/4*e^3 + 191/4*e^2 - 49/4*e - 17/2, -11/4*e^4 + 5/4*e^3 + 111/4*e^2 - 15/4*e - 15/2, 3/4*e^4 - 7/4*e^3 - 39/4*e^2 + 45/4*e + 37/2, -5/4*e^4 - 1/4*e^3 + 49/4*e^2 + 35/4*e - 5/2, 3/2*e^4 - 3/2*e^3 - 39/2*e^2 + 21/2*e + 33, -9/4*e^4 + 15/4*e^3 + 93/4*e^2 - 141/4*e - 21/2, 9/4*e^4 + 9/4*e^3 - 105/4*e^2 - 91/4*e + 37/2, 1/2*e^4 + 5/2*e^3 - 9/2*e^2 - 55/2*e + 3, -2*e^4 + 2*e^3 + 22*e^2 - 14*e - 20, -5/4*e^4 - 5/4*e^3 + 57/4*e^2 + 63/4*e - 9/2, -e^3 + 4*e^2 + 9*e - 20, -3/4*e^4 - 7/4*e^3 + 47/4*e^2 + 53/4*e - 47/2, -4*e^4 + 4*e^3 + 38*e^2 - 26*e - 6, 13/4*e^4 - 17/4*e^3 - 129/4*e^2 + 131/4*e + 19/2, 5*e^4 - 4*e^3 - 55*e^2 + 28*e + 44, 2*e^3 - 2*e^2 - 18*e + 6, -7/4*e^4 - 9/4*e^3 + 83/4*e^2 + 119/4*e - 33/2, 1/2*e^4 - 7/2*e^3 - 5/2*e^2 + 69/2*e - 3, -e^4 + 4*e^3 + 9*e^2 - 38*e - 6, -19/4*e^4 + 13/4*e^3 + 191/4*e^2 - 63/4*e - 39/2, -11/4*e^4 + 1/4*e^3 + 115/4*e^2 + 29/4*e - 19/2, 11/2*e^4 - 9/2*e^3 - 115/2*e^2 + 59/2*e + 29, -3*e^4 + 4*e^3 + 33*e^2 - 30*e - 32, -4*e^4 + 7*e^3 + 36*e^2 - 53*e, 3*e^4 - 5*e^3 - 31*e^2 + 35*e + 28, 3*e^4 - 29*e^2 - 6*e - 10, -9/2*e^4 + 7/2*e^3 + 97/2*e^2 - 37/2*e - 35, 1/4*e^4 + 7/4*e^3 + 3/4*e^2 - 77/4*e - 33/2, -11/4*e^4 + 13/4*e^3 + 119/4*e^2 - 111/4*e - 63/2, 3/2*e^4 - 5/2*e^3 - 23/2*e^2 + 27/2*e - 15, 9/4*e^4 - 1/4*e^3 - 85/4*e^2 - 13/4*e - 5/2, -e^4 - e^3 + 9*e^2 + 13*e + 4, -2*e^3 - 2*e^2 + 24*e + 6, 7*e^4 - 7*e^3 - 71*e^2 + 53*e + 10, -9/4*e^4 + 1/4*e^3 + 101/4*e^2 - 27/4*e - 43/2, -11/4*e^4 + 11/4*e^3 + 119/4*e^2 - 89/4*e - 65/2, -19/4*e^4 + 29/4*e^3 + 183/4*e^2 - 215/4*e - 39/2, 29/4*e^4 - 9/4*e^3 - 305/4*e^2 - 9/4*e + 99/2, -5/2*e^4 + 9/2*e^3 + 49/2*e^2 - 83/2*e + 7, -3/2*e^4 - 1/2*e^3 + 35/2*e^2 + 27/2*e - 25, -3/4*e^4 + 5/4*e^3 + 39/4*e^2 - 63/4*e - 55/2, -1/2*e^4 + 1/2*e^3 - 3/2*e^2 - 3/2*e + 33, 1/4*e^4 - 7/4*e^3 - 5/4*e^2 + 53/4*e - 19/2, 7/2*e^4 - 3/2*e^3 - 79/2*e^2 + 5/2*e + 37, -1/2*e^4 + 3/2*e^3 + 1/2*e^2 - 5/2*e + 35, -2*e^4 + e^3 + 22*e^2 - 15*e - 16, -1/2*e^4 + 7/2*e^3 + 5/2*e^2 - 53/2*e - 7, -15/4*e^4 - 7/4*e^3 + 159/4*e^2 + 101/4*e - 27/2, -9/4*e^4 + 1/4*e^3 + 101/4*e^2 + 41/4*e - 39/2, -7/4*e^4 - 7/4*e^3 + 79/4*e^2 + 133/4*e - 39/2, 2*e^4 - 2*e^3 - 22*e^2 + 12*e + 24, -11/4*e^4 + 9/4*e^3 + 103/4*e^2 - 67/4*e + 57/2, -3*e^4 + 4*e^3 + 33*e^2 - 38*e - 16, -3/2*e^4 + 1/2*e^3 + 35/2*e^2 - 19/2*e - 11, -3*e^4 + 3*e^3 + 31*e^2 - 23*e + 2, -5/4*e^4 - 1/4*e^3 + 57/4*e^2 + 51/4*e - 37/2, -3*e^4 + e^3 + 29*e^2 + 5*e - 2, -7/4*e^4 - 3/4*e^3 + 63/4*e^2 + 81/4*e + 33/2, -7/2*e^4 + 9/2*e^3 + 67/2*e^2 - 47/2*e - 11, -7*e^4 + e^3 + 71*e^2 + 15*e - 24, 8*e^4 - 10*e^3 - 82*e^2 + 76*e + 38, 13/4*e^4 - 19/4*e^3 - 125/4*e^2 + 137/4*e - 35/2, 11/4*e^4 - 1/4*e^3 - 135/4*e^2 + 3/4*e + 111/2, -25/4*e^4 + 17/4*e^3 + 229/4*e^2 - 83/4*e + 49/2, 11/2*e^4 - 9/2*e^3 - 103/2*e^2 + 39/2*e - 9, -5/2*e^4 + 7/2*e^3 + 49/2*e^2 - 49/2*e - 41, -5/4*e^4 + 17/4*e^3 + 57/4*e^2 - 175/4*e - 47/2, -1/4*e^4 - 21/4*e^3 + 33/4*e^2 + 183/4*e - 37/2, e^4 + 2*e^3 - 13*e^2 - 22*e + 22, 11/2*e^4 - 7/2*e^3 - 119/2*e^2 + 41/2*e + 35, e^4 - e^3 - 11*e^2 + e + 28, -e^4 + 5*e^3 + 9*e^2 - 49*e + 2, 1/4*e^4 + 9/4*e^3 - 5/4*e^2 - 107/4*e - 11/2, -e^4 - 3*e^3 + 13*e^2 + 23*e - 26, 4*e^4 - 3*e^3 - 42*e^2 + 23*e + 24, 21/4*e^4 - 9/4*e^3 - 241/4*e^2 + 55/4*e + 115/2, e^4 - 2*e^3 - 9*e^2 + 20*e - 28, 3/4*e^4 - 15/4*e^3 - 23/4*e^2 + 105/4*e - 19/2, 7/4*e^4 - 9/4*e^3 - 67/4*e^2 + 83/4*e + 11/2, -3/4*e^4 + 21/4*e^3 + 19/4*e^2 - 183/4*e - 23/2, 31/4*e^4 - 31/4*e^3 - 315/4*e^2 + 221/4*e + 69/2, -5/2*e^4 + 3/2*e^3 + 49/2*e^2 - 29/2*e + 7, 2*e^3 - 4*e^2 - 28*e + 20, -7/2*e^4 + 17/2*e^3 + 71/2*e^2 - 127/2*e - 25, -9/2*e^4 + 9/2*e^3 + 93/2*e^2 - 59/2*e - 23, 13/4*e^4 - 27/4*e^3 - 125/4*e^2 + 209/4*e + 33/2, -3/4*e^4 + 19/4*e^3 + 23/4*e^2 - 193/4*e + 23/2, 5*e^4 - 6*e^3 - 47*e^2 + 42*e - 6, -11/4*e^4 + 9/4*e^3 + 127/4*e^2 - 51/4*e - 67/2, 9/2*e^4 - 1/2*e^3 - 93/2*e^2 - 17/2*e + 29, 17/2*e^4 - 21/2*e^3 - 173/2*e^2 + 163/2*e + 31, -11/2*e^4 + 5/2*e^3 + 123/2*e^2 - 27/2*e - 35, -3/4*e^4 - 7/4*e^3 + 43/4*e^2 + 45/4*e - 3/2, 27/4*e^4 - 49/4*e^3 - 247/4*e^2 + 387/4*e + 27/2, -31/4*e^4 + 45/4*e^3 + 303/4*e^2 - 351/4*e - 55/2, -5/2*e^4 + 5/2*e^3 + 61/2*e^2 - 31/2*e - 51, -9/4*e^4 + 13/4*e^3 + 77/4*e^2 - 91/4*e - 3/2, -11/4*e^4 + 21/4*e^3 + 115/4*e^2 - 135/4*e - 31/2, 3*e^4 - e^3 - 27*e^2 - e - 20, 3*e^4 - e^3 - 29*e^2 + 5*e - 18, -31/4*e^4 + 19/4*e^3 + 331/4*e^2 - 97/4*e - 105/2, 13/4*e^4 - 5/4*e^3 - 129/4*e^2 + 23/4*e + 23/2, 19/4*e^4 - 13/4*e^3 - 195/4*e^2 + 31/4*e + 71/2, 17/4*e^4 - 15/4*e^3 - 185/4*e^2 + 141/4*e + 45/2, -3/2*e^4 + 7/2*e^3 + 27/2*e^2 - 57/2*e + 11, -5/2*e^4 + 17/2*e^3 + 37/2*e^2 - 163/2*e + 15, 19/4*e^4 - 35/4*e^3 - 199/4*e^2 + 285/4*e + 45/2, 4*e^4 - 6*e^3 - 38*e^2 + 38*e - 16, -1/4*e^4 - 11/4*e^3 + 13/4*e^2 + 113/4*e - 47/2, -1/2*e^4 + 5/2*e^3 + 17/2*e^2 - 59/2*e - 27, 15/2*e^4 - 13/2*e^3 - 159/2*e^2 + 83/2*e + 79, -7/2*e^4 + 1/2*e^3 + 71/2*e^2 + 9/2*e - 21, 1/2*e^4 + 9/2*e^3 - 13/2*e^2 - 107/2*e + 17, -21/4*e^4 + 27/4*e^3 + 193/4*e^2 - 233/4*e + 31/2, -2*e^4 - 3*e^3 + 24*e^2 + 35*e - 6]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 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;