/* 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, -6, 3, 10, -3, -3, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w - 1], [17, 17, -w^2 + 2*w + 1], [17, 17, -w^3 + w^2 + 4*w], [19, 19, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w - 4], [19, 19, w^2 - w - 1], [37, 37, w^4 - 2*w^3 - 3*w^2 + 3*w + 2], [37, 37, w^4 - 4*w^3 + w^2 + 7*w - 3], [53, 53, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 6], [53, 53, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 2], [64, 2, -2], [71, 71, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 1], [71, 71, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 12*w - 6], [71, 71, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 7], [73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 16*w^2 - 7*w + 3], [73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 6], [89, 89, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 4], [89, 89, w^5 - w^4 - 9*w^3 + 7*w^2 + 16*w - 6], [89, 89, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 3*w - 3], [89, 89, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 10*w - 6], [107, 107, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 12*w - 6], [107, 107, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 9*w - 5], [107, 107, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 6*w - 3], [107, 107, w^4 - 2*w^3 - 3*w^2 + 4*w + 1], [109, 109, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 13*w - 6], [109, 109, -w^4 + 4*w^3 - w^2 - 9*w + 3], [127, 127, w^4 - 3*w^3 + 5*w - 4], [127, 127, -w^5 + 3*w^4 + w^3 - 6*w^2 + w + 3], [127, 127, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 11*w - 6], [127, 127, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 7*w], [163, 163, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 5], [163, 163, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 2*w + 5], [163, 163, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 8*w + 4], [163, 163, w^5 - 4*w^4 + 10*w^2 - w - 4], [179, 179, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 7*w - 1], [179, 179, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 5*w - 7], [179, 179, -2*w^5 + 5*w^4 + 10*w^3 - 19*w^2 - 18*w + 10], [179, 179, -2*w^5 + 5*w^4 + 9*w^3 - 17*w^2 - 14*w + 9], [197, 197, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 3*w - 7], [197, 197, -w^5 + 4*w^4 - w^3 - 7*w^2 + 3*w - 2], [197, 197, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3], [197, 197, -2*w^5 + 8*w^4 - 2*w^3 - 17*w^2 + 10*w + 1], [197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2], [197, 197, w^5 - 4*w^4 + w^3 + 9*w^2 - 4*w - 2], [199, 199, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 10*w + 4], [199, 199, -w^4 + 3*w^3 + 2*w^2 - 9*w + 1], [233, 233, 2*w^5 - 4*w^4 - 12*w^3 + 17*w^2 + 19*w - 9], [233, 233, -w^5 + w^4 + 8*w^3 - 6*w^2 - 13*w + 6], [233, 233, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 9*w - 7], [233, 233, w^5 - 4*w^4 + 11*w^2 - 4*w - 2], [251, 251, -2*w^5 + 5*w^4 + 10*w^3 - 18*w^2 - 18*w + 7], [251, 251, w^3 - w^2 - 3*w - 2], [251, 251, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 3], [251, 251, -w^5 + 4*w^4 - 11*w^2 + 4*w + 3], [269, 269, w^5 - 3*w^4 - w^3 + 6*w^2 - 2*w - 3], [269, 269, 2*w^5 - 7*w^4 - 3*w^3 + 20*w^2 + 3*w - 8], [271, 271, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 7], [271, 271, -w^4 + 4*w^3 - w^2 - 10*w + 6], [289, 17, -w^5 + 3*w^4 + w^3 - 5*w^2 - 2], [289, 17, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 6*w - 4], [307, 307, 2*w^5 - 5*w^4 - 10*w^3 + 19*w^2 + 16*w - 8], [307, 307, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 6*w - 1], [307, 307, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 3*w + 2], [307, 307, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 7*w - 9], [361, 19, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 8*w - 5], [361, 19, w^5 - 3*w^4 - w^3 + 5*w^2 + 3], [379, 379, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 9*w - 5], [379, 379, -2*w + 3], [397, 397, -w^5 + 5*w^4 - 6*w^3 - 6*w^2 + 14*w - 5], [397, 397, -w^5 + 3*w^4 + w^3 - 4*w^2 - w - 5], [397, 397, 3*w^5 - 9*w^4 - 7*w^3 + 23*w^2 + 9*w - 6], [397, 397, -w^4 + 5*w^3 - 2*w^2 - 10*w + 1], [431, 431, -3*w^5 + 8*w^4 + 11*w^3 - 24*w^2 - 18*w + 9], [431, 431, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + w - 3], [431, 431, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 14*w + 8], [431, 431, -2*w^5 + 6*w^4 + 4*w^3 - 14*w^2 - 5*w], [433, 433, -3*w^5 + 9*w^4 + 8*w^3 - 25*w^2 - 11*w + 9], [433, 433, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 3*w + 7], [433, 433, 2*w^5 - 5*w^4 - 7*w^3 + 13*w^2 + 11*w - 4], [433, 433, -w^4 + 4*w^3 + w^2 - 10*w - 2], [449, 449, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 3], [449, 449, -w^4 + 4*w^3 - 2*w^2 - 5*w + 5], [449, 449, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 2], [449, 449, -w^3 + 5*w + 3], [449, 449, w^3 - 5*w], [449, 449, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - w - 6], [467, 467, -2*w^4 + 7*w^3 - 16*w + 7], [467, 467, -2*w^4 + 6*w^3 + w^2 - 10*w + 3], [487, 487, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9], [487, 487, 2*w^5 - 6*w^4 - 5*w^3 + 15*w^2 + 7*w - 3], [487, 487, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 8*w - 6], [487, 487, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 8*w + 1], [503, 503, 2*w^5 - 7*w^4 - 3*w^3 + 19*w^2 + 3*w - 6], [503, 503, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 4*w - 6], [503, 503, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 13*w - 7], [503, 503, -w^5 + 4*w^4 - 10*w^2 + 3*w - 1], [521, 521, w^5 - 4*w^4 + 11*w^2 - w - 6], [521, 521, w^5 - w^4 - 9*w^3 + 8*w^2 + 15*w - 6], [521, 521, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 9*w], [521, 521, -2*w^5 + 4*w^4 + 11*w^3 - 15*w^2 - 17*w + 9], [523, 523, 3*w^5 - 8*w^4 - 10*w^3 + 24*w^2 + 12*w - 10], [523, 523, -w^4 + 3*w^3 - w^2 - 4*w + 5], [523, 523, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 17*w + 7], [523, 523, -w^5 + 5*w^4 - 4*w^3 - 9*w^2 + 9*w + 1], [577, 577, -3*w^5 + 8*w^4 + 10*w^3 - 22*w^2 - 14*w + 7], [577, 577, -2*w^5 + 7*w^4 + w^3 - 15*w^2 + 2*w + 2], [593, 593, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 15*w - 10], [593, 593, w^3 - w^2 - 2*w - 2], [593, 593, -3*w^5 + 9*w^4 + 9*w^3 - 27*w^2 - 14*w + 10], [593, 593, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 8*w - 7], [613, 613, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 5], [613, 613, w^5 - w^4 - 9*w^3 + 8*w^2 + 13*w - 7], [613, 613, w^5 - 5*w^4 + 3*w^3 + 11*w^2 - 6*w - 2], [613, 613, -w^4 + 3*w^3 + w^2 - 5*w + 4], [613, 613, w^5 - w^4 - 9*w^3 + 7*w^2 + 14*w - 4], [613, 613, w^5 - 2*w^4 - 7*w^3 + 11*w^2 + 12*w - 8], [631, 631, 2*w^5 - 7*w^4 + 15*w^2 - 6*w - 2], [631, 631, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 12*w + 10], [631, 631, w^5 - 3*w^4 + w^3 + 3*w^2 - 6*w + 3], [631, 631, -3*w^5 + 8*w^4 + 13*w^3 - 29*w^2 - 21*w + 15], [683, 683, -w^5 + 4*w^4 - w^3 - 8*w^2 + 2*w - 1], [683, 683, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 11*w - 4], [701, 701, -w^5 + 3*w^4 + w^3 - 7*w^2 + 3*w + 2], [701, 701, 2*w^2 - 3*w - 4], [719, 719, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 13*w - 8], [719, 719, w^5 - 5*w^4 + 4*w^3 + 10*w^2 - 10*w - 1], [719, 719, w^5 - w^4 - 9*w^3 + 6*w^2 + 16*w - 2], [719, 719, 2*w^5 - 5*w^4 - 9*w^3 + 18*w^2 + 15*w - 10], [739, 739, -2*w^5 + 9*w^4 - 6*w^3 - 15*w^2 + 16*w - 7], [739, 739, 4*w^5 - 9*w^4 - 19*w^3 + 30*w^2 + 28*w - 14], [757, 757, 2*w^5 - 5*w^4 - 9*w^3 + 16*w^2 + 17*w - 7], [757, 757, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 6*w - 6], [757, 757, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 11*w - 6], [757, 757, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 1], [773, 773, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w], [773, 773, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 12*w - 6], [809, 809, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + 2*w - 3], [809, 809, -2*w^3 + 5*w^2 + 4*w - 6], [811, 811, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3], [811, 811, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 3*w - 6], [827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 4], [827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 5], [829, 829, 2*w^5 - 4*w^4 - 11*w^3 + 15*w^2 + 18*w - 9], [829, 829, 2*w^5 - 6*w^4 - 7*w^3 + 19*w^2 + 13*w - 7], [829, 829, -2*w^5 + 9*w^4 - 5*w^3 - 18*w^2 + 15*w], [829, 829, 3*w^5 - 9*w^4 - 9*w^3 + 28*w^2 + 11*w - 11], [829, 829, w^3 - 2*w^2 - 3*w - 1], [829, 829, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 20*w + 10], [863, 863, -w^4 + 4*w^3 + w^2 - 11*w], [863, 863, w^4 - 4*w^3 - w^2 + 11*w + 1], [919, 919, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 4*w + 1], [919, 919, -w^5 + 4*w^4 - w^3 - 8*w^2 + 6*w - 4], [919, 919, -w^4 + 5*w^3 - w^2 - 12*w + 1], [919, 919, -2*w^5 + 5*w^4 + 7*w^3 - 13*w^2 - 11*w + 6], [937, 937, 2*w^5 - 6*w^4 - 3*w^3 + 12*w^2 + 3*w], [937, 937, 2*w^5 - 4*w^4 - 10*w^3 + 12*w^2 + 16*w - 5], [937, 937, w^5 - 4*w^4 + 8*w^2 + w + 1], [937, 937, 2*w^4 - 8*w^3 + 2*w^2 + 16*w - 5], [953, 953, -w^5 + 4*w^4 - 11*w^2 + 3*w + 7], [953, 953, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 13*w - 7], [953, 953, -w^5 + 3*w^4 + w^3 - 7*w^2 + 4*w + 1], [953, 953, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w - 1], [971, 971, w^4 - w^3 - 5*w^2 + 2*w + 4], [971, 971, -w^4 + w^3 + 4*w^2 - w + 1], [971, 971, -2*w^5 + 5*w^4 + 10*w^3 - 20*w^2 - 15*w + 9], [971, 971, w^4 - 2*w^3 - 4*w^2 + 6*w + 1], [991, 991, -w - 3], [991, 991, -2*w^5 + 7*w^4 + 3*w^3 - 19*w^2 - 5*w + 6], [991, 991, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 5*w + 7], [991, 991, 2*w^5 - 7*w^4 - 2*w^3 + 16*w^2 + 2*w], [1009, 1009, 2*w^4 - 6*w^3 - 2*w^2 + 11*w], [1009, 1009, -w^5 + w^4 + 9*w^3 - 7*w^2 - 15*w + 6], [1061, 1061, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 4*w + 2], [1061, 1061, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + 2*w - 7], [1063, 1063, 2*w^5 - 5*w^4 - 10*w^3 + 18*w^2 + 18*w - 9], [1063, 1063, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 14*w + 8], [1063, 1063, -w^5 + 4*w^4 - 11*w^2 + 4*w + 5], [1063, 1063, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 14*w - 5], [1117, 1117, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 7*w + 5], [1117, 1117, -w^5 + 2*w^4 + 3*w^3 - 2*w^2 - 3*w - 3], [1117, 1117, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 5*w - 8], [1117, 1117, w^5 - 4*w^4 + 3*w^3 + 4*w^2 - 7*w + 4], [1151, 1151, -w^4 + 4*w^3 - 10*w - 1], [1151, 1151, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 3], [1151, 1151, 3*w^5 - 8*w^4 - 8*w^3 + 19*w^2 + 8*w - 3], [1151, 1151, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 12*w - 5], [1171, 1171, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 7*w + 1], [1171, 1171, 3*w^5 - 10*w^4 - 4*w^3 + 25*w^2 + 3*w - 7], [1171, 1171, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 8*w + 3], [1171, 1171, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 7*w + 6], [1187, 1187, -3*w^5 + 8*w^4 + 12*w^3 - 27*w^2 - 18*w + 11], [1187, 1187, w^4 - 2*w^3 - 2*w^2 + 4*w - 3], [1259, 1259, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 5], [1259, 1259, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 9*w - 7], [1277, 1277, -3*w^5 + 11*w^4 + w^3 - 25*w^2 + 2*w + 3], [1277, 1277, -w^4 + w^3 + 5*w^2 - w - 6], [1277, 1277, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w + 1], [1277, 1277, w^4 - 5*w^3 + 3*w^2 + 9*w - 7], [1277, 1277, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 4*w - 1], [1277, 1277, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 4], [1279, 1279, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 18*w + 11], [1279, 1279, -3*w^5 + 11*w^4 - 23*w^2 + 5*w + 2], [1279, 1279, w^3 + w^2 - 5*w - 2], [1279, 1279, -2*w^5 + 6*w^4 + 7*w^3 - 21*w^2 - 9*w + 9], [1279, 1279, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 9*w + 4], [1279, 1279, -2*w^3 + 5*w^2 + 3*w - 8], [1297, 1297, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 19*w + 7], [1297, 1297, -w^5 + w^4 + 10*w^3 - 9*w^2 - 19*w + 7], [1331, 11, -3*w^5 + 7*w^4 + 16*w^3 - 26*w^2 - 28*w + 9], [1331, 11, w^3 - w^2 - w - 3], [1367, 1367, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w], [1367, 1367, w^3 - 6*w], [1367, 1367, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 5], [1367, 1367, w^3 - 6*w - 2], [1369, 37, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 10*w - 8], [1369, 37, 3*w^5 - 9*w^4 - 8*w^3 + 25*w^2 + 10*w - 8], [1423, 1423, 3*w^5 - 10*w^4 - 5*w^3 + 26*w^2 + 6*w - 3], [1423, 1423, -2*w^4 + 7*w^3 - w^2 - 13*w + 4], [1459, 1459, -w^5 + 4*w^4 - 3*w^3 - 4*w^2 + 8*w - 5], [1459, 1459, 2*w^5 - 5*w^4 - 6*w^3 + 11*w^2 + 8*w], [1459, 1459, -2*w^5 + 7*w^4 + 4*w^3 - 22*w^2 - 2*w + 7], [1459, 1459, -3*w^5 + 8*w^4 + 14*w^3 - 29*w^2 - 24*w + 11], [1493, 1493, -3*w^5 + 9*w^4 + 9*w^3 - 25*w^2 - 16*w + 6], [1493, 1493, -2*w^5 + 3*w^4 + 15*w^3 - 14*w^2 - 27*w + 9], [1493, 1493, w^5 - 4*w^4 + 2*w^3 + 5*w^2 - 5*w + 3], [1493, 1493, -3*w^5 + 8*w^4 + 9*w^3 - 21*w^2 - 11*w + 7], [1531, 1531, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + w + 4], [1531, 1531, 2*w^5 - 6*w^4 - 6*w^3 + 17*w^2 + 11*w - 7], [1549, 1549, w^5 - 4*w^4 - 3*w^3 + 14*w^2 + 7*w - 4], [1549, 1549, -4*w^5 + 10*w^4 + 18*w^3 - 34*w^2 - 28*w + 15], [1567, 1567, -3*w^5 + 6*w^4 + 17*w^3 - 23*w^2 - 25*w + 9], [1567, 1567, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 12*w + 4], [1583, 1583, w^5 - 2*w^4 - 8*w^3 + 11*w^2 + 15*w - 7], [1583, 1583, w^5 - 4*w^4 - w^3 + 13*w^2 - w - 7], [1601, 1601, -w^4 + 4*w^3 - 3*w^2 - 6*w + 7], [1601, 1601, 3*w^5 - 8*w^4 - 9*w^3 + 22*w^2 + 10*w - 8], [1619, 1619, 2*w^5 - 5*w^4 - 11*w^3 + 19*w^2 + 21*w - 6], [1619, 1619, w^5 - 2*w^4 - 8*w^3 + 13*w^2 + 12*w - 12], [1619, 1619, -4*w^5 + 9*w^4 + 20*w^3 - 32*w^2 - 31*w + 13], [1619, 1619, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 10*w - 12], [1637, 1637, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 5*w - 3], [1637, 1637, w^5 - 4*w^4 + 2*w^3 + 6*w^2 - 5*w + 4], [1637, 1637, 4*w^5 - 10*w^4 - 17*w^3 + 31*w^2 + 29*w - 12], [1637, 1637, -w^5 + 5*w^4 - 5*w^3 - 8*w^2 + 15*w - 1], [1657, 1657, w^4 - 2*w^3 - w^2 + w - 4], [1657, 1657, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 7], [1693, 1693, w^5 - 2*w^4 - 9*w^3 + 14*w^2 + 15*w - 8], [1693, 1693, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + w], [1709, 1709, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 3], [1709, 1709, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 9], [1709, 1709, -w^5 + w^4 + 11*w^3 - 11*w^2 - 20*w + 9], [1709, 1709, -w^5 + w^4 + 7*w^3 - 3*w^2 - 12*w + 1], [1747, 1747, -w^5 + 5*w^4 - 3*w^3 - 9*w^2 + 7*w - 4], [1747, 1747, -3*w^5 + 11*w^4 - 24*w^2 + 5*w + 3], [1783, 1783, -w^4 + 3*w^3 + 2*w^2 - 5*w - 4], [1783, 1783, -2*w^5 + 7*w^4 + 2*w^3 - 18*w^2 - w + 4], [1871, 1871, -w^5 + 3*w^4 + 4*w^3 - 11*w^2 - 9*w + 6], [1871, 1871, 2*w^5 - 8*w^4 + w^3 + 17*w^2 - 3*w - 2], [1871, 1871, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 6], [1871, 1871, -3*w^5 + 11*w^4 - w^3 - 23*w^2 + 12*w - 3], [1873, 1873, -w^5 + w^4 + 8*w^3 - 5*w^2 - 11*w + 4], [1873, 1873, 3*w^5 - 6*w^4 - 15*w^3 + 19*w^2 + 22*w - 9], [1889, 1889, 2*w^5 - 5*w^4 - 8*w^3 + 14*w^2 + 13*w - 5], [1889, 1889, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 12*w + 8], [1889, 1889, 2*w^5 - 7*w^4 - w^3 + 16*w^2 - 3*w - 5], [1889, 1889, w^5 - 4*w^4 + w^3 + 8*w^2 - 4*w - 3], [1907, 1907, -3*w^5 + 6*w^4 + 18*w^3 - 25*w^2 - 28*w + 12], [1907, 1907, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 8*w - 1], [1979, 1979, 2*w^4 - 9*w^3 + 5*w^2 + 18*w - 11], [1979, 1979, 4*w^5 - 10*w^4 - 14*w^3 + 27*w^2 + 20*w - 8], [1997, 1997, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 10], [1997, 1997, -4*w^5 + 13*w^4 + 5*w^3 - 32*w^2 + w + 4], [1997, 1997, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 4], [1997, 1997, -4*w^5 + 11*w^4 + 15*w^3 - 35*w^2 - 24*w + 11]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, -3, -3, 2, 2, 2, 2, 9, 0, -7, -12, -3, 6, 11, -16, 12, -6, -6, -6, 0, 0, -18, 9, 2, 2, 11, 2, 11, 2, 11, 2, 2, -16, 15, 6, -21, 15, -6, -6, 12, 21, 12, 21, 20, -7, -12, -21, 24, -21, -24, 3, 3, -15, 0, 18, 2, 29, 11, 20, -7, 20, 20, -7, -7, -34, 20, 20, 20, 20, 2, 29, -18, 9, 36, -27, -25, 2, 2, 29, 24, 6, 24, 6, -30, 6, -24, 12, 11, 29, -16, 2, -12, -39, 42, -12, 21, 12, 30, -42, 20, -7, -34, -7, -25, 2, 18, 0, 9, 45, -16, -43, 2, 2, 29, -25, -16, 38, -7, 20, -42, -24, 0, -18, -30, 51, -12, 33, 38, 11, -7, -16, 47, -16, 15, 24, -45, 18, 47, -7, 6, -12, 2, 11, -16, -25, 2, -25, -9, 0, -25, -25, -52, 56, 2, -7, -52, 47, 57, 39, -51, 57, 0, -36, -36, 36, -61, 2, 47, 56, 29, 56, -6, -6, -61, -7, -34, 47, 20, -34, 20, -61, 15, 15, 24, 6, -52, 56, 56, -52, 45, -18, 15, -12, -33, 3, 57, 39, -42, -42, 2, 29, 29, 11, -16, 2, 2, -25, 30, -6, 24, -3, 60, -66, 2, 2, 38, -16, 20, 47, 38, -16, -6, -6, 66, 12, 29, 29, 20, -34, -25, 2, 24, -21, 66, -78, -45, 45, -36, 45, -30, 51, 6, -12, -70, 38, -70, -70, -60, -69, 39, 66, 20, 47, -25, 56, 30, 57, 84, 39, -34, 20, -54, -54, -36, 18, 24, -84, -42, 75, -36, 27, -18, -18]; 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;