/* 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, 3, 3, -4, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [11, 11, w^4 + w^3 - 4*w^2 - 3*w + 2], [23, 23, -w^4 + 3*w^2 + 1], [23, 23, -w^4 + 3*w^2 + w - 2], [23, 23, w^4 - w^3 - 3*w^2 + 3*w + 2], [23, 23, -w^4 + w^3 + 4*w^2 - 3*w - 1], [23, 23, -w^2 + w + 3], [32, 2, 2], [43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1], [43, 43, -w^4 + 2*w^2 + w + 1], [43, 43, w^3 + w^2 - 4*w - 2], [43, 43, 2*w^4 - w^3 - 7*w^2 + 3*w + 3], [43, 43, w^4 - w^3 - 4*w^2 + 4*w + 2], [67, 67, 2*w^4 - 7*w^2 + 2], [67, 67, w^4 - 2*w^3 - 3*w^2 + 6*w + 2], [67, 67, 2*w^4 - 7*w^2 - w + 4], [67, 67, w^4 - 2*w^3 - 4*w^2 + 6*w + 2], [67, 67, -w^4 + w^3 + 5*w^2 - 3*w - 3], [89, 89, w^3 + w^2 - 4*w - 1], [89, 89, -2*w^4 + w^3 + 7*w^2 - 3*w - 2], [89, 89, -w^4 + w^3 + 4*w^2 - 4*w - 3], [89, 89, -w^4 + 2*w^2 + w + 2], [89, 89, 2*w^4 - w^3 - 6*w^2 + 2*w + 2], [109, 109, -w^3 + 2*w^2 + 3*w - 3], [109, 109, w^4 - 4*w^2 - 2*w + 3], [109, 109, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3], [109, 109, 2*w^3 - 5*w - 1], [109, 109, -w^4 - w^3 + 5*w^2 + 2*w - 4], [131, 131, w^4 - 3*w^3 - 2*w^2 + 7*w], [131, 131, -w^4 + 2*w^3 + 5*w^2 - 7*w - 5], [131, 131, 2*w^4 - 2*w^3 - 6*w^2 + 3*w + 2], [131, 131, w^4 - 2*w^3 - 3*w^2 + 7*w], [131, 131, 2*w^4 + w^3 - 8*w^2 - 3*w + 4], [197, 197, 3*w^4 - w^3 - 10*w^2 + w + 5], [197, 197, 2*w^4 - 7*w^2 + w + 1], [197, 197, -3*w^4 + 2*w^3 + 10*w^2 - 5*w - 5], [197, 197, -2*w^4 - w^3 + 9*w^2 + 2*w - 5], [197, 197, -3*w^4 + 3*w^3 + 10*w^2 - 6*w - 5], [199, 199, w^4 + 2*w^3 - 5*w^2 - 5*w + 3], [199, 199, -2*w^4 + w^3 + 7*w^2 - 4*w - 2], [199, 199, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3], [199, 199, 2*w^4 - 8*w^2 + w + 4], [199, 199, 3*w^4 - w^3 - 10*w^2 + 2*w + 4], [241, 241, w^4 - 2*w^3 - w^2 + 3*w - 3], [241, 241, -w^4 + 3*w^3 + 2*w^2 - 9*w], [241, 241, 3*w^4 - 11*w^2 - 2*w + 6], [241, 241, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4], [241, 241, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6], [243, 3, -3], [263, 263, -w^4 + 4*w^2 + w + 1], [263, 263, 2*w^4 - w^3 - 8*w^2 + w + 2], [263, 263, 3*w^4 - 3*w^3 - 9*w^2 + 7*w + 2], [263, 263, -2*w^3 + w^2 + 4*w - 4], [263, 263, -3*w^2 + w + 5], [307, 307, 2*w^4 - w^3 - 6*w^2 + 3*w - 2], [307, 307, w^4 - 2*w^2 - 2*w - 4], [307, 307, -3*w^4 + 2*w^3 + 11*w^2 - 3*w - 6], [307, 307, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 6], [307, 307, -w^4 + 3*w^3 + 4*w^2 - 7*w - 3], [331, 331, -w^4 - 2*w^3 + 3*w^2 + 7*w - 1], [331, 331, -w^4 + 3*w^3 + 4*w^2 - 8*w - 4], [331, 331, w^4 - w^2 - 4], [331, 331, 3*w^4 - 2*w^3 - 9*w^2 + 5*w + 2], [331, 331, -3*w^4 + 2*w^3 + 11*w^2 - 6*w - 5], [353, 353, -2*w^4 + w^3 + 9*w^2 - 3*w - 5], [353, 353, -2*w^4 - w^3 + 7*w^2 + 4*w - 4], [353, 353, -w^4 + 3*w^3 + 3*w^2 - 9*w - 3], [353, 353, -3*w^4 + 11*w^2 + w - 7], [353, 353, w^4 - w^3 - 6*w^2 + 3*w + 4], [373, 373, w^3 + 2*w^2 - 5*w - 3], [373, 373, 2*w^4 - 3*w^3 - 6*w^2 + 6*w], [373, 373, -w^4 + w^3 + 3*w^2 - 3*w + 3], [373, 373, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 5], [373, 373, -w^4 + w^3 + 4*w^2 - 3*w - 6], [397, 397, -w^4 - w^3 + 6*w^2 + 3*w - 5], [397, 397, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 3], [397, 397, -w^4 + 5*w^2 + 2*w - 5], [397, 397, w^4 + 2*w^3 - 5*w^2 - 5*w + 4], [397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w - 2], [419, 419, 3*w^3 - w^2 - 8*w], [419, 419, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 1], [419, 419, w^4 - 2*w^3 - 5*w^2 + 7*w + 4], [419, 419, -2*w^4 - w^3 + 9*w^2 + 3*w - 7], [419, 419, -w^4 + 3*w^3 + w^2 - 8*w + 1], [439, 439, -w^4 + w^3 + 5*w^2 - 5*w - 5], [439, 439, 2*w^4 - 9*w^2 - 2*w + 8], [439, 439, -3*w^4 + 2*w^3 + 11*w^2 - 5*w - 3], [439, 439, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 4], [439, 439, -w^4 + 3*w^3 + 3*w^2 - 6*w - 3], [461, 461, -w^4 + 5*w^2 + 2*w - 6], [461, 461, -w^4 + 3*w^2 - w + 3], [461, 461, -w^4 - 2*w^3 + 5*w^2 + 5*w - 5], [461, 461, -2*w^4 + 7*w^2 + w - 6], [461, 461, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3], [463, 463, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 3], [463, 463, w^4 + w^3 - 3*w^2 - 2*w - 2], [463, 463, 2*w^4 - 7*w^2 - 2*w + 5], [463, 463, w^4 - 2*w^3 - 2*w^2 + 6*w + 1], [463, 463, -w^3 + 2*w^2 + w - 5], [571, 571, w^4 - w^3 - 3*w^2 + 2*w - 3], [571, 571, 3*w^4 - 2*w^3 - 10*w^2 + 6*w + 2], [571, 571, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 6], [571, 571, 2*w^4 - 5*w^2 - 2*w - 2], [571, 571, 3*w^4 - w^3 - 9*w^2 + 2*w + 3], [593, 593, 3*w^4 - w^3 - 9*w^2 + w + 3], [593, 593, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3], [593, 593, -2*w^4 + 2*w^3 + 7*w^2 - 7*w - 4], [593, 593, w^4 - w^2 - 2*w - 4], [593, 593, 3*w^4 - w^3 - 10*w^2 + 3*w + 2], [617, 617, -w^4 - w^3 + 6*w^2 + 4*w - 5], [617, 617, 3*w^3 - w^2 - 7*w], [617, 617, 2*w^4 - w^3 - 8*w^2 + 6], [617, 617, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 3], [617, 617, -w^4 - 2*w^3 + 6*w^2 + 5*w - 6], [659, 659, -w^4 + w^3 + 4*w^2 - 5*w - 4], [659, 659, 2*w^3 + w^2 - 7*w], [659, 659, -3*w^4 + 2*w^3 + 9*w^2 - 4*w - 4], [659, 659, 3*w^4 - w^3 - 11*w^2 + 3*w + 3], [659, 659, w^4 - w^2 - w - 5], [661, 661, -5*w^4 + 2*w^3 + 18*w^2 - 4*w - 9], [661, 661, -w^4 + 3*w^3 + 6*w^2 - 9*w - 8], [661, 661, w^4 + 2*w^3 - 2*w^2 - 7*w - 2], [661, 661, -w^4 + w^3 + 5*w^2 - 2*w - 9], [661, 661, -3*w^4 + 2*w^3 + 12*w^2 - 2*w - 8], [683, 683, -w^4 + w^3 + 4*w^2 - 5*w - 3], [683, 683, 2*w^3 + w^2 - 7*w - 1], [683, 683, w^4 - w^2 - w - 4], [683, 683, 3*w^4 - 2*w^3 - 9*w^2 + 4*w + 3], [683, 683, 3*w^4 - w^3 - 11*w^2 + 3*w + 4], [727, 727, -5*w^4 + 3*w^3 + 19*w^2 - 7*w - 9], [727, 727, -2*w^4 - 2*w^3 + 7*w^2 + 9*w - 4], [727, 727, 3*w^4 + 2*w^3 - 13*w^2 - 6*w + 6], [727, 727, w^4 - 2*w^2 + 3*w - 4], [727, 727, -2*w^4 + 5*w^3 + 6*w^2 - 14*w - 4], [769, 769, -2*w^4 - w^3 + 9*w^2 + 2*w - 7], [769, 769, -w^4 + 4*w^2 + 3*w - 4], [769, 769, -w^3 + 3*w^2 + 3*w - 4], [769, 769, 3*w^4 - 3*w^3 - 10*w^2 + 6*w + 3], [769, 769, 3*w^3 - 8*w - 2], [857, 857, w^4 - 5*w^3 - 2*w^2 + 14*w - 1], [857, 857, 4*w^4 - w^3 - 13*w^2 + 4], [857, 857, -2*w^4 + 4*w^3 + 6*w^2 - 10*w - 5], [857, 857, 4*w^4 + w^3 - 14*w^2 - 5*w + 6], [857, 857, 4*w^4 + w^3 - 16*w^2 - 4*w + 8], [859, 859, 3*w^4 - w^3 - 11*w^2 + 5*w + 5], [859, 859, -2*w^4 + 5*w^3 + 6*w^2 - 13*w - 1], [859, 859, -2*w^4 - 2*w^3 + 5*w^2 + 7*w + 1], [859, 859, -w^3 + w^2 + 6*w + 1], [859, 859, 4*w^4 - w^3 - 13*w^2 + 3*w + 4], [881, 881, 2*w^4 + w^3 - 11*w^2 - 3*w + 11], [881, 881, w^4 - 6*w^2 - 3*w + 5], [881, 881, -2*w^4 + 5*w^3 + 6*w^2 - 12*w - 1], [881, 881, -2*w^4 - 3*w^3 + 9*w^2 + 8*w - 4], [881, 881, 3*w^4 + w^3 - 12*w^2 - 3*w + 7], [947, 947, 2*w^4 + 3*w^3 - 7*w^2 - 8*w], [947, 947, 2*w^4 - 3*w^3 - 6*w^2 + 4*w], [947, 947, 2*w^4 + 2*w^3 - 7*w^2 - 8*w + 4], [947, 947, -w^4 + 2*w^3 - 7*w + 3], [947, 947, -4*w^4 + 5*w^3 + 14*w^2 - 12*w - 5], [967, 967, 2*w^4 - w^3 - 9*w^2 + 3*w + 4], [967, 967, -2*w^4 - w^3 + 7*w^2 + 4*w - 5], [967, 967, w^4 - 2*w^2 + w - 4], [967, 967, w^4 - w^3 - 2*w^2 - 3], [967, 967, -2*w^4 + 3*w^3 + 6*w^2 - 8*w - 4], [991, 991, -w^4 + 4*w^3 + 2*w^2 - 12*w - 1], [991, 991, w^4 + 3*w^3 - 3*w^2 - 11*w - 1], [991, 991, 2*w^3 - 2*w^2 - 5*w + 8], [991, 991, -2*w^4 + 3*w^3 + 4*w^2 - 5*w + 3], [991, 991, 4*w^4 - 4*w^3 - 11*w^2 + 9*w]]; primes := [ideal : I in primesArray]; heckePol := x^3 - 16*x - 16; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1/4*e^2, 1/2*e^2 - 8, 1/4*e^2 - 5/2*e - 2, -e^2 + 3*e + 12, -1/4*e^2 + 1/2*e + 4, 5/4*e^2 - 2*e - 13, 1/4*e^2 - 1/2*e + 4, -1/2*e^2 + e + 4, -1/2*e^2 - e + 8, -e^2 + 8, 3/4*e^2 - 4, -1/2*e^2 + e + 8, 3/4*e^2 + 1/2*e - 6, 1/2*e^2 - 2*e, 7/4*e^2 - 1/2*e - 18, 3/4*e^2 - 7/2*e - 8, 1, -1/4*e^2 + 1/2*e, 1/2*e^2 - 3*e - 6, -5/4*e^2 + 1/2*e + 14, -e^2 + 14, -9/4*e^2 + e + 22, -7/4*e^2 + 7/2*e + 20, -3/4*e^2 - e + 10, 2*e + 6, 2*e^2 - 2*e - 14, e, 3/2*e^2 - 5*e - 24, -1/4*e^2 + 3/2*e + 10, -1/4*e^2 - 2*e + 12, -1/2*e^2 + 3*e + 8, 3/4*e^2 + e - 2, -7/4*e^2 + 11/2*e + 14, 2*e^2 - 4*e - 22, -1/2*e^2 + 2*e + 6, 1/2*e^2 + 6, 3/2*e^2 - 4*e - 8, 11/4*e^2 - 7/2*e - 34, e - 12, -2*e^2 + 3*e + 12, 5/4*e^2 - 4*e - 32, e^2 - 5*e - 22, -e^2 - 3*e + 22, -7/2*e^2 + 6*e + 34, 7/4*e^2 - 5*e - 30, 3/4*e^2 + 7/2*e - 8, -11/4*e^2 + 9/2*e + 16, 5/4*e^2 + 7/2*e - 18, -9/4*e^2 + 6*e + 36, -7*e + 8, -11/4*e^2 + 3/2*e + 32, -15/4*e^2 + 9/2*e + 40, -3/4*e^2 + 1/2*e, 7/4*e^2 - 11/2*e - 38, 1/2*e^2 - 4*e - 16, 9/4*e^2 - 4*e - 8, 7/4*e^2 - 5/2*e - 24, 3/4*e^2 - 2*e - 32, -3*e - 12, -13/4*e^2 + 6*e + 28, 3*e^2 - 7*e - 36, 5/2*e^2 - e - 36, -e^2 + e - 14, 5/4*e^2 + 3*e - 22, -1/4*e^2 + 1/2*e + 14, -5/2*e^2 + 9*e + 46, -13/4*e^2 + 5*e + 42, 1/4*e^2 + 7/2*e - 14, 9/4*e^2 + 5/2*e - 30, -1/4*e^2 - 3*e + 6, -11/4*e^2 + 15/2*e + 30, -3/4*e^2 - 3/2*e + 20, 7/2*e^2 - 9*e - 38, 1/4*e^2 + 9/2*e - 2, 2*e^2 - 5*e - 10, -7/4*e^2 + e + 18, 3*e^2 - e - 42, -2*e^2 + 2*e + 24, 3/2*e^2 + 5*e - 16, -7/4*e^2 - 1/2*e + 28, 3/4*e^2 + 1/2*e + 16, 1/2*e^2 - 7*e - 8, -11/4*e^2 + 8*e + 20, -7/2*e^2 + 6*e + 52, 3/2*e^2 - 2*e + 8, 5/4*e^2 + 5/2*e - 30, 15/4*e^2 - 17/2*e - 38, e^2 - 6*e - 10, -e^2 - e - 2, 3/2*e^2 - 4*e - 6, -5/4*e^2 + 19/2*e + 8, e^2 - e - 10, -3*e^2 + 11*e + 44, -7/2*e^2 + e + 36, -4*e^2 + 4*e + 32, 13/4*e^2 - 6*e - 52, 17/4*e^2 - 11/2*e - 32, 2*e^2 - 2*e - 12, 15/4*e^2 - 15/2*e - 58, 11/4*e^2 - 23/2*e - 36, -7/4*e^2 + 1/2*e - 2, -3/2*e^2 - 3*e + 24, 9/2*e^2 - 7*e - 50, -3*e^2 + 8*e + 30, 8*e - 2, -11/4*e^2 + 9/2*e + 46, 11/4*e^2 + 7/2*e - 34, 7/4*e^2 + 3/2*e - 12, -7/4*e^2 - 3/2*e + 48, 9*e - 10, 5/4*e^2 - 11*e - 10, 7/4*e^2 - 11/2*e - 8, 5/2*e^2 - 4*e - 12, -7/4*e^2 - 1/2*e + 18, 13/4*e^2 - 2*e - 24, -1/2*e^2 + 12, -5*e^2 + 15*e + 60, 7/4*e^2 - 15/2*e - 40, 4*e^2 + 3*e - 50, 5/4*e^2 - 17/2*e - 26, -15/4*e^2 + 7*e + 38, 2*e^2 - 10*e - 26, -3/2*e^2 + 11*e + 8, -36, -7/4*e^2 - 3/2*e + 34, -3/2*e^2 - 2*e + 24, 3*e^2 - 48, 1/2*e^2 + e - 36, -3/2*e^2 - e - 4, 11/4*e^2 - 5/2*e - 48, 1/2*e^2 + 5*e - 24, -e^2 + 6*e + 20, 7/2*e^2 + 5*e - 54, 5/2*e^2 - 5*e - 30, -e^2 + 8*e - 2, -2*e^2 + e + 6, 11/4*e^2 - 1/2*e - 20, 5/2*e^2 - 5*e - 22, 2*e^2 + 3*e - 6, 3*e^2 + 3*e - 30, 3/2*e^2 + 3*e - 14, -3/4*e^2 + 1/2*e - 20, -3/2*e^2 + 2*e + 52, -1/2*e^2 + 5*e - 8, -21/4*e^2 + 29/2*e + 52, 2*e^2 + e, 5/2*e^2 - 12*e - 36, -3/2*e^2 + 8*e + 42, -13/2*e^2 + 8*e + 62, 13/4*e^2 + 7/2*e - 28, 2*e^2 - 6*e - 54, -e^2 - 3*e - 18, 4*e - 20, -19/4*e^2 + 8*e + 36, 9/4*e^2 - 6*e - 64, -e^2 + e, -13/4*e^2 - 3/2*e + 34, -4*e^2 + 10*e + 44, 19/4*e^2 - 9/2*e - 52, 9/2*e^2 - 12*e - 36, 2*e^2 + 8*e - 16, -7/2*e^2 - e + 24, e^2 - 2*e + 12, 7/4*e^2 - 2*e - 36, -2*e^2 + 2*e + 36, 4*e + 24, -3*e^2 - 6*e + 36]; 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;