/* 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![3, -4, -8, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w], [3, 3, w^3 - 2*w^2 - 6*w + 1], [9, 3, w^3 - 2*w^2 - 5*w + 1], [16, 2, 2], [17, 17, w^3 - w^2 - 8*w - 5], [17, 17, -2*w^3 + 4*w^2 + 11*w - 2], [17, 17, -w^3 + 3*w^2 + 2*w - 2], [17, 17, w^3 - 2*w^2 - 4*w + 1], [25, 5, w^3 - 3*w^2 - 3*w + 1], [25, 5, w^2 - 2*w - 7], [29, 29, w^3 - 2*w^2 - 6*w - 1], [29, 29, w - 2], [53, 53, w^3 - 3*w^2 - 4*w + 4], [53, 53, -w^3 + 3*w^2 + 4*w - 5], [61, 61, -w^3 + 2*w^2 + 7*w - 4], [61, 61, -4*w^3 + 11*w^2 + 14*w - 8], [101, 101, -w^3 + 2*w^2 + 7*w - 1], [103, 103, -w^3 + 3*w^2 + 2*w - 5], [103, 103, -w^3 + w^2 + 8*w + 2], [113, 113, w^3 - 3*w^2 - 3*w + 7], [113, 113, w^2 - 2*w - 1], [127, 127, w^3 - 2*w^2 - 4*w - 2], [127, 127, -2*w^3 + 4*w^2 + 11*w + 1], [131, 131, -3*w^3 + 7*w^2 + 14*w - 8], [131, 131, -w^2 + w + 8], [131, 131, 2*w^3 - 5*w^2 - 9*w + 1], [131, 131, -3*w^3 + 6*w^2 + 16*w + 1], [139, 139, -2*w^3 + 5*w^2 + 9*w - 2], [139, 139, -w^2 + w + 7], [157, 157, -2*w^3 + 5*w^2 + 7*w - 5], [157, 157, -2*w^3 + 3*w^2 + 13*w + 2], [157, 157, -2*w^3 + 5*w^2 + 6*w - 2], [157, 157, -3*w^3 + 5*w^2 + 19*w + 4], [169, 13, -2*w^3 + 4*w^2 + 10*w - 1], [173, 173, 3*w^3 - 8*w^2 - 10*w + 7], [173, 173, 2*w^3 - 2*w^2 - 15*w - 8], [179, 179, -3*w^2 + 6*w + 17], [179, 179, -2*w^3 + 3*w^2 + 12*w + 1], [179, 179, -3*w^3 + 7*w^2 + 13*w - 7], [179, 179, -2*w^3 + 3*w^2 + 14*w + 4], [191, 191, w^3 - 3*w^2 - w + 4], [191, 191, -2*w^3 + 3*w^2 + 14*w + 2], [199, 199, -w^3 + 8*w + 10], [199, 199, 5*w^3 - 15*w^2 - 14*w + 14], [211, 211, -w - 4], [211, 211, -w^3 + 2*w^2 + 6*w - 5], [211, 211, 2*w^3 - 3*w^2 - 12*w - 4], [211, 211, -3*w^3 + 7*w^2 + 13*w - 4], [257, 257, -3*w^3 + 8*w^2 + 10*w - 2], [257, 257, 2*w^3 - 2*w^2 - 16*w - 7], [263, 263, 3*w^3 - 6*w^2 - 17*w + 1], [263, 263, w^3 - 2*w^2 - 3*w - 1], [269, 269, 5*w^3 - 13*w^2 - 19*w + 7], [269, 269, -3*w^3 + 6*w^2 + 18*w - 1], [277, 277, w^3 - 4*w^2 - w + 5], [277, 277, -w^3 + 4*w^2 + w - 11], [283, 283, -w^3 + 5*w^2 - w - 14], [283, 283, 2*w^3 - 7*w^2 - 4*w + 10], [283, 283, -w^3 + w^2 + 9*w + 1], [283, 283, w^2 - 4*w - 5], [311, 311, -w^3 + w^2 + 9*w + 5], [311, 311, w^2 - 4*w - 1], [313, 313, -4*w^3 + 9*w^2 + 19*w - 10], [313, 313, -2*w^2 + 5*w + 11], [337, 337, -3*w^3 + 6*w^2 + 14*w + 2], [337, 337, -4*w^3 + 8*w^2 + 21*w + 1], [347, 347, -2*w^3 + 6*w^2 + 7*w - 10], [347, 347, -w^3 + 4*w^2 + 2*w - 7], [361, 19, -4*w^3 + 11*w^2 + 13*w - 10], [361, 19, -2*w^3 + 5*w^2 + 8*w - 10], [373, 373, 2*w^3 - 5*w^2 - 10*w + 2], [373, 373, -w^3 + 2*w^2 + 8*w - 2], [373, 373, -2*w^3 + 4*w^2 + 13*w - 1], [373, 373, -w^3 + 3*w^2 + 5*w - 8], [389, 389, -w^3 + 4*w^2 + 2*w - 10], [389, 389, -2*w^3 + 6*w^2 + 7*w - 7], [419, 419, -2*w^3 + 5*w^2 + 6*w - 1], [419, 419, -3*w^3 + 5*w^2 + 19*w + 5], [433, 433, w^2 - 4*w - 2], [433, 433, w^3 - w^2 - 9*w - 4], [439, 439, -w^3 + 4*w^2 + w - 2], [439, 439, w^3 - 4*w^2 - w + 14], [443, 443, -2*w^3 + 5*w^2 + 9*w + 1], [443, 443, -w^2 + w + 10], [467, 467, w^3 - w^2 - 8*w + 4], [467, 467, w^3 - 4*w^2 + 2], [491, 491, -4*w^3 + 12*w^2 + 10*w - 11], [491, 491, 2*w^3 - w^2 - 18*w - 11], [491, 491, 3*w^3 - 9*w^2 - 7*w + 11], [491, 491, 2*w^3 - 20*w - 19], [503, 503, w^3 - 3*w^2 - 4*w + 10], [503, 503, -w^3 + w^2 + 10*w - 2], [523, 523, -4*w^3 + 9*w^2 + 22*w - 7], [523, 523, w^3 - 3*w^2 - 7*w + 5], [529, 23, -2*w^3 + 4*w^2 + 10*w + 5], [529, 23, -2*w^3 + 4*w^2 + 10*w - 7], [547, 547, 2*w^3 - w^2 - 18*w - 14], [547, 547, 3*w^3 - 8*w^2 - 14*w + 8], [571, 571, 5*w^3 - 11*w^2 - 25*w + 5], [571, 571, -4*w^3 + 9*w^2 + 19*w - 11], [599, 599, -5*w^3 + 12*w^2 + 20*w - 4], [599, 599, 4*w^3 - 6*w^2 - 25*w - 11], [601, 601, -4*w^3 + 9*w^2 + 20*w - 5], [601, 601, w^3 - 9*w - 14], [601, 601, 3*w^3 - 8*w^2 - 11*w + 2], [601, 601, -2*w^3 + 3*w^2 + 14*w - 1], [641, 641, -2*w^3 + 3*w^2 + 15*w - 5], [641, 641, -4*w^3 + 8*w^2 + 23*w - 2], [647, 647, -3*w^3 + 8*w^2 + 13*w - 11], [647, 647, -w^3 + 4*w^2 + 3*w - 7], [659, 659, 3*w^3 - 7*w^2 - 14*w - 2], [659, 659, w^3 - w^2 - 6*w - 11], [673, 673, -3*w^3 + 7*w^2 + 11*w - 1], [673, 673, -4*w^3 + 7*w^2 + 24*w + 5], [677, 677, 5*w^3 - 12*w^2 - 21*w + 11], [677, 677, -2*w^3 + 6*w^2 + 5*w - 1], [677, 677, -2*w^3 + 6*w^2 + 8*w - 7], [677, 677, 2*w^3 - 6*w^2 - 8*w + 11], [719, 719, -2*w^3 + 6*w^2 + 10*w - 7], [719, 719, -4*w^3 + 10*w^2 + 20*w - 13], [727, 727, 3*w^3 - 10*w^2 - 8*w + 10], [727, 727, 3*w^3 - 8*w^2 - 13*w + 10], [727, 727, 2*w^3 - 8*w^2 - 3*w + 23], [727, 727, -w^3 + 4*w^2 + 3*w - 8], [751, 751, -7*w^3 + 18*w^2 + 25*w - 7], [751, 751, w^3 - 7*w^2 + 5*w + 26], [757, 757, -4*w^3 + 9*w^2 + 18*w - 7], [757, 757, -3*w^3 + 5*w^2 + 17*w + 1], [797, 797, 2*w^2 - 4*w - 5], [797, 797, 2*w^3 - 6*w^2 - 6*w + 11], [823, 823, -2*w^3 + 2*w^2 + 15*w + 14], [823, 823, -3*w^3 + 8*w^2 + 10*w - 1], [829, 829, -6*w^3 + 16*w^2 + 22*w - 7], [829, 829, 5*w^3 - 13*w^2 - 20*w + 8], [829, 829, -w^3 + 7*w + 4], [829, 829, -5*w^3 + 12*w^2 + 23*w - 14], [841, 29, -3*w^3 + 6*w^2 + 15*w - 2], [859, 859, -4*w^3 + 7*w^2 + 24*w - 1], [859, 859, -5*w^3 + 15*w^2 + 14*w - 13], [881, 881, 3*w^3 - 7*w^2 - 16*w + 4], [881, 881, -w^3 + 3*w^2 + 6*w - 7], [883, 883, -2*w^3 + 3*w^2 + 12*w + 10], [883, 883, -3*w^3 + 7*w^2 + 13*w + 2], [887, 887, 4*w^3 - 11*w^2 - 11*w + 11], [887, 887, 2*w^3 - 7*w^2 - 3*w + 7], [887, 887, 4*w^3 - 5*w^2 - 29*w - 10], [887, 887, 3*w^2 - 7*w - 16], [907, 907, -w^3 + 3*w^2 + 8*w - 5], [907, 907, -5*w^3 + 11*w^2 + 28*w - 8], [911, 911, -4*w^3 + 10*w^2 + 18*w - 17], [911, 911, 2*w^2 - 2*w - 1], [919, 919, 4*w^3 - 7*w^2 - 25*w - 4], [919, 919, 2*w^3 - 5*w^2 - 5*w + 1], [937, 937, -w^3 + 3*w^2 + w - 8], [937, 937, -2*w^3 + 3*w^2 + 14*w - 2], [953, 953, -2*w^3 + w^2 + 17*w + 19], [953, 953, 4*w^3 - 11*w^2 - 13*w + 4], [961, 31, 2*w^3 - 3*w^2 - 11*w - 8], [961, 31, w^3 - 4*w^2 + 2*w + 11], [971, 971, 2*w^2 - 5*w - 4], [971, 971, -w^3 + 4*w^2 - 11], [991, 991, -2*w^3 + 6*w^2 + 9*w - 8], [991, 991, -2*w^3 + 3*w^2 + 13*w - 1], [991, 991, 3*w^3 - 8*w^2 - 14*w + 11], [991, 991, -2*w^3 + 5*w^2 + 7*w - 8], [997, 997, -2*w^3 + 5*w^2 + 6*w + 1], [997, 997, -3*w^3 + 5*w^2 + 19*w + 7], [997, 997, -6*w^3 + 12*w^2 + 33*w - 4], [997, 997, 4*w^3 - 11*w^2 - 15*w + 11]]; primes := [ideal : I in primesArray]; heckePol := x^4 + 2*x^3 - 8*x^2 - 12*x + 6; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 1/3*e^3 + 1/3*e^2 - 3*e - 2, -1, 2/3*e^3 + 2/3*e^2 - 4*e - 3, e^2 + e - 4, -1/3*e^3 + 2/3*e^2 + 4*e - 2, 1/3*e^3 - 2/3*e^2 - 3*e + 4, 1/3*e^3 - 2/3*e^2 - 4*e + 4, -e^3 - e^2 + 6*e + 2, -e^3 - e^2 + 6*e + 2, -1/3*e^3 + 2/3*e^2 + 4*e, 1/3*e^3 - 2/3*e^2 - 4*e + 6, 4/3*e^3 + 1/3*e^2 - 12*e - 2, e^2 + 4*e - 4, 1/3*e^3 + 1/3*e^2 - 2*e + 4, 1/3*e^3 + 1/3*e^2 - 2*e + 4, -2/3*e^3 - 2/3*e^2 + 4*e + 6, 5/3*e^3 + 11/3*e^2 - 14*e - 14, 1/3*e^3 - 5/3*e^2 + 2*e + 14, 4/3*e^3 + 4/3*e^2 - 11*e - 4, 1/3*e^3 + 1/3*e^2 + e + 2, e^3 - 3*e^2 - 8*e + 20, 1/3*e^3 + 13/3*e^2 - 16, 1/3*e^3 + 4/3*e^2 - 4*e + 2, e^3 - e^2 - 8*e, 1/3*e^3 + 7/3*e^2 - 16, -1/3*e^3 - 4/3*e^2 + 4*e + 16, -5/3*e^3 - 2/3*e^2 + 10*e + 6, -5/3*e^3 - 8/3*e^2 + 10*e + 16, 4*e + 2, 4/3*e^3 + 4/3*e^2 - 12*e - 6, 2/3*e^3 + 2/3*e^2 - 6*e + 10, 2*e + 14, -4/3*e^3 - 4/3*e^2 + 8*e - 6, -2*e^3 - 4*e^2 + 14*e + 16, -4/3*e^3 + 2/3*e^2 + 6*e - 8, -7/3*e^3 - 4/3*e^2 + 14*e - 2, -1/3*e^3 + 5/3*e^2 - 2*e - 20, -5/3*e^3 - 11/3*e^2 + 14*e + 8, -7/3*e^3 - 10/3*e^2 + 14*e + 8, -4/3*e^3 + 2/3*e^2 + 12*e, -2*e^2 - 4*e + 12, 4/3*e^3 + 1/3*e^2 - 13*e + 8, -1/3*e^3 + 2/3*e^2 + 7*e + 8, -1/3*e^3 + 5/3*e^2 + 6*e + 4, e^3 - e^2 - 10*e + 16, e^2 + 9*e - 8, 3*e^3 + 2*e^2 - 27*e - 16, 4/3*e^3 + 4/3*e^2 - 14*e - 16, -2/3*e^3 - 2/3*e^2 + 10*e - 4, -2*e^2 + 20, 2*e^2, 4/3*e^3 + 7/3*e^2 - 11*e - 20, 1/3*e^3 - 2/3*e^2 + e - 4, 8/3*e^3 - 4/3*e^2 - 22*e + 4, 2/3*e^3 + 14/3*e^2 + 2*e - 24, 7/3*e^3 + 4/3*e^2 - 17*e - 8, 4/3*e^3 + 7/3*e^2 - 5*e - 12, -4/3*e^3 + 2/3*e^2 + 16*e + 2, 4/3*e^3 - 2/3*e^2 - 16*e + 6, -8/3*e^3 + 4/3*e^2 + 24*e - 6, -4*e^2 - 8*e + 18, 5/3*e^3 + 8/3*e^2 - 7*e - 4, 8/3*e^3 + 5/3*e^2 - 19*e, -2/3*e^3 - 2/3*e^2 + 13*e + 12, 7/3*e^3 + 7/3*e^2 - 23*e - 6, -3*e^3 - 2*e^2 + 21*e + 8, -2*e^3 - 3*e^2 + 9*e + 12, 11/3*e^3 + 2/3*e^2 - 20*e + 12, 13/3*e^3 + 22/3*e^2 - 28*e - 22, -2*e^3 + 17*e - 4, 1/3*e^3 - 8/3*e^2 - 5*e + 8, -2/3*e^3 + 7/3*e^2 + 7*e - 16, -1/3*e^3 - 7/3*e^2 - 3*e + 6, 1/3*e^3 - 23/3*e^2 - 9*e + 34, -2*e^3 + 6*e^2 + 19*e - 32, -2*e^2 + 2*e - 8, 2/3*e^3 + 8/3*e^2 - 6*e - 32, -2/3*e^3 - 2/3*e^2 - 20, -2*e^3 - 2*e^2 + 16*e - 12, -1/3*e^3 - 1/3*e^2 - 2*e + 2, -5/3*e^3 - 5/3*e^2 + 14*e + 10, -1/3*e^3 - 7/3*e^2 - 2*e + 10, -5/3*e^3 + 1/3*e^2 + 14*e - 2, -5/3*e^3 - 5/3*e^2 + 14*e + 18, -1/3*e^3 - 1/3*e^2 - 2*e + 10, 2/3*e^3 - 1/3*e^2 - 8*e + 20, -5/3*e^3 - 11/3*e^2 + 7*e - 2, -8/3*e^3 - 2/3*e^2 + 19*e - 16, -2/3*e^3 + 1/3*e^2 + 8*e + 18, 1/3*e^3 + 7/3*e^2 - 4*e + 14, -1/3*e^3 - 7/3*e^2 + 4*e + 38, 2/3*e^3 - 10/3*e^2 - 4*e + 40, 2/3*e^3 + 14/3*e^2 - 4*e, 14/3*e^3 + 14/3*e^2 - 28*e - 20, -2/3*e^3 - 2/3*e^2 + 4*e + 10, 5/3*e^3 + 17/3*e^2 - 2*e - 14, 13/3*e^3 + 1/3*e^2 - 34*e + 10, -13/3*e^3 - 7/3*e^2 + 26*e + 10, -13/3*e^3 - 19/3*e^2 + 26*e + 30, 10/3*e^3 + 10/3*e^2 - 30*e - 10, 10*e + 10, -3*e^3 + 25*e - 16, -11/3*e^3 - 17/3*e^2 + 21*e + 30, -4*e^3 - 2*e^2 + 25*e + 12, -2/3*e^3 - 11/3*e^2 - 3*e, -4*e^3 - 3*e^2 + 35*e + 28, -1/3*e^3 - 4/3*e^2 - 9*e + 16, 3*e^3 + 6*e^2 - 24*e - 18, e^3 - 2*e^2 + 24, 17/3*e^3 + 29/3*e^2 - 37*e - 30, 14/3*e^3 + 2/3*e^2 - 25*e + 16, -11/3*e^3 + 10/3*e^2 + 30*e - 14, -e^3 - 8*e^2 - 2*e + 40, e^3 + 5*e^2 - 8*e - 30, 1/3*e^3 - 11/3*e^2 + 14, -8/3*e^3 - 2/3*e^2 + 12*e - 12, -4*e^3 - 6*e^2 + 28*e + 16, 8/3*e^3 - 10/3*e^2 - 15*e + 32, 3*e^3 + 9*e^2 - 19*e - 30, 4*e^3 + 4*e^2 - 21*e + 8, 5*e^3 + 3*e^2 - 38*e - 16, 5*e^3 + 5*e^2 - 33*e + 2, 7/3*e^3 + 13/3*e^2 - 6*e - 20, -e^3 - 4*e^2 + 16*e + 36, 7/3*e^3 + 16/3*e^2 - 24*e - 14, -4/3*e^3 - 16/3*e^2 + 20, -4*e^3 + 32*e - 4, -e^3 - 7*e^2 + 5*e + 14, -4/3*e^3 + 14/3*e^2 + 9*e - 44, -11*e^2 - 12*e + 52, -4*e^3 + 7*e^2 + 36*e - 34, -14/3*e^3 - 14/3*e^2 + 38*e + 8, -4/3*e^3 - 4/3*e^2 - 2*e - 12, 6*e^3 + 2*e^2 - 44*e - 20, 10/3*e^3 + 22/3*e^2 - 12*e - 44, -4/3*e^3 - 4/3*e^2 + 8*e + 14, 13/3*e^3 + 13/3*e^2 - 40*e - 22, -1/3*e^3 - 1/3*e^2 + 16*e + 6, 8/3*e^3 + 2/3*e^2 - 20*e - 2, 4/3*e^3 + 10/3*e^2 - 4*e - 14, -11/3*e^3 + 7/3*e^2 + 32*e - 18, -1/3*e^3 - 19/3*e^2 - 8*e + 22, -e^3 - 5*e^2 - 10*e + 30, 1/3*e^3 - 2/3*e^2 - 5*e + 12, -19/3*e^3 - 7/3*e^2 + 54*e + 22, -2/3*e^3 + 1/3*e^2 + 7*e + 8, -19/3*e^3 - 7/3*e^2 + 39*e + 2, -6*e^3 - 10*e^2 + 35*e + 40, -4/3*e^3 - 4/3*e^2 + 12*e - 16, -4*e - 24, 7/3*e^3 - 11/3*e^2 - 18*e + 30, e^3 + 7*e^2 - 2*e - 22, 6*e^3 + 6*e^2 - 50*e - 34, 4/3*e^3 + 4/3*e^2 + 6*e - 6, 5*e^3 - e^2 - 46*e - 12, -1/3*e^3 + 17/3*e^2 + 18*e - 40, -2*e^3 + 26*e + 22, 8/3*e^3 + 2/3*e^2 - 30*e + 14, -20/3*e^3 - 23/3*e^2 + 48*e + 26, -4*e^3 - 3*e^2 + 16*e, 2*e^3 + 4*e^2 - 22*e - 14, 1/3*e^3 + 4/3*e^2 + 10*e + 12, -4/3*e^3 - 10/3*e^2 + 18*e + 26, 13/3*e^3 + 10/3*e^2 - 38*e - 2, -13/3*e^3 - 1/3*e^2 + 38*e + 8, -1/3*e^3 - 13/3*e^2 - 10*e + 24, 8/3*e^3 + 20/3*e^2 - 14*e - 12, 10/3*e^3 - 2/3*e^2 - 22*e + 24]; 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;