/* 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![25, 5, -11, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7], [5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4], [11, 11, w + 1], [11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2], [16, 2, 2], [19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1], [19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w], [19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w], [19, 19, w - 1], [29, 29, w^2 - w - 8], [29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2], [31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2], [31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3], [61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5], [61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15], [61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3], [61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13], [71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4], [71, 71, w^2 - 8], [79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22], [79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9], [81, 3, -3], [101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15], [101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16], [121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1], [131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8], [131, 131, w^2 - 2*w - 2], [149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5], [149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6], [151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3], [151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8], [151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2], [151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9], [179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6], [179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6], [211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10], [211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2], [229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11], [229, 229, 2*w^2 - 2*w - 9], [229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2], [229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3], [239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19], [239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3], [241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6], [241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1], [251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12], [251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15], [269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2], [269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9], [271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w], [271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2], [281, 281, w^3 - 3*w^2 - 4*w + 13], [281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6], [281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16], [281, 281, w^3 + w^2 - 8*w - 9], [289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5], [289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8], [311, 311, -w^2 + 2*w + 11], [311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1], [311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1], [311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10], [349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16], [349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7], [359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1], [359, 359, -w^3 + 3*w^2 + 5*w - 12], [359, 359, w^3 - w^2 - 7*w + 2], [359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11], [379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9], [379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7], [389, 389, w^2 - 3*w - 8], [389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16], [401, 401, -3*w^2 + 2*w + 23], [401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11], [409, 409, -w^3 + 2*w^2 + 3*w - 7], [409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2], [419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2], [419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w], [421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3], [421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3], [439, 439, -w^3 + 3*w^2 + 5*w - 16], [439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7], [439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11], [439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10], [449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5], [449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15], [461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4], [461, 461, 2*w^2 - 2*w - 11], [461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4], [479, 479, -w^3 + 3*w^2 + 6*w - 14], [479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3], [499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6], [499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17], [509, 509, -w^3 + 2*w^2 + 4*w - 6], [509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4], [521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7], [521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8], [529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15], [529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21], [541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10], [541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13], [541, 541, -2*w^2 + w + 13], [541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2], [569, 569, w^3 - 2*w^2 - 5*w + 7], [569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6], [571, 571, w^2 + w - 9], [571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12], [571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11], [571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4], [601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11], [601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1], [631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30], [631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15], [661, 661, -2*w^3 + 17*w + 12], [661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1], [691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35], [691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19], [701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6], [701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8], [709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18], [709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20], [709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5], [709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10], [719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12], [719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1], [719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14], [719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8], [739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5], [739, 739, w - 6], [761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11], [761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w], [769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8], [769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15], [769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18], [769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5], [809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14], [809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4], [809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5], [809, 809, w^3 - w^2 - 5*w + 4], [821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30], [821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15], [829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10], [829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19], [829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10], [829, 829, -w^3 + 4*w^2 + 5*w - 23], [839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4], [839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8], [841, 29, -w^3 + w^2 + 6*w - 4], [859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9], [859, 859, w^3 - 6*w - 3], [881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1], [881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14], [881, 881, -w^3 + 6*w + 2], [881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6], [911, 911, w^3 - w^2 - 5*w + 1], [911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2], [919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7], [919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5], [929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16], [929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7], [961, 31, w^3 - w^2 - 6*w + 2], [971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6], [971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w], [971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7], [971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10], [991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7], [991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]]; primes := [ideal : I in primesArray]; heckePol := x^4 - 6*x^2 + 3*x + 3; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1, -e^3 + 5*e - 1, e^3 + 2*e^2 - 4*e - 5, -e^3 - 3*e^2 + 4*e + 6, -e^3 + 3*e - 3, -e^3 - e^2 + 3*e - 2, 2*e^3 + e^2 - 9*e + 2, 1, -e^3 - e^2 + 6*e + 1, -3*e - 3, 2*e^3 + 3*e^2 - 5*e - 6, -3*e^3 - 5*e^2 + 11*e + 8, 3*e^3 + 3*e^2 - 18*e - 3, -e^3 - 3*e^2 + 3*e + 10, 4*e^3 + 2*e^2 - 19*e + 1, 4*e^3 + 3*e^2 - 16*e - 2, 2*e^3 + 2*e^2 - 9*e + 4, e^3 + e^2 - 2, e^3 - 10*e + 6, -5*e^3 - 6*e^2 + 18*e + 9, e^3 + 4*e^2 - 3*e - 11, 2*e^3 - 2*e^2 - 12*e + 7, 4*e^3 + 5*e^2 - 17*e - 13, -2*e^3 + e^2 + 16*e - 5, 2*e^3 + 5*e^2 - 5*e - 13, 3*e^3 + 3*e^2 - 10*e, -3*e^3 - e^2 + 14*e - 8, -5*e^3 - 9*e^2 + 23*e + 12, e^3 - 2*e^2 - 12*e + 14, 7*e^3 + 10*e^2 - 24*e - 14, e^2 + 2*e - 22, 6*e^3 + 9*e^2 - 24*e - 5, 4*e^3 + 5*e^2 - 16*e - 17, -4*e^3 + e^2 + 21*e - 9, -7*e^3 - 5*e^2 + 35*e, -e^3 - 10*e^2 - 3*e + 23, -e^3 + 6*e^2 + 8*e - 14, -6*e^3 - 8*e^2 + 27*e + 14, -4*e^3 - 5*e^2 + 13*e + 14, -9*e^3 - 11*e^2 + 30*e + 22, -2*e^3 - 5*e^2 + 5*e + 3, 2*e^2 + e - 6, -13*e^3 - 12*e^2 + 53*e + 2, 5*e^3 + 12*e^2 - 22*e - 24, e^3 + 2*e^2 - 16, e^3 - 6*e^2 - 14*e + 13, -8*e^3 - 10*e^2 + 30*e + 2, 7*e^3 + 2*e^2 - 32*e + 3, 5*e^3 + 8*e^2 - 16*e - 31, 2*e^2 - 3*e - 17, -8*e^3 - 9*e^2 + 25*e + 13, -e^3 + e^2 - 6*e - 8, -2*e^3 - 6*e^2 + 14*e + 18, e^3 - 12*e - 9, e^3 + 1, 8*e^3 + 3*e^2 - 39*e + 11, e^3 + 4*e^2 - 4*e - 27, -7*e^3 - 5*e^2 + 42*e, 5*e^3 + 8*e^2 - 16*e - 19, 8*e^3 + 13*e^2 - 34*e - 9, -8*e^3 - 6*e^2 + 39*e - 3, -2*e^3 - 8*e^2 + 10*e + 11, -6*e^3 - 9*e^2 + 26*e + 9, -10*e^3 - 5*e^2 + 52*e - 8, -5*e^3 - 6*e^2 + 19*e + 15, -7*e^3 - 7*e^2 + 33*e + 8, -2*e^3 - 5*e^2 + 7*e - 15, -5*e^3 + 43*e - 12, -2*e^3 - 5*e^2 + 19*e + 12, 4*e^3 + 5*e^2 - 15*e - 1, -8*e^3 - 7*e^2 + 36*e + 5, 4*e^3 + 7*e^2 - 21*e + 6, 7*e^3 + 13*e^2 - 31*e - 17, -4*e^3 - 6*e^2 + 21*e - 2, 2*e^3 - 2*e^2 - 19*e + 10, 4*e^3 - e^2 - 25*e + 15, -2*e^3 - 2*e^2 + 20*e, -2*e^3 - 5*e^2 - 4*e + 19, -4*e^3 - 4*e^2 + 18*e - 17, -4*e^3 - 5*e^2 + 19*e + 10, e^3 + 6*e^2 + 3*e - 17, 7*e^3 - e^2 - 43*e, -e^3 + 3*e^2 - e - 15, 14*e^3 + 7*e^2 - 62*e + 5, 5*e^3 + 9*e^2 - 19*e - 26, 7*e^3 + e^2 - 37*e - 3, e^3 - 2*e^2 - e + 9, -8*e^3 - 7*e^2 + 30*e + 9, 7*e^3 + 14*e^2 - 21*e - 33, -e^3 + 7*e^2 + 11*e - 7, 6*e^3 + 4*e^2 - 23*e + 19, 9*e^3 + 19*e^2 - 33*e - 23, -6*e^3 + 2*e^2 + 38*e - 5, 5*e^3 - 6*e^2 - 26*e + 33, 3*e^3 + e^2 - 9*e - 19, 8*e^3 + 10*e^2 - 30*e - 37, -e^3 - e^2 - 5*e - 5, -7*e^3 - 9*e^2 + 27*e - 11, -5*e^3 - 13*e^2 + 9*e + 32, 4*e^3 - 2*e^2 - 16*e + 10, 7*e^3 + 13*e^2 - 19*e - 25, -9*e^3 + e^2 + 49*e - 6, 9*e^3 + 17*e^2 - 31*e - 24, 6*e^3 + 10*e^2 - 14*e - 18, -3*e^3 + 5*e^2 + 30*e - 30, -13*e^3 - 17*e^2 + 48*e + 15, -14*e^3 - 22*e^2 + 52*e + 35, 7*e^3 + 8*e^2 - 30*e - 16, 3*e^3 + 12*e^2 - 9*e - 38, -7*e^3 - 6*e^2 + 37*e - 22, -3*e^3 - 3*e^2 + 22*e + 9, 2*e^3 - 6*e^2 - 25*e + 23, -e^3 - 9*e^2 - 10*e + 28, 9*e^3 + 23*e^2 - 35*e - 46, -5*e^3 - 15*e^2 + 12*e + 45, -3*e^3 - 14*e^2 - 4*e + 30, 6*e^3 + 11*e^2 - 15*e - 27, -7*e^3 + 3*e^2 + 38*e - 17, 8*e^3 + e^2 - 48*e + 4, 7*e^3 + 11*e^2 - 13*e - 22, -9*e^3 - 15*e^2 + 37*e + 13, 13*e^3 + 6*e^2 - 48*e + 24, -e^3 - 14*e^2 - 7*e + 31, 2*e^3 - 6*e^2 - 24*e + 9, -e^3 - 10*e^2 + 4*e + 28, -10*e^3 - 8*e^2 + 45*e - 7, 10*e^3 + 15*e^2 - 51*e - 11, 5*e^3 + 6*e^2 - 29*e, 12*e^3 + 10*e^2 - 56*e + 6, 8*e^3 + 10*e^2 - 30*e + 13, 5*e^3 + 5*e^2 - 19*e - 29, 4*e^3 + e^2 - 13*e - 5, -6*e^3 + 4*e^2 + 26*e - 34, -8*e^3 - 9*e^2 + 34*e - 5, 6*e^3 + 22*e^2 - 13*e - 54, 5*e^2 - 8*e - 18, -14*e^3 - 13*e^2 + 50*e - 6, -6*e^3 - 5*e^2 + 33*e - 8, -10*e^3 - e^2 + 49*e - 5, -2*e^3 + 10*e^2 + 17*e - 10, -12*e^3 - 4*e^2 + 53*e - 20, 5*e^3 - 4*e^2 - 16*e + 28, -7*e^3 - 14*e^2 + 28*e + 10, -7*e^3 - 4*e^2 + 43*e - 30, -2*e^3 + 3*e^2 + 21*e - 19, 10*e^3 + 17*e^2 - 36*e - 53, 3*e^3 + 5*e^2 - 25*e - 22, 11*e^3 + 14*e^2 - 42*e - 25, -e^3 - 7*e^2 + 10*e + 18, 14*e^3 + 13*e^2 - 58*e - 15, e^3 + 20*e^2 + 8*e - 48, -19*e^3 - 20*e^2 + 87*e + 6, -12*e^3 - 7*e^2 + 55*e - 15, e^3 + 5*e^2 + 13*e + 4, 9*e^2 + 7*e - 17, -e^3 - 6*e^2 - 11*e + 24, -10*e^3 - 15*e^2 + 20*e + 35, e^3 + 13*e^2 + 15*e - 42, 10*e^3 + 7*e^2 - 51*e - 11, 2*e^3 + 9*e + 8, -9*e^3 - 16*e^2 + 22*e + 47, 17*e^3 + 9*e^2 - 68*e + 12, -13*e^3 - 25*e^2 + 56*e + 36, 2*e^3 + 16*e^2 + 9*e - 40, -2*e^3 + e^2 + 4*e - 44]; 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;