/* 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, -1, -6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [5, 5, -w^3 + 5*w + 2], [5, 5, w - 1], [11, 11, -w^3 + 5*w], [13, 13, -w^3 + w^2 + 6*w - 3], [16, 2, 2], [19, 19, 2*w^3 - w^2 - 11*w + 2], [25, 5, -w^2 + w + 3], [27, 3, w^3 - w^2 - 5*w + 4], [31, 31, -w - 3], [37, 37, -w^3 - w^2 + 6*w + 4], [41, 41, w^3 + w^2 - 6*w - 5], [43, 43, w^3 - 7*w - 2], [47, 47, -2*w^3 + 11*w + 2], [61, 61, w^2 - 3], [67, 67, w^2 + w - 4], [79, 79, -4*w^3 + 2*w^2 + 22*w - 9], [97, 97, -3*w^3 + 2*w^2 + 19*w - 6], [97, 97, -w^3 + 6*w - 3], [97, 97, -3*w^3 + 16*w], [97, 97, -w^3 + w^2 + 4*w - 3], [101, 101, -2*w^3 + w^2 + 11*w - 6], [103, 103, w^3 - w^2 - 7*w], [109, 109, -w^3 - w^2 + 6*w + 2], [121, 11, 2*w^3 - 11*w + 1], [127, 127, -2*w^3 + w^2 + 13*w - 1], [127, 127, w^3 - w^2 - 6*w - 2], [157, 157, -2*w^3 + 2*w^2 + 10*w - 7], [157, 157, w - 4], [163, 163, 2*w^3 - w^2 - 13*w + 3], [173, 173, -2*w^3 + w^2 + 13*w - 6], [173, 173, 2*w^3 - 12*w + 1], [173, 173, 2*w^3 - 11*w + 2], [173, 173, w^2 + w - 5], [179, 179, -2*w^3 + 10*w + 3], [181, 181, -w^3 - w^2 + 7*w + 8], [191, 191, w^3 - 5*w - 5], [191, 191, 2*w^3 - w^2 - 9*w - 1], [193, 193, -3*w^3 + 3*w^2 + 19*w - 7], [197, 197, -w^3 + w^2 + 6*w - 7], [223, 223, w^3 + w^2 - 6*w + 1], [227, 227, 6*w^3 - 3*w^2 - 34*w + 9], [227, 227, w^3 + w^2 - 8*w], [233, 233, 2*w^3 - 13*w - 3], [233, 233, -w^3 + 2*w^2 + 4*w - 6], [239, 239, -2*w^3 + 10*w + 1], [239, 239, w^3 - 8*w - 2], [251, 251, -3*w^3 + w^2 + 18*w], [251, 251, -5*w^3 + 2*w^2 + 30*w - 11], [263, 263, -2*w^3 + w^2 + 10*w + 2], [269, 269, 3*w^3 - 18*w - 8], [271, 271, 2*w^3 + w^2 - 10*w - 2], [271, 271, -3*w^3 + 19*w], [277, 277, -w^3 + w^2 + 4*w - 8], [277, 277, 2*w^3 + w^2 - 14*w - 8], [281, 281, w^3 - 8*w], [307, 307, 3*w^3 - w^2 - 16*w + 2], [307, 307, 2*w^3 - w^2 - 9*w + 1], [307, 307, -2*w^3 + 2*w^2 + 13*w - 7], [307, 307, -w^3 + w^2 + 7*w - 6], [313, 313, -3*w^3 + w^2 + 15*w + 4], [317, 317, -4*w^3 + w^2 + 22*w - 3], [317, 317, 2*w^3 - w^2 - 10*w - 3], [337, 337, w^3 + w^2 - 4*w - 5], [349, 349, -2*w^3 - w^2 + 11*w + 3], [349, 349, -4*w^3 + 2*w^2 + 24*w - 11], [353, 353, 3*w^3 - w^2 - 17*w + 7], [353, 353, -3*w^3 - 2*w^2 + 17*w + 11], [367, 367, 2*w^3 - 11*w - 7], [373, 373, -w^3 + w^2 + 4*w - 5], [383, 383, -4*w^3 + 3*w^2 + 23*w - 9], [383, 383, 4*w^3 - w^2 - 22*w], [389, 389, w^2 - 2*w - 5], [401, 401, -3*w^3 + 2*w^2 + 16*w - 9], [401, 401, 2*w^3 - w^2 - 14*w - 1], [409, 409, 2*w^3 + 2*w^2 - 13*w - 9], [419, 419, -4*w^3 + 24*w + 7], [419, 419, 2*w^2 + 2*w - 7], [419, 419, -w^3 + 2*w^2 + 8*w - 6], [419, 419, 6*w^3 - 3*w^2 - 33*w + 13], [431, 431, 2*w^3 - w^2 - 12*w - 5], [439, 439, -3*w^3 + 2*w^2 + 18*w - 4], [439, 439, w^3 + 2*w^2 - 7*w - 3], [443, 443, 6*w^3 - 3*w^2 - 33*w + 7], [449, 449, -3*w^3 + w^2 + 14*w - 4], [457, 457, 2*w^3 - 9*w], [457, 457, 3*w^3 - w^2 - 16*w - 2], [479, 479, 3*w^3 - w^2 - 19*w + 4], [487, 487, -w^3 + w^2 + 4*w - 6], [491, 491, 3*w^3 + w^2 - 17*w - 5], [499, 499, 5*w^3 - 3*w^2 - 28*w + 8], [499, 499, 2*w^3 - w^2 - 12*w - 1], [503, 503, -2*w^3 - w^2 + 12*w + 3], [503, 503, w^3 - 3*w - 4], [509, 509, -3*w - 5], [509, 509, -4*w^3 + 3*w^2 + 25*w - 10], [523, 523, -2*w^3 + w^2 + 14*w - 6], [529, 23, 4*w^3 - 2*w^2 - 21*w + 5], [529, 23, -3*w^3 - w^2 + 15*w + 6], [557, 557, -2*w^3 + w^2 + 13*w + 2], [557, 557, 3*w^3 - 2*w^2 - 16*w + 3], [569, 569, 2*w^2 - w - 8], [587, 587, w^3 + 2*w^2 - 5*w - 16], [587, 587, -4*w^3 + w^2 + 25*w - 5], [599, 599, w^3 + 2*w^2 - 6*w - 5], [599, 599, -5*w^3 + 3*w^2 + 30*w - 15], [601, 601, w^3 - 6*w - 6], [607, 607, -w^3 + w^2 + 3*w - 4], [607, 607, -w^3 + 2*w - 3], [613, 613, 6*w^3 - 2*w^2 - 31*w + 11], [613, 613, w^3 + w^2 - 4*w - 6], [617, 617, 5*w^3 - w^2 - 30*w + 2], [617, 617, -2*w^3 + 2*w^2 + 9*w + 3], [619, 619, -w^3 + 2*w^2 + 6*w - 5], [619, 619, 2*w^3 - w^2 - 11*w - 3], [631, 631, -w^3 + 2*w^2 + 5*w - 5], [631, 631, -3*w^3 - w^2 + 18*w + 5], [647, 647, 6*w^3 - w^2 - 36*w + 2], [659, 659, 4*w^3 + w^2 - 24*w - 14], [661, 661, 2*w^2 - w - 4], [673, 673, 2*w^3 + w^2 - 14*w - 12], [677, 677, 3*w^3 + 2*w^2 - 16*w - 7], [677, 677, -3*w^3 + w^2 + 15*w + 1], [683, 683, 2*w^2 - 2*w - 9], [683, 683, 2*w^3 - 14*w + 1], [691, 691, 5*w^3 - 2*w^2 - 30*w + 6], [701, 701, -2*w^3 + w^2 + 12*w + 2], [701, 701, -5*w^3 + 2*w^2 + 26*w], [719, 719, 3*w^3 - 18*w + 1], [719, 719, 3*w^3 - w^2 - 19*w + 1], [727, 727, -3*w^2 - 2*w + 11], [727, 727, 2*w^3 + w^2 - 13*w - 2], [733, 733, 2*w^3 - 3*w^2 - 8*w + 5], [739, 739, -3*w^3 + 16*w - 6], [743, 743, -4*w^3 - w^2 + 25*w + 8], [751, 751, w^2 - 2*w - 7], [751, 751, -3*w^3 + 15*w + 5], [757, 757, 3*w^3 - 19*w - 2], [761, 761, 2*w^2 + w - 5], [769, 769, -2*w^3 + w^2 + 14*w - 9], [769, 769, 2*w^2 - w - 7], [773, 773, w^3 - w^2 - 5*w - 4], [773, 773, w^3 - 3*w - 6], [787, 787, 3*w^3 - 19*w - 3], [797, 797, 8*w^3 - 3*w^2 - 47*w + 10], [809, 809, 4*w^3 - 25*w - 1], [811, 811, w^3 - 9*w - 1], [821, 821, 6*w^3 - 3*w^2 - 35*w + 8], [823, 823, 2*w^3 + w^2 - 10*w + 1], [823, 823, -2*w^3 + 2*w^2 + 7*w - 4], [829, 829, -5*w^3 + w^2 + 27*w - 4], [829, 829, 5*w^3 - 2*w^2 - 26*w + 10], [839, 839, w^3 + 2*w^2 - 5*w - 7], [839, 839, 3*w^3 - w^2 - 15*w + 2], [841, 29, 3*w^3 + w^2 - 14*w - 2], [841, 29, 4*w^3 - 2*w^2 - 21*w + 10], [857, 857, -8*w^3 + 4*w^2 + 44*w - 11], [857, 857, -3*w^3 + 3*w^2 + 20*w - 9], [859, 859, w^3 + 2*w^2 - 6*w - 6], [877, 877, w^3 - 3*w^2 - 4*w + 14], [883, 883, 2*w^3 + w^2 - 10*w - 11], [887, 887, 3*w^3 + w^2 - 20*w - 2], [919, 919, 7*w^3 - 3*w^2 - 39*w + 9], [929, 929, 3*w^3 - 17*w + 2], [929, 929, -6*w^3 + 2*w^2 + 35*w - 10], [937, 937, w^3 + w^2 - 3*w - 5], [937, 937, -2*w^3 + 3*w^2 + 12*w - 12], [937, 937, -3*w^3 - 2*w^2 + 19*w + 8], [937, 937, 3*w^3 - 2*w^2 - 13*w + 3], [941, 941, w^3 + 2*w^2 - 6*w - 8], [941, 941, 5*w^3 - w^2 - 28*w - 3], [947, 947, -4*w^3 + 2*w^2 + 25*w - 6], [953, 953, w^3 - 7*w - 7], [953, 953, -3*w^3 + 2*w^2 + 18*w - 3], [967, 967, w^3 - 2*w^2 - 5*w - 3], [971, 971, w^2 + 2*w - 6], [977, 977, 2*w^3 - 11*w - 8], [997, 997, -3*w^3 + w^2 + 17*w + 2], [997, 997, -2*w^3 + 2*w^2 + 14*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^4 + 6*x^3 + 6*x^2 - 12*x - 12; K := NumberField(heckePol); heckeEigenvaluesArray := [1, e, -1, -e^3 - 4*e^2 + e + 6, e^3 + 4*e^2 - 8, 1/2*e^3 + e^2 - 3*e + 1, 1/2*e^3 + e^2 - 3*e - 4, 1/2*e^3 + 2*e^2 - 2*e - 8, -1/2*e^3 - e^2 + 4*e + 4, -e^3 - 5*e^2 - 2*e + 10, -1/2*e^3 - 2*e^2 + 2*e + 4, 2*e^2 + 2*e - 12, -e^3 - 5*e^2 - 2*e + 8, 1/2*e^3 + 2*e^2 + e - 6, 3/2*e^3 + 7*e^2 - 16, -e^3 - 3*e^2 + 6*e + 10, -2*e^3 - 8*e^2 - 3*e + 8, e^3 + 8*e^2 + 10*e - 14, -3*e^3 - 11*e^2 + 10, e^3 + 4*e^2 - 5*e - 14, -1/2*e^3 - 4*e^2 + 16, 3*e^3 + 12*e^2 - 4*e - 24, e^3 + 6*e^2 + 3*e - 14, -2*e^3 - 7*e^2 + 4, -e^2 - 6*e - 2, 2*e^3 + 8*e^2 - 6*e - 26, 3/2*e^3 + 8*e^2 - 20, -3*e^3 - 12*e^2 + 8, e^3 + 9*e^2 + 12*e - 14, e^3 + 6*e^2 - 22, 2*e^3 + 5*e^2 - 4*e, e^3 + e^2 - 8*e - 12, -e^3 - 9*e^2 - 10*e + 18, 2*e^3 + 5*e^2 - 10*e - 12, 3/2*e^3 + 5*e^2 - 6*e - 24, 4*e^2 + 15*e - 4, -2*e - 6, 5/2*e^3 + 7*e^2 - 9*e - 12, -1/2*e^3 - 5*e^2 - 5*e + 8, -2*e^3 - 7*e^2 + 12, 4*e^2 + 10*e - 2, 1/2*e^3 + 5*e^2 + 9*e - 12, -3*e^3 - 9*e^2 + 6*e + 12, 5*e^2 + 12*e - 12, 9/2*e^3 + 18*e^2 - 2*e - 24, -3/2*e^3 - 2*e^2 + 10*e - 12, -3*e^2 - 6*e + 12, -5/2*e^3 - 6*e^2 + 15*e + 18, 6*e^3 + 25*e^2 + 2*e - 42, -2*e^3 - 10*e^2 - 9*e + 18, -3/2*e^3 - 9*e^2 - 8*e, -e^3 - e^2 + 18*e + 10, -2*e^2 - 10*e + 2, -2*e^3 - 4*e^2 + 8*e - 4, -e^3 - 4*e^2 + e + 8, -2*e^3 - 10*e^2 + 6*e + 36, -7/2*e^3 - 17*e^2 - 8*e + 20, -2*e^3 - 10*e^2 - 4*e + 16, -2*e^3 - 11*e^2 - 2*e + 28, 2*e^3 + 12*e^2 + 16*e - 16, 3*e^3 + 12*e^2 - 6*e - 28, 5/2*e^3 + 11*e^2 + e - 24, 7/2*e^3 + 9*e^2 - 16*e - 12, 4*e^3 + 15*e^2 - 4*e - 20, -3*e^3 - 14*e^2 + 2*e + 46, 4*e^3 + 11*e^2 - 8*e + 4, -5*e^3 - 18*e^2 + 14*e + 42, -1/2*e^3 + 2*e^2 + 5*e - 18, 3*e^3 + 17*e^2 + 14*e - 28, -e^3 + 2*e^2 + 19*e + 8, 3/2*e^3 + 8*e^2 + 2*e - 36, 3*e^3 + 8*e^2 - 18*e - 24, 2*e^3 - 19*e + 6, -3*e^3 - 12*e^2 - 4*e + 24, -6*e^3 - 26*e^2 + 12*e + 66, -2*e^3 - 12*e^2 - 2*e + 44, 1/2*e^3 + 3*e^2 - 4*e, 7*e^3 + 28*e^2 - 2*e - 36, -5*e^3 - 13*e^2 + 20*e + 18, -1/2*e^3 - 5*e^2 - 6*e + 24, -1/2*e^3 - e^2 + 10*e, -3*e^3 - 8*e^2 + 9*e + 10, 2*e^3 + 9*e^2 + 4*e - 28, -11/2*e^3 - 23*e^2 - e + 36, -5/2*e^3 - 15*e^2 - 10*e + 24, -13/2*e^3 - 22*e^2 + 12*e + 28, 11/2*e^3 + 21*e^2 - e - 28, -5*e^3 - 14*e^2 + 16*e + 6, -7/2*e^3 - 16*e^2 - 5*e + 34, 4*e^3 + 16*e^2 - 6*e - 18, -3*e^3 - 4*e^2 + 28*e + 20, -2*e^3 - 14*e^2 - 6*e + 40, 2*e^3 + 16*e^2 + 16*e - 24, -6*e^3 - 21*e^2 + 4*e + 12, 9/2*e^3 + 20*e^2 + 10*e - 24, -e^3 - 7*e^2 - 4*e + 30, -2*e^3 - 13*e^2 - 16*e + 8, 13/2*e^3 + 27*e^2 - 9*e - 52, e^3 - 15*e - 14, -17/2*e^3 - 36*e^2 + 3*e + 66, -4*e^3 - 16*e^2 - 2*e + 24, 7/2*e^3 + 13*e^2 - e - 12, -4*e^3 - 12*e^2 + 16*e + 24, 4*e^3 + 9*e^2 - 24*e - 36, 11/2*e^3 + 26*e^2 + 4*e - 60, -e^2 - 6*e - 12, -6*e^2 - 6*e + 20, 7/2*e^3 + 20*e^2 + 15*e - 26, 5/2*e^3 + 7*e^2 - 12*e - 16, -13/2*e^3 - 29*e^2 + e + 56, 4*e^3 + 16*e^2 - 2*e - 40, -8*e^3 - 29*e^2 + 16*e + 60, 5*e^3 + 10*e^2 - 33*e - 30, -8*e^3 - 36*e^2 + 68, -9/2*e^3 - 14*e^2 + 13*e + 10, -9/2*e^3 - 18*e^2 - 7*e + 22, -3*e^3 - 11*e^2 + 6*e - 4, 4*e^3 + 22*e^2 + 14*e - 42, -e^3 - e^2 + 10*e - 18, 7/2*e^3 + 4*e^2 - 33*e - 2, -3/2*e^3 - 10*e^2 - 9*e - 2, -2*e^3 - 14*e^2 - 12*e + 42, -3*e^3 - 14*e^2 + 8*e + 48, -3*e^3 + 2*e^2 + 37*e - 6, -9*e^3 - 33*e^2 + 26*e + 84, -6*e^3 - 30*e^2 - 6*e + 34, 8*e^3 + 30*e^2 - 16*e - 72, 7/2*e^3 + 4*e^2 - 34*e - 12, 2*e^3 + 8*e^2 + 4*e - 24, 3*e^3 + 12*e^2 - 12*e - 42, -11/2*e^3 - 22*e^2 + 10*e + 52, -7*e^3 - 29*e^2 - 8*e + 40, -10*e^2 - 17*e + 44, e^2 + 16*e + 8, 6*e^3 + 23*e^2 - 14*e - 72, -9/2*e^3 - 23*e^2 - 14*e + 44, -7*e^3 - 30*e^2 + 10*e + 52, -4*e^3 - 10*e^2 + 22*e + 16, 6*e^3 + 12*e^2 - 31*e - 6, 6*e^3 + 24*e^2 - 4*e - 46, -15/2*e^3 - 26*e^2 + 21*e + 58, 6*e^3 + 28*e^2 + 16*e - 42, 4*e^3 + 19*e^2 + 4*e - 48, -4*e^3 - 16*e^2 + 10*e + 26, -e^3 + e^2 + 10*e - 6, -2*e^3 - 8*e^2 + 2*e + 48, 8*e^3 + 30*e^2 - 18*e - 82, -17/2*e^3 - 27*e^2 + 26*e + 60, e^3 + 13*e^2 + 22*e - 2, 5/2*e^3 + 13*e^2 + 15*e - 28, -8*e^2 - 18*e - 8, 11/2*e^3 + 28*e^2 + 13*e - 26, -1/2*e^3 + 6*e^2 + 29*e - 6, -1/2*e^3 - 6*e^2 + 3*e + 30, 8*e^3 + 36*e^2 + 15*e - 50, -e^3 - 10*e^2 - 15*e + 44, 3/2*e^3 + 8*e^2 - e - 42, 5*e^3 + 25*e^2 + 18*e - 54, -19/2*e^3 - 39*e^2 + 14*e + 80, -6*e^3 - 28*e^2 - e + 58, -2*e^3 - 4*e^2 + 16*e + 20, 7*e^3 + 25*e^2 - 10*e - 66, 7/2*e^3 + 19*e^2 + 12*e - 52, 4*e^3 + 17*e^2 - 20*e - 60, 7*e^3 + 28*e^2 + 4*e - 24, e^3 + e^2 - 6*e - 10, 2*e^3 + 10*e^2 - 3*e - 38, 8*e^2 + 20*e - 2, 7/2*e^3 + 15*e^2 - 28, -4*e^3 - 14*e^2 - 24, -3/2*e^3 - 7*e^2 + 9*e + 36, 4*e^3 + 16*e^2 - 12, 4*e^3 + 22*e^2 + 2*e - 36, 2*e^3 + 14*e^2 + 11*e - 36, 6*e^3 + 24*e^2 - 64, -4*e^3 - 17*e^2 - 6*e + 24, 2*e^3 + 15*e^2 + 4*e - 48, -13/2*e^3 - 14*e^2 + 37*e + 34, 4*e^3 + 16*e^2 - 16*e - 46]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; 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;