/* 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![2, 2, -5, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w], [3, 3, w + 1], [4, 2, -w^2 - w + 1], [13, 13, -w^2 + 3], [17, 17, -w + 3], [27, 3, w^3 - 2*w^2 - 3*w + 5], [31, 31, -w^2 - 2*w + 1], [41, 41, -w^2 + 5], [43, 43, w^3 - w^2 - 5*w - 1], [43, 43, w^3 - w^2 - 3*w + 1], [59, 59, -w - 3], [59, 59, -2*w^3 + 2*w^2 + 9*w - 5], [59, 59, -w^3 - 3*w^2 - w + 3], [59, 59, w^3 - 7*w + 1], [61, 61, -w^3 + w^2 + 2*w + 5], [61, 61, -w^3 - w^2 + 4*w + 3], [67, 67, 4*w^3 - 4*w^2 - 18*w + 7], [73, 73, -2*w^3 + 6*w - 3], [73, 73, -2*w + 3], [97, 97, w^3 - 2*w^2 - 5*w + 3], [101, 101, w^3 + w^2 - 5*w - 7], [101, 101, -w^3 - w^2 + 4*w + 5], [103, 103, 2*w^3 - 8*w - 1], [103, 103, w^2 - 2*w - 7], [109, 109, -w^3 + 2*w^2 + 3*w - 7], [113, 113, -w^3 + w^2 + 6*w - 3], [127, 127, -w^3 + w^2 + 6*w - 1], [131, 131, -2*w^3 + 3*w^2 + 10*w - 11], [137, 137, w^3 + w^2 - 6*w - 5], [139, 139, -3*w^3 + 2*w^2 + 15*w - 1], [149, 149, w^3 - 3*w^2 - 4*w + 11], [149, 149, -w^3 + 5*w - 1], [157, 157, -2*w^3 + 3*w^2 + 6*w - 1], [163, 163, 2*w^2 - 3], [163, 163, -2*w^3 + 2*w^2 + 7*w - 5], [167, 167, -2*w^3 + 2*w^2 + 8*w - 3], [167, 167, -2*w^2 + 2*w + 9], [173, 173, 2*w^3 - w^2 - 8*w + 3], [173, 173, 2*w^3 - 3*w^2 - 6*w + 3], [181, 181, 2*w^3 - 3*w^2 - 8*w + 5], [191, 191, -2*w^3 + w^2 + 10*w - 1], [193, 193, -w^3 + 3*w^2 + 2*w - 7], [193, 193, w^3 - 3*w - 3], [193, 193, 2*w^2 - 2*w - 3], [193, 193, w^3 - 7*w - 1], [197, 197, w^2 - 2*w - 5], [211, 211, w^2 - 4*w - 1], [229, 229, -w^3 + w^2 + 6*w - 7], [229, 229, -4*w^3 + 6*w^2 + 16*w - 13], [233, 233, 3*w^2 + 2*w - 5], [233, 233, 2*w^3 - 2*w^2 - 11*w + 7], [239, 239, -w^3 + w^2 + 7*w + 1], [241, 241, w^3 - 3*w^2 - 2*w + 9], [241, 241, w^3 + w^2 - 9*w - 5], [251, 251, 2*w^3 - 6*w + 1], [271, 271, 2*w^3 - w^2 - 8*w - 1], [271, 271, -2*w^2 + w + 7], [281, 281, 2*w^3 + 2*w^2 - 5*w - 1], [293, 293, w^3 + w^2 - 5*w - 1], [293, 293, -3*w + 7], [307, 307, -w^3 - 2*w^2 + 3*w + 5], [311, 311, -w^3 + 3*w^2 + 4*w - 7], [313, 313, 2*w^2 - 2*w - 5], [317, 317, -2*w^3 + 4*w^2 + 7*w - 13], [317, 317, -2*w^3 + 3*w^2 + 8*w - 11], [337, 337, 3*w^3 + 2*w^2 - 7*w + 1], [347, 347, 2*w^3 - 2*w^2 - 11*w + 1], [349, 349, -5*w^3 + 6*w^2 + 23*w - 13], [359, 359, 2*w^3 - 2*w^2 - 12*w + 9], [359, 359, -2*w^3 + 8*w + 5], [367, 367, -4*w^3 + 4*w^2 + 19*w - 5], [367, 367, 2*w^3 + w^2 - 10*w - 7], [383, 383, 2*w^3 - 2*w^2 - 7*w + 1], [389, 389, 2*w^2 - w - 5], [397, 397, 3*w^3 - 4*w^2 - 11*w + 9], [397, 397, 3*w^3 - 3*w^2 - 12*w + 7], [397, 397, 2*w^3 - 4*w^2 - 8*w + 15], [397, 397, -w^3 + 3*w^2 + w - 7], [409, 409, -w^3 - w^2 + 9*w - 1], [419, 419, 2*w^2 - 3*w - 7], [419, 419, 4*w^3 - 3*w^2 - 18*w + 3], [431, 431, -3*w^3 + w^2 + 17*w - 1], [431, 431, 3*w^3 - w^2 - 16*w - 1], [433, 433, -2*w^3 + 5*w^2 + 8*w - 19], [443, 443, 3*w - 5], [449, 449, -2*w^3 + 2*w^2 + 9*w - 9], [449, 449, -5*w^3 + 8*w^2 + 19*w - 17], [457, 457, -3*w^3 + 3*w^2 + 13*w - 9], [457, 457, -3*w^3 + 2*w^2 + 13*w + 1], [457, 457, -2*w^3 + w^2 + 8*w + 7], [457, 457, 3*w^3 - 3*w^2 - 15*w + 7], [461, 461, -2*w^3 + 2*w^2 + 9*w + 1], [479, 479, -w^3 + w^2 + 7*w - 9], [487, 487, -w^3 - w^2 + w - 3], [499, 499, -2*w^3 + 7*w - 3], [499, 499, -5*w^3 + 7*w^2 + 20*w - 15], [541, 541, -3*w^3 + 4*w^2 + 13*w - 7], [547, 547, -w^3 - w^2 + 3*w + 5], [557, 557, 4*w - 1], [563, 563, 2*w^3 - w^2 - 6*w - 1], [571, 571, w^3 - w^2 - w - 3], [587, 587, -2*w^3 + 2*w^2 + 10*w + 1], [593, 593, 2*w^3 - 2*w^2 - 6*w + 3], [613, 613, -5*w^3 + 5*w^2 + 23*w - 7], [613, 613, -w^3 + w^2 + 5*w - 7], [617, 617, -w - 5], [619, 619, -3*w^3 + 4*w^2 + 11*w - 7], [625, 5, -5], [643, 643, -3*w^3 + 3*w^2 + 14*w - 9], [643, 643, -3*w^2 - 4*w + 1], [653, 653, -w^3 + w^2 + 9*w + 3], [653, 653, -4*w^2 + 6*w + 11], [659, 659, -w^2 - 4*w + 1], [661, 661, -3*w^3 + 3*w^2 + 15*w - 5], [661, 661, w^3 + w^2 - 7*w - 11], [673, 673, 2*w^3 - 3*w^2 - 12*w + 1], [673, 673, -2*w^3 + w^2 + 10*w - 3], [683, 683, -w^3 - 2*w^2 + 5*w + 11], [691, 691, 4*w^3 - 3*w^2 - 18*w + 5], [709, 709, w^3 - 3*w - 7], [719, 719, 2*w^3 - 8*w + 1], [727, 727, -3*w^3 + 6*w^2 + 13*w - 19], [733, 733, w^3 - w^2 + 3], [739, 739, -w^3 + w^2 + 3*w - 7], [751, 751, -w^2 - 3], [773, 773, 2*w^3 - 2*w^2 - 11*w + 3], [787, 787, w^3 - w^2 - 3*w - 5], [797, 797, -2*w^3 + 2*w^2 + 5*w - 1], [797, 797, -7*w^3 + 8*w^2 + 31*w - 15], [809, 809, 4*w^3 - 6*w^2 - 17*w + 13], [809, 809, 3*w^2 - 4*w - 5], [811, 811, 2*w^2 + 4*w + 3], [811, 811, -3*w^3 + 3*w^2 + 12*w - 5], [821, 821, 2*w^3 - 10*w - 1], [821, 821, 6*w^3 - 10*w^2 - 24*w + 29], [823, 823, -5*w^3 + 3*w^2 + 24*w - 1], [823, 823, w^3 + w^2 + w + 3], [827, 827, -4*w^3 + 3*w^2 + 16*w - 7], [829, 829, 2*w^3 - 4*w^2 - 6*w + 11], [841, 29, w^3 - 3*w^2 - 5*w + 3], [841, 29, -3*w^3 + 3*w^2 + 13*w - 1], [857, 857, -4*w^3 + 6*w^2 + 13*w - 7], [857, 857, -3*w^3 + w^2 + 15*w + 1], [859, 859, -4*w^2 + 3*w + 17], [863, 863, -4*w^3 + 8*w^2 + 14*w - 21], [877, 877, -w^3 - w^2 + 6*w - 1], [881, 881, -5*w - 1], [883, 883, -4*w^3 + 4*w^2 + 21*w - 13], [883, 883, 2*w^3 - 11*w - 1], [887, 887, -3*w^3 - w^2 + 8*w - 5], [907, 907, w^3 - w^2 - w + 5], [911, 911, 2*w^2 - 5*w - 5], [919, 919, -w^3 + 3*w^2 + 6*w - 3], [919, 919, 5*w^2 + 4*w - 9], [929, 929, w^3 + 3*w^2 - 8*w - 17], [937, 937, -4*w^3 + 4*w^2 + 17*w - 7], [953, 953, -2*w^3 + 2*w^2 + 7*w + 5], [953, 953, 3*w^3 - 4*w^2 - 11*w + 3], [967, 967, 4*w^3 - 2*w^2 - 17*w + 3], [971, 971, w^3 - 3*w^2 - 8*w + 1], [983, 983, -2*w^3 - 2*w^2 + 12*w + 13], [991, 991, -2*w^3 - 2*w^2 + 8*w + 7], [991, 991, w^3 + w^2 - 10*w + 1]]; primes := [ideal : I in primesArray]; heckePol := x^5 - 2*x^4 - 9*x^3 + 16*x^2 + 16*x - 20; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1/2*e^4 + 7/2*e^2 - 2, -1, 2*e - 2, -1/2*e^4 + 7/2*e^2 + 2*e - 4, 1/2*e^4 - 11/2*e^2 + 12, -2*e - 2, 1/2*e^4 - 11/2*e^2 + 2*e + 10, -2*e^4 + e^3 + 14*e^2 - 5*e - 12, -1/2*e^4 + 2*e^3 + 7/2*e^2 - 10*e - 6, 3/2*e^4 - 25/2*e^2 + 16, -3/2*e^4 - 2*e^3 + 25/2*e^2 + 12*e - 18, 2*e^4 + e^3 - 18*e^2 - 5*e + 28, e^4 - 5*e^2 - 2*e, e^4 - 9*e^2 - 2*e + 18, -e^4 - 2*e^3 + 7*e^2 + 14*e - 8, -e^4 + 9*e^2 - 2*e - 16, e^4 + e^3 - 9*e^2 - 7*e + 18, -3/2*e^4 + 2*e^3 + 21/2*e^2 - 8*e - 4, e^4 + e^3 - 9*e^2 - 7*e + 22, e^4 - 9*e^2 - 2*e + 18, 4*e, e^4 + 2*e^3 - 9*e^2 - 12*e + 16, 3*e^4 - 23*e^2 - 2*e + 24, 4*e^2 - 16, 3/2*e^4 - 25/2*e^2 + 20, e^4 + 2*e^3 - 11*e^2 - 12*e + 18, -7/2*e^4 - 2*e^3 + 61/2*e^2 + 10*e - 40, -1/2*e^4 + 2*e^3 + 7/2*e^2 - 10*e - 2, e^4 + e^3 - 9*e^2 - 7*e + 4, -e^4 - 2*e^3 + 7*e^2 + 16*e - 6, 3*e^4 - 21*e^2 - 2*e + 22, -2*e^3 + 4*e^2 + 12*e - 10, -3/2*e^4 + 21/2*e^2 - 14, -3/2*e^4 + 17/2*e^2 + 2*e - 2, 3*e^4 + 2*e^3 - 23*e^2 - 12*e + 24, e^4 - 9*e^2 + 2*e + 12, -3*e^4 - 2*e^3 + 23*e^2 + 12*e - 22, -2*e^4 + 10*e^2 + 4*e + 14, e^4 + 2*e^3 - 11*e^2 - 14*e + 28, 3*e^4 - 23*e^2 + 2*e + 20, -1/2*e^4 + 2*e^3 + 3/2*e^2 - 14*e + 4, e^4 + 2*e^3 - 5*e^2 - 16*e + 2, -5/2*e^4 + 2*e^3 + 39/2*e^2 - 14*e - 16, 2*e^4 - e^3 - 18*e^2 + 5*e + 26, 2*e^4 + 2*e^3 - 18*e^2 - 14*e + 26, -7/2*e^4 - 2*e^3 + 61/2*e^2 + 10*e - 40, 2*e^4 - 18*e^2 + 30, -4*e^3 + 4*e^2 + 22*e - 14, -1/2*e^4 + 3/2*e^2 - 4, -3*e^4 - 2*e^3 + 27*e^2 + 12*e - 38, -2*e^4 + 18*e^2 - 20, -e^4 + 3*e^3 + 9*e^2 - 17*e - 18, -3/2*e^4 - 2*e^3 + 25/2*e^2 + 12*e - 6, -e^4 + e^3 + 13*e^2 - 3*e - 32, e^4 - 2*e^3 - 9*e^2 + 16*e + 8, -e^4 - 2*e^3 + 5*e^2 + 8*e + 8, -1/2*e^4 + 2*e^3 - 5/2*e^2 - 8*e + 30, 4*e^4 + 2*e^3 - 28*e^2 - 16*e + 26, -2*e^3 + 4*e^2 + 10*e - 18, e^4 - 4*e^3 - 9*e^2 + 30*e + 16, e^4 + 2*e^3 - 11*e^2 - 14*e + 8, -7/2*e^4 - 4*e^3 + 65/2*e^2 + 22*e - 48, 4*e^4 - 32*e^2 + 32, e^4 - 4*e^3 - 9*e^2 + 22*e + 10, -3/2*e^4 + 33/2*e^2 + 2*e - 30, -2*e^4 - 3*e^3 + 18*e^2 + 15*e - 36, -e^4 + 2*e^3 + 9*e^2 - 12*e + 2, 3*e^4 - 25*e^2 + 2*e + 26, 4*e^4 - 2*e^3 - 32*e^2 + 16*e + 30, 3*e^4 - 27*e^2 + 2*e + 32, 5*e^4 - 2*e^3 - 39*e^2 + 2*e + 52, 2*e^4 - 14*e^2 - 2*e + 14, 2*e^4 - 22*e^2 + 46, -e^4 - 2*e^3 + 3*e^2 + 8*e + 22, 3*e^4 - 27*e^2 - 2*e + 54, -e^4 - 4*e^3 + 5*e^2 + 30*e + 2, -3*e^4 - 6*e^3 + 29*e^2 + 34*e - 40, 3/2*e^4 + 2*e^3 - 21/2*e^2 - 10*e + 14, -5/2*e^4 + 39/2*e^2 + 4*e - 40, e^4 + 4*e^3 - 13*e^2 - 22*e + 24, -e^4 - 2*e^3 + 7*e^2 + 8*e - 10, 2*e^4 + 4*e^3 - 14*e^2 - 32*e + 4, 7/2*e^4 - 4*e^3 - 57/2*e^2 + 16*e + 42, -e^3 + 4*e^2 + e - 28, -9/2*e^4 - 2*e^3 + 63/2*e^2 + 8*e - 4, -3*e^4 + e^3 + 31*e^2 - 3*e - 54, 9/2*e^4 - 79/2*e^2 + 54, 7/2*e^4 + 2*e^3 - 69/2*e^2 - 12*e + 62, -3*e^4 + e^3 + 19*e^2 - 3*e + 2, e^4 - 2*e^3 - 5*e^2 + 20*e + 6, e^4 - 2*e^3 - 13*e^2 + 12*e + 26, 4*e^4 - 4*e^3 - 36*e^2 + 22*e + 42, -2*e^4 + 14*e^2 + 2*e - 22, -5/2*e^4 - 4*e^3 + 43/2*e^2 + 30*e - 28, -8*e^4 + 64*e^2 + 4*e - 80, e^4 + 2*e^3 - 17*e^2 - 16*e + 58, -2*e^4 - 4*e^3 + 14*e^2 + 20*e + 12, 2*e^4 + 2*e^3 - 10*e^2 - 14*e - 16, -4*e^3 + 4*e^2 + 24*e + 4, -e^4 + 3*e^3 + e^2 - 21*e + 16, -3/2*e^4 - 2*e^3 + 37/2*e^2 + 4*e - 40, -4*e^4 + 3*e^3 + 36*e^2 - 23*e - 38, -e^4 + e^2 + 6*e + 38, -e^4 + 2*e^3 - e^2 - 12*e + 26, -5/2*e^4 - 2*e^3 + 31/2*e^2 + 16*e + 22, 11/2*e^4 - 2*e^3 - 85/2*e^2 + 10*e + 46, 3*e^4 - 2*e^3 - 27*e^2 + 24*e + 38, e^4 + e^3 - e^2 - 7*e - 24, -7/2*e^4 + 45/2*e^2 - 2*e - 14, 2*e^4 - 8*e^3 - 18*e^2 + 44*e + 18, -4*e^4 - 4*e^3 + 32*e^2 + 28*e - 46, 1/2*e^4 - 2*e^3 + 1/2*e^2 + 2*e - 22, -3*e^4 + 23*e^2 + 14*e - 34, -e^4 + 4*e^3 + 11*e^2 - 28*e - 20, -9/2*e^4 + 91/2*e^2 + 2*e - 82, e^4 - e^3 + 3*e^2 - e - 34, -3*e^4 + 5*e^3 + 23*e^2 - 27*e - 44, -e^4 + 17*e^2 + 10*e - 60, -4*e^4 + 6*e^3 + 28*e^2 - 26*e - 22, -2*e^4 + 6*e^3 + 14*e^2 - 34*e - 4, 2*e^4 + 2*e^3 - 14*e^2 - 10*e + 8, -2*e^4 + 4*e^3 + 10*e^2 - 28*e + 2, -7/2*e^4 - 4*e^3 + 49/2*e^2 + 34*e - 16, 4*e^4 - 2*e^3 - 36*e^2 + 6*e + 68, -4*e^4 + 24*e^2 + 4*e - 10, -1/2*e^4 + 8*e^3 + 3/2*e^2 - 48*e + 8, 5*e^4 + 2*e^3 - 31*e^2 - 20*e - 2, -5*e^4 + 4*e^3 + 37*e^2 - 22*e - 22, 2*e^4 + 5*e^3 - 26*e^2 - 37*e + 58, -1/2*e^4 + 6*e^3 - 1/2*e^2 - 38*e + 2, -8*e^3 + 4*e^2 + 52*e - 4, 5*e^4 - 4*e^3 - 37*e^2 + 14*e + 16, -2*e^4 + 2*e^3 + 10*e^2 - 10*e - 10, 4*e^3 + 4*e^2 - 20*e - 2, e^4 - 6*e^3 - 13*e^2 + 24*e + 40, -2*e^4 + 2*e^3 + 18*e^2 - 16*e - 22, -8*e^4 - 3*e^3 + 64*e^2 + 19*e - 76, -6*e^4 + 50*e^2 - 54, -3*e^4 + 2*e^3 + 19*e^2 - 16*e - 22, -6*e^4 - 7*e^3 + 46*e^2 + 35*e - 22, 13/2*e^4 - 6*e^3 - 107/2*e^2 + 26*e + 62, 1/2*e^4 - 2*e^3 - 11/2*e^2 + 8*e + 42, e^4 + 3*e^3 - 5*e^2 - 9*e, 7*e^4 + 2*e^3 - 49*e^2 - 14*e + 36, 4*e^3 - 8*e^2 - 28*e + 26, 4*e^3 - 4*e^2 - 28*e + 34, -4*e^4 + 4*e^3 + 36*e^2 - 24*e - 32, -13/2*e^4 + 107/2*e^2 - 6*e - 48, -3*e^4 + 2*e^3 + 31*e^2 - 72, -19/2*e^4 - 4*e^3 + 165/2*e^2 + 34*e - 144, 9*e^4 - 6*e^3 - 67*e^2 + 30*e + 68, -2*e^3 - 8*e^2 + 6*e + 48, -2*e^4 + 22*e^2 + 2*e - 62, -6*e^4 - e^3 + 46*e^2 + 13*e - 50, -9/2*e^4 + 2*e^3 + 63/2*e^2 - 22*e - 14, 9/2*e^4 + 4*e^3 - 79/2*e^2 - 32*e + 58, -6*e^4 - 8*e^3 + 46*e^2 + 44*e - 30, -3*e^4 + 2*e^3 + 13*e^2 - 10*e, 1/2*e^4 + 2*e^3 - 23/2*e^2 - 18*e + 30, e^4 - 4*e^3 - 11*e^2 + 28*e + 16, 4*e^4 - 4*e^3 - 28*e^2 + 18*e + 14, 3*e^4 + 4*e^3 - 13*e^2 - 32*e - 16]; 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;