/* 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![6, 4, -6, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w - 2], [2, 2, w + 1], [3, 3, -w^2 + w + 3], [27, 3, -w^3 - 2*w^2 + 3*w + 5], [29, 29, w^3 + w^2 - 2*w - 1], [31, 31, -w^2 - w + 1], [31, 31, w^3 - 2*w^2 - 5*w + 7], [37, 37, -w^3 + 3*w + 1], [41, 41, -w^2 + w + 5], [41, 41, -w^3 + w^2 + 4*w - 1], [43, 43, -2*w - 1], [47, 47, w^2 + w - 5], [47, 47, 2*w^3 - 3*w^2 - 9*w + 11], [53, 53, w^3 + 2*w^2 - 5*w - 5], [59, 59, w^3 + w^2 - 4*w - 1], [59, 59, w^3 + w^2 - 4*w - 5], [71, 71, 2*w^3 - 4*w^2 - 10*w + 19], [71, 71, w^2 + 3*w + 1], [73, 73, w^3 - 5*w + 1], [89, 89, -4*w^3 + 7*w^2 + 19*w - 29], [97, 97, -w^3 + 2*w^2 + 3*w - 7], [101, 101, 2*w^3 - 12*w - 7], [103, 103, 2*w - 1], [103, 103, w^2 + w - 7], [107, 107, 2*w^3 - w^2 - 9*w + 1], [107, 107, 2*w^2 - 7], [109, 109, -w^3 + w^2 + 4*w + 1], [113, 113, -4*w^3 - 3*w^2 + 17*w + 13], [121, 11, -w^2 - w - 1], [121, 11, -w^3 + 2*w^2 + 5*w - 11], [139, 139, 2*w^3 - 10*w - 1], [139, 139, 2*w^3 - 10*w - 5], [149, 149, w^3 - w^2 - 6*w + 5], [157, 157, w^2 - 3*w - 1], [157, 157, 4*w^3 - 6*w^2 - 18*w + 25], [157, 157, 2*w^3 - w^2 - 9*w + 5], [157, 157, -w^3 - 2*w^2 + 3*w + 7], [163, 163, -w^3 + w^2 + 4*w - 7], [179, 179, w^3 - 3*w - 5], [181, 181, 3*w^3 - 4*w^2 - 13*w + 17], [191, 191, 2*w^3 - 8*w + 1], [191, 191, w^3 - 5*w - 5], [211, 211, 2*w + 5], [227, 227, w^3 + w^2 - 4*w - 7], [239, 239, 2*w^3 + 2*w^2 - 8*w - 5], [257, 257, 2*w^3 - 4*w^2 - 8*w + 19], [257, 257, w^3 + 2*w^2 - 5*w - 7], [263, 263, -2*w^3 - 2*w^2 + 6*w + 1], [269, 269, -2*w^3 + 4*w^2 + 8*w - 13], [277, 277, -3*w^2 + w + 13], [293, 293, -2*w^3 + 4*w^2 + 8*w - 17], [293, 293, -3*w^3 - 3*w^2 + 14*w + 11], [293, 293, 3*w^3 - 4*w^2 - 15*w + 19], [293, 293, 3*w^3 + w^2 - 14*w - 7], [307, 307, 2*w^3 - 3*w^2 - 5*w + 1], [307, 307, 2*w^3 - 4*w^2 - 4*w + 7], [311, 311, -3*w^3 + 3*w^2 + 14*w - 13], [313, 313, w^3 + 3*w^2 + 2*w + 1], [317, 317, -3*w^3 - 3*w^2 + 10*w + 5], [331, 331, -2*w^3 - 3*w^2 + 9*w + 13], [337, 337, -w^3 + 2*w^2 + 3*w - 11], [337, 337, w^3 + 2*w^2 - w - 5], [337, 337, -w^3 + 2*w^2 + 3*w - 1], [337, 337, w^3 - 3*w - 7], [347, 347, 3*w^3 - 4*w^2 - 15*w + 17], [347, 347, -w^3 + 2*w^2 + 5*w - 5], [349, 349, -2*w^3 + 2*w^2 + 8*w - 5], [353, 353, -2*w^3 + 3*w^2 + 3*w + 1], [361, 19, -5*w^3 - 4*w^2 + 21*w + 17], [361, 19, 3*w^3 + w^2 - 10*w + 1], [373, 373, -2*w^3 + w^2 + 11*w - 1], [379, 379, w^3 - w^2 - 6*w - 1], [379, 379, -w^3 + 7*w - 5], [389, 389, 2*w^3 - 2*w^2 - 8*w + 7], [397, 397, 2*w^3 + 5*w^2 - 5*w - 13], [409, 409, w^3 + w^2 - 6*w - 1], [421, 421, -2*w^3 + 2*w^2 + 10*w - 5], [433, 433, -w^3 - 3*w^2 + 6*w + 7], [439, 439, 3*w^2 - 3*w - 7], [443, 443, 3*w^3 + w^2 - 14*w - 5], [449, 449, -4*w^3 - 3*w^2 + 19*w + 17], [449, 449, -w^3 + 3*w^2 + 6*w - 17], [457, 457, w^3 + 2*w^2 + w + 1], [457, 457, -2*w^3 + w^2 + 9*w - 7], [461, 461, -w^3 + 7*w + 7], [461, 461, -4*w^2 + 17], [467, 467, -2*w^3 + w^2 + 11*w + 1], [479, 479, 5*w^3 - 8*w^2 - 25*w + 37], [487, 487, w^3 + w^2 - 8*w + 1], [487, 487, 7*w^3 - 12*w^2 - 31*w + 49], [487, 487, 2*w^3 + w^2 - 13*w - 11], [487, 487, 4*w^3 + 3*w^2 - 17*w - 11], [491, 491, -2*w^2 - 2*w + 1], [499, 499, 2*w^3 - 8*w - 1], [503, 503, -2*w^3 - w^2 + 11*w + 11], [503, 503, 3*w^3 - w^2 - 14*w + 1], [509, 509, -4*w^3 + 9*w^2 + 17*w - 37], [523, 523, -w^3 - 3*w^2 + 5], [541, 541, 6*w^3 + 6*w^2 - 20*w - 13], [541, 541, w^3 - 7*w + 1], [547, 547, w^2 - 3*w - 5], [547, 547, w^2 + 3*w - 1], [557, 557, -7*w^3 - 4*w^2 + 35*w + 25], [557, 557, -3*w^3 - 2*w^2 + 13*w + 7], [569, 569, -2*w^3 + 5*w^2 + 5*w - 11], [569, 569, w^3 + w^2 - 8*w - 5], [571, 571, -w^3 - 3*w^2 + 4*w + 11], [571, 571, -2*w^3 - 2*w^2 + 12*w + 13], [593, 593, -4*w^2 + 4*w + 7], [593, 593, 3*w^2 + w - 13], [599, 599, -2*w^3 + 2*w^2 + 6*w - 1], [601, 601, -w^3 + 3*w - 5], [607, 607, 2*w^3 - 12*w - 5], [607, 607, w^3 + w^2 + 1], [613, 613, -2*w^3 - 2*w^2 + 6*w + 7], [617, 617, w^3 - 7*w - 1], [619, 619, -2*w^3 + 12*w - 1], [625, 5, -5], [631, 631, -w^3 + w^2 + 2*w - 5], [641, 641, 2*w^3 - w^2 - 7*w - 1], [641, 641, 4*w^3 + 2*w^2 - 22*w - 19], [641, 641, -w^3 - 2*w^2 + 7*w + 11], [641, 641, 3*w^2 - 3*w - 11], [661, 661, -w^3 - w^2 + 6*w - 1], [661, 661, 3*w^3 + 2*w^2 - 11*w - 7], [673, 673, 3*w^3 - 2*w^2 - 13*w + 11], [673, 673, 2*w^3 - 6*w - 1], [683, 683, w^2 + 3*w - 5], [709, 709, 3*w^2 - w - 11], [719, 719, -2*w^3 + 6*w + 5], [719, 719, 2*w^3 + 5*w^2 - w - 5], [727, 727, 2*w^2 - 2*w - 11], [733, 733, 4*w^2 - 2*w - 19], [739, 739, -2*w^3 + 4*w^2 + 6*w - 13], [739, 739, w^3 - 9*w + 7], [739, 739, -2*w^3 + 10*w - 1], [739, 739, -2*w^3 - 2*w^2 + 12*w + 7], [751, 751, -4*w^3 - 3*w^2 + 15*w + 11], [751, 751, -2*w^3 - w^2 + 5*w - 1], [821, 821, 3*w^3 + 5*w^2 - 10*w - 11], [823, 823, -w^3 - w^2 + 8*w + 11], [839, 839, w^3 + 2*w^2 + w - 1], [853, 853, 2*w^3 + w^2 - 7*w - 7], [857, 857, w^3 + 4*w^2 - w - 13], [859, 859, 2*w - 7], [859, 859, -5*w^3 + 11*w^2 + 20*w - 43], [863, 863, 2*w^3 + 2*w^2 - 8*w - 1], [863, 863, -w^3 + 3*w^2 + 4*w - 7], [877, 877, -4*w^3 + 7*w^2 + 17*w - 31], [883, 883, 5*w^3 - 7*w^2 - 24*w + 31], [907, 907, 4*w^3 - 6*w^2 - 16*w + 19], [919, 919, 2*w^3 + 3*w^2 - 9*w - 11], [929, 929, w^3 - w^2 - 2*w - 5], [937, 937, -w^3 + 3*w^2 + 2*w - 11], [937, 937, -3*w^3 + 7*w^2 + 12*w - 29], [947, 947, 2*w^3 - 3*w^2 - 11*w + 13], [947, 947, 2*w^3 - w^2 - 11*w + 7], [961, 31, 2*w^3 - 4*w^2 - 2*w - 1], [967, 967, 3*w^3 + 4*w^2 - 9*w - 11], [971, 971, -4*w^3 - 4*w^2 + 18*w + 17], [991, 991, -w^3 + w^2 + 2*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^5 + 3*x^4 - 2*x^3 - 9*x^2 - 2*x + 2; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 0, e^4 + 2*e^3 - 3*e^2 - 4*e, -2*e^4 - 5*e^3 + 7*e^2 + 12*e - 6, e^4 + e^3 - 6*e^2 - 6*e + 4, 2*e^4 + 5*e^3 - 6*e^2 - 14*e - 4, e^4 + e^3 - 4*e^2 - 2*e + 4, 2*e^4 + 4*e^3 - 10*e^2 - 12*e + 4, -e^4 - 3*e^3 + 4*e^2 + 12*e - 4, -e^4 - 4*e^3 + 10*e + 4, 3*e^3 + 4*e^2 - 8*e - 2, -2*e^3 - e^2 + 8*e - 4, -3*e^4 - 6*e^3 + 9*e^2 + 10*e - 2, 3*e^3 + 5*e^2 - 10*e - 10, 5*e^4 + 10*e^3 - 18*e^2 - 28*e, 2*e^4 + 6*e^3 - 7*e^2 - 18*e + 6, e^4 - e^3 - 9*e^2 + 4*e + 10, -3*e^4 - 5*e^3 + 13*e^2 + 12*e - 10, -2*e^3 - 5*e^2 + 6*e + 6, -2*e^4 - 3*e^3 + 7*e^2 + 4*e - 6, 5*e^4 + 11*e^3 - 17*e^2 - 22*e + 14, -3*e^4 - 6*e^3 + 12*e^2 + 10*e - 12, -e^4 - 3*e^3 + 3*e^2 + 16*e - 2, -2*e^4 - 3*e^3 + 11*e^2 + 8*e - 6, -3*e^4 - 7*e^3 + 6*e^2 + 14*e, -5*e^4 - 12*e^3 + 16*e^2 + 36*e - 4, -3*e^4 - 3*e^3 + 16*e^2 + 12*e - 8, -3*e^4 - 5*e^3 + 8*e^2 + 10*e, -3*e^4 - 8*e^3 + 2*e^2 + 18*e + 8, -2*e^4 - 9*e^3 + 2*e^2 + 28*e + 4, -6*e^4 - 12*e^3 + 20*e^2 + 30*e - 6, e^4 - 2*e^3 - 8*e^2 + 12*e + 4, e^4 + 8*e^3 + 5*e^2 - 20*e - 2, 8*e^4 + 15*e^3 - 26*e^2 - 34*e + 2, 9*e^4 + 18*e^3 - 30*e^2 - 38*e + 12, -3*e^4 - 3*e^3 + 18*e^2 + 10*e - 20, -4*e^4 - 8*e^3 + 19*e^2 + 28*e - 14, 2*e^4 + 11*e^3 + e^2 - 36*e - 10, 2*e^4 + 9*e^3 + 2*e^2 - 20*e - 10, -4*e^4 - 3*e^3 + 21*e^2 + 8*e - 8, 7*e^4 + 14*e^3 - 28*e^2 - 38*e + 10, -10*e^4 - 19*e^3 + 35*e^2 + 46*e - 12, -5*e^4 - 12*e^3 + 18*e^2 + 40*e - 8, -5*e^4 - 12*e^3 + 12*e^2 + 24*e + 8, e^4 + 6*e^3 + 10*e^2 - 14*e - 22, 3*e^4 + 6*e^3 - 17*e^2 - 14*e + 20, -4*e^4 - 5*e^3 + 21*e^2 - 2*e - 30, -2*e^4 - 6*e^3 - 2*e^2 + 8*e + 12, -3*e^4 - 12*e^3 - e^2 + 28*e + 6, 2*e^4 + 4*e^3 - 8*e^2 - 10*e + 8, 2*e^4 + 5*e^3 - 8*e^2 - 22*e - 12, -5*e^4 - 9*e^3 + 23*e^2 + 26*e - 14, -4*e^4 - 10*e^3 + 9*e^2 + 22*e + 16, 8*e^4 + 18*e^3 - 23*e^2 - 48*e + 2, 7*e^4 + 16*e^3 - 20*e^2 - 44*e - 4, -e^3 + 2*e^2 + 14*e - 8, -6*e^4 - 14*e^3 + 15*e^2 + 32*e + 4, 4*e^4 + 3*e^3 - 16*e^2 - 2*e + 4, 3*e^4 + 6*e^3 - 7*e^2 - 4*e + 2, -11*e^4 - 19*e^3 + 36*e^2 + 38*e - 8, -4*e^4 - 6*e^3 + 22*e^2 + 6*e - 32, -2*e^3 - e^2 + 10*e + 6, 2*e^4 + 8*e^3 - 11*e^2 - 38*e + 12, 2*e^4 + 12*e^3 - 42*e, -4*e^4 - 11*e^3 + 2*e^2 + 14*e + 20, 3*e^4 - 2*e^3 - 22*e^2 + 12*e + 16, -3*e^4 - 3*e^3 + 11*e^2 + 2*e - 6, 8*e^4 + 21*e^3 - 22*e^2 - 64*e, -4*e^4 - 13*e^3 + 11*e^2 + 38*e - 10, -5*e^3 - 9*e^2 + 14*e - 6, -2*e^4 - e^3 + 20*e^2 + 6*e - 20, 3*e^4 + 14*e^3 - 8*e^2 - 54*e - 2, 2*e^4 + 5*e^3 - 6*e^2 - 14*e + 20, 11*e^4 + 25*e^3 - 33*e^2 - 58*e - 6, -10*e^4 - 23*e^3 + 22*e^2 + 52*e + 8, -8*e^4 - 16*e^3 + 24*e^2 + 34*e - 12, 4*e^4 + 10*e^3 + e^2 - 12*e - 26, 4*e^4 + 7*e^3 - 8*e^2 - 10*e - 28, -2*e^4 - 2*e^3 + 18*e^2 + 10*e - 34, -9*e^4 - 20*e^3 + 25*e^2 + 42*e - 2, 7*e^4 + 15*e^3 - 34*e^2 - 40*e + 28, 5*e^4 + 17*e^3 - 10*e^2 - 50*e - 12, 8*e^4 + 11*e^3 - 44*e^2 - 40*e + 36, 7*e^4 + 16*e^3 - 33*e^2 - 50*e + 24, -2*e^4 - 6*e^3 - 3*e^2 + 8*e + 2, -3*e^4 - 6*e^3 + 14*e^2 + 12*e + 8, -9*e^4 - 16*e^3 + 37*e^2 + 54*e - 18, 4*e^4 + 11*e^3 - 8*e^2 - 26*e - 20, -3*e^4 - 9*e^3 + 20*e + 10, -e^4 - 5*e^3 + e^2 + 20*e - 10, 9*e^4 + 23*e^3 - 20*e^2 - 56*e - 10, -9*e^4 - 22*e^3 + 36*e^2 + 64*e - 12, -7*e^4 - 22*e^3 + 21*e^2 + 64*e - 4, 3*e^4 - 5*e^3 - 30*e^2 + 14*e + 36, 9*e^4 + 19*e^3 - 30*e^2 - 50*e + 12, 7*e^4 + 14*e^3 - 17*e^2 - 34*e - 14, -10*e^4 - 22*e^3 + 31*e^2 + 60*e - 14, -9*e^4 - 20*e^3 + 31*e^2 + 60*e - 12, 6*e^4 - 37*e^2 + 12*e + 38, -5*e^4 - 18*e^3 - 2*e^2 + 52*e + 24, -5*e^4 - 7*e^3 + 19*e^2 + 24*e - 2, 3*e^4 + 8*e^3 - 17*e^2 - 30*e + 14, -7*e^4 - 14*e^3 + 26*e^2 + 40*e - 14, 9*e^4 + 12*e^3 - 41*e^2 - 22*e + 26, -7*e^4 - 8*e^3 + 23*e^2 + 6*e + 2, 2*e^4 + 6*e^3 - 2*e^2 - 14*e - 24, -10*e^4 - 16*e^3 + 42*e^2 + 40*e - 16, -7*e^4 - 24*e^3 + 15*e^2 + 62*e - 6, -2*e^4 - e^3 + 18*e^2 - 10*e - 38, -4*e^4 - 14*e^3 + 9*e^2 + 44*e + 14, -6*e^4 - e^3 + 39*e^2 - 2*e - 24, -e^4 - 4*e^3 + 10*e^2 + 26*e - 12, 6*e^4 + 3*e^3 - 35*e^2 + 6*e + 32, -5*e^4 + 35*e^2 - 14*e - 42, 10*e^4 + 26*e^3 - 23*e^2 - 66*e - 6, 3*e^4 - 3*e^3 - 15*e^2 + 24*e + 16, 14*e^4 + 26*e^3 - 43*e^2 - 66*e - 14, 5*e^4 + 14*e^3 - 11*e^2 - 34*e, -12*e^4 - 19*e^3 + 48*e^2 + 34*e - 24, 4*e^4 - 4*e^3 - 33*e^2 + 18*e + 30, -8*e^4 - 16*e^3 + 36*e^2 + 40*e - 40, -8*e^4 - 12*e^3 + 27*e^2 + 24*e + 10, 13*e^4 + 22*e^3 - 51*e^2 - 48*e + 18, -5*e^4 - 7*e^3 + 28*e^2 + 16*e - 44, -10*e^4 - 13*e^3 + 42*e^2 + 22*e - 14, 3*e^4 + 8*e^3 - 5*e^2 - 2*e + 4, 8*e^4 + 12*e^3 - 26*e^2 - 18*e + 16, 3*e^4 + 15*e^3 - 58*e - 24, 12*e^4 + 32*e^3 - 29*e^2 - 84*e - 18, 7*e^4 + 24*e^3 - 9*e^2 - 66*e - 14, -4*e^4 - 10*e^3 + 7*e^2 + 18*e - 6, 8*e^4 + 14*e^3 - 37*e^2 - 26*e + 30, e^4 + 12*e^3 + 11*e^2 - 34*e + 4, 13*e^4 + 31*e^3 - 33*e^2 - 56*e - 2, -e^4 + 3*e^3 + 14*e^2 - 14*e - 32, 4*e^4 + 18*e^3 - e^2 - 42*e - 2, 9*e^4 + 17*e^3 - 26*e^2 - 38*e + 4, -2*e^4 + 2*e^3 + 15*e^2 - 14*e - 18, 2*e^4 + 2*e^3 - e^2 + 14*e - 2, -4*e^4 - 16*e^3 + e^2 + 46*e - 6, -13*e^4 - 33*e^3 + 41*e^2 + 100*e - 10, -6*e^4 - 12*e^3 + 12*e^2 + 28*e + 28, -7*e^4 - 12*e^3 + 29*e^2 + 44*e - 6, -14*e^4 - 18*e^3 + 73*e^2 + 40*e - 70, 11*e^4 + 19*e^3 - 45*e^2 - 52*e + 10, e^4 + 7*e^3 - 6*e^2 - 38*e + 12, 5*e^4 + 13*e^3 - 32*e^2 - 52*e + 30, 8*e^4 + 29*e^3 - 8*e^2 - 76*e + 2, -6*e^4 - 5*e^3 + 34*e^2 + 4*e - 20, -17*e^4 - 30*e^3 + 67*e^2 + 78*e - 30, 9*e^4 + 5*e^3 - 47*e^2 + 12*e + 50, -7*e^4 - 5*e^3 + 32*e^2 - 4*e - 38, 4*e^4 + 6*e^3 + 2*e^2 + 16*e - 36, -3*e^4 - 14*e^3 + 2*e^2 + 56*e + 8, -7*e^4 - 14*e^3 + 26*e^2 + 32*e - 10, 5*e^4 + 8*e^3 - 26*e^2 - 28*e + 32, 5*e^4 + 20*e^3 - 16*e^2 - 68*e, 18*e^4 + 26*e^3 - 79*e^2 - 66*e + 36, -13*e^4 - 33*e^3 + 20*e^2 + 74*e + 24, -3*e^4 - e^3 + 30*e^2 - 2*e - 28, 4*e^3 + 13*e^2 + 6*e - 6]; 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;