/* 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, 10, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 1], [5, 5, w^3 + w^2 - 4*w + 1], [7, 7, -w^3 + 6*w - 2], [8, 2, w^3 - 7*w + 3], [11, 11, -w^3 + 6*w - 4], [17, 17, w^3 + w^2 - 6*w - 1], [17, 17, -w^2 - w + 3], [31, 31, 2*w^3 + w^2 - 13*w + 3], [37, 37, -3*w^3 - w^2 + 18*w - 3], [41, 41, w^2 + 2*w - 4], [47, 47, 2*w^3 + 2*w^2 - 11*w - 2], [47, 47, -3*w^3 - w^2 + 19*w - 6], [59, 59, -2*w^3 + 13*w - 6], [59, 59, -w^3 - w^2 + 5*w + 2], [59, 59, w^2 + w - 1], [59, 59, w^2 + 2*w - 6], [61, 61, -w^3 + w^2 + 8*w - 7], [67, 67, w^3 + 2*w^2 - 4*w - 4], [73, 73, -2*w^3 - w^2 + 12*w - 4], [81, 3, -3], [107, 107, 2*w^2 + 2*w - 11], [109, 109, 2*w^3 + w^2 - 11*w + 3], [113, 113, w^3 + w^2 - 7*w - 2], [113, 113, 2*w^3 - 12*w + 9], [125, 5, -4*w^3 - 3*w^2 + 24*w + 2], [131, 131, 5*w^3 + 3*w^2 - 31*w + 2], [151, 151, 4*w^3 + 2*w^2 - 26*w + 1], [151, 151, w^3 + 2*w^2 - 6*w - 10], [167, 167, w + 4], [167, 167, 4*w^3 + w^2 - 26*w + 10], [173, 173, -2*w^3 + w^2 + 13*w - 11], [179, 179, 4*w^3 + w^2 - 24*w + 4], [179, 179, 2*w^3 + 2*w^2 - 12*w - 3], [179, 179, -w^3 + 8*w - 4], [179, 179, w^3 + 2*w^2 - 4*w - 6], [181, 181, 6*w^3 + 3*w^2 - 39*w + 5], [197, 197, -w^3 + 4*w - 4], [199, 199, 2*w^3 - 12*w + 3], [199, 199, 3*w - 2], [223, 223, w^3 + w^2 - 7*w - 4], [223, 223, 2*w^3 + w^2 - 10*w + 4], [227, 227, 7*w^3 + 2*w^2 - 44*w + 12], [227, 227, -4*w^3 - w^2 + 25*w - 9], [229, 229, 8*w^3 + 4*w^2 - 51*w + 4], [229, 229, 3*w^3 + w^2 - 18*w + 5], [233, 233, -6*w^3 - 3*w^2 + 39*w - 7], [233, 233, -w^3 - 2*w^2 + 8*w + 2], [241, 241, 2*w^3 + w^2 - 10*w], [251, 251, w^3 + w^2 - 7*w - 6], [257, 257, -6*w^3 - w^2 + 39*w - 17], [263, 263, -2*w^3 + 13*w - 4], [263, 263, -w^3 + 6*w - 8], [269, 269, -8*w^3 - 4*w^2 + 50*w - 7], [269, 269, -w^3 + w^2 + 7*w - 6], [277, 277, 2*w^3 + 2*w^2 - 11*w + 4], [277, 277, w^3 + w^2 - 4*w - 3], [281, 281, -w^2 - 3*w + 5], [281, 281, 2*w^3 + 3*w^2 - 9*w - 3], [283, 283, -3*w^3 - 4*w^2 + 12*w + 2], [289, 17, -2*w^3 - w^2 + 13*w - 5], [293, 293, -w^3 - 2*w^2 + 6*w + 6], [293, 293, w^3 + w^2 - 8*w - 1], [307, 307, 2*w^3 - 11*w + 12], [317, 317, 2*w^3 - 14*w + 11], [317, 317, -2*w^3 - 2*w^2 + 8*w + 1], [337, 337, 2*w^2 + 4*w - 11], [343, 7, -5*w^3 - w^2 + 31*w - 10], [349, 349, 2*w^3 - 11*w + 8], [349, 349, w^2 + w - 9], [353, 353, 4*w^3 + 4*w^2 - 23*w - 8], [373, 373, -2*w - 3], [373, 373, -4*w^3 + 26*w - 13], [389, 389, 6*w^3 + 3*w^2 - 37*w + 1], [389, 389, -2*w^3 - w^2 + 14*w - 6], [401, 401, 2*w^3 - 12*w + 13], [421, 421, 2*w^2 + 2*w - 15], [421, 421, 3*w^3 + 2*w^2 - 20*w + 4], [431, 431, w^3 + 3*w^2 - 2*w - 9], [433, 433, 2*w^3 + 2*w^2 - 12*w + 3], [433, 433, -7*w^3 - 2*w^2 + 44*w - 16], [433, 433, -3*w^3 + 18*w - 14], [433, 433, -w^3 + w^2 + 9*w - 10], [439, 439, 5*w^3 + 3*w^2 - 31*w - 2], [443, 443, -w^3 - w^2 + 3*w - 4], [443, 443, -4*w^3 + 27*w - 14], [449, 449, w^3 + w^2 - 4*w - 5], [449, 449, w^3 + 3*w^2 - 3*w - 8], [449, 449, 2*w^3 - 11*w + 6], [449, 449, -3*w^3 - 2*w^2 + 18*w - 4], [461, 461, -3*w^3 + w^2 + 20*w - 15], [463, 463, 2*w^3 - 10*w + 7], [479, 479, -2*w^3 - 3*w^2 + 11*w + 9], [491, 491, w^3 + w^2 - 4*w + 5], [499, 499, -2*w^3 - 2*w^2 + 14*w - 3], [499, 499, -6*w^3 - 2*w^2 + 39*w - 10], [499, 499, 4*w^3 + w^2 - 24*w + 14], [499, 499, -5*w^3 - 3*w^2 + 30*w - 5], [503, 503, -2*w^3 + w^2 + 14*w - 18], [509, 509, 4*w^3 + 3*w^2 - 24*w], [509, 509, 4*w^3 + 2*w^2 - 23*w + 4], [547, 547, w^3 + 2*w^2 - 2*w - 6], [547, 547, 6*w^3 + 2*w^2 - 38*w + 7], [563, 563, -5*w^3 - w^2 + 33*w - 12], [569, 569, 2*w^3 - w^2 - 15*w + 13], [569, 569, -4*w^3 - 2*w^2 + 24*w - 7], [577, 577, -w^2 - 2], [577, 577, 3*w^3 - 20*w + 8], [587, 587, 6*w^3 + 4*w^2 - 36*w - 1], [587, 587, -w^3 + w^2 + 9*w - 8], [593, 593, 6*w^3 + 3*w^2 - 36*w + 2], [593, 593, -3*w^3 - w^2 + 21*w - 4], [599, 599, -2*w^3 - w^2 + 12*w - 10], [599, 599, -4*w^3 - 2*w^2 + 25*w - 6], [601, 601, -2*w^3 + 2*w^2 + 15*w - 16], [607, 607, w^3 - w^2 - 7*w + 4], [607, 607, -2*w^3 - 2*w^2 + 10*w + 3], [631, 631, 2*w^2 + 4*w - 7], [641, 641, -3*w^3 - 2*w^2 + 16*w], [643, 643, w^3 + w^2 - 3*w - 4], [643, 643, w^3 + 3*w^2 - 6*w - 13], [653, 653, -w^3 + 6*w + 2], [653, 653, -3*w^2 - 5*w + 11], [659, 659, -2*w^3 + 10*w - 5], [661, 661, w^3 - w^2 - 8*w + 3], [673, 673, 3*w^3 + 4*w^2 - 14*w - 2], [677, 677, 4*w^3 + 2*w^2 - 23*w + 6], [677, 677, 2*w^3 - 13*w + 14], [683, 683, 2*w^3 + w^2 - 13*w + 7], [701, 701, 8*w^3 + 4*w^2 - 51*w + 8], [701, 701, -4*w^3 - w^2 + 24*w - 12], [709, 709, -4*w^3 - w^2 + 27*w - 7], [709, 709, 3*w^3 + 3*w^2 - 18*w - 5], [709, 709, -3*w^3 + w^2 + 21*w - 14], [709, 709, w^2 - w - 7], [719, 719, -9*w^3 - 4*w^2 + 58*w - 8], [719, 719, 11*w^3 + 5*w^2 - 68*w + 9], [727, 727, -w^3 + 4*w - 6], [733, 733, 2*w^3 + w^2 - 13*w + 9], [739, 739, -2*w^2 - 3*w + 4], [743, 743, 3*w^3 + 3*w^2 - 19*w - 4], [751, 751, 2*w^3 + 3*w^2 - 10*w - 2], [761, 761, -12*w^3 - 5*w^2 + 75*w - 9], [761, 761, -3*w^3 - 3*w^2 + 17*w], [769, 769, -6*w^3 - 4*w^2 + 37*w], [769, 769, -8*w^3 - 3*w^2 + 51*w - 7], [773, 773, -w^2 - 2*w - 2], [787, 787, -5*w^3 - 5*w^2 + 28*w + 7], [811, 811, 7*w^3 + 2*w^2 - 42*w + 6], [811, 811, -2*w^3 - w^2 + 9*w - 7], [823, 823, -3*w^3 - 2*w^2 + 14*w - 10], [829, 829, 4*w^3 + 3*w^2 - 20*w + 6], [829, 829, -7*w^3 - 2*w^2 + 44*w - 14], [841, 29, w^3 + 4*w^2 - 2*w - 14], [841, 29, -w^2 + 10], [857, 857, -3*w^3 - w^2 + 20*w - 9], [859, 859, -w^3 + 2*w^2 + 10*w - 18], [863, 863, 11*w^3 + 5*w^2 - 69*w + 6], [863, 863, w^3 - 2*w^2 - 10*w + 12], [877, 877, 3*w^3 - 18*w + 10], [883, 883, 4*w^3 + 3*w^2 - 22*w], [887, 887, 3*w^3 + 3*w^2 - 15*w - 2], [887, 887, w^2 + 4*w - 10], [887, 887, w^3 + w^2 - 3*w - 8], [887, 887, w - 6], [907, 907, 6*w^3 + w^2 - 38*w + 16], [919, 919, -2*w^3 - 3*w^2 + 11*w + 1], [919, 919, 5*w^3 - w^2 - 34*w + 27], [937, 937, -w^3 - w^2 + 8*w + 3], [941, 941, -4*w^3 + 25*w - 20], [947, 947, -2*w^3 + w^2 + 8*w - 6], [967, 967, -7*w^3 - w^2 + 44*w - 23], [971, 971, 2*w^2 + 2*w - 3]]; primes := [ideal : I in primesArray]; heckePol := x^5 - 19*x^3 + 10*x^2 + 56*x - 16; K := NumberField(heckePol); heckeEigenvaluesArray := [0, e, -1/2*e^2 - 1/2*e + 5, 1/4*e^3 + 1/4*e^2 - 5/2*e + 1, 2, 1/4*e^4 - 17/4*e^2 + 2*e + 7, 1/4*e^4 + 1/4*e^3 - 7/2*e^2 + 4, 2*e + 2, -1/4*e^4 - 3/4*e^3 + 7/2*e^2 + 11/2*e - 7, -1/4*e^4 - 1/4*e^3 + 4*e^2 + 1/2*e - 5, 1/4*e^4 - 17/4*e^2 + e + 9, 1/2*e^2 + 1/2*e + 1, 1/2*e^2 - 3/2*e - 5, -1/2*e^4 - 1/2*e^3 + 8*e^2 - e - 14, -e^2 - 3*e + 10, -1/4*e^4 + 1/2*e^3 + 19/4*e^2 - 8*e - 11, -1/4*e^4 - 3/4*e^3 + 4*e^2 + 8*e - 10, -1/2*e^3 - 3/2*e^2 + 6*e + 8, -1/2*e^3 + 19/2*e - 1, -1/4*e^4 + 1/2*e^3 + 19/4*e^2 - 9*e - 9, -1/4*e^4 - 1/2*e^3 + 15/4*e^2 + 2*e - 7, -1/4*e^4 - 1/4*e^3 + 7/2*e^2 - 2*e + 6, 1/2*e^3 - 1/2*e^2 - 5*e + 8, 1/4*e^4 + 5/4*e^3 - 3/2*e^2 - 13*e - 4, -1/2*e^3 - 5/2*e^2 + 4*e + 16, -1/2*e^4 - 1/2*e^3 + 7*e^2 - 6, 1/4*e^4 - 1/2*e^3 - 23/4*e^2 + 9*e + 9, -1/2*e^4 - 1/2*e^3 + 9*e^2 + 4*e - 16, e^3 + e^2 - 14*e + 2, 1/2*e^4 + 1/2*e^3 - 15/2*e^2 - 1/2*e + 23, -1/4*e^4 - 3/4*e^3 + 7/2*e^2 + 15/2*e - 11, -1/2*e^4 - 1/2*e^3 + 9*e^2 + 2*e - 20, 1/2*e^3 + 1/2*e^2 - 7*e - 6, e^2 - e - 8, -1/2*e^3 - 1/2*e^2 + 9*e - 4, 1/4*e^4 + 1/4*e^3 - 9/2*e^2 - e + 8, 1/4*e^4 + 1/2*e^3 - 11/4*e^2 - 4*e + 5, 1/2*e^3 + 3/2*e^2 - 2*e - 6, 1/2*e^4 - 19/2*e^2 + e + 26, 1/4*e^4 - 13/4*e^2 - 3, -1/2*e^3 - 5/2*e^2 + e + 16, -1/2*e^4 - 1/2*e^3 + 9*e^2 - 20, -1/2*e^4 + e^3 + 21/2*e^2 - 15*e - 24, -3/4*e^4 - 3/2*e^3 + 45/4*e^2 + 10*e - 17, -1/4*e^4 + 1/4*e^3 + 2*e^2 - 7*e + 10, -1/4*e^4 - 7/4*e^3 + e^2 + 20*e, -1/2*e^4 - e^3 + 13/2*e^2 + 8*e - 14, 3/4*e^4 - 47/4*e^2 + 9*e + 15, 1/4*e^4 - 21/4*e^2 + 6*e + 9, 1/2*e^3 - 2*e^2 - 19/2*e + 15, -1/2*e^3 - 3/2*e^2 + 10, 1/4*e^4 + 1/2*e^3 - 11/4*e^2 - 5*e + 5, 1/4*e^4 + 1/4*e^3 - 9/2*e^2 - 5*e + 28, -1/4*e^4 - 1/4*e^3 + 6*e^2 + 13/2*e - 23, -e^4 - e^3 + 17*e^2 + 3*e - 34, -e^4 - 2*e^3 + 15*e^2 + 15*e - 32, e^3 + 2*e^2 - 11*e, -e^4 - e^3 + 16*e^2 + 2*e - 26, 1/2*e^4 - 19/2*e^2 + 5*e + 24, -2*e^2 + 4*e + 26, -1/2*e^4 + 17/2*e^2 - 6*e - 32, -1/4*e^4 - e^3 + 5/4*e^2 + 8*e + 9, 1/4*e^4 + 2*e^3 - 1/4*e^2 - 23*e - 3, 1/4*e^4 - 7/4*e^3 - 13/2*e^2 + 23*e + 20, -1/2*e^4 + 15/2*e^2 - 6*e + 2, -1/2*e^3 - 1/2*e^2 + 10*e + 12, e^4 + 1/2*e^3 - 17*e^2 + 9/2*e + 29, -1/2*e^4 + 15/2*e^2 - 7*e - 12, -1/4*e^4 + e^3 + 25/4*e^2 - 19*e - 13, 3/4*e^4 + e^3 - 47/4*e^2 - 8*e + 25, -3/4*e^4 + 1/2*e^3 + 57/4*e^2 - 11*e - 21, 3/4*e^4 + 3/4*e^3 - 13*e^2 - 21/2*e + 27, -1/2*e^3 + 1/2*e^2 + 9*e - 8, -1/4*e^4 + 1/4*e^3 + 6*e^2 - 3*e - 14, -1/2*e^4 - e^3 + 13/2*e^2 + 3*e + 2, 1/2*e^4 + 2*e^3 - 15/2*e^2 - 18*e + 12, -1/4*e^4 + 3/4*e^3 + 7/2*e^2 - 9*e + 8, -1/4*e^4 - e^3 + 9/4*e^2 + 2*e + 7, 3/4*e^4 + 3/4*e^3 - 12*e^2 - 11/2*e + 41, -1/2*e^3 + e^2 + 25/2*e - 17, -1/2*e^4 + 15/2*e^2 - 9*e - 2, -1/4*e^4 + 1/4*e^3 + 6*e^2 - 7*e - 14, e^4 + 3/2*e^3 - 31/2*e^2 - 9*e + 36, -e^4 - e^3 + 14*e^2 - 12, 1/2*e^4 + 1/2*e^3 - 5*e^2 - 4*e - 18, 1/4*e^4 - 17/4*e^2 + 6*e - 13, -3/4*e^4 - 1/2*e^3 + 53/4*e^2 - 23, 1/2*e^4 + e^3 - 13/2*e^2 - 3*e - 8, 1/2*e^4 - e^3 - 21/2*e^2 + 25*e + 30, 1/4*e^4 - 1/4*e^3 - 5*e^2 + 2*e + 20, 1/2*e^3 + 3*e^2 - 5/2*e - 31, 1/4*e^4 - 3/2*e^3 - 31/4*e^2 + 20*e + 23, 5/4*e^4 + 3/2*e^3 - 79/4*e^2 - 4*e + 23, 1/2*e^4 - 17/2*e^2 + 10*e + 12, -e^4 + 15*e^2 - 12*e - 30, 1/2*e^4 - 1/2*e^3 - 17/2*e^2 + 23/2*e + 9, -1/2*e^3 - 3/2*e^2 + 4*e - 2, 1/2*e^4 + 3*e^3 - 11/2*e^2 - 32*e + 12, 3/4*e^4 + 1/2*e^3 - 45/4*e^2 + 8*e + 17, 3/4*e^4 + 5/4*e^3 - 12*e^2 - 15*e + 44, 1/2*e^4 + 5/2*e^3 - 3*e^2 - 22*e - 20, e^4 + 1/2*e^3 - 37/2*e^2 + 7*e + 40, 1/2*e^4 - 1/2*e^3 - 9*e^2 + 9*e + 6, 3/4*e^4 + 3/4*e^3 - 12*e^2 + 1/2*e + 33, 1/4*e^4 + 1/4*e^3 - 7/2*e^2 + 6*e - 14, e^3 + 2*e^2 - 15*e - 34, 3/4*e^4 + 3/4*e^3 - 23/2*e^2 - 3*e + 20, -3/4*e^4 - 3/2*e^3 + 49/4*e^2 + 15*e - 35, -1/2*e^4 - 2*e^3 + 9/2*e^2 + 22*e - 18, 1/2*e^4 - 1/2*e^3 - 10*e^2 + 15*e + 16, -1/4*e^4 - 7/4*e^3 + 3/2*e^2 + 25/2*e - 1, 3/4*e^4 + 1/2*e^3 - 45/4*e^2 - 3*e - 1, -e^4 - 1/2*e^3 + 37/2*e^2 - 3*e - 44, 3/4*e^4 + 9/4*e^3 - 10*e^2 - 18*e + 16, -1/2*e^4 + 11/2*e^2 - 5*e + 12, -1/2*e^4 - e^3 + 15/2*e^2 + 4*e - 20, 1/2*e^4 + 3*e^3 - 3/2*e^2 - 32*e - 18, -1/2*e^4 - 2*e^3 + 13/2*e^2 + 20*e - 38, 3/4*e^4 - 1/2*e^3 - 53/4*e^2 + 10*e + 35, -4*e^2 - 4*e + 36, -1/2*e^4 - 1/2*e^3 + 8*e^2 - 4*e - 16, -1/2*e^4 + e^3 + 17/2*e^2 - 15*e - 18, -3/2*e^3 + e^2 + 47/2*e - 17, -3/4*e^4 + 1/4*e^3 + 27/2*e^2 - 4*e - 6, e^4 + e^3 - 16*e^2 - e + 36, -1/4*e^4 + 5/4*e^3 + 13/2*e^2 - 47/2*e - 5, -e^4 - e^3 + 17*e^2 + 6*e - 50, -e^4 - e^3 + 16*e^2 - 8, 3/4*e^4 + 7/4*e^3 - 14*e^2 - 29/2*e + 41, -3/4*e^4 - 1/4*e^3 + 21/2*e^2 - 23/2*e - 19, -3/4*e^4 - 3/4*e^3 + 31/2*e^2 + 9*e - 48, 1/2*e^4 - 3/2*e^3 - 8*e^2 + 28*e + 26, -e^4 - 2*e^3 + 12*e^2 + 15*e - 24, -1/4*e^4 + 3/2*e^3 + 27/4*e^2 - 21*e - 21, 3/2*e^3 + 5/2*e^2 - 14*e - 38, e^4 + 1/2*e^3 - 35/2*e^2 + 50, e^4 + 3/2*e^3 - 31/2*e^2 - 9*e + 32, 1/4*e^4 - 3/4*e^3 - 15/2*e^2 + 13*e + 32, -1/2*e^4 - e^3 + 23/2*e^2 + 10*e - 48, e^3 + 3*e^2 - 10*e - 8, 1/4*e^4 + 1/2*e^3 - 15/4*e^2 - 2*e + 5, 7/4*e^4 + 7/4*e^3 - 53/2*e^2 - 6*e + 50, -1/2*e^4 - 1/2*e^3 + 6*e^2 + 2*e - 16, -e^4 + 17*e^2 - 21*e - 32, -1/4*e^4 - 3/4*e^3 + 3*e^2 + 9*e - 16, 1/2*e^4 - 9/2*e^2 + 10*e - 20, 1/4*e^4 + 2*e^3 - 5/4*e^2 - 26*e + 17, 1/2*e^4 - 9*e^2 - 1/2*e + 31, -1/2*e^4 + 17/2*e^2 + 4*e - 6, -3/4*e^4 + e^3 + 63/4*e^2 - 19*e - 49, 1/4*e^4 - 3/4*e^3 - 13/2*e^2 + 8*e + 20, -3/4*e^4 - 3/4*e^3 + 8*e^2 - 5/2*e + 3, 3/2*e^4 + e^3 - 39/2*e^2 + 6*e + 12, 1/2*e^4 - 1/2*e^3 - 4*e^2 + 17*e - 16, 1/4*e^4 + 3/4*e^3 - 2*e^2 - 2*e + 8, 1/4*e^4 - e^3 - 5/4*e^2 + 19*e - 1, -3*e^2 + 3*e + 16, -1/2*e^3 + 27/2*e - 17, 3/2*e^4 + e^3 - 55/2*e^2 - e + 42, -e^4 - e^3 + 17*e^2 + 9*e - 38, -1/2*e^4 - e^3 + 19/2*e^2 + 14*e - 34, -1/4*e^4 - 2*e^3 - 7/4*e^2 + 27*e + 33, 5/4*e^4 - 1/2*e^3 - 91/4*e^2 + 17*e + 37, -1/2*e^4 - 3/2*e^3 + 6*e^2 + 18*e, 1/2*e^4 + e^3 - 19/2*e^2 - 8*e + 26, e^4 + e^3 - 14*e^2 - 4*e + 8, -1/2*e^4 + 1/2*e^3 + 27/2*e^2 - 21/2*e - 31, 1/2*e^4 - 21/2*e^2 + 2*e + 14, -3*e^3 - 4*e^2 + 35*e + 36, 1/2*e^4 - 14*e^2 + 5/2*e + 53, -3/2*e^4 - 1/2*e^3 + 23*e^2 - 7*e - 38, 3/2*e^4 - 26*e^2 + 51/2*e + 55]; 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;