/* 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![-5, -9, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w - 1], [5, 5, w], [7, 7, w^2 - 2*w - 8], [7, 7, -2*w^2 + 3*w + 16], [7, 7, -w^2 + 2*w + 6], [11, 11, w^2 - 2*w - 2], [19, 19, -w + 2], [23, 23, -w^2 + 3*w + 1], [25, 5, w^2 - w - 9], [27, 3, 3], [29, 29, -w^2 - w + 1], [29, 29, w^2 - 2*w - 4], [29, 29, -w^2 + w + 11], [43, 43, 2*w^2 - 3*w - 18], [47, 47, -2*w + 7], [53, 53, w^2 - w - 3], [61, 61, w^2 + 2*w + 2], [67, 67, -2*w^2 + 5*w + 8], [79, 79, w^2 - 3*w - 9], [97, 97, -w^2 - 2*w + 2], [109, 109, -2*w^2 + 4*w + 11], [109, 109, -3*w^2 + 4*w + 26], [109, 109, -3*w^2 + 4*w + 28], [113, 113, -4*w^2 + 8*w + 27], [121, 11, 5*w^2 - 8*w - 38], [127, 127, -3*w^2 + 4*w + 24], [131, 131, -4*w^2 + 5*w + 38], [139, 139, 2*w - 3], [149, 149, -w^2 + 3*w - 1], [151, 151, w^2 - 4*w - 6], [157, 157, 2*w^2 - 3*w - 12], [167, 167, -3*w^2 + 6*w + 22], [173, 173, -3*w^2 + 8*w + 8], [181, 181, w^2 - 2*w - 12], [181, 181, w^2 - 12], [181, 181, -4*w - 1], [191, 191, 2*w^2 - 6*w - 9], [193, 193, -8*w^2 + 12*w + 67], [197, 197, 2*w^2 - 3*w - 4], [199, 199, 2*w^2 - 2*w - 11], [211, 211, 4*w^2 - 6*w - 29], [223, 223, 3*w^2 - 5*w - 27], [227, 227, -4*w^2 + 7*w + 28], [229, 229, 3*w^2 - 8*w - 14], [229, 229, w^2 - 4*w - 8], [229, 229, -2*w - 7], [233, 233, -5*w^2 + 7*w + 43], [241, 241, 2*w^2 - 4*w - 7], [251, 251, w^2 + 2], [251, 251, -2*w^2 + 2*w + 1], [251, 251, 2*w^2 - 5*w - 2], [263, 263, 2*w^2 - 5*w - 14], [269, 269, -w^2 + 5*w - 1], [271, 271, -2*w^2 - w + 2], [277, 277, -3*w^2 + 7*w + 7], [281, 281, -5*w^2 + 10*w + 34], [293, 293, -2*w^2 + 5*w + 16], [311, 311, w^2 + w + 3], [313, 313, 2*w^2 + w - 4], [317, 317, -7*w^2 + 13*w + 49], [331, 331, w^2 - 5*w - 7], [337, 337, -w^2 + 2*w - 2], [337, 337, 7*w^2 - 12*w - 52], [337, 337, -2*w^2 + 3], [347, 347, -10*w^2 + 17*w + 76], [349, 349, 2*w^2 - 2*w - 9], [353, 353, 2*w^2 - 3*w - 8], [359, 359, -4*w^2 + 5*w + 36], [361, 19, w^2 + w - 7], [367, 367, -4*w^2 + 8*w + 29], [373, 373, 2*w^2 - 6*w - 11], [373, 373, 8*w^2 - 14*w - 59], [379, 379, -2*w^2 + w + 4], [383, 383, -3*w^2 + 2*w + 34], [383, 383, 5*w^2 - 7*w - 41], [383, 383, 3*w^2 - 7*w - 19], [397, 397, -7*w^2 + 13*w + 51], [431, 431, -8*w^2 + 14*w + 63], [433, 433, -2*w^2 + 9*w + 8], [443, 443, -6*w^2 + 10*w + 51], [443, 443, 3*w - 4], [443, 443, w^2 - 7*w - 3], [449, 449, -7*w^2 + 10*w + 62], [461, 461, -6*w^2 + 11*w + 46], [461, 461, 2*w^2 - 4*w - 1], [461, 461, w^2 - 5*w - 9], [463, 463, 7*w + 6], [463, 463, -5*w^2 + 8*w + 36], [463, 463, 3*w^2 - 2*w - 16], [467, 467, 3*w^2 - 7*w - 9], [467, 467, 4*w - 1], [467, 467, 6*w^2 - 11*w - 48], [479, 479, 2*w^2 - 4*w - 21], [479, 479, w^2 + 7*w + 7], [479, 479, w^2 - 2*w - 14], [487, 487, w^2 - 5*w - 11], [487, 487, 3*w^2 - 3*w - 23], [487, 487, w^2 + 3*w - 3], [491, 491, w^2 - 8*w - 4], [499, 499, -6*w^2 + 14*w + 31], [509, 509, -w - 8], [523, 523, 2*w^2 + 1], [529, 23, 2*w^2 - 5*w - 22], [541, 541, -w^2 + 5*w - 7], [547, 547, 2*w^2 - 6*w - 13], [563, 563, 2*w^2 - 9*w - 4], [569, 569, 4*w^2 - 7*w - 26], [569, 569, -3*w^2 + 5*w + 3], [569, 569, 4*w^2 - 6*w - 27], [577, 577, 3*w^2 - 7*w - 21], [577, 577, -8*w^2 + 12*w + 69], [577, 577, -2*w^2 + w + 26], [587, 587, w^2 - 6*w - 8], [593, 593, -2*w^2 + w + 16], [593, 593, 3*w^2 - 5*w - 17], [593, 593, 2*w^2 - 8*w - 9], [601, 601, -6*w^2 + 8*w + 53], [607, 607, -2*w - 9], [613, 613, w^2 + 2*w - 6], [617, 617, 5*w^2 - 11*w - 31], [619, 619, -4*w^2 + 6*w + 37], [641, 641, 3*w^2 - 6*w - 14], [647, 647, -2*w^2 + 3*w + 22], [653, 653, 2*w^2 - 7*w - 22], [653, 653, 2*w^2 + w + 2], [653, 653, -3*w^2 + 5*w + 29], [659, 659, 4*w^2 - 11*w - 18], [659, 659, 3*w^2 - 6*w - 8], [659, 659, 4*w^2 - 8*w - 23], [661, 661, w^2 + w - 11], [677, 677, -w^2 - 4], [701, 701, -2*w^2 - 5*w + 2], [727, 727, -2*w^2 + w + 18], [733, 733, -9*w^2 + 13*w + 79], [733, 733, -7*w^2 + 10*w + 56], [733, 733, 2*w^2 - 13], [739, 739, 6*w^2 - 10*w - 43], [739, 739, 4*w^2 - 4*w - 29], [739, 739, -8*w^2 + 15*w + 58], [751, 751, -11*w^2 + 19*w + 87], [757, 757, 3*w^2 - 10*w - 12], [761, 761, 4*w - 3], [769, 769, 3*w^2 - w - 13], [811, 811, 5*w^2 - 9*w - 33], [823, 823, 2*w^2 - 7*w - 12], [853, 853, -7*w - 8], [853, 853, -3*w^2 + 4*w + 2], [853, 853, 2*w^2 + 5*w + 6], [857, 857, 10*w^2 - 15*w - 82], [857, 857, 3*w^2 - 8*w - 18], [857, 857, -4*w^2 + 3*w + 44], [859, 859, -w^2 + 4*w - 6], [877, 877, -4*w^2 + 13*w + 14], [881, 881, 2*w^2 - 9*w - 2], [887, 887, 3*w - 14], [907, 907, -11*w^2 + 21*w + 77], [919, 919, 12*w^2 - 21*w - 92], [929, 929, w^2 - 18], [937, 937, 9*w + 4], [941, 941, -2*w^2 + 2*w - 1], [947, 947, 4*w^2 - 9*w - 18], [953, 953, -7*w^2 + 14*w + 48], [977, 977, w^2 - 2*w - 16], [983, 983, 4*w^2 - 7*w - 24], [991, 991, -3*w^2 - 2*w + 4]]; primes := [ideal : I in primesArray]; heckePol := x^6 - 8*x^4 - 2*x^3 + 14*x^2 + 2*x - 6; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 1, -e^4 + 6*e^2 + 2*e - 5, -e^5 + 7*e^3 + 2*e^2 - 8*e, -e^4 + e^3 + 6*e^2 - 2*e - 3, e^5 - 6*e^3 - 4*e^2 + 4*e + 6, 2*e^5 - e^4 - 13*e^3 + 2*e^2 + 12*e - 3, 2*e^5 + e^4 - 14*e^3 - 12*e^2 + 12*e + 9, -2*e^4 + 14*e^2 + 2*e - 12, e^5 - 2*e^4 - 5*e^3 + 8*e^2 + 4*e - 2, 2*e^4 - e^3 - 14*e^2 + 2*e + 15, -2*e^5 + 2*e^4 + 13*e^3 - 8*e^2 - 12*e + 9, -2*e^5 + 14*e^3 + 4*e^2 - 16*e - 3, 2*e^2 - 2*e - 2, -2*e^5 + e^4 + 11*e^3 + 2*e^2 - 4*e - 3, -e^3 + 10*e, -3*e^5 + 2*e^4 + 20*e^3 - 4*e^2 - 22*e + 4, 2*e^5 + 2*e^4 - 15*e^3 - 16*e^2 + 14*e + 12, -2*e^5 + 2*e^4 + 11*e^3 - 8*e^2 - 6*e + 10, 2*e^5 - 4*e^4 - 12*e^3 + 18*e^2 + 14*e - 12, 4*e^4 - 3*e^3 - 26*e^2 + 2*e + 27, -2*e^5 + 2*e^4 + 14*e^3 - 8*e^2 - 16*e + 6, e^5 + 4*e^4 - 9*e^3 - 28*e^2 + 4*e + 26, 4*e^4 - 30*e^2 - 6*e + 27, 2*e^4 - 2*e^3 - 10*e^2 + 6*e + 18, -e^5 + 2*e^4 + 5*e^3 - 10*e^2 - 4*e + 6, -e^5 - 4*e^4 + 10*e^3 + 28*e^2 - 18*e - 24, -2*e^5 + e^4 + 13*e^3 + 4*e^2 - 16*e - 15, 2*e^5 + 6*e^4 - 14*e^3 - 44*e^2 + 6*e + 36, 3*e^5 - 23*e^3 - 4*e^2 + 28*e - 4, 6*e^5 - 4*e^4 - 41*e^3 + 10*e^2 + 44*e - 6, -2*e^5 + 3*e^4 + 12*e^3 - 10*e^2 - 10*e + 3, 4*e^5 + 4*e^4 - 31*e^3 - 28*e^2 + 34*e + 18, 3*e^5 - 4*e^4 - 18*e^3 + 16*e^2 + 18*e - 20, -2*e^5 + 2*e^4 + 14*e^3 - 6*e^2 - 28*e - 6, -e^5 + 3*e^3 + 6*e^2 + 8*e - 6, 4*e^4 - 4*e^3 - 20*e^2 + 12*e + 18, -2*e^5 - 4*e^4 + 17*e^3 + 26*e^2 - 16*e - 15, 2*e^4 - 3*e^3 - 14*e^2 + 6*e + 21, -4*e^5 + 7*e^4 + 25*e^3 - 36*e^2 - 34*e + 33, 2*e^5 - 4*e^4 - 14*e^3 + 22*e^2 + 24*e - 12, 3*e^5 + 2*e^4 - 21*e^3 - 24*e^2 + 20*e + 22, 2*e^5 - 4*e^4 - 10*e^3 + 14*e^2 + 2*e, -2*e^4 + e^3 + 6*e^2 + 4*e + 10, -2*e^4 - 2*e^3 + 18*e^2 + 8*e - 29, e^5 + 2*e^4 - 10*e^3 - 10*e^2 + 12*e - 4, -4*e^5 + 4*e^4 + 27*e^3 - 18*e^2 - 36*e + 15, 3*e^5 + 4*e^4 - 23*e^3 - 26*e^2 + 24*e + 6, -4*e^5 - 4*e^4 + 27*e^3 + 40*e^2 - 22*e - 36, 3*e^4 - 20*e^2 - 8*e + 9, 2*e^5 - 15*e^3 - 6*e^2 + 24*e, e^5 - 2*e^4 - 4*e^3 + 2*e + 30, -5*e^5 + 6*e^4 + 29*e^3 - 28*e^2 - 26*e + 36, 2*e^5 - 10*e^4 - 9*e^3 + 56*e^2 + 18*e - 42, -5*e^5 + 28*e^3 + 18*e^2 - 10*e - 24, -3*e^5 + 2*e^4 + 23*e^3 - 12*e^2 - 32*e + 24, -3*e^5 + 2*e^4 + 21*e^3 - 4*e^2 - 28*e + 6, 2*e^5 + 6*e^4 - 11*e^3 - 50*e^2 - 4*e + 48, -2*e^5 + 2*e^4 + 13*e^3 - 10*e^2 - 4*e + 15, 4*e^5 - 4*e^4 - 29*e^3 + 12*e^2 + 46*e + 3, 6*e^5 - 46*e^3 - 10*e^2 + 58*e - 2, 2*e^4 - 4*e^3 - 4*e^2 + 8*e - 10, -2*e^5 - 2*e^4 + 15*e^3 + 10*e^2 - 16*e + 12, -e^5 + 4*e^4 - 2*e^3 - 12*e^2 + 28*e + 12, -8*e^5 + e^4 + 55*e^3 + 16*e^2 - 60*e - 3, 8*e^5 - 4*e^4 - 53*e^3 + 4*e^2 + 52*e + 3, -e^5 - 6*e^4 + 12*e^3 + 44*e^2 - 16*e - 48, 3*e^5 + 6*e^4 - 24*e^3 - 38*e^2 + 22*e + 18, -2*e^5 + 10*e^4 + 12*e^3 - 60*e^2 - 24*e + 54, 2*e^5 - 16*e^3 - 6*e^2 + 14*e + 20, -2*e^5 - 6*e^4 + 17*e^3 + 42*e^2 - 16*e - 38, 6*e^5 + 4*e^4 - 41*e^3 - 44*e^2 + 30*e + 39, 4*e^5 - 10*e^4 - 24*e^3 + 48*e^2 + 30*e - 30, e^5 + 8*e^4 - 7*e^3 - 62*e^2 - 8*e + 60, e^5 - 4*e^4 - 5*e^3 + 20*e^2 + 6*e - 6, 6*e^5 - 4*e^4 - 47*e^3 + 18*e^2 + 72*e - 12, -10*e^5 + 2*e^4 + 61*e^3 + 20*e^2 - 46*e - 26, -8*e^5 - e^4 + 54*e^3 + 26*e^2 - 42*e - 27, -2*e^4 - 2*e^3 + 24*e^2 + 8*e - 49, -2*e^5 - 3*e^4 + 16*e^3 + 24*e^2 - 18*e - 33, 4*e^4 + 3*e^3 - 30*e^2 - 28*e + 36, -8*e^5 - 2*e^4 + 56*e^3 + 36*e^2 - 56*e - 36, 5*e^5 - 33*e^3 - 6*e^2 + 30*e - 12, 2*e^5 + 6*e^4 - 10*e^3 - 50*e^2 - 14*e + 48, -6*e^5 - 4*e^4 + 46*e^3 + 36*e^2 - 54*e - 27, -3*e^5 + 4*e^4 + 21*e^3 - 18*e^2 - 22*e + 18, -7*e^5 + 6*e^4 + 49*e^3 - 20*e^2 - 72*e + 4, 6*e^5 - 2*e^4 - 44*e^3 + 2*e^2 + 60*e - 12, 2*e^5 - 7*e^4 - 15*e^3 + 42*e^2 + 22*e - 21, -2*e^5 - e^4 + 19*e^3 + 10*e^2 - 38*e - 15, -7*e^5 + 2*e^4 + 56*e^3 - 4*e^2 - 80*e + 6, 6*e^5 - 46*e^3 - 16*e^2 + 62*e + 6, 2*e^5 + 3*e^4 - 18*e^3 - 16*e^2 + 26*e - 3, -10*e^5 + 74*e^3 + 16*e^2 - 82*e - 6, -4*e^5 + 14*e^4 + 20*e^3 - 80*e^2 - 30*e + 78, 4*e^5 - 12*e^4 - 23*e^3 + 64*e^2 + 38*e - 40, 4*e^5 - e^4 - 25*e^3 - 4*e^2 + 22*e + 21, -6*e^5 + 6*e^4 + 40*e^3 - 20*e^2 - 46*e + 18, -3*e^5 - 4*e^4 + 25*e^3 + 34*e^2 - 40*e - 30, 2*e^5 + 5*e^4 - 21*e^3 - 30*e^2 + 38*e + 9, 3*e^5 - 6*e^4 - 15*e^3 + 32*e^2 + 4*e - 36, -2*e^5 - 10*e^4 + 16*e^3 + 66*e^2 - 2*e - 50, 9*e^5 - 61*e^3 - 26*e^2 + 58*e + 16, 9*e^5 + 6*e^4 - 58*e^3 - 66*e^2 + 40*e + 62, -4*e^5 + 6*e^4 + 22*e^3 - 18*e^2 - 14*e - 2, -2*e^5 + 10*e^4 + 8*e^3 - 54*e^2 - 14*e + 42, 4*e^5 + 4*e^4 - 37*e^3 - 28*e^2 + 66*e + 21, -5*e^5 + 12*e^4 + 29*e^3 - 60*e^2 - 32*e + 54, 8*e^5 - 2*e^4 - 53*e^3 - 10*e^2 + 56*e - 12, -8*e^5 + 4*e^4 + 49*e^3 - 4*e^2 - 34*e + 10, 6*e^5 - 2*e^4 - 40*e^3 - 6*e^2 + 54*e + 16, -6*e^5 + 44*e^3 + 12*e^2 - 54*e - 16, -8*e^5 + 15*e^4 + 50*e^3 - 80*e^2 - 66*e + 69, -8*e^5 - 2*e^4 + 55*e^3 + 40*e^2 - 58*e - 42, -3*e^5 - 4*e^4 + 25*e^3 + 36*e^2 - 42*e - 42, 9*e^5 - 2*e^4 - 67*e^3 + 2*e^2 + 86*e, 8*e^5 + 4*e^4 - 53*e^3 - 50*e^2 + 34*e + 45, 10*e^5 + 5*e^4 - 74*e^3 - 58*e^2 + 84*e + 41, -3*e^5 - 6*e^4 + 20*e^3 + 44*e^2 + 4*e - 54, -3*e^5 - 12*e^4 + 30*e^3 + 78*e^2 - 32*e - 66, -4*e^5 + 2*e^4 + 29*e^3 - 6*e^2 - 28*e + 20, 6*e^5 - 2*e^4 - 45*e^3 - 6*e^2 + 64*e + 15, -2*e^5 - 12*e^4 + 16*e^3 + 80*e^2 - 66, 2*e^5 + 2*e^4 - 18*e^3 - 10*e^2 + 42*e - 3, 8*e^5 - 4*e^4 - 53*e^3 + 6*e^2 + 52*e, 2*e^5 - 4*e^4 - 12*e^3 + 30*e^2 + 10*e - 45, 8*e^5 - 7*e^4 - 50*e^3 + 24*e^2 + 52*e - 45, 4*e^5 - 4*e^4 - 21*e^3 + 16*e^2 - 2*e - 18, -8*e^5 - 2*e^4 + 62*e^3 + 24*e^2 - 80*e - 6, 3*e^5 + 14*e^4 - 29*e^3 - 92*e^2 + 22*e + 72, -2*e^5 + 8*e^3 + 8*e^2 + 8*e - 15, 12*e^5 - 2*e^4 - 78*e^3 - 16*e^2 + 68*e - 6, -4*e^5 + 4*e^4 + 30*e^3 - 22*e^2 - 48*e + 24, 2*e^5 + 4*e^4 - 24*e^3 - 20*e^2 + 52*e + 1, 2*e^5 - 4*e^4 - 18*e^3 + 18*e^2 + 42*e - 12, 14*e^5 - 4*e^4 - 96*e^3 - 10*e^2 + 92*e - 6, -3*e^5 - 2*e^4 + 16*e^3 + 32*e^2 + 4*e - 36, 8*e^5 + 6*e^4 - 62*e^3 - 46*e^2 + 70*e + 18, -6*e^5 + 7*e^4 + 44*e^3 - 28*e^2 - 78*e + 11, 4*e^5 + 3*e^4 - 24*e^3 - 38*e^2 + 16*e + 37, -4*e^5 - 4*e^4 + 26*e^3 + 46*e^2 - 22*e - 67, -9*e^3 + 12*e^2 + 34*e - 21, 7*e^5 + 6*e^4 - 46*e^3 - 56*e^2 + 20*e + 48, 2*e^5 - 4*e^4 - 4*e^3 + 16*e^2 - 20*e - 6, -6*e^5 + e^4 + 34*e^3 + 14*e^2 - 18*e - 13, -8*e^5 - 8*e^4 + 60*e^3 + 74*e^2 - 54*e - 55, -9*e^5 + 8*e^4 + 61*e^3 - 36*e^2 - 76*e + 40, 6*e^5 - 4*e^4 - 30*e^3 - 2*e^2 + 12*e + 19, 8*e^5 - 2*e^4 - 57*e^3 + 4*e^2 + 68*e - 33, 16*e^5 - 10*e^4 - 104*e^3 + 28*e^2 + 94*e - 48, -5*e^5 + 8*e^4 + 37*e^3 - 36*e^2 - 66*e + 30, 4*e^5 + e^4 - 28*e^3 - 12*e^2 + 24*e - 13, -13*e^5 + 12*e^4 + 78*e^3 - 50*e^2 - 60*e + 76, -4*e^5 - 4*e^4 + 26*e^3 + 34*e^2 - 12*e - 18, 10*e^5 + 7*e^4 - 70*e^3 - 60*e^2 + 60*e + 39, -8*e^5 + 9*e^4 + 48*e^3 - 36*e^2 - 46*e + 31, 12*e^5 - e^4 - 86*e^3 - 14*e^2 + 94*e + 13, 8*e^4 - 12*e^3 - 38*e^2 + 38*e + 18, 2*e^5 - 12*e^4 - 4*e^3 + 66*e^2 + 2*e - 54, 4*e^5 - 12*e^4 - 24*e^3 + 64*e^2 + 32*e - 51, -5*e^5 - 4*e^4 + 41*e^3 + 32*e^2 - 46*e + 6, 6*e^5 - 10*e^4 - 38*e^3 + 54*e^2 + 46*e - 57, -14*e^5 - 2*e^4 + 98*e^3 + 34*e^2 - 98*e - 3, e^4 + 3*e^3 + 4*e^2 - 16*e - 33, 6*e^5 + 5*e^4 - 41*e^3 - 48*e^2 + 26*e + 63]; 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;