/* 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, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w^3 - w^2 - 6*w - 2], [3, 3, -w^3 + 2*w^2 + 4*w - 2], [4, 2, w^3 - w^2 - 5*w - 2], [17, 17, -w^3 + w^2 + 6*w - 1], [17, 17, -w + 2], [29, 29, -w^3 + 2*w^2 + 4*w - 4], [29, 29, w^3 - 2*w^2 - 4*w], [31, 31, -w^2 + 2], [31, 31, -w^3 + 7*w + 5], [41, 41, -2*w^3 + 2*w^2 + 13*w - 2], [41, 41, 3*w^3 - 4*w^2 - 17*w + 5], [67, 67, 3*w^3 - 4*w^2 - 17*w + 1], [67, 67, -w^2 + 4*w + 2], [83, 83, 2*w^2 - 5*w - 2], [83, 83, -w^3 + 8*w + 6], [83, 83, w^3 - 2*w^2 - 2*w - 2], [83, 83, w^3 - 2*w^2 - 6*w], [97, 97, -3*w^3 + 2*w^2 + 17*w + 7], [97, 97, w^3 - 4*w^2 + 3*w + 1], [97, 97, -5*w^3 + 8*w^2 + 24*w - 8], [97, 97, 2*w^3 - 2*w^2 - 13*w - 4], [101, 101, -w^3 + 6*w + 6], [101, 101, -w^3 + 6*w + 2], [103, 103, -4*w^3 + 3*w^2 + 26*w + 12], [103, 103, -w^3 + 2*w^2 + 9*w + 3], [107, 107, 2*w^3 - 2*w^2 - 11*w], [107, 107, -w^3 + 2*w^2 + 3*w - 3], [107, 107, w^3 - w^2 - 4*w - 3], [107, 107, 2*w^3 - 3*w^2 - 10*w], [121, 11, -2*w^3 + 3*w^2 + 8*w + 2], [149, 149, -3*w^3 + 5*w^2 + 14*w - 3], [149, 149, -2*w^3 + 4*w^2 + 7*w - 4], [157, 157, 2*w^3 - 2*w^2 - 10*w - 1], [157, 157, 2*w^3 - 3*w^2 - 8*w + 2], [157, 157, 2*w^3 - 2*w^2 - 10*w - 3], [157, 157, -3*w^3 + 4*w^2 + 15*w + 1], [163, 163, -2*w^2 + 4*w + 5], [163, 163, 2*w^3 - 4*w^2 - 10*w + 5], [169, 13, 2*w^3 - 3*w^2 - 12*w + 2], [169, 13, -w^3 + 9*w + 3], [173, 173, 3*w^3 - 2*w^2 - 17*w - 9], [173, 173, -2*w^3 + 14*w + 7], [197, 197, 2*w^3 - 2*w^2 - 12*w + 1], [197, 197, 3*w^3 - 3*w^2 - 18*w - 1], [199, 199, 3*w^3 - 4*w^2 - 18*w + 8], [199, 199, 4*w^3 - 5*w^2 - 24*w + 6], [223, 223, w^3 - 2*w^2 - 3*w - 3], [223, 223, 2*w^3 - 3*w^2 - 10*w + 6], [227, 227, -w^3 + 2*w^2 + 6*w - 8], [227, 227, 3*w + 4], [227, 227, 3*w^3 - 3*w^2 - 18*w - 7], [227, 227, -2*w^3 + 3*w^2 + 12*w - 6], [229, 229, -w - 4], [229, 229, w^2 - 2*w + 2], [229, 229, -w^3 + w^2 + 6*w + 5], [229, 229, -w^3 + 2*w^2 + 5*w - 7], [233, 233, -2*w^3 + 3*w^2 + 8*w - 4], [233, 233, -3*w^3 + 4*w^2 + 15*w + 3], [281, 281, -2*w^3 + 3*w^2 + 10*w + 2], [281, 281, -w^3 + 2*w^2 + 3*w - 5], [289, 17, -2*w^3 + 2*w^2 + 14*w + 5], [293, 293, w^2 - 6], [293, 293, w^3 - 7*w - 1], [331, 331, -6*w^3 + 10*w^2 + 27*w - 6], [331, 331, 2*w^3 - 2*w^2 - 12*w - 7], [347, 347, 3*w^3 - 2*w^2 - 18*w - 6], [347, 347, -4*w^3 + 2*w^2 + 23*w + 10], [347, 347, 2*w^3 - 6*w^2 - w + 4], [347, 347, -4*w^3 + 3*w^2 + 24*w + 8], [359, 359, -3*w^3 + 4*w^2 + 17*w + 1], [359, 359, w^2 - 4*w - 4], [359, 359, w^3 - 8*w], [359, 359, -w^3 + 2*w^2 + 6*w - 6], [367, 367, 2*w^2 - 3*w - 4], [367, 367, -w^3 + 3*w^2 + 4*w - 7], [379, 379, 3*w^3 - 3*w^2 - 16*w - 3], [379, 379, 2*w^3 - 2*w^2 - 9*w - 2], [461, 461, -2*w^3 + 5*w^2 + 2*w + 2], [461, 461, 4*w^3 - 4*w^2 - 26*w - 11], [463, 463, -2*w^3 + 13*w + 8], [463, 463, -w^3 - w^2 + 6*w + 7], [487, 487, 2*w^3 - 17*w - 8], [487, 487, -w^3 + 10*w + 4], [491, 491, -5*w^3 + 3*w^2 + 30*w + 13], [491, 491, -5*w^3 + 4*w^2 + 28*w + 14], [491, 491, 3*w^3 - 8*w^2 - 4*w + 4], [491, 491, 7*w^3 - 12*w^2 - 33*w + 15], [499, 499, -2*w^3 + 2*w^2 + 12*w - 5], [499, 499, 2*w^2 - 7*w], [529, 23, -w^3 + 3*w^2 + 4*w - 5], [529, 23, 2*w^2 - 3*w - 6], [557, 557, -2*w^3 + 3*w^2 + 12*w + 4], [557, 557, w^3 - 9*w - 9], [569, 569, 4*w^3 - 5*w^2 - 22*w + 4], [569, 569, -3*w^3 + 2*w^2 + 20*w + 4], [577, 577, 3*w^3 - 3*w^2 - 20*w + 3], [577, 577, -4*w^3 + 8*w^2 + 16*w - 9], [577, 577, 7*w^3 - 11*w^2 - 34*w + 9], [577, 577, -4*w^3 + 6*w^2 + 22*w - 11], [593, 593, -6*w^3 + 10*w^2 + 29*w - 14], [593, 593, -w^3 + 4*w^2 - w - 7], [619, 619, -3*w^3 + 5*w^2 + 12*w - 1], [619, 619, -4*w^3 + 8*w^2 + 18*w - 15], [625, 5, -5], [631, 631, -3*w^3 + 8*w^2 + 6*w - 4], [631, 631, -2*w^3 + 8*w^2 - 3*w - 8], [643, 643, 3*w^3 - 4*w^2 - 14*w + 2], [643, 643, -3*w^3 + 4*w^2 + 14*w + 2], [661, 661, w^3 - 8*w - 10], [661, 661, -3*w^3 + 2*w^2 + 21*w + 3], [661, 661, w^3 - 2*w^2 - 6*w - 4], [661, 661, 2*w^3 - 3*w^2 - 14*w + 4], [677, 677, -9*w^3 + 12*w^2 + 50*w - 10], [677, 677, 3*w^3 - 4*w^2 - 17*w - 7], [691, 691, 3*w^3 - 2*w^2 - 21*w - 9], [691, 691, 2*w^3 - 3*w^2 - 14*w - 2], [701, 701, -9*w^3 + 13*w^2 + 48*w - 15], [701, 701, 2*w^3 - 6*w^2 + w - 2], [709, 709, 2*w^2 - 2*w - 5], [709, 709, 2*w^2 - 2*w - 7], [709, 709, 4*w^3 - 5*w^2 - 20*w - 4], [709, 709, -3*w^3 + 6*w^2 + 14*w - 10], [727, 727, 7*w^3 - 10*w^2 - 36*w + 8], [727, 727, 3*w^2 - 6*w - 8], [743, 743, 9*w^3 - 12*w^2 - 51*w + 13], [743, 743, w^3 - 2*w^2 - w - 5], [743, 743, -4*w^3 + 4*w^2 + 27*w - 8], [743, 743, -3*w^3 + 4*w^2 + 21*w + 9], [751, 751, -w^3 + 10*w + 2], [751, 751, 3*w^3 - 4*w^2 - 18*w + 2], [761, 761, 4*w^3 - 4*w^2 - 23*w - 2], [761, 761, -3*w^3 + 2*w^2 + 19*w + 5], [809, 809, -3*w^3 + 5*w^2 + 10*w + 5], [809, 809, -w^3 - w^2 + 10*w + 3], [821, 821, -w^3 + 3*w^2 + 2*w - 9], [821, 821, 2*w^2 - 4*w - 9], [823, 823, w^3 - 6*w^2 + 9*w + 1], [823, 823, -3*w^3 + 2*w^2 + 15*w + 5], [827, 827, 3*w^3 - 3*w^2 - 18*w + 1], [827, 827, 2*w^3 - 16*w - 11], [827, 827, 3*w^3 - 24*w - 14], [827, 827, 3*w - 4], [829, 829, w^2 - 6*w - 2], [829, 829, 2*w^3 - 2*w^2 - 10*w - 9], [829, 829, 5*w^3 - 6*w^2 - 29*w - 1], [829, 829, w^3 + 2*w^2 - 14*w - 6], [841, 29, -2*w^3 + 2*w^2 + 14*w + 3], [857, 857, -3*w^3 + 6*w^2 + 11*w - 7], [857, 857, -4*w^3 + 7*w^2 + 18*w - 4], [859, 859, -w^3 + w^2 + 6*w - 5], [859, 859, w - 6], [883, 883, 5*w^3 - 5*w^2 - 26*w - 11], [883, 883, -3*w^3 + 3*w^2 + 14*w + 5], [887, 887, 5*w^3 - 2*w^2 - 34*w - 16], [887, 887, -w^3 + 4*w^2 - 10], [887, 887, 4*w^3 - 2*w^2 - 26*w - 11], [887, 887, 2*w^3 - 8*w^2 + 4*w + 7], [907, 907, 6*w^3 - 8*w^2 - 34*w + 7], [907, 907, -w^3 + w^2 + 8*w - 7], [941, 941, 2*w^3 - 2*w^2 - 8*w - 5], [941, 941, -3*w^3 + 23*w + 11], [953, 953, 2*w^3 - 10*w - 3], [953, 953, -4*w^3 + 2*w^2 + 24*w + 15], [961, 31, -2*w^3 + 2*w^2 + 14*w - 7], [991, 991, -3*w^3 + 4*w^2 + 13*w + 3], [991, 991, 4*w^3 - 5*w^2 - 20*w + 2]]; primes := [ideal : I in primesArray]; heckePol := x^9 + 6*x^8 + 5*x^7 - 31*x^6 - 61*x^5 + 9*x^4 + 79*x^3 + 26*x^2 - 21*x - 9; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1/3*e^8 - e^7 + 7/3*e^6 + 22/3*e^5 - 14/3*e^4 - 14*e^3 + 14/3*e^2 + 22/3*e - 1, 5/3*e^8 + 7*e^7 - 11/3*e^6 - 131/3*e^5 - 83/3*e^4 + 55*e^3 + 128/3*e^2 - 50/3*e - 11, 1, -2*e^8 - 10*e^7 - 2*e^6 + 57*e^5 + 72*e^4 - 50*e^3 - 95*e^2 + 7*e + 21, -e^8 - 5*e^7 + 33*e^5 + 34*e^4 - 50*e^3 - 60*e^2 + 23*e + 18, -2*e^7 - 5*e^6 + 12*e^5 + 29*e^4 - 14*e^3 - 27*e^2 + 7*e, 8/3*e^8 + 13*e^7 + 1/3*e^6 - 233/3*e^5 - 248/3*e^4 + 84*e^3 + 341/3*e^2 - 71/3*e - 29, 8/3*e^8 + 15*e^7 + 19/3*e^6 - 263/3*e^5 - 356/3*e^4 + 86*e^3 + 458/3*e^2 - 44/3*e - 35, -5*e^8 - 24*e^7 + e^6 + 146*e^5 + 147*e^4 - 170*e^3 - 214*e^2 + 48*e + 57, -e^8 - 3*e^7 + 5*e^6 + 17*e^5 - 2*e^4 - 13*e^3 + 2*e^2 - e, -7/3*e^8 - 7*e^7 + 37/3*e^6 + 124/3*e^5 - 35/3*e^4 - 41*e^3 + 74/3*e^2 + 1/3*e - 14, -10/3*e^8 - 13*e^7 + 37/3*e^6 + 259/3*e^5 + 76/3*e^4 - 130*e^3 - 157/3*e^2 + 148/3*e + 16, -7*e^7 - 23*e^6 + 34*e^5 + 146*e^4 - 3*e^3 - 187*e^2 - 13*e + 39, e^8 + 6*e^7 + 3*e^6 - 37*e^5 - 49*e^4 + 46*e^3 + 62*e^2 - 18*e - 12, 2*e^6 + 3*e^5 - 13*e^4 - 15*e^3 + 14*e^2 + 4*e + 3, 2*e^8 + 13*e^7 + 10*e^6 - 78*e^5 - 129*e^4 + 87*e^3 + 193*e^2 - 29*e - 57, 2/3*e^8 - 3*e^7 - 53/3*e^6 + 52/3*e^5 + 271/3*e^4 - 19*e^3 - 322/3*e^2 + 46/3*e + 22, 14/3*e^8 + 22*e^7 - 14/3*e^6 - 419/3*e^5 - 347/3*e^4 + 186*e^3 + 539/3*e^2 - 197/3*e - 56, -1/3*e^8 - e^7 + 4/3*e^6 + 16/3*e^5 + 7/3*e^4 - 3*e^3 - 16/3*e^2 - 5/3*e - 5, 5/3*e^8 + 13*e^7 + 46/3*e^6 - 224/3*e^5 - 437/3*e^4 + 73*e^3 + 560/3*e^2 - 80/3*e - 44, -e^8 - 4*e^7 + 4*e^6 + 27*e^5 + 6*e^4 - 41*e^3 - 19*e^2 + 9*e + 9, -5*e^8 - 24*e^7 - 2*e^6 + 138*e^5 + 164*e^4 - 125*e^3 - 227*e^2 + 13*e + 54, -19/3*e^8 - 34*e^7 - 29/3*e^6 + 604/3*e^5 + 754/3*e^4 - 208*e^3 - 1018/3*e^2 + 118/3*e + 82, 17/3*e^8 + 28*e^7 + 4/3*e^6 - 503/3*e^5 - 539/3*e^4 + 183*e^3 + 746/3*e^2 - 146/3*e - 62, -4*e^8 - 17*e^7 + 8*e^6 + 104*e^5 + 70*e^4 - 118*e^3 - 101*e^2 + 18*e + 24, -e^8 - 5*e^7 + 34*e^5 + 37*e^4 - 56*e^3 - 75*e^2 + 30*e + 24, 7*e^8 + 35*e^7 + 5*e^6 - 206*e^5 - 242*e^4 + 212*e^3 + 329*e^2 - 53*e - 87, 7*e^8 + 33*e^7 - 3*e^6 - 199*e^5 - 196*e^4 + 222*e^3 + 289*e^2 - 56*e - 81, -1/3*e^8 - 3*e^7 - 11/3*e^6 + 52/3*e^5 + 94/3*e^4 - 17*e^3 - 100/3*e^2 + 16/3*e + 4, -5*e^8 - 30*e^7 - 19*e^6 + 173*e^5 + 272*e^4 - 161*e^3 - 370*e^2 + 33*e + 93, e^8 + 4*e^7 - 3*e^6 - 21*e^5 - 2*e^4 + 10*e^3 - 30*e^2 + 3*e + 24, -10/3*e^8 - 13*e^7 + 31/3*e^6 + 247/3*e^5 + 118/3*e^4 - 105*e^3 - 235/3*e^2 + 49/3*e + 28, -2*e^8 - 15*e^7 - 16*e^6 + 85*e^5 + 157*e^4 - 73*e^3 - 194*e^2 + 6*e + 50, -1/3*e^8 + 16/3*e^6 + 4/3*e^5 - 86/3*e^4 - 9*e^3 + 170/3*e^2 + 25/3*e - 23, -4*e^8 - 20*e^7 - 2*e^6 + 120*e^5 + 137*e^4 - 133*e^3 - 200*e^2 + 42*e + 50, -22/3*e^8 - 33*e^7 + 19/3*e^6 + 592/3*e^5 + 559/3*e^4 - 213*e^3 - 853/3*e^2 + 145/3*e + 82, 8/3*e^8 + 14*e^7 + 10/3*e^6 - 239/3*e^5 - 290/3*e^4 + 66*e^3 + 338/3*e^2 - 2/3*e - 17, 8/3*e^8 + 22*e^7 + 82/3*e^6 - 386/3*e^5 - 770/3*e^4 + 131*e^3 + 1061/3*e^2 - 104/3*e - 92, -10/3*e^8 - 21*e^7 - 50/3*e^6 + 358/3*e^5 + 613/3*e^4 - 108*e^3 - 805/3*e^2 + 91/3*e + 76, e^8 - 2*e^7 - 21*e^6 + 6*e^5 + 100*e^4 + 16*e^3 - 120*e^2 - 8*e + 30, -2*e^8 - 4*e^7 + 18*e^6 + 31*e^5 - 50*e^4 - 65*e^3 + 47*e^2 + 34*e - 21, -7*e^8 - 25*e^7 + 28*e^6 + 154*e^5 + 28*e^4 - 181*e^3 - 31*e^2 + 29*e - 9, 7*e^8 + 34*e^7 - 209*e^5 - 221*e^4 + 256*e^3 + 341*e^2 - 95*e - 99, e^8 - 14*e^6 - e^5 + 57*e^4 + 2*e^3 - 65*e^2 + 12*e + 20, 3*e^8 + 20*e^7 + 19*e^6 - 109*e^5 - 208*e^4 + 75*e^3 + 270*e^2 - 7*e - 67, -4/3*e^8 - 3*e^7 + 31/3*e^6 + 58/3*e^5 - 77/3*e^4 - 27*e^3 + 80/3*e^2 + 25/3*e + 7, -4/3*e^8 - 15*e^7 - 83/3*e^6 + 238/3*e^5 + 625/3*e^4 - 44*e^3 - 739/3*e^2 - 17/3*e + 55, -e^7 - 6*e^6 + 2*e^5 + 45*e^4 + 13*e^3 - 89*e^2 - 11*e + 39, 3*e^8 + 11*e^7 - 9*e^6 - 67*e^5 - 35*e^4 + 82*e^3 + 57*e^2 - 35*e - 15, -3*e^8 - 13*e^7 + 6*e^6 + 80*e^5 + 47*e^4 - 98*e^3 - 49*e^2 + 31*e + 3, -6*e^8 - 28*e^7 + 4*e^6 + 172*e^5 + 161*e^4 - 210*e^3 - 250*e^2 + 79*e + 78, -10/3*e^8 - 24*e^7 - 74/3*e^6 + 412/3*e^5 + 766/3*e^4 - 121*e^3 - 985/3*e^2 + 46/3*e + 70, -22/3*e^8 - 32*e^7 + 31/3*e^6 + 586/3*e^5 + 472/3*e^4 - 229*e^3 - 664/3*e^2 + 196/3*e + 46, 5/3*e^8 + 4*e^7 - 38/3*e^6 - 83/3*e^5 + 79/3*e^4 + 40*e^3 - 43/3*e^2 + 19/3*e - 11, 20/3*e^8 + 30*e^7 - 20/3*e^6 - 530/3*e^5 - 452/3*e^4 + 181*e^3 + 539/3*e^2 - 128/3*e - 26, 2*e^8 + 10*e^7 - 63*e^5 - 65*e^4 + 80*e^3 + 110*e^2 - 17*e - 45, -2*e^8 - 7*e^7 + 6*e^6 + 38*e^5 + 19*e^4 - 23*e^3 - 15*e^2 - 14*e - 6, -e^8 - 3*e^7 + 4*e^6 + 13*e^5 - e^4 + 9*e^3 + 17*e^2 - 22*e + 6, 3*e^8 + 10*e^7 - 13*e^6 - 58*e^5 - 6*e^4 + 49*e^3 + 11*e^2 + 13*e - 15, -28/3*e^8 - 45*e^7 + 16/3*e^6 + 844/3*e^5 + 751/3*e^4 - 360*e^3 - 1078/3*e^2 + 382/3*e + 94, -2*e^8 - 13*e^7 - 10*e^6 + 76*e^5 + 125*e^4 - 79*e^3 - 177*e^2 + 31*e + 39, -3*e^8 - 15*e^7 - 4*e^6 + 86*e^5 + 121*e^4 - 75*e^3 - 179*e^2 - 2*e + 33, -e^8 - 7*e^7 - 5*e^6 + 48*e^5 + 73*e^4 - 80*e^3 - 136*e^2 + 44*e + 38, -12*e^8 - 54*e^7 + 12*e^6 + 324*e^5 + 287*e^4 - 354*e^3 - 408*e^2 + 92*e + 119, -5*e^7 - 18*e^6 + 18*e^5 + 106*e^4 + 30*e^3 - 101*e^2 - 19*e + 9, -2*e^8 - 5*e^7 + 14*e^6 + 29*e^5 - 33*e^4 - 24*e^3 + 58*e^2 - 5*e - 36, -11*e^8 - 48*e^7 + 16*e^6 + 292*e^5 + 235*e^4 - 330*e^3 - 350*e^2 + 56*e + 96, -2*e^8 - 18*e^7 - 26*e^6 + 100*e^5 + 224*e^4 - 76*e^3 - 290*e^2 + 10*e + 66, 17*e^8 + 78*e^7 - 11*e^6 - 459*e^5 - 432*e^4 + 460*e^3 + 562*e^2 - 75*e - 132, 6*e^8 + 33*e^7 + 14*e^6 - 189*e^5 - 265*e^4 + 171*e^3 + 340*e^2 - 32*e - 90, -7*e^8 - 29*e^7 + 17*e^6 + 178*e^5 + 103*e^4 - 202*e^3 - 148*e^2 + 22*e + 30, 6*e^8 + 24*e^7 - 14*e^6 - 142*e^5 - 93*e^4 + 139*e^3 + 142*e^2 - e - 51, -28/3*e^8 - 47*e^7 - 17/3*e^6 + 859/3*e^5 + 985/3*e^4 - 337*e^3 - 1468/3*e^2 + 310/3*e + 145, 20/3*e^8 + 36*e^7 + 34/3*e^6 - 635/3*e^5 - 812/3*e^4 + 210*e^3 + 1058/3*e^2 - 89/3*e - 74, -34/3*e^8 - 46*e^7 + 97/3*e^6 + 877/3*e^5 + 427/3*e^4 - 388*e^3 - 703/3*e^2 + 394/3*e + 76, -1/3*e^8 - 6*e^7 - 35/3*e^6 + 103/3*e^5 + 241/3*e^4 - 30*e^3 - 289/3*e^2 + 16/3*e + 31, 11*e^8 + 52*e^7 - 10*e^6 - 323*e^5 - 269*e^4 + 401*e^3 + 383*e^2 - 124*e - 105, -17*e^8 - 85*e^7 - 8*e^6 + 507*e^5 + 568*e^4 - 541*e^3 - 798*e^2 + 128*e + 210, -10*e^8 - 41*e^7 + 23*e^6 + 253*e^5 + 159*e^4 - 317*e^3 - 251*e^2 + 128*e + 80, e^8 - 9*e^7 - 48*e^6 + 36*e^5 + 275*e^4 + 32*e^3 - 354*e^2 - 31*e + 80, 47/3*e^8 + 79*e^7 + 34/3*e^6 - 1412/3*e^5 - 1646/3*e^4 + 511*e^3 + 2324/3*e^2 - 437/3*e - 212, -37/3*e^8 - 51*e^7 + 79/3*e^6 + 925/3*e^5 + 610/3*e^4 - 348*e^3 - 877/3*e^2 + 265/3*e + 76, 6*e^8 + 33*e^7 + 17*e^6 - 185*e^5 - 286*e^4 + 149*e^3 + 377*e^2 + e - 96, 8*e^8 + 39*e^7 - 4*e^6 - 244*e^5 - 221*e^4 + 309*e^3 + 324*e^2 - 93*e - 78, -16*e^7 - 48*e^6 + 90*e^5 + 306*e^4 - 80*e^3 - 402*e^2 + 28*e + 102, 14*e^8 + 73*e^7 + 16*e^6 - 436*e^5 - 523*e^4 + 479*e^3 + 715*e^2 - 143*e - 195, -46/3*e^8 - 64*e^7 + 94/3*e^6 + 1168/3*e^5 + 796/3*e^4 - 448*e^3 - 1183/3*e^2 + 328/3*e + 103, 5/3*e^8 + 4*e^7 - 23/3*e^6 - 38/3*e^5 + 7/3*e^4 - 41*e^3 - 40/3*e^2 + 175/3*e - 5, 3*e^8 + 15*e^7 + 3*e^6 - 92*e^5 - 122*e^4 + 116*e^3 + 212*e^2 - 61*e - 70, 2*e^7 + 9*e^6 - 9*e^5 - 66*e^4 - 2*e^3 + 119*e^2 + 4*e - 49, 5*e^7 + 20*e^6 - 14*e^5 - 120*e^4 - 56*e^3 + 129*e^2 + 66*e - 18, 8*e^8 + 45*e^7 + 17*e^6 - 277*e^5 - 358*e^4 + 344*e^3 + 523*e^2 - 131*e - 153, -6*e^8 - 30*e^7 - 4*e^6 + 176*e^5 + 206*e^4 - 177*e^3 - 279*e^2 + 43*e + 69, 2*e^8 + 18*e^7 + 24*e^6 - 107*e^5 - 218*e^4 + 111*e^3 + 310*e^2 - 24*e - 90, 6*e^8 + 42*e^7 + 36*e^6 - 252*e^5 - 416*e^4 + 277*e^3 + 587*e^2 - 66*e - 160, -16/3*e^8 - 33*e^7 - 68/3*e^6 + 607/3*e^5 + 976/3*e^4 - 251*e^3 - 1555/3*e^2 + 304/3*e + 160, -40/3*e^8 - 54*e^7 + 112/3*e^6 + 1030/3*e^5 + 529/3*e^4 - 455*e^3 - 913/3*e^2 + 412/3*e + 91, e^8 - 5*e^7 - 33*e^6 + 16*e^5 + 177*e^4 + 44*e^3 - 227*e^2 - 50*e + 71, -12*e^7 - 33*e^6 + 70*e^5 + 204*e^4 - 74*e^3 - 249*e^2 + 34*e + 66, e^8 + 8*e^7 + 10*e^6 - 48*e^5 - 100*e^4 + 53*e^3 + 155*e^2 - 13*e - 60, -43/3*e^8 - 76*e^7 - 65/3*e^6 + 1348/3*e^5 + 1723/3*e^4 - 469*e^3 - 2422/3*e^2 + 346/3*e + 220, 11/3*e^8 + 24*e^7 + 43/3*e^6 - 482/3*e^5 - 656/3*e^4 + 253*e^3 + 1061/3*e^2 - 353/3*e - 113, 2*e^8 + 17*e^7 + 23*e^6 - 103*e^5 - 217*e^4 + 129*e^3 + 329*e^2 - 75*e - 103, -4*e^8 - 21*e^7 - 7*e^6 + 122*e^5 + 167*e^4 - 119*e^3 - 233*e^2 + 39*e + 56, -2*e^8 - 16*e^7 - 20*e^6 + 93*e^5 + 198*e^4 - 91*e^3 - 308*e^2 + 24*e + 92, -4*e^8 - 15*e^7 + 13*e^6 + 96*e^5 + 47*e^4 - 132*e^3 - 104*e^2 + 36*e + 23, 5*e^8 + 14*e^7 - 35*e^6 - 99*e^5 + 70*e^4 + 180*e^3 - 73*e^2 - 112*e + 26, -4/3*e^8 - 4*e^7 + 40/3*e^6 + 112/3*e^5 - 116/3*e^4 - 99*e^3 + 53/3*e^2 + 199/3*e + 19, -25/3*e^8 - 34*e^7 + 67/3*e^6 + 652/3*e^5 + 361/3*e^4 - 300*e^3 - 646/3*e^2 + 400/3*e + 49, -1/3*e^8 + 4*e^7 + 46/3*e^6 - 65/3*e^5 - 236/3*e^4 + 20*e^3 + 212/3*e^2 - 68/3*e - 8, 41/3*e^8 + 60*e^7 - 47/3*e^6 - 1067/3*e^5 - 941/3*e^4 + 374*e^3 + 1349/3*e^2 - 257/3*e - 122, -e^8 + 3*e^7 + 26*e^6 - 7*e^5 - 129*e^4 - 39*e^3 + 142*e^2 + 30*e - 51, 11*e^8 + 58*e^7 + 14*e^6 - 352*e^5 - 442*e^4 + 404*e^3 + 682*e^2 - 120*e - 201, -12*e^8 - 52*e^7 + 16*e^6 + 311*e^5 + 270*e^4 - 325*e^3 - 412*e^2 + 33*e + 107, -3*e^8 - 11*e^7 + 6*e^6 + 55*e^5 + 38*e^4 - 18*e^3 + 6*e^2 - 37, -e^8 - 15*e^7 - 35*e^6 + 74*e^5 + 244*e^4 - 17*e^3 - 283*e^2 - 15*e + 48, 20*e^8 + 82*e^7 - 49*e^6 - 506*e^5 - 294*e^4 + 606*e^3 + 435*e^2 - 146*e - 108, -7/3*e^8 + 5*e^7 + 157/3*e^6 - 44/3*e^5 - 794/3*e^4 - 47*e^3 + 1076/3*e^2 + 112/3*e - 104, 5/3*e^8 + 13*e^7 + 31/3*e^6 - 260/3*e^5 - 368/3*e^4 + 129*e^3 + 584/3*e^2 - 80/3*e - 68, 11/3*e^8 + 16*e^7 - 8/3*e^6 - 257/3*e^5 - 239/3*e^4 + 48*e^3 + 176/3*e^2 + 52/3*e + 1, -1/3*e^8 + e^7 + 16/3*e^6 - 44/3*e^5 - 116/3*e^4 + 49*e^3 + 332/3*e^2 - 47/3*e - 62, -3*e^8 - 29*e^7 - 48*e^6 + 154*e^5 + 389*e^4 - 89*e^3 - 491*e^2 - 6*e + 107, 25*e^8 + 110*e^7 - 34*e^6 - 664*e^5 - 542*e^4 + 740*e^3 + 786*e^2 - 168*e - 217, 7*e^8 + 32*e^7 - 7*e^6 - 198*e^5 - 173*e^4 + 243*e^3 + 254*e^2 - 79*e - 45, 10*e^8 + 41*e^7 - 21*e^6 - 249*e^5 - 169*e^4 + 295*e^3 + 246*e^2 - 110*e - 63, 25*e^8 + 118*e^7 - 9*e^6 - 705*e^5 - 697*e^4 + 753*e^3 + 969*e^2 - 147*e - 243, 3*e^8 + 24*e^7 + 27*e^6 - 142*e^5 - 265*e^4 + 154*e^3 + 378*e^2 - 57*e - 126, 7*e^8 + 27*e^7 - 26*e^6 - 180*e^5 - 64*e^4 + 268*e^3 + 173*e^2 - 81*e - 76, 6*e^8 + 14*e^7 - 50*e^6 - 103*e^5 + 133*e^4 + 192*e^3 - 145*e^2 - 75*e + 32, 8*e^8 + 40*e^7 + 3*e^6 - 239*e^5 - 256*e^4 + 256*e^3 + 330*e^2 - 54*e - 78, -14*e^8 - 49*e^7 + 60*e^6 + 318*e^5 + 62*e^4 - 445*e^3 - 196*e^2 + 138*e + 84, -e^8 + 8*e^7 + 41*e^6 - 49*e^5 - 251*e^4 + 70*e^3 + 410*e^2 - 57*e - 147, 2*e^8 - 3*e^7 - 40*e^6 - e^5 + 181*e^4 + 86*e^3 - 190*e^2 - 72*e + 39, 5*e^8 + 13*e^7 - 39*e^6 - 87*e^5 + 105*e^4 + 126*e^3 - 150*e^2 - 21*e + 42, 11*e^8 + 60*e^7 + 25*e^6 - 347*e^5 - 491*e^4 + 327*e^3 + 668*e^2 - 60*e - 183, -1/3*e^8 - 7*e^7 - 47/3*e^6 + 127/3*e^5 + 334/3*e^4 - 58*e^3 - 448/3*e^2 + 190/3*e + 40, 8/3*e^8 + 18*e^7 + 25/3*e^6 - 362/3*e^5 - 410/3*e^4 + 190*e^3 + 605/3*e^2 - 263/3*e - 77, 3*e^8 + 14*e^7 - 2*e^6 - 87*e^5 - 79*e^4 + 117*e^3 + 123*e^2 - 71*e - 60, -4*e^8 - 17*e^7 + e^6 + 88*e^5 + 101*e^4 - 39*e^3 - 82*e^2 - 19*e - 9, -10*e^7 - 36*e^6 + 42*e^5 + 227*e^4 + 33*e^3 - 278*e^2 - 57*e + 45, -2*e^8 - 17*e^7 - 22*e^6 + 101*e^5 + 203*e^4 - 122*e^3 - 282*e^2 + 94*e + 87, -4*e^8 - 15*e^7 + 16*e^6 + 101*e^5 + 26*e^4 - 162*e^3 - 72*e^2 + 77*e + 41, 2/3*e^8 + 6*e^7 + 7/3*e^6 - 155/3*e^5 - 146/3*e^4 + 123*e^3 + 371/3*e^2 - 197/3*e - 65, -20*e^8 - 94*e^7 + 9*e^6 + 566*e^5 + 553*e^4 - 629*e^3 - 800*e^2 + 154*e + 215, 20/3*e^8 + 24*e^7 - 74/3*e^6 - 458/3*e^5 - 161/3*e^4 + 196*e^3 + 341/3*e^2 - 89/3*e - 47, -14*e^8 - 48*e^7 + 62*e^6 + 306*e^5 + 41*e^4 - 407*e^3 - 140*e^2 + 111*e + 59, -2*e^8 - 3*e^7 + 13*e^6 + 6*e^5 - 25*e^4 + 45*e^3 + 36*e^2 - 43*e - 3, 4*e^8 + 17*e^7 - 8*e^6 - 105*e^5 - 70*e^4 + 131*e^3 + 98*e^2 - 69*e - 12, 10*e^8 + 37*e^7 - 31*e^6 - 219*e^5 - 103*e^4 + 229*e^3 + 153*e^2 - 59*e - 25, -e^8 + 7*e^7 + 35*e^6 - 39*e^5 - 190*e^4 + 37*e^3 + 225*e^2 - 24*e - 22, -37/3*e^8 - 68*e^7 - 89/3*e^6 + 1186/3*e^5 + 1699/3*e^4 - 379*e^3 - 2347/3*e^2 + 217/3*e + 196, -52/3*e^8 - 68*e^7 + 139/3*e^6 + 1225/3*e^5 + 661/3*e^4 - 445*e^3 - 871/3*e^2 + 292/3*e + 55, 7*e^7 + 25*e^6 - 35*e^5 - 169*e^4 + 17*e^3 + 270*e^2 - 13*e - 102, -2*e^8 - 22*e^7 - 31*e^6 + 144*e^5 + 269*e^4 - 215*e^3 - 402*e^2 + 110*e + 129, -10*e^8 - 47*e^7 + 5*e^6 + 282*e^5 + 267*e^4 - 310*e^3 - 365*e^2 + 74*e + 108, 8*e^8 + 44*e^7 + 15*e^6 - 259*e^5 - 330*e^4 + 268*e^3 + 425*e^2 - 87*e - 96, 11/3*e^8 + 34*e^7 + 154/3*e^6 - 593/3*e^5 - 1346/3*e^4 + 202*e^3 + 1931/3*e^2 - 233/3*e - 176, 17/3*e^8 + 30*e^7 + 31/3*e^6 - 527/3*e^5 - 713/3*e^4 + 172*e^3 + 974/3*e^2 - 44/3*e - 104, 17*e^8 + 77*e^7 - 13*e^6 - 451*e^5 - 420*e^4 + 451*e^3 + 551*e^2 - 102*e - 108, 8*e^8 + 40*e^7 + 4*e^6 - 234*e^5 - 256*e^4 + 235*e^3 + 307*e^2 - 67*e - 63, 5*e^8 + 26*e^7 + 3*e^6 - 156*e^5 - 162*e^4 + 173*e^3 + 188*e^2 - 50*e - 33, 23*e^8 + 117*e^7 + 14*e^6 - 707*e^5 - 793*e^4 + 806*e^3 + 1140*e^2 - 236*e - 324, -7*e^8 - 39*e^7 - 16*e^6 + 233*e^5 + 313*e^4 - 255*e^3 - 409*e^2 + 75*e + 71, -6*e^8 - 23*e^7 + 13*e^6 + 135*e^5 + 113*e^4 - 132*e^3 - 231*e^2 + e + 98, 8*e^8 + 61*e^7 + 61*e^6 - 369*e^5 - 633*e^4 + 432*e^3 + 873*e^2 - 159*e - 250]; 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;