/* 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, 0, -5, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w], [2, 2, w + 1], [13, 13, -w^2 + w + 3], [13, 13, w^2 + w - 3], [19, 19, -w^3 + 3*w + 1], [19, 19, -w^3 + 3*w - 1], [43, 43, -w^2 + w - 1], [43, 43, w^2 + w + 1], [49, 7, w^3 + w^2 - 6*w - 3], [49, 7, w^3 - w^2 - 6*w + 3], [53, 53, 2*w^3 - w^2 - 9*w + 3], [53, 53, 2*w^3 + w^2 - 9*w - 3], [59, 59, w^3 - w^2 - 4*w + 1], [59, 59, -w^3 - w^2 + 4*w + 1], [67, 67, 3*w^3 - 13*w + 1], [67, 67, -w^3 + w^2 + 6*w - 5], [81, 3, -3], [83, 83, -2*w^3 - w^2 + 9*w + 7], [83, 83, 4*w^3 - 18*w - 1], [89, 89, -2*w^3 + 10*w + 1], [89, 89, -w^2 + w + 5], [89, 89, w^2 + w - 5], [89, 89, 2*w^3 - 10*w + 1], [101, 101, -2*w^3 + 8*w + 1], [101, 101, 2*w^3 - 8*w + 1], [103, 103, w^3 + 2*w^2 - 5*w - 7], [103, 103, w^3 + w^2 - 6*w - 7], [103, 103, -w^2 - 3*w + 1], [103, 103, -w^3 + 2*w^2 + 5*w - 7], [127, 127, -w^3 + 2*w^2 + 5*w - 5], [127, 127, w^3 + 3*w^2 - 4*w - 13], [127, 127, -w^3 + 3*w^2 + 4*w - 13], [127, 127, w^3 + 2*w^2 - 5*w - 5], [149, 149, -2*w^3 - w^2 + 7*w - 1], [149, 149, 2*w^3 - w^2 - 7*w - 1], [151, 151, -w^3 + w^2 + 2*w - 3], [151, 151, 2*w^3 - 10*w + 3], [151, 151, -2*w^3 + 10*w + 3], [151, 151, w^3 + w^2 - 2*w - 3], [157, 157, w^3 + 2*w^2 - 3*w - 1], [157, 157, -w^3 + 2*w^2 + 3*w - 1], [169, 13, 2*w^2 - 3], [179, 179, 2*w^3 + w^2 - 7*w - 3], [179, 179, -2*w^3 + w^2 + 7*w - 3], [229, 229, -w^3 + 2*w^2 + 3*w - 7], [229, 229, w^3 + 2*w^2 - 3*w - 7], [251, 251, -2*w^3 + 2*w^2 + 8*w - 9], [251, 251, -2*w^3 - 2*w^2 + 8*w + 9], [263, 263, 2*w^3 - w^2 - 3*w + 1], [263, 263, -2*w^3 + 2*w^2 + 10*w - 5], [263, 263, -2*w^3 - 2*w^2 + 10*w + 5], [263, 263, -2*w^3 - w^2 + 3*w + 1], [289, 17, 2*w^2 - 5], [293, 293, 4*w^3 - w^2 - 19*w + 7], [293, 293, -4*w^3 - w^2 + 19*w + 7], [307, 307, -w^3 + 3*w - 5], [307, 307, w^3 - 3*w - 5], [331, 331, 4*w^3 - 2*w^2 - 18*w + 7], [331, 331, 5*w^3 - 3*w^2 - 22*w + 11], [349, 349, -3*w^3 + 13*w + 3], [349, 349, -3*w^3 + 13*w - 3], [361, 19, 2*w^2 - 11], [373, 373, w^3 + 2*w^2 + w - 3], [373, 373, -w^3 + 2*w^2 - w - 3], [383, 383, -w^3 + 5*w + 5], [383, 383, -3*w^3 + 2*w^2 + 11*w - 5], [383, 383, 3*w^3 + 2*w^2 - 11*w - 5], [383, 383, w^3 - 5*w + 5], [389, 389, w^3 - 4*w^2 + w + 7], [389, 389, -2*w^3 - w^2 + 7*w + 7], [421, 421, -w^3 - w^2 - 2*w + 1], [421, 421, w^3 - w^2 + 2*w + 1], [433, 433, 4*w^3 - 2*w^2 - 16*w + 5], [433, 433, -4*w + 1], [433, 433, 4*w + 1], [433, 433, -5*w^3 + 3*w^2 + 24*w - 13], [443, 443, -w^3 + w^2 + 3], [443, 443, -4*w^2 + 6*w + 7], [457, 457, 3*w^3 + w^2 - 12*w - 3], [457, 457, -2*w^3 + 6*w - 1], [457, 457, 2*w^3 - 6*w - 1], [457, 457, 3*w^3 - w^2 - 12*w + 3], [461, 461, w^3 - 3*w^2 - 2*w + 7], [461, 461, w^3 + 3*w^2 - 2*w - 7], [463, 463, -2*w^3 + 3*w^2 + 9*w - 11], [463, 463, 2*w^3 - w^2 - 5*w + 1], [463, 463, -2*w^3 - w^2 + 5*w + 1], [463, 463, -2*w^3 - 3*w^2 + 9*w + 11], [467, 467, w^3 - 7*w + 1], [467, 467, -w^3 + 7*w + 1], [491, 491, 2*w^2 - 2*w - 7], [491, 491, 2*w^2 + 2*w - 7], [509, 509, w^3 + 3*w^2 - 4*w - 11], [509, 509, -w^3 + 3*w^2 + 4*w - 11], [523, 523, 2*w^3 - w^2 - 11*w + 3], [523, 523, 2*w^3 + w^2 - 11*w - 3], [529, 23, 3*w^2 - w - 15], [529, 23, -3*w^2 - w + 15], [557, 557, 3*w^2 + w - 13], [557, 557, -3*w^2 + w + 13], [563, 563, -3*w^3 + 3*w^2 + 12*w - 7], [563, 563, 2*w^3 - 2*w^2 - 8*w + 3], [569, 569, w^3 + 3*w^2 - 6*w - 9], [569, 569, 4*w^3 - w^2 - 17*w + 1], [569, 569, -4*w^3 - w^2 + 17*w + 1], [569, 569, -w^3 + 3*w^2 + 6*w - 9], [587, 587, -w^3 + w^2 + 2*w - 5], [587, 587, w^3 + w^2 - 2*w - 5], [593, 593, 3*w^3 - 2*w^2 - 15*w + 5], [593, 593, -w^3 - 2*w^2 + 5*w + 1], [593, 593, w^3 - 2*w^2 - 5*w + 1], [593, 593, -3*w^3 - 2*w^2 + 15*w + 5], [599, 599, w^3 + w^2 - 3], [599, 599, -w^2 - w - 3], [599, 599, -w^2 + w - 3], [599, 599, -w^3 + w^2 - 3], [613, 613, -w^3 + w^2 + 6*w - 9], [613, 613, w^3 + w^2 - 6*w - 9], [625, 5, -5], [659, 659, -2*w^3 + 4*w^2 + 4*w - 1], [659, 659, 2*w^3 + 4*w^2 - 4*w - 1], [661, 661, w^3 + 3*w^2 - 6*w - 7], [661, 661, -2*w^3 + 6*w^2 - 7], [701, 701, -w^3 - 2*w^2 + 3*w + 9], [701, 701, -w^3 + 2*w^2 + 3*w - 9], [733, 733, w^3 + 2*w^2 - 7*w - 1], [733, 733, -4*w^3 + w^2 + 19*w - 11], [739, 739, -w^3 - w^2 + 2*w + 9], [739, 739, -w^3 + w^2 + 2*w - 9], [757, 757, 3*w^3 - w^2 - 10*w + 5], [757, 757, -3*w^3 - w^2 + 10*w + 5], [773, 773, -4*w^3 + 5*w^2 + 19*w - 23], [773, 773, -7*w^3 + 6*w^2 + 33*w - 29], [797, 797, 3*w^3 - 13*w + 5], [797, 797, 3*w^3 - 13*w - 5], [829, 829, -2*w^3 - w^2 + 9*w - 1], [829, 829, 2*w^3 - w^2 - 9*w - 1], [859, 859, 3*w^3 - w^2 - 14*w + 1], [859, 859, -3*w^3 - w^2 + 14*w + 1], [863, 863, -5*w^3 + 21*w + 3], [863, 863, -3*w^3 - w^2 + 12*w - 1], [863, 863, 3*w^3 - w^2 - 12*w - 1], [863, 863, -5*w^3 + 21*w - 3], [883, 883, -2*w^3 + 4*w^2 + 8*w - 17], [883, 883, 2*w^3 + 4*w^2 - 8*w - 17], [953, 953, -w^3 - w^2 + 8*w + 3], [953, 953, 4*w^3 - 2*w^2 - 16*w + 7], [953, 953, -3*w^3 + w^2 + 16*w - 7], [953, 953, w^3 - w^2 - 8*w + 3], [961, 31, -4*w^3 + 2*w^2 + 12*w + 3], [961, 31, 3*w^3 - w^2 - 10*w - 1], [971, 971, 3*w^3 - 4*w^2 - 13*w + 15], [971, 971, -4*w^3 + 3*w^2 + 19*w - 17], [977, 977, -2*w^3 + w^2 + 9*w + 3], [977, 977, -4*w^3 + w^2 + 19*w - 9], [977, 977, 3*w^3 - 15*w + 5], [977, 977, 2*w^3 + w^2 - 9*w + 3]]; primes := [ideal : I in primesArray]; heckePol := x^9 + 3*x^8 - 7*x^7 - 25*x^6 + 11*x^5 + 65*x^4 + 10*x^3 - 50*x^2 - 20*x - 1; K := NumberField(heckePol); heckeEigenvaluesArray := [-e^5 - e^4 + 6*e^3 + 4*e^2 - 8*e - 3, e, e^8 + 2*e^7 - 8*e^6 - 16*e^5 + 18*e^4 + 39*e^3 - 7*e^2 - 27*e - 5, e^5 + 2*e^4 - 5*e^3 - 7*e^2 + 5*e, -e^8 - e^7 + 8*e^6 + 5*e^5 - 20*e^4 - 4*e^3 + 15*e^2 - 4*e - 3, 1, -2*e^8 - 3*e^7 + 17*e^6 + 22*e^5 - 44*e^4 - 46*e^3 + 32*e^2 + 23*e + 1, e^8 + 4*e^7 - 6*e^6 - 29*e^5 + 10*e^4 + 62*e^3 - 2*e^2 - 37*e - 6, -e^8 + 9*e^6 - 3*e^5 - 27*e^4 + 11*e^3 + 28*e^2 - 3*e - 6, -3*e^7 - 4*e^6 + 22*e^5 + 23*e^4 - 48*e^3 - 35*e^2 + 27*e + 8, 2*e^8 + 2*e^7 - 17*e^6 - 11*e^5 + 46*e^4 + 16*e^3 - 35*e^2 - 8*e - 7, 2*e^8 + 3*e^7 - 17*e^6 - 21*e^5 + 45*e^4 + 39*e^3 - 36*e^2 - 13*e + 1, -2*e^8 - 6*e^7 + 15*e^6 + 48*e^5 - 36*e^4 - 121*e^3 + 22*e^2 + 93*e + 19, e^8 + e^7 - 9*e^6 - 8*e^5 + 23*e^4 + 19*e^3 - 12*e^2 - 13*e - 3, e^7 + e^6 - 10*e^5 - 8*e^4 + 30*e^3 + 17*e^2 - 27*e - 11, e^6 + 2*e^5 - 3*e^4 - 8*e^3 - 5*e^2 + 8*e + 5, -e^8 + e^7 + 14*e^6 - 4*e^5 - 52*e^4 + e^3 + 53*e^2 + 6*e - 3, 4*e^8 + 6*e^7 - 33*e^6 - 40*e^5 + 90*e^4 + 81*e^3 - 82*e^2 - 52*e + 3, -e^8 - e^7 + 10*e^6 + 11*e^5 - 28*e^4 - 35*e^3 + 17*e^2 + 30*e + 6, -3*e^8 - 6*e^7 + 27*e^6 + 50*e^5 - 77*e^4 - 131*e^3 + 63*e^2 + 108*e + 10, 2*e^7 + 2*e^6 - 17*e^5 - 10*e^4 + 49*e^3 + 10*e^2 - 47*e - 3, 4*e^8 + 6*e^7 - 40*e^6 - 54*e^5 + 122*e^4 + 145*e^3 - 102*e^2 - 109*e - 15, -e^8 - 3*e^7 + 8*e^6 + 29*e^5 - 14*e^4 - 84*e^3 - 7*e^2 + 68*e + 8, -3*e^8 - 2*e^7 + 32*e^6 + 19*e^5 - 102*e^4 - 49*e^3 + 93*e^2 + 27*e - 1, -e^8 - 2*e^7 + 10*e^6 + 18*e^5 - 35*e^4 - 51*e^3 + 45*e^2 + 41*e - 10, -e^8 - 3*e^7 + 11*e^6 + 29*e^5 - 38*e^4 - 83*e^3 + 42*e^2 + 67*e - 3, -2*e^8 - 6*e^7 + 13*e^6 + 44*e^5 - 24*e^4 - 101*e^3 + 5*e^2 + 74*e + 15, 2*e^8 + 4*e^7 - 16*e^6 - 29*e^5 + 37*e^4 + 59*e^3 - 17*e^2 - 28*e - 11, 2*e^8 + e^7 - 24*e^6 - 17*e^5 + 84*e^4 + 63*e^3 - 89*e^2 - 53*e + 8, e^8 + e^7 - 11*e^6 - 9*e^5 + 37*e^4 + 26*e^3 - 38*e^2 - 30*e + 3, e^8 + 4*e^7 - 4*e^6 - 28*e^5 - 2*e^4 + 60*e^3 + 8*e^2 - 40*e + 6, -2*e^8 - 4*e^7 + 17*e^6 + 36*e^5 - 42*e^4 - 107*e^3 + 20*e^2 + 106*e + 12, 3*e^7 + 4*e^6 - 29*e^5 - 28*e^4 + 86*e^3 + 49*e^2 - 73*e - 14, e^8 - 10*e^6 + 33*e^4 + 5*e^3 - 36*e^2 - 17*e - 1, 3*e^8 + 3*e^7 - 28*e^6 - 20*e^5 + 85*e^4 + 38*e^3 - 83*e^2 - 14*e + 6, -2*e^8 - 5*e^7 + 19*e^6 + 51*e^5 - 47*e^4 - 150*e^3 + 9*e^2 + 116*e + 26, -2*e^8 - 5*e^7 + 13*e^6 + 34*e^5 - 24*e^4 - 69*e^3 + 7*e^2 + 45*e + 7, -4*e^8 - 7*e^7 + 32*e^6 + 52*e^5 - 79*e^4 - 124*e^3 + 51*e^2 + 99*e + 19, 3*e^8 + 5*e^7 - 27*e^6 - 41*e^5 + 73*e^4 + 99*e^3 - 49*e^2 - 64*e - 16, -e^8 + 11*e^6 - 3*e^5 - 40*e^4 + 20*e^3 + 56*e^2 - 29*e - 25, -3*e^8 - 8*e^7 + 24*e^6 + 65*e^5 - 55*e^4 - 161*e^3 + 17*e^2 + 113*e + 23, 2*e^8 - e^7 - 21*e^6 + 11*e^5 + 69*e^4 - 41*e^3 - 75*e^2 + 52*e + 8, 5*e^8 + 11*e^7 - 41*e^6 - 85*e^5 + 104*e^4 + 204*e^3 - 63*e^2 - 150*e - 33, -e^8 - 3*e^7 + 4*e^6 + 15*e^5 + 2*e^4 - 11*e^3 - 14*e^2 - 12*e - 5, -4*e^8 - 7*e^7 + 30*e^6 + 45*e^5 - 72*e^4 - 84*e^3 + 53*e^2 + 40*e + 7, 2*e^8 + e^7 - 20*e^6 - 4*e^5 + 64*e^4 - 2*e^3 - 57*e^2 + 9*e - 15, -7*e^7 - 10*e^6 + 55*e^5 + 57*e^4 - 134*e^3 - 77*e^2 + 94*e + 9, 2*e^8 + 7*e^7 - 14*e^6 - 54*e^5 + 31*e^4 + 128*e^3 - 16*e^2 - 89*e - 24, -4*e^8 - 9*e^7 + 34*e^6 + 73*e^5 - 92*e^4 - 184*e^3 + 73*e^2 + 131*e + 5, -2*e^8 + e^7 + 25*e^6 + e^5 - 83*e^4 - 24*e^3 + 72*e^2 + 31*e - 4, -3*e^8 - 6*e^7 + 19*e^6 + 37*e^5 - 30*e^4 - 68*e^3 - 4*e^2 + 46*e + 18, e^8 + 6*e^7 - 5*e^6 - 46*e^5 + 8*e^4 + 112*e^3 - 4*e^2 - 86*e - 17, -e^7 - 2*e^6 + 9*e^5 + 14*e^4 - 22*e^3 - 26*e^2 + 3*e + 8, 4*e^8 + 10*e^7 - 35*e^6 - 90*e^5 + 88*e^4 + 243*e^3 - 43*e^2 - 177*e - 26, 5*e^8 + 5*e^7 - 54*e^6 - 55*e^5 + 174*e^4 + 175*e^3 - 155*e^2 - 155*e - 24, 4*e^8 + 9*e^7 - 32*e^6 - 71*e^5 + 75*e^4 + 177*e^3 - 32*e^2 - 139*e - 28, -3*e^8 - 8*e^7 + 24*e^6 + 60*e^5 - 63*e^4 - 136*e^3 + 49*e^2 + 87*e, e^7 + 3*e^6 - 14*e^5 - 25*e^4 + 60*e^3 + 51*e^2 - 78*e - 13, 2*e^8 + 7*e^7 - 13*e^6 - 56*e^5 + 18*e^4 + 141*e^3 + 18*e^2 - 112*e - 21, -3*e^8 - 10*e^7 + 18*e^6 + 80*e^5 - 23*e^4 - 207*e^3 - 16*e^2 + 172*e + 25, 2*e^7 + 3*e^6 - 22*e^5 - 29*e^4 + 75*e^3 + 67*e^2 - 84*e - 25, 2*e^7 - 2*e^6 - 24*e^5 + 2*e^4 + 70*e^3 + 22*e^2 - 45*e - 13, e^8 - 7*e^6 + 3*e^5 + 8*e^4 - 9*e^3 + 20*e^2 - 7*e - 10, 6*e^8 + 8*e^7 - 55*e^6 - 63*e^5 + 163*e^4 + 152*e^3 - 160*e^2 - 99*e + 16, -8*e^8 - 13*e^7 + 72*e^6 + 104*e^5 - 200*e^4 - 246*e^3 + 158*e^2 + 155*e + 1, -3*e^8 - 4*e^7 + 33*e^6 + 37*e^5 - 114*e^4 - 107*e^3 + 109*e^2 + 95*e + 19, -6*e^8 - 14*e^7 + 43*e^6 + 107*e^5 - 88*e^4 - 251*e^3 + 37*e^2 + 161*e + 14, 4*e^8 + 10*e^7 - 38*e^6 - 94*e^5 + 110*e^4 + 277*e^3 - 76*e^2 - 246*e - 40, 5*e^8 + 12*e^7 - 44*e^6 - 101*e^5 + 124*e^4 + 258*e^3 - 97*e^2 - 181*e - 21, -e^8 + 2*e^7 + 19*e^6 - 4*e^5 - 85*e^4 - 15*e^3 + 119*e^2 + 14*e - 32, -2*e^8 + 5*e^7 + 34*e^6 - 33*e^5 - 146*e^4 + 68*e^3 + 178*e^2 - 45*e - 10, 4*e^8 + 6*e^7 - 36*e^6 - 50*e^5 + 101*e^4 + 136*e^3 - 77*e^2 - 123*e - 15, 5*e^8 + 9*e^7 - 30*e^6 - 46*e^5 + 49*e^4 + 49*e^3 - 27*e^2 + 9*e + 25, -3*e^8 + 32*e^6 + 4*e^5 - 104*e^4 - 30*e^3 + 103*e^2 + 56*e - 12, 3*e^8 + e^7 - 33*e^6 - 12*e^5 + 110*e^4 + 34*e^3 - 106*e^2 - 10*e - 2, -11*e^8 - 26*e^7 + 85*e^6 + 197*e^5 - 203*e^4 - 455*e^3 + 127*e^2 + 309*e + 43, 3*e^8 - e^7 - 39*e^6 - 11*e^5 + 139*e^4 + 78*e^3 - 138*e^2 - 90*e - 17, -2*e^7 + 9*e^6 + 38*e^5 - 47*e^4 - 154*e^3 + 49*e^2 + 168*e + 12, -2*e^7 + e^6 + 21*e^5 - e^4 - 71*e^3 - 28*e^2 + 75*e + 38, 7*e^8 + 12*e^7 - 62*e^6 - 95*e^5 + 175*e^4 + 229*e^3 - 151*e^2 - 158*e - 15, -2*e^8 - 6*e^7 + 11*e^6 + 51*e^5 - e^4 - 139*e^3 - 49*e^2 + 120*e + 35, -4*e^8 - 8*e^7 + 36*e^6 + 69*e^5 - 96*e^4 - 172*e^3 + 58*e^2 + 109*e + 35, 4*e^8 + 15*e^7 - 29*e^6 - 129*e^5 + 50*e^4 + 338*e^3 + 23*e^2 - 262*e - 50, e^7 + 4*e^6 - 11*e^5 - 34*e^4 + 38*e^3 + 79*e^2 - 25*e - 31, 4*e^8 + 6*e^7 - 37*e^6 - 43*e^5 + 109*e^4 + 77*e^3 - 98*e^2 - 10*e + 3, 4*e^8 + 5*e^7 - 36*e^6 - 32*e^5 + 112*e^4 + 64*e^3 - 125*e^2 - 54*e + 19, 9*e^7 + 5*e^6 - 81*e^5 - 29*e^4 + 220*e^3 + 44*e^2 - 166*e - 24, 3*e^8 + 4*e^7 - 23*e^6 - 28*e^5 + 46*e^4 + 60*e^3 - 2*e^2 - 43*e - 35, e^8 - e^7 - 15*e^6 + 2*e^5 + 61*e^4 + 16*e^3 - 68*e^2 - 28*e - 11, 3*e^8 + 4*e^7 - 21*e^6 - 25*e^5 + 40*e^4 + 57*e^3 - 5*e^2 - 68*e - 34, 7*e^8 + 13*e^7 - 59*e^6 - 95*e^5 + 168*e^4 + 213*e^3 - 169*e^2 - 145*e + 12, -3*e^8 - 12*e^7 + 15*e^6 + 97*e^5 - 3*e^4 - 257*e^3 - 54*e^2 + 216*e + 41, -6*e^8 - 10*e^7 + 61*e^6 + 96*e^5 - 186*e^4 - 269*e^3 + 152*e^2 + 209*e + 22, e^8 - e^7 - 9*e^6 + 18*e^5 + 35*e^4 - 61*e^3 - 64*e^2 + 36*e + 34, -e^8 - 9*e^7 - e^6 + 68*e^5 + 28*e^4 - 164*e^3 - 50*e^2 + 131*e + 14, 3*e^8 - 28*e^6 + 5*e^5 + 81*e^4 - 22*e^3 - 67*e^2 + 15*e - 2, 13*e^8 + 28*e^7 - 104*e^6 - 220*e^5 + 249*e^4 + 532*e^3 - 135*e^2 - 384*e - 63, 7*e^8 + 10*e^7 - 60*e^6 - 73*e^5 + 154*e^4 + 149*e^3 - 99*e^2 - 67*e - 17, e^7 + 3*e^6 + 8*e^5 + 4*e^4 - 58*e^3 - 58*e^2 + 58*e + 37, 5*e^8 + 4*e^7 - 54*e^6 - 40*e^5 + 184*e^4 + 118*e^3 - 206*e^2 - 91*e + 29, -4*e^8 - 3*e^7 + 30*e^6 + 3*e^5 - 72*e^4 + 52*e^3 + 58*e^2 - 92*e - 11, 5*e^8 + 19*e^7 - 30*e^6 - 146*e^5 + 46*e^4 + 349*e^3 - e^2 - 241*e - 28, -e^8 - 9*e^7 - 9*e^6 + 55*e^5 + 76*e^4 - 93*e^3 - 118*e^2 + 41*e + 22, -e^8 + 5*e^7 + 10*e^6 - 62*e^5 - 43*e^4 + 215*e^3 + 81*e^2 - 197*e - 42, -5*e^8 - 12*e^7 + 42*e^6 + 96*e^5 - 113*e^4 - 240*e^3 + 87*e^2 + 181*e + 19, -3*e^7 - 7*e^6 + 25*e^5 + 42*e^4 - 80*e^3 - 71*e^2 + 91*e + 37, 3*e^8 + 7*e^7 - 31*e^6 - 75*e^5 + 93*e^4 + 240*e^3 - 67*e^2 - 223*e - 32, 3*e^8 + 3*e^7 - 24*e^6 - 13*e^5 + 57*e^4 + 5*e^3 - 29*e^2 + 5*e - 24, -4*e^8 - 11*e^7 + 18*e^6 + 71*e^5 + e^4 - 150*e^3 - 60*e^2 + 126*e + 29, -3*e^8 + 34*e^6 + 12*e^5 - 99*e^4 - 61*e^3 + 43*e^2 + 68*e + 51, 10*e^8 + 20*e^7 - 81*e^6 - 153*e^5 + 205*e^4 + 366*e^3 - 141*e^2 - 268*e - 39, -e^8 + 3*e^7 + 24*e^6 - 3*e^5 - 109*e^4 - 43*e^3 + 131*e^2 + 54*e + 2, -12*e^8 - 21*e^7 + 99*e^6 + 152*e^5 - 263*e^4 - 336*e^3 + 213*e^2 + 221*e + 17, 2*e^8 + 12*e^7 - 7*e^6 - 98*e^5 - 20*e^4 + 246*e^3 + 84*e^2 - 182*e - 44, -e^8 - 8*e^7 - 3*e^6 + 63*e^5 + 48*e^4 - 164*e^3 - 94*e^2 + 135*e + 31, 3*e^8 + e^7 - 34*e^6 - 24*e^5 + 104*e^4 + 103*e^3 - 67*e^2 - 104*e - 44, 8*e^8 + 13*e^7 - 82*e^6 - 132*e^5 + 246*e^4 + 400*e^3 - 181*e^2 - 348*e - 57, -3*e^8 - 5*e^7 + 24*e^6 + 37*e^5 - 56*e^4 - 83*e^3 + 19*e^2 + 62*e + 37, e^8 + 2*e^7 - 7*e^6 - 19*e^5 + 9*e^4 + 66*e^3 + 19*e^2 - 76*e - 41, 5*e^8 + 4*e^7 - 52*e^6 - 35*e^5 + 166*e^4 + 81*e^3 - 154*e^2 - 29*e - 4, e^8 - 2*e^7 - 6*e^6 + 18*e^5 - e^4 - 50*e^3 + 22*e^2 + 42*e + 12, 2*e^8 + 8*e^7 - 11*e^6 - 62*e^5 + 22*e^4 + 159*e^3 - 34*e^2 - 137*e, -3*e^8 - 5*e^7 + 29*e^6 + 57*e^5 - 74*e^4 - 191*e^3 + 30*e^2 + 180*e + 17, -5*e^8 - 8*e^7 + 42*e^6 + 61*e^5 - 109*e^4 - 152*e^3 + 75*e^2 + 128*e + 21, e^8 - 3*e^7 - 13*e^6 + 36*e^5 + 58*e^4 - 126*e^3 - 86*e^2 + 136*e + 23, -e^8 - 10*e^7 - 4*e^6 + 72*e^5 + 48*e^4 - 152*e^3 - 75*e^2 + 78*e + 19, -4*e^8 - e^7 + 36*e^6 - e^5 - 105*e^4 + 24*e^3 + 113*e^2 - 19*e - 37, -6*e^8 - 16*e^7 + 41*e^6 + 115*e^5 - 89*e^4 - 261*e^3 + 58*e^2 + 195*e + 10, -10*e^8 - 14*e^7 + 86*e^6 + 100*e^5 - 229*e^4 - 201*e^3 + 178*e^2 + 89*e - 5, 2*e^8 + 2*e^7 - 15*e^6 - 16*e^5 + 33*e^4 + 50*e^3 - 17*e^2 - 50*e - 4, 6*e^8 + 5*e^7 - 63*e^6 - 55*e^5 + 194*e^4 + 170*e^3 - 166*e^2 - 142*e + 7, -3*e^8 - 4*e^7 + 31*e^6 + 44*e^5 - 93*e^4 - 146*e^3 + 55*e^2 + 142*e + 56, -2*e^8 + 2*e^7 + 18*e^6 - 25*e^5 - 57*e^4 + 75*e^3 + 75*e^2 - 49*e - 25, -8*e^8 - 2*e^7 + 87*e^6 + 24*e^5 - 284*e^4 - 69*e^3 + 272*e^2 + 44*e - 14, -e^8 + e^7 + 8*e^6 - 22*e^5 - 22*e^4 + 102*e^3 + 25*e^2 - 115*e - 11, 2*e^8 - 18*e^6 + 8*e^5 + 55*e^4 - 44*e^3 - 57*e^2 + 55*e + 14, 4*e^8 + 6*e^7 - 46*e^6 - 69*e^5 + 159*e^4 + 233*e^3 - 165*e^2 - 213*e - 7, -e^8 - 11*e^7 - e^6 + 95*e^5 + 36*e^4 - 256*e^3 - 83*e^2 + 198*e + 56, 5*e^8 + 12*e^7 - 32*e^6 - 79*e^5 + 56*e^4 + 150*e^3 - 13*e^2 - 64*e - 16, -12*e^8 - 22*e^7 + 100*e^6 + 157*e^5 - 265*e^4 - 317*e^3 + 225*e^2 + 152*e - 28, -7*e^8 - 15*e^7 + 52*e^6 + 103*e^5 - 115*e^4 - 203*e^3 + 56*e^2 + 112*e + 23, 5*e^8 + 10*e^7 - 39*e^6 - 84*e^5 + 83*e^4 + 228*e^3 - 21*e^2 - 199*e - 42, 8*e^8 + 21*e^7 - 62*e^6 - 163*e^5 + 154*e^4 + 377*e^3 - 122*e^2 - 234*e - 5, -13*e^8 - 22*e^7 + 118*e^6 + 184*e^5 - 325*e^4 - 464*e^3 + 240*e^2 + 326*e + 30, 2*e^8 + 15*e^7 - 8*e^6 - 130*e^5 - 9*e^4 + 358*e^3 + 53*e^2 - 307*e - 52, -9*e^8 - 18*e^7 + 66*e^6 + 120*e^5 - 165*e^4 - 245*e^3 + 165*e^2 + 149*e - 13, -7*e^8 - 6*e^7 + 65*e^6 + 52*e^5 - 173*e^4 - 135*e^3 + 91*e^2 + 86*e + 54, 8*e^8 + 14*e^7 - 71*e^6 - 110*e^5 + 200*e^4 + 271*e^3 - 160*e^2 - 211*e - 16, -4*e^8 - 7*e^7 + 34*e^6 + 62*e^5 - 87*e^4 - 174*e^3 + 62*e^2 + 133*e - 9, 3*e^8 + 4*e^7 - 28*e^6 - 25*e^5 + 95*e^4 + 32*e^3 - 135*e^2 + 18*e + 58, -15*e^8 - 25*e^7 + 124*e^6 + 181*e^5 - 326*e^4 - 393*e^3 + 259*e^2 + 239*e + 16, 12*e^8 + 19*e^7 - 110*e^6 - 148*e^5 + 332*e^4 + 353*e^3 - 331*e^2 - 238*e + 23, -5*e^8 - 3*e^7 + 63*e^6 + 41*e^5 - 225*e^4 - 128*e^3 + 219*e^2 + 89*e + 12, 6*e^8 + 11*e^7 - 42*e^6 - 71*e^5 + 83*e^4 + 128*e^3 - 22*e^2 - 48*e - 44, -18*e^8 - 33*e^7 + 151*e^6 + 261*e^5 - 377*e^4 - 633*e^3 + 216*e^2 + 449*e + 78, -7*e^8 - 15*e^7 + 52*e^6 + 98*e^5 - 129*e^4 - 181*e^3 + 105*e^2 + 87*e + 9, -13*e^8 - 22*e^7 + 115*e^6 + 183*e^5 - 313*e^4 - 469*e^3 + 236*e^2 + 355*e + 37]; 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;