/* 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, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [5, 5, -w^3 + 5*w + 2], [5, 5, w - 1], [11, 11, -w^3 + 5*w], [13, 13, -w^3 + w^2 + 6*w - 3], [16, 2, 2], [19, 19, 2*w^3 - w^2 - 11*w + 2], [25, 5, -w^2 + w + 3], [27, 3, w^3 - w^2 - 5*w + 4], [31, 31, -w - 3], [37, 37, -w^3 - w^2 + 6*w + 4], [41, 41, w^3 + w^2 - 6*w - 5], [43, 43, w^3 - 7*w - 2], [47, 47, -2*w^3 + 11*w + 2], [61, 61, w^2 - 3], [67, 67, w^2 + w - 4], [79, 79, -4*w^3 + 2*w^2 + 22*w - 9], [97, 97, -3*w^3 + 2*w^2 + 19*w - 6], [97, 97, -w^3 + 6*w - 3], [97, 97, -3*w^3 + 16*w], [97, 97, -w^3 + w^2 + 4*w - 3], [101, 101, -2*w^3 + w^2 + 11*w - 6], [103, 103, w^3 - w^2 - 7*w], [109, 109, -w^3 - w^2 + 6*w + 2], [121, 11, 2*w^3 - 11*w + 1], [127, 127, -2*w^3 + w^2 + 13*w - 1], [127, 127, w^3 - w^2 - 6*w - 2], [157, 157, -2*w^3 + 2*w^2 + 10*w - 7], [157, 157, w - 4], [163, 163, 2*w^3 - w^2 - 13*w + 3], [173, 173, -2*w^3 + w^2 + 13*w - 6], [173, 173, 2*w^3 - 12*w + 1], [173, 173, 2*w^3 - 11*w + 2], [173, 173, w^2 + w - 5], [179, 179, -2*w^3 + 10*w + 3], [181, 181, -w^3 - w^2 + 7*w + 8], [191, 191, w^3 - 5*w - 5], [191, 191, 2*w^3 - w^2 - 9*w - 1], [193, 193, -3*w^3 + 3*w^2 + 19*w - 7], [197, 197, -w^3 + w^2 + 6*w - 7], [223, 223, w^3 + w^2 - 6*w + 1], [227, 227, 6*w^3 - 3*w^2 - 34*w + 9], [227, 227, w^3 + w^2 - 8*w], [233, 233, 2*w^3 - 13*w - 3], [233, 233, -w^3 + 2*w^2 + 4*w - 6], [239, 239, -2*w^3 + 10*w + 1], [239, 239, w^3 - 8*w - 2], [251, 251, -3*w^3 + w^2 + 18*w], [251, 251, -5*w^3 + 2*w^2 + 30*w - 11], [263, 263, -2*w^3 + w^2 + 10*w + 2], [269, 269, 3*w^3 - 18*w - 8], [271, 271, 2*w^3 + w^2 - 10*w - 2], [271, 271, -3*w^3 + 19*w], [277, 277, -w^3 + w^2 + 4*w - 8], [277, 277, 2*w^3 + w^2 - 14*w - 8], [281, 281, w^3 - 8*w], [307, 307, 3*w^3 - w^2 - 16*w + 2], [307, 307, 2*w^3 - w^2 - 9*w + 1], [307, 307, -2*w^3 + 2*w^2 + 13*w - 7], [307, 307, -w^3 + w^2 + 7*w - 6], [313, 313, -3*w^3 + w^2 + 15*w + 4], [317, 317, -4*w^3 + w^2 + 22*w - 3], [317, 317, 2*w^3 - w^2 - 10*w - 3], [337, 337, w^3 + w^2 - 4*w - 5], [349, 349, -2*w^3 - w^2 + 11*w + 3], [349, 349, -4*w^3 + 2*w^2 + 24*w - 11], [353, 353, 3*w^3 - w^2 - 17*w + 7], [353, 353, -3*w^3 - 2*w^2 + 17*w + 11], [367, 367, 2*w^3 - 11*w - 7], [373, 373, -w^3 + w^2 + 4*w - 5], [383, 383, -4*w^3 + 3*w^2 + 23*w - 9], [383, 383, 4*w^3 - w^2 - 22*w], [389, 389, w^2 - 2*w - 5], [401, 401, -3*w^3 + 2*w^2 + 16*w - 9], [401, 401, 2*w^3 - w^2 - 14*w - 1], [409, 409, 2*w^3 + 2*w^2 - 13*w - 9], [419, 419, -4*w^3 + 24*w + 7], [419, 419, 2*w^2 + 2*w - 7], [419, 419, -w^3 + 2*w^2 + 8*w - 6], [419, 419, 6*w^3 - 3*w^2 - 33*w + 13], [431, 431, 2*w^3 - w^2 - 12*w - 5], [439, 439, -3*w^3 + 2*w^2 + 18*w - 4], [439, 439, w^3 + 2*w^2 - 7*w - 3], [443, 443, 6*w^3 - 3*w^2 - 33*w + 7], [449, 449, -3*w^3 + w^2 + 14*w - 4], [457, 457, 2*w^3 - 9*w], [457, 457, 3*w^3 - w^2 - 16*w - 2], [479, 479, 3*w^3 - w^2 - 19*w + 4], [487, 487, -w^3 + w^2 + 4*w - 6], [491, 491, 3*w^3 + w^2 - 17*w - 5], [499, 499, 5*w^3 - 3*w^2 - 28*w + 8], [499, 499, 2*w^3 - w^2 - 12*w - 1], [503, 503, -2*w^3 - w^2 + 12*w + 3], [503, 503, w^3 - 3*w - 4], [509, 509, -3*w - 5], [509, 509, -4*w^3 + 3*w^2 + 25*w - 10], [523, 523, -2*w^3 + w^2 + 14*w - 6], [529, 23, 4*w^3 - 2*w^2 - 21*w + 5], [529, 23, -3*w^3 - w^2 + 15*w + 6], [557, 557, -2*w^3 + w^2 + 13*w + 2], [557, 557, 3*w^3 - 2*w^2 - 16*w + 3], [569, 569, 2*w^2 - w - 8], [587, 587, w^3 + 2*w^2 - 5*w - 16], [587, 587, -4*w^3 + w^2 + 25*w - 5], [599, 599, w^3 + 2*w^2 - 6*w - 5], [599, 599, -5*w^3 + 3*w^2 + 30*w - 15], [601, 601, w^3 - 6*w - 6], [607, 607, -w^3 + w^2 + 3*w - 4], [607, 607, -w^3 + 2*w - 3], [613, 613, 6*w^3 - 2*w^2 - 31*w + 11], [613, 613, w^3 + w^2 - 4*w - 6], [617, 617, 5*w^3 - w^2 - 30*w + 2], [617, 617, -2*w^3 + 2*w^2 + 9*w + 3], [619, 619, -w^3 + 2*w^2 + 6*w - 5], [619, 619, 2*w^3 - w^2 - 11*w - 3], [631, 631, -w^3 + 2*w^2 + 5*w - 5], [631, 631, -3*w^3 - w^2 + 18*w + 5], [647, 647, 6*w^3 - w^2 - 36*w + 2], [659, 659, 4*w^3 + w^2 - 24*w - 14], [661, 661, 2*w^2 - w - 4], [673, 673, 2*w^3 + w^2 - 14*w - 12], [677, 677, 3*w^3 + 2*w^2 - 16*w - 7], [677, 677, -3*w^3 + w^2 + 15*w + 1], [683, 683, 2*w^2 - 2*w - 9], [683, 683, 2*w^3 - 14*w + 1], [691, 691, 5*w^3 - 2*w^2 - 30*w + 6], [701, 701, -2*w^3 + w^2 + 12*w + 2], [701, 701, -5*w^3 + 2*w^2 + 26*w], [719, 719, 3*w^3 - 18*w + 1], [719, 719, 3*w^3 - w^2 - 19*w + 1], [727, 727, -3*w^2 - 2*w + 11], [727, 727, 2*w^3 + w^2 - 13*w - 2], [733, 733, 2*w^3 - 3*w^2 - 8*w + 5], [739, 739, -3*w^3 + 16*w - 6], [743, 743, -4*w^3 - w^2 + 25*w + 8], [751, 751, w^2 - 2*w - 7], [751, 751, -3*w^3 + 15*w + 5], [757, 757, 3*w^3 - 19*w - 2], [761, 761, 2*w^2 + w - 5], [769, 769, -2*w^3 + w^2 + 14*w - 9], [769, 769, 2*w^2 - w - 7], [773, 773, w^3 - w^2 - 5*w - 4], [773, 773, w^3 - 3*w - 6], [787, 787, 3*w^3 - 19*w - 3], [797, 797, 8*w^3 - 3*w^2 - 47*w + 10], [809, 809, 4*w^3 - 25*w - 1], [811, 811, w^3 - 9*w - 1], [821, 821, 6*w^3 - 3*w^2 - 35*w + 8], [823, 823, 2*w^3 + w^2 - 10*w + 1], [823, 823, -2*w^3 + 2*w^2 + 7*w - 4], [829, 829, -5*w^3 + w^2 + 27*w - 4], [829, 829, 5*w^3 - 2*w^2 - 26*w + 10], [839, 839, w^3 + 2*w^2 - 5*w - 7], [839, 839, 3*w^3 - w^2 - 15*w + 2], [841, 29, 3*w^3 + w^2 - 14*w - 2], [841, 29, 4*w^3 - 2*w^2 - 21*w + 10], [857, 857, -8*w^3 + 4*w^2 + 44*w - 11], [857, 857, -3*w^3 + 3*w^2 + 20*w - 9], [859, 859, w^3 + 2*w^2 - 6*w - 6], [877, 877, w^3 - 3*w^2 - 4*w + 14], [883, 883, 2*w^3 + w^2 - 10*w - 11], [887, 887, 3*w^3 + w^2 - 20*w - 2], [919, 919, 7*w^3 - 3*w^2 - 39*w + 9], [929, 929, 3*w^3 - 17*w + 2], [929, 929, -6*w^3 + 2*w^2 + 35*w - 10], [937, 937, w^3 + w^2 - 3*w - 5], [937, 937, -2*w^3 + 3*w^2 + 12*w - 12], [937, 937, -3*w^3 - 2*w^2 + 19*w + 8], [937, 937, 3*w^3 - 2*w^2 - 13*w + 3], [941, 941, w^3 + 2*w^2 - 6*w - 8], [941, 941, 5*w^3 - w^2 - 28*w - 3], [947, 947, -4*w^3 + 2*w^2 + 25*w - 6], [953, 953, w^3 - 7*w - 7], [953, 953, -3*w^3 + 2*w^2 + 18*w - 3], [967, 967, w^3 - 2*w^2 - 5*w - 3], [971, 971, w^2 + 2*w - 6], [977, 977, 2*w^3 - 11*w - 8], [997, 997, -3*w^3 + w^2 + 17*w + 2], [997, 997, -2*w^3 + 2*w^2 + 14*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^6 + 2*x^5 - 12*x^4 - 8*x^3 + 47*x^2 - 33*x + 5; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1, 1, -e^5 - 3*e^4 + 8*e^3 + 16*e^2 - 23*e, 2*e^3 + e^2 - 16*e + 8, e^5 + 3*e^4 - 9*e^3 - 16*e^2 + 32*e - 10, e^5 + 3*e^4 - 9*e^3 - 16*e^2 + 32*e - 8, e^5 + 3*e^4 - 12*e^3 - 19*e^2 + 51*e - 17, -2*e^5 - 6*e^4 + 16*e^3 + 29*e^2 - 53*e + 16, -e^4 - 2*e^3 + 8*e^2 + 9*e - 10, -e^4 - e^3 + 7*e^2 - 2*e - 5, e^5 + 3*e^4 - 6*e^3 - 14*e^2 + 11*e + 7, e^4 - 3*e^3 - 12*e^2 + 26*e - 4, -e^5 - e^4 + 15*e^3 + 5*e^2 - 56*e + 22, e^5 + 5*e^4 - 9*e^3 - 32*e^2 + 47*e - 12, -e^5 - 4*e^4 + 10*e^3 + 24*e^2 - 49*e + 11, -e^5 - 3*e^4 + 11*e^3 + 17*e^2 - 45*e + 20, -2*e^4 - 4*e^3 + 10*e^2 + 8*e + 5, 2*e^5 + 5*e^4 - 12*e^3 - 16*e^2 + 19*e - 12, e^5 + 3*e^4 - 7*e^3 - 10*e^2 + 24*e - 21, -e^5 - 3*e^4 + 10*e^3 + 15*e^2 - 42*e + 8, e^5 + 3*e^4 - 10*e^3 - 16*e^2 + 42*e - 14, 3*e^5 + 11*e^4 - 25*e^3 - 60*e^2 + 105*e - 29, -3*e^5 - 7*e^4 + 27*e^3 + 31*e^2 - 82*e + 33, -e^5 - e^4 + 7*e^3 - e^2, -5*e^5 - 12*e^4 + 47*e^3 + 57*e^2 - 149*e + 48, 3*e^5 + 7*e^4 - 25*e^3 - 28*e^2 + 68*e - 27, e^5 + 4*e^4 - 6*e^3 - 22*e^2 + 17*e + 8, -e^5 - 2*e^4 + 14*e^3 + 15*e^2 - 57*e + 9, -e^4 - 2*e^3 + 4*e^2 + 5*e + 1, e^5 + 3*e^4 - 7*e^3 - 17*e^2 + 10*e + 8, 2*e^5 + 5*e^4 - 15*e^3 - 18*e^2 + 42*e - 17, -2*e^5 - 3*e^4 + 27*e^3 + 17*e^2 - 101*e + 37, -e^5 - 5*e^4 + 8*e^3 + 32*e^2 - 39*e + 1, -e^5 - 4*e^4 + 5*e^3 + 20*e^2 - 15*e - 6, -2*e^5 - 8*e^4 + 11*e^3 + 38*e^2 - 42*e + 19, 2*e^4 + 12*e^3 - 6*e^2 - 62*e + 36, -e^5 - 5*e^4 + 14*e^3 + 40*e^2 - 75*e + 7, -e^5 - 5*e^4 + 8*e^3 + 31*e^2 - 43*e - 3, 4*e^5 + 15*e^4 - 30*e^3 - 79*e^2 + 120*e - 34, -e^5 - 5*e^4 + 11*e^3 + 31*e^2 - 70*e + 18, -3*e^5 - 5*e^4 + 31*e^3 + 20*e^2 - 91*e + 21, 3*e^5 + 10*e^4 - 28*e^3 - 55*e^2 + 113*e - 32, e^4 + 5*e^3 - 4*e^2 - 22*e + 14, 2*e^3 + 3*e^2 - 15*e - 6, e^4 - 5*e^3 - 13*e^2 + 38*e - 14, -e^5 + e^4 + 18*e^3 - 12*e^2 - 71*e + 46, -2*e^4 + 15*e^2 - 23*e - 7, 2*e^5 + 2*e^4 - 22*e^3 - 9*e^2 + 49*e - 8, -5*e^5 - 13*e^4 + 44*e^3 + 65*e^2 - 136*e + 21, -4*e^5 - 11*e^4 + 38*e^3 + 60*e^2 - 127*e + 25, 4*e^5 + 9*e^4 - 45*e^3 - 53*e^2 + 154*e - 41, -e^5 - 7*e^4 + 8*e^3 + 43*e^2 - 67*e + 24, 3*e^4 - 20*e^2 + 31*e - 17, -4*e^5 - 12*e^4 + 32*e^3 + 65*e^2 - 98*e + 3, 3*e^5 + 13*e^4 - 14*e^3 - 70*e^2 + 43*e + 19, 2*e^5 + 5*e^4 - 22*e^3 - 33*e^2 + 73*e - 11, 3*e^5 + 7*e^4 - 26*e^3 - 35*e^2 + 66*e - 11, e^5 - 2*e^4 - 18*e^3 + 12*e^2 + 45*e - 22, 2*e^5 + 6*e^4 - 17*e^3 - 35*e^2 + 49*e + 7, 7*e^5 + 15*e^4 - 70*e^3 - 72*e^2 + 216*e - 74, 2*e^5 + 2*e^4 - 18*e^3 + 2*e^2 + 33*e - 29, -e^5 - 6*e^4 + 4*e^3 + 37*e^2 - 19*e - 10, -2*e^5 - 6*e^4 + 21*e^3 + 34*e^2 - 84*e + 17, -2*e^5 - 4*e^4 + 19*e^3 + 17*e^2 - 59*e + 21, 4*e^5 + 14*e^4 - 44*e^3 - 88*e^2 + 193*e - 60, e^5 - e^4 - 15*e^3 + 10*e^2 + 41*e - 24, -2*e^4 - 6*e^3 + 10*e^2 + 27*e - 5, 2*e^5 + 4*e^4 - 22*e^3 - 16*e^2 + 79*e - 29, e^5 + 3*e^4 - 13*e^3 - 23*e^2 + 52*e - 10, 4*e^5 + 8*e^4 - 48*e^3 - 46*e^2 + 169*e - 61, e^5 + 3*e^4 - 15*e^3 - 18*e^2 + 75*e - 32, 4*e^5 + 8*e^4 - 46*e^3 - 39*e^2 + 168*e - 68, 5*e^5 + 11*e^4 - 51*e^3 - 58*e^2 + 160*e - 41, 6*e^5 + 12*e^4 - 61*e^3 - 52*e^2 + 198*e - 67, -5*e^5 - 7*e^4 + 62*e^3 + 32*e^2 - 212*e + 79, -2*e^5 - 5*e^4 + 15*e^3 + 15*e^2 - 44*e + 32, -2*e^5 - e^4 + 17*e^3 - 10*e^2 - 21*e + 13, e^5 - 11*e^3 + 4*e^2 + 11*e - 8, 2*e^5 + 4*e^4 - 14*e^3 - 9*e^2 + 26*e - 5, -3*e^5 - 5*e^4 + 27*e^3 + 14*e^2 - 71*e + 36, -6*e^5 - 16*e^4 + 56*e^3 + 80*e^2 - 191*e + 53, 3*e^4 + 5*e^3 - 10*e^2 + 7*e - 28, 7*e^4 + 15*e^3 - 39*e^2 - 39*e + 24, e^5 - 2*e^4 - 19*e^3 + 9*e^2 + 55*e - 2, -4*e^5 - 15*e^4 + 33*e^3 + 82*e^2 - 135*e + 40, -4*e^5 - 13*e^4 + 38*e^3 + 76*e^2 - 141*e + 24, e^5 - 3*e^4 - 19*e^3 + 21*e^2 + 40*e - 30, -2*e^5 - 3*e^4 + 20*e^3 + 15*e^2 - 49*e - 14, -5*e^5 - 14*e^4 + 49*e^3 + 77*e^2 - 169*e + 54, 4*e^5 + 9*e^4 - 44*e^3 - 41*e^2 + 163*e - 73, 6*e^5 + 19*e^4 - 60*e^3 - 112*e^2 + 234*e - 59, 8*e^5 + 22*e^4 - 72*e^3 - 107*e^2 + 240*e - 81, -6*e^5 - 15*e^4 + 54*e^3 + 73*e^2 - 165*e + 49, -e^5 + 8*e^3 - 5*e^2 + 9*e - 9, 2*e^5 + 6*e^4 - 29*e^3 - 39*e^2 + 139*e - 57, -3*e^5 - 5*e^4 + 35*e^3 + 25*e^2 - 115*e + 47, 3*e^5 + 4*e^4 - 36*e^3 - 18*e^2 + 112*e - 50, 8*e^5 + 25*e^4 - 77*e^3 - 140*e^2 + 296*e - 86, 2*e^5 + 4*e^4 - 13*e^3 - 13*e^2 + 11*e + 3, -e^5 - 6*e^4 + 15*e^3 + 45*e^2 - 98*e + 23, 4*e^5 + 11*e^4 - 46*e^3 - 63*e^2 + 193*e - 77, -4*e^5 - 12*e^4 + 18*e^3 + 49*e^2 - 2*e - 37, 11*e^3 + 4*e^2 - 87*e + 48, e^5 + 5*e^4 - 13*e^3 - 46*e^2 + 58*e + 21, 6*e^5 + 15*e^4 - 67*e^3 - 86*e^2 + 251*e - 83, 5*e^5 + 16*e^4 - 50*e^3 - 95*e^2 + 203*e - 35, 6*e^5 + 11*e^4 - 60*e^3 - 45*e^2 + 181*e - 53, 4*e^5 + 11*e^4 - 35*e^3 - 57*e^2 + 111*e - 4, 4*e^5 + 14*e^4 - 24*e^3 - 61*e^2 + 70*e - 18, -4*e^5 - 8*e^4 + 28*e^3 + 27*e^2 - 40*e, -6*e^5 - 20*e^4 + 53*e^3 + 117*e^2 - 188*e + 24, -5*e^5 - 14*e^4 + 42*e^3 + 70*e^2 - 125*e + 32, 6*e^5 + 13*e^4 - 69*e^3 - 71*e^2 + 242*e - 90, -3*e^5 - 13*e^4 + 21*e^3 + 76*e^2 - 83*e - 9, 11*e^5 + 30*e^4 - 94*e^3 - 148*e^2 + 289*e - 60, -2*e^5 - 6*e^4 + 15*e^3 + 27*e^2 - 38*e + 28, 4*e^5 + 16*e^4 - 22*e^3 - 84*e^2 + 69*e + 2, 5*e^5 + 12*e^4 - 42*e^3 - 56*e^2 + 121*e - 4, -5*e^5 - 10*e^4 + 54*e^3 + 50*e^2 - 179*e + 44, 2*e^5 + 4*e^4 - 16*e^3 - 6*e^2 + 51*e - 36, -6*e^5 - 15*e^4 + 52*e^3 + 67*e^2 - 148*e + 64, -3*e^5 - 13*e^4 + 20*e^3 + 71*e^2 - 85*e + 32, 3*e^5 + 6*e^4 - 42*e^3 - 41*e^2 + 164*e - 56, 5*e^5 + 7*e^4 - 64*e^3 - 33*e^2 + 217*e - 92, -5*e^5 - 17*e^4 + 50*e^3 + 105*e^2 - 203*e + 47, -4*e^5 - 13*e^4 + 41*e^3 + 75*e^2 - 167*e + 53, -3*e^5 - 4*e^4 + 51*e^3 + 31*e^2 - 208*e + 70, e^4 + 5*e^3 - e^2 - 9*e - 3, -2*e^5 - 5*e^4 + 21*e^3 + 17*e^2 - 88*e + 48, 7*e^5 + 17*e^4 - 66*e^3 - 92*e^2 + 195*e - 40, -3*e^4 + 2*e^3 + 24*e^2 - 36*e + 17, 2*e^5 + 8*e^4 - 27*e^3 - 52*e^2 + 151*e - 51, 3*e^5 + 13*e^4 - 24*e^3 - 77*e^2 + 116*e - 6, 6*e^5 + 21*e^4 - 59*e^3 - 123*e^2 + 252*e - 68, -8*e^5 - 25*e^4 + 70*e^3 + 130*e^2 - 265*e + 96, -2*e^5 - 2*e^4 + 16*e^3 + 5*e^2 - 7*e - 46, -2*e^5 - 6*e^4 + 19*e^3 + 38*e^2 - 70*e + 2, 2*e^4 + 7*e^3 - 11*e^2 - 41*e + 5, 3*e^5 + 8*e^4 - 42*e^3 - 52*e^2 + 188*e - 77, -6*e^5 - 16*e^4 + 58*e^3 + 83*e^2 - 194*e + 77, -3*e^5 - 7*e^4 + 25*e^3 + 35*e^2 - 67*e - 7, -8*e^5 - 25*e^4 + 62*e^3 + 129*e^2 - 190*e + 9, -7*e^5 - 20*e^4 + 52*e^3 + 91*e^2 - 151*e + 47, 2*e^4 + 9*e^3 - 4*e^2 - 28*e - 5, 4*e^5 + 13*e^4 - 28*e^3 - 68*e^2 + 82*e - 7, 4*e^5 + 14*e^4 - 24*e^3 - 69*e^2 + 63*e, -10*e^5 - 32*e^4 + 88*e^3 + 173*e^2 - 317*e + 78, -3*e^4 + 3*e^3 + 28*e^2 - 50*e + 5, -9*e^5 - 29*e^4 + 73*e^3 + 149*e^2 - 253*e + 64, -7*e^5 - 23*e^4 + 55*e^3 + 123*e^2 - 189*e + 25, -10*e^5 - 25*e^4 + 98*e^3 + 125*e^2 - 338*e + 96, -e^5 - 8*e^4 - 5*e^3 + 36*e^2 + 13*e + 3, 6*e^5 + 16*e^4 - 46*e^3 - 78*e^2 + 110*e - 9, -3*e^5 - 5*e^4 + 28*e^3 + 26*e^2 - 60*e - 29, 8*e^4 + 14*e^3 - 42*e^2 - 20*e - 11, -3*e^5 - 16*e^4 + 10*e^3 + 84*e^2 - 43*e - 8, 3*e^5 + 10*e^4 - 26*e^3 - 48*e^2 + 97*e - 61, e^5 + 8*e^4 + 5*e^3 - 36*e^2 - 10*e - 1, -e^5 - 7*e^4 + 9*e^3 + 52*e^2 - 52*e - 15, -8*e^5 - 31*e^4 + 68*e^3 + 179*e^2 - 290*e + 46, -5*e^5 - 8*e^4 + 55*e^3 + 41*e^2 - 158*e + 35, 7*e^5 + 24*e^4 - 59*e^3 - 133*e^2 + 225*e - 33, 8*e^5 + 27*e^4 - 65*e^3 - 146*e^2 + 229*e - 50, e^5 - 3*e^4 - 22*e^3 + 28*e^2 + 85*e - 55, -6*e^5 - 16*e^4 + 74*e^3 + 102*e^2 - 304*e + 89, 5*e^5 + 13*e^4 - 46*e^3 - 65*e^2 + 151*e - 56, -8*e^5 - 13*e^4 + 88*e^3 + 53*e^2 - 280*e + 102, -5*e^5 - 15*e^4 + 40*e^3 + 80*e^2 - 119*e - 10, -8*e^5 - 20*e^4 + 68*e^3 + 90*e^2 - 199*e + 52, -e^5 + 3*e^4 + 12*e^3 - 36*e^2 - 14*e + 49, -13*e^5 - 31*e^4 + 131*e^3 + 162*e^2 - 428*e + 133, 6*e^5 + 13*e^4 - 56*e^3 - 54*e^2 + 168*e - 59, 8*e^5 + 21*e^4 - 92*e^3 - 119*e^2 + 372*e - 127, -2*e^5 - 6*e^4 + 26*e^3 + 33*e^2 - 128*e + 27, -6*e^5 - 23*e^4 + 53*e^3 + 145*e^2 - 209*e + 16, -6*e^5 - 17*e^4 + 54*e^3 + 90*e^2 - 165*e + 35, 3*e^5 + 6*e^4 - 29*e^3 - 33*e^2 + 66*e - 19, -4*e^5 - 14*e^4 + 29*e^3 + 78*e^2 - 99*e - 17]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; 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;