/* 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, -2, 7, 2, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [29, 29, -9*w^5 + 3*w^4 + 64*w^3 + 26*w^2 - 40*w - 10], [29, 29, w^5 - 7*w^3 - 5*w^2 + 2*w + 2], [29, 29, w^4 - w^3 - 6*w^2 + 2], [29, 29, 5*w^5 - w^4 - 36*w^3 - 19*w^2 + 21*w + 9], [29, 29, -w^5 + w^4 + 7*w^3 - 2*w^2 - 6*w + 1], [29, 29, 2*w^5 - 15*w^3 - 10*w^2 + 11*w + 5], [41, 41, 5*w^5 - w^4 - 36*w^3 - 18*w^2 + 21*w + 5], [41, 41, -5*w^5 + 2*w^4 + 36*w^3 + 11*w^2 - 25*w - 2], [41, 41, 6*w^5 - w^4 - 44*w^3 - 23*w^2 + 30*w + 8], [41, 41, 13*w^5 - 4*w^4 - 93*w^3 - 39*w^2 + 59*w + 16], [41, 41, -4*w^5 + 30*w^3 + 19*w^2 - 19*w - 8], [41, 41, w^5 - 7*w^3 - 6*w^2 + 2*w + 3], [49, 7, -5*w^5 + w^4 + 36*w^3 + 19*w^2 - 22*w - 6], [64, 2, -2], [71, 71, -8*w^5 + w^4 + 58*w^3 + 34*w^2 - 34*w - 16], [71, 71, -6*w^5 + 2*w^4 + 42*w^3 + 18*w^2 - 23*w - 6], [71, 71, -8*w^5 + 2*w^4 + 58*w^3 + 27*w^2 - 38*w - 10], [71, 71, 4*w^5 - 30*w^3 - 19*w^2 + 20*w + 8], [71, 71, -10*w^5 + 3*w^4 + 72*w^3 + 30*w^2 - 48*w - 10], [71, 71, -8*w^5 + 2*w^4 + 58*w^3 + 26*w^2 - 37*w - 8], [125, 5, 4*w^5 - w^4 - 29*w^3 - 13*w^2 + 19*w + 2], [139, 139, -w^5 - w^4 + 8*w^3 + 12*w^2 - 3*w - 5], [139, 139, -11*w^5 + 4*w^4 + 78*w^3 + 29*w^2 - 49*w - 11], [139, 139, -12*w^5 + 3*w^4 + 86*w^3 + 41*w^2 - 52*w - 18], [139, 139, -4*w^5 + w^4 + 28*w^3 + 15*w^2 - 14*w - 7], [139, 139, 8*w^5 - 2*w^4 - 58*w^3 - 27*w^2 + 39*w + 10], [139, 139, 7*w^5 - 2*w^4 - 51*w^3 - 21*w^2 + 35*w + 7], [169, 13, -8*w^5 + 2*w^4 + 57*w^3 + 28*w^2 - 34*w - 13], [169, 13, w^5 - w^4 - 6*w^3 + w^2 + 2*w - 2], [169, 13, -7*w^5 + w^4 + 51*w^3 + 29*w^2 - 32*w - 12], [181, 181, -5*w^5 + 36*w^3 + 26*w^2 - 18*w - 12], [181, 181, -6*w^5 + 44*w^3 + 30*w^2 - 23*w - 13], [181, 181, 9*w^5 - 4*w^4 - 64*w^3 - 19*w^2 + 44*w + 7], [181, 181, 2*w^4 - w^3 - 14*w^2 - 2*w + 8], [181, 181, -12*w^5 + 4*w^4 + 85*w^3 + 35*w^2 - 53*w - 16], [181, 181, -11*w^5 + 3*w^4 + 78*w^3 + 37*w^2 - 45*w - 18], [211, 211, -10*w^5 + 3*w^4 + 71*w^3 + 32*w^2 - 43*w - 15], [211, 211, -7*w^5 + 3*w^4 + 49*w^3 + 16*w^2 - 31*w - 5], [211, 211, 8*w^5 - w^4 - 58*w^3 - 34*w^2 + 35*w + 15], [211, 211, -w^4 + w^3 + 7*w^2 - 2*w - 4], [211, 211, 6*w^5 - 2*w^4 - 42*w^3 - 17*w^2 + 23*w + 6], [211, 211, w^3 - w^2 - 5*w], [239, 239, w^3 - w^2 - 5*w + 1], [239, 239, w^4 - w^3 - 7*w^2 + 2*w + 5], [239, 239, 8*w^5 - w^4 - 58*w^3 - 34*w^2 + 35*w + 14], [239, 239, 7*w^5 - 3*w^4 - 49*w^3 - 16*w^2 + 31*w + 6], [239, 239, 10*w^5 - 3*w^4 - 71*w^3 - 32*w^2 + 43*w + 14], [239, 239, -11*w^5 + 2*w^4 + 80*w^3 + 42*w^2 - 50*w - 18], [251, 251, 6*w^5 - w^4 - 44*w^3 - 23*w^2 + 31*w + 8], [251, 251, 11*w^5 - 2*w^4 - 79*w^3 - 43*w^2 + 47*w + 18], [251, 251, 10*w^5 - w^4 - 73*w^3 - 44*w^2 + 45*w + 18], [251, 251, -11*w^5 + 4*w^4 + 78*w^3 + 29*w^2 - 48*w - 13], [251, 251, 6*w^5 - 2*w^4 - 42*w^3 - 17*w^2 + 23*w + 5], [251, 251, -16*w^5 + 4*w^4 + 115*w^3 + 54*w^2 - 70*w - 22], [281, 281, 8*w^5 - 3*w^4 - 57*w^3 - 20*w^2 + 36*w + 7], [281, 281, 3*w^5 - w^4 - 22*w^3 - 8*w^2 + 18*w + 3], [281, 281, 2*w^5 - 2*w^4 - 13*w^3 + 2*w^2 + 9*w], [281, 281, w^5 + w^4 - 8*w^3 - 11*w^2 + 3*w + 4], [281, 281, w^5 - w^4 - 7*w^3 + 2*w^2 + 8*w - 2], [281, 281, -13*w^5 + 4*w^4 + 93*w^3 + 39*w^2 - 58*w - 15], [349, 349, 7*w^5 - 2*w^4 - 51*w^3 - 21*w^2 + 36*w + 5], [349, 349, 7*w^5 - w^4 - 51*w^3 - 28*w^2 + 32*w + 8], [349, 349, 3*w^5 - w^4 - 21*w^3 - 8*w^2 + 12*w], [349, 349, 10*w^5 - 2*w^4 - 72*w^3 - 38*w^2 + 45*w + 17], [349, 349, 7*w^5 - 2*w^4 - 50*w^3 - 23*w^2 + 33*w + 12], [349, 349, 17*w^5 - 5*w^4 - 122*w^3 - 53*w^2 + 79*w + 21], [379, 379, 2*w^5 - 14*w^3 - 11*w^2 + 7*w + 4], [379, 379, -10*w^5 + 3*w^4 + 71*w^3 + 32*w^2 - 42*w - 15], [379, 379, -15*w^5 + 5*w^4 + 106*w^3 + 44*w^2 - 63*w - 17], [379, 379, 2*w^5 - 3*w^4 - 12*w^3 + 9*w^2 + 8*w - 4], [379, 379, 4*w^5 - 29*w^3 - 20*w^2 + 14*w + 10], [379, 379, 13*w^5 - 2*w^4 - 94*w^3 - 53*w^2 + 55*w + 24], [419, 419, 3*w^5 - 2*w^4 - 20*w^3 - 4*w^2 + 12*w + 4], [419, 419, 3*w^5 - 22*w^3 - 16*w^2 + 15*w + 8], [419, 419, -11*w^5 + 4*w^4 + 77*w^3 + 31*w^2 - 45*w - 14], [419, 419, -19*w^5 + 5*w^4 + 137*w^3 + 62*w^2 - 87*w - 24], [419, 419, -16*w^5 + 5*w^4 + 114*w^3 + 48*w^2 - 72*w - 17], [419, 419, 17*w^5 - 5*w^4 - 121*w^3 - 54*w^2 + 73*w + 23], [421, 421, 4*w^5 - w^4 - 29*w^3 - 13*w^2 + 20*w + 1], [421, 421, 4*w^5 - 2*w^4 - 29*w^3 - 6*w^2 + 22*w + 1], [421, 421, -8*w^5 + 3*w^4 + 57*w^3 + 20*w^2 - 37*w - 10], [421, 421, -6*w^5 + 45*w^3 + 30*w^2 - 31*w - 14], [421, 421, 5*w^5 + w^4 - 38*w^3 - 31*w^2 + 22*w + 14], [421, 421, -6*w^5 + 3*w^4 + 41*w^3 + 12*w^2 - 24*w - 5], [449, 449, -9*w^5 + w^4 + 66*w^3 + 38*w^2 - 43*w - 16], [449, 449, 2*w^5 + w^4 - 16*w^3 - 17*w^2 + 11*w + 9], [449, 449, 10*w^5 - w^4 - 74*w^3 - 43*w^2 + 49*w + 18], [449, 449, 17*w^5 - 6*w^4 - 121*w^3 - 46*w^2 + 78*w + 16], [449, 449, -11*w^5 + 2*w^4 + 79*w^3 + 44*w^2 - 47*w - 21], [449, 449, 6*w^5 - w^4 - 44*w^3 - 22*w^2 + 28*w + 5], [461, 461, -6*w^5 + 3*w^4 + 42*w^3 + 10*w^2 - 28*w - 4], [461, 461, -15*w^5 + 5*w^4 + 107*w^3 + 43*w^2 - 70*w - 16], [461, 461, -10*w^5 + 2*w^4 + 73*w^3 + 36*w^2 - 46*w - 13], [461, 461, -9*w^5 + 2*w^4 + 66*w^3 + 31*w^2 - 46*w - 10], [461, 461, 12*w^5 - 4*w^4 - 85*w^3 - 34*w^2 + 50*w + 13], [461, 461, -18*w^5 + 4*w^4 + 130*w^3 + 64*w^2 - 81*w - 28], [491, 491, 17*w^5 - 5*w^4 - 121*w^3 - 54*w^2 + 73*w + 21], [491, 491, -10*w^5 + 4*w^4 + 70*w^3 + 26*w^2 - 43*w - 11], [491, 491, -6*w^5 + 3*w^4 + 41*w^3 + 12*w^2 - 23*w - 3], [491, 491, 2*w^5 - 16*w^3 - 9*w^2 + 16*w + 3], [491, 491, 20*w^5 - 5*w^4 - 144*w^3 - 68*w^2 + 90*w + 29], [491, 491, -6*w^5 + w^4 + 44*w^3 + 24*w^2 - 31*w - 13], [601, 601, -10*w^5 + 2*w^4 + 73*w^3 + 36*w^2 - 47*w - 12], [601, 601, -7*w^5 + 2*w^4 + 49*w^3 + 24*w^2 - 26*w - 10], [601, 601, -15*w^5 + 3*w^4 + 108*w^3 + 56*w^2 - 64*w - 24], [601, 601, -3*w^5 + 3*w^4 + 20*w^3 - 5*w^2 - 15*w + 3], [601, 601, 9*w^5 - 2*w^4 - 66*w^3 - 31*w^2 + 45*w + 11], [601, 601, 14*w^5 - 2*w^4 - 102*w^3 - 57*w^2 + 63*w + 25], [631, 631, 5*w^5 - 2*w^4 - 35*w^3 - 13*w^2 + 20*w + 9], [631, 631, -10*w^5 + 4*w^4 + 71*w^3 + 24*w^2 - 47*w - 6], [631, 631, -5*w^5 + w^4 + 36*w^3 + 18*w^2 - 20*w - 9], [631, 631, -21*w^5 + 5*w^4 + 151*w^3 + 73*w^2 - 93*w - 31], [631, 631, 8*w^5 - 2*w^4 - 57*w^3 - 27*w^2 + 34*w + 8], [631, 631, -14*w^5 + 3*w^4 + 102*w^3 + 50*w^2 - 68*w - 19], [659, 659, 14*w^5 - 3*w^4 - 102*w^3 - 49*w^2 + 67*w + 19], [659, 659, 7*w^5 - 2*w^4 - 50*w^3 - 22*w^2 + 30*w + 5], [659, 659, -6*w^5 + 2*w^4 + 43*w^3 + 17*w^2 - 29*w - 10], [659, 659, -18*w^5 + 6*w^4 + 128*w^3 + 52*w^2 - 80*w - 21], [659, 659, 14*w^5 - 2*w^4 - 102*w^3 - 57*w^2 + 64*w + 24], [659, 659, -16*w^5 + 5*w^4 + 114*w^3 + 48*w^2 - 72*w - 18], [701, 701, -7*w^5 + 4*w^4 + 49*w^3 + 9*w^2 - 34*w - 4], [701, 701, 15*w^5 - 3*w^4 - 108*w^3 - 57*w^2 + 65*w + 26], [701, 701, -12*w^5 + 2*w^4 + 87*w^3 + 48*w^2 - 55*w - 22], [701, 701, 7*w^5 - 3*w^4 - 49*w^3 - 16*w^2 + 28*w + 5], [701, 701, -8*w^5 + 2*w^4 + 57*w^3 + 27*w^2 - 32*w - 12], [701, 701, -9*w^5 + 3*w^4 + 64*w^3 + 26*w^2 - 38*w - 12], [729, 3, -3], [769, 769, -12*w^5 + w^4 + 88*w^3 + 53*w^2 - 54*w - 21], [769, 769, 14*w^5 - 2*w^4 - 102*w^3 - 57*w^2 + 62*w + 25], [769, 769, -3*w^5 + 23*w^3 + 13*w^2 - 17*w - 1], [769, 769, 9*w^5 - 2*w^4 - 65*w^3 - 32*w^2 + 43*w + 12], [769, 769, -13*w^5 + 4*w^4 + 94*w^3 + 38*w^2 - 63*w - 14], [769, 769, 16*w^5 - 4*w^4 - 116*w^3 - 53*w^2 + 75*w + 21], [811, 811, 20*w^5 - 7*w^4 - 142*w^3 - 55*w^2 + 89*w + 21], [811, 811, 11*w^5 - 2*w^4 - 80*w^3 - 42*w^2 + 53*w + 17], [811, 811, -w^5 + 7*w^3 + 4*w^2 - w + 1], [811, 811, 9*w^5 - 2*w^4 - 64*w^3 - 34*w^2 + 35*w + 16], [811, 811, 5*w^5 - 2*w^4 - 35*w^3 - 13*w^2 + 20*w + 3], [811, 811, -7*w^5 + 52*w^3 + 34*w^2 - 31*w - 15], [839, 839, -14*w^5 + 5*w^4 + 99*w^3 + 38*w^2 - 60*w - 16], [839, 839, -20*w^5 + 6*w^4 + 143*w^3 + 62*w^2 - 90*w - 25], [839, 839, 7*w^5 - w^4 - 50*w^3 - 29*w^2 + 26*w + 12], [839, 839, -19*w^5 + 5*w^4 + 137*w^3 + 62*w^2 - 88*w - 22], [839, 839, -14*w^5 + 3*w^4 + 101*w^3 + 50*w^2 - 62*w - 21], [839, 839, -w^5 - w^4 + 9*w^3 + 11*w^2 - 10*w - 6], [881, 881, -14*w^5 + 3*w^4 + 102*w^3 + 49*w^2 - 67*w - 18], [881, 881, 12*w^5 - 5*w^4 - 84*w^3 - 29*w^2 + 52*w + 13], [881, 881, -3*w^5 + 2*w^4 + 21*w^3 + 2*w^2 - 15*w - 3], [881, 881, -3*w^5 + 21*w^3 + 16*w^2 - 8*w - 8], [881, 881, -17*w^5 + 4*w^4 + 123*w^3 + 58*w^2 - 79*w - 22], [881, 881, 19*w^5 - 5*w^4 - 136*w^3 - 64*w^2 + 83*w + 26], [911, 911, -6*w^5 + 2*w^4 + 44*w^3 + 15*w^2 - 32*w - 2], [911, 911, -15*w^5 + 4*w^4 + 107*w^3 + 51*w^2 - 66*w - 23], [911, 911, 13*w^5 - 4*w^4 - 94*w^3 - 37*w^2 + 63*w + 13], [911, 911, -15*w^5 + 3*w^4 + 109*w^3 + 55*w^2 - 71*w - 20], [911, 911, -7*w^5 + 3*w^4 + 48*w^3 + 17*w^2 - 25*w - 6], [911, 911, 12*w^5 - 4*w^4 - 86*w^3 - 34*w^2 + 55*w + 13]]; primes := [ideal : I in primesArray]; heckePol := x^2 - 4*x - 8; K := NumberField(heckePol); heckeEigenvaluesArray := [-1, -1/2*e + 1, e, -2*e + 2, -e + 9, e, 2*e - 4, 1/2*e - 3, 2*e - 4, -e - 2, 2*e - 4, -3*e + 7, -2*e - 1, 5/2*e - 3, e + 5, -2, -3/2*e - 1, -e + 14, -e + 14, -3/2*e - 1, 2*e - 8, 3*e - 10, 2*e + 6, 1/2*e + 7, 4*e - 3, -3*e + 17, 3*e - 10, 7/2*e - 11, -5*e + 10, -e - 8, 2*e - 12, -4*e + 15, -13/2*e + 9, 2*e - 12, e + 4, 2*e - 12, e + 20, -e + 6, -3*e - 8, 15/2*e - 15, -1/2*e + 21, -3/2*e - 9, -e + 11, -13/2*e + 7, -3*e - 3, 2*e - 14, -3*e - 3, e + 2, 2*e - 2, 3/2*e - 17, 9/2*e - 19, -7/2*e + 17, 3*e - 18, 6*e - 20, 1/2*e + 1, -3*e - 12, -2*e - 5, -2*e + 18, 5*e - 2, -3*e - 12, 2*e + 8, -2*e - 20, -e + 10, 3*e - 8, e + 24, 4*e - 24, -3*e + 12, e + 17, e - 6, -3*e - 11, 7*e - 10, 10, 3*e - 26, 2*e + 13, -5/2*e - 7, -5*e + 10, -3*e + 24, 2*e - 10, 9*e - 14, 9/2*e - 11, -8, e + 22, -8, -8, -e + 8, -3*e - 6, 3*e - 10, e - 1, -3*e - 6, 6*e - 12, 4*e - 23, -e + 11, -4*e - 10, 5*e - 16, 7*e - 2, 4*e, -e + 36, 20, 3*e - 28, -6*e + 24, -e + 13, -26, -9*e + 16, -5/2*e + 27, e - 6, -13, e - 6, -15/2*e + 15, -4*e - 11, -19/2*e + 31, -7*e + 14, -10*e + 16, 7*e - 26, 10*e - 28, -7*e + 14, -2*e + 26, e - 22, 2*e + 8, -3*e - 4, -4*e + 35, -9/2*e + 25, -2*e + 8, 10*e - 23, -5*e + 10, -5*e + 10, -9*e + 28, 8*e, e - 9, -8*e + 20, 7*e - 36, -2*e + 16, -3/2*e - 15, 5*e - 27, -13*e + 22, -7*e + 18, 13*e - 26, -7*e + 18, -3*e + 23, 3*e - 4, -10*e + 20, e - 18, -3*e, 4*e + 26, 5*e - 36, -3*e, -3*e - 18, -9*e + 32, 8*e - 10, e + 10, -4*e - 2, 6*e - 24, -4*e - 4, 24, 6*e - 3, -5*e + 35, -9/2*e + 27, 7*e - 42]; 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;