/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([-1, -2, 7, 2, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([71,71,3*w^5 - w^4 - 21*w^3 - 9*w^2 + 13*w + 2]) primes_array = [ [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 = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 28 K. = NumberField(heckePol) hecke_eigenvalues_array = [-e, -1/2*e + 7, 0, 0, e, 0, -e - 2, -1/2*e + 5, e - 2, -2, -2, e - 2, 1/2*e - 1, -2*e + 5, -2, e - 2, 12, 2*e - 2, -e - 2, -1, 1/2*e - 1, 3*e + 6, 1/2*e - 1, 1/2*e + 13, -8, 2*e - 8, -e + 6, 3/2*e + 1, -2*e - 6, 3/2*e + 1, -2*e - 10, -e + 4, -e + 18, -e - 10, -2*e - 10, 1/2*e + 11, -3*e - 10, 4, -5/2*e + 11, 7/2*e - 3, 4*e + 4, -1/2*e + 11, -3*e + 4, e - 10, -e + 18, 3/2*e - 3, -5*e + 4, -4*e + 4, e - 12, e + 16, 1/2*e + 23, -2*e - 12, -e + 2, 16, e + 14, 0, -4*e, 5/2*e - 7, -3*e + 14, 3/2*e + 21, -5*e - 4, 1/2*e + 17, 2*e - 18, -3/2*e + 17, 2*e - 4, -2*e - 18, 3*e - 4, 3*e - 18, -3*e - 18, -e + 24, 4*e + 10, 2*e - 4, 3*e + 18, -e - 10, -10, 6*e + 4, 4*e - 10, -7/2*e - 3, -20, e - 20, -e + 8, -2*e - 6, 1/2*e + 15, 2*e - 6, 4*e + 4, -2*e + 18, 18, 11/2*e + 11, 7/2*e + 11, -e + 4, -5/2*e - 17, -5*e + 4, 7/2*e - 3, -2*e + 18, -e + 4, -4*e - 10, e + 16, e + 16, 1/2*e + 23, 1/2*e - 5, 3*e - 26, -3*e + 16, 6*e + 10, 2*e + 10, -5/2*e + 3, -2*e + 10, -3*e - 4, -2*e - 4, -5*e - 2, 4*e - 16, 2*e - 16, -9/2*e - 9, -2*e - 16, 11/2*e + 5, -3*e - 6, 36, -6*e - 6, 2*e - 20, 7/2*e - 13, -6, 4*e - 22, 3*e + 6, 6, 4*e + 6, -4*e - 22, -3*e + 6, 5*e - 24, 5/2*e + 23, 2*e - 12, 5*e - 26, 5/2*e + 37, -e - 26, e - 26, -2, 2*e - 30, -8*e + 12, 1/2*e + 5, 4*e + 12, 4*e + 26, -2*e - 34, -9/2*e - 27, -19/2*e + 1, -17/2*e + 1, 9/2*e + 29, 3*e - 6, 9*e - 8, -1/2*e + 13, -e + 20, -2*e - 22, 6*e - 22, -4*e + 34, 7*e + 10, 5*e - 4, -13/2*e + 3, -2*e - 32, -11/2*e + 17, -7*e - 4] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([71,71,3*w^5 - w^4 - 21*w^3 - 9*w^2 + 13*w + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]