/* 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([2, 1, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 4, w^2 - 2*w - 3]) primes_array = [ [2, 2, w],\ [2, 2, -1/2*w^3 + w^2 + 3/2*w],\ [11, 11, 1/2*w^3 - w^2 - 5/2*w + 4],\ [11, 11, -w^3 + w^2 + 6*w + 1],\ [13, 13, -w^3 + w^2 + 6*w - 3],\ [19, 19, -1/2*w^3 + w^2 + 5/2*w],\ [23, 23, -1/2*w^3 + 2*w^2 - 1/2*w - 2],\ [31, 31, -w^3 + w^2 + 6*w - 1],\ [31, 31, 1/2*w^3 - 7/2*w],\ [43, 43, 1/2*w^3 - w^2 - 9/2*w + 2],\ [67, 67, w^2 - w - 5],\ [67, 67, -1/2*w^3 + 11/2*w + 2],\ [73, 73, w^2 + w + 1],\ [79, 79, 1/2*w^3 + w^2 - 5/2*w - 2],\ [81, 3, -3],\ [83, 83, 2*w^3 - 2*w^2 - 12*w + 5],\ [83, 83, -1/2*w^3 - w^2 + 1/2*w + 2],\ [89, 89, -3/2*w^3 + 2*w^2 + 17/2*w - 2],\ [89, 89, 3/2*w^3 - 2*w^2 - 21/2*w + 2],\ [97, 97, 1/2*w^3 - w^2 - 1/2*w - 2],\ [97, 97, -w^3 + w^2 + 8*w + 1],\ [103, 103, 2*w^3 - 2*w^2 - 14*w + 1],\ [103, 103, w^2 - w - 3],\ [107, 107, -3/2*w^3 + 2*w^2 + 21/2*w - 4],\ [109, 109, 5/2*w^3 - 3*w^2 - 33/2*w + 4],\ [109, 109, -1/2*w^3 + w^2 + 9/2*w - 4],\ [113, 113, -w^3 + w^2 + 8*w - 3],\ [113, 113, 1/2*w^3 - w^2 - 1/2*w + 2],\ [121, 11, 5/2*w^3 - 2*w^2 - 31/2*w + 2],\ [127, 127, -3/2*w^3 + w^2 + 23/2*w],\ [127, 127, -1/2*w^3 + 11/2*w],\ [131, 131, w^2 - 3*w - 3],\ [137, 137, 1/2*w^3 - w^2 - 5/2*w + 6],\ [139, 139, -3/2*w^3 + 2*w^2 + 13/2*w + 2],\ [151, 151, -1/2*w^3 + 2*w^2 + 3/2*w - 10],\ [157, 157, 7/2*w^3 - 5*w^2 - 47/2*w + 10],\ [157, 157, 7/2*w^3 - 4*w^2 - 45/2*w + 4],\ [167, 167, 1/2*w^3 + w^2 - 5/2*w - 4],\ [167, 167, 2*w + 5],\ [173, 173, -w^3 + w^2 + 4*w - 7],\ [181, 181, w^3 - 2*w^2 - 9*w + 1],\ [193, 193, -1/2*w^3 + 2*w^2 + 3/2*w - 8],\ [197, 197, -1/2*w^3 - w^2 + 13/2*w + 2],\ [197, 197, -w^3 + w^2 + 4*w - 1],\ [197, 197, w^3 - 2*w^2 - 7*w + 7],\ [197, 197, -2*w^3 + 3*w^2 + 13*w - 11],\ [199, 199, -2*w^3 + 3*w^2 + 11*w - 5],\ [239, 239, -w^3 + 2*w^2 + 5*w - 1],\ [239, 239, 7/2*w^3 - 4*w^2 - 45/2*w + 6],\ [241, 241, 1/2*w^3 - 2*w^2 - 7/2*w + 8],\ [257, 257, 3/2*w^3 - 2*w^2 - 17/2*w - 4],\ [257, 257, 1/2*w^3 - 7/2*w - 6],\ [257, 257, 7/2*w^3 - 5*w^2 - 43/2*w + 10],\ [257, 257, 3*w^3 - 4*w^2 - 19*w + 7],\ [269, 269, 1/2*w^3 + w^2 + 3/2*w + 2],\ [269, 269, -2*w^3 + 3*w^2 + 11*w - 11],\ [277, 277, 5/2*w^3 - 4*w^2 - 35/2*w + 8],\ [281, 281, -3/2*w^3 + w^2 + 23/2*w - 2],\ [283, 283, -w^3 - w^2 + 8*w + 11],\ [283, 283, -3*w^2 + 3*w + 19],\ [311, 311, 3/2*w^3 - 2*w^2 - 17/2*w - 2],\ [331, 331, -4*w^3 + 7*w^2 + 25*w - 21],\ [337, 337, -5/2*w^3 + 2*w^2 + 35/2*w + 4],\ [337, 337, -2*w^3 + 3*w^2 + 11*w - 9],\ [347, 347, 2*w^3 - 4*w^2 - 12*w + 13],\ [347, 347, 5/2*w^3 - 5*w^2 - 29/2*w + 20],\ [347, 347, -1/2*w^3 - 1/2*w - 6],\ [347, 347, -1/2*w^3 + 3*w^2 + 1/2*w - 16],\ [349, 349, 1/2*w^3 + w^2 - 9/2*w - 4],\ [353, 353, -1/2*w^3 + 2*w^2 + 7/2*w],\ [359, 359, 3/2*w^3 - w^2 - 19/2*w + 2],\ [373, 373, -2*w^3 + 2*w^2 + 12*w + 1],\ [373, 373, -1/2*w^3 + w^2 + 9/2*w - 6],\ [379, 379, w^2 - w - 9],\ [379, 379, -2*w^3 + 3*w^2 + 13*w - 5],\ [383, 383, w^2 + w - 5],\ [389, 389, 9/2*w^3 - 6*w^2 - 59/2*w + 14],\ [397, 397, 2*w^2 - 2*w - 15],\ [401, 401, -1/2*w^3 + 2*w^2 + 3/2*w - 4],\ [419, 419, -3/2*w^3 + w^2 + 15/2*w - 2],\ [419, 419, w^3 + w^2 - 8*w - 15],\ [431, 431, -5/2*w^3 + 4*w^2 + 27/2*w - 16],\ [431, 431, 2*w^3 - 4*w^2 - 10*w + 11],\ [443, 443, 1/2*w^3 - 3*w^2 - 5/2*w + 14],\ [449, 449, 1/2*w^3 + w^2 - 9/2*w - 6],\ [457, 457, -w^3 + 2*w^2 + 7*w - 9],\ [461, 461, 4*w^3 - 5*w^2 - 27*w + 7],\ [487, 487, 2*w^2 - 9],\ [499, 499, 3/2*w^3 - w^2 - 15/2*w - 2],\ [499, 499, 3/2*w^3 - 4*w^2 - 17/2*w + 16],\ [503, 503, -5/2*w^3 + 3*w^2 + 37/2*w],\ [509, 509, -3*w^3 + 3*w^2 + 18*w + 7],\ [521, 521, -1/2*w^3 - 2*w^2 + 11/2*w + 18],\ [521, 521, -5/2*w^3 + 4*w^2 + 31/2*w - 8],\ [523, 523, -1/2*w^3 + 2*w^2 + 11/2*w],\ [523, 523, -1/2*w^3 - w^2 + 9/2*w],\ [547, 547, -7/2*w^3 + 6*w^2 + 45/2*w - 18],\ [547, 547, -w^3 - w^2 + 12*w + 5],\ [547, 547, 3/2*w^3 - w^2 - 19/2*w + 4],\ [547, 547, w^3 - w^2 - 6*w - 5],\ [557, 557, 1/2*w^3 + w^2 - 9/2*w - 12],\ [557, 557, 3/2*w^3 - 2*w^2 - 21/2*w],\ [557, 557, 2*w^3 - 2*w^2 - 12*w + 1],\ [557, 557, w^3 - 3*w^2 - 6*w + 11],\ [569, 569, w^3 - 2*w^2 - 7*w + 1],\ [577, 577, w^3 + w^2 - 6*w - 5],\ [599, 599, -3/2*w^3 + 3*w^2 + 15/2*w - 10],\ [607, 607, -5/2*w^3 + 3*w^2 + 25/2*w + 4],\ [613, 613, w^3 - 9*w + 1],\ [619, 619, 3*w^2 - w - 17],\ [619, 619, w^3 - 3*w^2 - 6*w + 13],\ [625, 5, -5],\ [631, 631, 11/2*w^3 - 7*w^2 - 71/2*w + 16],\ [631, 631, -3/2*w^3 + 2*w^2 + 13/2*w],\ [643, 643, w^2 - 3*w - 5],\ [653, 653, 2*w^2 - 2*w - 11],\ [659, 659, 5*w^3 - 3*w^2 - 36*w - 9],\ [659, 659, -w^3 + 7*w - 1],\ [673, 673, 5/2*w^3 - 2*w^2 - 31/2*w - 4],\ [677, 677, 13/2*w^3 - 6*w^2 - 87/2*w + 2],\ [691, 691, -3/2*w^3 + 2*w^2 + 13/2*w - 2],\ [701, 701, -w^3 + 3*w^2 + 4*w - 13],\ [701, 701, w^3 - 2*w^2 - 9*w + 3],\ [709, 709, -w^2 - 5*w - 1],\ [739, 739, 3/2*w^3 - w^2 - 15/2*w],\ [743, 743, -2*w^3 + w^2 + 15*w + 1],\ [751, 751, -2*w^3 + 4*w^2 + 12*w - 11],\ [757, 757, -6*w^3 + 9*w^2 + 39*w - 23],\ [757, 757, -w^3 - w^2 + 8*w + 3],\ [757, 757, 13/2*w^3 - 8*w^2 - 83/2*w + 16],\ [757, 757, 3*w^3 - 4*w^2 - 17*w + 3],\ [761, 761, 7/2*w^3 - 5*w^2 - 47/2*w + 12],\ [769, 769, -3*w^2 + 7*w + 3],\ [773, 773, -3/2*w^3 - w^2 + 19/2*w + 4],\ [787, 787, -4*w^3 + 5*w^2 + 23*w - 17],\ [787, 787, 1/2*w^3 - w^2 - 9/2*w - 4],\ [809, 809, -2*w^3 + 2*w^2 + 16*w + 3],\ [809, 809, -4*w^3 + 5*w^2 + 25*w - 9],\ [811, 811, 3*w^3 - 3*w^2 - 20*w + 5],\ [811, 811, w^3 - 3*w^2 - 6*w + 17],\ [823, 823, 5/2*w^3 - w^2 - 33/2*w - 10],\ [829, 829, 5/2*w^3 - 3*w^2 - 33/2*w + 2],\ [829, 829, -3*w^3 + 2*w^2 + 23*w - 1],\ [839, 839, -11/2*w^3 + 7*w^2 + 67/2*w - 18],\ [853, 853, -w^3 + 2*w^2 + 3*w - 5],\ [853, 853, -5*w^3 + 7*w^2 + 32*w - 15],\ [857, 857, 3/2*w^3 - 5*w^2 - 15/2*w + 26],\ [859, 859, -w^3 + 3*w^2 + 4*w - 17],\ [863, 863, 4*w^3 - 4*w^2 - 26*w + 3],\ [877, 877, w^3 - 5*w - 5],\ [887, 887, 3/2*w^3 - w^2 - 11/2*w + 8],\ [907, 907, -9/2*w^3 + 7*w^2 + 53/2*w - 24],\ [919, 919, 7/2*w^3 - 6*w^2 - 41/2*w + 22],\ [929, 929, -2*w^3 + 3*w^2 + 11*w - 1],\ [937, 937, 2*w^3 - 4*w^2 - 12*w + 7],\ [947, 947, 9/2*w^3 - 5*w^2 - 61/2*w + 10],\ [953, 953, -1/2*w^3 + 2*w^2 + 3/2*w + 2],\ [961, 31, -1/2*w^3 + 2*w^2 - 1/2*w - 6],\ [977, 977, 5/2*w^3 - 2*w^2 - 31/2*w],\ [983, 983, -5/2*w^3 + 6*w^2 + 27/2*w - 24],\ [983, 983, -3/2*w^3 + 2*w^2 + 21/2*w - 10],\ [983, 983, -w^3 + 4*w^2 + 7*w - 15],\ [983, 983, -1/2*w^3 + 3*w^2 - 3/2*w - 4],\ [997, 997, 1/2*w^3 - w^2 - 5/2*w - 4],\ [997, 997, 1/2*w^3 + w^2 - 17/2*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 10*x^4 + 24*x^2 - 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 0, e^4 - 7*e^2 + 6, -2*e^3 + 10*e, -e^5 + 8*e^3 - 13*e, -e^5 + 10*e^3 - 25*e, e^4 - 5*e^2, 2*e^3 - 10*e, 4*e, -e^5 + 10*e^3 - 21*e, -e^4 + 5*e^2 + 4, -4*e^3 + 20*e, 2, 2*e^4 - 12*e^2 + 2, -e^4 + 7*e^2 - 8, e^5 - 10*e^3 + 29*e, e^4 - 11*e^2 + 18, -2*e^5 + 16*e^3 - 24*e, 2*e^5 - 18*e^3 + 34*e, 2*e^4 - 12*e^2 + 4, 2*e^3 - 16*e, 2*e^3 - 6*e, 2*e^2 - 2, -e^5 + 8*e^3 - 7*e, 2*e^5 - 22*e^3 + 54*e, 4*e^2 - 14, 2*e^5 - 18*e^3 + 42*e, 4*e^2 - 18, 2*e^5 - 20*e^3 + 44*e, e^5 - 8*e^3 + 15*e, -2*e^5 + 18*e^3 - 32*e, 3*e^4 - 17*e^2 + 6, -e^4 + 11*e^2 - 12, -3*e^5 + 28*e^3 - 57*e, -e^4 + 5*e^2 + 16, -2*e^5 + 14*e^3 - 18*e, -4*e^3 + 22*e, 4*e^2 - 12, 2*e^4 - 12*e^2 + 18, 2*e^2 + 12, 3*e^5 - 30*e^3 + 65*e, -6*e^2 + 16, 3*e^5 - 28*e^3 + 59*e, 6*e, -e^4 + e^2 + 18, e^4 - 5*e^2 - 18, -4*e^5 + 38*e^3 - 78*e, 2*e^3 - 2*e, -2*e^5 + 14*e^3 - 16*e, 4*e^4 - 24*e^2 + 22, 2*e^4 - 14*e^2 + 6, -e^4 + 5*e^2 + 18, e^5 - 10*e^3 + 23*e, 4*e^5 - 40*e^3 + 90*e, -3*e^4 + 11*e^2 + 18, 2*e^4 - 14*e^2 + 18, -4*e^5 + 36*e^3 - 66*e, -3*e^5 + 28*e^3 - 63*e, -2*e^4 + 6*e^2 + 16, -e^4 + 3*e^2 - 2, e^5 - 10*e^3 + 25*e, 2*e^4 - 12*e^2 - 2, -4*e^5 + 36*e^3 - 70*e, 4*e^4 - 20*e^2 + 2, -4*e^4 + 22*e^2 - 6, 6*e^2 - 18, 12, -2*e^2 + 6, e^4 - 11*e^2 + 20, 2*e^5 - 24*e^3 + 64*e, -3*e^5 + 22*e^3 - 35*e, -2*e^3 + 12*e, -2*e^2 + 16, -3*e^4 + 19*e^2 - 8, 6*e^3 - 26*e, 4*e^4 - 30*e^2 + 42, 2*e^3, 4*e^4 - 30*e^2 + 32, -4*e^2 + 6, -4*e^5 + 32*e^3 - 56*e, 2*e^4 - 6*e^2 - 24, 4*e^4 - 22*e^2 - 6, -6*e^4 + 40*e^2 - 42, 4*e^4 - 26*e^2 + 18, -4*e^4 + 28*e^2 - 30, -5*e^4 + 33*e^2 - 38, 3*e^5 - 26*e^3 + 57*e, -3*e^4 + 17*e^2 - 2, -4*e^5 + 32*e^3 - 56*e, 2*e^4 - 6*e^2 - 8, -2*e^5 + 22*e^3 - 56*e, 4*e^5 - 32*e^3 + 54*e, -6*e^4 + 32*e^2, e^5 - 6*e^3 - e, 3*e^5 - 36*e^3 + 109*e, -2*e^5 + 22*e^3 - 56*e, -3*e^4 + 15*e^2 + 28, -7*e^5 + 60*e^3 - 105*e, -2*e^5 + 24*e^3 - 78*e, -6*e^4 + 34*e^2 - 8, 5*e^4 - 33*e^2 + 30, 4*e^5 - 38*e^3 + 68*e, -2*e^5 + 22*e^3 - 54*e, 5*e^4 - 33*e^2 + 30, -e^5 + 35*e, -3*e^4 + 25*e^2 - 16, 4*e^4 - 34*e^2 + 30, -4*e^5 + 46*e^3 - 130*e, -4*e^5 + 32*e^3 - 58*e, 8*e^4 - 48*e^2 + 28, 4*e^4 - 20*e^2 - 28, 3*e^4 - 9*e^2 - 20, 4*e^5 - 26*e^3 + 26*e, -e^5 + 14*e^3 - 45*e, -4*e^4 + 18*e^2 + 22, -18, 3*e^5 - 22*e^3 + 39*e, 6*e^3 - 30*e, 4*e^5 - 44*e^3 + 118*e, 2*e^3 - 20*e, 5*e^5 - 46*e^3 + 97*e, -18, -4*e^5 + 36*e^3 - 86*e, 4*e^5 - 42*e^3 + 120*e, 4*e^5 - 38*e^3 + 90*e, 7*e^5 - 56*e^3 + 89*e, 4*e^4 - 18*e^2 - 34, -2*e^5 + 14*e^3 - 2*e, -e^5 + 14*e^3 - 47*e, 4*e^5 - 36*e^3 + 58*e, 4*e^5 - 38*e^3 + 72*e, e^5 - 10*e^3 + 19*e, 4*e^5 - 30*e^3 + 32*e, 3*e^5 - 28*e^3 + 67*e, 8*e^4 - 48*e^2 + 20, -4*e^4 + 28*e^2 - 20, 8*e^5 - 70*e^3 + 124*e, 12*e^3 - 70*e, -5*e^5 + 44*e^3 - 83*e, 5*e^4 - 39*e^2 + 26, -8, -5*e^5 + 46*e^3 - 83*e, 7*e^5 - 68*e^3 + 139*e, -7*e^5 + 60*e^3 - 101*e, -4*e^4 + 12*e^2 + 46, 6*e^5 - 60*e^3 + 124*e, -5*e^4 + 21*e^2 + 6, -6*e^4 + 36*e^2 - 10, 6*e^5 - 54*e^3 + 100*e, -6*e^2 + 28, 2*e^4 - 20*e^2 + 42, 3*e^4 - 29*e^2 + 26, -e^4 + 5*e^2 - 8, -4*e^5 + 48*e^3 - 138*e, -3*e^5 + 20*e^3 - 7*e, 2*e^5 - 24*e^3 + 70*e, 2*e^4 - 8*e^2, -2*e^4 + 20*e^2 - 32, -6*e^5 + 50*e^3 - 78*e, 5*e^4 - 25*e^2, -7*e^4 + 39*e^2 + 12, 3*e^4 - 33*e^2 + 42, -4*e^4 + 12*e^2 + 24, 2*e^4 - 16*e^2 - 4, -2*e^5 + 26*e^3 - 90*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, -1/2*w^3 + w^2 + 3/2*w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]