/* 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([10, -8, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([14, 14, 2*w^2 + 3*w - 9]) primes_array = [ [2, 2, -w + 2],\ [2, 2, -w + 1],\ [5, 5, -w^2 - w + 5],\ [7, 7, w^2 + w - 9],\ [13, 13, -w^2 - w + 7],\ [19, 19, w^2 - w - 1],\ [25, 5, -w^2 + w + 3],\ [27, 3, 3],\ [31, 31, -w^2 + w + 11],\ [43, 43, -3*w^2 - w + 21],\ [47, 47, 2*w - 1],\ [49, 7, 4*w^2 + 2*w - 29],\ [61, 61, 2*w^2 + 2*w - 9],\ [71, 71, w^2 + w - 11],\ [71, 71, -w^2 - 3*w + 7],\ [71, 71, 2*w^2 + 2*w - 13],\ [79, 79, w^2 + 3*w - 1],\ [79, 79, w^2 + w - 1],\ [79, 79, -w^2 - 3*w - 1],\ [83, 83, w^2 - w - 7],\ [97, 97, 4*w^2 + 4*w - 23],\ [101, 101, -w^2 + w + 9],\ [101, 101, w^2 + 3*w - 3],\ [101, 101, 2*w - 7],\ [107, 107, 2*w^2 + 2*w - 17],\ [107, 107, 5*w^2 + 7*w - 21],\ [107, 107, w^2 - 3*w - 1],\ [109, 109, -2*w - 1],\ [109, 109, 4*w - 7],\ [109, 109, 4*w^2 + 2*w - 27],\ [113, 113, 2*w^2 - 11],\ [127, 127, -2*w^2 + 2*w + 7],\ [131, 131, -2*w - 3],\ [137, 137, 2*w + 7],\ [139, 139, -5*w^2 - 3*w + 33],\ [149, 149, -w^2 - 3*w + 13],\ [157, 157, w^2 - 5*w + 3],\ [163, 163, -4*w^2 + 33],\ [163, 163, 7*w^2 + 3*w - 53],\ [163, 163, -8*w^2 - 4*w + 57],\ [169, 13, w^2 + 3*w - 9],\ [173, 173, 3*w^2 + w - 19],\ [179, 179, 6*w^2 + 2*w - 43],\ [193, 193, -6*w + 7],\ [197, 197, -5*w^2 - w + 41],\ [197, 197, 4*w^2 + 4*w - 21],\ [197, 197, 3*w^2 + 3*w - 19],\ [199, 199, 2*w^2 - 9],\ [211, 211, w^2 + 3*w - 11],\ [223, 223, w^2 + 5*w - 7],\ [223, 223, 5*w^2 + 5*w - 27],\ [227, 227, -2*w^2 + 2*w + 3],\ [229, 229, 2*w^2 - 7],\ [233, 233, -4*w - 11],\ [239, 239, 4*w^2 + 6*w - 19],\ [251, 251, 5*w^2 + 5*w - 29],\ [269, 269, w^2 + w - 13],\ [277, 277, -w^2 - w - 1],\ [283, 283, 2*w^2 + 4*w - 3],\ [293, 293, -4*w^2 + 31],\ [313, 313, 3*w^2 - w - 27],\ [337, 337, 2*w^2 + 2*w - 21],\ [347, 347, -2*w^2 - 6*w + 13],\ [353, 353, -5*w^2 - 3*w + 37],\ [353, 353, 2*w^2 + 4*w - 13],\ [353, 353, -w^2 + w - 3],\ [359, 359, 2*w - 9],\ [361, 19, -7*w^2 - 3*w + 49],\ [367, 367, w^2 - 5*w + 9],\ [373, 373, 3*w^2 + w - 27],\ [379, 379, -w^2 - 5*w - 7],\ [397, 397, -w^2 - 5*w + 11],\ [401, 401, 6*w^2 + 4*w - 41],\ [401, 401, -9*w^2 - 3*w + 71],\ [401, 401, 6*w - 13],\ [409, 409, 2*w^2 - 2*w + 1],\ [419, 419, 4*w^2 + 4*w - 19],\ [421, 421, -w^2 - 3*w - 3],\ [433, 433, 2*w^2 + 6*w - 1],\ [433, 433, 9*w^2 + 5*w - 63],\ [433, 433, 3*w^2 + 3*w - 23],\ [443, 443, -3*w^2 - 7*w + 7],\ [449, 449, -3*w^2 - 3*w + 7],\ [457, 457, -2*w^2 - 6*w + 9],\ [463, 463, -3*w^2 - w + 9],\ [479, 479, -w^2 - 3*w + 17],\ [487, 487, -w^2 + 5*w - 1],\ [499, 499, -14*w^2 - 4*w + 109],\ [509, 509, 4*w - 1],\ [509, 509, -5*w^2 - w + 33],\ [509, 509, -3*w^2 - 5*w + 17],\ [521, 521, -15*w^2 - 5*w + 111],\ [523, 523, -2*w^2 + 6*w - 7],\ [523, 523, 2*w^2 - 21],\ [523, 523, 2*w + 9],\ [541, 541, 5*w^2 + 3*w - 31],\ [547, 547, -w^2 - 5*w + 17],\ [557, 557, -3*w^2 + w + 23],\ [563, 563, -4*w^2 - 8*w + 9],\ [563, 563, 2*w^2 - 2*w - 13],\ [563, 563, -2*w^2 + 6*w + 1],\ [569, 569, -4*w - 13],\ [593, 593, 8*w - 9],\ [599, 599, w^2 + 5*w - 13],\ [601, 601, w^2 + 7*w - 13],\ [613, 613, 3*w^2 + w - 11],\ [641, 641, 4*w^2 + 2*w - 23],\ [643, 643, -w^2 - w - 3],\ [647, 647, -3*w^2 - 7*w + 17],\ [653, 653, 3*w^2 - w - 13],\ [659, 659, w^2 + 7*w - 9],\ [661, 661, 2*w^2 + 4*w - 17],\ [677, 677, -6*w^2 + 49],\ [677, 677, -11*w^2 - 5*w + 81],\ [677, 677, -14*w^2 - 6*w + 103],\ [683, 683, 2*w^2 + 4*w - 19],\ [691, 691, w^2 - 3*w + 7],\ [691, 691, -8*w + 13],\ [691, 691, 3*w^2 - 5*w - 3],\ [709, 709, 4*w^2 + 4*w - 27],\ [719, 719, -w^2 - 7*w - 11],\ [727, 727, -13*w^2 - 3*w + 99],\ [739, 739, -5*w^2 - w + 43],\ [739, 739, 4*w + 9],\ [739, 739, -w^2 + 3*w + 19],\ [743, 743, 2*w^2 - 4*w - 7],\ [751, 751, 7*w^2 + 5*w - 47],\ [757, 757, -8*w^2 + 69],\ [761, 761, w^2 - 3*w - 7],\ [773, 773, -2*w^2 - 4*w + 1],\ [809, 809, -10*w^2 - 4*w + 71],\ [811, 811, w^2 + 5*w - 3],\ [811, 811, -8*w^2 - 10*w + 37],\ [811, 811, 6*w^2 + 6*w - 31],\ [823, 823, 2*w^2 - 2*w - 17],\ [823, 823, 3*w^2 - w - 11],\ [823, 823, -3*w^2 - 5*w + 19],\ [829, 829, 7*w^2 + 5*w - 43],\ [853, 853, -3*w^2 + 3*w + 7],\ [857, 857, 3*w^2 + 7*w - 3],\ [857, 857, 4*w^2 - 2*w - 19],\ [857, 857, 3*w^2 - w - 9],\ [859, 859, 2*w^2 + 2*w - 23],\ [859, 859, -10*w^2 - 6*w + 69],\ [859, 859, -3*w^2 - 5*w + 31],\ [877, 877, -10*w^2 - 2*w + 81],\ [883, 883, 7*w^2 + 7*w - 41],\ [883, 883, 3*w^2 + w - 29],\ [883, 883, 8*w^2 + 6*w - 49],\ [887, 887, 16*w^2 + 6*w - 117],\ [907, 907, 5*w^2 + 5*w - 23],\ [911, 911, w^2 - 3*w - 13],\ [919, 919, 5*w^2 + 5*w - 19],\ [919, 919, 3*w^2 + 3*w - 31],\ [919, 919, 4*w^2 + 2*w - 21],\ [937, 937, w^2 - 5*w - 1],\ [947, 947, -7*w^2 - 3*w + 47],\ [953, 953, 7*w^2 + 7*w - 37],\ [961, 31, -5*w^2 + w + 47],\ [967, 967, 4*w^2 + 4*w - 33],\ [967, 967, -5*w^2 - w + 31],\ [967, 967, -6*w^2 + 47],\ [977, 977, -7*w^2 - w + 59],\ [983, 983, -w^2 - 5*w - 1],\ [983, 983, 8*w - 17],\ [983, 983, 5*w^2 + 3*w - 29],\ [991, 991, 6*w^2 + 8*w - 31],\ [991, 991, -11*w^2 - 3*w + 79],\ [991, 991, w^2 + 7*w + 13],\ [997, 997, -4*w - 7],\ [997, 997, w^2 - 3*w - 11],\ [997, 997, -w^2 + 5*w - 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + x^2 - 5*x - 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1, -e^2 + 3, -1, e^2 - 5, -2, -e^2 - 2*e + 1, -2, 2*e - 4, 2*e + 4, -2*e^2 - 2*e + 6, 3*e^2 + 2*e - 11, e^2 - 5, 6*e, -4*e, -4*e, 2*e + 8, 2*e^2 - 4*e - 10, 4*e^2 + 2*e - 16, -2*e^2 - 4*e + 6, 3*e^2 - 2*e - 11, -2*e^2 - 2*e + 6, -2*e^2 - 2*e + 6, 4*e^2 + 2*e - 12, 2*e^2 - 2*e - 6, -2*e^2 - 4*e + 12, -2*e^2 + 12, -2*e - 4, e^2 - 17, -e^2 - 5, e^2 - 4*e - 9, -2*e^2 + 10, 2*e^2 - 4*e - 6, 4*e^2 + 6*e - 12, 2*e^2 + 2*e - 14, -e^2 - 9, 4*e^2 - 8*e - 22, -2*e^2 + 4*e + 20, 4*e^2 + 2*e - 20, -4*e^2 + 4*e + 10, e^2 + 4*e - 17, -e^2 - 9, 4*e^2 - 6, 2*e^2 + 6*e - 22, -6*e^2 - 2*e + 18, -5*e^2 + 4*e + 15, -e^2 - 12*e + 3, -8, -2, -2*e + 8, -6*e^2 - 4*e + 10, 2*e^2 - 2*e - 6, -5*e^2 + 19, -2*e^2 + 2*e + 18, 12, 18, 4*e^2 + 8*e - 6, 4*e^2 + 4*e - 22, -10*e^2 - 4*e + 26, -2*e^2 - 6*e + 6, -2*e^2 - 2*e + 14, -4*e^2 + 8*e + 14, 6*e^2 + 6*e - 30, -3*e^2 - 9, -7*e^2 - 8*e + 15, 2*e^2 - 10*e - 6, -4*e^2 - 8*e + 24, -e^2 - 10*e - 11, 8*e^2 - 28, 2*e^2 + 6*e - 10, 6*e^2 + 16*e - 22, 5*e^2 - 4*e - 17, -e^2 + 6*e + 21, 4*e^2 + 8*e - 18, 5*e^2 + 4*e - 9, -2*e + 20, 2*e^2 - 2*e - 18, 2*e + 8, -4*e^2 + 12*e + 14, 5*e^2 - 4*e - 17, 5*e^2 + 8*e - 29, 4*e^2 + 8*e - 30, 6, -6*e^2 + 10*e + 38, 6*e^2 + 12*e - 14, 6*e^2 - 8*e - 30, 6*e + 20, 2*e^2 - 8*e - 4, 10*e, 5*e^2 + 12*e - 9, e^2 - 4*e - 9, 3*e^2 - 10*e - 15, -12*e^2 + 44, -8*e^2 + 8*e + 50, -12*e^2 - 4*e + 44, -9*e^2 + 4*e + 31, 4*e^2 + 6*e - 8, -12*e^2 + 42, 12*e^2 + 8*e - 30, 2*e^2 + 12*e, -2*e^2 + 4*e + 12, -10*e^2 + 6*e + 42, 6*e^2 - 6*e - 18, 2*e^2 + 12*e - 6, -3*e^2 - 8*e + 7, 3*e^2 - 12*e - 17, e^2 + 20*e + 3, 4*e^2 - 4*e - 10, 2*e^2 + 8*e + 6, e^2 + 4*e - 45, 6*e^2 - 48, 3*e^2 + 8*e - 17, -4*e - 18, 7*e^2 + 8*e - 45, 13*e^2 + 8*e - 45, 6*e^2 + 12*e - 12, -8*e + 26, 10*e^2 + 2*e - 26, 4*e^2 + 10*e - 20, -e^2 + 8*e - 5, 6*e^2 - 6*e - 18, 4*e^2 - 16*e - 16, -16*e^2 - 4*e + 50, 4*e^2 - 12*e - 16, -2*e^2 + 4*e + 20, 8*e^2 + 2*e - 48, -10*e^2 + 8*e + 34, -6*e^2 - 2*e + 26, -4*e^2 - 6, -4*e^2 + 6*e + 24, -5*e^2 + 6*e + 9, -8*e^2 - 16*e + 32, -2*e^2 - 16*e - 8, -6*e^2 - 10*e + 10, -4*e - 16, 2*e^2 + 12*e - 38, -10*e^2 + 4*e + 46, -e^2 + 12*e - 5, 13*e^2 - 4*e - 53, -4*e^2 - 2*e, 7*e^2 + 18*e - 39, e^2 - 4*e - 9, -2*e^2 - 12*e + 14, -6*e^2 + 10*e + 34, -12*e^2 - 4*e + 46, 12*e^2 + 4*e - 34, 12*e^2 - 6*e - 56, 6*e^2 - 4*e - 40, -8*e^2 + 16*e + 58, -8*e^2 - 4*e + 48, 8*e^2 + 6*e - 44, 2*e^2 - 10*e - 18, -10*e^2 + 58, -2*e^2 + 4*e + 2, -12*e^2 + 4*e + 40, 6*e^2 + 14*e - 10, 12*e^2 - 10*e - 60, 5*e^2 - 8*e + 3, 4*e^2 - 8*e - 46, 2*e^2 + 12*e - 14, 2*e^2 - 4*e - 2, 8*e^2 + 12*e - 4, -12*e^2 + 6*e + 60, -14*e^2 - 10*e + 42, 10*e^2 - 18, 8*e^2 + 20*e - 48, -10*e^2 + 8*e + 70, -10*e^2 - 4*e - 2, 4*e^2 + 2*e - 4, -12*e^2 - 6*e + 44, -4*e^2 - 4*e + 38, 4*e^2 + 8*e - 22] 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, -w + 1])] = 1 AL_eigenvalues[ZF.ideal([7, 7, w^2 + w - 9])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]