/* 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([41, 13, -12, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([19, 19, w + 1]) primes_array = [ [9, 3, -w^3 + 3*w^2 + 5*w - 15],\ [9, 3, -w^3 + 8*w + 8],\ [16, 2, 2],\ [19, 19, w + 1],\ [19, 19, -w^2 + 6],\ [19, 19, -w^2 + 2*w + 5],\ [19, 19, -w + 2],\ [25, 5, 2*w^2 - 2*w - 13],\ [29, 29, -w^2 + 9],\ [29, 29, -w^2 + 2*w + 6],\ [29, 29, w^2 - 7],\ [29, 29, -w^2 + 2*w + 8],\ [31, 31, -2*w^2 + w + 12],\ [31, 31, 2*w^2 - 3*w - 11],\ [41, 41, -w],\ [41, 41, -w + 1],\ [49, 7, w^3 + 2*w^2 - 10*w - 20],\ [49, 7, w^3 - 5*w^2 - 3*w + 27],\ [61, 61, 2*w^2 - 3*w - 14],\ [61, 61, 2*w^2 - w - 15],\ [71, 71, -w^3 + w^2 + 7*w - 5],\ [71, 71, w^3 - 2*w^2 - 6*w + 2],\ [79, 79, w^3 - 2*w^2 - 6*w + 4],\ [79, 79, 3*w^2 - 2*w - 18],\ [89, 89, 2*w^3 - 6*w^2 - 11*w + 33],\ [89, 89, -4*w^2 + 3*w + 25],\ [101, 101, w^3 + w^2 - 9*w - 12],\ [101, 101, w^3 - 4*w^2 - 4*w + 19],\ [109, 109, -w^3 + 3*w^2 + 6*w - 19],\ [109, 109, -2*w^2 + w + 18],\ [121, 11, -3*w^2 + 3*w + 19],\ [121, 11, 3*w^2 - 3*w - 20],\ [131, 131, 2*w^2 - 3*w - 15],\ [131, 131, 2*w^2 - w - 16],\ [139, 139, 3*w^2 - 4*w - 21],\ [139, 139, -w^3 - w^2 + 7*w + 13],\ [151, 151, 4*w^2 - 3*w - 27],\ [151, 151, 2*w^3 - w^2 - 13*w - 8],\ [151, 151, w^3 + 3*w^2 - 13*w - 27],\ [151, 151, 4*w^2 - 5*w - 26],\ [179, 179, w^3 - 3*w^2 - 5*w + 11],\ [179, 179, 3*w^3 - 11*w^2 - 14*w + 61],\ [179, 179, 2*w^3 - 2*w^2 - 12*w + 3],\ [179, 179, w^3 - 8*w - 4],\ [181, 181, -w - 4],\ [181, 181, w^3 - 5*w^2 - 3*w + 25],\ [181, 181, w^3 + 2*w^2 - 10*w - 18],\ [181, 181, w - 5],\ [229, 229, w^2 - 2*w - 11],\ [229, 229, w^2 - 12],\ [239, 239, 6*w^2 - 7*w - 42],\ [239, 239, 2*w^3 - 2*w^2 - 14*w - 3],\ [241, 241, w^3 - 4*w^2 - 4*w + 18],\ [241, 241, 3*w^2 - 2*w - 16],\ [269, 269, 3*w^2 - 4*w - 22],\ [269, 269, -w^3 + 6*w + 8],\ [281, 281, w^3 - w^2 - 9*w + 4],\ [281, 281, 2*w^3 - 9*w^2 - 8*w + 49],\ [311, 311, -2*w^3 - w^2 + 17*w + 19],\ [311, 311, 2*w^3 - 7*w^2 - 9*w + 33],\ [331, 331, w^3 + 2*w^2 - 9*w - 21],\ [331, 331, w^3 - 5*w^2 - 2*w + 27],\ [349, 349, 2*w^3 - w^2 - 15*w - 4],\ [349, 349, -2*w^3 + 5*w^2 + 11*w - 18],\ [359, 359, -w^3 + 6*w^2 - 31],\ [359, 359, -w^3 - 3*w^2 + 9*w + 26],\ [379, 379, 2*w^3 - 5*w^2 - 11*w + 20],\ [379, 379, 2*w^3 - 14*w - 11],\ [401, 401, w^3 - 3*w^2 - 4*w + 16],\ [401, 401, -2*w^3 + 4*w^2 + 11*w - 16],\ [401, 401, 2*w^3 - 2*w^2 - 13*w - 3],\ [401, 401, w^3 - 7*w - 10],\ [409, 409, w^3 - 4*w^2 - 4*w + 17],\ [409, 409, 2*w^3 - 6*w^2 - 9*w + 22],\ [409, 409, -2*w^3 + 15*w + 9],\ [409, 409, -w^3 + 3*w^2 + 6*w - 12],\ [419, 419, -w^3 + 5*w^2 + 3*w - 24],\ [419, 419, w^3 + 2*w^2 - 10*w - 17],\ [421, 421, w^3 + 3*w^2 - 12*w - 25],\ [421, 421, -w^3 - w^2 + 9*w + 7],\ [431, 431, -w^3 - 2*w^2 + 12*w + 20],\ [431, 431, w^3 + 3*w^2 - 10*w - 28],\ [431, 431, -w^3 + 6*w^2 + w - 34],\ [431, 431, -w^3 + 5*w^2 + 5*w - 29],\ [439, 439, w^3 - 7*w^2 - 2*w + 39],\ [439, 439, w^3 + 4*w^2 - 13*w - 31],\ [449, 449, -w^3 - 4*w^2 + 13*w + 34],\ [449, 449, 6*w^2 - 5*w - 36],\ [461, 461, -w^3 + 5*w^2 + w - 25],\ [461, 461, 2*w^3 - 15*w - 12],\ [461, 461, -2*w^3 + 6*w^2 + 9*w - 25],\ [461, 461, w^3 + 2*w^2 - 8*w - 20],\ [479, 479, -w^3 - 4*w^2 + 12*w + 31],\ [479, 479, 2*w^3 - 15*w - 16],\ [479, 479, -2*w^3 + 6*w^2 + 9*w - 29],\ [479, 479, w^3 - 7*w^2 - w + 38],\ [491, 491, w^2 - 3*w - 7],\ [491, 491, -2*w^3 + 3*w^2 + 11*w - 2],\ [491, 491, 4*w^2 - 2*w - 29],\ [491, 491, w^2 + w - 9],\ [499, 499, 5*w^2 - 6*w - 33],\ [499, 499, 5*w^2 - 4*w - 34],\ [509, 509, 2*w - 3],\ [509, 509, -2*w - 1],\ [541, 541, -w^3 - 4*w^2 + 12*w + 29],\ [541, 541, -w^3 + 7*w^2 + w - 36],\ [599, 599, -w^3 - w^2 + 8*w + 7],\ [599, 599, w^3 - 4*w^2 - 3*w + 13],\ [619, 619, w^2 - 3*w - 6],\ [619, 619, w^2 + w - 8],\ [641, 641, -2*w^3 + 16*w + 11],\ [641, 641, -2*w^3 + 6*w^2 + 10*w - 25],\ [659, 659, 2*w^3 + w^2 - 16*w - 16],\ [659, 659, 2*w^3 - 7*w^2 - 8*w + 29],\ [691, 691, -w^3 + 5*w^2 + 3*w - 23],\ [691, 691, -w^3 + 3*w^2 + 6*w - 11],\ [701, 701, w^3 - w^2 - 9*w + 3],\ [701, 701, 2*w^3 - w^2 - 14*w - 5],\ [709, 709, 2*w^3 - 3*w^2 - 11*w + 6],\ [719, 719, w^2 - 2*w - 12],\ [719, 719, -w^2 + 13],\ [739, 739, -2*w^3 + 8*w^2 + 7*w - 36],\ [739, 739, w^2 + 2*w - 13],\ [751, 751, -2*w^3 + 5*w^2 + 10*w - 20],\ [751, 751, 2*w^3 - w^2 - 14*w - 7],\ [761, 761, w^3 + 4*w^2 - 12*w - 32],\ [761, 761, -w^3 + 7*w^2 + w - 39],\ [769, 769, -2*w^3 - 4*w^2 + 23*w + 45],\ [769, 769, 2*w^3 - 5*w^2 - 9*w + 21],\ [769, 769, 2*w^3 + 2*w^2 - 18*w - 31],\ [769, 769, 2*w^3 - 10*w^2 - 9*w + 62],\ [811, 811, -w^3 + 10*w + 5],\ [811, 811, -5*w^2 + 8*w + 30],\ [811, 811, w^3 - 8*w^2 + 2*w + 43],\ [811, 811, 2*w^3 - 2*w^2 - 13*w + 2],\ [829, 829, -2*w^3 - 2*w^2 + 18*w + 25],\ [829, 829, -2*w^3 + 8*w^2 + 8*w - 39],\ [859, 859, -w^3 - 4*w^2 + 12*w + 33],\ [859, 859, -w^3 + 7*w^2 + w - 40],\ [919, 919, w^3 - 4*w^2 - 3*w + 24],\ [919, 919, -w^3 + 5*w^2 + w - 26],\ [929, 929, 5*w^2 - 7*w - 28],\ [929, 929, -w^3 + 2*w^2 + 6*w - 13],\ [929, 929, w^3 - w^2 - 7*w - 6],\ [929, 929, -2*w^3 + 2*w^2 + 14*w - 5],\ [941, 941, -2*w^3 + 3*w^2 + 11*w - 1],\ [941, 941, 2*w^3 - 3*w^2 - 11*w + 11],\ [961, 31, 5*w^2 - 5*w - 33],\ [971, 971, -w^3 - 3*w^2 + 11*w + 22],\ [971, 971, 7*w^2 - 5*w - 45],\ [971, 971, 7*w^2 - 9*w - 43],\ [971, 971, w^3 - 6*w^2 - 2*w + 29],\ [991, 991, w^3 - 3*w^2 - 3*w + 15],\ [991, 991, -w^3 + 5*w^2 + 2*w - 29]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-2, -2, -7, 1, -1, -1, -1, -1, 0, 6, 0, 9, -7, -4, -3, 6, -4, 2, 8, -10, 12, 0, -1, 2, 6, 18, 0, 6, -7, 11, -4, 8, 18, 6, 14, -13, -22, -13, 2, -10, -15, 6, 15, -21, -10, -10, -1, -19, -4, -1, -18, -6, 14, -10, 6, 27, 15, -18, 15, 0, -28, 32, 29, -22, 9, 30, -4, 11, 21, -6, 0, -30, 5, -31, -19, -28, 27, 9, -22, 20, -15, 0, -36, -24, 26, 35, -33, 36, 15, 6, 0, -6, 18, 18, -36, 0, -24, 21, 3, -21, -28, 29, -42, 6, -16, -34, -6, 33, -10, -13, 12, 36, -18, -36, -37, -22, 0, -51, -19, 24, 0, -40, 8, 11, -13, -9, -24, -13, -1, 26, -34, 2, 38, -28, -16, -16, 2, 14, -4, -16, -16, 12, -18, 48, 0, 54, -30, 23, 24, 6, -42, -15, 2, 5] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([19, 19, w + 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]