/* 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([9, 3, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([45, 15, -w - 3]) primes_array = [ [5, 5, -1/3*w^3 - 2/3*w^2 + 7/3*w + 4],\ [5, 5, 1/3*w^3 - 4/3*w^2 - 1/3*w + 3],\ [9, 3, -w],\ [9, 3, 1/3*w^3 - 1/3*w^2 - 7/3*w + 1],\ [16, 2, 2],\ [19, 19, 2/3*w^3 - 2/3*w^2 - 11/3*w],\ [19, 19, -1/3*w^3 + 1/3*w^2 + 1/3*w + 1],\ [31, 31, 2/3*w^3 + 1/3*w^2 - 11/3*w - 2],\ [31, 31, 2/3*w^3 - 5/3*w^2 - 5/3*w + 5],\ [41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 6],\ [41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 3],\ [61, 61, 1/3*w^3 - 4/3*w^2 + 2/3*w + 4],\ [61, 61, 2/3*w^3 + 1/3*w^2 - 14/3*w - 2],\ [71, 71, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],\ [71, 71, 1/3*w^3 + 2/3*w^2 - 7/3*w],\ [89, 89, -1/3*w^3 + 1/3*w^2 + 10/3*w - 3],\ [89, 89, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],\ [101, 101, 1/3*w^3 - 1/3*w^2 - 7/3*w - 3],\ [101, 101, -2/3*w^3 + 5/3*w^2 + 11/3*w - 4],\ [101, 101, 2/3*w^3 - 5/3*w^2 - 8/3*w + 3],\ [101, 101, w - 4],\ [109, 109, 2/3*w^3 + 1/3*w^2 - 8/3*w - 4],\ [109, 109, -w^3 + 2*w^2 + 4*w - 4],\ [121, 11, w^3 - w^2 - 4*w + 1],\ [121, 11, -w^3 + w^2 + 4*w - 2],\ [131, 131, 4/3*w^3 - 1/3*w^2 - 22/3*w - 1],\ [131, 131, -w^3 + 7*w + 2],\ [139, 139, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],\ [139, 139, 2*w - 1],\ [139, 139, 1/3*w^3 + 5/3*w^2 - 7/3*w - 10],\ [139, 139, -2/3*w^3 + 8/3*w^2 + 5/3*w - 5],\ [149, 149, w^3 + w^2 - 7*w - 11],\ [149, 149, -w^3 - w^2 + 5*w + 4],\ [151, 151, -2/3*w^3 - 4/3*w^2 + 14/3*w + 5],\ [151, 151, 2/3*w^3 - 8/3*w^2 - 2/3*w + 9],\ [169, 13, -2/3*w^3 - 1/3*w^2 + 11/3*w],\ [169, 13, -2/3*w^3 + 5/3*w^2 + 5/3*w - 7],\ [179, 179, 4/3*w^3 - 7/3*w^2 - 13/3*w + 6],\ [179, 179, -4/3*w^3 + 1/3*w^2 + 19/3*w + 1],\ [181, 181, 5/3*w^3 - 8/3*w^2 - 20/3*w + 5],\ [181, 181, -1/3*w^3 + 1/3*w^2 + 10/3*w - 1],\ [181, 181, -4/3*w^3 + 1/3*w^2 + 16/3*w + 3],\ [191, 191, w^2 - 3*w - 1],\ [191, 191, -1/3*w^3 + 4/3*w^2 + 10/3*w - 3],\ [199, 199, -w^3 + 3*w^2 + w - 7],\ [199, 199, -4/3*w^3 + 1/3*w^2 + 19/3*w - 1],\ [211, 211, -1/3*w^3 - 5/3*w^2 + 4/3*w + 8],\ [211, 211, w^3 - 3*w^2 - 4*w + 8],\ [229, 229, -4/3*w^3 + 4/3*w^2 + 13/3*w - 4],\ [229, 229, -5/3*w^3 + 5/3*w^2 + 23/3*w - 5],\ [229, 229, 1/3*w^3 - 7/3*w^2 + 2/3*w + 10],\ [229, 229, -2/3*w^3 + 2/3*w^2 + 2/3*w - 3],\ [239, 239, 4/3*w^3 - 1/3*w^2 - 19/3*w - 2],\ [239, 239, -2/3*w^3 + 2/3*w^2 + 11/3*w - 6],\ [251, 251, 2*w^2 - w - 8],\ [251, 251, -1/3*w^3 - 2/3*w^2 + 7/3*w - 2],\ [251, 251, 1/3*w^3 - 4/3*w^2 - 1/3*w + 9],\ [251, 251, -1/3*w^3 + 7/3*w^2 + 1/3*w - 7],\ [269, 269, -5/3*w^3 + 8/3*w^2 + 17/3*w - 8],\ [269, 269, -5/3*w^3 + 11/3*w^2 + 17/3*w - 12],\ [271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 3],\ [271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 2],\ [311, 311, -1/3*w^3 + 7/3*w^2 - 5/3*w - 8],\ [311, 311, -w^3 + 5*w - 1],\ [331, 331, -2/3*w^3 - 1/3*w^2 + 8/3*w + 6],\ [331, 331, -w^3 + 2*w^2 + 4*w - 2],\ [349, 349, -w^3 + 2*w^2 + w + 1],\ [349, 349, 1/3*w^3 - 1/3*w^2 - 7/3*w - 4],\ [349, 349, -4/3*w^3 + 13/3*w^2 + 10/3*w - 11],\ [349, 349, w - 5],\ [361, 19, -4/3*w^3 + 4/3*w^2 + 16/3*w - 3],\ [379, 379, 7/3*w^3 - 1/3*w^2 - 43/3*w - 9],\ [379, 379, 2*w^3 - 3*w^2 - 9*w + 8],\ [401, 401, 1/3*w^3 + 2/3*w^2 - 7/3*w - 8],\ [401, 401, -1/3*w^3 + 10/3*w^2 - 2/3*w - 15],\ [401, 401, -4/3*w^3 + 4/3*w^2 + 13/3*w - 3],\ [401, 401, -1/3*w^3 + 4/3*w^2 - 5/3*w + 3],\ [409, 409, -5/3*w^3 + 8/3*w^2 + 20/3*w - 7],\ [409, 409, 4/3*w^3 - 1/3*w^2 - 16/3*w - 1],\ [419, 419, 4/3*w^3 - 16/3*w^2 - 1/3*w + 10],\ [419, 419, 5/3*w^3 + 7/3*w^2 - 35/3*w - 17],\ [421, 421, -2*w^3 + 4*w^2 + 7*w - 10],\ [421, 421, -5/3*w^3 + 8/3*w^2 + 29/3*w - 7],\ [421, 421, 1/3*w^3 + 2/3*w^2 + 5/3*w - 4],\ [421, 421, -5/3*w^3 - 1/3*w^2 + 23/3*w + 5],\ [431, 431, -2/3*w^3 + 5/3*w^2 + 11/3*w - 1],\ [431, 431, -w^2 - w + 8],\ [439, 439, -2/3*w^3 + 11/3*w^2 - 1/3*w - 8],\ [439, 439, -2/3*w^3 + 11/3*w^2 + 8/3*w - 15],\ [479, 479, w^3 + 2*w^2 - 6*w - 10],\ [479, 479, -5/3*w^3 + 5/3*w^2 + 14/3*w],\ [479, 479, -5/3*w^3 + 5/3*w^2 + 23/3*w - 3],\ [479, 479, -4/3*w^3 + 4/3*w^2 + 13/3*w - 2],\ [521, 521, -w^3 + 2*w^2 + 5*w + 1],\ [521, 521, 1/3*w^3 + 2/3*w^2 - 1/3*w - 10],\ [541, 541, 1/3*w^3 + 5/3*w^2 + 2/3*w - 7],\ [541, 541, 5/3*w^3 - 5/3*w^2 - 26/3*w],\ [569, 569, -w - 5],\ [569, 569, 1/3*w^3 - 1/3*w^2 - 7/3*w + 6],\ [571, 571, w^3 - 4*w - 8],\ [571, 571, -1/3*w^3 + 7/3*w^2 + 7/3*w - 4],\ [599, 599, 5/3*w^3 - 5/3*w^2 - 14/3*w + 4],\ [599, 599, -4/3*w^3 + 13/3*w^2 + 13/3*w - 15],\ [599, 599, 2*w^3 - 3*w^2 - 8*w + 5],\ [599, 599, 7/3*w^3 - 7/3*w^2 - 34/3*w + 6],\ [601, 601, -4/3*w^3 + 13/3*w^2 + 13/3*w - 13],\ [601, 601, -w^3 + 8*w + 5],\ [619, 619, -5/3*w^3 + 11/3*w^2 + 17/3*w - 13],\ [619, 619, 4/3*w^3 + 2/3*w^2 - 19/3*w - 2],\ [631, 631, w^3 - w^2 - 7*w + 1],\ [631, 631, 3*w - 2],\ [641, 641, -2/3*w^3 + 5/3*w^2 + 2/3*w - 7],\ [641, 641, -1/3*w^3 + 7/3*w^2 - 5/3*w - 9],\ [661, 661, -5/3*w^3 + 11/3*w^2 + 20/3*w - 9],\ [661, 661, w^3 + w^2 - 4*w - 7],\ [691, 691, -5/3*w^3 + 8/3*w^2 + 17/3*w - 6],\ [691, 691, 2*w^2 - 3*w - 10],\ [691, 691, 5/3*w^3 - 2/3*w^2 - 23/3*w - 1],\ [691, 691, 1/3*w^3 + 5/3*w^2 - 13/3*w - 3],\ [701, 701, 2/3*w^3 + 4/3*w^2 - 20/3*w - 11],\ [701, 701, -2/3*w^3 - 1/3*w^2 + 17/3*w + 9],\ [709, 709, -2/3*w^3 - 7/3*w^2 + 17/3*w + 11],\ [709, 709, -1/3*w^3 + 10/3*w^2 - 2/3*w - 8],\ [709, 709, -2/3*w^3 + 11/3*w^2 - 1/3*w - 10],\ [709, 709, 3*w^2 - 2*w - 14],\ [719, 719, 7/3*w^3 + 8/3*w^2 - 46/3*w - 21],\ [719, 719, -2*w^3 + 7*w^2 + 2*w - 13],\ [719, 719, 2/3*w^3 + 7/3*w^2 - 14/3*w - 9],\ [719, 719, -w^3 + 4*w^2 + 2*w - 13],\ [761, 761, 1/3*w^3 + 5/3*w^2 - 1/3*w - 8],\ [761, 761, 5/3*w^3 - 2/3*w^2 - 14/3*w + 1],\ [761, 761, -8/3*w^3 + 11/3*w^2 + 38/3*w - 11],\ [761, 761, -4/3*w^3 + 10/3*w^2 + 19/3*w - 9],\ [809, 809, -2/3*w^3 + 11/3*w^2 + 5/3*w - 15],\ [809, 809, 3*w^2 - w - 8],\ [811, 811, 2/3*w^3 + 7/3*w^2 - 14/3*w - 11],\ [811, 811, 1/3*w^3 + 8/3*w^2 - 13/3*w - 10],\ [811, 811, 1/3*w^3 - 10/3*w^2 + 5/3*w + 11],\ [811, 811, -w^3 + 4*w^2 + 2*w - 11],\ [821, 821, 2*w^3 - w^2 - 12*w - 5],\ [821, 821, 1/3*w^3 + 5/3*w^2 - 7/3*w - 14],\ [829, 829, -2/3*w^3 + 11/3*w^2 + 5/3*w - 10],\ [829, 829, 3*w^2 - w - 13],\ [839, 839, 4/3*w^3 - 1/3*w^2 - 25/3*w + 1],\ [839, 839, -2/3*w^3 + 5/3*w^2 - 1/3*w - 6],\ [841, 29, 5/3*w^3 - 5/3*w^2 - 20/3*w + 1],\ [841, 29, -5/3*w^3 + 5/3*w^2 + 20/3*w - 4],\ [859, 859, -1/3*w^3 - 8/3*w^2 + 13/3*w + 11],\ [859, 859, -1/3*w^3 + 10/3*w^2 - 5/3*w - 10],\ [859, 859, 2/3*w^3 - 8/3*w^2 + 1/3*w + 9],\ [859, 859, -2/3*w^3 + 5/3*w^2 + 2/3*w - 8],\ [881, 881, -4/3*w^3 + 16/3*w^2 + 4/3*w - 11],\ [881, 881, 1/3*w^3 - 13/3*w^2 - 4/3*w + 18],\ [911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 14],\ [911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 9],\ [929, 929, 4/3*w^3 - 7/3*w^2 - 22/3*w + 4],\ [929, 929, 1/3*w^3 + 2/3*w^2 + 2/3*w - 6],\ [961, 31, 5/3*w^3 - 5/3*w^2 - 20/3*w + 2],\ [971, 971, -2/3*w^3 + 11/3*w^2 + 2/3*w - 13],\ [971, 971, 1/3*w^3 + 8/3*w^2 - 10/3*w - 9],\ [991, 991, -4/3*w^3 + 10/3*w^2 + 16/3*w - 17],\ [991, 991, -2*w^3 + 4*w^2 + 11*w - 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, 2, -1, -4, -1, -4, -6, 2, -10, -6, 0, -4, -14, -6, 8, 16, 14, -14, -6, 0, 10, -10, -2, -4, 2, -2, -8, -8, 8, -16, 10, 10, 4, -4, 12, -6, 14, 4, -8, 2, -24, 6, -8, 10, -28, 16, 0, 12, -22, 22, 20, 16, -24, 22, -14, 12, -6, -12, -10, -28, 20, -14, -8, -6, -24, 4, -22, 2, 30, -22, -10, 34, 26, 12, -14, 2, -6, 12, -34, -12, -16, 22, 26, 2, -20, -8, 6, -20, -32, 42, -20, 12, 40, 10, 18, -36, -32, -38, -22, 12, 16, 34, -32, 20, -36, 22, 22, -20, 12, 8, 32, -10, 42, -12, 50, 28, -20, 12, -28, 10, 8, 10, -20, 26, 4, 32, 24, 40, 34, 2, -6, 38, 44, 34, 36, -42, -22, 16, 38, 38, -18, 4, 10, 20, 40, -30, -50, -12, -32, -42, 44, -26, -30, -40, -48, 14, -6, 10, 12, -12, -40, 16] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, -1/3*w^3 - 2/3*w^2 + 7/3*w + 4])] = -1 AL_eigenvalues[ZF.ideal([9, 3, -w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]