/* 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, -3, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([68, 34, -w^3 + w^2 + 5*w - 2]) primes_array = [ [4, 2, w^3 - 2*w^2 - 2*w + 1],\ [7, 7, -w^3 + 3*w^2 + w - 3],\ [7, 7, -w^2 + w + 2],\ [17, 17, -w^3 + 3*w^2 - 3],\ [17, 17, -w^3 + w^2 + 4*w],\ [25, 5, -w^3 + 3*w^2 + 2*w - 2],\ [25, 5, -2*w^3 + 4*w^2 + 5*w - 1],\ [41, 41, -w^3 + 2*w^2 + 4*w - 2],\ [47, 47, -2*w^3 + 5*w^2 + 4*w - 4],\ [47, 47, 2*w^3 - 4*w^2 - 5*w],\ [49, 7, w^2 - 4*w - 1],\ [71, 71, 2*w - 3],\ [71, 71, -w^3 + w^2 + 6*w - 2],\ [73, 73, -w^3 + 3*w^2 + 3*w - 5],\ [73, 73, 2*w^3 - 5*w^2 - 5*w + 4],\ [73, 73, -w^3 + 3*w^2 - 5],\ [73, 73, w^3 - w^2 - 4*w + 2],\ [79, 79, 2*w^3 - 3*w^2 - 6*w + 2],\ [79, 79, 2*w^3 - 3*w^2 - 5*w],\ [81, 3, -3],\ [89, 89, -w^3 + w^2 + 5*w - 3],\ [89, 89, w - 4],\ [97, 97, -3*w^3 + 7*w^2 + 6*w - 5],\ [97, 97, -2*w^3 + 3*w^2 + 7*w - 2],\ [103, 103, -w - 3],\ [103, 103, -w^3 + 2*w^2 + 3*w - 5],\ [103, 103, -w^3 + 4*w^2 - w - 5],\ [103, 103, -3*w^3 + 6*w^2 + 7*w - 3],\ [113, 113, w^3 - 4*w^2 + w + 4],\ [113, 113, 2*w^2 - 3*w - 4],\ [113, 113, 3*w^3 - 6*w^2 - 7*w + 4],\ [113, 113, -2*w^3 + 6*w^2 + 2*w - 7],\ [137, 137, -w^3 + 4*w^2 - 4],\ [137, 137, w^3 - 4*w^2 + 6],\ [151, 151, -3*w^3 + 8*w^2 + 3*w - 4],\ [151, 151, -2*w^3 + 2*w^2 + 7*w + 4],\ [167, 167, -2*w^3 + 3*w^2 + 6*w - 3],\ [167, 167, -2*w^3 + 5*w^2 + 2*w - 6],\ [191, 191, w^3 - 2*w^2 - w + 5],\ [191, 191, -w^3 + 5*w^2 - 3*w - 3],\ [193, 193, -w^3 + 2*w^2 + 5*w - 2],\ [193, 193, -2*w^3 + 4*w^2 + 7*w - 4],\ [199, 199, -2*w^3 + 5*w^2 + 4*w - 2],\ [199, 199, w^2 - 5],\ [223, 223, -w^3 + 8*w - 2],\ [223, 223, -2*w^3 + 2*w^2 + 9*w],\ [223, 223, -3*w^3 + 5*w^2 + 11*w - 1],\ [223, 223, -w^3 + 4*w^2 + w - 8],\ [233, 233, -3*w^3 + 7*w^2 + 6*w - 4],\ [233, 233, w^3 - w^2 - 2*w - 3],\ [239, 239, -w^3 + 6*w],\ [239, 239, 2*w^3 - 5*w^2 - 4*w + 1],\ [257, 257, -3*w^3 + 7*w^2 + 7*w - 5],\ [257, 257, 3*w^3 - 7*w^2 - 5*w + 1],\ [263, 263, w^3 - w^2 - 4*w - 5],\ [263, 263, -2*w^3 + 3*w^2 + 9*w - 4],\ [281, 281, -w^3 + 3*w^2 + 4*w - 6],\ [281, 281, 3*w^3 - 7*w^2 - 8*w + 5],\ [289, 17, -3*w^3 + 6*w^2 + 6*w - 4],\ [311, 311, -3*w^3 + 5*w^2 + 11*w - 3],\ [311, 311, w - 5],\ [313, 313, -2*w^3 + 6*w^2 + w - 9],\ [313, 313, -w^3 + 3*w^2 + 3*w - 7],\ [359, 359, 3*w^3 - 6*w^2 - 8*w],\ [359, 359, -w^2 + 3*w - 4],\ [359, 359, w^3 - w^2 - 5*w - 5],\ [359, 359, w^3 - 2*w^2 - 4],\ [383, 383, -4*w^3 + 8*w^2 + 9*w - 3],\ [383, 383, -w^3 + 5*w^2 - 3*w - 7],\ [401, 401, -3*w^3 + 8*w^2 + 3*w - 9],\ [401, 401, -3*w^3 + 7*w^2 + 6*w - 3],\ [401, 401, w^3 - w^2 - 2*w - 4],\ [401, 401, -2*w^3 + 2*w^2 + 7*w - 1],\ [439, 439, -4*w^3 + 9*w^2 + 7*w - 6],\ [439, 439, 3*w^3 - 5*w^2 - 7*w + 1],\ [449, 449, -4*w^3 + 9*w^2 + 9*w - 6],\ [449, 449, 2*w^3 - 2*w^2 - 11*w - 2],\ [449, 449, -w^3 + 5*w^2 - 3*w - 11],\ [449, 449, -3*w^3 + 5*w^2 + 10*w - 5],\ [457, 457, -2*w^3 + 2*w^2 + 9*w - 1],\ [457, 457, w^3 - 4*w^2 + 3*w + 5],\ [463, 463, 3*w^2 - 5*w - 6],\ [463, 463, -w^3 + 5*w^2 - 3*w - 5],\ [479, 479, -2*w^3 + 3*w^2 + 9*w - 2],\ [479, 479, -w^3 + w^2 + 7*w - 1],\ [487, 487, -2*w^2 + 5*w + 6],\ [487, 487, -w^3 + 7*w - 2],\ [487, 487, -w^3 + 6*w - 2],\ [487, 487, -w^3 + 4*w^2 - 2*w - 8],\ [503, 503, w^3 - 4*w - 6],\ [503, 503, -3*w^3 + 8*w^2 + 4*w - 4],\ [521, 521, 3*w^2 - 4*w - 7],\ [521, 521, 2*w^3 - 7*w^2 + 6],\ [529, 23, -w^3 + 2*w^2 + 2*w - 6],\ [529, 23, w^3 - 2*w^2 - 2*w - 4],\ [569, 569, 2*w^3 - 7*w^2 - 2*w + 7],\ [569, 569, 4*w^3 - 7*w^2 - 10*w + 4],\ [569, 569, 3*w^3 - 4*w^2 - 13*w],\ [569, 569, 4*w^3 - 8*w^2 - 11*w + 3],\ [577, 577, 4*w^2 - 7*w - 10],\ [577, 577, w^3 + w^2 - 6*w - 9],\ [593, 593, w^3 - 4*w^2 + 2*w - 2],\ [593, 593, 3*w^3 - 10*w^2 + w + 11],\ [601, 601, -2*w^3 + 4*w^2 + 8*w - 3],\ [601, 601, -2*w^3 + 4*w^2 + 8*w - 5],\ [617, 617, 5*w^3 - 12*w^2 - 11*w + 11],\ [617, 617, 3*w^2 - 4*w - 5],\ [617, 617, -2*w^3 + 7*w^2 - 8],\ [617, 617, 5*w^3 - 10*w^2 - 13*w + 11],\ [631, 631, -3*w^2 + 8*w + 5],\ [631, 631, 2*w^3 - 7*w^2 - w + 10],\ [631, 631, 5*w^3 - 11*w^2 - 10*w + 9],\ [631, 631, 2*w^3 - w^2 - 12*w],\ [641, 641, -3*w^3 + 8*w^2 + 5*w - 5],\ [641, 641, 2*w^2 - w - 7],\ [647, 647, -2*w^3 + 2*w^2 + 11*w - 1],\ [647, 647, 2*w^3 - 2*w^2 - 11*w - 1],\ [647, 647, -3*w^3 + 9*w^2 + 4*w - 10],\ [647, 647, w^3 - 5*w^2 + 6],\ [673, 673, -5*w^3 + 11*w^2 + 12*w - 8],\ [673, 673, 3*w^3 - 5*w^2 - 10*w + 6],\ [719, 719, -3*w^3 + 4*w^2 + 10*w - 2],\ [719, 719, -3*w^3 + 8*w^2 + 2*w - 8],\ [727, 727, -w^3 + 5*w^2 - 2*w - 6],\ [727, 727, -w^3 + 5*w^2 - 2*w - 7],\ [743, 743, -5*w^3 + 10*w^2 + 13*w - 4],\ [743, 743, -4*w^3 + 9*w^2 + 10*w - 6],\ [743, 743, 2*w^3 - 2*w^2 - 10*w + 3],\ [743, 743, w^2 + 2*w - 5],\ [751, 751, -w^3 + 4*w^2 - 3*w - 6],\ [751, 751, 2*w^3 - 2*w^2 - 9*w + 2],\ [761, 761, 4*w^3 - 9*w^2 - 6*w + 4],\ [761, 761, 3*w^3 - 3*w^2 - 12*w - 1],\ [761, 761, 3*w^3 - 9*w^2 + 8],\ [761, 761, 4*w^3 - 7*w^2 - 10*w + 1],\ [769, 769, 4*w^3 - 7*w^2 - 10*w + 3],\ [769, 769, -4*w^3 + 7*w^2 + 14*w - 5],\ [769, 769, 4*w^3 - 9*w^2 - 6*w + 6],\ [769, 769, -w^2 + 6*w],\ [809, 809, -2*w^3 + 2*w^2 + 7*w - 3],\ [809, 809, -3*w^3 + 8*w^2 + 3*w - 11],\ [823, 823, 3*w^3 - 8*w^2 - 7*w + 6],\ [823, 823, -2*w^3 + 6*w^2 + 5*w - 10],\ [839, 839, -3*w^3 + 5*w^2 + 12*w - 1],\ [839, 839, -w^3 + w^2 + 8*w - 4],\ [887, 887, -4*w^3 + 9*w^2 + 5*w - 6],\ [887, 887, -5*w^3 + 9*w^2 + 13*w - 5],\ [919, 919, -4*w^3 + 6*w^2 + 14*w - 3],\ [919, 919, -5*w^3 + 12*w^2 + 9*w - 8],\ [929, 929, -2*w^3 + 3*w^2 + 10*w - 5],\ [929, 929, -2*w^3 + 3*w^2 + 10*w],\ [937, 937, -3*w^3 + 7*w^2 + 8*w - 3],\ [937, 937, -w^3 + 3*w^2 + 4*w - 8],\ [961, 31, -4*w^3 + 8*w^2 + 8*w - 5],\ [961, 31, -4*w^3 + 8*w^2 + 8*w - 3],\ [967, 967, 4*w^3 - 8*w^2 - 7*w + 1],\ [967, 967, -5*w^3 + 10*w^2 + 11*w - 3],\ [977, 977, 3*w^3 - 8*w^2 - w + 7],\ [977, 977, -4*w^3 + 6*w^2 + 13*w - 3],\ [991, 991, w^2 - w - 8],\ [991, 991, -3*w^3 + 5*w^2 + 12*w - 5]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 20 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -1/2*e - 1, -1, 4, e + 4, e + 4, -1/2*e - 1, 1/2*e - 9, -2, 1/2*e + 3, -e + 10, 3*e - 2, -e - 4, 4, -e + 8, -1/2*e + 1, -e + 2, -2*e - 6, -e - 2, -3/2*e - 1, -1/2*e + 7, -e - 4, -e - 4, -e + 14, 0, -5/2*e - 3, 2*e, -3*e - 4, -3*e - 4, -e + 2, 10, -2, 20, -1/2*e - 9, 2*e, 2*e + 4, -4*e, 12, -2*e - 4, -3*e - 4, 3/2*e - 1, 16, -2*e, -e - 6, -e - 14, 4*e + 4, 2*e - 4, e, 4*e + 2, 1/2*e + 1, -3*e - 16, -3*e - 12, -1/2*e + 19, -2*e + 12, e - 8, -e - 8, 6*e + 4, -8, 28, 9/2*e - 13, 7/2*e + 9, 3*e + 16, -2*e + 8, e - 16, 7/2*e - 7, 2*e - 4, -4*e - 14, 2*e + 12, -5*e + 8, 1/2*e + 3, -5*e - 8, 3*e + 6, -6*e + 4, 8, 1/2*e - 15, 30, 5/2*e - 5, 3/2*e - 7, -2*e - 10, -6*e + 2, 11/2*e + 15, 2*e - 24, -4*e + 4, 9/2*e - 5, -3*e + 28, -5/2*e + 21, -2*e - 12, -11/2*e - 7, -5*e - 6, -6*e + 8, -1/2*e - 11, 9/2*e + 7, 13/2*e + 5, -e - 20, -3/2*e + 21, -8, 3/2*e - 21, -3*e - 10, 6*e - 14, -5/2*e - 27, -4*e - 22, 2*e + 26, 3*e + 4, -15/2*e - 3, 22, -5*e - 16, -20, e - 14, -10*e, -e - 40, -9/2*e + 9, -11/2*e + 3, -9/2*e - 23, 7*e - 8, e + 10, 3*e - 10, 13/2*e - 15, -3*e - 22, 3*e + 20, -4*e + 8, -5/2*e + 13, 4*e - 12, -1/2*e - 33, -2*e - 12, -2*e + 28, 5*e - 26, 8*e - 2, -6*e - 4, 5*e - 12, 3/2*e + 15, -2*e + 30, 7/2*e + 29, -6*e - 14, 4*e + 34, 6*e + 10, 6*e + 18, 8*e - 18, 2*e + 30, 6, 5*e - 20, 9*e - 10, -3*e - 18, -3*e - 30, -5*e - 2, -e + 6, e + 22, 5*e - 24, -1/2*e + 9, -4, 4*e + 6, -8*e + 22, -2*e - 18, -19/2*e + 5, 6*e - 14, 11/2*e + 29, e - 18, 8*e - 10, 3*e + 16, -5*e + 18, 3*e + 38] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, w^3 - 2*w^2 - 2*w + 1])] = 1 AL_eigenvalues[ZF.ideal([17, 17, -w^3 + 3*w^2 - 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]