/* 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, 0, -4, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([71,71,-w^3 + w^2 + 2*w - 3]) primes_array = [ [2, 2, w^3 - 4*w + 1],\ [9, 3, -w^2 + 2],\ [23, 23, w^3 - w^2 - 4*w + 1],\ [23, 23, w^2 - w - 3],\ [23, 23, -w^2 - w + 3],\ [23, 23, -w^3 - w^2 + 4*w + 1],\ [25, 5, w^3 - 5*w + 1],\ [25, 5, w^3 - 5*w - 1],\ [47, 47, -3*w^3 + 2*w^2 + 12*w - 8],\ [47, 47, -2*w^3 - w^2 + 6*w],\ [47, 47, 2*w^3 - w^2 - 6*w],\ [47, 47, w^2 + w - 5],\ [49, 7, 2*w^3 - 6*w - 1],\ [49, 7, -2*w^3 + 6*w - 1],\ [71, 71, 2*w^3 - w^2 - 7*w + 1],\ [71, 71, 3*w^3 - w^2 - 11*w + 2],\ [71, 71, 4*w^3 - 2*w^2 - 14*w + 5],\ [71, 71, -3*w^3 + 2*w^2 + 10*w - 4],\ [73, 73, -w^3 - 2*w^2 + 3*w + 5],\ [73, 73, -w^3 + 2*w^2 + 3*w - 3],\ [73, 73, w^3 + 2*w^2 - 3*w - 3],\ [73, 73, w^3 - 2*w^2 - 3*w + 5],\ [97, 97, 2*w^2 + w - 4],\ [97, 97, w^3 + 2*w^2 - 4*w - 4],\ [97, 97, -w^3 + 2*w^2 + 4*w - 4],\ [97, 97, 2*w^2 - w - 4],\ [121, 11, 2*w^3 - 5*w],\ [121, 11, 3*w^3 - 10*w],\ [167, 167, -2*w^3 + 3*w^2 + 5*w - 7],\ [167, 167, -2*w^3 - w^2 + 10*w - 2],\ [167, 167, 3*w^3 - 11*w + 3],\ [167, 167, 3*w^3 - 2*w^2 - 10*w + 8],\ [169, 13, 2*w^3 - 9*w],\ [169, 13, w^3 - 6*w],\ [191, 191, w^2 - 2*w - 4],\ [191, 191, 2*w^3 + w^2 - 8*w],\ [191, 191, -2*w^3 + w^2 + 8*w],\ [191, 191, w^2 + 2*w - 4],\ [193, 193, -w - 4],\ [193, 193, -w^3 + 4*w - 4],\ [193, 193, w^3 - 4*w - 4],\ [193, 193, w - 4],\ [239, 239, 4*w^3 - 3*w^2 - 16*w + 12],\ [239, 239, -5*w^3 + w^2 + 19*w - 6],\ [239, 239, 4*w^3 - 16*w + 3],\ [239, 239, w^3 - 2*w^2 - 5*w + 9],\ [241, 241, 2*w^3 + w^2 - 10*w - 4],\ [241, 241, 2*w^3 - w^2 - 10*w],\ [241, 241, -2*w^3 - w^2 + 10*w],\ [241, 241, -2*w^3 + w^2 + 10*w - 4],\ [263, 263, -2*w^3 + 2*w^2 + 6*w - 7],\ [263, 263, -4*w^3 + 3*w^2 + 13*w - 9],\ [263, 263, -w^3 + 2*w^2 + 2*w - 6],\ [263, 263, -3*w^3 + 3*w^2 + 9*w - 8],\ [289, 17, 3*w^3 - 9*w + 1],\ [289, 17, -3*w^3 + 9*w + 1],\ [311, 311, 2*w^3 + w^2 - 8*w + 2],\ [311, 311, -w^2 - 2*w + 6],\ [311, 311, 3*w^3 - 2*w^2 - 13*w + 9],\ [311, 311, -2*w^3 + w^2 + 8*w + 2],\ [313, 313, -w^3 - w^2 + 4*w - 3],\ [313, 313, w^2 - w - 7],\ [313, 313, w^2 + w - 7],\ [313, 313, w^3 - w^2 - 4*w - 3],\ [337, 337, 3*w^3 - 8*w - 2],\ [337, 337, -4*w^3 + 13*w - 2],\ [337, 337, 4*w^3 - 13*w - 2],\ [337, 337, 3*w^3 - 8*w + 2],\ [359, 359, w^3 + 2*w^2 - 6*w - 6],\ [359, 359, -2*w^3 + 2*w^2 + 9*w - 2],\ [359, 359, 2*w^3 + 2*w^2 - 9*w - 2],\ [359, 359, -w^3 + 2*w^2 + 6*w - 6],\ [361, 19, -w^3 + 5*w - 5],\ [361, 19, w^3 - 5*w - 5],\ [383, 383, -w^3 + w^2 - 3],\ [383, 383, 4*w^3 + w^2 - 15*w - 1],\ [383, 383, -4*w^3 + w^2 + 15*w - 1],\ [383, 383, w^3 + w^2 - 3],\ [409, 409, 3*w^3 + w^2 - 8*w - 1],\ [409, 409, 4*w^3 - w^2 - 13*w + 3],\ [409, 409, 4*w^3 + w^2 - 13*w - 3],\ [409, 409, -3*w^3 + w^2 + 8*w - 1],\ [431, 431, -3*w^3 - 2*w^2 + 11*w + 3],\ [431, 431, -w^3 + 2*w^2 + w - 5],\ [431, 431, w^3 + 2*w^2 - w - 5],\ [431, 431, 3*w^3 - 2*w^2 - 11*w + 3],\ [433, 433, 3*w^3 - 13*w + 1],\ [433, 433, w^3 + w^2 - 4*w - 7],\ [433, 433, -w^3 + w^2 + 4*w - 7],\ [433, 433, 3*w^3 - 13*w - 1],\ [457, 457, 2*w^3 - 9*w - 6],\ [457, 457, w^3 - 6*w - 6],\ [457, 457, -w^3 + 6*w - 6],\ [457, 457, -2*w^3 + 9*w - 6],\ [479, 479, -2*w^3 + w^2 + 4*w - 4],\ [479, 479, 4*w^3 + w^2 - 14*w],\ [479, 479, -4*w^3 + w^2 + 14*w],\ [479, 479, 2*w^3 + w^2 - 4*w - 4],\ [503, 503, -3*w^3 - 2*w^2 + 10*w + 2],\ [503, 503, 2*w^3 + 2*w^2 - 5*w - 6],\ [503, 503, -2*w^3 + 2*w^2 + 5*w - 6],\ [503, 503, 3*w^3 - 2*w^2 - 10*w + 2],\ [577, 577, 3*w^3 + w^2 - 11*w - 8],\ [577, 577, -w^3 + w^2 + 6*w - 9],\ [577, 577, -4*w^3 + 4*w^2 + 15*w - 12],\ [577, 577, -4*w^3 + w^2 + 12*w],\ [599, 599, 2*w^3 + 2*w^2 - 8*w - 1],\ [599, 599, 2*w^2 - 2*w - 7],\ [599, 599, 2*w^2 + 2*w - 7],\ [599, 599, -2*w^3 + 2*w^2 + 8*w - 1],\ [601, 601, -w^3 + 3*w^2 + 4*w - 7],\ [601, 601, 3*w^2 - w - 5],\ [601, 601, 3*w^2 + w - 5],\ [601, 601, w^3 + 3*w^2 - 4*w - 7],\ [647, 647, -6*w^3 + 3*w^2 + 21*w - 7],\ [647, 647, -5*w^3 + 3*w^2 + 17*w - 6],\ [647, 647, 4*w^3 - 2*w^2 - 14*w + 3],\ [647, 647, 5*w^3 - 2*w^2 - 18*w + 4],\ [673, 673, -5*w^3 + 2*w^2 + 16*w - 6],\ [673, 673, 3*w^3 - w^2 - 14*w + 5],\ [673, 673, -3*w^3 - w^2 + 14*w + 5],\ [673, 673, 5*w^3 + 2*w^2 - 16*w - 6],\ [719, 719, 3*w^3 - w^2 - 13*w - 2],\ [719, 719, 6*w^3 - w^2 - 23*w + 7],\ [719, 719, -5*w^3 + 20*w - 4],\ [719, 719, -3*w^3 - w^2 + 13*w - 2],\ [743, 743, -3*w^3 + w^2 + 11*w + 2],\ [743, 743, w^3 + w^2 - w - 6],\ [743, 743, -w^3 + w^2 + w - 6],\ [743, 743, 3*w^3 + w^2 - 11*w + 2],\ [769, 769, -4*w^3 + 17*w - 2],\ [769, 769, -2*w^3 + w^2 + 8*w - 8],\ [769, 769, 2*w^3 + w^2 - 8*w - 8],\ [769, 769, 4*w^3 - 17*w - 2],\ [839, 839, w^3 + 3*w^2 - 6*w - 9],\ [839, 839, -4*w^3 - 2*w^2 + 16*w + 3],\ [839, 839, 4*w^3 - 2*w^2 - 16*w + 3],\ [839, 839, -w^3 + 3*w^2 + 6*w - 9],\ [841, 29, -4*w^3 + w^2 + 12*w - 2],\ [841, 29, 4*w^3 + w^2 - 12*w - 2],\ [863, 863, 2*w^2 + 3*w - 6],\ [863, 863, 3*w^3 - 2*w^2 - 12*w + 2],\ [863, 863, 3*w^3 + 2*w^2 - 12*w - 2],\ [863, 863, 2*w^2 - 3*w - 6],\ [887, 887, w^3 + 2*w^2 - 6*w - 8],\ [887, 887, 2*w^2 - 2*w - 9],\ [887, 887, 2*w^2 + 2*w - 9],\ [887, 887, -w^3 + 2*w^2 + 6*w - 8],\ [911, 911, -3*w^3 - 3*w^2 + 11*w + 4],\ [911, 911, w^3 + 3*w^2 - w - 8],\ [911, 911, -w^3 + 3*w^2 + w - 8],\ [911, 911, 3*w^3 - 3*w^2 - 11*w + 4],\ [937, 937, -w^3 - w^2 + 8*w + 1],\ [937, 937, 4*w^3 - w^2 - 17*w + 3],\ [937, 937, -4*w^3 - w^2 + 17*w + 3],\ [937, 937, w^3 - w^2 - 8*w + 1],\ [961, 31, -4*w^3 + 12*w - 1],\ [961, 31, 4*w^3 - 12*w - 1],\ [983, 983, w^2 + 3*w - 7],\ [983, 983, 3*w^3 - w^2 - 12*w - 3],\ [983, 983, 3*w^3 + w^2 - 12*w + 3],\ [983, 983, w^2 - 3*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + 3*x^2 - 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e - 2, 2*e^2 + 2*e - 9, -e - 3, 4*e^2 + 5*e - 9, -5*e^2 - 7*e + 6, -5*e^2 - 6*e + 8, 4*e^2 + 9*e - 4, 3*e^2 + 7*e - 9, -e^2 - 5*e - 6, -e^2 - 2*e, -2*e^2 - 2*e + 3, e^2 - 7, e^2 + 3*e - 10, -3*e^2 - 6*e, 1, 5*e^2 + 12*e - 3, 5*e^2 + 3*e - 15, -6*e^2 - 15*e + 2, -1, -4*e^2 - 3*e + 5, -3*e^2 + 8, -3*e^2 - 9*e + 5, 5*e^2 + 12*e - 7, e^2 + 3*e + 2, -3*e^2 - 6*e + 8, -2*e^2 - 12*e - 4, -11*e^2 - 15*e + 17, e^2 + 5*e - 12, -e^2 - e - 9, -6*e^2 - 4*e + 24, 10*e^2 + 14*e - 15, 4*e^2 + 12*e - 1, 7*e^2 + 9*e - 19, 2*e^2 - 2*e - 6, -4*e^2 + 7*e + 18, 7*e^2 + e - 24, e^2 + 10*e - 6, 4*e^2 + 9*e + 8, 6*e^2 + 9*e - 16, 8*e^2 + 9*e - 13, -12*e^2 - 12*e + 29, -12*e^2 - 25*e + 18, e^2 - e + 12, -9*e^2 - 7*e + 33, e^2 + 5*e - 3, 10*e^2 + 9*e - 34, 4*e^2 + 3*e - 1, 2*e^2 + 9*e + 2, 5*e^2 + 12*e - 1, -7*e^2 - 8*e + 24, 5*e^2 + 4*e - 18, -2*e^2 + e, 11*e^2 + 7*e - 27, -3*e^2 - 12*e + 8, 14*e^2 + 30*e - 13, -4*e^2 + 2*e + 3, -5*e^2 + 5*e + 27, -11*e^2 - 31*e + 3, 3*e^2 + 11*e - 3, 16*e^2 + 18*e - 25, 3*e^2 - 3*e - 22, -13*e^2 - 18*e + 14, 15*e^2 + 33*e - 16, 11*e^2 + 21*e - 19, -12*e^2 - 9*e + 26, -4*e^2 - 3*e + 5, -8*e^2 - 21*e - 1, -e^2 - 12*e - 18, 16*e^2 + 21*e - 18, -8*e^2 - 21*e - 9, 4*e^2 + 6*e + 24, 2*e^2 - 3*e + 5, -2*e^2 - 9*e - 16, -11*e^2 - 16*e + 21, -11*e^2 - 22*e + 21, 14*e^2 + 17*e - 30, 4*e^2 + 20*e + 3, e^2 - 3*e + 2, 7*e^2 + 15*e - 7, 9*e^2 + 3*e - 16, -11*e^2 - 15*e + 14, -8*e^2 - 12*e + 9, -9*e^2 - 30*e, 4*e^2 - 9*e - 12, 4*e^2 + 15*e - 3, 15*e^2 + 27*e - 16, 4*e^2 + 3*e - 16, -11*e^2 - 21*e + 20, -5*e^2 + 3*e + 5, -5*e^2 + 3*e + 29, 3*e - 13, e^2 - 6*e + 14, 2*e^2 - 10, 9*e^2 + e - 24, 7*e^2 - 8*e - 30, 10*e - 6, -13*e^2 - 29*e + 12, 7*e^2 + 18*e + 6, -5*e^2 - 21*e + 6, 7*e^2 + 3*e - 27, 5*e^2 + 12*e + 21, -13*e^2 - 3*e + 41, -9*e^2 - 33*e - 4, 16*e^2 + 33*e - 1, 4*e^2 + 9*e + 14, -9*e^2 - 22*e - 6, 11*e^2 + 14*e - 30, 6*e^2 + 20*e - 6, -9*e^2 - 7*e + 3, 11*e^2 + 18*e - 7, 3*e^2 - 15*e - 25, 18*e^2 + 12*e - 40, -17*e^2 - 30*e + 26, -11*e^2 - 27*e - 12, -11*e^2 - 21*e + 9, -10*e^2 - 3*e + 24, 2*e^2 - 9*e - 15, -2*e^2 + 3*e - 7, 19*e^2 + 36*e - 34, -2*e^2 - 9*e + 35, 4*e^2 - 9*e - 28, 7*e^2 + 21*e - 21, 7*e^2 + 6*e - 12, -15*e^2 - 18*e + 9, -12*e^2 - 21*e + 24, 31*e^2 + 53*e - 24, 2*e - 9, 20*e^2 + 23*e - 24, -27*e^2 - 43*e + 33, -14*e^2 + 38, -17*e^2 - 30*e + 23, 13*e^2 + 21*e - 19, -2*e^2 + 12*e + 2, 15*e^2 + 10*e - 51, -3*e^2 - 2*e + 21, -14*e^2 - 23*e, 13*e^2 + 40*e - 9, 22*e^2 + 30*e - 40, 4*e^2 + 15*e + 29, -12*e^2 - 21*e + 21, 3*e^2 + 3*e + 33, e^2 + 12*e + 18, -e^2 - 6*e - 21, -12*e^2 - 4*e + 18, -11*e^2 - 7*e + 60, -12*e^2 - 31*e - 18, 15*e^2 + 32*e, 7*e + 27, -12*e^2 - 32*e + 15, -11*e^2 - 26*e + 24, 9*e^2 + 16*e - 36, 3*e^2 - 12*e - 31, -3*e^2 + 3*e + 11, 27*e^2 + 48*e - 34, -6*e^2 - 30*e - 1, 16*e^2 - 46, -5*e^2 - 3*e + 38, -4*e^2 - 2*e + 33, -4*e^2 - 8*e - 21, -27*e^2 - 47*e + 12, e^2 - 14*e - 3] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([71,71,-w^3 + w^2 + 2*w - 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]