/* 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, 1, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([1, 1, 1]) primes_array = [ [4, 2, -1/2*w^3 + 3*w + 5/2],\ [4, 2, w^3 - w^2 - 6*w],\ [13, 13, -1/2*w^3 + w^2 + 3*w - 1/2],\ [13, 13, 1/2*w^3 - w^2 - 3*w + 7/2],\ [13, 13, -1/2*w^3 + 4*w - 1/2],\ [13, 13, -1/2*w^3 + 4*w + 5/2],\ [17, 17, 1/2*w^3 - w^2 - w + 3/2],\ [47, 47, -w^2 + 2],\ [47, 47, -w^3 + 2*w^2 + 5*w - 5],\ [47, 47, -3/2*w^3 + w^2 + 10*w + 3/2],\ [47, 47, -1/2*w^3 + 5*w - 1/2],\ [67, 67, 3/2*w^3 - 2*w^2 - 8*w + 3/2],\ [67, 67, 1/2*w^3 - 2*w - 5/2],\ [67, 67, -1/2*w^3 + w^2 + w - 5/2],\ [67, 67, -3/2*w^3 + w^2 + 9*w - 1/2],\ [81, 3, -3],\ [89, 89, w^2 - 2*w - 4],\ [89, 89, 3/2*w^3 - 2*w^2 - 9*w + 3/2],\ [89, 89, -1/2*w^3 + w^2 + 4*w - 7/2],\ [89, 89, -w^3 + 7*w + 1],\ [101, 101, -1/2*w^3 + 3*w - 5/2],\ [101, 101, -1/2*w^3 + w^2 + 2*w - 11/2],\ [101, 101, -w^3 + w^2 + 6*w + 3],\ [101, 101, w - 4],\ [103, 103, 3/2*w^3 - 3*w^2 - 7*w + 7/2],\ [103, 103, -w^2 + 8],\ [103, 103, -1/2*w^3 - w^2 + 3*w + 11/2],\ [103, 103, -1/2*w^3 - w^2 + 3*w + 13/2],\ [137, 137, 1/2*w^3 - 2*w - 7/2],\ [137, 137, -1/2*w^3 + w^2 + w - 7/2],\ [137, 137, 3/2*w^3 - 2*w^2 - 8*w + 1/2],\ [137, 137, -3/2*w^3 + w^2 + 9*w - 3/2],\ [149, 149, 3/2*w^3 - 9*w - 13/2],\ [149, 149, 7/2*w^3 - 3*w^2 - 21*w - 3/2],\ [149, 149, 2*w^3 - 2*w^2 - 12*w - 1],\ [149, 149, -w^2 + 6],\ [157, 157, -5/2*w^3 + 3*w^2 + 13*w - 9/2],\ [157, 157, -2*w^3 + 3*w^2 + 12*w - 4],\ [157, 157, 1/2*w^3 - 2*w^2 - 2*w + 15/2],\ [157, 157, 5/2*w^3 - 2*w^2 - 14*w - 3/2],\ [191, 191, 3/2*w^3 - 2*w^2 - 8*w - 1/2],\ [191, 191, -3/2*w^3 + w^2 + 9*w - 5/2],\ [191, 191, 1/2*w^3 - 2*w - 9/2],\ [191, 191, -1/2*w^3 + w^2 + w - 9/2],\ [239, 239, -5/2*w^3 + 2*w^2 + 16*w + 1/2],\ [239, 239, 2*w^3 - 13*w - 8],\ [239, 239, -7/2*w^3 + 3*w^2 + 20*w - 1/2],\ [239, 239, -1/2*w^3 + 4*w + 11/2],\ [251, 251, -5/2*w^3 + 3*w^2 + 13*w - 1/2],\ [251, 251, -2*w^3 + w^2 + 14*w + 2],\ [251, 251, 3/2*w^3 - w^2 - 7*w - 7/2],\ [251, 251, -w^3 + 9*w + 1],\ [271, 271, -w^3 + 7*w - 1],\ [271, 271, -3/2*w^3 + 2*w^2 + 9*w + 1/2],\ [271, 271, w^2 - 2*w - 6],\ [271, 271, -1/2*w^3 + w^2 + 4*w - 11/2],\ [293, 293, -1/2*w^3 + 2*w^2 + 2*w - 13/2],\ [293, 293, -3/2*w^3 + w^2 + 11*w - 3/2],\ [293, 293, -1/2*w^3 + 2*w^2 + 2*w - 7/2],\ [293, 293, -3/2*w^3 + w^2 + 11*w + 3/2],\ [307, 307, w^3 - w^2 - 8*w + 1],\ [307, 307, 2*w^3 - 2*w^2 - 13*w + 2],\ [307, 307, -1/2*w^3 + 2*w^2 + w - 9/2],\ [307, 307, -1/2*w^3 - w^2 + 4*w + 9/2],\ [353, 353, -3/2*w^3 + 2*w^2 + 8*w - 15/2],\ [353, 353, 11/2*w^3 - 4*w^2 - 34*w - 7/2],\ [353, 353, 6*w^3 - 4*w^2 - 36*w - 7],\ [353, 353, 9/2*w^3 - 3*w^2 - 26*w - 5/2],\ [361, 19, 1/2*w^3 + w^2 - 3*w - 15/2],\ [361, 19, -2*w^3 + 2*w^2 + 10*w - 7],\ [373, 373, -1/2*w^3 + 2*w^2 + 2*w - 9/2],\ [373, 373, -3/2*w^3 + w^2 + 11*w + 1/2],\ [373, 373, -1/2*w^3 + 2*w^2 + 2*w - 11/2],\ [373, 373, 3/2*w^3 - w^2 - 11*w + 1/2],\ [409, 409, 2*w^3 - 3*w^2 - 10*w + 2],\ [409, 409, -w^3 + 5*w + 5],\ [409, 409, -1/2*w^3 + w^2 - 7/2],\ [409, 409, 5/2*w^3 - 2*w^2 - 15*w + 5/2],\ [421, 421, -2*w^3 + w^2 + 12*w],\ [421, 421, 5/2*w^3 - 3*w^2 - 14*w + 3/2],\ [421, 421, -w^3 + 2*w^2 + 3*w - 5],\ [421, 421, 1/2*w^3 - w - 7/2],\ [443, 443, 5/2*w^3 - 3*w^2 - 15*w + 3/2],\ [443, 443, 1/2*w^3 - w^2 - 5*w + 9/2],\ [443, 443, -1/2*w^3 + 2*w^2 - 13/2],\ [443, 443, -3/2*w^3 + 10*w + 3/2],\ [463, 463, 4*w^3 - 4*w^2 - 22*w + 5],\ [463, 463, 3*w^3 - 4*w^2 - 14*w + 2],\ [463, 463, -5/2*w^3 + 2*w^2 + 14*w - 7/2],\ [463, 463, 3/2*w^3 - w^2 - 7*w - 9/2],\ [509, 509, -7/2*w^3 + 2*w^2 + 20*w + 3/2],\ [509, 509, 5/2*w^3 - w^2 - 16*w - 5/2],\ [509, 509, -5/2*w^3 + 4*w^2 + 10*w - 13/2],\ [509, 509, 9/2*w^3 - 5*w^2 - 25*w + 7/2],\ [523, 523, 3*w^3 - 3*w^2 - 16*w + 3],\ [523, 523, 2*w^3 - 2*w^2 - 9*w + 2],\ [523, 523, 5/2*w^3 - 3*w^2 - 12*w + 3/2],\ [523, 523, -5/2*w^3 + 2*w^2 + 13*w + 3/2],\ [557, 557, 3*w^3 - 3*w^2 - 16*w + 1],\ [557, 557, 5/2*w^3 - 2*w^2 - 13*w + 1/2],\ [557, 557, -2*w^3 + 2*w^2 + 9*w],\ [557, 557, -5/2*w^3 + 3*w^2 + 12*w - 7/2],\ [577, 577, -w^3 - w^2 + 6*w + 9],\ [577, 577, 7/2*w^3 - 3*w^2 - 21*w - 7/2],\ [577, 577, 1/2*w^3 - 2*w^2 - 3*w + 17/2],\ [577, 577, -5/2*w^3 + 3*w^2 + 13*w - 19/2],\ [599, 599, -4*w^3 + 4*w^2 + 22*w - 1],\ [599, 599, w^3 + w^2 - 8*w - 7],\ [599, 599, -5/2*w^3 + 3*w^2 + 16*w - 11/2],\ [599, 599, 3/2*w^3 - w^2 - 7*w - 13/2],\ [613, 613, w^3 - 5*w - 9],\ [613, 613, -2*w^3 + 3*w^2 + 10*w + 2],\ [613, 613, -1/2*w^3 + w^2 - 15/2],\ [613, 613, -5/2*w^3 + 2*w^2 + 15*w - 13/2],\ [625, 5, -5],\ [647, 647, 2*w^3 - 3*w^2 - 8*w + 6],\ [647, 647, -3*w^3 + 2*w^2 + 17*w - 1],\ [647, 647, 3/2*w^3 - w^2 - 6*w - 7/2],\ [647, 647, -7/2*w^3 + 4*w^2 + 19*w - 3/2],\ [659, 659, w^3 + w^2 - 6*w - 5],\ [659, 659, -2*w^3 + 4*w^2 + 9*w - 8],\ [659, 659, -7/2*w^3 + 3*w^2 + 22*w - 1/2],\ [659, 659, 1/2*w^3 - 7*w + 1/2],\ [701, 701, 3*w^3 - 2*w^2 - 18*w - 8],\ [701, 701, 7/2*w^3 - 3*w^2 - 21*w - 9/2],\ [701, 701, -1/2*w^3 - 2*w^2 + 3*w + 19/2],\ [701, 701, 15/2*w^3 - 5*w^2 - 45*w - 19/2],\ [727, 727, 2*w^3 - 3*w^2 - 12*w + 2],\ [727, 727, w^3 - 2*w^2 - 7*w + 7],\ [727, 727, 1/2*w^3 + w^2 - 6*w - 13/2],\ [727, 727, 3/2*w^3 - 11*w - 1/2],\ [761, 761, 3*w^3 - 4*w^2 - 18*w + 4],\ [761, 761, -w^3 + 2*w^2 + 8*w - 6],\ [761, 761, 2*w^2 - 4*w - 7],\ [761, 761, -2*w^3 + 14*w + 3],\ [769, 769, 5/2*w^3 - 3*w^2 - 15*w + 1/2],\ [769, 769, -1/2*w^3 + w^2 + 5*w - 11/2],\ [769, 769, 3/2*w^3 - 10*w - 1/2],\ [769, 769, -1/2*w^3 + 2*w^2 - 15/2],\ [829, 829, 7/2*w^3 - 4*w^2 - 20*w + 5/2],\ [829, 829, -5/2*w^3 + 4*w^2 + 12*w - 7/2],\ [829, 829, 7/2*w^3 - 3*w^2 - 21*w + 7/2],\ [829, 829, 3/2*w^3 - 3*w^2 - 5*w + 13/2],\ [863, 863, 3*w^3 - 2*w^2 - 19*w + 1],\ [863, 863, w^2 - 4*w - 4],\ [863, 863, -5/2*w^3 + 4*w^2 + 13*w - 9/2],\ [863, 863, 1/2*w^3 + w^2 - 2*w - 13/2],\ [883, 883, -4*w^3 + 4*w^2 + 22*w - 3],\ [883, 883, 3*w^3 - 2*w^2 - 16*w - 2],\ [883, 883, -3*w^3 + 4*w^2 + 14*w - 4],\ [883, 883, 2*w^3 - 2*w^2 - 8*w + 1],\ [919, 919, 1/2*w^3 - w - 11/2],\ [919, 919, -5/2*w^3 + 3*w^2 + 14*w + 1/2],\ [919, 919, w^3 - 2*w^2 - 3*w + 7],\ [919, 919, -2*w^3 + w^2 + 12*w - 2],\ [953, 953, 7/2*w^3 - 3*w^2 - 19*w + 3/2],\ [953, 953, 5/2*w^3 - 2*w^2 - 12*w - 3/2],\ [953, 953, 7/2*w^3 - 4*w^2 - 18*w + 5/2],\ [953, 953, -5/2*w^3 + 3*w^2 + 11*w - 7/2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 2*x - 7 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e + 2, e - 3, e - 3, -e - 1, -e - 1, -6, 0, 0, 0, 0, -2*e + 6, -2*e + 6, 2*e + 2, 2*e + 2, 10, 3*e - 9, -3*e - 3, -3*e - 3, 3*e - 9, 3*e + 3, 3*e + 3, -3*e + 9, -3*e + 9, 2*e - 10, -2*e - 6, -2*e - 6, 2*e - 10, 3*e - 9, -3*e - 3, 3*e - 9, -3*e - 3, 6, 6, 6, 6, 4*e - 6, -4*e + 2, 4*e - 6, -4*e + 2, -6*e + 6, 6*e - 6, -6*e + 6, 6*e - 6, -6*e + 6, -6*e + 6, 6*e - 6, 6*e - 6, 12, 12, 12, 12, -2*e + 18, 2*e + 14, -2*e + 18, 2*e + 14, -18, -18, -18, -18, 2*e + 2, 2*e + 2, -2*e + 6, -2*e + 6, -6, -6, -6, -6, 2, 2, -e + 23, e + 21, -e + 23, e + 21, 4*e - 18, 4*e - 18, -4*e - 10, -4*e - 10, 9*e - 19, -9*e - 1, 9*e - 19, -9*e - 1, -6*e + 18, -6*e + 18, 6*e + 6, 6*e + 6, -2*e + 18, 2*e + 14, 2*e + 14, -2*e + 18, 12*e - 6, -12*e + 18, 12*e - 6, -12*e + 18, -4*e - 16, -4*e - 16, 4*e - 24, 4*e - 24, -3*e - 15, 3*e - 21, -3*e - 15, 3*e - 21, -9*e + 11, 9*e - 7, -9*e + 11, 9*e - 7, -6*e + 30, -6*e + 30, 6*e + 18, 6*e + 18, 8*e + 14, 8*e + 14, -8*e + 30, -8*e + 30, 50, -6*e + 30, -6*e + 30, 6*e + 18, 6*e + 18, 12*e - 24, 12*e - 24, -12*e, -12*e, 3*e - 21, -3*e - 15, -3*e - 15, 3*e - 21, -8*e, -8*e, 8*e - 16, 8*e - 16, 9*e - 15, 9*e - 15, -9*e + 3, -9*e + 3, 5*e - 19, 5*e - 19, -5*e - 9, -5*e - 9, 8*e - 10, -8*e + 6, 8*e - 10, -8*e + 6, -6*e + 6, -6*e + 6, 6*e - 6, 6*e - 6, 20, 20, 20, 20, -4*e + 44, -4*e + 44, 4*e + 36, 4*e + 36, -15*e + 9, 15*e - 21, 15*e - 21, -15*e + 9] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]