/* 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, 4, -4, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([61, 61, 4*w^3 + w^2 - 13*w - 1]) primes_array = [ [5, 5, -w^2 + 1],\ [9, 3, w^3 + w^2 - 4*w - 3],\ [16, 2, 2],\ [29, 29, -w^3 - w^2 + 2*w + 3],\ [29, 29, -w^2 + w + 3],\ [29, 29, w^3 - w^2 - 4*w + 2],\ [29, 29, 2*w^3 + w^2 - 7*w],\ [31, 31, -2*w + 1],\ [31, 31, 2*w^2 - 5],\ [31, 31, 2*w^3 + 2*w^2 - 6*w - 3],\ [31, 31, 2*w^3 - 8*w + 1],\ [59, 59, w^3 + w^2 - 2*w - 5],\ [59, 59, -w^3 + 2*w^2 + 4*w - 5],\ [59, 59, -3*w^3 + 10*w - 4],\ [59, 59, -2*w^3 - w^2 + 7*w - 2],\ [61, 61, 4*w^3 + w^2 - 13*w - 1],\ [61, 61, 2*w^3 - w^2 - 5*w + 2],\ [61, 61, -3*w^3 - w^2 + 8*w],\ [61, 61, 3*w^3 - w^2 - 10*w + 5],\ [89, 89, w^3 + w^2 - w - 4],\ [89, 89, 2*w^2 - w - 6],\ [89, 89, -3*w^3 - 2*w^2 + 10*w + 1],\ [89, 89, -2*w^3 + w^2 + 8*w - 2],\ [121, 11, -w^3 + 3*w + 3],\ [121, 11, 3*w^3 - 9*w + 1],\ [149, 149, 3*w^3 + 2*w^2 - 9*w - 5],\ [149, 149, w^3 + 2*w^2 - 2*w - 7],\ [149, 149, w^3 + w^2 - 5*w - 4],\ [149, 149, -3*w^3 + 11*w - 4],\ [151, 151, 3*w^3 - w^2 - 9*w + 3],\ [151, 151, 3*w^3 - 8*w + 2],\ [151, 151, 4*w^3 + w^2 - 12*w - 1],\ [151, 151, -4*w^3 + 13*w - 3],\ [179, 179, -4*w^3 - w^2 + 14*w - 1],\ [179, 179, 3*w^3 + 2*w^2 - 8*w - 4],\ [179, 179, -w^3 + w^2 + w - 3],\ [179, 179, -2*w^3 + 2*w^2 + 7*w - 5],\ [181, 181, 4*w^3 + w^2 - 12*w],\ [181, 181, -3*w^3 + 8*w - 1],\ [181, 181, 4*w^3 - 13*w + 2],\ [181, 181, -3*w^3 + w^2 + 9*w - 4],\ [211, 211, 2*w^3 + 3*w^2 - 6*w - 6],\ [211, 211, -w^3 + 6*w - 2],\ [211, 211, 2*w^3 - 9*w + 2],\ [211, 211, 2*w^3 + 3*w^2 - 6*w - 5],\ [239, 239, w^3 + 3*w^2 - 2*w - 10],\ [239, 239, 2*w^3 + w^2 - 9*w - 4],\ [239, 239, 2*w^3 + w^2 - 9*w + 3],\ [239, 239, -w^3 - 3*w^2 + 2*w + 3],\ [241, 241, 5*w^3 + w^2 - 16*w - 1],\ [241, 241, -2*w^3 - 4*w^2 + 5*w + 9],\ [241, 241, 3*w^3 + w^2 - 13*w + 1],\ [241, 241, 5*w^3 - 16*w],\ [269, 269, -3*w^3 - 2*w^2 + 7*w + 4],\ [269, 269, -2*w^3 - 3*w^2 + 5*w + 8],\ [269, 269, -3*w^3 - 2*w^2 + 9*w + 7],\ [269, 269, 5*w^3 + 2*w^2 - 17*w - 1],\ [271, 271, 5*w^3 + w^2 - 16*w],\ [271, 271, -2*w^3 - w^2 + 6*w + 6],\ [271, 271, 4*w^3 + w^2 - 11*w],\ [271, 271, 3*w^3 - w^2 - 8*w + 3],\ [331, 331, -w^3 + 2*w^2 + 6*w - 6],\ [331, 331, -2*w^3 - 3*w^2 + 7*w + 5],\ [331, 331, 2*w^3 + 3*w^2 - 8*w - 5],\ [331, 331, 3*w^3 + 3*w^2 - 11*w - 5],\ [359, 359, 2*w^3 + 2*w^2 - 9*w - 5],\ [359, 359, 3*w^3 + 2*w^2 - 12*w],\ [359, 359, w^3 + 3*w^2 - w - 9],\ [359, 359, 3*w^2 + 2*w - 5],\ [361, 19, -w^3 + 3*w + 4],\ [361, 19, 4*w^3 - 12*w + 1],\ [389, 389, w^3 - 3*w^2 - 3*w + 5],\ [389, 389, 2*w^3 + 2*w^2 - 5*w - 7],\ [389, 389, 4*w^3 + 3*w^2 - 12*w - 7],\ [389, 389, -4*w^3 + 15*w - 5],\ [419, 419, -3*w^3 - 3*w^2 + 8*w + 7],\ [419, 419, -4*w^3 - 2*w^2 + 13*w - 1],\ [419, 419, -4*w^3 - w^2 + 15*w - 2],\ [419, 419, w^3 - 3*w^2 - 4*w + 6],\ [421, 421, w^2 - 7],\ [421, 421, -w^3 + 4*w + 4],\ [421, 421, 4*w^3 - 11*w],\ [421, 421, 4*w^3 - w^2 - 12*w + 6],\ [449, 449, 3*w^3 + 2*w^2 - 11*w + 1],\ [449, 449, -w^3 - 2*w^2 + 5*w + 7],\ [449, 449, -w^3 + 2*w^2 + 5*w - 2],\ [449, 449, w^3 + 2*w^2 - w - 8],\ [479, 479, 2*w^3 + 3*w^2 - 5*w - 9],\ [479, 479, -3*w^2 - w + 4],\ [479, 479, -3*w^3 - w^2 + 12*w - 3],\ [479, 479, w^3 + w^2 - 6*w - 4],\ [509, 509, 4*w^3 + 3*w^2 - 11*w - 6],\ [509, 509, -w^3 + w^2 - 3],\ [509, 509, -2*w^3 + 3*w^2 + 7*w - 7],\ [509, 509, -5*w^3 - w^2 + 18*w - 2],\ [541, 541, -4*w^3 + 11*w - 2],\ [541, 541, -4*w^3 + w^2 + 12*w - 4],\ [541, 541, 5*w^3 - 16*w + 3],\ [541, 541, 5*w^3 + w^2 - 15*w],\ [569, 569, -4*w^3 - w^2 + 14*w - 3],\ [569, 569, 4*w^3 + 2*w^2 - 15*w],\ [569, 569, w^3 - 3*w^2 - 5*w + 6],\ [569, 569, 2*w^3 + 3*w^2 - 4*w - 8],\ [571, 571, -4*w^3 - w^2 + 17*w + 2],\ [571, 571, 3*w^3 + 4*w^2 - 10*w - 10],\ [571, 571, w^3 + 5*w^2 - 2*w - 14],\ [571, 571, -6*w^3 + 20*w - 3],\ [599, 599, -2*w^3 + w^2 + 4*w - 5],\ [599, 599, -4*w^3 - 2*w^2 + 11*w + 5],\ [599, 599, 3*w^3 - 2*w^2 - 10*w + 4],\ [599, 599, -5*w^3 - w^2 + 17*w - 3],\ [601, 601, w^3 - 7*w + 1],\ [601, 601, 3*w^3 + 4*w^2 - 9*w - 8],\ [601, 601, 3*w^3 - 13*w + 3],\ [601, 601, w^3 + 4*w^2 - 3*w - 8],\ [631, 631, 6*w^3 + w^2 - 19*w - 1],\ [631, 631, -3*w^3 + w^2 + 13*w - 3],\ [631, 631, 5*w^3 + w^2 - 14*w + 2],\ [631, 631, -5*w^3 + w^2 + 16*w - 7],\ [659, 659, -6*w^3 - 2*w^2 + 19*w - 2],\ [659, 659, -5*w^3 + w^2 + 17*w - 2],\ [659, 659, -4*w^3 + 15*w - 6],\ [659, 659, -4*w^3 - 3*w^2 + 12*w + 8],\ [661, 661, -5*w^3 + 2*w^2 + 16*w - 7],\ [661, 661, 6*w^3 + 2*w^2 - 17*w - 2],\ [661, 661, -7*w^3 - w^2 + 23*w - 2],\ [661, 661, -4*w^3 + w^2 + 10*w - 4],\ [691, 691, -6*w^3 + 19*w - 6],\ [691, 691, -5*w^3 + w^2 + 15*w - 2],\ [691, 691, 5*w^3 - 14*w + 5],\ [691, 691, 2*w^3 + 4*w^2 - 7*w - 10],\ [719, 719, -7*w^3 - 2*w^2 + 24*w],\ [719, 719, -2*w^3 + 2*w^2 + 3*w - 5],\ [719, 719, -5*w^3 - 3*w^2 + 13*w + 5],\ [719, 719, -4*w^3 + 3*w^2 + 14*w - 9],\ [751, 751, 4*w^3 + 4*w^2 - 14*w - 5],\ [751, 751, 5*w^3 + 4*w^2 - 17*w - 8],\ [751, 751, -3*w^3 + 2*w^2 + 13*w - 9],\ [751, 751, 6*w^3 + w^2 - 19*w],\ [809, 809, w^3 + 3*w^2 - w - 11],\ [809, 809, -5*w^3 + 18*w - 6],\ [809, 809, -6*w^3 - w^2 + 20*w - 4],\ [809, 809, 5*w^3 + 3*w^2 - 15*w - 7],\ [811, 811, 5*w^3 + 4*w^2 - 18*w - 6],\ [811, 811, 7*w^3 + 2*w^2 - 22*w - 1],\ [811, 811, 4*w^3 - 2*w^2 - 11*w + 6],\ [811, 811, 5*w^3 + w^2 - 13*w],\ [839, 839, -4*w^3 - 4*w^2 + 11*w + 9],\ [839, 839, 5*w^3 + w^2 - 19*w + 3],\ [839, 839, -6*w^3 - 2*w^2 + 21*w],\ [839, 839, -5*w^3 - 3*w^2 + 16*w],\ [929, 929, 5*w^3 + w^2 - 17*w + 4],\ [929, 929, 3*w^3 - 2*w^2 - 10*w + 3],\ [929, 929, 4*w^3 + 2*w^2 - 11*w - 6],\ [929, 929, 4*w^3 + 2*w^2 - 15*w + 1],\ [991, 991, 3*w^3 + w^2 - 14*w + 1],\ [991, 991, 2*w^3 + 5*w^2 - 5*w - 11],\ [991, 991, 2*w^3 + 5*w^2 - 5*w - 10],\ [991, 991, 3*w^2 + 4*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 4*x + 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e - 4, -2*e + 5, -4*e + 8, -4*e + 8, 3*e - 12, -4*e + 8, 4*e - 6, -4*e + 8, 4*e - 6, -4*e + 8, -e - 4, -8*e + 16, e + 8, e + 8, -1, 8*e - 14, 0, -8*e + 14, 11*e - 24, -5*e + 4, 2*e - 16, -3*e + 16, 8*e - 16, -2, -6*e, e - 20, -6*e, 5*e + 4, -4*e + 4, 4*e - 10, 4*e - 10, 4*e - 10, 8*e - 8, 15*e - 28, -e, 8*e - 8, -8*e + 10, -8*e + 10, 8*e - 18, 8*e - 18, 4*e - 12, 4*e - 12, -4*e + 2, 12*e - 26, -9*e + 16, 7*e - 12, 5*e - 24, 7*e - 12, -4*e + 32, -6, -6, 16*e - 34, e - 16, 5*e + 8, -4*e + 16, -4*e + 16, 24, -4*e, -4*e, 4*e - 14, -12*e + 12, -8*e + 36, -8*e + 36, -8*e + 36, -2*e, -2*e, 9*e + 4, 2*e + 24, -10, -10, -20*e + 48, e - 12, -6*e + 8, 8*e - 32, 8*e, -6*e + 40, 22*e - 40, -3*e - 4, -8*e + 2, 4*e + 12, -12, -8*e + 2, -14*e + 24, 4*e + 8, 9*e - 24, 9*e - 24, 21*e - 44, -4*e - 8, -7*e + 36, 16, -e - 20, 15*e - 48, -8*e, 5*e + 16, -8*e - 2, -16*e + 12, -8*e - 2, 24*e - 58, 16*e - 40, 7*e - 32, 7*e - 32, -12*e + 40, 4*e - 24, 20*e - 52, -8*e + 28, -8*e + 28, -6*e - 16, -11*e + 16, 19*e - 52, 5*e - 12, 16*e - 46, -4*e + 20, 16*e - 46, -20*e + 48, 4*e + 36, 12, -16*e + 40, 12, 22*e - 32, -15*e + 56, -10*e + 24, 15*e - 12, 42, 8*e - 34, 24*e - 62, -8*e - 6, -8*e + 24, -4*e + 48, 4*e - 28, 8*e - 4, 3*e - 20, 24, 14*e - 16, -9*e + 32, 4*e - 30, -4*e - 16, -4*e - 16, -32*e + 64, -5*e + 28, -30*e + 64, -12*e + 48, -3*e + 40, -24*e + 48, -12*e + 58, -4*e - 18, -4*e - 18, 12*e - 24, -25*e + 64, 3*e - 16, -2*e + 16, -14*e + 40, -14*e + 40, -18*e + 16, 0, 4*e + 24, -4*e + 38, 12*e - 52, 4*e + 24] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([61, 61, 4*w^3 + w^2 - 13*w - 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]