/* 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, -3, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([109,109,-w^3 + 3*w^2 - w - 4]) primes_array = [ [11, 11, -w^3 + 2*w^2 + w - 3],\ [11, 11, w^3 - 3*w],\ [16, 2, 2],\ [19, 19, -w^3 + 2*w + 2],\ [19, 19, 2*w^3 - 3*w^2 - 4*w + 2],\ [25, 5, 2*w^3 - 2*w^2 - 4*w + 1],\ [29, 29, w^3 - w^2 - 4*w + 1],\ [31, 31, w^3 - 4*w + 1],\ [31, 31, -w^2 + 2*w + 3],\ [41, 41, 2*w^2 - w - 3],\ [41, 41, -w^3 + 3*w^2 + w - 4],\ [49, 7, 2*w^3 - 3*w^2 - 5*w + 2],\ [49, 7, w^2 + w - 3],\ [61, 61, 2*w^3 - 3*w^2 - 4*w],\ [61, 61, -3*w^3 + 4*w^2 + 7*w - 3],\ [79, 79, 2*w^3 - 4*w^2 - 3*w + 2],\ [79, 79, w^3 + w^2 - 3*w - 5],\ [81, 3, -3],\ [89, 89, -3*w^3 + 4*w^2 + 5*w - 3],\ [89, 89, 3*w^3 - 2*w^2 - 7*w],\ [101, 101, -3*w^3 + 5*w^2 + 7*w - 5],\ [101, 101, -2*w^2 - w + 4],\ [109, 109, 3*w^3 - 5*w^2 - 5*w + 3],\ [109, 109, -2*w^3 + 7*w + 1],\ [109, 109, 4*w^3 - 5*w^2 - 9*w + 3],\ [109, 109, -2*w^3 + 5*w + 4],\ [121, 11, 3*w^3 - 3*w^2 - 6*w + 1],\ [131, 131, -3*w^3 + 2*w^2 + 8*w - 2],\ [131, 131, w^3 + w^2 - 5*w],\ [139, 139, -w^3 + 2*w^2 + 2*w - 6],\ [139, 139, 2*w^3 - 5*w^2 - 2*w + 6],\ [139, 139, 3*w^3 - 5*w^2 - 6*w + 4],\ [139, 139, -2*w^3 + 2*w^2 + 7*w - 2],\ [149, 149, -3*w^3 + 4*w^2 + 7*w - 2],\ [149, 149, w^3 - w^2 - 3*w - 3],\ [149, 149, 3*w^3 - 2*w^2 - 9*w + 2],\ [149, 149, w - 4],\ [169, 13, -4*w^3 + 5*w^2 + 8*w - 4],\ [169, 13, 3*w^3 - 2*w^2 - 6*w],\ [179, 179, -4*w^3 + 3*w^2 + 10*w - 3],\ [179, 179, -4*w^3 + 4*w^2 + 9*w - 3],\ [179, 179, -2*w^3 + w^2 + 8*w - 1],\ [179, 179, 4*w^3 - 3*w^2 - 8*w - 2],\ [191, 191, -w^3 + 4*w^2 - 6],\ [191, 191, -4*w^3 + 4*w^2 + 9*w - 2],\ [211, 211, 2*w^3 - w^2 - 8*w],\ [211, 211, -w^3 + 4*w^2 + w - 5],\ [229, 229, w^3 - 3*w^2 + w + 5],\ [229, 229, 2*w^3 - 7*w],\ [251, 251, 3*w^3 - 4*w^2 - 7*w + 1],\ [251, 251, w^3 - w - 4],\ [269, 269, 3*w^3 - 6*w^2 - 3*w + 7],\ [269, 269, 3*w^3 - 9*w - 2],\ [271, 271, -3*w^3 + w^2 + 10*w],\ [271, 271, 4*w^3 - 5*w^2 - 9*w + 2],\ [281, 281, -3*w^3 + w^2 + 11*w - 1],\ [281, 281, -2*w^2 + 5*w + 4],\ [281, 281, -3*w^3 + 3*w^2 + 10*w - 1],\ [281, 281, 3*w^3 - 4*w^2 - 7*w - 1],\ [311, 311, -w^3 + 7*w + 2],\ [311, 311, 3*w^2 - 7],\ [331, 331, w^3 + 2*w^2 - 3*w - 7],\ [331, 331, w^3 - w^2 - w + 5],\ [359, 359, -4*w^3 + 5*w^2 + 6*w - 4],\ [359, 359, -5*w^3 + 4*w^2 + 12*w - 2],\ [361, 19, -4*w^3 + 4*w^2 + 8*w - 3],\ [379, 379, -3*w^3 + 10*w + 2],\ [379, 379, 2*w^3 - 5*w^2 + 6],\ [389, 389, -5*w^3 + 6*w^2 + 10*w - 3],\ [389, 389, w^3 + 3*w^2 - 6*w - 7],\ [409, 409, -4*w^3 + 4*w^2 + 7*w],\ [409, 409, -4*w^2 + 4*w + 5],\ [421, 421, -2*w^3 + 2*w^2 + 8*w - 3],\ [421, 421, w^3 + 3*w^2 - 6*w - 6],\ [431, 431, 4*w^3 - 6*w^2 - 10*w + 5],\ [431, 431, 3*w^3 - 5*w^2 - 6*w + 2],\ [431, 431, -4*w^3 + 3*w^2 + 8*w - 2],\ [431, 431, -5*w^3 + 6*w^2 + 10*w - 6],\ [449, 449, 4*w^3 - 5*w^2 - 6*w + 2],\ [449, 449, -w^3 + 2*w^2 + w - 7],\ [461, 461, 5*w^3 - 4*w^2 - 12*w + 1],\ [461, 461, w^3 + 2*w^2 - 2*w - 6],\ [479, 479, w^3 + 3*w^2 - 5*w - 7],\ [479, 479, 2*w^3 - 6*w^2 - w + 6],\ [491, 491, 3*w^2 - 5*w - 6],\ [491, 491, -2*w^3 - w^2 + 9*w + 1],\ [499, 499, 4*w^3 - 3*w^2 - 12*w + 4],\ [499, 499, 4*w^3 - 5*w^2 - 10*w + 2],\ [499, 499, -5*w^3 + 5*w^2 + 11*w - 4],\ [499, 499, -4*w^3 + 4*w^2 + 7*w - 3],\ [509, 509, -5*w^3 + 6*w^2 + 10*w - 4],\ [509, 509, -4*w^2 + 3*w + 8],\ [509, 509, -3*w^3 + w^2 + 10*w - 1],\ [509, 509, 5*w^3 - 6*w^2 - 12*w + 3],\ [521, 521, -5*w^3 + 6*w^2 + 10*w - 5],\ [521, 521, 2*w^3 - 6*w^2 - w + 7],\ [521, 521, w^3 - 5*w^2 + 9],\ [521, 521, w^3 + 3*w^2 - 5*w - 6],\ [529, 23, 3*w^3 - 5*w^2 - 6*w + 1],\ [529, 23, -2*w^3 - w^2 + 8*w + 1],\ [541, 541, 3*w^3 - 5*w^2 - 8*w + 3],\ [541, 541, -w^3 + 3*w^2 + 4*w - 7],\ [569, 569, -w^3 + 7*w - 1],\ [569, 569, -3*w^3 + 2*w^2 + 11*w],\ [571, 571, w^3 - 4*w - 6],\ [571, 571, 4*w^3 - 5*w^2 - 11*w + 2],\ [571, 571, -5*w^3 + 5*w^2 + 11*w - 3],\ [571, 571, -2*w^3 + 6*w^2 + 2*w - 9],\ [599, 599, 3*w^3 - 6*w^2 - 6*w + 4],\ [599, 599, 3*w^2 - 8],\ [601, 601, -4*w^3 + 2*w^2 + 12*w - 1],\ [601, 601, -5*w^3 + 7*w^2 + 11*w - 5],\ [619, 619, -w^3 + 5*w^2 - w - 6],\ [619, 619, 4*w^2 - 3*w - 7],\ [631, 631, 2*w^3 - 3*w^2 - 5*w - 3],\ [631, 631, 5*w^3 - 3*w^2 - 13*w + 2],\ [631, 631, 2*w^3 + w^2 - 11*w - 2],\ [631, 631, 6*w^3 - 6*w^2 - 14*w + 3],\ [641, 641, 2*w^2 - w - 9],\ [641, 641, -w^3 + 3*w^2 + 2*w - 10],\ [659, 659, -4*w^3 + w^2 + 14*w + 1],\ [659, 659, 5*w^3 - 6*w^2 - 11*w + 1],\ [709, 709, 6*w^3 - 7*w^2 - 14*w + 3],\ [709, 709, 6*w^3 - 8*w^2 - 15*w + 7],\ [709, 709, 2*w^3 - 5*w^2 - 5*w + 2],\ [709, 709, -4*w^3 + 3*w^2 + 11*w - 5],\ [739, 739, 5*w^3 - 7*w^2 - 7*w + 5],\ [739, 739, -w^3 - 3*w^2 + 6*w + 1],\ [751, 751, 5*w^3 - 3*w^2 - 11*w + 1],\ [751, 751, 5*w^3 - 6*w^2 - 13*w + 3],\ [769, 769, -w^2 + 2*w + 7],\ [769, 769, -4*w^3 + 6*w^2 + 11*w - 4],\ [809, 809, -5*w^3 + 4*w^2 + 13*w - 5],\ [809, 809, 2*w^3 + w^2 - 8*w],\ [811, 811, -w^3 - 4*w^2 + 7*w + 6],\ [811, 811, 3*w^3 - 5*w^2 - 7*w],\ [811, 811, -2*w^3 - w^2 + 9*w],\ [811, 811, -5*w^3 + 5*w^2 + 9*w - 5],\ [821, 821, -w^3 + 3*w^2 + 4*w - 8],\ [821, 821, -3*w^3 + 7*w + 6],\ [821, 821, -3*w^3 + w^2 + 6*w + 5],\ [821, 821, 5*w^3 - 8*w^2 - 9*w + 5],\ [829, 829, -w^3 + w^2 + w - 6],\ [829, 829, 5*w^3 - 6*w^2 - 9*w + 3],\ [839, 839, -4*w^3 + 2*w^2 + 14*w - 3],\ [839, 839, 5*w^3 - 7*w^2 - 7*w + 4],\ [841, 29, -w^2 + 6*w + 1],\ [859, 859, -4*w^3 + 2*w^2 + 13*w - 2],\ [859, 859, 5*w^3 - 6*w^2 - 12*w + 2],\ [881, 881, -2*w^3 + 4*w^2 + w - 9],\ [881, 881, -6*w^3 + 9*w^2 + 12*w - 7],\ [911, 911, 5*w^3 - 6*w^2 - 11*w],\ [911, 911, -w^3 + 4*w^2 + 4*w - 8],\ [929, 929, 5*w^3 - 8*w^2 - 11*w + 7],\ [929, 929, -4*w^3 + 13*w + 3],\ [929, 929, 3*w^3 - 7*w^2 - w + 8],\ [929, 929, -6*w^3 + 5*w^2 + 16*w - 5],\ [941, 941, -4*w^3 + w^2 + 11*w],\ [941, 941, -2*w^3 - w^2 + 10*w - 1],\ [941, 941, -5*w^3 + 5*w^2 + 9*w],\ [941, 941, -5*w^2 + 5*w + 6],\ [961, 31, -5*w^3 + 5*w^2 + 10*w - 3],\ [971, 971, -2*w^3 - 3*w^2 + 8*w + 7],\ [971, 971, 3*w^3 - 8*w^2 - 2*w + 9]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 12 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, e, 1/2*e, -2*e, -1/2*e + 4, -e - 6, 5/2*e, -e, e - 2, -2*e, -e - 1, -e - 4, 6, e + 6, -1/2*e + 2, 2*e + 1, 3/2*e - 4, e - 7, -1/2*e - 14, 2*e, 3*e, -3*e + 6, -3*e + 3, e + 10, -1, -2*e - 9, -3/2*e - 14, -2*e + 2, -7/2*e - 2, -5*e + 2, -3*e, -3, 6*e + 2, e + 6, -3/2*e + 14, 4*e + 3, 5/2*e - 12, 2*e - 8, 7/2*e - 4, e + 3, -9/2*e - 8, e + 3, 3*e + 12, 3*e + 2, 1/2*e - 12, -5*e + 8, e - 9, 6*e + 4, -2*e + 12, 3*e - 15, 7/2*e - 10, -3/2*e + 2, 3*e + 14, -14, -4*e - 10, 4*e - 8, -2*e - 24, 2*e - 6, -4, 4*e, e + 3, 5*e + 8, -4*e + 17, -e - 24, 4*e + 4, -12, 3*e + 14, -2*e + 8, -e + 6, -2*e + 18, 2*e + 1, 6*e + 8, -6*e + 10, 5*e + 10, 7*e - 4, -11/2*e + 14, 7/2*e - 6, 14, 7*e - 8, e + 20, 1/2*e - 28, 2*e + 20, e - 16, -3*e + 32, -4*e + 12, -7*e + 4, -4*e - 2, -3*e + 8, -7*e - 10, e - 18, -6*e - 12, -2*e + 28, 4*e, 4*e + 22, e + 37, e + 4, -4*e - 2, -5*e - 12, -4*e + 6, -7*e + 20, -6*e - 2, -e - 7, -2*e, 4*e + 16, 22, -11/2*e - 22, -e - 32, -17/2*e + 14, -9*e + 4, 7*e - 1, 4*e - 18, -4*e - 10, 5*e - 12, -2*e - 38, 4*e - 32, -8*e + 2, -15/2*e + 18, 2*e + 25, 5*e + 21, 4*e, -7*e - 4, -2*e - 20, -9*e + 4, 4*e + 2, 9*e + 8, 7*e - 1, 7*e + 7, 1/2*e - 36, -6*e - 12, 26, -12*e - 10, -11*e + 11, -2*e + 20, 21/2*e + 2, 12*e - 3, 5/2*e - 10, 3*e + 17, -13/2*e + 10, -e + 31, -5*e - 20, 2*e - 16, -8*e + 16, -e + 6, -5*e + 32, 9*e + 6, -2*e - 16, 8*e - 13, e - 32, -6*e + 8, -1/2*e - 36, 7*e + 28, -e - 33, -e + 22, 4*e - 20, 6*e, -19/2*e + 10, -14*e + 9, 4*e - 30, 7*e + 33, -6*e + 24, -5*e - 32, 5*e + 22, -7*e, 12*e + 14] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([109,109,-w^3 + 3*w^2 - w - 4])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]