/* 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([11, 0, -7, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([20,10,-w^2 + w + 5]) primes_array = [ [4, 2, -w^2 + w + 3],\ [5, 5, w + 1],\ [5, 5, -w^3 + w^2 + 4*w - 4],\ [11, 11, w],\ [29, 29, w^3 - 2*w^2 - 3*w + 7],\ [29, 29, -w^3 - 2*w^2 + 3*w + 7],\ [31, 31, -w^3 + w^2 + 4*w - 2],\ [31, 31, -w^3 - w^2 + 4*w + 2],\ [41, 41, w^3 + 2*w^2 - 4*w - 6],\ [41, 41, w^3 - 5*w + 2],\ [49, 7, -w^3 + w^2 + 4*w - 1],\ [49, 7, w^3 + w^2 - 4*w - 1],\ [59, 59, -3*w^2 - w + 10],\ [59, 59, -3*w^2 + w + 10],\ [61, 61, -2*w^3 + 2*w^2 + 7*w - 5],\ [61, 61, 2*w^3 + w^2 - 8*w - 6],\ [71, 71, 2*w^2 + w - 9],\ [71, 71, 2*w^2 - w - 9],\ [81, 3, -3],\ [101, 101, 2*w^2 - w - 10],\ [101, 101, 2*w^2 + w - 10],\ [109, 109, 2*w^3 + w^2 - 8*w - 3],\ [109, 109, -2*w^3 + w^2 + 8*w - 3],\ [121, 11, -w^2 + 7],\ [131, 131, 4*w^2 - 2*w - 15],\ [131, 131, 2*w^2 + 2*w - 3],\ [131, 131, 2*w^2 - 2*w - 3],\ [131, 131, -w^3 - w^2 + 3*w + 7],\ [139, 139, w^3 + 4*w^2 - 5*w - 16],\ [139, 139, w^3 + 3*w^2 - 6*w - 10],\ [139, 139, 2*w^3 - w^2 - 8*w + 1],\ [139, 139, -4*w^2 - w + 12],\ [149, 149, 2*w^3 - w^2 - 8*w + 4],\ [149, 149, 2*w^3 + w^2 - 8*w - 4],\ [151, 151, 3*w^2 - 3*w - 8],\ [151, 151, 2*w^3 - 4*w^2 - 8*w + 17],\ [151, 151, w^2 - 2*w - 7],\ [151, 151, w^3 + 2*w^2 - 6*w - 8],\ [179, 179, w^3 - w^2 - 5*w + 1],\ [179, 179, -w^3 - w^2 + 5*w + 1],\ [191, 191, 2*w^3 - w^2 - 7*w + 2],\ [191, 191, 3*w^3 - 2*w^2 - 10*w + 2],\ [199, 199, w^3 - 4*w^2 - 3*w + 13],\ [199, 199, -w^3 - 4*w^2 + 3*w + 13],\ [211, 211, -w^3 + w^2 + 6*w - 5],\ [211, 211, w^3 - 5*w - 5],\ [211, 211, -w^3 + 5*w - 5],\ [211, 211, w^3 + w^2 - 6*w - 5],\ [229, 229, w^3 - 4*w^2 - 2*w + 14],\ [229, 229, 2*w^3 + 3*w^2 - 7*w - 12],\ [229, 229, 2*w^3 - 3*w^2 - 7*w + 12],\ [229, 229, 2*w^3 + w^2 - 7*w - 8],\ [241, 241, w^3 + w^2 - 6*w - 4],\ [241, 241, -w^3 + w^2 + 6*w - 4],\ [251, 251, 2*w^2 - w - 2],\ [251, 251, 2*w^2 + w - 2],\ [271, 271, 2*w^3 - 2*w^2 - 8*w + 5],\ [271, 271, -4*w^2 - w + 14],\ [271, 271, -4*w^2 + w + 14],\ [271, 271, 2*w^3 + 2*w^2 - 8*w - 5],\ [281, 281, -2*w^3 - w^2 + 9*w + 2],\ [281, 281, 2*w^3 - w^2 - 9*w + 2],\ [311, 311, -w^3 + 3*w - 5],\ [311, 311, w^3 - 3*w - 5],\ [331, 331, -3*w^3 + 3*w^2 + 12*w - 13],\ [331, 331, 3*w^3 + 3*w^2 - 12*w - 13],\ [349, 349, 2*w^3 + 2*w^2 - 7*w - 10],\ [349, 349, -2*w^3 + 2*w^2 + 7*w - 10],\ [361, 19, 4*w^2 - 15],\ [361, 19, -4*w^2 + 13],\ [379, 379, -w^3 + 3*w^2 + 3*w - 13],\ [379, 379, w^3 + 3*w^2 - 3*w - 13],\ [409, 409, 3*w^3 - 2*w^2 - 11*w + 2],\ [409, 409, w^3 + 3*w^2 - 6*w - 9],\ [419, 419, -w^3 + w^2 + 6*w + 2],\ [419, 419, -3*w^3 + 5*w^2 + 13*w - 19],\ [421, 421, -w^3 + 3*w^2 + 7*w - 13],\ [421, 421, 2*w^3 - 2*w^2 - 9*w + 6],\ [421, 421, -2*w^3 - 2*w^2 + 9*w + 6],\ [421, 421, w^3 + 3*w^2 - 7*w - 13],\ [439, 439, 3*w^2 + 2*w - 13],\ [439, 439, 3*w^3 + 2*w^2 - 12*w - 10],\ [439, 439, -3*w^3 + 2*w^2 + 12*w - 10],\ [439, 439, 3*w^2 - 2*w - 13],\ [449, 449, 2*w^3 - 4*w^2 - 9*w + 14],\ [449, 449, -3*w^3 + 3*w^2 + 11*w - 9],\ [449, 449, 3*w^3 + 3*w^2 - 11*w - 9],\ [449, 449, w^3 - 2*w^2 - 5*w + 2],\ [461, 461, -w - 5],\ [461, 461, w - 5],\ [499, 499, w^3 - 5*w^2 - w + 17],\ [499, 499, -w^3 - 5*w^2 + w + 17],\ [509, 509, w^3 + 5*w^2 - 4*w - 16],\ [509, 509, w^2 - 2*w - 10],\ [509, 509, w^2 + 2*w - 10],\ [509, 509, w^3 - w^2 - 5*w + 9],\ [541, 541, 3*w^3 - 10*w - 6],\ [541, 541, -2*w^3 + 5*w^2 + 5*w - 16],\ [569, 569, 3*w^3 + 2*w^2 - 13*w - 6],\ [569, 569, -2*w^3 + 4*w^2 + 7*w - 10],\ [599, 599, w^3 + 2*w^2 - 7*w - 10],\ [599, 599, -w^3 + 2*w^2 + 7*w - 10],\ [601, 601, -3*w^3 - w^2 + 12*w + 6],\ [601, 601, 3*w^3 - w^2 - 12*w + 6],\ [619, 619, 3*w^3 - 12*w + 2],\ [619, 619, -3*w^3 + 12*w + 2],\ [631, 631, -w^3 + w^2 + 2*w - 8],\ [631, 631, w^3 + w^2 - 2*w - 8],\ [641, 641, w^3 + w^2 - 7*w - 1],\ [641, 641, -3*w^3 - 2*w^2 + 12*w + 4],\ [641, 641, 3*w^3 - 2*w^2 - 12*w + 4],\ [641, 641, w^3 - w^2 - 7*w + 1],\ [659, 659, -2*w^3 + w^2 + 10*w - 1],\ [659, 659, -w^3 + 5*w^2 + 4*w - 19],\ [659, 659, w^3 + 5*w^2 - 4*w - 19],\ [659, 659, 2*w^3 + w^2 - 10*w - 1],\ [691, 691, 3*w^2 - 2*w - 14],\ [691, 691, 3*w^2 + 2*w - 14],\ [701, 701, w^3 + 4*w^2 - 7*w - 12],\ [701, 701, 3*w^3 - w^2 - 12*w - 1],\ [719, 719, -w^3 + 7*w - 3],\ [719, 719, w^3 - 7*w - 3],\ [739, 739, 3*w^3 + w^2 - 12*w - 3],\ [739, 739, w^3 + 2*w^2 - 3*w - 1],\ [739, 739, -w^3 + 2*w^2 + 3*w - 1],\ [739, 739, -3*w^3 + w^2 + 12*w - 3],\ [751, 751, -2*w^3 + 6*w^2 + 7*w - 20],\ [751, 751, 2*w^3 + 6*w^2 - 7*w - 20],\ [761, 761, 2*w^3 + 6*w^2 - 8*w - 21],\ [761, 761, -2*w^3 + 6*w^2 + 8*w - 21],\ [769, 769, -2*w^3 + 6*w^2 + 8*w - 23],\ [769, 769, w^2 + 2*w + 3],\ [809, 809, 2*w^3 + 3*w^2 - 7*w - 14],\ [809, 809, -2*w^3 + 3*w^2 + 7*w - 14],\ [821, 821, -3*w^3 - w^2 + 11*w + 1],\ [821, 821, 3*w^3 - w^2 - 11*w + 1],\ [829, 829, 3*w^3 - w^2 - 12*w + 4],\ [829, 829, w^3 - 4*w^2 - 6*w + 12],\ [829, 829, -w^3 - 4*w^2 + 6*w + 12],\ [829, 829, -3*w^3 - w^2 + 12*w + 4],\ [839, 839, -w^3 + 3*w - 6],\ [839, 839, w^3 - 3*w - 6],\ [841, 29, 5*w^2 - 19],\ [859, 859, w^3 + 2*w^2 - 7*w - 9],\ [859, 859, -w^3 + 2*w^2 + 7*w - 9],\ [911, 911, -2*w^3 + 4*w^2 + 9*w - 13],\ [911, 911, 2*w^3 + 4*w^2 - 9*w - 13],\ [919, 919, 2*w^3 + 4*w^2 - 7*w - 17],\ [919, 919, 3*w^3 + 2*w^2 - 11*w - 6],\ [919, 919, 3*w^3 - 2*w^2 - 11*w + 6],\ [919, 919, 2*w^3 - 4*w^2 - 7*w + 17],\ [941, 941, -2*w^3 + 2*w^2 + 7*w - 15],\ [941, 941, 2*w^3 + 2*w^2 - 7*w - 15],\ [961, 31, -2*w^2 + 13],\ [971, 971, w^3 + 2*w^2 - 7*w - 7],\ [971, 971, -w^3 + 2*w^2 + 7*w - 7],\ [991, 991, 3*w^3 + 2*w^2 - 11*w - 7],\ [991, 991, -3*w^3 + 2*w^2 + 11*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + x - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, e, 1, -e + 1, -3*e - 3, e + 7, 2*e + 3, -2, 8, 2*e - 4, -1, -2*e - 6, e + 3, -e - 2, -3*e - 5, -e, 4, 2*e + 9, -2*e - 8, -e + 6, 3*e - 1, -e - 3, -2*e - 14, 3, 5*e, -4, -3*e - 3, -e + 2, e - 19, -5*e, -e - 7, 4, e - 9, e - 9, e - 7, -4*e + 6, -3*e, -2*e - 6, e - 13, 5*e + 14, e + 16, 2*e + 10, 5*e, 6*e + 11, 6*e + 6, -5*e - 13, 9*e + 5, -9*e - 6, 3*e - 9, 2*e + 14, 7*e + 1, -9*e - 5, 4, 2*e - 8, e - 9, 3*e + 13, 5*e + 4, -2*e + 12, 17, -2*e + 12, 2*e - 19, 6*e - 9, -4*e - 4, -4*e + 13, -3*e + 10, -4*e + 16, -6*e + 12, -9*e - 4, 6*e - 14, 8*e + 8, -2*e - 16, -2*e + 18, 2, -3*e - 14, e + 23, -9*e - 2, -5*e + 10, 12*e + 10, 3*e + 13, e - 9, -5*e + 11, -6*e, 8*e + 18, -3*e + 16, -4*e - 2, -8*e - 12, -8*e + 5, -10*e - 17, e - 20, -2*e - 2, 7*e + 16, -4*e - 20, 5*e + 4, e - 6, -5*e - 4, 5*e + 4, -e - 30, -7*e - 11, e + 3, -42, 2*e + 44, -12, -e + 13, -10, -e + 14, 12*e + 4, -e - 25, -10*e + 12, 8*e - 18, 8*e - 18, 30, -5*e + 26, -6*e - 18, 2*e - 32, -9*e, -e - 31, 13*e + 19, -4*e - 32, -9*e - 9, 3*e + 4, 2*e + 11, 7*e + 15, 3*e + 8, -4*e + 16, 9*e - 11, 2*e - 20, 12*e, -10*e - 4, 6*e - 22, -20, e - 1, 12*e + 18, 5*e - 2, -6*e + 30, -7*e - 3, -4*e + 30, -17*e - 3, 13*e - 13, 3*e - 21, 12*e + 10, 6*e - 12, 10*e - 2, 9, 2*e - 36, -7*e - 16, -8*e + 8, -9*e - 20, 12*e - 2, -13*e - 5, 2*e - 27, 16*e + 8, 5*e - 23, 3*e + 6, 2*e - 36, -17*e + 3, 11*e + 5, 3*e - 12, -6*e + 8] 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^2 - w + 3])] = -1 AL_eigenvalues[ZF.ideal([5,5,-w + 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]