/* 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, 10, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([22, 22, w^2 - 3]) primes_array = [ [2, 2, -w + 1],\ [5, 5, w^3 + w^2 - 4*w + 1],\ [7, 7, -w^3 + 6*w - 2],\ [8, 2, w^3 - 7*w + 3],\ [11, 11, -w^3 + 6*w - 4],\ [17, 17, w^3 + w^2 - 6*w - 1],\ [17, 17, -w^2 - w + 3],\ [31, 31, 2*w^3 + w^2 - 13*w + 3],\ [37, 37, -3*w^3 - w^2 + 18*w - 3],\ [41, 41, w^2 + 2*w - 4],\ [47, 47, 2*w^3 + 2*w^2 - 11*w - 2],\ [47, 47, -3*w^3 - w^2 + 19*w - 6],\ [59, 59, -2*w^3 + 13*w - 6],\ [59, 59, -w^3 - w^2 + 5*w + 2],\ [59, 59, w^2 + w - 1],\ [59, 59, w^2 + 2*w - 6],\ [61, 61, -w^3 + w^2 + 8*w - 7],\ [67, 67, w^3 + 2*w^2 - 4*w - 4],\ [73, 73, -2*w^3 - w^2 + 12*w - 4],\ [81, 3, -3],\ [107, 107, 2*w^2 + 2*w - 11],\ [109, 109, 2*w^3 + w^2 - 11*w + 3],\ [113, 113, w^3 + w^2 - 7*w - 2],\ [113, 113, 2*w^3 - 12*w + 9],\ [125, 5, -4*w^3 - 3*w^2 + 24*w + 2],\ [131, 131, 5*w^3 + 3*w^2 - 31*w + 2],\ [151, 151, 4*w^3 + 2*w^2 - 26*w + 1],\ [151, 151, w^3 + 2*w^2 - 6*w - 10],\ [167, 167, w + 4],\ [167, 167, 4*w^3 + w^2 - 26*w + 10],\ [173, 173, -2*w^3 + w^2 + 13*w - 11],\ [179, 179, 4*w^3 + w^2 - 24*w + 4],\ [179, 179, 2*w^3 + 2*w^2 - 12*w - 3],\ [179, 179, -w^3 + 8*w - 4],\ [179, 179, w^3 + 2*w^2 - 4*w - 6],\ [181, 181, 6*w^3 + 3*w^2 - 39*w + 5],\ [197, 197, -w^3 + 4*w - 4],\ [199, 199, 2*w^3 - 12*w + 3],\ [199, 199, 3*w - 2],\ [223, 223, w^3 + w^2 - 7*w - 4],\ [223, 223, 2*w^3 + w^2 - 10*w + 4],\ [227, 227, 7*w^3 + 2*w^2 - 44*w + 12],\ [227, 227, -4*w^3 - w^2 + 25*w - 9],\ [229, 229, 8*w^3 + 4*w^2 - 51*w + 4],\ [229, 229, 3*w^3 + w^2 - 18*w + 5],\ [233, 233, -6*w^3 - 3*w^2 + 39*w - 7],\ [233, 233, -w^3 - 2*w^2 + 8*w + 2],\ [241, 241, 2*w^3 + w^2 - 10*w],\ [251, 251, w^3 + w^2 - 7*w - 6],\ [257, 257, -6*w^3 - w^2 + 39*w - 17],\ [263, 263, -2*w^3 + 13*w - 4],\ [263, 263, -w^3 + 6*w - 8],\ [269, 269, -8*w^3 - 4*w^2 + 50*w - 7],\ [269, 269, -w^3 + w^2 + 7*w - 6],\ [277, 277, 2*w^3 + 2*w^2 - 11*w + 4],\ [277, 277, w^3 + w^2 - 4*w - 3],\ [281, 281, -w^2 - 3*w + 5],\ [281, 281, 2*w^3 + 3*w^2 - 9*w - 3],\ [283, 283, -3*w^3 - 4*w^2 + 12*w + 2],\ [289, 17, -2*w^3 - w^2 + 13*w - 5],\ [293, 293, -w^3 - 2*w^2 + 6*w + 6],\ [293, 293, w^3 + w^2 - 8*w - 1],\ [307, 307, 2*w^3 - 11*w + 12],\ [317, 317, 2*w^3 - 14*w + 11],\ [317, 317, -2*w^3 - 2*w^2 + 8*w + 1],\ [337, 337, 2*w^2 + 4*w - 11],\ [343, 7, -5*w^3 - w^2 + 31*w - 10],\ [349, 349, 2*w^3 - 11*w + 8],\ [349, 349, w^2 + w - 9],\ [353, 353, 4*w^3 + 4*w^2 - 23*w - 8],\ [373, 373, -2*w - 3],\ [373, 373, -4*w^3 + 26*w - 13],\ [389, 389, 6*w^3 + 3*w^2 - 37*w + 1],\ [389, 389, -2*w^3 - w^2 + 14*w - 6],\ [401, 401, 2*w^3 - 12*w + 13],\ [421, 421, 2*w^2 + 2*w - 15],\ [421, 421, 3*w^3 + 2*w^2 - 20*w + 4],\ [431, 431, w^3 + 3*w^2 - 2*w - 9],\ [433, 433, 2*w^3 + 2*w^2 - 12*w + 3],\ [433, 433, -7*w^3 - 2*w^2 + 44*w - 16],\ [433, 433, -3*w^3 + 18*w - 14],\ [433, 433, -w^3 + w^2 + 9*w - 10],\ [439, 439, 5*w^3 + 3*w^2 - 31*w - 2],\ [443, 443, -w^3 - w^2 + 3*w - 4],\ [443, 443, -4*w^3 + 27*w - 14],\ [449, 449, w^3 + w^2 - 4*w - 5],\ [449, 449, w^3 + 3*w^2 - 3*w - 8],\ [449, 449, 2*w^3 - 11*w + 6],\ [449, 449, -3*w^3 - 2*w^2 + 18*w - 4],\ [461, 461, -3*w^3 + w^2 + 20*w - 15],\ [463, 463, 2*w^3 - 10*w + 7],\ [479, 479, -2*w^3 - 3*w^2 + 11*w + 9],\ [491, 491, w^3 + w^2 - 4*w + 5],\ [499, 499, -2*w^3 - 2*w^2 + 14*w - 3],\ [499, 499, -6*w^3 - 2*w^2 + 39*w - 10],\ [499, 499, 4*w^3 + w^2 - 24*w + 14],\ [499, 499, -5*w^3 - 3*w^2 + 30*w - 5],\ [503, 503, -2*w^3 + w^2 + 14*w - 18],\ [509, 509, 4*w^3 + 3*w^2 - 24*w],\ [509, 509, 4*w^3 + 2*w^2 - 23*w + 4],\ [547, 547, w^3 + 2*w^2 - 2*w - 6],\ [547, 547, 6*w^3 + 2*w^2 - 38*w + 7],\ [563, 563, -5*w^3 - w^2 + 33*w - 12],\ [569, 569, 2*w^3 - w^2 - 15*w + 13],\ [569, 569, -4*w^3 - 2*w^2 + 24*w - 7],\ [577, 577, -w^2 - 2],\ [577, 577, 3*w^3 - 20*w + 8],\ [587, 587, 6*w^3 + 4*w^2 - 36*w - 1],\ [587, 587, -w^3 + w^2 + 9*w - 8],\ [593, 593, 6*w^3 + 3*w^2 - 36*w + 2],\ [593, 593, -3*w^3 - w^2 + 21*w - 4],\ [599, 599, -2*w^3 - w^2 + 12*w - 10],\ [599, 599, -4*w^3 - 2*w^2 + 25*w - 6],\ [601, 601, -2*w^3 + 2*w^2 + 15*w - 16],\ [607, 607, w^3 - w^2 - 7*w + 4],\ [607, 607, -2*w^3 - 2*w^2 + 10*w + 3],\ [631, 631, 2*w^2 + 4*w - 7],\ [641, 641, -3*w^3 - 2*w^2 + 16*w],\ [643, 643, w^3 + w^2 - 3*w - 4],\ [643, 643, w^3 + 3*w^2 - 6*w - 13],\ [653, 653, -w^3 + 6*w + 2],\ [653, 653, -3*w^2 - 5*w + 11],\ [659, 659, -2*w^3 + 10*w - 5],\ [661, 661, w^3 - w^2 - 8*w + 3],\ [673, 673, 3*w^3 + 4*w^2 - 14*w - 2],\ [677, 677, 4*w^3 + 2*w^2 - 23*w + 6],\ [677, 677, 2*w^3 - 13*w + 14],\ [683, 683, 2*w^3 + w^2 - 13*w + 7],\ [701, 701, 8*w^3 + 4*w^2 - 51*w + 8],\ [701, 701, -4*w^3 - w^2 + 24*w - 12],\ [709, 709, -4*w^3 - w^2 + 27*w - 7],\ [709, 709, 3*w^3 + 3*w^2 - 18*w - 5],\ [709, 709, -3*w^3 + w^2 + 21*w - 14],\ [709, 709, w^2 - w - 7],\ [719, 719, -9*w^3 - 4*w^2 + 58*w - 8],\ [719, 719, 11*w^3 + 5*w^2 - 68*w + 9],\ [727, 727, -w^3 + 4*w - 6],\ [733, 733, 2*w^3 + w^2 - 13*w + 9],\ [739, 739, -2*w^2 - 3*w + 4],\ [743, 743, 3*w^3 + 3*w^2 - 19*w - 4],\ [751, 751, 2*w^3 + 3*w^2 - 10*w - 2],\ [761, 761, -12*w^3 - 5*w^2 + 75*w - 9],\ [761, 761, -3*w^3 - 3*w^2 + 17*w],\ [769, 769, -6*w^3 - 4*w^2 + 37*w],\ [769, 769, -8*w^3 - 3*w^2 + 51*w - 7],\ [773, 773, -w^2 - 2*w - 2],\ [787, 787, -5*w^3 - 5*w^2 + 28*w + 7],\ [811, 811, 7*w^3 + 2*w^2 - 42*w + 6],\ [811, 811, -2*w^3 - w^2 + 9*w - 7],\ [823, 823, -3*w^3 - 2*w^2 + 14*w - 10],\ [829, 829, 4*w^3 + 3*w^2 - 20*w + 6],\ [829, 829, -7*w^3 - 2*w^2 + 44*w - 14],\ [841, 29, w^3 + 4*w^2 - 2*w - 14],\ [841, 29, -w^2 + 10],\ [857, 857, -3*w^3 - w^2 + 20*w - 9],\ [859, 859, -w^3 + 2*w^2 + 10*w - 18],\ [863, 863, 11*w^3 + 5*w^2 - 69*w + 6],\ [863, 863, w^3 - 2*w^2 - 10*w + 12],\ [877, 877, 3*w^3 - 18*w + 10],\ [883, 883, 4*w^3 + 3*w^2 - 22*w],\ [887, 887, 3*w^3 + 3*w^2 - 15*w - 2],\ [887, 887, w^2 + 4*w - 10],\ [887, 887, w^3 + w^2 - 3*w - 8],\ [887, 887, w - 6],\ [907, 907, 6*w^3 + w^2 - 38*w + 16],\ [919, 919, -2*w^3 - 3*w^2 + 11*w + 1],\ [919, 919, 5*w^3 - w^2 - 34*w + 27],\ [937, 937, -w^3 - w^2 + 8*w + 3],\ [941, 941, -4*w^3 + 25*w - 20],\ [947, 947, -2*w^3 + w^2 + 8*w - 6],\ [967, 967, -7*w^3 - w^2 + 44*w - 23],\ [971, 971, 2*w^2 + 2*w - 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 4*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -e - 2, e, 1, 2*e + 1, 2*e + 5, 2*e + 2, -4, -2*e, -2*e + 4, 2*e + 13, e + 8, -4*e - 14, 2*e + 6, -4*e - 10, -6*e - 9, 3*e + 2, 2*e + 1, 2*e + 12, 2*e - 11, e, -e - 10, 4*e + 8, -2*e - 8, -2*e - 20, 8*e + 15, -10*e - 20, 6*e + 10, -10*e - 20, 2*e + 20, 2*e + 2, -2*e - 12, -6*e - 20, -2*e + 8, -2*e - 6, -10, -4*e + 4, -2*e - 5, 5*e - 4, -8*e - 8, -9*e - 18, -8*e - 5, -8*e - 18, -12*e - 30, -e + 14, 5*e + 10, 12*e + 22, -1, -6*e - 8, 7*e, -7*e - 24, -4*e - 20, 4*e + 4, -4*e - 10, -8*e - 8, -4*e + 4, 4*e + 2, -12*e - 24, -4, 7*e + 20, 6*e - 3, -e + 18, 8*e, 8*e + 8, 4*e - 11, -8*e - 6, -2*e - 33, -8*e - 28, -12*e - 16, -10*e - 38, 6*e - 8, -4*e + 8, -14*e - 36, -8*e - 34, 14*e + 17, -8*e - 14, 10*e + 16, 26, -6*e - 16, 16*e + 36, 11*e + 16, -5*e - 12, 2*e + 40, 6*e - 1, -9*e - 12, 32, -4*e - 16, 3*e - 18, 9*e + 10, 12*e + 4, 4*e - 22, 2*e + 3, -2*e - 14, -6*e - 27, 6*e + 16, 20*e + 36, -9*e + 6, 4*e + 4, 18*e + 39, -20, 4*e + 22, -9*e - 14, -7*e - 44, -12*e - 20, -16*e - 12, -16*e - 40, 15*e + 48, -9*e - 30, -16*e - 21, 10*e + 22, 6*e - 12, -5*e + 6, -4*e + 2, -3*e + 22, 12*e + 6, -10*e - 10, -4*e + 2, 2*e + 14, 6*e - 20, -13*e - 26, 6*e + 26, -14*e - 8, 5*e + 18, 4*e - 22, 18*e + 42, 2*e - 11, -16*e - 50, 8*e - 6, 8*e + 2, 16, 4*e + 11, -2*e + 10, -12*e - 18, -2*e + 38, 21, -14*e - 14, -4*e + 20, 12*e + 18, 11*e + 32, 10*e + 28, -2*e - 16, -2*e - 32, 12*e + 18, -18*e - 15, -22*e - 45, 42, 2*e + 26, 16*e + 32, -16*e - 51, 28*e + 56, 12*e + 20, -3*e + 12, -12*e - 22, -2*e + 16, -4*e - 14, -12*e + 8, 26*e + 62, 4*e + 8, -12*e - 35, -10*e - 30, 8*e - 20, 10*e + 37, 22*e + 38, -10*e - 36, -22*e - 37, 12*e + 37, -12*e - 20, -12*e + 9, 8*e + 34, -2*e + 12, 8*e + 4] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, -w + 1])] = 1 AL_eigenvalues[ZF.ideal([11, 11, -w^3 + 6*w - 4])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]