/* 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([5, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([8, 2, 2]) primes_array = [ [3, 3, w - 2],\ [5, 5, w],\ [5, 5, -w + 3],\ [8, 2, 2],\ [9, 3, w^2 + w - 4],\ [13, 13, w + 3],\ [17, 17, -w^2 + w + 3],\ [23, 23, w^2 - 2],\ [23, 23, w^2 - 3],\ [23, 23, -w^2 + 8],\ [29, 29, w - 4],\ [37, 37, w^2 + w - 8],\ [41, 41, w^2 + 2*w - 4],\ [47, 47, 2*w^2 + w - 8],\ [59, 59, -2*w^2 - 3*w + 6],\ [61, 61, -2*w - 1],\ [67, 67, -2*w - 3],\ [79, 79, 2*w^2 - 9],\ [109, 109, w^2 + 2*w - 6],\ [109, 109, 2*w^2 + w - 14],\ [109, 109, -w^2 - w - 1],\ [113, 113, 2*w^2 + w - 12],\ [127, 127, w^2 - 3*w - 2],\ [131, 131, -4*w^2 - 5*w + 9],\ [137, 137, w^2 + 2*w - 7],\ [137, 137, 3*w + 8],\ [137, 137, 2*w^2 + w - 17],\ [139, 139, 3*w^2 + 2*w - 11],\ [149, 149, w - 6],\ [151, 151, w^2 + 2*w - 9],\ [157, 157, -w^2 - 2*w + 13],\ [157, 157, 2*w^2 - 7],\ [163, 163, w^2 + 3*w + 3],\ [163, 163, 2*w^2 - 3*w - 3],\ [163, 163, 2*w^2 - w - 7],\ [167, 167, 3*w^2 + w - 13],\ [169, 13, 2*w^2 + 5*w - 1],\ [173, 173, w^2 + 2*w + 2],\ [179, 179, 4*w^2 + 2*w - 19],\ [181, 181, w^2 + 3*w - 6],\ [191, 191, 3*w^2 + 2*w - 14],\ [193, 193, w^2 - 3*w - 3],\ [197, 197, 3*w^2 - 2*w - 13],\ [211, 211, -w - 6],\ [223, 223, -w^2 + w - 3],\ [223, 223, -3*w - 2],\ [223, 223, 2*w^2 - 2*w - 3],\ [229, 229, 2*w^2 - w - 4],\ [233, 233, 2*w + 7],\ [239, 239, -3*w - 4],\ [239, 239, w^2 - 11],\ [239, 239, 4*w^2 - w - 21],\ [241, 241, 3*w^2 + w - 19],\ [251, 251, -w^2 - 4*w - 1],\ [257, 257, w - 7],\ [263, 263, 3*w^2 + w - 12],\ [269, 269, 3*w^2 - 4*w - 11],\ [271, 271, 4*w^2 + 6*w - 11],\ [277, 277, -4*w^2 - 3*w + 17],\ [283, 283, 5*w^2 + 4*w - 18],\ [289, 17, 3*w^2 + 2*w - 9],\ [307, 307, -2*w^2 - w + 18],\ [307, 307, w^2 + 2*w + 3],\ [307, 307, -w^2 - 3],\ [313, 313, 3*w^2 - 13],\ [317, 317, 3*w^2 - 2*w - 12],\ [331, 331, w^2 + 3*w - 9],\ [331, 331, w^2 + 3*w - 11],\ [331, 331, w^2 + 3*w + 4],\ [343, 7, -7],\ [347, 347, 2*w^2 + 2*w - 13],\ [353, 353, 4*w^2 + 5*w - 13],\ [359, 359, 3*w^2 + 6*w - 1],\ [379, 379, w^2 - 3*w - 6],\ [383, 383, 3*w^2 - 23],\ [383, 383, w^2 - 12],\ [383, 383, 3*w^2 + 3*w - 13],\ [401, 401, -6*w^2 - 7*w + 16],\ [409, 409, -4*w^2 + 3*w + 21],\ [421, 421, w^2 - 3*w - 9],\ [433, 433, w^2 - 3*w - 8],\ [433, 433, w^2 + 4*w - 8],\ [433, 433, 3*w^2 - w - 12],\ [439, 439, -4*w - 11],\ [449, 449, 2*w^2 + 3*w - 19],\ [457, 457, w^2 - 4*w - 3],\ [461, 461, 5*w^2 + 6*w - 16],\ [461, 461, -2*w^2 + 3*w + 16],\ [461, 461, 2*w^2 + 4*w - 9],\ [467, 467, 3*w^2 - 4*w - 12],\ [479, 479, 3*w^2 + w - 9],\ [487, 487, 5*w^2 + 2*w - 22],\ [491, 491, 3*w^2 - 11],\ [499, 499, 5*w^2 + w - 24],\ [503, 503, 3*w^2 + 2*w - 17],\ [503, 503, 3*w^2 + w - 7],\ [503, 503, 2*w^2 - 4*w - 7],\ [509, 509, 3*w - 11],\ [521, 521, 3*w^2 - 3*w - 16],\ [523, 523, -w - 8],\ [541, 541, 6*w^2 + 7*w - 19],\ [547, 547, 3*w^2 + 4*w - 12],\ [557, 557, 2*w^2 - 3*w - 12],\ [563, 563, 4*w^2 + 3*w - 12],\ [563, 563, 4*w^2 + w - 17],\ [563, 563, -w^2 - 5*w - 3],\ [569, 569, w^2 - 4*w - 4],\ [577, 577, w^2 - w - 13],\ [587, 587, 4*w^2 + 3*w - 8],\ [587, 587, 2*w^2 - 3*w - 13],\ [587, 587, 2*w^2 + 3*w - 18],\ [593, 593, 4*w - 13],\ [599, 599, w - 9],\ [613, 613, -w^2 + 3*w - 7],\ [613, 613, w^2 + 5*w + 2],\ [613, 613, 3*w^2 - 3*w - 17],\ [617, 617, w^2 + 5*w + 8],\ [641, 641, 4*w^2 + 3*w - 11],\ [641, 641, w^2 + w - 14],\ [641, 641, 3*w^2 - 2*w - 9],\ [647, 647, 2*w^2 + 3*w - 13],\ [653, 653, 3*w^2 - 8],\ [659, 659, 5*w^2 + 4*w - 21],\ [659, 659, 2*w^2 - 4*w - 19],\ [659, 659, -2*w^2 - 6*w + 9],\ [677, 677, -6*w - 13],\ [691, 691, 6*w - 11],\ [701, 701, 3*w^2 - 3*w - 19],\ [719, 719, 2*w^2 + 3*w - 16],\ [727, 727, 3*w^2 + 4*w - 13],\ [733, 733, 3*w^2 - w - 8],\ [739, 739, 2*w^2 + 4*w - 11],\ [739, 739, 4*w^2 - 3*w - 24],\ [739, 739, 3*w^2 - w - 6],\ [743, 743, 4*w^2 - 2*w - 17],\ [751, 751, -w - 9],\ [761, 761, 5*w^2 + 5*w - 19],\ [769, 769, w^2 - 4*w - 6],\ [769, 769, -5*w - 1],\ [769, 769, 2*w^2 - 19],\ [773, 773, 2*w^2 - 6*w - 3],\ [797, 797, 4*w^2 - 17],\ [809, 809, -7*w^2 - 4*w + 29],\ [811, 811, 6*w^2 + 3*w - 29],\ [823, 823, 4*w^2 + 2*w - 13],\ [823, 823, -w^2 - 4*w - 7],\ [823, 823, 4*w^2 + 4*w - 17],\ [827, 827, -2*w^2 - 3*w - 2],\ [829, 829, -w^2 + w - 6],\ [829, 829, -w^2 + 4*w - 9],\ [829, 829, -2*w^2 + 3*w + 21],\ [839, 839, 6*w^2 - 2*w - 31],\ [841, 29, w^2 + 3*w + 6],\ [853, 853, -7*w^2 - 6*w + 27],\ [859, 859, 6*w - 1],\ [881, 881, w^2 + 5*w - 16],\ [907, 907, 2*w^2 + 7*w + 7],\ [919, 919, w^2 - 4*w - 9],\ [953, 953, -4*w^2 - w + 32],\ [967, 967, 5*w^2 - 5*w - 18],\ [971, 971, 3*w^2 - 4*w - 24],\ [977, 977, 2*w^2 + 4*w + 3],\ [997, 997, w^2 + 5*w - 12],\ [997, 997, 5*w^2 - 3*w - 22],\ [997, 997, 3*w^2 + 8*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 3*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 2*e + 2, -2, -1, e + 4, -2*e - 4, 3*e + 5, -6*e - 10, 4*e + 2, -6*e - 10, -2*e - 8, 2*e, -e + 4, 2*e + 6, -11*e - 14, 8, e - 3, -6*e - 4, -4*e - 16, -10*e - 20, 4*e + 6, e - 15, -4*e - 14, 3*e + 9, -e - 11, -11*e - 22, -e + 9, 3*e + 7, 8*e + 12, -22, -8*e, 4*e - 2, -3*e - 8, 3*e + 3, 7*e + 2, -2*e - 10, 4*e + 6, 4*e + 12, 5*e - 5, 12, 2*e + 26, 3*e + 3, 14*e + 14, 15*e + 17, -10*e - 6, -10*e - 6, -6*e - 28, -6*e - 14, -11*e - 13, 0, 4*e + 16, 8*e + 22, 25*e + 38, -12*e - 20, -15*e - 27, 18*e + 26, 6*e + 34, 10*e + 12, 24*e + 34, 3*e + 18, -11*e - 29, e + 1, 4*e + 8, -13*e - 25, 13*e + 38, 16*e + 26, -e + 9, 12*e + 20, 5*e + 13, 8*e - 12, -9*e - 24, 3*e + 26, 12*e + 18, 5*e, -2*e - 34, 12*e + 24, 12*e + 22, -7*e - 10, -5*e - 35, 16*e + 16, -28*e - 46, 3*e - 4, -9*e - 32, 14*e + 6, -8*e - 22, 27*e + 36, -28*e - 40, 18*e + 10, -22*e - 40, 11*e + 6, 26*e + 34, 6*e + 16, -15*e - 42, -5*e, -18*e - 46, 14*e + 10, -6*e - 10, -8*e - 12, 31*e + 37, 5*e - 26, 4*e - 26, 3*e, -22*e - 16, -21*e - 45, -13*e + 2, -15*e - 44, -9*e + 19, 4*e + 18, -19*e - 3, -4*e - 8, -21*e - 47, 35*e + 54, -20*e - 10, 36, 18*e + 26, 22*e + 22, -11*e - 27, -5*e - 37, 31*e + 57, -12*e - 26, 2, 28*e + 38, 13*e + 47, 9*e - 19, 9*e + 36, -14*e - 18, -5*e - 3, 2, 14*e + 46, 12*e, 4*e - 8, 17*e + 23, -9*e + 9, -24*e - 36, 4*e + 40, -2*e + 20, 3*e + 5, 4*e - 34, 5*e + 50, -9*e + 4, -20*e - 44, 6*e - 4, 9*e + 6, -15*e + 8, -26*e - 58, -16*e - 48, 12*e + 4, -e - 16, -2*e - 8, 18*e + 2, 10*e - 10, -28*e - 32, 25*e + 42, 6, -5*e - 40, 9*e + 48, 7*e - 14, -32*e - 68, -8*e - 46, -8*e - 24, -9*e - 14, 32*e + 50, 38*e + 60, 30*e + 32, 12*e + 36] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([8, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]