/* 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([12, -9, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 3, -w^2 + 9]) primes_array = [ [2, 2, w - 2],\ [3, 3, -w + 3],\ [3, 3, w - 1],\ [4, 2, w^2 + w - 7],\ [19, 19, w + 1],\ [23, 23, w^2 - 2*w - 1],\ [31, 31, -2*w^2 + 19],\ [31, 31, -w^2 + 5],\ [31, 31, -3*w + 5],\ [41, 41, w^2 + 2*w - 7],\ [43, 43, w^2 - 11],\ [47, 47, 3*w - 7],\ [53, 53, -3*w^2 - 6*w + 11],\ [59, 59, 2*w - 1],\ [67, 67, 2*w^2 + w - 19],\ [67, 67, 3*w^2 + 2*w - 25],\ [67, 67, w - 5],\ [73, 73, -4*w^2 - 3*w + 29],\ [73, 73, 2*w^2 - w - 11],\ [73, 73, w^2 + 2*w - 11],\ [79, 79, 2*w^2 + 2*w - 11],\ [83, 83, 2*w^2 - 13],\ [89, 89, -2*w + 7],\ [97, 97, -2*w^2 + 4*w - 1],\ [101, 101, -2*w - 5],\ [103, 103, 2*w^2 + w - 13],\ [107, 107, -2*w^2 - 2*w + 17],\ [109, 109, -4*w^2 - 2*w + 31],\ [113, 113, w^2 + 2*w - 1],\ [125, 5, -5],\ [127, 127, 2*w^2 + 3*w - 13],\ [137, 137, -2*w^2 + 7*w - 7],\ [139, 139, 2*w^2 + w - 7],\ [149, 149, -6*w^2 - 9*w + 31],\ [157, 157, w^2 - 4*w + 1],\ [163, 163, 2*w^2 + 5*w - 5],\ [173, 173, 5*w^2 + 4*w - 37],\ [179, 179, 2*w^2 + w - 11],\ [179, 179, 3*w^2 - 2*w - 17],\ [179, 179, -4*w^2 - w + 37],\ [191, 191, -6*w^2 - 9*w + 29],\ [193, 193, -2*w^2 - 4*w + 5],\ [223, 223, -2*w^2 + 2*w + 7],\ [227, 227, 6*w^2 + 4*w - 47],\ [233, 233, 2*w^2 + 4*w - 13],\ [233, 233, -4*w^2 - 4*w + 25],\ [233, 233, 4*w + 13],\ [239, 239, -4*w^2 - 8*w + 13],\ [239, 239, 2*w^2 + 2*w - 5],\ [239, 239, -w^2 + 8*w - 11],\ [241, 241, 3*w - 1],\ [251, 251, w^2 + 4*w - 19],\ [263, 263, -2*w^2 + 2*w + 25],\ [263, 263, -5*w^2 - 4*w + 35],\ [263, 263, 3*w^2 - 25],\ [269, 269, 6*w^2 + 3*w - 47],\ [269, 269, -w^2 - 1],\ [269, 269, 2*w^2 - 7],\ [277, 277, 2*w^2 - w - 5],\ [277, 277, -5*w^2 - 6*w + 31],\ [277, 277, 2*w^2 - 2*w - 11],\ [281, 281, 4*w^2 + 5*w - 25],\ [307, 307, 4*w^2 + 2*w - 37],\ [317, 317, -w - 7],\ [317, 317, 3*w^2 + 2*w - 19],\ [317, 317, -2*w^2 + 2*w + 5],\ [331, 331, 2*w^2 + 3*w - 19],\ [337, 337, w^2 - 4*w + 7],\ [343, 7, -7],\ [347, 347, 4*w^2 + 2*w - 29],\ [349, 349, 3*w^2 - 19],\ [361, 19, w^2 - 2*w - 7],\ [367, 367, 4*w^2 + 5*w - 19],\ [367, 367, -6*w + 11],\ [373, 373, -4*w^2 + w + 25],\ [379, 379, -5*w^2 - 10*w + 19],\ [383, 383, 4*w^2 + 6*w - 23],\ [401, 401, -3*w + 11],\ [401, 401, -3*w - 1],\ [401, 401, -3*w - 7],\ [409, 409, -w^2 - 4*w + 13],\ [419, 419, 2*w^2 - w - 17],\ [421, 421, -3*w^2 + 31],\ [431, 431, w^2 + 4*w - 1],\ [449, 449, -w^2 + 4*w + 1],\ [457, 457, -w^2 + 2*w + 13],\ [461, 461, 4*w^2 - 4*w - 19],\ [463, 463, 2*w^2 + 4*w - 23],\ [467, 467, 2*w^2 - 5*w + 5],\ [479, 479, 3*w + 11],\ [491, 491, 8*w^2 + 9*w - 49],\ [499, 499, -6*w^2 - 6*w + 41],\ [499, 499, 2*w^2 + 2*w - 23],\ [499, 499, 2*w^2 + 5*w - 29],\ [523, 523, -4*w^2 + w + 41],\ [529, 23, 6*w^2 + 7*w - 35],\ [547, 547, -2*w^2 + 6*w + 1],\ [557, 557, 3*w^2 + 2*w - 13],\ [563, 563, -w^2 + 4*w + 23],\ [569, 569, 10*w^2 + 8*w - 73],\ [587, 587, 2*w^2 - 2*w - 13],\ [593, 593, -6*w^2 - 8*w + 31],\ [601, 601, 4*w^2 + 5*w - 41],\ [601, 601, 10*w^2 + 16*w - 49],\ [601, 601, w^2 - 2*w + 5],\ [613, 613, -8*w^2 - 7*w + 55],\ [617, 617, -8*w^2 - 12*w + 37],\ [617, 617, 2*w^2 + 3*w - 25],\ [617, 617, 3*w^2 - 17],\ [641, 641, 4*w^2 - 35],\ [643, 643, 4*w^2 + 4*w - 37],\ [659, 659, -3*w^2 - 4*w + 23],\ [661, 661, -w^2 - 4*w - 5],\ [673, 673, 4*w^2 + 4*w - 31],\ [683, 683, 2*w^2 + 4*w - 19],\ [683, 683, 2*w^2 + w - 23],\ [683, 683, 3*w^2 + 6*w - 19],\ [701, 701, -6*w^2 - 12*w + 23],\ [701, 701, -2*w^2 + 7*w - 1],\ [701, 701, 3*w^2 + 2*w - 31],\ [709, 709, 5*w^2 + 4*w - 41],\ [719, 719, -6*w^2 - 2*w + 55],\ [719, 719, w^2 + 2*w - 17],\ [719, 719, 3*w^2 + 4*w - 29],\ [727, 727, -9*w^2 - 12*w + 49],\ [733, 733, 13*w^2 + 20*w - 65],\ [739, 739, 11*w^2 + 6*w - 91],\ [743, 743, -2*w^2 - 8*w + 35],\ [751, 751, 3*w^2 - 4*w - 13],\ [769, 769, 2*w^2 - 1],\ [797, 797, 2*w^2 - 8*w + 11],\ [811, 811, 4*w^2 - w - 29],\ [821, 821, 2*w^2 + 5*w - 17],\ [827, 827, -3*w^2 - 6*w + 7],\ [827, 827, -11*w^2 - 8*w + 83],\ [827, 827, -5*w^2 - 6*w + 25],\ [829, 829, -4*w^2 + 41],\ [829, 829, 4*w^2 + 8*w - 23],\ [829, 829, 8*w - 19],\ [839, 839, 2*w^2 + 6*w - 7],\ [853, 853, 10*w^2 + 6*w - 77],\ [859, 859, -5*w^2 - 2*w + 47],\ [863, 863, 12*w^2 + 19*w - 59],\ [877, 877, 8*w^2 + 6*w - 61],\ [907, 907, -4*w - 1],\ [907, 907, 9*w^2 + 4*w - 71],\ [907, 907, 2*w^2 + 7*w - 17],\ [919, 919, -10*w^2 - 5*w + 79],\ [929, 929, 2*w^2 + 7*w - 11],\ [937, 937, -4*w^2 + 6*w + 11],\ [941, 941, 3*w^2 - 11],\ [947, 947, 6*w - 5],\ [967, 967, -2*w^2 + 5*w + 5],\ [971, 971, 5*w^2 - 2*w - 29],\ [991, 991, 4*w^2 + 2*w - 25],\ [997, 997, 3*w^2 - 2*w - 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 - 2*x^4 - 6*x^3 + 10*x^2 + 8*x - 7 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 0, e^4 - 6*e^2 - e + 6, e^4 - e^3 - 5*e^2 + 3*e + 3, 2*e^4 + 2*e^3 - 12*e^2 - 10*e + 6, -e^3 + 3*e^2 + 2*e - 8, 4*e^2 - 2*e - 10, 2*e^4 - 10*e^2, 2*e^4 - e^3 - 9*e^2 + 4, 2*e^4 - 12*e^2 - 2*e + 14, -4*e^3 + 2*e^2 + 16*e - 2, -2*e^4 - 2*e^3 + 10*e^2 + 8*e + 2, -2*e^4 + 12*e^2 + 4*e - 12, 3*e^3 - 3*e^2 - 10*e + 10, -e^4 + 2*e^3 + 4*e^2 - 7*e + 2, -2*e^2 - 2*e + 8, -2*e^4 + 12*e^2 - 14, -4*e^4 + 2*e^3 + 18*e^2 - 4*e - 6, -6*e^4 + 2*e^3 + 32*e^2 - 4*e - 22, -4*e^4 - 3*e^3 + 23*e^2 + 18*e - 16, 2*e^3 - 4*e^2 - 2*e + 12, -3*e^4 - 2*e^3 + 20*e^2 + 13*e - 20, 5*e^4 - 26*e^2 - 5*e + 12, -4*e^4 + 2*e^3 + 20*e^2 - 8, 6*e^4 - 3*e^3 - 27*e^2 + 4*e + 6, -e^4 - 4*e^3 + 6*e^2 + 19*e, -7*e^4 + 38*e^2 + 5*e - 20, 4*e^3 - 16*e - 2, -2*e^4 + 8*e^2 + 8*e + 4, -4*e^3 + 4*e^2 + 16*e + 2, 2*e^4 + 6*e^3 - 12*e^2 - 28*e, -e^4 + 6*e^2 - 3*e - 8, -5*e^4 - 4*e^3 + 30*e^2 + 21*e - 18, 2*e^3 - 4*e^2 - 10*e + 18, -4*e^4 - e^3 + 25*e^2 + 6*e - 20, 4*e^4 - 2*e^3 - 20*e^2 + 6, 2*e^4 - 2*e^3 - 12*e^2 + 6*e + 16, -2*e^4 + 2*e^3 + 12*e^2 - 4*e - 12, 4*e^4 + 3*e^3 - 21*e^2 - 22*e + 4, -e^4 + 4*e^3 + 2*e^2 - 17*e + 4, 2*e^4 - 10*e^2 - 6*e + 14, 2*e^4 + 5*e^3 - 9*e^2 - 32*e - 4, 2*e^4 - e^3 - 11*e^2 - 4*e + 18, 7*e^3 + e^2 - 30*e - 10, -4*e^4 - 8*e^3 + 32*e^2 + 34*e - 32, e^4 - 10*e^2 - 3*e + 18, e^4 + 2*e^3 - 4*e^2 - 15*e - 2, -2*e^4 + 2*e^3 + 10*e^2 - 2*e - 8, -e^3 + 5*e^2 + 2*e - 18, -3*e^4 - 2*e^3 + 20*e^2 + 11*e - 14, -e^4 - 6*e^3 + 12*e^2 + 21*e - 24, 3*e^4 - 2*e^3 - 20*e^2 + 13*e + 22, 2*e^4 + 2*e^3 - 10*e^2 - 12*e + 10, 4*e^4 + 6*e^3 - 20*e^2 - 36*e + 6, 2*e^4 - 20*e^2 - 6*e + 32, 4*e^2 + 2*e, 2*e^4 - 3*e^3 - 11*e^2 + 16*e + 6, -4*e^4 + 4*e^3 + 20*e^2 - 16*e - 10, -6*e^4 + 34*e^2 + 12*e - 30, -4*e^4 - 2*e^3 + 22*e^2 + 12*e - 30, 3*e^4 - 18*e^2 + e + 12, -6*e^4 - 4*e^3 + 40*e^2 + 14*e - 38, -7*e^4 + 6*e^3 + 32*e^2 - 19*e - 12, -2*e^4 - 3*e^3 + 19*e^2 + 8*e - 24, 6*e^4 - e^3 - 31*e^2 - 4*e + 20, 2*e^4 + 2*e^3 - 16*e^2 - 16*e + 22, 6*e^2 - 2*e - 16, 3*e^4 + 10*e^3 - 20*e^2 - 43*e + 4, -2*e^4 + e^3 + 3*e^2 + 10, 2*e^4 - 6*e^3 - 10*e^2 + 32*e + 10, -8*e^4 + 4*e^3 + 36*e^2 - 6, -10*e^4 - 2*e^3 + 50*e^2 + 20*e - 8, 4*e^4 + 5*e^3 - 21*e^2 - 30*e + 6, 2*e^4 - 3*e^3 - 5*e^2 - 6, -3*e^4 + 18*e^2 + 5*e - 2, -9*e^4 - 4*e^3 + 54*e^2 + 23*e - 40, 2*e^4 + 5*e^3 - 9*e^2 - 24*e + 6, -4*e^3 + 2*e^2 + 24*e - 20, 12*e^4 - 2*e^3 - 60*e^2 - 12*e + 36, 10*e^4 - 4*e^3 - 52*e^2 + 4*e + 28, -8*e^4 - 2*e^3 + 38*e^2 + 26*e - 24, 4*e^4 - 4*e^3 - 24*e^2 + 28*e + 24, -3*e^4 + 2*e^3 + 20*e^2 - 9*e - 36, 2*e^4 - e^3 - 15*e^2 + 42, 2*e^4 + 6*e^3 - 16*e^2 - 28*e + 26, 2*e + 4, 4*e^4 + 6*e^3 - 28*e^2 - 32*e + 24, -e^4 + 10*e^3 - 35*e - 10, -6*e^4 - 2*e^3 + 36*e^2 + 20*e - 52, -12*e^4 + 6*e^3 + 60*e^2 - 8*e - 30, -4*e^3 + 24*e + 8, -4*e^2 + 16, -6*e^4 + 6*e^3 + 26*e^2 - 12*e - 10, -6*e^3 + 2*e^2 + 26*e - 18, -7*e^3 + e^2 + 30*e + 16, 8*e^4 + 2*e^3 - 48*e^2 - 22*e + 46, 6*e^4 - 2*e^3 - 28*e^2 - 6*e + 18, 7*e^4 - 50*e^2 - 5*e + 66, 4*e^4 - 2*e^3 - 22*e^2 + 8*e + 16, -6*e^4 - 2*e^3 + 20*e^2 + 26*e + 24, -4*e^4 + 8*e^3 + 24*e^2 - 38*e - 10, 4*e^4 - 9*e^3 - 19*e^2 + 22*e + 20, -15*e^4 + 4*e^3 + 74*e^2 + e - 26, -2*e^4 - 2*e^3 + 6*e^2 + 4*e + 16, 16*e^4 - 94*e^2 - 10*e + 70, -11*e^4 + 66*e^2 - e - 48, -6*e^4 - e^3 + 35*e^2 + 20*e - 50, 5*e^4 - 4*e^3 - 30*e^2 + 5*e + 30, -6*e^4 + 38*e^2 + 6*e - 24, 8*e^3 - 8*e^2 - 28*e + 10, 6*e^4 - 8*e^3 - 34*e^2 + 32*e + 24, 3*e^4 + 2*e^3 - 32*e^2 + e + 50, -4*e^4 + 10*e^2 + 2*e + 34, 4*e^3 - 2*e^2 - 14*e + 6, -6*e^4 + 16*e^3 + 34*e^2 - 60*e - 28, 12*e^4 - 6*e^3 - 64*e^2 + 26*e + 44, -10*e^4 + 6*e^3 + 56*e^2 - 8*e - 48, -4*e^4 + 24*e^2 - 6*e - 4, -8*e^4 + e^3 + 47*e^2 - 10*e - 32, 16*e^4 + 4*e^3 - 96*e^2 - 24*e + 78, 13*e^4 - 4*e^3 - 78*e^2 + 21*e + 66, -8*e^4 - 8*e^3 + 48*e^2 + 40*e - 8, 4*e^3 - 22*e + 18, 4*e^4 - 12*e^2 - 4*e - 4, -5*e^4 - 6*e^3 + 24*e^2 + 43*e - 12, -4*e^4 + 4*e^3 + 32*e^2 - 8*e - 42, 4*e^3 - 2*e^2 - 26*e + 28, 4*e^4 + 3*e^3 - 31*e^2 - 22*e + 34, 2*e^4 - 6*e^3 - 6*e^2 + 24*e + 10, 16*e^4 + 6*e^3 - 88*e^2 - 48*e + 60, 2*e^4 - 2*e^3 + 6*e^2 - 10*e - 46, 11*e^4 - 8*e^3 - 54*e^2 + 31*e + 36, 3*e^4 - 6*e^3 - 12*e^2 + 31*e + 2, -6*e^4 + 6*e^3 + 44*e^2 - 24*e - 48, -17*e^4 - 2*e^3 + 84*e^2 + 33*e - 26, -12*e^4 + 8*e^3 + 64*e^2 - 32*e - 40, -6*e^4 - 2*e^3 + 40*e^2 + 10*e - 56, 8*e^4 - 9*e^3 - 31*e^2 + 34*e + 4, -4*e^4 + 2*e^3 + 24*e^2 + 6*e - 14, -14*e^4 - e^3 + 71*e^2 + 24*e - 36, 4*e^3 - 18*e^2 - 4*e + 44, 2*e^4 + 6*e^3 - 28*e^2 - 32*e + 48, -12*e^4 + 8*e^3 + 64*e^2 - 16*e - 52, -8*e^4 + 12*e^3 + 32*e^2 - 46*e - 28, 4*e^4 - 6*e^3 - 12*e^2 + 8*e + 2, -8*e^4 + 2*e^3 + 44*e^2 - 8*e - 10, -4*e^4 + 20*e^2 + 12, 4*e^4 - 8*e^3 - 12*e^2 + 28*e + 4, -6*e^4 + 4*e^3 + 26*e^2 - 6*e + 12, 14*e^4 + 2*e^3 - 76*e^2 - 30*e + 52, -10*e^4 - 4*e^3 + 50*e^2 + 36*e - 10, -18*e^4 + 2*e^3 + 102*e^2 + 10*e - 76, 9*e^4 + 6*e^3 - 52*e^2 - 27*e + 28, 10*e^4 - 2*e^3 - 58*e^2 + 2*e + 36, 4*e^4 + 3*e^3 - 33*e^2 - 2*e + 32, 12*e^4 + 6*e^3 - 58*e^2 - 56*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([3, 3, -w + 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]