/* 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([-48, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([6,6,w - 7]) primes_array = [ [2, 2, -9*w - 58],\ [2, 2, -9*w + 67],\ [3, 3, -2*w + 15],\ [3, 3, 2*w + 13],\ [7, 7, 186*w - 1385],\ [7, 7, -186*w - 1199],\ [23, 23, -38*w - 245],\ [23, 23, 38*w - 283],\ [25, 5, 5],\ [31, 31, 16*w - 119],\ [31, 31, 16*w + 103],\ [43, 43, 4*w + 25],\ [43, 43, -4*w + 29],\ [59, 59, 12*w - 89],\ [59, 59, -12*w - 77],\ [67, 67, 92*w + 593],\ [67, 67, 92*w - 685],\ [83, 83, 204*w - 1519],\ [83, 83, 204*w + 1315],\ [97, 97, -24*w + 179],\ [97, 97, -24*w - 155],\ [101, 101, 428*w + 2759],\ [101, 101, 428*w - 3187],\ [107, 107, -298*w + 2219],\ [107, 107, -298*w - 1921],\ [109, 109, 818*w - 6091],\ [109, 109, 818*w + 5273],\ [121, 11, -11],\ [131, 131, 2*w - 19],\ [131, 131, -2*w - 17],\ [137, 137, 6*w + 37],\ [137, 137, -6*w + 43],\ [139, 139, 150*w + 967],\ [139, 139, 150*w - 1117],\ [151, 151, -42*w - 271],\ [151, 151, -42*w + 313],\ [157, 157, 2*w - 7],\ [157, 157, -2*w - 5],\ [169, 13, -13],\ [179, 179, 1692*w + 10907],\ [179, 179, 1692*w - 12599],\ [181, 181, 132*w - 983],\ [181, 181, 132*w + 851],\ [191, 191, 576*w + 3713],\ [191, 191, 576*w - 4289],\ [193, 193, 2*w - 1],\ [197, 197, 28*w - 209],\ [197, 197, -28*w - 181],\ [199, 199, 6*w + 41],\ [199, 199, 6*w - 47],\ [211, 211, -78*w + 581],\ [211, 211, -78*w - 503],\ [229, 229, 70*w - 521],\ [229, 229, 70*w + 451],\ [239, 239, 1320*w - 9829],\ [239, 239, 1320*w + 8509],\ [241, 241, 14*w + 89],\ [241, 241, -14*w + 103],\ [257, 257, 18*w - 133],\ [257, 257, 18*w + 115],\ [277, 277, -12*w - 79],\ [277, 277, 12*w - 91],\ [289, 17, -17],\ [293, 293, 6*w - 41],\ [293, 293, 6*w + 35],\ [311, 311, 240*w + 1547],\ [311, 311, 240*w - 1787],\ [317, 317, -4*w - 31],\ [317, 317, -4*w + 35],\ [331, 331, 4*w - 23],\ [331, 331, -4*w - 19],\ [337, 337, 2194*w + 14143],\ [337, 337, 2194*w - 16337],\ [359, 359, 10*w - 77],\ [359, 359, 10*w + 67],\ [361, 19, -19],\ [379, 379, -6*w + 49],\ [379, 379, -6*w - 43],\ [383, 383, 2*w - 25],\ [383, 383, -2*w - 23],\ [389, 389, -1804*w + 13433],\ [389, 389, -1804*w - 11629],\ [409, 409, 1078*w + 6949],\ [409, 409, 1078*w - 8027],\ [449, 449, 258*w - 1921],\ [449, 449, 258*w + 1663],\ [461, 461, 594*w - 4423],\ [461, 461, 594*w + 3829],\ [467, 467, -82*w - 529],\ [467, 467, -82*w + 611],\ [479, 479, -50*w - 323],\ [479, 479, 50*w - 373],\ [487, 487, 8*w - 55],\ [487, 487, 8*w + 47],\ [523, 523, -522*w + 3887],\ [523, 523, -522*w - 3365],\ [547, 547, -4*w - 13],\ [547, 547, 4*w - 17],\ [563, 563, 84*w + 541],\ [563, 563, 84*w - 625],\ [571, 571, 124*w - 923],\ [571, 571, 124*w + 799],\ [577, 577, 912*w + 5879],\ [577, 577, 912*w - 6791],\ [587, 587, 374*w + 2411],\ [587, 587, 374*w - 2785],\ [593, 593, -136*w + 1013],\ [593, 593, -136*w - 877],\ [607, 607, 1208*w - 8995],\ [607, 607, -1208*w - 7787],\ [641, 641, 40*w - 299],\ [641, 641, -40*w - 259],\ [643, 643, 2324*w - 17305],\ [643, 643, 2324*w + 14981],\ [677, 677, 966*w + 6227],\ [677, 677, 966*w - 7193],\ [691, 691, 4*w - 11],\ [691, 691, -4*w - 7],\ [709, 709, -58*w - 373],\ [709, 709, 58*w - 431],\ [751, 751, 18*w - 137],\ [751, 751, 18*w + 119],\ [769, 769, -24*w - 157],\ [769, 769, 24*w - 181],\ [773, 773, 294*w + 1895],\ [773, 773, 294*w - 2189],\ [797, 797, -3198*w - 20615],\ [797, 797, 3198*w - 23813],\ [821, 821, 54*w - 401],\ [821, 821, -54*w - 347],\ [827, 827, 1414*w + 9115],\ [827, 827, 1414*w - 10529],\ [839, 839, -634*w + 4721],\ [839, 839, -634*w - 4087],\ [841, 29, -29],\ [853, 853, -22*w + 161],\ [853, 853, 22*w + 139],\ [857, 857, -104*w + 775],\ [857, 857, -104*w - 671],\ [881, 881, -8*w + 67],\ [881, 881, -8*w - 59],\ [911, 911, 120*w - 893],\ [911, 911, 120*w + 773],\ [919, 919, 486*w + 3133],\ [919, 919, 486*w - 3619],\ [929, 929, 320*w - 2383],\ [929, 929, 320*w + 2063],\ [937, 937, 406*w + 2617],\ [937, 937, 406*w - 3023],\ [941, 941, 1710*w + 11023],\ [941, 941, 1710*w - 12733],\ [947, 947, 14*w + 95],\ [947, 947, 14*w - 109],\ [953, 953, -6*w - 25],\ [953, 953, 6*w - 31],\ [967, 967, 6*w - 55],\ [967, 967, -6*w - 49],\ [971, 971, -3050*w - 19661],\ [971, 971, 3050*w - 22711],\ [977, 977, 1024*w - 7625],\ [977, 977, -1024*w - 6601],\ [983, 983, -22*w - 145],\ [983, 983, 22*w - 167],\ [997, 997, 742*w - 5525],\ [997, 997, 742*w + 4783]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - x^2 - 5*x + 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 1, -e, 1, 1, -e^2 - e + 6, 2*e^2 - e - 5, 3, -e^2 + 2*e + 1, -2*e^2 + 8, 3*e^2 - e - 6, 2*e^2 + 2*e - 8, -e^2 + e + 8, -e^2 + 6, 2*e^2 + 3*e - 9, -3*e^2 + e + 15, -e^2 + 4*e + 9, e^2 + e - 12, 3*e^2 + e - 9, -3*e^2 + 3*e + 4, -4*e^2 - 4*e + 20, -2*e^2 + 6*e + 11, 4*e - 1, -3*e^2 + 4*e + 11, 2*e^2 + 3*e - 10, -3*e^2 + 3*e + 2, -6*e + 2, 3*e^2 - 3*e - 8, -2*e^2 - 4, -e^2 + 3*e - 8, 2*e^2 - 4*e - 8, -3*e^2 + 5*e + 3, e + 4, e^2 - 3*e - 14, e^2 - 8*e - 8, e^2 - 3*e - 16, -3*e^2 + 2*e + 4, -7*e^2 + 25, -2*e^2 - e + 16, 2*e^2 - e - 10, 2*e^2 + 2*e - 19, 3*e^2 - 5*e - 12, 3*e^2 - 2*e + 7, -e^2 - 2*e + 11, -5*e^2 - 6*e + 17, 6*e^2 - 2*e - 22, 6*e^2 - 24, e^2 - 3*e - 12, 3*e^2 + e - 12, 3*e^2 + 2*e - 22, e^2 - e + 17, e^2 - e - 25, 5*e + 12, e^2 + e - 6, -5*e^2 + 3*e + 31, e^2 - 6*e + 11, -6*e^2 + 23, -4*e^2 - 7*e + 26, 7*e^2 + 6*e - 26, -e^2 - 3*e + 3, 2*e^2 + 3*e - 15, e^2 - 4*e + 8, 2*e^2 - 11*e - 10, -4*e^2 + 11*e + 10, 3*e^2 + e - 23, -2*e^2 - 9*e + 7, 2*e^2 - 2*e - 8, -7*e^2 - 3*e + 26, 6*e^2 + 2*e - 8, 5*e^2 - 5*e - 20, -7*e^2 - 4*e + 36, -6*e^2 - 10*e + 37, 2*e^2 + 3*e + 10, -5*e^2 + 7*e + 20, -5*e^2 - 2*e + 5, -5*e^2 + 3*e - 1, -5*e^2 + 19, 11*e^2 + 4*e - 39, 4*e^2 - e - 27, -8*e^2 + 2*e + 9, 5*e^2 - 13*e - 22, 10*e^2 - 6*e - 32, -e^2 - 4*e + 18, -6*e^2 - 8*e + 33, -2*e^2 - 8*e + 23, 5*e^2 - 6*e - 18, 12*e^2 + 3*e - 47, 10*e^2 - 3*e - 25, 6*e^2 + 7*e - 20, 7*e^2 + 2*e - 28, -3*e^2 - 15*e + 20, -7*e^2 - 5*e + 26, 3*e^2 - 7*e + 6, 7*e^2 + 8*e - 40, -2*e^2 - 10*e + 5, 9*e^2 - 19, e^2 - 4*e + 5, 6*e^2 - 4*e - 19, 8*e^2 + 10*e - 42, e^2 - 10*e - 3, 12*e + 11, -11*e^2 + 9*e + 28, -3*e^2 + 5*e - 5, -6*e^2 + 6*e + 19, -3*e^2 + 9*e + 28, 9*e^2 - 5*e - 24, -11*e^2 + 9*e + 36, 2*e^2 - 2*e + 1, 10*e^2 + 9*e - 41, 6*e^2 - 7*e - 20, -15*e^2 + 5*e + 34, 4*e^2 - 11*e - 12, -11*e^2 + 6*e + 18, -9*e^2 - 6*e + 50, 11*e^2 + 11*e - 42, -8*e^2 + 7*e + 29, 15*e^2 - 59, 7*e - 1, 9*e^2 - 7*e - 36, e^2 - 9*e + 7, -8*e^2 - 8*e + 43, -2*e - 40, 9*e^2 - 4*e - 48, 13*e^2 + 2*e - 46, -6*e^2 + 17*e + 14, -e - 42, -4*e^2 + 8*e + 40, 3*e^2 - 9*e + 14, -11*e^2 - 3*e + 41, 13*e^2 + 12*e - 59, -10*e^2 - 6*e + 26, 4*e^2 + 11*e - 25, -e^2 - 11*e - 14, 15*e^2 - 3*e - 56, -8*e^2 + 14*e + 18, -5*e^2 - 14*e + 24, 5*e^2 + 3*e + 2, 17*e^2 - 4*e - 46, 17*e^2 + 6*e - 62, -4*e^2 - 10*e + 10, -5*e^2 - 19*e + 25, -6*e^2 + 13*e + 38, 5*e^2 + 10*e - 10, -9*e^2 + 7*e + 35, -3*e^2 - 19*e + 10, -12*e^2 - 5*e + 59, -11*e^2 + 14*e + 42, 11*e^2 - 9*e - 30, -13*e^2 + 14*e + 40, -3*e^2 + 10*e + 15, -9*e^2 + 4*e + 38, -9*e^2 + 6*e + 10, 4*e^2 - 13*e - 8, 9*e^2 + 9*e - 50, -e^2 - 6*e - 20, -10*e^2 - 10*e + 31, -11*e^2 + 9*e + 60, -4*e^2 - 23*e + 20, 14*e^2 + 13*e - 61, 10*e^2 + e - 53, -12*e^2 - e + 32, -7*e^2 + 11*e + 20, -11*e^2 + 19*e + 39, -5*e^2 + 7*e + 21, -3*e^2 - 19*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([2,2,9*w - 67])] = -1 AL_eigenvalues[ZF.ideal([3,3,2*w + 13])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]