/* 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([-4, -9, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 4, -w]) primes_array = [ [2, 2, w + 2],\ [3, 3, w + 1],\ [4, 2, -w^2 + 3*w + 3],\ [9, 3, -w^2 + 2*w + 7],\ [13, 13, w + 3],\ [13, 13, -w + 3],\ [13, 13, -w + 1],\ [17, 17, -w^2 - 2*w + 1],\ [23, 23, w^2 - 2*w - 5],\ [31, 31, 2*w + 3],\ [31, 31, -2*w^2 + 3*w + 15],\ [31, 31, 3*w + 7],\ [37, 37, 3*w^2 - 4*w - 27],\ [41, 41, -2*w^2 + 7*w + 1],\ [59, 59, w^2 - 3],\ [61, 61, 4*w^2 - 12*w - 11],\ [71, 71, w^2 - 2*w - 11],\ [97, 97, 3*w + 5],\ [101, 101, 2*w^2 - 6*w - 7],\ [103, 103, 2*w^2 - 3*w - 19],\ [107, 107, -3*w^2 + 10*w + 3],\ [109, 109, -w - 5],\ [109, 109, 5*w^2 - 8*w - 41],\ [109, 109, 2*w^2 - 2*w - 21],\ [113, 113, -4*w - 3],\ [125, 5, -5],\ [127, 127, 2*w^2 - 2*w - 17],\ [131, 131, 2*w - 3],\ [137, 137, 2*w^2 - 5*w - 5],\ [137, 137, w^2 - 4*w - 13],\ [137, 137, 2*w - 5],\ [139, 139, 2*w^2 - 3*w - 13],\ [151, 151, 2*w^2 - 9],\ [163, 163, 2*w^2 - 5*w - 11],\ [173, 173, -2*w^2 + 8*w - 1],\ [179, 179, w^2 - 15],\ [181, 181, -w^2 + 11],\ [191, 191, 3*w - 1],\ [193, 193, -6*w^2 + 16*w + 23],\ [197, 197, 2*w^2 - 8*w + 3],\ [199, 199, 4*w^2 - 5*w - 35],\ [211, 211, 2*w - 9],\ [223, 223, 3*w^2 - 8*w - 13],\ [227, 227, 3*w^2 + 2*w - 11],\ [227, 227, w^2 - 4*w - 15],\ [227, 227, w - 7],\ [229, 229, -4*w^2 + 15*w - 1],\ [239, 239, -4*w - 5],\ [257, 257, 2*w^2 + w - 9],\ [263, 263, -w^2 - 6*w - 7],\ [269, 269, w^2 - 4*w - 7],\ [277, 277, 4*w^2 - 7*w - 33],\ [281, 281, 6*w^2 - 17*w - 19],\ [281, 281, 3*w^2 - 8*w - 9],\ [281, 281, -5*w - 1],\ [289, 17, -2*w^2 + 9*w - 5],\ [293, 293, -3*w^2 + 8*w + 7],\ [293, 293, 2*w^2 + 5*w - 1],\ [293, 293, 6*w^2 - 10*w - 47],\ [307, 307, 2*w^2 - 2*w - 3],\ [313, 313, 2*w^2 - 4*w - 7],\ [317, 317, 3*w^2 - 6*w - 17],\ [317, 317, 2*w^2 - 7*w - 7],\ [317, 317, 2*w^2 + 2*w - 7],\ [331, 331, 2*w^2 + 7*w + 7],\ [337, 337, w^2 - 4*w - 9],\ [337, 337, 3*w^2 - 4*w - 29],\ [337, 337, w^2 - 6*w - 5],\ [343, 7, -7],\ [347, 347, 2*w^2 + 4*w - 3],\ [349, 349, 2*w^2 - w - 25],\ [359, 359, -2*w^2 - w + 5],\ [379, 379, 4*w^2 - 9*w - 23],\ [379, 379, -5*w^2 + 18*w + 3],\ [379, 379, 2*w^2 - 3*w - 21],\ [383, 383, -w^2 + 6*w - 7],\ [397, 397, 2*w^2 - 7],\ [401, 401, 9*w^2 - 28*w - 21],\ [419, 419, 3*w^2 - 6*w - 25],\ [419, 419, 4*w^2 - 7*w - 27],\ [419, 419, 4*w^2 - 6*w - 29],\ [433, 433, 8*w^2 - 13*w - 61],\ [433, 433, -6*w^2 + 23*w - 3],\ [433, 433, -w^2 - 3],\ [443, 443, -2*w^2 + 5*w + 15],\ [449, 449, -7*w^2 + 12*w + 51],\ [457, 457, 2*w^2 - 2*w - 9],\ [457, 457, -w^2 - 6*w - 1],\ [457, 457, 6*w^2 - 11*w - 43],\ [461, 461, -5*w^2 + 6*w + 45],\ [463, 463, 2*w^2 - 5*w - 17],\ [463, 463, 3*w - 5],\ [463, 463, 2*w^2 - 3*w - 7],\ [467, 467, -3*w^2 + 10*w + 9],\ [479, 479, 3*w - 7],\ [487, 487, 7*w^2 - 20*w - 23],\ [487, 487, -4*w - 11],\ [487, 487, 5*w + 7],\ [509, 509, 2*w^2 - 2*w - 5],\ [523, 523, -8*w^2 + 28*w + 7],\ [529, 23, 2*w^2 - w - 7],\ [541, 541, 3*w^2 + 2*w - 9],\ [547, 547, -6*w - 1],\ [557, 557, 2*w^2 - 13],\ [563, 563, w - 9],\ [569, 569, -8*w^2 + 27*w + 9],\ [571, 571, 5*w^2 - 14*w - 15],\ [577, 577, w^2 + 4*w + 7],\ [587, 587, 4*w^2 - 4*w - 27],\ [593, 593, 2*w^2 + 5*w + 5],\ [599, 599, -2*w - 9],\ [599, 599, -w^2 + 6*w - 1],\ [599, 599, 2*w^2 - w - 19],\ [607, 607, 4*w - 13],\ [617, 617, -5*w - 13],\ [617, 617, -w^2 - 2*w - 5],\ [617, 617, 4*w^2 - 16*w + 5],\ [619, 619, 3*w^2 - 8*w - 15],\ [643, 643, -6*w - 11],\ [647, 647, -w^2 + 17],\ [647, 647, 2*w^2 - 9*w - 7],\ [647, 647, 2*w^2 - 6*w - 21],\ [659, 659, 4*w^2 - 8*w - 23],\ [661, 661, 2*w - 11],\ [673, 673, 3*w^2 - 13],\ [677, 677, 5*w^2 - 16*w - 13],\ [677, 677, -3*w^2 - 4*w + 3],\ [677, 677, 5*w^2 - 6*w - 43],\ [691, 691, 6*w^2 - 12*w - 37],\ [701, 701, -3*w^2 + 12*w - 1],\ [709, 709, -6*w^2 + 21*w + 7],\ [719, 719, -w^2 + 6*w - 3],\ [733, 733, -w - 9],\ [733, 733, 2*w^2 + w - 11],\ [733, 733, -2*w^2 + 9*w - 3],\ [739, 739, 6*w^2 - 17*w - 21],\ [743, 743, -2*w^2 - 6*w + 1],\ [751, 751, 4*w^2 - 8*w - 29],\ [751, 751, -w^2 - 8*w - 11],\ [751, 751, 4*w^2 - 21],\ [757, 757, 2*w^2 - 9*w - 1],\ [761, 761, -4*w^2 + 10*w + 15],\ [809, 809, w^2 + 2*w - 19],\ [811, 811, -7*w - 17],\ [823, 823, 3*w^2 - 17],\ [827, 827, -9*w - 19],\ [857, 857, 4*w^2 + 14*w + 9],\ [857, 857, 4*w^2 - 16*w + 3],\ [857, 857, 4*w^2 - 6*w - 27],\ [859, 859, -2*w^2 + 5*w - 1],\ [863, 863, 12*w^2 - 37*w - 29],\ [877, 877, 3*w^2 - 6*w - 7],\ [877, 877, 3*w^2 - 6*w - 31],\ [877, 877, 3*w^2 - 6*w - 13],\ [907, 907, 8*w^2 - 14*w - 57],\ [907, 907, 4*w^2 - 7*w - 25],\ [907, 907, 8*w^2 - 24*w - 19],\ [911, 911, -3*w - 11],\ [919, 919, -16*w^2 + 53*w + 23],\ [929, 929, 2*w^2 - 6*w - 19],\ [937, 937, w^2 - 6*w - 15],\ [937, 937, 3*w^2 - 4*w - 17],\ [937, 937, -3*w^2 + 12*w + 1],\ [941, 941, -2*w^2 + 6*w + 15],\ [947, 947, -2*w^2 - 3*w + 7],\ [953, 953, 2*w^2 - 25],\ [971, 971, 3*w^2 - 10*w - 11],\ [971, 971, 3*w^2 - 12*w + 5],\ [971, 971, -4*w^2 + 10*w + 9],\ [977, 977, 2*w^2 - 8*w - 9],\ [983, 983, -5*w^2 + 8*w + 35],\ [991, 991, 4*w^2 + 5*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - x^2 - 9*x + 12 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -e^2 - e + 7, -e^2 - e + 7, -3, e^2 - 7, e^2 - 3, -2*e^2 - 2*e + 15, e^2 + 2*e - 8, -e^2 + e + 6, e - 2, -e^2 - 2*e + 14, 2*e^2 - 10, -1, 3*e^2 + 4*e - 26, 2*e^2 + 2*e - 7, -2*e^2 - 3*e + 24, 11, e^2 + 5, 3*e^2 + e - 22, -2*e^2 - 6*e + 14, -3*e^2 + 25, -2*e^2 + e + 18, 2*e^2 + 6*e - 11, 2*e^2 + 3*e - 17, -e^2 - 2*e + 6, e^2 + e - 18, 3*e^2 + 4*e - 16, 2*e^2 + 5*e - 13, 4*e^2 + 3*e - 29, -4*e + 3, -6*e + 2, 3*e^2 - e - 20, e^2 - e - 8, -e^2 + 4*e + 9, 3*e^2 + e - 6, 5*e^2 + 4*e - 35, 5*e + 2, e^2 + 2*e - 18, 2*e^2 + e - 19, 8, -e^2 - 4*e + 4, -3*e^2 - 6*e + 12, -6, e^2 - e - 14, -e^2 - e + 26, 3*e^2 + 3*e - 11, 7*e^2 + 3*e - 34, 6*e^2 + 4*e - 37, 6*e^2 + 2*e - 42, -e^2 - 3*e - 14, -7*e^2 - 12*e + 61, -e^2 + 6*e - 1, 4*e^2 - 17, -5*e^2 - 2*e + 34, -3*e^2 - 2*e + 23, -8*e^2 - 8*e + 57, -5*e + 5, -4*e - 18, 3*e^2 + 10*e - 22, 3*e^2 - 2*e - 26, -2*e^2 - e + 17, -5*e^2 - e + 29, 5*e^2 + e - 42, -2*e^2 + 8*e + 18, -2*e^2 - 4*e + 22, 8*e^2 + 4*e - 41, 2*e + 19, -2*e^2 + e - 4, 7*e^2 - 46, 4*e^2 + 2*e - 26, 3*e^2 + 5*e - 8, 3*e^2 + 4*e - 42, -3*e^2 + 7*e + 24, 6*e^2 + 11*e - 44, -2*e^2 + 3*e + 14, -4*e^2 - 8*e + 33, -e^2 + 4*e + 27, -5*e^2 - 9*e + 32, -4*e^2 - 3*e + 44, e^2 + 9*e + 6, 11*e^2 + 3*e - 62, -3*e^2 + 4*e + 19, e^2 + 7*e - 25, -3*e^2 - 12*e + 20, -e^2 + e + 6, -2*e^2 - 7*e + 7, 5*e^2 - 7*e - 38, 6*e^2 + 4*e - 37, 3*e^2 - 3*e - 3, -10*e^2 - 5*e + 50, 4*e^2 + e - 18, -5*e^2 - 4*e + 34, 2*e^2 - 5*e - 36, 2*e^2 + 13*e - 16, 7*e^2 + 5*e - 32, 4*e^2 + 4*e - 4, -5*e^2 - 7*e + 40, -e^2 - 2*e + 10, e^2 - 4*e - 4, -4*e^2 - 8*e + 35, -5*e^2 - 12*e + 38, -11*e^2 - 9*e + 74, -8*e^2 - 17*e + 65, -10*e^2 - 11*e + 62, 4*e^2 + e - 14, -6*e^2 + 6*e + 40, e^2 + 7*e - 5, -8*e^2 - 4*e + 52, e^2 - 5*e - 9, 3*e^2 + 8*e - 22, 3*e^2 + 9*e - 34, -9*e^2 + 48, -2*e^2 + 2*e - 6, 5*e^2 + e - 13, 2*e^2 - 8*e - 18, -9*e^2 + e + 47, 6*e^2 - 38, 8*e^2 + 13*e - 48, 10*e^2 + 6*e - 58, 2*e^2 + 5*e - 16, -5*e^2 + 6*e + 40, 10*e^2 + 3*e - 58, e^2 + 8*e - 23, -4*e^2 - 12*e + 19, -6*e^2 + 42, 2*e^2 - 9*e - 11, -5*e^2 - 11*e + 45, -2*e^2 + 3*e + 4, 2*e^2 - 6*e - 15, 6*e^2 + 5*e - 43, -9*e^2 - 9*e + 66, -e^2 + 8*e - 3, -4*e^2 - 15*e + 34, 9*e^2 + 19*e - 71, 3*e^2 + 12*e - 16, e^2 - 11*e + 10, -3*e^2 + 12*e + 26, 11*e^2 + 15*e - 86, -5*e^2 + 2*e + 44, -7*e^2 - 3*e + 14, -6*e^2 + 9*e + 39, 13*e^2 + 14*e - 89, 4*e^2 + 2*e - 10, -7*e^2 - 6*e + 74, -e - 8, 7*e^2 + e - 30, -8*e^2 - 17*e + 67, 2*e^2 + 12*e - 17, 4*e + 10, 3*e^2 - e - 18, -4*e^2 - 11*e + 9, -5*e^2 - 15*e + 21, -3*e^2 - 11*e + 2, 6*e^2 + 13*e - 50, -11*e^2 - 6*e + 58, -5*e^2 - e + 10, 11*e^2 + 18*e - 82, -2*e^2 - 5*e - 10, e^2 + 11*e - 6, 4*e^2 - 7*e - 57, e^2 + e + 18, -11*e^2 - 9*e + 71, -10*e^2 - 8*e + 61, -7*e^2 - 2*e + 18, 4*e^2 - 11*e - 22, -8*e^2 - 15*e + 84, -7*e^2 - 14*e + 60, -7*e^2 - 2*e + 36, 2*e + 39, -4*e^2 - 2*e - 18, -5*e^2 - 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([2, 2, w + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]