/* 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([6, 4, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([1, 1, 1]) primes_array = [ [2, 2, w - 2],\ [2, 2, w + 1],\ [3, 3, -w^2 + w + 3],\ [27, 3, -w^3 - 2*w^2 + 3*w + 5],\ [29, 29, w^3 + w^2 - 2*w - 1],\ [31, 31, -w^2 - w + 1],\ [31, 31, w^3 - 2*w^2 - 5*w + 7],\ [37, 37, -w^3 + 3*w + 1],\ [41, 41, -w^2 + w + 5],\ [41, 41, -w^3 + w^2 + 4*w - 1],\ [43, 43, -2*w - 1],\ [47, 47, w^2 + w - 5],\ [47, 47, 2*w^3 - 3*w^2 - 9*w + 11],\ [53, 53, w^3 + 2*w^2 - 5*w - 5],\ [59, 59, w^3 + w^2 - 4*w - 1],\ [59, 59, w^3 + w^2 - 4*w - 5],\ [71, 71, 2*w^3 - 4*w^2 - 10*w + 19],\ [71, 71, w^2 + 3*w + 1],\ [73, 73, w^3 - 5*w + 1],\ [89, 89, -4*w^3 + 7*w^2 + 19*w - 29],\ [97, 97, -w^3 + 2*w^2 + 3*w - 7],\ [101, 101, 2*w^3 - 12*w - 7],\ [103, 103, 2*w - 1],\ [103, 103, w^2 + w - 7],\ [107, 107, 2*w^3 - w^2 - 9*w + 1],\ [107, 107, 2*w^2 - 7],\ [109, 109, -w^3 + w^2 + 4*w + 1],\ [113, 113, -4*w^3 - 3*w^2 + 17*w + 13],\ [121, 11, -w^2 - w - 1],\ [121, 11, -w^3 + 2*w^2 + 5*w - 11],\ [139, 139, 2*w^3 - 10*w - 1],\ [139, 139, 2*w^3 - 10*w - 5],\ [149, 149, w^3 - w^2 - 6*w + 5],\ [157, 157, w^2 - 3*w - 1],\ [157, 157, 4*w^3 - 6*w^2 - 18*w + 25],\ [157, 157, 2*w^3 - w^2 - 9*w + 5],\ [157, 157, -w^3 - 2*w^2 + 3*w + 7],\ [163, 163, -w^3 + w^2 + 4*w - 7],\ [179, 179, w^3 - 3*w - 5],\ [181, 181, 3*w^3 - 4*w^2 - 13*w + 17],\ [191, 191, 2*w^3 - 8*w + 1],\ [191, 191, w^3 - 5*w - 5],\ [211, 211, 2*w + 5],\ [227, 227, w^3 + w^2 - 4*w - 7],\ [239, 239, 2*w^3 + 2*w^2 - 8*w - 5],\ [257, 257, 2*w^3 - 4*w^2 - 8*w + 19],\ [257, 257, w^3 + 2*w^2 - 5*w - 7],\ [263, 263, -2*w^3 - 2*w^2 + 6*w + 1],\ [269, 269, -2*w^3 + 4*w^2 + 8*w - 13],\ [277, 277, -3*w^2 + w + 13],\ [293, 293, -2*w^3 + 4*w^2 + 8*w - 17],\ [293, 293, -3*w^3 - 3*w^2 + 14*w + 11],\ [293, 293, 3*w^3 - 4*w^2 - 15*w + 19],\ [293, 293, 3*w^3 + w^2 - 14*w - 7],\ [307, 307, 2*w^3 - 3*w^2 - 5*w + 1],\ [307, 307, 2*w^3 - 4*w^2 - 4*w + 7],\ [311, 311, -3*w^3 + 3*w^2 + 14*w - 13],\ [313, 313, w^3 + 3*w^2 + 2*w + 1],\ [317, 317, -3*w^3 - 3*w^2 + 10*w + 5],\ [331, 331, -2*w^3 - 3*w^2 + 9*w + 13],\ [337, 337, -w^3 + 2*w^2 + 3*w - 11],\ [337, 337, w^3 + 2*w^2 - w - 5],\ [337, 337, -w^3 + 2*w^2 + 3*w - 1],\ [337, 337, w^3 - 3*w - 7],\ [347, 347, 3*w^3 - 4*w^2 - 15*w + 17],\ [347, 347, -w^3 + 2*w^2 + 5*w - 5],\ [349, 349, -2*w^3 + 2*w^2 + 8*w - 5],\ [353, 353, -2*w^3 + 3*w^2 + 3*w + 1],\ [361, 19, -5*w^3 - 4*w^2 + 21*w + 17],\ [361, 19, 3*w^3 + w^2 - 10*w + 1],\ [373, 373, -2*w^3 + w^2 + 11*w - 1],\ [379, 379, w^3 - w^2 - 6*w - 1],\ [379, 379, -w^3 + 7*w - 5],\ [389, 389, 2*w^3 - 2*w^2 - 8*w + 7],\ [397, 397, 2*w^3 + 5*w^2 - 5*w - 13],\ [409, 409, w^3 + w^2 - 6*w - 1],\ [421, 421, -2*w^3 + 2*w^2 + 10*w - 5],\ [433, 433, -w^3 - 3*w^2 + 6*w + 7],\ [439, 439, 3*w^2 - 3*w - 7],\ [443, 443, 3*w^3 + w^2 - 14*w - 5],\ [449, 449, -4*w^3 - 3*w^2 + 19*w + 17],\ [449, 449, -w^3 + 3*w^2 + 6*w - 17],\ [457, 457, w^3 + 2*w^2 + w + 1],\ [457, 457, -2*w^3 + w^2 + 9*w - 7],\ [461, 461, -w^3 + 7*w + 7],\ [461, 461, -4*w^2 + 17],\ [467, 467, -2*w^3 + w^2 + 11*w + 1],\ [479, 479, 5*w^3 - 8*w^2 - 25*w + 37],\ [487, 487, w^3 + w^2 - 8*w + 1],\ [487, 487, 7*w^3 - 12*w^2 - 31*w + 49],\ [487, 487, 2*w^3 + w^2 - 13*w - 11],\ [487, 487, 4*w^3 + 3*w^2 - 17*w - 11],\ [491, 491, -2*w^2 - 2*w + 1],\ [499, 499, 2*w^3 - 8*w - 1],\ [503, 503, -2*w^3 - w^2 + 11*w + 11],\ [503, 503, 3*w^3 - w^2 - 14*w + 1],\ [509, 509, -4*w^3 + 9*w^2 + 17*w - 37],\ [523, 523, -w^3 - 3*w^2 + 5],\ [541, 541, 6*w^3 + 6*w^2 - 20*w - 13],\ [541, 541, w^3 - 7*w + 1],\ [547, 547, w^2 - 3*w - 5],\ [547, 547, w^2 + 3*w - 1],\ [557, 557, -7*w^3 - 4*w^2 + 35*w + 25],\ [557, 557, -3*w^3 - 2*w^2 + 13*w + 7],\ [569, 569, -2*w^3 + 5*w^2 + 5*w - 11],\ [569, 569, w^3 + w^2 - 8*w - 5],\ [571, 571, -w^3 - 3*w^2 + 4*w + 11],\ [571, 571, -2*w^3 - 2*w^2 + 12*w + 13],\ [593, 593, -4*w^2 + 4*w + 7],\ [593, 593, 3*w^2 + w - 13],\ [599, 599, -2*w^3 + 2*w^2 + 6*w - 1],\ [601, 601, -w^3 + 3*w - 5],\ [607, 607, 2*w^3 - 12*w - 5],\ [607, 607, w^3 + w^2 + 1],\ [613, 613, -2*w^3 - 2*w^2 + 6*w + 7],\ [617, 617, w^3 - 7*w - 1],\ [619, 619, -2*w^3 + 12*w - 1],\ [625, 5, -5],\ [631, 631, -w^3 + w^2 + 2*w - 5],\ [641, 641, 2*w^3 - w^2 - 7*w - 1],\ [641, 641, 4*w^3 + 2*w^2 - 22*w - 19],\ [641, 641, -w^3 - 2*w^2 + 7*w + 11],\ [641, 641, 3*w^2 - 3*w - 11],\ [661, 661, -w^3 - w^2 + 6*w - 1],\ [661, 661, 3*w^3 + 2*w^2 - 11*w - 7],\ [673, 673, 3*w^3 - 2*w^2 - 13*w + 11],\ [673, 673, 2*w^3 - 6*w - 1],\ [683, 683, w^2 + 3*w - 5],\ [709, 709, 3*w^2 - w - 11],\ [719, 719, -2*w^3 + 6*w + 5],\ [719, 719, 2*w^3 + 5*w^2 - w - 5],\ [727, 727, 2*w^2 - 2*w - 11],\ [733, 733, 4*w^2 - 2*w - 19],\ [739, 739, -2*w^3 + 4*w^2 + 6*w - 13],\ [739, 739, w^3 - 9*w + 7],\ [739, 739, -2*w^3 + 10*w - 1],\ [739, 739, -2*w^3 - 2*w^2 + 12*w + 7],\ [751, 751, -4*w^3 - 3*w^2 + 15*w + 11],\ [751, 751, -2*w^3 - w^2 + 5*w - 1],\ [821, 821, 3*w^3 + 5*w^2 - 10*w - 11],\ [823, 823, -w^3 - w^2 + 8*w + 11],\ [839, 839, w^3 + 2*w^2 + w - 1],\ [853, 853, 2*w^3 + w^2 - 7*w - 7],\ [857, 857, w^3 + 4*w^2 - w - 13],\ [859, 859, 2*w - 7],\ [859, 859, -5*w^3 + 11*w^2 + 20*w - 43],\ [863, 863, 2*w^3 + 2*w^2 - 8*w - 1],\ [863, 863, -w^3 + 3*w^2 + 4*w - 7],\ [877, 877, -4*w^3 + 7*w^2 + 17*w - 31],\ [883, 883, 5*w^3 - 7*w^2 - 24*w + 31],\ [907, 907, 4*w^3 - 6*w^2 - 16*w + 19],\ [919, 919, 2*w^3 + 3*w^2 - 9*w - 11],\ [929, 929, w^3 - w^2 - 2*w - 5],\ [937, 937, -w^3 + 3*w^2 + 2*w - 11],\ [937, 937, -3*w^3 + 7*w^2 + 12*w - 29],\ [947, 947, 2*w^3 - 3*w^2 - 11*w + 13],\ [947, 947, 2*w^3 - w^2 - 11*w + 7],\ [961, 31, 2*w^3 - 4*w^2 - 2*w - 1],\ [967, 967, 3*w^3 + 4*w^2 - 9*w - 11],\ [971, 971, -4*w^3 - 4*w^2 + 18*w + 17],\ [991, 991, -w^3 + w^2 + 2*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e + 1, -2, -4*e, -4*e - 4, -4, -4, 4, -4*e, 4*e + 4, -4*e - 2, -4*e + 2, 0, 4*e + 4, 4*e, 8*e + 4, -8*e + 4, 12, 12, 4*e + 12, -2*e + 4, 2*e + 12, -12*e, 4*e + 8, -4*e + 4, 4*e - 8, 2*e - 4, 8*e + 4, 10*e - 4, 4*e - 8, 8*e - 2, -8, -2*e + 4, 18, 2*e - 8, 8*e - 4, 4*e - 16, 0, -4*e + 18, 8*e + 10, -8*e + 2, 4*e - 18, 4*e + 16, -12, -8*e + 6, -8*e + 18, 2*e, -8*e, -10*e - 8, -12*e + 4, -4*e + 20, 10*e + 8, -16*e - 2, -4*e + 4, -4*e + 16, -8*e - 8, 4*e + 2, 4*e - 4, 20*e, 0, 6*e - 24, -4*e + 20, 8*e - 14, -12*e + 12, -16*e + 4, 4*e + 16, 6*e - 24, 10*e + 20, -18*e, -12*e - 4, -8*e + 4, 8*e - 2, -20, 4*e, -4*e - 28, -4*e - 16, 6*e + 8, -24*e + 4, 12*e - 2, -16*e + 16, 14*e - 12, 4*e + 36, 20*e - 4, -2, 4*e + 20, 8*e + 12, 8*e + 8, 8*e - 24, -18, 8*e - 20, -22, -24, 16*e + 2, -8, 24*e, -4*e, 4*e - 32, 12*e - 10, 12*e + 12, 20, 12*e - 8, -12, -8*e + 2, -12*e + 12, 4*e + 20, 4*e + 24, 20, 24, -8*e - 6, 14*e + 24, 16*e - 14, -18*e, 24*e + 6, -4*e - 20, 20*e + 20, 8*e + 6, -8, -6, 12*e - 12, 4*e - 20, 16*e - 4, 18*e, -6*e, -6*e, -14, 24*e - 6, -4*e - 16, 8*e + 8, -6*e + 28, -8*e + 24, 8*e + 12, -4*e + 16, -16*e - 14, -24, -12*e + 20, 12*e, -32, 8*e - 16, -12*e - 8, 20*e + 12, 16*e + 20, 24*e + 12, -12*e + 24, 22*e - 20, 24*e + 20, 24*e - 16, -20*e - 22, -16*e - 22, -18*e + 24, -12*e + 24, -12*e - 24, -28*e - 10, -32*e + 12, 20*e - 20, 8*e - 34, 16*e, 20*e + 12, 8*e - 14, -24, -16*e + 16, 16] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]