/* 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([3, -3, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([17, 17, -w^3 + 5*w + 2]) primes_array = [ [3, 3, w],\ [5, 5, w - 1],\ [7, 7, w^3 - w^2 - 4*w + 1],\ [11, 11, -w^2 + w + 2],\ [13, 13, -w^3 + w^2 + 3*w - 1],\ [16, 2, 2],\ [17, 17, -w^3 + 5*w + 2],\ [29, 29, -w^2 + w + 1],\ [37, 37, -w^3 + 2*w^2 + 4*w - 4],\ [41, 41, -w^3 + w^2 + 5*w + 1],\ [41, 41, w^2 - 5],\ [47, 47, w^3 - w^2 - 5*w - 2],\ [47, 47, -w^3 + 2*w^2 + 4*w - 1],\ [53, 53, w^3 - 4*w - 4],\ [53, 53, 2*w^3 - 2*w^2 - 9*w + 1],\ [61, 61, -w^3 + w^2 + 6*w - 2],\ [71, 71, w^3 - w^2 - 6*w - 2],\ [103, 103, 2*w^3 - 2*w^2 - 8*w + 1],\ [103, 103, -3*w^3 + 3*w^2 + 13*w - 5],\ [109, 109, 2*w^3 - w^2 - 8*w - 4],\ [109, 109, 2*w^3 - w^2 - 9*w - 1],\ [113, 113, -2*w^3 + w^2 + 11*w - 1],\ [113, 113, 2*w^3 - w^2 - 10*w - 4],\ [125, 5, 2*w^3 - 3*w^2 - 10*w + 4],\ [131, 131, -2*w^3 + w^2 + 9*w - 1],\ [139, 139, 2*w^3 - 4*w^2 - 7*w + 5],\ [139, 139, 3*w^3 - w^2 - 16*w - 5],\ [149, 149, w^2 + w - 4],\ [149, 149, 3*w^3 - 3*w^2 - 13*w + 1],\ [151, 151, w^2 - 2*w - 8],\ [157, 157, w^3 - w^2 - 7*w - 2],\ [157, 157, -2*w^2 + w + 7],\ [167, 167, 2*w^3 - 2*w^2 - 7*w + 5],\ [173, 173, -w^3 + 6*w - 2],\ [173, 173, -w^3 + 2*w^2 + 5*w - 7],\ [181, 181, -3*w^3 + 2*w^2 + 14*w + 1],\ [191, 191, -w^3 + 2*w^2 + 2*w - 5],\ [191, 191, -w^3 + 2*w^2 + 6*w - 5],\ [193, 193, -w^3 + 3*w^2 + 4*w - 4],\ [193, 193, 2*w^3 + 3*w^2 - 13*w - 20],\ [197, 197, w^3 - 2*w - 2],\ [199, 199, 2*w^3 - w^2 - 11*w - 2],\ [211, 211, -2*w^3 + 9*w + 5],\ [223, 223, -w^3 + w^2 + 4*w - 5],\ [223, 223, w^2 - 3*w - 2],\ [229, 229, 2*w^3 - 2*w^2 - 7*w + 1],\ [241, 241, 3*w^2 - 2*w - 13],\ [241, 241, -w^3 + w^2 + 5*w + 4],\ [257, 257, -3*w^3 + w^2 + 15*w + 8],\ [263, 263, -3*w^3 + 5*w^2 + 9*w - 7],\ [263, 263, 2*w^3 - w^2 - 9*w + 2],\ [263, 263, -w^3 + 4*w^2 + 2*w - 16],\ [263, 263, w^3 - w^2 - 2*w - 2],\ [271, 271, -3*w - 1],\ [281, 281, 2*w^3 - 3*w^2 - 8*w + 2],\ [281, 281, 3*w^3 - 4*w^2 - 12*w + 4],\ [283, 283, w^3 - 5*w - 7],\ [293, 293, -2*w^3 + w^2 + 10*w - 1],\ [307, 307, w^3 - 3*w^2 - 3*w + 8],\ [307, 307, 3*w^3 - 15*w - 10],\ [311, 311, 2*w^3 - 3*w^2 - 7*w + 7],\ [313, 313, -w^3 + 7*w + 2],\ [337, 337, 3*w^3 - 17*w - 10],\ [337, 337, 2*w^3 - 2*w^2 - 9*w - 5],\ [343, 7, -2*w^3 + w^2 + 8*w + 2],\ [347, 347, -3*w^3 + 17*w + 14],\ [347, 347, -3*w^3 - w^2 + 17*w + 13],\ [349, 349, 2*w^3 - w^2 - 8*w - 1],\ [349, 349, -w^3 + 3*w^2 + 4*w - 8],\ [353, 353, -w^3 - w^2 + 7*w + 4],\ [359, 359, -2*w^3 + 9*w + 10],\ [367, 367, 3*w^3 - 4*w^2 - 9*w + 4],\ [367, 367, -w^3 + w^2 + 4*w + 4],\ [367, 367, -w^3 + 3*w^2 + 3*w - 7],\ [373, 373, -w^2 + w - 2],\ [373, 373, -2*w^3 + 2*w^2 + 11*w - 2],\ [379, 379, -2*w^3 + 4*w^2 + 7*w - 8],\ [379, 379, -w^2 - 2],\ [389, 389, -w^3 - 2*w^2 + 5*w + 11],\ [397, 397, -2*w^3 + 4*w^2 + 7*w - 7],\ [401, 401, w^3 - 3*w^2 - 2*w + 10],\ [401, 401, -2*w^3 + 11*w + 4],\ [401, 401, 3*w^3 - 2*w^2 - 14*w - 4],\ [401, 401, -3*w^3 + 4*w^2 + 15*w - 5],\ [421, 421, -w^3 + w^2 + 7*w - 1],\ [421, 421, -2*w^3 + 2*w^2 + 9*w - 7],\ [433, 433, w^2 - w - 8],\ [439, 439, -w^3 + 3*w^2 + 4*w - 7],\ [443, 443, -2*w^3 + w^2 + 10*w - 2],\ [449, 449, w^2 + 2*w - 4],\ [449, 449, 2*w^3 - 2*w^2 - 10*w - 1],\ [461, 461, 2*w^3 - 2*w^2 - 6*w + 5],\ [461, 461, -2*w^3 + 4*w^2 + 7*w - 13],\ [463, 463, -3*w^3 + 3*w^2 + 12*w - 4],\ [463, 463, w - 5],\ [467, 467, -3*w^3 + w^2 + 16*w + 8],\ [467, 467, 2*w^2 - 3*w - 10],\ [479, 479, 2*w^2 - 4*w - 1],\ [479, 479, -w^3 + 2*w^2 + 2*w - 7],\ [487, 487, -3*w^3 + 5*w^2 + 12*w - 10],\ [503, 503, w^3 + 2*w^2 - 8*w - 8],\ [509, 509, -3*w^3 + 5*w^2 + 11*w - 5],\ [521, 521, 2*w^2 - 3*w - 11],\ [523, 523, -2*w^3 - w^2 + 13*w + 11],\ [523, 523, -2*w^3 + 4*w^2 + 8*w - 7],\ [541, 541, 3*w^3 - 2*w^2 - 13*w + 1],\ [563, 563, w^3 - 3*w - 5],\ [563, 563, -w^3 + 3*w^2 + 2*w - 11],\ [569, 569, -2*w^3 + 4*w^2 + 9*w - 4],\ [569, 569, 3*w^2 - w - 14],\ [571, 571, -w^3 + 8*w - 1],\ [577, 577, 3*w^3 - 2*w^2 - 16*w + 1],\ [587, 587, -4*w^3 + 5*w^2 + 18*w - 7],\ [587, 587, 2*w^3 + w^2 - 10*w - 7],\ [599, 599, 2*w^3 + w^2 - 11*w - 8],\ [607, 607, w^3 - 3*w + 4],\ [613, 613, 3*w^3 - w^2 - 14*w - 5],\ [613, 613, 3*w^2 - w - 11],\ [619, 619, 2*w^2 - 5*w - 8],\ [631, 631, -2*w^3 - 2*w^2 + 12*w + 19],\ [647, 647, 2*w^3 - 2*w^2 - 11*w - 2],\ [653, 653, 3*w^3 - 5*w^2 - 8*w + 7],\ [677, 677, -2*w^3 + 4*w^2 + 5*w - 8],\ [683, 683, 2*w^3 + w^2 - 12*w - 8],\ [683, 683, -w^3 + 3*w^2 - 8],\ [683, 683, -2*w^3 + 3*w^2 + 11*w - 8],\ [683, 683, -2*w^3 + 3*w^2 + 10*w - 2],\ [691, 691, w^2 - 4*w - 1],\ [709, 709, 3*w^3 - 2*w^2 - 13*w - 2],\ [709, 709, w^3 - 2*w^2 - 3*w - 2],\ [727, 727, -w^3 + 4*w^2 + 2*w - 7],\ [727, 727, -w^3 + 8*w + 5],\ [733, 733, 3*w^2 - 2*w - 10],\ [733, 733, -w^3 + 5*w + 8],\ [757, 757, 3*w^3 - 14*w - 8],\ [761, 761, -w^3 + 3*w^2 + 5*w - 10],\ [773, 773, -2*w^3 + 3*w^2 + 9*w - 1],\ [787, 787, -w^3 + 4*w - 4],\ [797, 797, w^2 + 2*w - 5],\ [797, 797, w^3 + w^2 - 5*w - 1],\ [809, 809, w^3 + 3*w^2 - 9*w - 17],\ [811, 811, -4*w^2 + 3*w + 17],\ [811, 811, w^3 - 2*w^2 - 8*w + 2],\ [823, 823, -4*w^3 + 4*w^2 + 17*w - 4],\ [857, 857, 3*w^2 - w - 16],\ [881, 881, -2*w^3 + 9*w + 1],\ [883, 883, w^3 - w^2 - 4*w - 5],\ [887, 887, 3*w^2 + w - 10],\ [911, 911, w^3 - 3*w - 7],\ [919, 919, -w^3 + w^2 + 3*w - 7],\ [919, 919, 3*w^3 - 2*w^2 - 12*w + 2],\ [929, 929, 3*w^3 - w^2 - 16*w - 2],\ [929, 929, w^3 + w^2 - 6*w - 2],\ [941, 941, -3*w^3 + 5*w^2 + 14*w - 14],\ [941, 941, -w^3 + 2*w^2 + 6*w - 8],\ [947, 947, 3*w^3 - 3*w^2 - 13*w - 5],\ [947, 947, 3*w^3 - 4*w^2 - 12*w + 2],\ [953, 953, -3*w^3 + 5*w^2 + 10*w - 10],\ [961, 31, -w^3 + 4*w^2 + 3*w - 13],\ [961, 31, -4*w^3 + 3*w^2 + 17*w - 4],\ [967, 967, -w^3 - 4*w^2 + 9*w + 19],\ [967, 967, w^3 + 5*w^2 - 9*w - 26],\ [971, 971, -3*w^3 + 6*w^2 + 13*w - 7],\ [971, 971, w^3 + 2*w^2 - 7*w - 7],\ [977, 977, 2*w^3 - 3*w^2 - 4*w - 2],\ [983, 983, 3*w^3 - 5*w^2 - 7*w + 7],\ [991, 991, 2*w^2 - w - 13],\ [991, 991, w^3 - 8*w - 4]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + x^2 - 6*x - 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -3, e - 1, -3, -e^2 + e + 5, -e^2 - 2*e + 5, 1, 3*e, e - 4, -3, e^2 - 3*e - 9, -e^2 + 3, -3*e^2 - 3*e + 12, -2*e^2 - 3*e + 9, 3*e^2 - 15, -1, -3, 2*e^2 - 2*e - 16, -5*e + 5, -3*e^2 - 2*e + 14, 2*e^2 - 3*e - 22, e^2 + 6, 4*e^2 + 3*e - 18, e^2 + 3*e - 12, 2*e^2 - 3, -2*e^2 - 8*e + 8, e^2 + 6*e + 2, -e^2 - 3*e, -2*e^2 - 3*e - 3, 5*e^2 - 16, e^2 - 5*e - 13, -e^2 - 2*e + 11, 5*e^2 - 18, -e^2 - 3*e + 9, -e^2, -e^2 - 5*e + 2, -e^2 + 9*e + 9, 7*e^2 + 3*e - 24, -3*e^2 + 4*e + 17, e - 1, -e^2 - 3, 3*e^2 + 4*e - 25, -3*e^2 - 2*e + 14, e^2 + 4*e + 5, 3*e^2 - 1, -6*e^2 - 8*e + 20, -e^2 - 2*e + 2, 4*e^2 + 7*e - 10, -4*e^2 + 3*e + 15, 2*e^2 + 3*e + 6, -4*e^2 + 3*e + 24, -2*e^2 - 3, -3*e^2 + 6*e + 18, -e^2 - 5*e + 5, 4*e^2 + 3*e - 12, -3*e^2 + 9*e + 21, -2*e + 23, e^2 - 9*e - 9, -e^2 + 7*e + 5, 2*e^2 - 5*e - 7, 2*e^2 + 9*e - 6, 4*e^2 - 11*e - 28, 6*e^2 + 4*e - 16, -4*e^2 - 8*e + 23, 5*e^2 + e - 10, 3*e^2 - 24, e^2 - 3*e + 9, -5*e^2 - 2*e + 17, -8*e^2 - 5*e + 29, 2*e^2 - 3*e - 21, -4*e^2 - 3*e - 9, -2*e^2 + e + 8, -e^2 - 8*e + 11, -5*e^2 - 5*e + 17, -3*e^2 + 4*e + 29, 5*e^2 + 3*e - 10, 6*e^2 + 13*e - 25, 5*e^2 + 3*e - 4, 3*e^2 - 6*e - 6, 5*e^2 - 2*e - 16, -3*e^2 - 6*e, e^2 - 33, -e^2 + 9*e - 3, 7*e^2 + 12*e - 30, e - 13, 2*e^2 + 6*e - 16, 2*e^2 + 3*e - 13, 10*e^2 + 10*e - 37, -e^2 + 36, 5*e^2 - 3*e - 21, 6*e + 3, -4*e^2 + 3*e + 12, -3*e^2 - 9*e + 27, 5*e^2 + 3*e - 19, -3*e^2 - 9*e + 32, -11*e^2 - 9*e + 36, e^2 + 3*e - 21, -10*e^2 - 6*e + 33, -5*e^2 + 9, e - 13, -3*e^2 + 3*e + 6, 2*e^2 - 12*e - 18, 6*e^2 + 3*e - 24, 10*e^2 + 9*e - 28, -e^2 + e + 8, -2*e^2 - 3*e - 10, -3*e^2 - 6*e + 21, -9*e^2 - 6*e + 30, -e^2 - 6*e - 3, 8*e^2 - 6*e - 42, -7*e^2 + e + 38, -8*e + 11, 4*e^2 + 3*e - 30, -5*e^2 - 3*e - 12, 6*e^2 + 3*e - 30, 4*e^2 + 13*e - 22, -4*e^2 + 4*e + 11, 9*e^2 - 3*e - 43, e^2 + 7*e + 5, 2*e^2 + 12*e - 13, 8*e^2 - 45, -2*e^2 + 15*e + 18, 10*e^2 + 6*e - 30, 4*e^2 + 6*e + 9, -3*e^2 + 9*e + 21, 9*e^2 + 9*e - 30, -3*e - 30, 6*e^2 + 3*e - 37, 2*e^2 - 2*e - 1, -e^2 - 9*e - 25, -9*e^2 + 4*e + 29, 8*e^2 - 6*e - 43, 4*e^2 + 15*e - 25, -2*e^2 + 10*e + 29, 6*e^2 + e - 19, 3*e^2 - 3*e - 3, e^2 + 18, -3*e^2 + 4*e + 14, -9*e^2 - 6*e + 21, -e^2 + 6*e, 11*e^2 + 3*e - 33, e^2 - 2*e - 46, 3*e^2 - 11*e - 16, e^2 + e - 19, -2*e^2 - 12*e + 9, -6*e^2 + 24, 4*e^2 + 10*e - 4, 13*e^2 + 3*e - 51, -12*e^2 - 12*e + 39, 4*e^2 + 12*e - 19, -8*e^2 + e + 29, -e^2 - 3*e + 42, 5*e^2, 2*e^2 + 3*e - 30, 3*e^2 + 3, -3*e^2 + 15*e + 27, -6*e^2 - 3*e + 6, -14*e^2 + 3*e + 66, -7*e^2 - 8*e + 29, 3*e^2 - 17*e - 31, -7*e^2 - 18*e + 35, -10*e^2 + e + 44, e^2 + 6*e - 42, -13*e^2 - 15*e + 48, -4*e^2 + 6*e + 30, -3*e^2 + 15*e + 9, e^2 + 7*e + 26, -8*e^2 - 11*e + 5] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([17, 17, -w^3 + 5*w + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]