/* 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([1, 5, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [5, 5, w^3 - 4*w],\ [7, 7, w + 1],\ [7, 7, -w^3 + 4*w - 1],\ [13, 13, -w^2 + 3],\ [16, 2, 2],\ [17, 17, w^3 + w^2 - 4*w - 2],\ [17, 17, -w^3 + 5*w - 2],\ [19, 19, -w^2 - w + 4],\ [19, 19, -w^2 - w + 1],\ [29, 29, 2*w^3 - w^2 - 9*w + 5],\ [41, 41, -w^3 + w^2 + 5*w - 3],\ [43, 43, w^3 - w^2 - 4*w + 2],\ [43, 43, w^3 - 6*w],\ [47, 47, -w^3 - w^2 + 5*w],\ [49, 7, w^2 + 2*w - 2],\ [59, 59, 2*w^3 - 8*w + 3],\ [67, 67, w^2 - w - 4],\ [79, 79, w^3 - w^2 - 4*w + 1],\ [81, 3, -3],\ [97, 97, w^3 + w^2 - 5*w + 1],\ [107, 107, -2*w^3 + 3*w^2 + 10*w - 12],\ [109, 109, 2*w^3 - w^2 - 9*w + 3],\ [125, 5, -3*w^3 - w^2 + 12*w + 1],\ [127, 127, -2*w^3 + w^2 + 10*w - 8],\ [137, 137, -w^3 + w^2 + 3*w - 5],\ [139, 139, w^3 - w^2 - 5*w + 1],\ [149, 149, 2*w^2 - w - 6],\ [149, 149, 2*w^3 + w^2 - 9*w],\ [163, 163, w^3 - 5*w - 3],\ [163, 163, 2*w^3 - 7*w],\ [173, 173, -2*w^3 + 11*w - 3],\ [173, 173, -w^3 + w^2 + 4*w - 8],\ [179, 179, 2*w^3 - w^2 - 8*w + 5],\ [181, 181, 2*w^3 - w^2 - 8*w],\ [181, 181, 2*w^3 - 9*w - 3],\ [191, 191, 2*w^2 - 5],\ [191, 191, -2*w^2 - w + 10],\ [193, 193, w^3 + w^2 - 3*w - 4],\ [193, 193, -w^3 + 2*w^2 + 3*w - 3],\ [197, 197, -2*w^3 + 9*w - 4],\ [197, 197, w^2 - 7],\ [199, 199, 3*w^3 - 13*w + 1],\ [199, 199, w^3 + 2*w^2 - 5*w - 7],\ [227, 227, -3*w^3 + w^2 + 13*w - 4],\ [227, 227, -w^3 + 2*w^2 + 7*w - 6],\ [241, 241, -2*w^3 + 7*w - 1],\ [257, 257, -2*w^3 + w^2 + 7*w - 5],\ [257, 257, 2*w^3 - w^2 - 7*w + 2],\ [263, 263, -w^3 + 7*w - 3],\ [269, 269, -2*w^3 + 2*w^2 + 11*w - 8],\ [271, 271, w^2 + 3*w - 2],\ [271, 271, w^2 + 2*w - 7],\ [277, 277, -2*w^3 + 11*w - 2],\ [281, 281, 2*w^3 - 11*w + 1],\ [283, 283, 2*w^3 - 9*w + 5],\ [283, 283, -2*w^3 + w^2 + 11*w - 7],\ [289, 17, -w^3 + 2*w^2 + 4*w - 4],\ [293, 293, -2*w^3 - w^2 + 7*w + 3],\ [307, 307, -3*w^3 + 2*w^2 + 13*w - 9],\ [311, 311, -w^3 + w^2 + 3*w - 7],\ [313, 313, -w^3 + 2*w^2 + 3*w - 2],\ [313, 313, w^2 - w - 8],\ [317, 317, -2*w^3 - w^2 + 9*w - 1],\ [331, 331, w^3 + 2*w^2 - 5*w - 3],\ [337, 337, w^3 + 2*w^2 - 6*w - 4],\ [337, 337, w^3 - 2*w^2 - 6*w + 5],\ [347, 347, 2*w^3 + 2*w^2 - 9*w - 3],\ [347, 347, -2*w^3 + w^2 + 7*w - 4],\ [349, 349, w^2 - 2*w - 4],\ [353, 353, -w^3 + 2*w^2 + 2*w - 6],\ [359, 359, -w^3 + w^2 + 7*w - 6],\ [359, 359, -w^2 - w - 2],\ [361, 19, 2*w^3 - w^2 - 11*w],\ [367, 367, -2*w^3 + 2*w^2 + 8*w - 5],\ [373, 373, w^3 + w^2 - 2*w - 4],\ [379, 379, 2*w^3 - w^2 - 11*w + 4],\ [389, 389, 3*w^2 + w - 11],\ [397, 397, 3*w^3 - 13*w + 5],\ [401, 401, w - 5],\ [409, 409, 3*w^3 - 12*w + 4],\ [409, 409, 3*w^3 + 2*w^2 - 14*w - 4],\ [421, 421, 2*w^3 - w^2 - 10*w + 1],\ [421, 421, w^3 + 2*w^2 - 5*w - 4],\ [439, 439, -2*w^3 + 10*w - 5],\ [443, 443, -2*w^3 + 12*w - 5],\ [443, 443, -4*w^3 + w^2 + 18*w - 9],\ [443, 443, -2*w^3 + w^2 + 7*w - 9],\ [443, 443, 3*w^3 + 2*w^2 - 14*w - 5],\ [449, 449, w^3 + 2*w^2 - 2*w - 6],\ [449, 449, -2*w^3 + w^2 + 11*w - 5],\ [461, 461, 2*w^3 + 2*w^2 - 9*w - 4],\ [467, 467, -w^3 + w^2 + 7*w - 5],\ [487, 487, 2*w^2 + 4*w - 7],\ [491, 491, -w^3 + w^2 + 7*w - 4],\ [503, 503, 2*w^3 + w^2 - 9*w + 2],\ [521, 521, w^2 - 3*w - 3],\ [523, 523, -w^3 + 2*w^2 + 5*w - 4],\ [523, 523, -w^3 + 8*w - 5],\ [563, 563, w^3 + 3*w^2 - w - 8],\ [563, 563, w^3 - 2*w^2 - 8*w + 4],\ [563, 563, -w^3 + w^2 + 2*w - 5],\ [563, 563, -3*w^3 - w^2 + 10*w - 2],\ [577, 577, 3*w^3 - 16*w],\ [587, 587, w^3 + 2*w^2 - 6],\ [587, 587, -w^3 + 4*w - 6],\ [601, 601, -w - 5],\ [607, 607, w^3 - 2*w^2 - 4*w + 1],\ [613, 613, 4*w^3 - 17*w],\ [617, 617, -3*w^3 + w^2 + 14*w - 2],\ [617, 617, -4*w^3 + 17*w - 4],\ [619, 619, -3*w^3 + 12*w - 2],\ [643, 643, 3*w^3 - w^2 - 12*w + 3],\ [643, 643, -2*w^3 - w^2 + 11*w - 2],\ [647, 647, 2*w^3 + w^2 - 9*w + 3],\ [653, 653, w^3 + w^2 - 7*w - 4],\ [661, 661, -w^3 + 4*w^2 + 6*w - 15],\ [661, 661, 3*w^3 - 2*w^2 - 12*w + 10],\ [677, 677, -3*w^2 - 2*w + 12],\ [691, 691, -2*w^3 + 3*w^2 + 11*w - 10],\ [691, 691, -3*w^3 + 16*w - 3],\ [701, 701, 3*w^3 - 13*w + 6],\ [719, 719, -3*w^2 + w + 10],\ [727, 727, w^3 - 3*w^2 - 4*w + 6],\ [743, 743, -w^3 + 2*w^2 + 5*w - 3],\ [761, 761, -w^3 + 3*w^2 + 3*w - 12],\ [761, 761, 3*w^3 - 2*w^2 - 14*w + 7],\ [769, 769, -3*w^3 + 3*w^2 + 16*w - 13],\ [769, 769, w^3 + 2*w^2 - 7*w - 6],\ [773, 773, w^3 - 8*w + 4],\ [773, 773, 4*w^2 + w - 16],\ [787, 787, w^2 + 4*w - 2],\ [797, 797, -2*w^3 + 3*w^2 + 10*w - 10],\ [797, 797, -w^3 - w^2 + 4*w - 4],\ [797, 797, w^3 + 2*w^2 - 3*w - 9],\ [797, 797, -3*w^3 + 3*w^2 + 11*w - 10],\ [821, 821, w^2 + w - 9],\ [827, 827, w^3 - 8*w + 1],\ [827, 827, -2*w^3 + w^2 + 9*w - 11],\ [827, 827, 3*w^3 - 14*w + 5],\ [827, 827, 2*w^3 + w^2 - 7*w - 5],\ [829, 829, -2*w^3 - 2*w^2 + 9*w - 1],\ [839, 839, -3*w^3 + 15*w - 5],\ [853, 853, -4*w^3 + 15*w - 8],\ [853, 853, -w^3 + 2*w^2 + 6*w - 3],\ [859, 859, -w^3 + w^2 + w - 4],\ [859, 859, w^3 + w^2 - w - 5],\ [881, 881, -w^3 + w^2 + 2*w - 6],\ [881, 881, w^3 + w^2 - 2*w - 6],\ [883, 883, w^3 + 3*w^2 - 3*w - 6],\ [887, 887, w^3 + 3*w^2 - 3*w - 8],\ [919, 919, 2*w^3 - w^2 - 13*w],\ [929, 929, -w^3 - 3*w^2 + 2*w + 10],\ [929, 929, w^3 + w^2 - 5*w - 8],\ [941, 941, w^3 + 2*w^2 - 3*w - 10],\ [953, 953, -w^3 + 5*w - 7],\ [953, 953, -2*w^3 + 3*w^2 + 9*w - 9],\ [953, 953, -2*w^3 - w^2 + 10*w - 3],\ [953, 953, -3*w^3 + 11*w - 1],\ [961, 31, -w^3 + 3*w^2 + 2*w - 10],\ [961, 31, -3*w^2 + 2*w + 7],\ [967, 967, w^3 + 3*w^2 - 3*w - 7],\ [967, 967, 4*w^3 + w^2 - 20*w + 1],\ [983, 983, 2*w^3 - w^2 - 12*w + 8],\ [983, 983, 2*w^3 + w^2 - 6*w - 4],\ [991, 991, 2*w^3 - 12*w + 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 22*x^2 + 100 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1/10*e^3 + 6/5*e, e, e, -1/10*e^3 + 11/5*e, -1, -3/10*e^3 + 18/5*e, e^2 - 12, e^2 - 10, -1/5*e^3 + 17/5*e, -1/2*e^3 + 6*e, 3/10*e^3 - 23/5*e, 2/5*e^3 - 24/5*e, 10, -e^2 + 10, e^2 - 14, -e, 0, -1/5*e^3 + 12/5*e, -e^2 + 10, -4, 1/5*e^3 - 22/5*e, -3/10*e^3 + 33/5*e, -1/10*e^3 - 4/5*e, -2*e^2 + 18, -3/10*e^3 + 8/5*e, -1/5*e^3 + 7/5*e, 1/2*e^3 - 9*e, 3*e^2 - 42, 5*e^2 - 54, 4/5*e^3 - 43/5*e, -2*e^2 + 30, e^2 - 26, -e^3 + 10*e, 1/10*e^3 - 16/5*e, 2*e^2 - 14, -4/5*e^3 + 53/5*e, -12, -3*e^2 + 46, -7/10*e^3 + 52/5*e, 2*e^2 - 32, -6, -6/5*e^3 + 67/5*e, 6/5*e^3 - 72/5*e, 1/5*e^3 - 32/5*e, -4*e^2 + 42, 11/10*e^3 - 86/5*e, 7/10*e^3 - 67/5*e, 1/10*e^3 - 11/5*e, -4*e^2 + 44, e^2 + 2, -4*e^2 + 52, 0, e^2 - 24, e^2 + 14, 4*e^2 - 40, e^2 - 18, -3/10*e^3 + 18/5*e, 3*e^2 - 34, 7/5*e^3 - 94/5*e, -4/5*e^3 + 43/5*e, -3/10*e^3 + 33/5*e, -3*e^2 + 30, -e^2 + 8, 3/5*e^3 - 61/5*e, -3/10*e^3 + 48/5*e, -21/10*e^3 + 131/5*e, 8/5*e^3 - 116/5*e, 1/5*e^3 - 42/5*e, -17/10*e^3 + 87/5*e, -11/10*e^3 + 76/5*e, 2*e^2 - 40, 2*e^2 - 40, 3/2*e^3 - 21*e, 2/5*e^3 - 34/5*e, -3*e^2 + 22, 6*e^2 - 72, -21/10*e^3 + 121/5*e, 2*e^2 - 36, 2*e^2 + 2, -3/10*e^3 + 8/5*e, 1/10*e^3 - 1/5*e, 11/10*e^3 - 101/5*e, 9/10*e^3 - 54/5*e, -4*e^2 + 50, 4*e^2 - 50, 5*e^2 - 46, 6/5*e^3 - 47/5*e, -8/5*e^3 + 101/5*e, 3*e^2 - 14, -3*e^2 + 30, 13/10*e^3 - 123/5*e, 4*e^2 - 26, -2*e^2 + 4, -6*e^2 + 60, -4*e^2 + 46, -1/2*e^3 + 8*e, -7/5*e^3 + 104/5*e, -2*e^2 + 40, -2*e^2 + 46, 5*e^2 - 62, -e^3 + 7*e, 3/5*e^3 - 1/5*e, 2*e^2 + 8, 6*e^2 - 68, 4*e^2 - 42, -2*e^2 + 50, -6/5*e^3 + 67/5*e, -1/2*e^3 + 11*e, -17/10*e^3 + 127/5*e, -1/2*e^3 + 20*e, -9/5*e^3 + 88/5*e, -7/5*e^3 + 94/5*e, 2*e^2 - 28, 4*e^2 - 16, 1/2*e^3 + 3*e, -23/10*e^3 + 168/5*e, 1/2*e^3 - 13*e, -e^2 - 6, -11/5*e^3 + 142/5*e, 2*e^2 + 20, e^2 - 8, 3/5*e^3 - 31/5*e, -1/5*e^3 - 8/5*e, 2*e^3 - 18*e, -21/10*e^3 + 141/5*e, 9/10*e^3 - 54/5*e, 5*e^2 - 60, 3/10*e^3 + 7/5*e, 6*e^2 - 52, 13/10*e^3 - 98/5*e, -3*e^2 + 62, -1/2*e^3, 3*e^2 - 40, -3*e^2 + 38, -13/10*e^3 + 103/5*e, 9*e^2 - 94, -2*e^2 + 38, -10*e^2 + 108, -4*e^2 + 30, -7*e^2 + 86, 2*e^2 + 22, 5*e^2 - 58, 5/2*e^3 - 32*e, 3/2*e^3 - 19*e, -4/5*e^3 + 3/5*e, -2*e^2 + 36, -11/10*e^3 + 66/5*e, e^2 - 32, -12/5*e^3 + 139/5*e, 4/5*e^3 - 58/5*e, -32, e^2 - 18, 7*e^2 - 62, -2*e^2 + 50, -6*e^2 + 46, 3/10*e^3 + 37/5*e, 2*e^2 - 58, 3/10*e^3 + 37/5*e, 19/10*e^3 - 79/5*e, -13/10*e^3 + 78/5*e, -e^3 + 15*e, 4*e^2 - 36, -5*e^2 + 46, 3*e^2 - 58, -2*e^2 + 66] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([16, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]