/* 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, 3, -1, -5, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([85, 85, 3*w^4 - w^3 - 14*w^2 + 3*w + 5]) primes_array = [ [5, 5, -2*w^4 + w^3 + 9*w^2 - 2*w - 3],\ [17, 17, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],\ [17, 17, w^2 - 2],\ [23, 23, w^3 - 3*w],\ [29, 29, 2*w^4 - w^3 - 10*w^2 + 2*w + 4],\ [32, 2, 2],\ [37, 37, -w^4 + w^3 + 4*w^2 - 2*w],\ [41, 41, -3*w^4 + 2*w^3 + 14*w^2 - 5*w - 7],\ [43, 43, -3*w^4 + w^3 + 14*w^2 - 3*w - 6],\ [47, 47, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],\ [53, 53, 2*w^4 - w^3 - 8*w^2 + 2*w + 1],\ [53, 53, -2*w^4 + w^3 + 10*w^2 - 4*w - 6],\ [59, 59, -w^4 + 4*w^2 + 1],\ [59, 59, -3*w^4 + 2*w^3 + 14*w^2 - 6*w - 8],\ [61, 61, 4*w^4 - 2*w^3 - 18*w^2 + 5*w + 7],\ [61, 61, 3*w^4 - w^3 - 15*w^2 + 2*w + 7],\ [73, 73, -4*w^4 + 2*w^3 + 18*w^2 - 7*w - 6],\ [83, 83, -2*w^4 + 9*w^2 + 2*w - 4],\ [83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6],\ [97, 97, -2*w^4 + w^3 + 8*w^2 - 3*w + 1],\ [97, 97, -w^3 + w^2 + 4*w - 1],\ [101, 101, 2*w^4 - 9*w^2 - w + 5],\ [103, 103, 2*w^4 - 2*w^3 - 9*w^2 + 8*w + 2],\ [107, 107, -3*w^4 + w^3 + 15*w^2 - 4*w - 8],\ [109, 109, -3*w^4 + w^3 + 13*w^2 - 2*w - 4],\ [113, 113, 4*w^4 - 3*w^3 - 18*w^2 + 8*w + 8],\ [121, 11, -w^4 - w^3 + 5*w^2 + 4*w - 2],\ [131, 131, -4*w^4 + w^3 + 18*w^2 - 5],\ [139, 139, 2*w^3 - w^2 - 8*w + 1],\ [139, 139, -2*w^4 + w^3 + 9*w^2 - 3],\ [149, 149, 5*w^4 - 2*w^3 - 23*w^2 + 4*w + 11],\ [149, 149, w^4 - 4*w^2 - 3],\ [149, 149, 4*w^4 - 3*w^3 - 19*w^2 + 10*w + 8],\ [151, 151, 2*w^4 - w^3 - 9*w^2 + 5*w + 4],\ [157, 157, -2*w^4 - w^3 + 10*w^2 + 5*w - 4],\ [157, 157, 5*w^4 - 3*w^3 - 24*w^2 + 8*w + 13],\ [163, 163, -2*w^4 + 11*w^2 + w - 7],\ [167, 167, w^4 - 3*w^2 - 2*w - 3],\ [167, 167, -3*w^4 + 3*w^3 + 14*w^2 - 10*w - 9],\ [169, 13, 3*w^4 - 2*w^3 - 14*w^2 + 6*w + 3],\ [191, 191, -3*w^4 + w^3 + 15*w^2 - w - 6],\ [193, 193, 3*w^4 - 2*w^3 - 14*w^2 + 4*w + 5],\ [199, 199, -5*w^4 + 2*w^3 + 23*w^2 - 3*w - 10],\ [223, 223, -4*w^4 + w^3 + 20*w^2 - w - 9],\ [223, 223, -w^4 + 3*w^2 + 2],\ [233, 233, w^3 - 4*w - 4],\ [241, 241, -3*w^4 + 2*w^3 + 15*w^2 - 7*w - 7],\ [243, 3, -3],\ [251, 251, -3*w^4 + 3*w^3 + 13*w^2 - 8*w - 6],\ [257, 257, -4*w^4 + w^3 + 20*w^2 - 2*w - 9],\ [257, 257, -3*w^4 + 2*w^3 + 15*w^2 - 5*w - 11],\ [269, 269, 2*w^4 - 11*w^2 - w + 6],\ [271, 271, w^4 + w^3 - 5*w^2 - 6*w + 2],\ [271, 271, 3*w^4 - w^3 - 15*w^2 + 8],\ [277, 277, -w^4 - w^3 + 6*w^2 + 3*w - 5],\ [283, 283, -5*w^4 + 3*w^3 + 23*w^2 - 10*w - 10],\ [293, 293, -w^4 + w^3 + 6*w^2 - 3*w - 8],\ [307, 307, -w^4 - w^3 + 6*w^2 + 4*w - 2],\ [311, 311, -w^3 + 2*w^2 + 3*w - 2],\ [311, 311, 2*w^4 - w^3 - 10*w^2 + 5*w + 7],\ [313, 313, -5*w^4 + 4*w^3 + 23*w^2 - 12*w - 11],\ [313, 313, -w^4 - w^3 + 5*w^2 + 3*w - 3],\ [347, 347, -w^4 + 2*w^3 + 3*w^2 - 7*w + 1],\ [353, 353, -3*w^4 + 14*w^2 + 3*w - 7],\ [353, 353, -3*w^4 + 3*w^3 + 13*w^2 - 9*w - 7],\ [367, 367, -w^4 + w^3 + 5*w^2 - 6*w - 4],\ [367, 367, -3*w^4 + w^3 + 15*w^2 - 3*w - 5],\ [379, 379, 5*w^4 - 2*w^3 - 23*w^2 + 6*w + 9],\ [383, 383, 4*w^4 - 2*w^3 - 19*w^2 + 7*w + 11],\ [397, 397, w^4 - 3*w^2 - w - 3],\ [397, 397, -2*w^3 + w^2 + 8*w],\ [401, 401, 2*w^4 - 2*w^3 - 8*w^2 + 5*w],\ [421, 421, -4*w^4 + 2*w^3 + 17*w^2 - 5*w - 6],\ [421, 421, -w^4 + 2*w^3 + 6*w^2 - 8*w - 5],\ [431, 431, 5*w^4 - 2*w^3 - 22*w^2 + 5*w + 7],\ [431, 431, -w^4 - w^3 + 6*w^2 + 5*w - 4],\ [439, 439, 4*w^4 - 2*w^3 - 19*w^2 + 7*w + 10],\ [443, 443, 4*w^4 - w^3 - 19*w^2 + 3*w + 8],\ [457, 457, -3*w^4 + 3*w^3 + 15*w^2 - 10*w - 9],\ [461, 461, -w^3 + 6*w],\ [461, 461, 3*w^4 - 3*w^3 - 14*w^2 + 9*w + 8],\ [463, 463, 5*w^4 - 3*w^3 - 23*w^2 + 7*w + 8],\ [499, 499, 6*w^4 - 3*w^3 - 27*w^2 + 7*w + 11],\ [509, 509, 2*w^4 - w^3 - 8*w^2 + w - 1],\ [509, 509, 3*w^4 - 2*w^3 - 13*w^2 + 6*w + 7],\ [521, 521, w^3 - 6*w + 1],\ [523, 523, 4*w^4 - 2*w^3 - 18*w^2 + 3*w + 7],\ [541, 541, w^2 + w - 5],\ [547, 547, 5*w^4 - 2*w^3 - 25*w^2 + 4*w + 12],\ [547, 547, -6*w^4 + 3*w^3 + 27*w^2 - 10*w - 9],\ [557, 557, -4*w^4 + 3*w^3 + 18*w^2 - 10*w - 9],\ [557, 557, -4*w^4 + w^3 + 19*w^2 - 3*w - 6],\ [557, 557, 3*w^4 - 2*w^3 - 14*w^2 + 8*w + 7],\ [563, 563, -w^4 + 3*w^3 + 4*w^2 - 11*w + 1],\ [569, 569, w^2 + 3*w - 2],\ [577, 577, -5*w^4 + 2*w^3 + 23*w^2 - 6*w - 8],\ [587, 587, -2*w^4 + 10*w^2 + 3*w - 6],\ [593, 593, 4*w^4 - 3*w^3 - 19*w^2 + 8*w + 11],\ [601, 601, -4*w^4 + w^3 + 18*w^2 - 3*w - 6],\ [601, 601, -5*w^4 + 3*w^3 + 24*w^2 - 10*w - 13],\ [601, 601, -w^4 + 5*w^2 + 3*w - 3],\ [607, 607, -3*w^4 + 2*w^3 + 15*w^2 - 8*w - 9],\ [613, 613, 3*w^4 - 14*w^2 - 2*w + 8],\ [617, 617, -4*w^4 + w^3 + 19*w^2 - 3*w - 7],\ [619, 619, w^3 + w^2 - 3*w - 5],\ [625, 5, 4*w^4 - 2*w^3 - 19*w^2 + 3*w + 8],\ [631, 631, 3*w^4 - w^3 - 12*w^2 - 1],\ [641, 641, -w^4 + 2*w^3 + 3*w^2 - 7*w],\ [647, 647, 2*w^2 - w - 5],\ [647, 647, -2*w^3 + w^2 + 8*w - 5],\ [647, 647, 5*w^4 - 3*w^3 - 23*w^2 + 9*w + 12],\ [661, 661, -8*w^4 + 3*w^3 + 39*w^2 - 7*w - 19],\ [677, 677, -7*w^4 + 4*w^3 + 34*w^2 - 11*w - 17],\ [691, 691, w^2 + 2*w - 4],\ [691, 691, -5*w^4 + 2*w^3 + 23*w^2 - 3*w - 8],\ [691, 691, -7*w^4 + 5*w^3 + 32*w^2 - 14*w - 15],\ [701, 701, w - 4],\ [709, 709, -3*w^4 + 3*w^3 + 14*w^2 - 8*w - 7],\ [709, 709, 5*w^4 - 2*w^3 - 24*w^2 + 5*w + 9],\ [709, 709, -w^4 + w^3 + 5*w^2 - 3*w - 7],\ [719, 719, -3*w^4 + 13*w^2 + w - 7],\ [719, 719, -3*w^4 + 14*w^2 - 4],\ [727, 727, -7*w^4 + 4*w^3 + 32*w^2 - 10*w - 16],\ [727, 727, -3*w^4 + w^3 + 14*w^2 - 5*w - 8],\ [733, 733, -3*w^4 + 2*w^3 + 15*w^2 - 8*w - 10],\ [733, 733, -3*w^4 + 3*w^3 + 13*w^2 - 10*w - 7],\ [733, 733, 8*w^4 - 4*w^3 - 38*w^2 + 9*w + 15],\ [739, 739, 4*w^4 - 3*w^3 - 20*w^2 + 9*w + 11],\ [739, 739, -w^4 + w^3 + 5*w^2 - 3*w + 1],\ [743, 743, -5*w^4 + 4*w^3 + 24*w^2 - 13*w - 16],\ [743, 743, w^4 - 7*w^2 - 2*w + 5],\ [751, 751, -w^4 + 2*w^3 + 3*w^2 - 7*w - 1],\ [757, 757, 5*w^4 - 2*w^3 - 22*w^2 + 5*w + 8],\ [757, 757, w^4 + w^3 - 6*w^2 - 5*w + 2],\ [761, 761, 8*w^4 - 3*w^3 - 38*w^2 + 8*w + 18],\ [761, 761, 3*w^4 - w^3 - 14*w^2 + 2*w + 10],\ [761, 761, -4*w^4 + 3*w^3 + 19*w^2 - 7*w - 10],\ [769, 769, -w^4 + 2*w^3 + 5*w^2 - 8*w - 1],\ [773, 773, 9*w^4 - 4*w^3 - 41*w^2 + 9*w + 15],\ [773, 773, 4*w^4 - 2*w^3 - 18*w^2 + 3*w + 6],\ [773, 773, 2*w^2 - 5],\ [787, 787, -w^4 + 6*w^2 - w - 7],\ [787, 787, w^4 - w^3 - 6*w^2 + w + 6],\ [821, 821, 2*w^4 - w^3 - 9*w^2 + 4*w + 8],\ [823, 823, 6*w^4 - w^3 - 27*w^2 - w + 8],\ [827, 827, -3*w^4 + 2*w^3 + 16*w^2 - 6*w - 9],\ [827, 827, -8*w^4 + 5*w^3 + 37*w^2 - 16*w - 17],\ [839, 839, 2*w^4 - 10*w^2 + w + 8],\ [853, 853, 2*w^4 - 2*w^3 - 11*w^2 + 7*w + 7],\ [857, 857, 5*w^4 - 2*w^3 - 24*w^2 + 7*w + 14],\ [877, 877, -w^4 + 2*w^3 + 5*w^2 - 10*w - 6],\ [883, 883, 3*w^4 - 3*w^3 - 15*w^2 + 12*w + 10],\ [887, 887, -4*w^4 + w^3 + 17*w^2 - w - 7],\ [907, 907, 2*w^4 - 11*w^2 - w + 2],\ [907, 907, -2*w^4 - w^3 + 10*w^2 + 4*w - 5],\ [911, 911, -9*w^4 + 4*w^3 + 43*w^2 - 9*w - 19],\ [919, 919, 4*w^4 - w^3 - 20*w^2 + w + 8],\ [937, 937, w^4 - 7*w^2 - 2*w + 4],\ [947, 947, 2*w^4 + 2*w^3 - 10*w^2 - 9*w + 3],\ [953, 953, -2*w^3 + 5*w + 4],\ [961, 31, -4*w^4 + w^3 + 19*w^2 - w - 4],\ [967, 967, 8*w^4 - 5*w^3 - 37*w^2 + 13*w + 18],\ [967, 967, 3*w^4 - w^3 - 13*w^2 - 2*w + 3],\ [971, 971, -4*w^4 + 3*w^3 + 21*w^2 - 10*w - 15],\ [971, 971, -3*w^4 + 12*w^2 + 3*w - 4],\ [971, 971, w^4 - 6*w^2 + 3*w + 6],\ [983, 983, -7*w^4 + 2*w^3 + 34*w^2 - 5*w - 16],\ [983, 983, w^4 - 7*w^2 - w + 8],\ [991, 991, -4*w^4 + 4*w^3 + 19*w^2 - 15*w - 12],\ [991, 991, -8*w^4 + 3*w^3 + 37*w^2 - 5*w - 15],\ [991, 991, -w^4 + w^3 + 7*w^2 - 3*w - 7],\ [997, 997, -8*w^4 + 5*w^3 + 37*w^2 - 16*w - 16]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -2, -1, 5, 1, -5, 5, -5, 0, -11, 5, 9, -13, 7, 11, 6, -6, 12, 3, -9, 12, -9, 8, -8, 9, 9, 17, 16, -5, -1, -8, 18, -18, 16, -22, -6, -9, 12, 3, 10, -9, -14, -5, -2, -2, -21, 15, 2, -1, 8, -16, 10, 25, -8, -13, -16, 4, 25, -29, 31, 22, -25, 30, 12, 10, 6, -19, 14, -33, 7, -3, 23, 13, -30, 36, 12, 20, 12, -26, 0, -12, -32, 11, -8, 11, 12, 4, 22, -22, -27, 38, -8, -16, -6, 0, -38, -44, -12, 15, 24, 14, -22, -29, 10, -3, 32, 33, 25, -37, -8, -20, -20, -43, 23, -25, -36, 6, 50, -24, -35, 6, -27, 27, 10, -34, -6, 6, 20, 12, 24, 19, 16, 16, 22, 22, -10, 12, -30, 24, -31, 6, -11, -2, 0, -40, 0, -51, -6, -28, -36, -52, 43, 52, 30, 33, -59, -18, -9, -38, -18, 11, 10, -44, 60, -20, -18, -27, -27, -34, 38, 5, -46] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, -2*w^4 + w^3 + 9*w^2 - 2*w - 3])] = -1 AL_eigenvalues[ZF.ideal([17, 17, w^2 - 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]