/* 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([-25, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25,25,-w + 1]) primes_array = [ [4, 2, 2],\ [5, 5, -w + 5],\ [5, 5, -w - 4],\ [9, 3, 3],\ [13, 13, w + 3],\ [13, 13, w - 4],\ [17, 17, w + 6],\ [17, 17, -w + 7],\ [19, 19, w + 2],\ [19, 19, w - 3],\ [23, 23, w + 1],\ [23, 23, -w + 2],\ [31, 31, -w - 7],\ [31, 31, w - 8],\ [37, 37, 2*w - 9],\ [37, 37, 2*w + 7],\ [43, 43, 4*w + 17],\ [43, 43, 4*w - 21],\ [47, 47, -w - 8],\ [47, 47, w - 9],\ [49, 7, -7],\ [71, 71, 3*w - 14],\ [71, 71, -5*w + 29],\ [79, 79, 5*w + 21],\ [79, 79, 5*w - 26],\ [97, 97, 2*w - 3],\ [97, 97, -2*w - 1],\ [101, 101, 2*w - 1],\ [107, 107, -w - 11],\ [107, 107, w - 12],\ [121, 11, -11],\ [131, 131, -w - 12],\ [131, 131, w - 13],\ [137, 137, 3*w - 11],\ [137, 137, -3*w - 8],\ [157, 157, -w - 13],\ [157, 157, w - 14],\ [179, 179, -4*w - 13],\ [179, 179, 4*w - 17],\ [181, 181, 7*w + 29],\ [181, 181, 7*w - 36],\ [193, 193, 3*w - 22],\ [193, 193, -3*w - 19],\ [197, 197, -3*w - 4],\ [197, 197, 3*w - 7],\ [211, 211, -9*w + 52],\ [211, 211, 5*w - 23],\ [223, 223, 2*w - 19],\ [223, 223, -2*w - 17],\ [227, 227, 3*w - 2],\ [227, 227, 3*w - 1],\ [233, 233, 6*w - 29],\ [233, 233, 6*w + 23],\ [239, 239, 9*w - 47],\ [239, 239, 9*w + 38],\ [251, 251, -5*w + 22],\ [251, 251, 5*w + 17],\ [281, 281, -w - 17],\ [281, 281, w - 18],\ [283, 283, -4*w - 9],\ [283, 283, 4*w - 13],\ [307, 307, 7*w - 34],\ [307, 307, 7*w + 27],\ [317, 317, -w - 18],\ [317, 317, w - 19],\ [359, 359, 5*w - 19],\ [359, 359, -5*w - 14],\ [367, 367, -11*w + 64],\ [367, 367, 7*w - 33],\ [373, 373, -3*w - 23],\ [373, 373, 3*w - 26],\ [379, 379, 4*w - 7],\ [379, 379, -4*w - 3],\ [383, 383, 2*w - 23],\ [383, 383, -2*w - 21],\ [409, 409, 10*w + 41],\ [409, 409, 10*w - 51],\ [421, 421, 5*w - 17],\ [421, 421, -5*w - 12],\ [449, 449, -5*w - 11],\ [449, 449, 5*w - 16],\ [491, 491, 5*w - 36],\ [491, 491, -5*w - 31],\ [499, 499, 5*w - 14],\ [499, 499, -5*w - 9],\ [509, 509, 6*w - 23],\ [509, 509, -6*w - 17],\ [521, 521, -5*w - 8],\ [521, 521, 5*w - 13],\ [541, 541, 5*w - 12],\ [541, 541, -5*w - 7],\ [557, 557, 4*w - 33],\ [557, 557, -4*w - 29],\ [563, 563, 9*w + 34],\ [563, 563, 9*w - 43],\ [569, 569, 15*w + 64],\ [569, 569, 15*w - 79],\ [587, 587, -7*w - 22],\ [587, 587, 7*w - 29],\ [593, 593, 14*w + 59],\ [593, 593, 14*w - 73],\ [601, 601, -5*w - 3],\ [601, 601, 5*w - 8],\ [607, 607, 13*w - 67],\ [607, 607, 13*w + 54],\ [619, 619, -5*w - 1],\ [619, 619, 5*w - 6],\ [631, 631, 5*w - 2],\ [631, 631, 5*w - 3],\ [643, 643, 3*w - 31],\ [643, 643, -3*w - 28],\ [653, 653, -6*w - 13],\ [653, 653, 6*w - 19],\ [677, 677, -w - 26],\ [677, 677, w - 27],\ [683, 683, 2*w - 29],\ [683, 683, -2*w - 27],\ [691, 691, 6*w - 43],\ [691, 691, -6*w - 37],\ [701, 701, -5*w - 34],\ [701, 701, 5*w - 39],\ [727, 727, 13*w + 53],\ [727, 727, 13*w - 66],\ [743, 743, 7*w + 41],\ [743, 743, 19*w + 82],\ [761, 761, -16*w + 93],\ [761, 761, 10*w - 47],\ [787, 787, -w - 28],\ [787, 787, w - 29],\ [809, 809, -6*w - 7],\ [809, 809, 6*w - 13],\ [821, 821, 4*w - 37],\ [821, 821, -4*w - 33],\ [827, 827, 12*w - 59],\ [827, 827, 12*w + 47],\ [829, 829, 3*w - 34],\ [829, 829, -3*w - 31],\ [839, 839, 16*w - 83],\ [839, 839, 16*w + 67],\ [841, 29, -29],\ [853, 853, 14*w + 57],\ [853, 853, 14*w - 71],\ [857, 857, 7*w - 23],\ [857, 857, -7*w - 16],\ [887, 887, 8*w - 31],\ [887, 887, -8*w - 23],\ [929, 929, -5*w - 37],\ [929, 929, 5*w - 42],\ [967, 967, -w - 31],\ [967, 967, w - 32],\ [977, 977, 17*w + 71],\ [977, 977, 17*w - 88],\ [991, 991, 8*w - 29],\ [991, 991, -8*w - 21],\ [997, 997, -7*w - 12],\ [997, 997, 7*w - 19]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 3*x^3 - 9*x^2 - 27*x - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 1/4*e^3 + 1/2*e^2 - 7/4*e - 3, 0, 1/2*e^2 - 1/2*e - 6, -1/2*e^3 - 1/2*e^2 + 4*e + 3, 1/2*e^3 + 1/2*e^2 - 4*e - 3, 3/4*e^3 + e^2 - 31/4*e - 7, -3/4*e^3 - e^2 + 31/4*e + 7, e^2 - 7, e^2 - 7, -1/2*e^3 - e^2 + 13/2*e + 9, 1/2*e^3 + e^2 - 13/2*e - 9, -e + 1, -e + 1, 1/4*e^3 - 13/4*e - 4, -1/4*e^3 + 13/4*e + 4, -2*e - 4, 2*e + 4, e^3 + e^2 - 9*e - 11, -e^3 - e^2 + 9*e + 11, -3/4*e^3 - 1/2*e^2 + 29/4*e, e^3 - 8*e + 5, e^3 - 8*e + 5, 3/2*e^3 + 2*e^2 - 33/2*e - 17, 3/2*e^3 + 2*e^2 - 33/2*e - 17, 3/4*e^3 + 1/2*e^2 - 37/4*e, -3/4*e^3 - 1/2*e^2 + 37/4*e, -2*e^3 - 2*e^2 + 24*e + 22, 3/2*e^3 - 33/2*e - 3, -3/2*e^3 + 33/2*e + 3, 5/4*e^3 + 4*e^2 - 49/4*e - 16, -1/2*e^3 - e^2 + 13/2*e + 15, -1/2*e^3 - e^2 + 13/2*e + 15, e^2 + 2*e - 1, -e^2 - 2*e + 1, -3/4*e^3 - 2*e^2 + 19/4*e + 5, 3/4*e^3 + 2*e^2 - 19/4*e - 5, -1/2*e^3 - 3*e^2 - 1/2*e + 20, -1/2*e^3 - 3*e^2 - 1/2*e + 20, e^3 + 3/2*e^2 - 27/2*e - 7, e^3 + 3/2*e^2 - 27/2*e - 7, e^3 + 1/2*e^2 - 25/2*e - 3, -e^3 - 1/2*e^2 + 25/2*e + 3, 2*e^3 + 1/2*e^2 - 39/2*e - 1, -2*e^3 - 1/2*e^2 + 39/2*e + 1, 3/2*e^3 + 4*e^2 - 37/2*e - 27, 3/2*e^3 + 4*e^2 - 37/2*e - 27, -1/2*e^3 - 2*e^2 + 5/2*e + 4, 1/2*e^3 + 2*e^2 - 5/2*e - 4, 1/2*e^3 + e^2 - 1/2*e - 3, -1/2*e^3 - e^2 + 1/2*e + 3, -3/4*e^3 - 5/2*e^2 + 37/4*e + 28, 3/4*e^3 + 5/2*e^2 - 37/4*e - 28, 1/2*e^3 + e^2 - 1/2*e - 15, 1/2*e^3 + e^2 - 1/2*e - 15, e^3 - 14*e - 7, e^3 - 14*e - 7, 3/2*e^3 + 7/2*e^2 - 20*e - 35, 3/2*e^3 + 7/2*e^2 - 20*e - 35, -4*e^3 - 6*e^2 + 36*e + 40, 4*e^3 + 6*e^2 - 36*e - 40, -e^2 - 2*e + 11, e^2 + 2*e - 11, 1/4*e^3 + 5/2*e^2 - 15/4*e - 17, -1/4*e^3 - 5/2*e^2 + 15/4*e + 17, -3*e + 3, -3*e + 3, 4*e^3 + 5*e^2 - 40*e - 41, -4*e^3 - 5*e^2 + 40*e + 41, 5/4*e^3 + 1/2*e^2 - 59/4*e + 7, -5/4*e^3 - 1/2*e^2 + 59/4*e - 7, 3/2*e^3 - 23/2*e + 16, 3/2*e^3 - 23/2*e + 16, e^3 - 3*e^2 - 14*e + 20, -e^3 + 3*e^2 + 14*e - 20, -1/2*e^3 - 1/2*e^2 + 5*e - 14, -1/2*e^3 - 1/2*e^2 + 5*e - 14, -13/4*e^3 - 6*e^2 + 109/4*e + 47, -13/4*e^3 - 6*e^2 + 109/4*e + 47, -5/4*e^3 + 37/4*e + 17, -5/4*e^3 + 37/4*e + 17, -9*e - 9, -9*e - 9, -1/2*e^3 - 3*e^2 + 5/2*e + 31, -1/2*e^3 - 3*e^2 + 5/2*e + 31, -1/4*e^3 - e^2 + 9/4*e + 8, -1/4*e^3 - e^2 + 9/4*e + 8, 1/4*e^3 + 2*e^2 - 25/4*e - 21, 1/4*e^3 + 2*e^2 - 25/4*e - 21, -1/4*e^3 + 7/2*e^2 + 7/4*e - 25, -1/4*e^3 + 7/2*e^2 + 7/4*e - 25, -7/2*e^3 - 11/2*e^2 + 29*e + 42, 7/2*e^3 + 11/2*e^2 - 29*e - 42, -2*e^3 - 4*e^2 + 23*e + 9, 2*e^3 + 4*e^2 - 23*e - 9, -1/4*e^3 - 5/2*e^2 + 39/4*e + 23, -1/4*e^3 - 5/2*e^2 + 39/4*e + 23, -3*e^3 - 2*e^2 + 32*e + 5, 3*e^3 + 2*e^2 - 32*e - 5, -3/2*e^2 - 21/2*e + 18, 3/2*e^2 + 21/2*e - 18, 9/4*e^3 + 9/2*e^2 - 71/4*e - 9, 9/4*e^3 + 9/2*e^2 - 71/4*e - 9, -7/2*e^3 - 3*e^2 + 63/2*e + 23, 7/2*e^3 + 3*e^2 - 63/2*e - 23, -1/2*e^3 + 4*e^2 + 11/2*e - 9, -1/2*e^3 + 4*e^2 + 11/2*e - 9, -1/2*e^3 - e^2 + 5/2*e - 11, -1/2*e^3 - e^2 + 5/2*e - 11, -e^3 + 3*e^2 + 13*e - 19, e^3 - 3*e^2 - 13*e + 19, -3/2*e^3 - 11/2*e^2 + 19*e + 34, 3/2*e^3 + 11/2*e^2 - 19*e - 34, 2*e^3 + e^2 - 20*e - 21, -2*e^3 - e^2 + 20*e + 21, -9/2*e^3 - 6*e^2 + 93/2*e + 48, 9/2*e^3 + 6*e^2 - 93/2*e - 48, 2*e^3 + 5*e^2 - 29*e - 28, 2*e^3 + 5*e^2 - 29*e - 28, 7/4*e^3 + e^2 - 27/4*e + 7, 7/4*e^3 + e^2 - 27/4*e + 7, 3/2*e^3 - 5*e^2 - 39/2*e + 31, -3/2*e^3 + 5*e^2 + 39/2*e - 31, -9/2*e^3 - 9*e^2 + 93/2*e + 63, 9/2*e^3 + 9*e^2 - 93/2*e - 63, 5/4*e^3 + 3/2*e^2 - 43/4*e - 32, 5/4*e^3 + 3/2*e^2 - 43/4*e - 32, -e^3 + 2*e^2 + 12*e - 11, e^3 - 2*e^2 - 12*e + 11, 5/4*e^3 - 1/2*e^2 - 59/4*e - 12, 5/4*e^3 - 1/2*e^2 - 59/4*e - 12, 7/4*e^3 + e^2 - 87/4*e + 4, 7/4*e^3 + e^2 - 87/4*e + 4, 5*e^2 + 7*e - 44, -5*e^2 - 7*e + 44, -9/4*e^3 - e^2 + 125/4*e + 27, -9/4*e^3 - e^2 + 125/4*e + 27, 11/2*e^3 + 5*e^2 - 101/2*e - 36, 11/2*e^3 + 5*e^2 - 101/2*e - 36, 3*e^3 - 23*e + 38, 1/4*e^3 + 5*e^2 - 57/4*e - 52, -1/4*e^3 - 5*e^2 + 57/4*e + 52, 13/4*e^3 + 7*e^2 - 129/4*e - 41, -13/4*e^3 - 7*e^2 + 129/4*e + 41, -4*e^3 - 5*e^2 + 37*e + 24, 4*e^3 + 5*e^2 - 37*e - 24, -3/4*e^3 - 4*e^2 + 19/4*e + 31, -3/4*e^3 - 4*e^2 + 19/4*e + 31, 7/2*e^3 + 6*e^2 - 81/2*e - 47, -7/2*e^3 - 6*e^2 + 81/2*e + 47, 1/2*e^3 + 19/2*e^2 + 3*e - 55, -1/2*e^3 - 19/2*e^2 - 3*e + 55, 2*e^3 + 3*e^2 - 17*e - 20, 2*e^3 + 3*e^2 - 17*e - 20, 13/4*e^3 + 5/2*e^2 - 139/4*e - 29, -13/4*e^3 - 5/2*e^2 + 139/4*e + 29] 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,w + 4])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]