/* 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, -1, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 4, -w^2 + 3]) primes_array = [ [2, 2, w^3 - w^2 - 6*w - 2],\ [3, 3, -w^3 + 2*w^2 + 4*w - 2],\ [4, 2, w^3 - w^2 - 5*w - 2],\ [17, 17, -w^3 + w^2 + 6*w - 1],\ [17, 17, -w + 2],\ [29, 29, -w^3 + 2*w^2 + 4*w - 4],\ [29, 29, w^3 - 2*w^2 - 4*w],\ [31, 31, -w^2 + 2],\ [31, 31, -w^3 + 7*w + 5],\ [41, 41, -2*w^3 + 2*w^2 + 13*w - 2],\ [41, 41, 3*w^3 - 4*w^2 - 17*w + 5],\ [67, 67, 3*w^3 - 4*w^2 - 17*w + 1],\ [67, 67, -w^2 + 4*w + 2],\ [83, 83, 2*w^2 - 5*w - 2],\ [83, 83, -w^3 + 8*w + 6],\ [83, 83, w^3 - 2*w^2 - 2*w - 2],\ [83, 83, w^3 - 2*w^2 - 6*w],\ [97, 97, -3*w^3 + 2*w^2 + 17*w + 7],\ [97, 97, w^3 - 4*w^2 + 3*w + 1],\ [97, 97, -5*w^3 + 8*w^2 + 24*w - 8],\ [97, 97, 2*w^3 - 2*w^2 - 13*w - 4],\ [101, 101, -w^3 + 6*w + 6],\ [101, 101, -w^3 + 6*w + 2],\ [103, 103, -4*w^3 + 3*w^2 + 26*w + 12],\ [103, 103, -w^3 + 2*w^2 + 9*w + 3],\ [107, 107, 2*w^3 - 2*w^2 - 11*w],\ [107, 107, -w^3 + 2*w^2 + 3*w - 3],\ [107, 107, w^3 - w^2 - 4*w - 3],\ [107, 107, 2*w^3 - 3*w^2 - 10*w],\ [121, 11, -2*w^3 + 3*w^2 + 8*w + 2],\ [149, 149, -3*w^3 + 5*w^2 + 14*w - 3],\ [149, 149, -2*w^3 + 4*w^2 + 7*w - 4],\ [157, 157, 2*w^3 - 2*w^2 - 10*w - 1],\ [157, 157, 2*w^3 - 3*w^2 - 8*w + 2],\ [157, 157, 2*w^3 - 2*w^2 - 10*w - 3],\ [157, 157, -3*w^3 + 4*w^2 + 15*w + 1],\ [163, 163, -2*w^2 + 4*w + 5],\ [163, 163, 2*w^3 - 4*w^2 - 10*w + 5],\ [169, 13, 2*w^3 - 3*w^2 - 12*w + 2],\ [169, 13, -w^3 + 9*w + 3],\ [173, 173, 3*w^3 - 2*w^2 - 17*w - 9],\ [173, 173, -2*w^3 + 14*w + 7],\ [197, 197, 2*w^3 - 2*w^2 - 12*w + 1],\ [197, 197, 3*w^3 - 3*w^2 - 18*w - 1],\ [199, 199, 3*w^3 - 4*w^2 - 18*w + 8],\ [199, 199, 4*w^3 - 5*w^2 - 24*w + 6],\ [223, 223, w^3 - 2*w^2 - 3*w - 3],\ [223, 223, 2*w^3 - 3*w^2 - 10*w + 6],\ [227, 227, -w^3 + 2*w^2 + 6*w - 8],\ [227, 227, 3*w + 4],\ [227, 227, 3*w^3 - 3*w^2 - 18*w - 7],\ [227, 227, -2*w^3 + 3*w^2 + 12*w - 6],\ [229, 229, -w - 4],\ [229, 229, w^2 - 2*w + 2],\ [229, 229, -w^3 + w^2 + 6*w + 5],\ [229, 229, -w^3 + 2*w^2 + 5*w - 7],\ [233, 233, -2*w^3 + 3*w^2 + 8*w - 4],\ [233, 233, -3*w^3 + 4*w^2 + 15*w + 3],\ [281, 281, -2*w^3 + 3*w^2 + 10*w + 2],\ [281, 281, -w^3 + 2*w^2 + 3*w - 5],\ [289, 17, -2*w^3 + 2*w^2 + 14*w + 5],\ [293, 293, w^2 - 6],\ [293, 293, w^3 - 7*w - 1],\ [331, 331, -6*w^3 + 10*w^2 + 27*w - 6],\ [331, 331, 2*w^3 - 2*w^2 - 12*w - 7],\ [347, 347, 3*w^3 - 2*w^2 - 18*w - 6],\ [347, 347, -4*w^3 + 2*w^2 + 23*w + 10],\ [347, 347, 2*w^3 - 6*w^2 - w + 4],\ [347, 347, -4*w^3 + 3*w^2 + 24*w + 8],\ [359, 359, -3*w^3 + 4*w^2 + 17*w + 1],\ [359, 359, w^2 - 4*w - 4],\ [359, 359, w^3 - 8*w],\ [359, 359, -w^3 + 2*w^2 + 6*w - 6],\ [367, 367, 2*w^2 - 3*w - 4],\ [367, 367, -w^3 + 3*w^2 + 4*w - 7],\ [379, 379, 3*w^3 - 3*w^2 - 16*w - 3],\ [379, 379, 2*w^3 - 2*w^2 - 9*w - 2],\ [461, 461, -2*w^3 + 5*w^2 + 2*w + 2],\ [461, 461, 4*w^3 - 4*w^2 - 26*w - 11],\ [463, 463, -2*w^3 + 13*w + 8],\ [463, 463, -w^3 - w^2 + 6*w + 7],\ [487, 487, 2*w^3 - 17*w - 8],\ [487, 487, -w^3 + 10*w + 4],\ [491, 491, -5*w^3 + 3*w^2 + 30*w + 13],\ [491, 491, -5*w^3 + 4*w^2 + 28*w + 14],\ [491, 491, 3*w^3 - 8*w^2 - 4*w + 4],\ [491, 491, 7*w^3 - 12*w^2 - 33*w + 15],\ [499, 499, -2*w^3 + 2*w^2 + 12*w - 5],\ [499, 499, 2*w^2 - 7*w],\ [529, 23, -w^3 + 3*w^2 + 4*w - 5],\ [529, 23, 2*w^2 - 3*w - 6],\ [557, 557, -2*w^3 + 3*w^2 + 12*w + 4],\ [557, 557, w^3 - 9*w - 9],\ [569, 569, 4*w^3 - 5*w^2 - 22*w + 4],\ [569, 569, -3*w^3 + 2*w^2 + 20*w + 4],\ [577, 577, 3*w^3 - 3*w^2 - 20*w + 3],\ [577, 577, -4*w^3 + 8*w^2 + 16*w - 9],\ [577, 577, 7*w^3 - 11*w^2 - 34*w + 9],\ [577, 577, -4*w^3 + 6*w^2 + 22*w - 11],\ [593, 593, -6*w^3 + 10*w^2 + 29*w - 14],\ [593, 593, -w^3 + 4*w^2 - w - 7],\ [619, 619, -3*w^3 + 5*w^2 + 12*w - 1],\ [619, 619, -4*w^3 + 8*w^2 + 18*w - 15],\ [625, 5, -5],\ [631, 631, -3*w^3 + 8*w^2 + 6*w - 4],\ [631, 631, -2*w^3 + 8*w^2 - 3*w - 8],\ [643, 643, 3*w^3 - 4*w^2 - 14*w + 2],\ [643, 643, -3*w^3 + 4*w^2 + 14*w + 2],\ [661, 661, w^3 - 8*w - 10],\ [661, 661, -3*w^3 + 2*w^2 + 21*w + 3],\ [661, 661, w^3 - 2*w^2 - 6*w - 4],\ [661, 661, 2*w^3 - 3*w^2 - 14*w + 4],\ [677, 677, -9*w^3 + 12*w^2 + 50*w - 10],\ [677, 677, 3*w^3 - 4*w^2 - 17*w - 7],\ [691, 691, 3*w^3 - 2*w^2 - 21*w - 9],\ [691, 691, 2*w^3 - 3*w^2 - 14*w - 2],\ [701, 701, -9*w^3 + 13*w^2 + 48*w - 15],\ [701, 701, 2*w^3 - 6*w^2 + w - 2],\ [709, 709, 2*w^2 - 2*w - 5],\ [709, 709, 2*w^2 - 2*w - 7],\ [709, 709, 4*w^3 - 5*w^2 - 20*w - 4],\ [709, 709, -3*w^3 + 6*w^2 + 14*w - 10],\ [727, 727, 7*w^3 - 10*w^2 - 36*w + 8],\ [727, 727, 3*w^2 - 6*w - 8],\ [743, 743, 9*w^3 - 12*w^2 - 51*w + 13],\ [743, 743, w^3 - 2*w^2 - w - 5],\ [743, 743, -4*w^3 + 4*w^2 + 27*w - 8],\ [743, 743, -3*w^3 + 4*w^2 + 21*w + 9],\ [751, 751, -w^3 + 10*w + 2],\ [751, 751, 3*w^3 - 4*w^2 - 18*w + 2],\ [761, 761, 4*w^3 - 4*w^2 - 23*w - 2],\ [761, 761, -3*w^3 + 2*w^2 + 19*w + 5],\ [809, 809, -3*w^3 + 5*w^2 + 10*w + 5],\ [809, 809, -w^3 - w^2 + 10*w + 3],\ [821, 821, -w^3 + 3*w^2 + 2*w - 9],\ [821, 821, 2*w^2 - 4*w - 9],\ [823, 823, w^3 - 6*w^2 + 9*w + 1],\ [823, 823, -3*w^3 + 2*w^2 + 15*w + 5],\ [827, 827, 3*w^3 - 3*w^2 - 18*w + 1],\ [827, 827, 2*w^3 - 16*w - 11],\ [827, 827, 3*w^3 - 24*w - 14],\ [827, 827, 3*w - 4],\ [829, 829, w^2 - 6*w - 2],\ [829, 829, 2*w^3 - 2*w^2 - 10*w - 9],\ [829, 829, 5*w^3 - 6*w^2 - 29*w - 1],\ [829, 829, w^3 + 2*w^2 - 14*w - 6],\ [841, 29, -2*w^3 + 2*w^2 + 14*w + 3],\ [857, 857, -3*w^3 + 6*w^2 + 11*w - 7],\ [857, 857, -4*w^3 + 7*w^2 + 18*w - 4],\ [859, 859, -w^3 + w^2 + 6*w - 5],\ [859, 859, w - 6],\ [883, 883, 5*w^3 - 5*w^2 - 26*w - 11],\ [883, 883, -3*w^3 + 3*w^2 + 14*w + 5],\ [887, 887, 5*w^3 - 2*w^2 - 34*w - 16],\ [887, 887, -w^3 + 4*w^2 - 10],\ [887, 887, 4*w^3 - 2*w^2 - 26*w - 11],\ [887, 887, 2*w^3 - 8*w^2 + 4*w + 7],\ [907, 907, 6*w^3 - 8*w^2 - 34*w + 7],\ [907, 907, -w^3 + w^2 + 8*w - 7],\ [941, 941, 2*w^3 - 2*w^2 - 8*w - 5],\ [941, 941, -3*w^3 + 23*w + 11],\ [953, 953, 2*w^3 - 10*w - 3],\ [953, 953, -4*w^3 + 2*w^2 + 24*w + 15],\ [961, 31, -2*w^3 + 2*w^2 + 14*w - 7],\ [991, 991, -3*w^3 + 4*w^2 + 13*w + 3],\ [991, 991, 4*w^3 - 5*w^2 - 20*w + 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 44*x^2 + 416 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, 1/4*e^2 - 5, 1/4*e^2 - 6, -e, e, -e, e, 1/4*e^2 - 1, 1/4*e^2 - 1, 1/4*e^3 - 6*e, -1/4*e^3 + 6*e, -1/4*e^2 + 13, -1/4*e^2 + 13, -1/4*e^3 + 7*e, 1/4*e^3 - 7*e, 1/4*e^3 - 7*e, -1/4*e^3 + 7*e, -1/4*e^2 + 15, -1/4*e^2 + 15, 3/4*e^2 - 17, 3/4*e^2 - 17, 1/4*e^3 - 6*e, -1/4*e^3 + 6*e, e^2 - 28, e^2 - 28, 1/4*e^3 - 7*e, -1/2*e^3 + 12*e, -1/4*e^3 + 7*e, 1/2*e^3 - 12*e, -e^2 + 10, -3*e, 3*e, 1/4*e^2 - 15, -3/4*e^2 + 21, 1/4*e^2 - 15, -3/4*e^2 + 21, -e^2 + 24, -e^2 + 24, -e^2 + 26, -e^2 + 26, 1/4*e^3 - 8*e, -1/4*e^3 + 8*e, 1/4*e^3 - 6*e, -1/4*e^3 + 6*e, e^2 - 12, e^2 - 12, -1/4*e^2 + 17, -1/4*e^2 + 17, -1/4*e^3 + 3*e, 2*e, -2*e, 1/4*e^3 - 3*e, 3/4*e^2 - 25, -5/4*e^2 + 27, 3/4*e^2 - 25, -5/4*e^2 + 27, 1/4*e^3 - 6*e, -1/4*e^3 + 6*e, -1/4*e^3 + 6*e, 1/4*e^3 - 6*e, -34, -1/2*e^3 + 15*e, 1/2*e^3 - 15*e, -5/4*e^2 + 17, -5/4*e^2 + 17, -1/4*e^3 + 3*e, 2*e, -2*e, 1/4*e^3 - 3*e, 4*e, -4*e, 1/4*e^3 - 5*e, -1/4*e^3 + 5*e, -5/4*e^2 + 37, -5/4*e^2 + 37, 7/4*e^2 - 43, 7/4*e^2 - 43, -7*e, 7*e, 5/4*e^2 - 21, 5/4*e^2 - 21, -11/4*e^2 + 67, -11/4*e^2 + 67, 1/4*e^3 - 7*e, -1/4*e^3 + 7*e, -1/4*e^3 + 7*e, 1/4*e^3 - 7*e, e^2, e^2, 9/4*e^2 - 43, 9/4*e^2 - 43, 1/4*e^3 - 10*e, -1/4*e^3 + 10*e, 5*e, -5*e, 9/4*e^2 - 55, -7/4*e^2 + 25, -7/4*e^2 + 25, 9/4*e^2 - 55, -1/4*e^3 + 2*e, 1/4*e^3 - 2*e, -13/4*e^2 + 73, -13/4*e^2 + 73, -3/4*e^2 + 37, 5/4*e^2 - 29, 5/4*e^2 - 29, -3/4*e^2 - 9, -3/4*e^2 - 9, -11/4*e^2 + 69, 9/4*e^2 - 47, -11/4*e^2 + 69, 9/4*e^2 - 47, -1/2*e^3 + 7*e, 1/2*e^3 - 7*e, -3/4*e^2 - 1, -3/4*e^2 - 1, -1/4*e^3 + 2*e, 1/4*e^3 - 2*e, 7/4*e^2 - 49, 7/4*e^2 - 49, -17/4*e^2 + 91, -17/4*e^2 + 91, 5/4*e^2 + 3, 5/4*e^2 + 3, -3/4*e^3 + 23*e, -1/4*e^3 + 13*e, 3/4*e^3 - 23*e, 1/4*e^3 - 13*e, 19/4*e^2 - 99, 19/4*e^2 - 99, 1/2*e^3 - 7*e, -1/2*e^3 + 7*e, -1/4*e^3 + 6*e, 1/4*e^3 - 6*e, -e^3 + 23*e, e^3 - 23*e, -1/4*e^2 - 7, -1/4*e^2 - 7, -1/4*e^3 + 3*e, -1/2*e^3 + 20*e, 1/2*e^3 - 20*e, 1/4*e^3 - 3*e, -17/4*e^2 + 99, -17/4*e^2 + 95, -17/4*e^2 + 99, -17/4*e^2 + 95, 2*e^2 - 82, -1/2*e^3 + 21*e, 1/2*e^3 - 21*e, -15/4*e^2 + 83, -15/4*e^2 + 83, -e^2 + 24, -e^2 + 24, 1/4*e^3 - 13*e, -1/4*e^3 + 13*e, -1/4*e^3 + 13*e, 1/4*e^3 - 13*e, 5*e^2 - 96, 5*e^2 - 96, -3/4*e^3 + 20*e, 3/4*e^3 - 20*e, e^3 - 23*e, -e^3 + 23*e, 1/4*e^2 + 49, -e^2 - 12, -e^2 - 12] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, w^3 - w^2 - 6*w - 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]