/* 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([31, 0, -12, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([20, 10, -1/2*w^3 - 1/2*w^2 + 7/2*w + 7/2]) primes_array = [ [4, 2, -1/2*w^3 - 1/2*w^2 + 7/2*w + 11/2],\ [5, 5, -1/2*w^3 + w^2 + 5/2*w - 4],\ [5, 5, -1/2*w^2 - w + 3/2],\ [9, 3, -1/2*w^3 + 1/2*w^2 + 7/2*w - 9/2],\ [9, 3, -1/2*w^3 - 1/2*w^2 + 7/2*w + 9/2],\ [19, 19, -1/2*w^2 + w + 5/2],\ [19, 19, 1/2*w^2 + w - 5/2],\ [29, 29, -3/2*w^2 - w + 17/2],\ [29, 29, -3/2*w^2 + w + 17/2],\ [31, 31, 1/2*w^3 - 7/2*w],\ [59, 59, 1/2*w^2 + w - 11/2],\ [59, 59, 1/2*w^2 - w - 11/2],\ [61, 61, -1/2*w^3 + w^2 + 7/2*w - 5],\ [61, 61, 1/2*w^3 + w^2 - 7/2*w - 5],\ [71, 71, -1/2*w^3 - 2*w^2 + 11/2*w + 13],\ [71, 71, 3/2*w^3 + 2*w^2 - 23/2*w - 18],\ [79, 79, 1/2*w^3 + 1/2*w^2 - 7/2*w - 1/2],\ [79, 79, -2*w^2 + w + 10],\ [79, 79, 2*w^2 + w - 10],\ [79, 79, 1/2*w^3 - 1/2*w^2 - 7/2*w + 1/2],\ [89, 89, 1/2*w^3 - 2*w^2 - 5/2*w + 11],\ [89, 89, -1/2*w^2 + w + 15/2],\ [109, 109, -w^3 + 7*w + 3],\ [109, 109, 1/2*w^3 - 2*w^2 - 9/2*w + 12],\ [109, 109, w^3 + w^2 - 8*w - 11],\ [109, 109, w^3 - 7*w + 3],\ [121, 11, 3/2*w^2 - 19/2],\ [121, 11, -1/2*w^2 + 13/2],\ [131, 131, -1/2*w^3 + 3/2*w^2 + 7/2*w - 15/2],\ [131, 131, -1/2*w^3 + 9/2*w + 3],\ [179, 179, -w^3 - 5/2*w^2 + 9*w + 35/2],\ [179, 179, 1/2*w^3 + 2*w^2 - 7/2*w - 13],\ [179, 179, -1/2*w^3 + 2*w^2 + 7/2*w - 13],\ [179, 179, 3/2*w^3 + 4*w^2 - 27/2*w - 29],\ [181, 181, -1/2*w^3 + w^2 + 5/2*w - 7],\ [181, 181, 1/2*w^3 + w^2 - 5/2*w - 7],\ [191, 191, 2*w^2 + w - 12],\ [191, 191, 2*w^2 - w - 12],\ [211, 211, w^3 - w^2 - 7*w + 4],\ [211, 211, -w^3 - w^2 + 7*w + 4],\ [229, 229, 3/2*w^2 + w - 23/2],\ [229, 229, 3/2*w^2 - w - 23/2],\ [241, 241, 1/2*w^3 + 3/2*w^2 - 9/2*w - 17/2],\ [241, 241, -1/2*w^3 + 3/2*w^2 + 9/2*w - 17/2],\ [251, 251, w^3 - 5*w + 3],\ [251, 251, 3/2*w^3 - 23/2*w + 2],\ [251, 251, 5/2*w^2 + 3*w - 23/2],\ [251, 251, w^3 + 3*w^2 - 9*w - 24],\ [269, 269, 1/2*w^2 + 2*w - 9/2],\ [269, 269, 1/2*w^2 - 2*w - 9/2],\ [271, 271, 1/2*w^3 - 1/2*w^2 - 3/2*w - 3/2],\ [271, 271, w^3 + 3/2*w^2 - 6*w - 15/2],\ [271, 271, 1/2*w^3 + 3*w^2 - 9/2*w - 16],\ [271, 271, -1/2*w^3 - 2*w^2 + 5/2*w + 13],\ [281, 281, w^3 - 5/2*w^2 - 8*w + 33/2],\ [281, 281, w^2 + 2*w - 6],\ [281, 281, w^2 - 2*w - 6],\ [281, 281, w^3 + 5/2*w^2 - 8*w - 33/2],\ [311, 311, -5/2*w^3 - 3*w^2 + 37/2*w + 30],\ [311, 311, 2*w^2 - 2*w - 13],\ [331, 331, -1/2*w^3 + 7/2*w - 5],\ [331, 331, -w^3 + 5/2*w^2 + 7*w - 29/2],\ [331, 331, w^3 + 5/2*w^2 - 7*w - 29/2],\ [331, 331, 1/2*w^3 - 7/2*w - 5],\ [349, 349, 1/2*w^3 - 2*w^2 - 11/2*w + 16],\ [349, 349, w^3 + 3*w^2 - 6*w - 19],\ [349, 349, 1/2*w^3 - w^2 - 1/2*w - 1],\ [349, 349, 1/2*w^3 + 2*w^2 - 11/2*w - 16],\ [359, 359, 1/2*w^3 + 3/2*w^2 - 11/2*w - 15/2],\ [359, 359, w^3 - 4*w^2 - 2*w + 16],\ [361, 19, -1/2*w^2 + 15/2],\ [379, 379, 1/2*w^3 + 5/2*w^2 - 11/2*w - 37/2],\ [379, 379, -w^3 + 1/2*w^2 + 6*w - 15/2],\ [389, 389, w^3 - 1/2*w^2 - 7*w + 5/2],\ [389, 389, w^3 + 1/2*w^2 - 7*w - 5/2],\ [401, 401, 1/2*w^3 + 3/2*w^2 - 5/2*w - 21/2],\ [401, 401, -1/2*w^3 + 3/2*w^2 + 5/2*w - 21/2],\ [409, 409, -2*w^3 - 5*w^2 + 18*w + 38],\ [409, 409, 1/2*w^3 - w^2 - 13/2*w + 1],\ [419, 419, -1/2*w^3 + 1/2*w^2 + 5/2*w - 19/2],\ [419, 419, 1/2*w^3 + 1/2*w^2 - 5/2*w - 19/2],\ [431, 431, 1/2*w^3 - 3/2*w^2 - 7/2*w + 11/2],\ [431, 431, 1/2*w^3 + 3/2*w^2 - 7/2*w - 11/2],\ [439, 439, -1/2*w^3 + 5/2*w^2 + 5/2*w - 27/2],\ [439, 439, -1/2*w^3 - 5/2*w^2 + 5/2*w + 27/2],\ [449, 449, w^3 - 1/2*w^2 - 7*w + 7/2],\ [449, 449, w^3 + 1/2*w^2 - 7*w - 7/2],\ [461, 461, w^3 - 7/2*w^2 - 5*w + 37/2],\ [461, 461, -1/2*w^2 + 2*w + 21/2],\ [479, 479, -w^3 + 4*w^2 + 4*w - 20],\ [479, 479, 1/2*w^3 - 3/2*w + 8],\ [491, 491, 1/2*w^3 - 5/2*w^2 - 3/2*w + 29/2],\ [491, 491, 1/2*w^3 + 1/2*w^2 - 1/2*w - 7/2],\ [491, 491, 1/2*w^3 + w^2 - 11/2*w - 8],\ [491, 491, -1/2*w^3 + w^2 + 5/2*w - 12],\ [499, 499, 3*w^2 - w - 21],\ [499, 499, 1/2*w^3 + 5/2*w^2 - 11/2*w - 29/2],\ [499, 499, w^3 + 5*w^2 - 11*w - 32],\ [499, 499, 3*w^2 + w - 21],\ [509, 509, -w^3 + 3/2*w^2 + 4*w - 3/2],\ [509, 509, 1/2*w^3 + 3*w^2 - 3/2*w - 18],\ [529, 23, 1/2*w^3 - 2*w^2 - 11/2*w + 11],\ [529, 23, 1/2*w^3 + 2*w^2 - 11/2*w - 11],\ [569, 569, 1/2*w^3 + w^2 - 11/2*w - 3],\ [569, 569, -1/2*w^3 + w^2 + 11/2*w - 3],\ [571, 571, 5/2*w^2 + 2*w - 33/2],\ [571, 571, 2*w^3 - 5/2*w^2 - 16*w + 49/2],\ [571, 571, 2*w^3 + 5/2*w^2 - 16*w - 49/2],\ [571, 571, 5/2*w^2 - 2*w - 33/2],\ [599, 599, 1/2*w^3 - 1/2*w^2 - 11/2*w - 3/2],\ [599, 599, -1/2*w^3 - 1/2*w^2 + 11/2*w - 3/2],\ [601, 601, 2*w^3 + 3/2*w^2 - 15*w - 35/2],\ [601, 601, 5/2*w^3 + 3*w^2 - 39/2*w - 29],\ [619, 619, 5/2*w^3 + 5/2*w^2 - 37/2*w - 51/2],\ [619, 619, w^3 - 1/2*w^2 - 5*w + 11/2],\ [619, 619, 3/2*w^2 - 2*w - 23/2],\ [619, 619, w^3 + 3*w^2 - 10*w - 21],\ [631, 631, -5/2*w^3 + 33/2*w + 13],\ [631, 631, 3/2*w^3 - 5*w^2 - 13/2*w + 22],\ [631, 631, -2*w^3 - 2*w^2 + 14*w + 23],\ [631, 631, -5/2*w^2 - 5*w + 13/2],\ [641, 641, -1/2*w^3 + 1/2*w^2 + 11/2*w - 5/2],\ [641, 641, 1/2*w^3 + 1/2*w^2 - 11/2*w - 5/2],\ [659, 659, -3/2*w^3 - w^2 + 23/2*w + 14],\ [659, 659, 3/2*w^3 + 5/2*w^2 - 19/2*w - 27/2],\ [661, 661, -1/2*w^3 + 1/2*w^2 + 11/2*w - 9/2],\ [661, 661, w^3 + w^2 - 8*w - 5],\ [661, 661, -1/2*w^3 + 2*w^2 + 7/2*w - 8],\ [661, 661, -1/2*w^3 - 1/2*w^2 + 11/2*w + 9/2],\ [691, 691, 1/2*w^3 - 3/2*w^2 - 3/2*w + 19/2],\ [691, 691, 1/2*w^3 + 3/2*w^2 - 3/2*w - 19/2],\ [709, 709, 1/2*w^3 - 2*w^2 - 3/2*w + 12],\ [709, 709, 1/2*w^3 + 2*w^2 - 3/2*w - 12],\ [751, 751, 3/2*w^3 - w^2 - 21/2*w + 11],\ [751, 751, 3/2*w^3 + w^2 - 21/2*w - 11],\ [761, 761, 2*w^3 + 11/2*w^2 - 17*w - 85/2],\ [761, 761, -3/2*w^3 - 4*w^2 + 17/2*w + 23],\ [769, 769, w^3 - 7/2*w^2 - 8*w + 45/2],\ [769, 769, -1/2*w^3 - 5/2*w^2 + 5/2*w + 35/2],\ [769, 769, -1/2*w^3 + 5/2*w^2 + 5/2*w - 35/2],\ [769, 769, w^3 + 7/2*w^2 - 8*w - 45/2],\ [809, 809, w^3 - 7/2*w^2 - 7*w + 41/2],\ [809, 809, w^3 + 7/2*w^2 - 7*w - 41/2],\ [811, 811, -1/2*w^3 - 9/2*w^2 + 13/2*w + 59/2],\ [811, 811, 3*w^3 + 5*w^2 - 24*w - 45],\ [821, 821, -2*w^3 - 7/2*w^2 + 12*w + 33/2],\ [821, 821, w^3 - 4*w^2 - w + 13],\ [829, 829, -w^3 + 1/2*w^2 + 8*w - 1/2],\ [829, 829, w^3 + 1/2*w^2 - 8*w - 1/2],\ [839, 839, w^3 - 3/2*w^2 - 7*w + 13/2],\ [839, 839, w^3 + 3/2*w^2 - 7*w - 13/2],\ [841, 29, 5/2*w^2 - 27/2],\ [859, 859, 1/2*w^3 - 7/2*w - 6],\ [859, 859, -3/2*w^3 + 2*w^2 + 21/2*w - 15],\ [859, 859, 3/2*w^3 + 2*w^2 - 21/2*w - 15],\ [859, 859, 1/2*w^3 - 7/2*w + 6],\ [881, 881, -3/2*w^3 + 21/2*w + 8],\ [881, 881, -2*w^3 + 1/2*w^2 + 13*w + 15/2],\ [919, 919, -3/2*w^3 + 5/2*w^2 + 15/2*w - 19/2],\ [919, 919, 3/2*w^3 + w^2 - 21/2*w - 4],\ [929, 929, 1/2*w^3 + w^2 - 9/2*w - 1],\ [929, 929, 1/2*w^3 - w^2 - 9/2*w + 1],\ [941, 941, 2*w^3 + 7*w^2 - 19*w - 51],\ [941, 941, 5/2*w^2 - 3*w - 33/2],\ [961, 31, -w^2 + 12],\ [971, 971, 3/2*w^2 - 3*w - 25/2],\ [971, 971, w^3 + 7/2*w^2 - 6*w - 39/2],\ [991, 991, 1/2*w^3 - 3/2*w^2 - 13/2*w + 13/2],\ [991, 991, w^3 + 5/2*w^2 - 5*w - 29/2],\ [991, 991, w^3 - 5/2*w^2 - 5*w + 29/2],\ [991, 991, -3/2*w^3 + 2*w^2 + 17/2*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 6*x + 7 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, 1, e, 2*e + 3, -e - 4, -4*e - 11, -e - 5, 3*e + 13, e + 8, 2*e + 5, -3*e - 16, 9*e + 28, 3*e + 3, -6*e - 14, 4*e + 14, -4*e - 21, -4, -4*e - 8, -5, e + 3, 3*e + 15, 5*e + 6, e - 5, -2*e - 2, 2*e + 2, -6*e - 18, e + 14, -5*e - 9, 6*e + 16, 3*e + 3, 9*e + 32, -8*e - 30, -8*e - 30, -12*e - 36, 3*e + 19, -9*e - 39, 11*e + 26, -3*e + 5, 5*e + 34, -4*e, -8*e - 14, -2*e - 6, -11*e - 31, e + 6, -14*e - 40, -7*e - 23, 8*e + 36, 14*e + 37, -4*e + 3, 4*e - 3, -4*e - 19, -2, -5*e + 7, 4*e + 22, -6*e - 28, 9*e + 29, -10, -4*e - 39, e - 23, 4*e - 12, 12*e + 22, -4*e - 29, -9*e - 28, -4*e - 4, 9*e + 16, -3*e - 32, -5*e - 9, -5*e - 19, -4*e - 2, -4*e + 17, -2*e - 17, -11*e - 45, -e - 13, -4*e - 11, -10*e - 45, 15*e + 37, 15*e + 37, 15*e + 34, 10*e + 46, -9*e - 19, 28, -11*e - 40, 9*e, -2*e - 18, 8*e + 42, -8*e - 30, -10*e - 50, 4, -e - 16, 16*e + 38, 15*e + 39, -20*e - 59, 15*e + 42, -22, 3*e - 5, -13*e - 22, 12*e + 46, 16*e + 66, 11*e + 24, -22*e - 69, 5*e + 23, 8*e + 52, 18*e + 48, 6*e + 32, -19*e - 61, 16*e + 35, -17*e - 56, -4*e - 21, -14*e - 54, -16*e - 42, 6*e + 52, -23*e - 67, -8*e, 17*e + 57, -6*e + 5, -4*e + 17, 20*e + 53, -12*e - 58, 2*e - 4, 20*e + 48, 29*e + 83, 3*e + 11, -9*e - 26, 3*e + 15, 4*e - 8, e + 2, 10, 22*e + 66, -20*e - 60, -6*e + 5, 20*e + 56, 10*e + 55, -9*e - 3, 5*e + 48, -e - 38, -4*e - 47, -16*e - 35, -9*e - 21, -5*e - 7, -5*e, -2*e + 7, -12*e - 36, 2*e - 22, -20*e - 50, -3*e - 11, 11*e + 34, 3*e + 1, -9*e - 17, -28*e - 72, -10, e + 30, -6*e - 50, e - 29, -21*e - 41, 6*e + 21, -25*e - 59, -28*e - 88, -35*e - 108, 16*e + 31, 19*e + 53, 5*e - 19, 6*e + 36, 9*e + 17, -18*e - 82, 7*e - 5, 25*e + 78, -4*e - 12, -4*e - 56, -2*e - 7, -10*e - 6, -e + 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([4,2,-1/2*w^3-1/2*w^2+7/2*w+11/2])] = 1 AL_eigenvalues[ZF.ideal([5,5,-1/2*w^3+w^2+5/2*w-4])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]