/* 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, 12, -11, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [4, 2, -1/2*w^2 - 1/2*w + 3/2],\ [4, 2, -1/2*w^2 + 3/2*w + 1/2],\ [5, 5, w - 3],\ [11, 11, 1/2*w^3 - 4*w - 7/2],\ [11, 11, -1/2*w^3 + 3/2*w^2 + 5/2*w - 7],\ [19, 19, -w^2 + 2*w + 4],\ [19, 19, -w^2 + 5],\ [31, 31, w],\ [31, 31, w + 3],\ [31, 31, -w + 4],\ [31, 31, w - 1],\ [41, 41, -w^2 + 2],\ [41, 41, 1/2*w^3 - 5/2*w^2 - 3/2*w + 12],\ [59, 59, 1/2*w^3 + 1/2*w^2 - 9/2*w - 5],\ [59, 59, -1/2*w^3 + 2*w^2 + 2*w - 17/2],\ [79, 79, -w^2 + 8],\ [79, 79, w^2 - 2*w - 7],\ [81, 3, -3],\ [101, 101, -w^3 + 1/2*w^2 + 15/2*w + 3/2],\ [101, 101, w^3 - 5/2*w^2 - 11/2*w + 17/2],\ [109, 109, 1/2*w^3 - w^2 - 5*w + 19/2],\ [121, 11, 3/2*w^2 - 3/2*w - 19/2],\ [131, 131, 2*w^2 - w - 11],\ [131, 131, -w^3 + 8*w + 6],\ [131, 131, w^3 - 3*w^2 - 5*w + 13],\ [131, 131, 2*w^2 - 3*w - 10],\ [139, 139, -w^2 + 2*w + 10],\ [139, 139, -w^2 + 11],\ [149, 149, w^3 - 3*w^2 - 3*w + 11],\ [149, 149, -w^3 - w^2 + 11*w + 17],\ [151, 151, 2*w^2 - 3*w - 12],\ [151, 151, -2*w^2 + w + 13],\ [169, 13, -1/2*w^3 + 3/2*w^2 + 5/2*w - 3],\ [169, 13, 1/2*w^3 - 2*w^2 - 3*w + 19/2],\ [179, 179, w^3 - 5*w^2 + 19],\ [179, 179, -w^3 - 2*w^2 + 7*w + 15],\ [181, 181, w^2 - 3*w - 6],\ [181, 181, w^2 + w - 8],\ [191, 191, -w - 4],\ [191, 191, -1/2*w^3 + 4*w + 15/2],\ [191, 191, 1/2*w^3 - 3/2*w^2 - 5/2*w + 11],\ [191, 191, w - 5],\ [199, 199, w^2 - 10],\ [199, 199, w^2 - 2*w - 9],\ [229, 229, -1/2*w^3 + 2*w^2 + 4*w - 17/2],\ [229, 229, w^3 - 3/2*w^2 - 9/2*w + 11/2],\ [241, 241, 1/2*w^3 - 2*w^2 + w - 1/2],\ [241, 241, -1/2*w^3 - 1/2*w^2 + 3/2*w - 1],\ [251, 251, w^3 - 12*w - 14],\ [251, 251, 1/2*w^3 - w^2 - 5*w + 3/2],\ [269, 269, -1/2*w^3 + 5/2*w^2 + 3/2*w - 10],\ [269, 269, 1/2*w^3 + w^2 - 5*w - 13/2],\ [271, 271, 3/2*w^3 + 1/2*w^2 - 23/2*w - 10],\ [271, 271, -w^3 - 1/2*w^2 + 9/2*w - 1/2],\ [289, 17, -1/2*w^3 + 2*w^2 + 2*w - 13/2],\ [289, 17, 1/2*w^3 + 1/2*w^2 - 9/2*w - 3],\ [311, 311, w^3 + 3/2*w^2 - 19/2*w - 33/2],\ [311, 311, 1/2*w^2 + 3/2*w - 11/2],\ [311, 311, 1/2*w^2 - 5/2*w - 7/2],\ [311, 311, -w^3 + 9/2*w^2 + 7/2*w - 47/2],\ [349, 349, w^3 - 7*w - 8],\ [349, 349, 1/2*w^2 + 3/2*w - 17/2],\ [349, 349, -1/2*w^2 + 5/2*w + 13/2],\ [349, 349, -w^3 + 3*w^2 + 4*w - 14],\ [359, 359, 1/2*w^3 + w^2 - 6*w - 15/2],\ [359, 359, -1/2*w^3 + 5/2*w^2 + 5/2*w - 12],\ [361, 19, 2*w^2 - 2*w - 13],\ [379, 379, w^3 - 5/2*w^2 - 7/2*w + 19/2],\ [379, 379, -w^3 + 1/2*w^2 + 11/2*w + 9/2],\ [389, 389, -w^3 + 4*w^2 + 4*w - 18],\ [389, 389, 2*w^3 - 8*w^2 - 5*w + 30],\ [409, 409, 3/2*w^2 + 1/2*w - 11/2],\ [409, 409, 1/2*w^2 - 5/2*w - 15/2],\ [409, 409, 1/2*w^3 - 8*w - 23/2],\ [409, 409, -3/2*w^2 + 7/2*w + 7/2],\ [419, 419, -1/2*w^3 + 1/2*w^2 + 11/2*w - 6],\ [419, 419, w^3 + w^2 - 8*w - 9],\ [421, 421, -w^2 - 2*w + 2],\ [421, 421, -1/2*w^3 + 2*w^2 + 4*w - 21/2],\ [421, 421, -1/2*w^3 - 1/2*w^2 + 13/2*w + 5],\ [421, 421, 3/2*w^3 - 9/2*w^2 - 15/2*w + 19],\ [431, 431, -w^3 + 2*w^2 + 6*w - 6],\ [431, 431, 1/2*w^3 + 1/2*w^2 - 13/2*w - 7],\ [431, 431, -1/2*w^3 + 2*w^2 + 4*w - 25/2],\ [431, 431, w^3 - w^2 - 7*w + 1],\ [439, 439, 1/2*w^3 + 3/2*w^2 - 7/2*w - 12],\ [439, 439, 1/2*w^3 + 2*w^2 - w - 13/2],\ [439, 439, -1/2*w^3 + 7/2*w^2 - 9/2*w - 5],\ [439, 439, 1/2*w^3 - 3*w^2 + w + 27/2],\ [449, 449, -1/2*w^3 + 3/2*w^2 + 3/2*w - 9],\ [449, 449, 1/2*w^3 - 3*w - 13/2],\ [491, 491, w^3 - 3/2*w^2 - 13/2*w + 5/2],\ [491, 491, w^3 - 3/2*w^2 - 13/2*w + 9/2],\ [499, 499, -w^3 + 4*w^2 + 3*w - 19],\ [499, 499, -1/2*w^3 - 2*w^2 + 7*w + 25/2],\ [499, 499, -1/2*w^3 + 7/2*w^2 + 3/2*w - 17],\ [499, 499, -w^3 - w^2 + 8*w + 13],\ [509, 509, 1/2*w^3 - 3*w^2 + 33/2],\ [509, 509, -w^3 + 7/2*w^2 + 9/2*w - 29/2],\ [509, 509, w^3 + 1/2*w^2 - 17/2*w - 15/2],\ [509, 509, 1/2*w^3 + 3/2*w^2 - 9/2*w - 14],\ [521, 521, 1/2*w^3 - 7*w + 7/2],\ [521, 521, 1/2*w^3 - 3/2*w^2 - 11/2*w + 3],\ [529, 23, -2*w^3 + 15/2*w^2 + 13/2*w - 59/2],\ [529, 23, 2*w^3 + 3/2*w^2 - 31/2*w - 35/2],\ [569, 569, w^3 - 2*w^2 - 6*w + 2],\ [569, 569, w^3 - w^2 - 7*w + 5],\ [571, 571, 1/2*w^3 + 2*w^2 - 6*w - 35/2],\ [571, 571, w^3 + w^2 - 5*w - 3],\ [571, 571, w^3 - 4*w^2 + 6],\ [571, 571, -1/2*w^3 + 7/2*w^2 + 1/2*w - 21],\ [599, 599, -w^3 + 1/2*w^2 + 19/2*w + 1/2],\ [599, 599, -w^3 + 5/2*w^2 + 15/2*w - 19/2],\ [601, 601, 1/2*w^3 + 2*w^2 - 7*w - 33/2],\ [601, 601, -3*w^2 + 2*w + 15],\ [631, 631, -1/2*w^3 + 3/2*w^2 + 9/2*w - 7],\ [631, 631, -w^3 + 2*w^2 + 5*w - 4],\ [641, 641, w^3 - 11*w - 11],\ [641, 641, -2*w^3 - w^2 + 16*w + 17],\ [659, 659, 2*w^3 + 3/2*w^2 - 27/2*w - 25/2],\ [659, 659, -w^3 + w^2 + 8*w - 3],\ [659, 659, -1/2*w^3 + 5/2*w^2 + 3/2*w - 8],\ [659, 659, -2*w^3 + 15/2*w^2 + 9/2*w - 45/2],\ [661, 661, 1/2*w^3 - 6*w - 7/2],\ [661, 661, 3/2*w^3 + 3*w^2 - 13*w - 55/2],\ [661, 661, -w^3 + 4*w^2 - w - 3],\ [661, 661, -1/2*w^3 + 3/2*w^2 + 9/2*w - 9],\ [691, 691, -1/2*w^3 - 5/2*w^2 + 9/2*w + 17],\ [691, 691, 3/2*w^3 - 7/2*w^2 - 17/2*w + 18],\ [701, 701, w^2 - 13],\ [701, 701, -w^2 + 2*w + 12],\ [709, 709, 1/2*w^3 + w^2 - 4*w - 23/2],\ [709, 709, -1/2*w^3 + 5/2*w^2 + 1/2*w - 14],\ [719, 719, -2*w^2 + 15],\ [719, 719, 2*w^2 - 4*w - 13],\ [739, 739, -w^3 + 2*w^2 + 9*w - 15],\ [739, 739, 3/2*w^3 + w^2 - 12*w - 25/2],\ [751, 751, w^3 + 2*w^2 - 9*w - 16],\ [751, 751, -3*w - 8],\ [751, 751, 2*w^3 + 2*w^2 - 19*w - 28],\ [751, 751, -w^3 + 5*w^2 + 2*w - 22],\ [761, 761, -1/2*w^3 + w^2 + 5*w - 7/2],\ [761, 761, 1/2*w^3 - 1/2*w^2 - 11/2*w + 2],\ [769, 769, 5/2*w^3 + 5/2*w^2 - 35/2*w - 21],\ [769, 769, -5/2*w^3 + 10*w^2 + 5*w - 67/2],\ [809, 809, w^3 + 2*w^2 - 7*w - 16],\ [809, 809, 3/2*w^3 - 4*w^2 - 8*w + 27/2],\ [809, 809, -3/2*w^3 + 1/2*w^2 + 23/2*w + 3],\ [809, 809, 1/2*w^3 - 1/2*w^2 - 3/2*w - 8],\ [821, 821, -5/2*w^2 - 3/2*w + 23/2],\ [821, 821, -2*w^3 + 7*w^2 + 5*w - 20],\ [839, 839, 4*w + 9],\ [839, 839, -w^3 - w^2 + 12*w + 19],\ [841, 29, 5/2*w^2 - 5/2*w - 33/2],\ [841, 29, -5/2*w^2 + 5/2*w + 27/2],\ [859, 859, -w^3 + 7*w + 9],\ [859, 859, w^3 - 3*w^2 - 4*w + 15],\ [911, 911, 3/2*w^3 + 3/2*w^2 - 17/2*w - 7],\ [911, 911, -3/2*w^3 + 6*w^2 + w - 25/2],\ [919, 919, w^3 + w^2 - 4*w - 3],\ [919, 919, 4*w + 7],\ [929, 929, -3/2*w^2 - 1/2*w + 37/2],\ [929, 929, -3/2*w^2 + 7/2*w + 33/2],\ [941, 941, -w^3 - 2*w^2 + 9*w + 17],\ [941, 941, w^3 - 5*w^2 - 2*w + 23],\ [971, 971, 3/2*w^3 + 1/2*w^2 - 21/2*w - 10],\ [971, 971, -1/2*w^3 - 3/2*w^2 - 1/2*w + 2],\ [991, 991, -1/2*w^3 - 1/2*w^2 + 15/2*w + 1],\ [991, 991, -1/2*w^3 + 2*w^2 + 5*w - 15/2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 3*x - 9 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, -1, 0, e, e, e + 1, e + 1, 2*e - 4, -2*e + 2, -2*e + 2, 2*e - 4, -e + 3, -e + 3, -e - 6, -e - 6, 10, 10, 4*e - 2, 2*e + 6, 2*e + 6, -20, 2, -5*e + 9, 3*e - 3, 3*e - 3, -5*e + 9, e - 14, e - 14, 2*e - 18, 2*e - 18, -6*e + 8, -6*e + 8, -5*e + 10, -5*e + 10, 3*e + 3, 3*e + 3, 2, 2, 12, 24, 24, 12, -2*e - 2, -2*e - 2, 6*e - 14, 6*e - 14, 5*e - 16, 5*e - 16, -5*e + 12, -5*e + 12, -2*e - 12, -2*e - 12, 2, 2, 3*e + 13, 3*e + 13, -4*e, 2*e + 6, 2*e + 6, -4*e, -6*e + 4, -4*e + 16, -4*e + 16, -6*e + 4, 2*e + 12, 2*e + 12, 2, -3*e - 23, -3*e - 23, 30, 30, -7*e - 2, -e + 34, -e + 34, -7*e - 2, -7*e + 18, -7*e + 18, 2, 2*e - 4, 2*e - 4, 2, -6*e + 18, 12, 12, -6*e + 18, 2*e - 8, 10, 10, 2*e - 8, 9*e - 6, 9*e - 6, -5*e - 3, -5*e - 3, 9*e - 11, -3*e + 22, -3*e + 22, 9*e - 11, 4*e - 6, -2*e + 18, -2*e + 18, 4*e - 6, -7*e + 12, -7*e + 12, -e - 11, -e - 11, -e + 9, -e + 9, 3*e + 32, -5*e + 17, -5*e + 17, 3*e + 32, 6*e - 24, 6*e - 24, -3*e - 4, -3*e - 4, -16, -16, 3*e + 12, 3*e + 12, e - 39, -4*e + 36, -4*e + 36, e - 39, -2*e - 28, -4*e - 10, -4*e - 10, -2*e - 28, -e - 7, -e - 7, 24, 24, 4*e + 4, 4*e + 4, -2*e - 12, -2*e - 12, 4*e + 4, 4*e + 4, 6*e - 40, 4*e - 22, 4*e - 22, 6*e - 40, 5*e - 3, 5*e - 3, 5*e - 5, 5*e - 5, -5*e - 15, 5*e + 30, 5*e + 30, -5*e - 15, 12, 12, 30, 30, -5*e + 32, -3*e + 41, -3*e + 22, -3*e + 22, -18, -18, -6*e + 34, -6*e + 34, 9*e - 21, 9*e - 21, 4*e - 42, 4*e - 42, e + 15, e + 15, -6*e + 38, -6*e + 38] 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^2 - 1/2*w + 3/2])] = 1 AL_eigenvalues[ZF.ideal([4, 2, -1/2*w^2 + 3/2*w + 1/2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]