/* 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([19, 2, -10, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([19, 19, w]) primes_array = [ [9, 3, 1/3*w^3 - 2/3*w^2 - 8/3*w + 10/3],\ [9, 3, w + 1],\ [11, 11, w^2 - w - 6],\ [11, 11, 2/3*w^3 - 7/3*w^2 - 7/3*w + 23/3],\ [16, 2, 2],\ [19, 19, w],\ [19, 19, 1/3*w^3 - 2/3*w^2 - 8/3*w + 7/3],\ [19, 19, 1/3*w^3 - 2/3*w^2 - 2/3*w + 10/3],\ [19, 19, -2/3*w^3 + 4/3*w^2 + 13/3*w - 17/3],\ [25, 5, 2/3*w^3 - 4/3*w^2 - 10/3*w + 11/3],\ [31, 31, w + 3],\ [31, 31, -1/3*w^3 + 2/3*w^2 + 8/3*w - 16/3],\ [31, 31, 2/3*w^3 - 7/3*w^2 - 10/3*w + 23/3],\ [31, 31, -1/3*w^3 + 5/3*w^2 + 5/3*w - 25/3],\ [59, 59, -2/3*w^3 + 1/3*w^2 + 10/3*w + 1/3],\ [59, 59, 5/3*w^3 - 13/3*w^2 - 25/3*w + 47/3],\ [61, 61, 1/3*w^3 + 1/3*w^2 - 11/3*w - 8/3],\ [61, 61, -2/3*w^3 + 7/3*w^2 + 4/3*w - 26/3],\ [71, 71, 2/3*w^3 - 4/3*w^2 - 13/3*w + 5/3],\ [71, 71, -w^3 + 3*w^2 + 5*w - 10],\ [79, 79, -1/3*w^3 + 2/3*w^2 + 11/3*w - 16/3],\ [79, 79, -4/3*w^3 + 11/3*w^2 + 20/3*w - 37/3],\ [89, 89, -2/3*w^3 + 7/3*w^2 + 10/3*w - 20/3],\ [89, 89, -1/3*w^3 + 5/3*w^2 + 5/3*w - 28/3],\ [101, 101, 4/3*w^3 - 8/3*w^2 - 23/3*w + 25/3],\ [101, 101, w^3 - 2*w^2 - 7*w + 4],\ [121, 11, 1/3*w^3 - 2/3*w^2 - 5/3*w + 16/3],\ [131, 131, -5/3*w^3 + 13/3*w^2 + 25/3*w - 41/3],\ [131, 131, 5/3*w^3 - 16/3*w^2 - 22/3*w + 56/3],\ [149, 149, 5/3*w^3 - 13/3*w^2 - 22/3*w + 44/3],\ [149, 149, -5/3*w^3 + 16/3*w^2 + 19/3*w - 59/3],\ [179, 179, 1/3*w^3 - 8/3*w^2 + 1/3*w + 43/3],\ [179, 179, w^3 - 4*w^2 - 3*w + 13],\ [181, 181, 2/3*w^3 - 7/3*w^2 - 4/3*w + 29/3],\ [181, 181, 2/3*w^3 - 10/3*w^2 - 4/3*w + 35/3],\ [181, 181, -2/3*w^3 + 10/3*w^2 + 4/3*w - 47/3],\ [181, 181, -w^3 + 3*w^2 + 4*w - 13],\ [191, 191, -4/3*w^3 + 11/3*w^2 + 17/3*w - 43/3],\ [191, 191, -4/3*w^3 + 14/3*w^2 + 17/3*w - 61/3],\ [191, 191, 5/3*w^3 - 10/3*w^2 - 28/3*w + 23/3],\ [191, 191, 2/3*w^3 - 1/3*w^2 - 13/3*w + 2/3],\ [199, 199, -4/3*w^3 + 8/3*w^2 + 17/3*w - 28/3],\ [199, 199, -5/3*w^3 + 10/3*w^2 + 28/3*w - 35/3],\ [211, 211, 2/3*w^3 - 7/3*w^2 - 13/3*w + 35/3],\ [211, 211, -2/3*w^3 + 7/3*w^2 + 13/3*w - 20/3],\ [229, 229, -w^3 + 3*w^2 + 3*w - 11],\ [229, 229, -2/3*w^3 + 1/3*w^2 + 16/3*w + 1/3],\ [241, 241, -2*w^2 + w + 10],\ [241, 241, -5/3*w^3 + 16/3*w^2 + 22/3*w - 59/3],\ [241, 241, w^3 - w^2 - 6*w + 3],\ [241, 241, -5/3*w^3 + 13/3*w^2 + 22/3*w - 50/3],\ [251, 251, w^3 - 4*w^2 - 3*w + 16],\ [251, 251, -1/3*w^3 + 8/3*w^2 - 1/3*w - 34/3],\ [289, 17, 7/3*w^3 - 20/3*w^2 - 29/3*w + 70/3],\ [289, 17, -w^3 + 7*w + 4],\ [311, 311, -w^3 + 4*w^2 + 3*w - 15],\ [311, 311, -1/3*w^3 + 8/3*w^2 - 1/3*w - 37/3],\ [331, 331, -1/3*w^3 + 2/3*w^2 - 1/3*w + 5/3],\ [331, 331, -w^3 + 2*w^2 + 7*w - 3],\ [349, 349, -2/3*w^3 + 10/3*w^2 + 4/3*w - 41/3],\ [359, 359, -4/3*w^3 + 8/3*w^2 + 17/3*w - 25/3],\ [359, 359, -5/3*w^3 + 10/3*w^2 + 28/3*w - 32/3],\ [379, 379, -5/3*w^3 + 16/3*w^2 + 22/3*w - 74/3],\ [379, 379, 7/3*w^3 - 17/3*w^2 - 35/3*w + 46/3],\ [389, 389, 4/3*w^3 - 11/3*w^2 - 20/3*w + 31/3],\ [389, 389, w^3 - 3*w^2 - 6*w + 9],\ [401, 401, -1/3*w^3 + 8/3*w^2 + 2/3*w - 37/3],\ [401, 401, 4/3*w^3 - 14/3*w^2 - 17/3*w + 52/3],\ [419, 419, 2*w^2 - w - 13],\ [419, 419, 5/3*w^3 - 10/3*w^2 - 28/3*w + 29/3],\ [419, 419, 4/3*w^3 - 8/3*w^2 - 17/3*w + 22/3],\ [419, 419, 5/3*w^3 - 16/3*w^2 - 22/3*w + 50/3],\ [421, 421, 1/3*w^3 - 8/3*w^2 - 2/3*w + 34/3],\ [421, 421, -4/3*w^3 + 14/3*w^2 + 17/3*w - 55/3],\ [431, 431, -4/3*w^3 + 11/3*w^2 + 14/3*w - 40/3],\ [431, 431, w^3 - w^2 - 7*w + 2],\ [439, 439, 5/3*w^3 - 7/3*w^2 - 31/3*w + 14/3],\ [439, 439, -1/3*w^3 + 8/3*w^2 + 2/3*w - 49/3],\ [449, 449, 2*w^3 - 5*w^2 - 10*w + 14],\ [449, 449, -2/3*w^3 + 1/3*w^2 + 16/3*w - 2/3],\ [449, 449, w^3 - 3*w^2 - 3*w + 12],\ [449, 449, w^3 - w^2 - 5*w - 2],\ [461, 461, -1/3*w^3 + 8/3*w^2 - 7/3*w - 43/3],\ [461, 461, 4/3*w^3 - 8/3*w^2 - 26/3*w + 37/3],\ [491, 491, 2/3*w^3 - 10/3*w^2 - 7/3*w + 44/3],\ [491, 491, -2*w^3 + 5*w^2 + 9*w - 17],\ [491, 491, -1/3*w^3 - 1/3*w^2 + 11/3*w + 26/3],\ [491, 491, w^3 - 4*w^2 - 4*w + 15],\ [499, 499, -2/3*w^3 + 7/3*w^2 + 4/3*w - 35/3],\ [499, 499, 1/3*w^3 + 1/3*w^2 - 11/3*w + 1/3],\ [509, 509, 4/3*w^3 - 5/3*w^2 - 23/3*w + 1/3],\ [509, 509, 2*w^3 - 5*w^2 - 9*w + 14],\ [521, 521, -w^3 + 8*w + 5],\ [521, 521, -2*w^3 + 6*w^2 + 7*w - 20],\ [529, 23, 7/3*w^3 - 20/3*w^2 - 32/3*w + 79/3],\ [529, 23, 2/3*w^3 + 2/3*w^2 - 13/3*w - 10/3],\ [569, 569, -1/3*w^3 + 8/3*w^2 + 2/3*w - 52/3],\ [569, 569, 4/3*w^3 - 14/3*w^2 - 17/3*w + 37/3],\ [571, 571, 4/3*w^3 - 11/3*w^2 - 23/3*w + 34/3],\ [571, 571, 4/3*w^3 - 11/3*w^2 - 20/3*w + 28/3],\ [599, 599, -5/3*w^3 + 10/3*w^2 + 22/3*w - 20/3],\ [599, 599, 2*w^3 - 4*w^2 - 11*w + 9],\ [601, 601, 2*w^3 - 5*w^2 - 8*w + 18],\ [601, 601, 5/3*w^3 - 7/3*w^2 - 31/3*w + 20/3],\ [619, 619, 4/3*w^3 - 8/3*w^2 - 29/3*w + 34/3],\ [619, 619, -1/3*w^3 + 11/3*w^2 - 4/3*w - 46/3],\ [641, 641, 2*w^3 - 5*w^2 - 9*w + 15],\ [641, 641, -1/3*w^3 + 2/3*w^2 + 2/3*w - 22/3],\ [641, 641, 7/3*w^3 - 23/3*w^2 - 35/3*w + 82/3],\ [641, 641, -2/3*w^3 + 13/3*w^2 + 10/3*w - 62/3],\ [659, 659, -4/3*w^3 + 11/3*w^2 + 20/3*w - 22/3],\ [659, 659, -1/3*w^3 - 1/3*w^2 + 5/3*w + 26/3],\ [691, 691, -4/3*w^3 + 11/3*w^2 + 20/3*w - 25/3],\ [691, 691, -5/3*w^3 + 13/3*w^2 + 28/3*w - 41/3],\ [709, 709, -w^2 + 2*w + 11],\ [709, 709, -2/3*w^3 - 5/3*w^2 + 16/3*w + 28/3],\ [719, 719, -1/3*w^3 + 5/3*w^2 + 5/3*w - 40/3],\ [719, 719, -2/3*w^3 + 7/3*w^2 + 10/3*w - 8/3],\ [739, 739, 5/3*w^3 - 4/3*w^2 - 34/3*w - 1/3],\ [739, 739, 10/3*w^3 - 26/3*w^2 - 50/3*w + 91/3],\ [739, 739, 2*w^3 - 3*w^2 - 10*w + 2],\ [739, 739, 4/3*w^3 - 2/3*w^2 - 20/3*w - 5/3],\ [751, 751, 1/3*w^3 - 2/3*w^2 - 8/3*w - 11/3],\ [751, 751, w - 6],\ [761, 761, 1/3*w^3 - 5/3*w^2 + 1/3*w + 40/3],\ [761, 761, 2/3*w^3 + 5/3*w^2 - 13/3*w - 22/3],\ [769, 769, 1/3*w^3 - 5/3*w^2 - 14/3*w + 28/3],\ [769, 769, -10/3*w^3 + 26/3*w^2 + 50/3*w - 85/3],\ [811, 811, 1/3*w^3 + 7/3*w^2 - 11/3*w - 50/3],\ [811, 811, 7/3*w^3 - 11/3*w^2 - 44/3*w + 16/3],\ [821, 821, -4/3*w^3 + 8/3*w^2 + 26/3*w - 1/3],\ [821, 821, 4/3*w^3 - 5/3*w^2 - 20/3*w + 7/3],\ [821, 821, -3*w^3 + 8*w^2 + 15*w - 27],\ [821, 821, -7/3*w^3 + 17/3*w^2 + 35/3*w - 55/3],\ [829, 829, -5/3*w^3 + 16/3*w^2 + 25/3*w - 53/3],\ [829, 829, -1/3*w^3 + 8/3*w^2 + 5/3*w - 43/3],\ [839, 839, -3*w^3 + 8*w^2 + 13*w - 23],\ [839, 839, -3*w^3 + 8*w^2 + 14*w - 31],\ [841, 29, -5/3*w^3 + 10/3*w^2 + 25/3*w - 23/3],\ [841, 29, -5/3*w^3 + 10/3*w^2 + 25/3*w - 32/3],\ [859, 859, 2/3*w^3 - 7/3*w^2 - 16/3*w + 11/3],\ [859, 859, -8/3*w^3 + 22/3*w^2 + 37/3*w - 68/3],\ [881, 881, -5/3*w^3 + 16/3*w^2 + 19/3*w - 65/3],\ [881, 881, 1/3*w^3 + 4/3*w^2 - 11/3*w - 17/3],\ [911, 911, -2*w^3 + 4*w^2 + 13*w - 16],\ [911, 911, -7/3*w^3 + 14/3*w^2 + 41/3*w - 31/3],\ [919, 919, 2/3*w^3 - 1/3*w^2 - 19/3*w + 8/3],\ [919, 919, -2/3*w^3 + 7/3*w^2 + 1/3*w - 35/3],\ [941, 941, 2/3*w^3 - 1/3*w^2 - 10/3*w - 16/3],\ [941, 941, -5/3*w^3 + 13/3*w^2 + 25/3*w - 32/3],\ [941, 941, -1/3*w^3 + 11/3*w^2 - 1/3*w - 34/3],\ [941, 941, -2*w^3 + 7*w^2 + 8*w - 32]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-5, 1, 0, 0, -7, -1, -2, 4, -2, -2, -4, 8, 10, 4, -6, 6, -5, 1, 0, 6, 8, 14, -9, -3, 9, 3, 13, -6, 12, -15, 15, 18, 6, -13, -11, -23, -7, 18, 0, 18, 0, 14, 20, 22, -14, -13, -7, 11, 5, 1, -23, 24, 0, 25, -11, 30, 0, 28, -26, 26, -12, -6, 16, 4, -6, 18, -3, 21, 30, -6, 30, -30, 10, 10, -6, -6, -2, -14, -9, 6, 6, -3, -3, 9, 36, -30, 18, 42, 34, 16, 33, -15, 18, -30, 17, 29, 30, -42, 8, 8, 0, -24, -34, 26, -28, 26, -33, -45, -21, -3, 30, 6, -8, 16, 41, 41, 36, 6, 46, 26, -2, -52, 26, -40, 6, 18, 5, 23, 38, 44, -3, -27, 21, 21, -25, -13, 12, -42, -2, 10, 38, -22, 30, -18, -42, 30, 4, 4, 3, -33, -45, 33] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([19, 19, w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]