/* 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([9, 0, -11, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([36,6,1/6*w^3 + 1/2*w^2 - 11/6*w - 1]) primes_array = [ [4, 2, 1/6*w^3 - 7/3*w - 3/2],\ [4, 2, 1/6*w^3 - 7/3*w + 3/2],\ [9, 3, -1/3*w^3 + 11/3*w],\ [9, 3, w],\ [19, 19, w + 2],\ [19, 19, -1/3*w^3 + 11/3*w - 2],\ [19, 19, -1/3*w^3 + 11/3*w + 2],\ [19, 19, -w + 2],\ [25, 5, 1/3*w^3 - 8/3*w],\ [49, 7, 2/3*w^3 - 19/3*w],\ [49, 7, -1/3*w^3 + 5/3*w],\ [59, 59, 1/6*w^3 + 1/2*w^2 - 11/6*w],\ [59, 59, 1/2*w^2 + 1/2*w - 11/2],\ [59, 59, 1/2*w^2 - 1/2*w - 11/2],\ [59, 59, 1/6*w^3 - 1/2*w^2 - 11/6*w],\ [89, 89, 2/3*w^3 + 1/2*w^2 - 35/6*w - 7/2],\ [89, 89, -5/6*w^3 + 26/3*w - 3/2],\ [89, 89, 1/6*w^3 - 1/2*w^2 + 1/6*w - 2],\ [89, 89, -1/6*w^3 - w^2 + 7/3*w + 5/2],\ [101, 101, 1/6*w^3 + 1/2*w^2 - 17/6*w + 1],\ [101, 101, -1/3*w^3 - 1/2*w^2 + 25/6*w + 13/2],\ [101, 101, -1/3*w^3 + 1/2*w^2 + 25/6*w - 13/2],\ [101, 101, 1/6*w^3 - 1/2*w^2 - 17/6*w - 1],\ [121, 11, 1/2*w^3 - 4*w - 1/2],\ [121, 11, -1/2*w^3 + 4*w - 1/2],\ [149, 149, 1/6*w^3 + 1/2*w^2 - 5/6*w - 5],\ [149, 149, -1/3*w^3 + 1/2*w^2 + 19/6*w - 1/2],\ [149, 149, 1/3*w^3 + 1/2*w^2 - 19/6*w - 1/2],\ [149, 149, -1/6*w^3 + 1/2*w^2 + 5/6*w - 5],\ [151, 151, -5/6*w^3 - 1/2*w^2 + 49/6*w + 3],\ [151, 151, 1/3*w^3 + w^2 - 8/3*w - 7],\ [151, 151, 1/2*w^3 + 1/2*w^2 - 13/2*w - 1],\ [151, 151, 5/6*w^3 - 1/2*w^2 - 49/6*w + 3],\ [169, 13, -1/3*w^3 + 14/3*w + 2],\ [169, 13, -1/3*w^3 + 14/3*w - 2],\ [179, 179, -1/3*w^3 + 17/3*w - 6],\ [179, 179, -2/3*w^3 + 25/3*w + 6],\ [179, 179, -2/3*w^3 + 25/3*w - 6],\ [179, 179, -1/3*w^3 + 17/3*w + 6],\ [191, 191, 1/2*w^3 + 1/2*w^2 - 9/2*w - 1],\ [191, 191, 1/3*w^3 + 1/2*w^2 - 13/6*w - 9/2],\ [191, 191, -1/3*w^3 + 1/2*w^2 + 13/6*w - 9/2],\ [191, 191, -1/2*w^3 + 1/2*w^2 + 9/2*w - 1],\ [229, 229, 5/6*w^3 - 1/2*w^2 - 43/6*w + 2],\ [229, 229, -2/3*w^3 - 1/2*w^2 + 29/6*w + 7/2],\ [229, 229, 2/3*w^3 - 1/2*w^2 - 29/6*w + 7/2],\ [229, 229, 5/6*w^3 + 1/2*w^2 - 43/6*w - 2],\ [239, 239, -1/3*w^3 + w^2 + 11/3*w - 7],\ [239, 239, 1/6*w^3 + w^2 - 7/3*w - 9/2],\ [239, 239, w^2 - w - 4],\ [239, 239, -1/3*w^3 - w^2 + 11/3*w + 7],\ [251, 251, 1/3*w^3 + w^2 - 8/3*w - 5],\ [251, 251, 1/3*w^3 + w^2 - 8/3*w - 6],\ [251, 251, -1/3*w^3 + w^2 + 8/3*w - 6],\ [251, 251, -1/3*w^3 + w^2 + 8/3*w - 5],\ [271, 271, 1/6*w^3 + w^2 - 7/3*w - 19/2],\ [271, 271, 2/3*w^3 - 3/2*w^2 - 29/6*w + 17/2],\ [271, 271, -w^3 + w^2 + 8*w - 4],\ [271, 271, w^3 - 10*w - 2],\ [281, 281, 5/6*w^3 - 1/2*w^2 - 37/6*w + 5],\ [281, 281, -1/6*w^3 - 2/3*w - 5/2],\ [281, 281, 1/3*w^3 + 1/2*w^2 - 25/6*w - 17/2],\ [281, 281, 5/6*w^3 - 26/3*w + 5/2],\ [289, 17, 1/6*w^3 + w^2 - 7/3*w - 11/2],\ [331, 331, -5/6*w^3 + 32/3*w + 17/2],\ [331, 331, -2*w + 5],\ [331, 331, -2*w - 5],\ [331, 331, -2/3*w^3 + 22/3*w + 5],\ [349, 349, 5/6*w^3 - 1/2*w^2 - 43/6*w],\ [349, 349, w^3 - 11*w + 2],\ [349, 349, 1/3*w^3 - w^2 - 2/3*w + 4],\ [349, 349, -5/6*w^3 - 1/2*w^2 + 43/6*w],\ [359, 359, -w - 5],\ [359, 359, -1/3*w^3 + 11/3*w - 5],\ [359, 359, 1/3*w^3 - 11/3*w - 5],\ [359, 359, w - 5],\ [389, 389, -w^3 - 1/2*w^2 + 21/2*w + 5/2],\ [389, 389, -7/6*w^3 + w^2 + 31/3*w - 13/2],\ [389, 389, 7/6*w^3 + w^2 - 31/3*w - 13/2],\ [389, 389, 5/6*w^3 - w^2 - 17/3*w + 9/2],\ [409, 409, -3/2*w^3 - 3/2*w^2 + 27/2*w + 11],\ [409, 409, 1/3*w^3 + w^2 - 8/3*w - 11],\ [409, 409, -2/3*w^3 + 25/3*w + 8],\ [409, 409, -3/2*w^3 + 3/2*w^2 + 27/2*w - 11],\ [421, 421, 1/6*w^3 + 1/2*w^2 + 1/6*w - 4],\ [421, 421, 1/3*w^3 + 1/2*w^2 - 13/6*w - 13/2],\ [421, 421, -1/3*w^3 + 1/2*w^2 + 13/6*w - 13/2],\ [421, 421, 1/2*w^3 + 1/2*w^2 - 9/2*w + 1],\ [461, 461, 2/3*w^3 - 1/2*w^2 - 23/6*w - 1/2],\ [461, 461, 1/6*w^3 - 1/2*w^2 - 11/6*w + 8],\ [461, 461, -1/6*w^3 - 1/2*w^2 + 11/6*w + 8],\ [461, 461, 2/3*w^3 - 13/3*w - 2],\ [491, 491, 7/6*w^3 + 2*w^2 - 28/3*w - 21/2],\ [491, 491, 1/6*w^3 - 10/3*w - 13/2],\ [491, 491, 7/6*w^3 + 2*w^2 - 28/3*w - 23/2],\ [491, 491, -7/6*w^3 + 2*w^2 + 28/3*w - 21/2],\ [509, 509, 1/2*w^3 - w^2 - 5*w + 7/2],\ [509, 509, -2/3*w^3 + w^2 + 25/3*w - 4],\ [509, 509, -5/6*w^3 + 26/3*w - 7/2],\ [509, 509, 1/3*w^3 + w^2 - 17/3*w - 7],\ [529, 23, -2/3*w^3 + 1/2*w^2 + 53/6*w - 19/2],\ [529, 23, 7/6*w^3 - 3/2*w^2 - 59/6*w + 11],\ [569, 569, w^3 - w^2 - 9*w + 5],\ [569, 569, 2/3*w^3 - w^2 - 13/3*w + 6],\ [569, 569, 2/3*w^3 + w^2 - 13/3*w - 6],\ [569, 569, w^3 + w^2 - 9*w - 5],\ [599, 599, 2/3*w^3 - 1/2*w^2 - 23/6*w + 3/2],\ [599, 599, -7/6*w^3 + 1/2*w^2 + 65/6*w - 4],\ [599, 599, -7/6*w^3 - 1/2*w^2 + 65/6*w + 4],\ [599, 599, 2/3*w^3 + 1/2*w^2 - 23/6*w - 3/2],\ [631, 631, 7/6*w^3 + 2*w^2 - 22/3*w - 17/2],\ [631, 631, 7/6*w^3 + w^2 - 31/3*w - 9/2],\ [631, 631, -7/6*w^3 + w^2 + 31/3*w - 9/2],\ [631, 631, -w^3 + 11*w - 1],\ [659, 659, w^3 - 9*w - 1],\ [659, 659, 2/3*w^3 - 13/3*w - 1],\ [659, 659, -2/3*w^3 + 13/3*w - 1],\ [659, 659, 1/3*w^3 + 1/2*w^2 - 19/6*w - 17/2],\ [661, 661, -1/6*w^3 - 1/2*w^2 + 35/6*w - 6],\ [661, 661, w^3 + 3/2*w^2 - 17/2*w - 23/2],\ [661, 661, -5/3*w^3 + w^2 + 52/3*w - 14],\ [661, 661, -1/2*w^3 + 1/2*w^2 + 15/2*w - 8],\ [701, 701, -2/3*w^3 + w^2 + 19/3*w - 4],\ [701, 701, 5/6*w^3 - 26/3*w + 9/2],\ [701, 701, -5/6*w^3 + 26/3*w + 9/2],\ [701, 701, 2/3*w^3 + w^2 - 19/3*w - 4],\ [739, 739, 1/3*w^3 - 17/3*w - 3],\ [739, 739, 2/3*w^3 - 25/3*w + 3],\ [739, 739, 2/3*w^3 - 25/3*w - 3],\ [739, 739, -1/3*w^3 + 17/3*w - 3],\ [761, 761, -1/2*w^3 + w^2 + 3*w - 13/2],\ [761, 761, 5/6*w^3 - w^2 - 23/3*w + 9/2],\ [761, 761, 5/6*w^3 + w^2 - 23/3*w - 9/2],\ [761, 761, 1/2*w^3 + w^2 - 3*w - 13/2],\ [769, 769, 2/3*w^3 + 1/2*w^2 - 35/6*w + 5/2],\ [769, 769, 1/2*w^3 + 1/2*w^2 - 7/2*w - 8],\ [769, 769, -1/2*w^3 + 1/2*w^2 + 7/2*w - 8],\ [769, 769, 2/3*w^3 - 1/2*w^2 - 35/6*w - 5/2],\ [829, 829, w^2 + w - 10],\ [829, 829, 1/3*w^3 + w^2 - 11/3*w - 1],\ [829, 829, -1/3*w^3 + w^2 + 11/3*w - 1],\ [829, 829, w^2 - w - 10],\ [841, 29, 5/6*w^3 - 20/3*w - 3/2],\ [841, 29, 5/6*w^3 - 20/3*w + 3/2],\ [859, 859, 1/6*w^3 + 1/2*w^2 - 23/6*w - 6],\ [859, 859, -2/3*w^3 - 1/2*w^2 + 47/6*w - 1/2],\ [859, 859, -2/3*w^3 + 1/2*w^2 + 47/6*w + 1/2],\ [859, 859, -1/6*w^3 + 1/2*w^2 + 23/6*w - 6],\ [919, 919, 1/2*w^3 + w^2 - 5*w - 5/2],\ [919, 919, 1/6*w^3 + w^2 - 1/3*w - 17/2],\ [919, 919, -1/6*w^3 + w^2 + 1/3*w - 17/2],\ [919, 919, -1/2*w^3 + w^2 + 5*w - 5/2],\ [961, 31, 1/3*w^3 - 8/3*w - 6],\ [961, 31, 5/6*w^3 - 20/3*w - 1/2],\ [971, 971, 1/2*w^3 + 3/2*w^2 - 7/2*w - 8],\ [971, 971, 2/3*w^3 - 3/2*w^2 - 35/6*w + 17/2],\ [971, 971, 1/3*w^3 - 1/2*w^2 - 37/6*w - 3/2],\ [971, 971, 1/2*w^3 - 3/2*w^2 - 7/2*w + 8]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, 1, -1, -2, -8, -4, 8, 0, -6, -6, -2, -4, -12, -4, 4, -6, 18, 14, -2, -6, -10, 10, -2, -10, -10, 22, -10, 10, 6, 4, -16, -8, 8, 2, 10, -16, 0, -20, 16, -12, 0, -24, 8, -22, 10, -6, -26, -16, 12, -24, -16, -16, -8, 20, 0, 16, 8, -12, 20, -2, 2, 6, 26, 26, 4, 20, 28, -28, -10, 10, -26, -6, 36, 16, -24, -32, -6, -6, 10, 10, 22, -22, -26, -14, 22, -38, -6, 10, 6, -2, -10, -30, 24, -44, 12, 4, 2, -34, 14, 14, -10, 10, -22, 42, -6, -22, 8, 0, 8, -40, 24, -8, 32, -16, -20, -36, 36, -4, 2, 26, 22, -10, -6, 2, 34, 2, 16, 4, -28, 12, -10, -26, -10, -50, -50, 30, -10, 30, -6, -14, -50, 30, 22, 14, -20, -20, 4, -20, -48, 16, -24, -24, -30, -2, 20, 12, -12, -44] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4,2,1/6*w^3 - 7/3*w + 3/2])] = -1 AL_eigenvalues[ZF.ideal([9,3,-1/3*w^3 + 11/3*w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]