/* 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, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([27,9,w^3 - 2*w^2 - 4*w + 2]) primes_array = [ [3, 3, -w^3 + w^2 + 5*w],\ [3, 3, w - 1],\ [9, 3, w^3 - w^2 - 4*w],\ [13, 13, -2*w^3 + w^2 + 11*w + 3],\ [13, 13, w^3 - 2*w^2 - 4*w + 3],\ [16, 2, 2],\ [23, 23, -w^2 + 2],\ [23, 23, -w^3 + 2*w^2 + 4*w - 4],\ [49, 7, -w^3 + 2*w^2 + 5*w - 2],\ [49, 7, 2*w^3 - 2*w^2 - 9*w - 1],\ [53, 53, w^3 - 3*w^2 + 3],\ [53, 53, 2*w^3 - w^2 - 8*w - 3],\ [53, 53, -w^3 + w^2 + 6*w - 1],\ [61, 61, -w - 3],\ [61, 61, -w^3 + w^2 + 5*w - 4],\ [79, 79, w^3 - 2*w^2 - 4*w + 1],\ [79, 79, w^2 - 5],\ [101, 101, w^3 - 6*w],\ [101, 101, w^2 - 2*w - 4],\ [103, 103, 4*w^3 - 3*w^2 - 20*w - 3],\ [103, 103, w^3 - 2*w^2 - 1],\ [113, 113, 5*w^3 - 3*w^2 - 25*w - 6],\ [113, 113, 2*w - 3],\ [113, 113, w^3 - 3*w^2 - w + 4],\ [113, 113, 2*w^3 - 2*w^2 - 10*w - 1],\ [121, 11, -w^2 + 2*w - 2],\ [121, 11, -w^3 + 6*w + 6],\ [127, 127, 2*w^3 - 3*w^2 - 6*w + 3],\ [127, 127, -3*w^3 + 2*w^2 + 14*w + 1],\ [139, 139, 3*w^3 - 4*w^2 - 12*w + 6],\ [139, 139, 2*w^3 - w^2 - 8*w],\ [157, 157, -w^3 + 7*w + 5],\ [157, 157, w^3 - 7*w + 2],\ [173, 173, w^2 - 3*w - 3],\ [173, 173, -2*w^3 + w^2 + 11*w],\ [179, 179, -2*w^3 + w^2 + 8*w + 1],\ [179, 179, -3*w^3 + 4*w^2 + 12*w - 5],\ [181, 181, -2*w^3 + w^2 + 11*w + 6],\ [181, 181, w^2 - 3*w + 3],\ [191, 191, w^2 + w - 4],\ [191, 191, 2*w^3 - 3*w^2 - 9*w + 3],\ [199, 199, -w^3 - w^2 + 7*w + 6],\ [199, 199, -2*w^3 + 3*w^2 + 8*w - 2],\ [199, 199, -3*w^3 + 3*w^2 + 13*w],\ [199, 199, w^3 - 4*w - 4],\ [211, 211, -2*w^3 + 3*w^2 + 7*w - 6],\ [211, 211, w^3 - w^2 - 7*w + 2],\ [211, 211, 2*w^3 - w^2 - 9*w + 1],\ [211, 211, 2*w^3 - 2*w^2 - 11*w + 1],\ [233, 233, w^2 - 8],\ [233, 233, w^3 - 2*w^2 - 4*w - 2],\ [251, 251, 2*w^2 - 3*w - 4],\ [251, 251, 2*w^3 - 3*w^2 - 11*w + 4],\ [257, 257, -4*w^3 + 2*w^2 + 22*w + 5],\ [257, 257, 4*w^3 - 5*w^2 - 17*w + 8],\ [263, 263, 3*w^3 - 2*w^2 - 15*w],\ [263, 263, -w^3 + 2*w^2 + w - 3],\ [269, 269, -w^3 + w^2 + 2*w - 4],\ [269, 269, -w^3 - w^2 + 5*w + 4],\ [269, 269, 2*w^3 - 4*w^2 - 7*w + 7],\ [269, 269, -2*w^3 + 4*w^2 + 5*w],\ [277, 277, -4*w^3 + 3*w^2 + 19*w - 1],\ [277, 277, 2*w^3 - 3*w^2 - 5*w + 4],\ [283, 283, 3*w^3 - 3*w^2 - 13*w + 3],\ [283, 283, 2*w^3 - 2*w^2 - 7*w + 2],\ [289, 17, -w^3 + w^2 + 4*w - 5],\ [289, 17, w^3 - w^2 - 4*w - 4],\ [313, 313, -2*w^3 + 3*w^2 + 7*w - 9],\ [313, 313, -2*w^3 + w^2 + 9*w - 4],\ [337, 337, -2*w^3 + 3*w^2 + 8*w - 1],\ [337, 337, w^3 - 4*w - 5],\ [373, 373, -2*w^3 + w^2 + 10*w - 1],\ [373, 373, -w^3 + 2*w^2 + 2*w - 5],\ [389, 389, w^3 + w^2 - 7*w - 11],\ [389, 389, -2*w^2 + 3*w - 2],\ [419, 419, -w^3 + 3*w^2 + 4*w - 7],\ [419, 419, -w^3 + 3*w^2 + 4*w - 5],\ [433, 433, -2*w^3 + 2*w^2 + 6*w - 5],\ [433, 433, -4*w^3 + 4*w^2 + 18*w - 7],\ [433, 433, 2*w^3 - 3*w^2 - 11*w + 1],\ [433, 433, 7*w^3 - 4*w^2 - 35*w - 8],\ [443, 443, -3*w^3 + w^2 + 15*w + 3],\ [443, 443, -2*w^3 + 4*w^2 + 5*w - 6],\ [467, 467, 6*w^3 - 4*w^2 - 30*w - 3],\ [467, 467, -2*w^3 + 4*w^2 + 2*w - 3],\ [467, 467, 3*w - 4],\ [467, 467, 3*w^3 - 3*w^2 - 15*w - 1],\ [491, 491, -w^3 + 4*w^2 - 3*w + 2],\ [491, 491, -5*w^3 + 2*w^2 + 27*w + 13],\ [503, 503, 3*w^3 - 18*w - 11],\ [503, 503, -3*w^2 + 6*w + 1],\ [523, 523, w^3 - 3*w^2 - 3*w + 6],\ [523, 523, -3*w^3 + 16*w + 9],\ [523, 523, 2*w^2 - w - 5],\ [523, 523, 2*w^3 - 5*w^2 - 4*w + 5],\ [529, 23, 2*w^3 - 2*w^2 - 8*w - 5],\ [547, 547, -w^3 + 2*w^2 + w - 4],\ [547, 547, -2*w^3 + w^2 + 11*w - 2],\ [547, 547, -3*w^3 + 2*w^2 + 15*w - 1],\ [547, 547, w^2 - 3*w - 5],\ [563, 563, -3*w^3 + 4*w^2 + 10*w - 7],\ [563, 563, -3*w^3 + 3*w^2 + 15*w + 4],\ [569, 569, 2*w^3 - w^2 - 11*w - 7],\ [569, 569, w^2 - 3*w + 4],\ [571, 571, -3*w^3 + 2*w^2 + 12*w + 5],\ [571, 571, 4*w^3 - 5*w^2 - 16*w + 1],\ [601, 601, -5*w^3 + 4*w^2 + 26*w + 4],\ [601, 601, -w^2 + 6*w - 4],\ [607, 607, 2*w^3 - 4*w^2 - 6*w + 1],\ [607, 607, 2*w^3 - 2*w^2 - 11*w - 5],\ [607, 607, w^3 - w^2 - 7*w + 8],\ [607, 607, 2*w^3 - 10*w - 9],\ [625, 5, -5],\ [641, 641, -4*w^3 + 5*w^2 + 19*w - 13],\ [641, 641, -4*w^3 + 4*w^2 + 21*w - 1],\ [653, 653, 3*w^3 - 2*w^2 - 14*w + 2],\ [653, 653, 4*w^3 - 7*w^2 - 17*w + 10],\ [653, 653, -2*w^3 + w^2 + 14*w + 6],\ [653, 653, -2*w^3 + 3*w^2 + 6*w - 6],\ [659, 659, 3*w - 5],\ [659, 659, 3*w^3 - 3*w^2 - 15*w - 2],\ [673, 673, 2*w^3 - 4*w^2 - 9*w + 4],\ [673, 673, 3*w^3 - 3*w^2 - 16*w + 3],\ [673, 673, -w^3 + 2*w^2 + 4*w - 9],\ [673, 673, -w^3 + 3*w^2 + 5*w - 9],\ [677, 677, -3*w^3 + 4*w^2 + 15*w - 3],\ [677, 677, -w^3 + 2*w^2 + 7*w - 6],\ [701, 701, w^2 - 3*w - 6],\ [701, 701, 2*w^3 - w^2 - 11*w + 3],\ [719, 719, w^3 - w^2 - 8*w + 7],\ [719, 719, -4*w^3 + 5*w^2 + 21*w - 11],\ [797, 797, -2*w^3 + w^2 + 13*w + 2],\ [797, 797, -2*w^3 + w^2 + 13*w - 1],\ [809, 809, -w^3 + w^2 + 6*w - 7],\ [809, 809, w^3 - w^2 - 6*w - 5],\ [823, 823, -w^3 + 2*w^2 + 6*w - 8],\ [823, 823, 2*w^3 - 3*w^2 - 10*w],\ [823, 823, -7*w^3 + 5*w^2 + 34*w + 6],\ [823, 823, 6*w^3 - 6*w^2 - 27*w + 1],\ [829, 829, -w^3 + 4*w^2 + w - 11],\ [829, 829, 4*w^3 - 3*w^2 - 18*w - 3],\ [841, 29, -3*w^3 + 3*w^2 + 12*w - 1],\ [841, 29, -3*w^3 + 3*w^2 + 12*w - 2],\ [857, 857, -3*w^3 + 2*w^2 + 12*w - 1],\ [857, 857, 2*w^2 - 9],\ [857, 857, -4*w^3 + 5*w^2 + 16*w - 7],\ [857, 857, 2*w^3 - 4*w^2 - 8*w + 3],\ [881, 881, -3*w^3 + 2*w^2 + 12*w + 2],\ [881, 881, 4*w^3 - 5*w^2 - 16*w + 4],\ [883, 883, -4*w^3 + 4*w^2 + 22*w + 1],\ [883, 883, -2*w^3 + 2*w^2 + 14*w - 7],\ [887, 887, 8*w^3 - 6*w^2 - 41*w - 4],\ [887, 887, -w^3 + 3*w^2 - 5*w + 1],\ [907, 907, -3*w^3 + 4*w^2 + 10*w - 4],\ [907, 907, 4*w^3 - 3*w^2 - 18*w],\ [907, 907, w^3 - w^2 - 5*w - 5],\ [907, 907, w - 6],\ [911, 911, -3*w^3 + 6*w^2 + 7*w],\ [911, 911, 5*w^3 - 2*w^2 - 25*w - 13],\ [911, 911, 2*w^3 - w^2 - 8*w - 6],\ [911, 911, -3*w^3 + 4*w^2 + 12*w],\ [919, 919, -2*w^3 + 3*w^2 + 11*w - 10],\ [919, 919, -w^3 - w^2 + 8*w + 10]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-1, 0, 2, 4, -4, 1, 6, 0, -2, -2, 6, 0, -6, 10, 8, 14, 4, -18, 0, -8, 4, 18, 6, -6, -6, 4, -4, 2, -2, 16, -8, -10, 2, -6, 18, -12, 12, 14, -10, 0, 24, 22, -16, -10, 10, -4, 14, -14, 10, 6, 6, 18, -24, 6, 12, -24, -12, 12, -30, 6, 0, 22, -26, -16, -14, 2, -2, 26, -14, 4, 8, -4, 4, 18, -6, -24, 0, 16, 2, -26, 8, 12, 12, 12, 0, 12, -6, 18, 36, -24, 24, 4, -44, 28, -20, -10, 32, -44, 32, -8, -12, -24, 42, -18, 4, -10, 22, 2, 40, 8, 8, -32, -14, 30, 18, -12, -6, -18, 36, 12, -30, -2, 14, -34, 10, -12, 6, 18, -18, 12, 0, -18, 42, 24, 24, -40, -40, -40, -40, 20, 46, -46, 10, -18, 0, 6, -54, -54, -30, 20, -20, 18, 12, -16, 10, -50, 26, 48, 48, 18, 24, -32, -16] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3,3,-w + 1])] = 1 AL_eigenvalues[ZF.ideal([3,3,w^3 - w^2 - 5*w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]