/* 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, -8, 8, 6, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([41,41,w^5 - 5*w^3 + 2*w^2 + 5*w - 5]) primes_array = [ [7, 7, -w^5 + 5*w^3 - 5*w - 1],\ [27, 3, -2*w^5 + 10*w^3 - w^2 - 10*w + 2],\ [41, 41, -w^5 + 6*w^3 - w^2 - 7*w + 2],\ [41, 41, w^4 - w^3 - 4*w^2 + 3*w + 1],\ [41, 41, -2*w^5 + 12*w^3 - 2*w^2 - 17*w + 5],\ [41, 41, w^5 - 5*w^3 + 2*w^2 + 5*w - 5],\ [41, 41, -w^4 - 2*w^3 + 4*w^2 + 6*w - 3],\ [41, 41, -2*w^5 + 10*w^3 - w^2 - 10*w + 3],\ [43, 43, -w^5 + w^4 + 6*w^3 - 5*w^2 - 9*w + 4],\ [43, 43, -w^4 - w^3 + 4*w^2 + 4*w - 3],\ [43, 43, -w^4 + 3*w^2 + 1],\ [43, 43, -w^3 + w^2 + 4*w - 2],\ [43, 43, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 3],\ [43, 43, -w^2 - w + 3],\ [64, 2, -2],\ [83, 83, -w^5 + 6*w^3 - w^2 - 10*w + 4],\ [83, 83, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 17*w + 6],\ [83, 83, w^5 - 6*w^3 + 2*w^2 + 8*w - 3],\ [83, 83, w^5 - 4*w^3 + w - 1],\ [83, 83, -2*w^5 + 11*w^3 - w^2 - 13*w + 1],\ [83, 83, w^5 - 5*w^3 + 2*w^2 + 6*w - 6],\ [125, 5, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 2],\ [125, 5, -w^5 - w^4 + 5*w^3 + 4*w^2 - 6*w - 1],\ [127, 127, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 4],\ [127, 127, -w^5 + 7*w^3 - 2*w^2 - 12*w + 5],\ [127, 127, -2*w^5 + w^4 + 11*w^3 - 5*w^2 - 13*w + 5],\ [127, 127, -2*w^3 + w^2 + 7*w - 2],\ [127, 127, -w^5 + 6*w^3 - 7*w],\ [127, 127, -w^4 - 2*w^3 + 4*w^2 + 7*w - 3],\ [167, 167, -w^5 - w^4 + 4*w^3 + 4*w^2 - 1],\ [167, 167, -2*w^4 + 7*w^2 + w - 3],\ [167, 167, w^5 - w^4 - 4*w^3 + 3*w^2 + w + 2],\ [167, 167, w^5 + w^4 - 4*w^3 - 4*w^2 + 3*w - 1],\ [167, 167, 2*w^5 - 11*w^3 + w^2 + 12*w - 3],\ [167, 167, 3*w^5 + w^4 - 16*w^3 - w^2 + 19*w - 6],\ [169, 13, w^5 - 4*w^3 + w^2 + 2*w - 3],\ [169, 13, -2*w^5 + 11*w^3 - 2*w^2 - 13*w + 4],\ [169, 13, -w^5 + 7*w^3 - w^2 - 11*w + 4],\ [211, 211, 3*w^5 - w^4 - 17*w^3 + 7*w^2 + 23*w - 9],\ [211, 211, -3*w^3 + 10*w - 1],\ [211, 211, w^5 + w^4 - 7*w^3 - 4*w^2 + 12*w - 1],\ [211, 211, 3*w^5 - w^4 - 16*w^3 + 6*w^2 + 18*w - 5],\ [211, 211, -2*w^5 + 12*w^3 - 3*w^2 - 16*w + 6],\ [211, 211, w^3 - 2*w^2 - 4*w + 3],\ [251, 251, 3*w^5 - 18*w^3 + 3*w^2 + 26*w - 8],\ [251, 251, 2*w^5 - 2*w^4 - 13*w^3 + 9*w^2 + 20*w - 7],\ [251, 251, w^5 + 2*w^4 - 5*w^3 - 8*w^2 + 6*w + 2],\ [251, 251, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 11*w],\ [251, 251, 2*w^4 - w^3 - 8*w^2 + 3*w + 3],\ [251, 251, w^4 + 3*w^3 - 5*w^2 - 9*w + 3],\ [293, 293, -2*w^5 - 2*w^4 + 10*w^3 + 7*w^2 - 11*w],\ [293, 293, -3*w^5 - w^4 + 16*w^3 + 2*w^2 - 18*w + 2],\ [293, 293, 2*w^5 - 13*w^3 + w^2 + 20*w - 5],\ [293, 293, w^5 - 2*w^4 - 7*w^3 + 8*w^2 + 12*w - 5],\ [293, 293, -2*w^4 - w^3 + 9*w^2 + 2*w - 7],\ [293, 293, -w^5 - w^4 + 4*w^3 + 5*w^2 - 3*w - 3],\ [337, 337, -w^5 + w^4 + 5*w^3 - 5*w^2 - 6*w + 3],\ [337, 337, -w^5 + 7*w^3 - 2*w^2 - 12*w + 4],\ [337, 337, -w^5 + 6*w^3 - 7*w - 1],\ [337, 337, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 13*w - 6],\ [337, 337, -w^4 - 2*w^3 + 4*w^2 + 7*w - 4],\ [337, 337, w^5 + w^4 - 5*w^3 - 2*w^2 + 5*w - 4],\ [379, 379, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 13*w - 4],\ [379, 379, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 4],\ [379, 379, 3*w^5 - 16*w^3 + w^2 + 18*w - 5],\ [379, 379, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 10*w - 7],\ [379, 379, -2*w^5 - 2*w^4 + 11*w^3 + 6*w^2 - 15*w + 1],\ [379, 379, w^5 - 7*w^3 + 3*w^2 + 13*w - 8],\ [419, 419, 2*w^5 - 10*w^3 + 3*w^2 + 11*w - 9],\ [419, 419, -3*w^5 + w^4 + 18*w^3 - 7*w^2 - 25*w + 8],\ [419, 419, -w^4 + 2*w^3 + 3*w^2 - 6*w + 3],\ [419, 419, -2*w^5 + 9*w^3 - w^2 - 6*w + 4],\ [419, 419, -w^5 + w^4 + 8*w^3 - 4*w^2 - 14*w + 5],\ [419, 419, 2*w^5 + w^4 - 11*w^3 - 2*w^2 + 12*w - 1],\ [421, 421, -w^3 - 2*w^2 + 4*w + 3],\ [421, 421, -3*w^5 + w^4 + 17*w^3 - 6*w^2 - 23*w + 9],\ [421, 421, 3*w^5 + w^4 - 17*w^3 - w^2 + 23*w - 7],\ [421, 421, 3*w^5 - 15*w^3 + w^2 + 16*w - 2],\ [421, 421, -2*w^4 - 3*w^3 + 9*w^2 + 9*w - 6],\ [421, 421, w^5 - w^4 - 6*w^3 + 5*w^2 + 6*w - 5],\ [461, 461, 2*w^5 - w^4 - 13*w^3 + 7*w^2 + 20*w - 8],\ [461, 461, -2*w^5 + w^4 + 10*w^3 - 5*w^2 - 9*w + 6],\ [461, 461, 2*w^4 + w^3 - 9*w^2 - 3*w + 4],\ [461, 461, w^5 - w^4 - 8*w^3 + 4*w^2 + 15*w - 5],\ [461, 461, 2*w^5 + w^4 - 10*w^3 - w^2 + 11*w - 7],\ [461, 461, w^5 - w^4 - 4*w^3 + 4*w^2 + 3*w],\ [463, 463, w^5 - 2*w^4 - 7*w^3 + 8*w^2 + 10*w - 6],\ [463, 463, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 8],\ [463, 463, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w + 1],\ [463, 463, -3*w^5 - w^4 + 15*w^3 + 3*w^2 - 15*w],\ [463, 463, -w^4 + 3*w^2 - 3],\ [463, 463, -w^4 - w^3 + 4*w^2 + 4*w + 1],\ [503, 503, w^5 - 7*w^3 + 12*w - 3],\ [503, 503, -3*w^5 - w^4 + 17*w^3 + w^2 - 23*w + 6],\ [503, 503, 2*w^5 - 10*w^3 + 3*w^2 + 10*w - 7],\ [503, 503, -w^5 + w^4 + 8*w^3 - 5*w^2 - 15*w + 8],\ [503, 503, -3*w^5 + 15*w^3 - w^2 - 16*w + 3],\ [503, 503, -2*w^5 + w^4 + 10*w^3 - 5*w^2 - 10*w + 7],\ [547, 547, 2*w^5 + w^4 - 11*w^3 - 4*w^2 + 14*w - 1],\ [547, 547, 3*w^5 - w^4 - 17*w^3 + 7*w^2 + 21*w - 9],\ [547, 547, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 21*w + 5],\ [547, 547, 2*w^4 + w^3 - 9*w^2 - 2*w + 4],\ [547, 547, 2*w^5 + w^4 - 10*w^3 - 3*w^2 + 12*w - 3],\ [547, 547, -w^5 - w^4 + 6*w^3 + 4*w^2 - 10*w - 3],\ [587, 587, 3*w^5 - w^4 - 16*w^3 + 7*w^2 + 19*w - 10],\ [587, 587, -2*w^5 - 2*w^4 + 9*w^3 + 7*w^2 - 7*w - 3],\ [587, 587, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 8*w - 7],\ [587, 587, 2*w^5 + w^4 - 12*w^3 - w^2 + 16*w - 5],\ [587, 587, -3*w^5 + 17*w^3 - w^2 - 22*w + 5],\ [587, 587, w^5 + w^4 - 7*w^3 - 3*w^2 + 13*w],\ [631, 631, -w^5 + 2*w^4 + 8*w^3 - 10*w^2 - 14*w + 9],\ [631, 631, 3*w^5 + w^4 - 16*w^3 - w^2 + 20*w - 6],\ [631, 631, -w^5 + 2*w^4 + 7*w^3 - 9*w^2 - 10*w + 6],\ [631, 631, 2*w^5 + 2*w^4 - 11*w^3 - 6*w^2 + 14*w - 1],\ [631, 631, -w^5 - w^4 + 6*w^3 + 5*w^2 - 8*w - 3],\ [631, 631, 3*w^5 - 16*w^3 + 2*w^2 + 20*w - 6],\ [673, 673, -w^5 - w^4 + 5*w^3 + w^2 - 5*w + 5],\ [673, 673, -w^4 + 6*w^2 - 6],\ [673, 673, 2*w^5 - 13*w^3 + 3*w^2 + 21*w - 8],\ [673, 673, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 6],\ [673, 673, -3*w^5 + w^4 + 16*w^3 - 5*w^2 - 18*w + 5],\ [673, 673, -2*w^4 - 3*w^3 + 8*w^2 + 10*w - 5],\ [757, 757, 2*w^4 - w^3 - 8*w^2 + 3*w + 5],\ [757, 757, -3*w^5 + 18*w^3 - 3*w^2 - 26*w + 10],\ [757, 757, -w^5 + 6*w^3 - w^2 - 6*w + 4],\ [757, 757, -2*w^5 - w^4 + 10*w^3 + 4*w^2 - 11*w - 4],\ [757, 757, -3*w^5 + 15*w^3 - w^2 - 15*w + 1],\ [757, 757, w^5 + w^4 - 5*w^3 - 4*w^2 + 7*w + 4],\ [797, 797, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 3],\ [797, 797, -w^5 - w^4 + 7*w^3 + 4*w^2 - 11*w],\ [797, 797, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 10*w + 4],\ [797, 797, 3*w^5 + w^4 - 17*w^3 - w^2 + 22*w - 5],\ [797, 797, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 14*w],\ [797, 797, 2*w^5 - 12*w^3 + w^2 + 18*w - 6],\ [839, 839, w^5 + w^4 - 5*w^3 - w^2 + 4*w - 5],\ [839, 839, -4*w^5 + w^4 + 22*w^3 - 6*w^2 - 27*w + 8],\ [839, 839, 2*w^5 + w^4 - 13*w^3 - w^2 + 21*w - 6],\ [839, 839, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 2],\ [839, 839, w^5 + 2*w^4 - 2*w^3 - 8*w^2 - 5*w + 5],\ [839, 839, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 6*w + 8],\ [841, 29, 2*w^5 - 13*w^3 + 2*w^2 + 19*w - 6],\ [841, 29, -3*w^5 + 17*w^3 - 3*w^2 - 21*w + 7],\ [841, 29, 3*w^5 - 16*w^3 + 3*w^2 + 18*w - 7],\ [881, 881, 3*w^5 + w^4 - 17*w^3 - w^2 + 22*w - 7],\ [881, 881, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 15*w - 8],\ [881, 881, -w^5 + 7*w^3 - 2*w^2 - 11*w + 8],\ [881, 881, -2*w^5 - w^4 + 10*w^3 + 2*w^2 - 10*w],\ [881, 881, 2*w^5 - w^4 - 11*w^3 + 6*w^2 + 13*w - 4],\ [881, 881, 2*w^5 - 12*w^3 + w^2 + 16*w - 6],\ [883, 883, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 19*w - 4],\ [883, 883, -2*w^5 + w^4 + 14*w^3 - 6*w^2 - 22*w + 8],\ [883, 883, w^5 + w^4 - 3*w^3 - 5*w^2 + 5],\ [883, 883, -4*w^5 + 22*w^3 - 4*w^2 - 27*w + 10],\ [883, 883, -4*w^5 + w^4 + 23*w^3 - 8*w^2 - 30*w + 12],\ [883, 883, 3*w^5 + 2*w^4 - 16*w^3 - 6*w^2 + 19*w - 1],\ [967, 967, -w^5 + 2*w^4 + 7*w^3 - 9*w^2 - 13*w + 5],\ [967, 967, -3*w^5 + 16*w^3 - 3*w^2 - 20*w + 9],\ [967, 967, 2*w^5 - 13*w^3 + 2*w^2 + 18*w - 6],\ [967, 967, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 15*w - 8],\ [967, 967, 3*w^5 - 16*w^3 + 4*w^2 + 18*w - 9],\ [967, 967, 2*w^5 + 2*w^4 - 11*w^3 - 6*w^2 + 13*w]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-2, -9, 6, -3, -3, 1, -6, -6, 10, 1, -8, 1, 7, -8, -11, -9, -6, -12, -15, -9, 0, 18, -3, -16, 2, -10, 5, 11, 8, 15, -6, 21, 3, -12, 0, -19, 20, 5, 10, -23, -5, 7, 16, -14, -21, 0, 21, -15, 12, 18, 30, 12, 9, 18, -21, 0, 19, -32, -8, -11, 13, -23, -2, -29, 13, -20, 4, -26, 36, -33, 36, -24, -39, -30, 38, -25, -19, 26, -10, -19, 12, -6, 18, -30, -27, 24, 8, -1, -19, -19, 20, 17, 36, -30, -27, -30, -6, -9, -7, -4, 35, 26, -22, 20, 24, -42, -18, 18, 24, -15, -16, -37, 14, -1, -25, 29, 1, -44, 46, -26, 1, 28, 7, -20, -14, 16, -29, -38, -12, -54, 6, 0, -27, -54, 3, 30, 36, -51, -24, 21, -5, -14, -23, -15, -54, 15, 18, 42, 30, 4, -44, 46, -8, 34, 52, 32, -4, -10, 47, 14, -4] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([41,41,w^5 - 5*w^3 + 2*w^2 + 5*w - 5])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]