Some Research On The Mean Value Of Fourier Coefficients Of Automorphic L-functions

Let Hk*be the set of all normalized Hecke promitive eigencuspforms of weight k for the full modular group SL(2,Z).Let f∈Hk*,and λf(n)denote the n-th normalized Fourier coefficient of L(f,s).Let λsymjf(n)be the n-th normalized Fourier cofficient of L(symjf,s).We use the method mentioned in[5]and[14]to get some new results which improve the previous results.In this paper,it is proved that (?) where (?)j is a suitable constant,Pj(t)is a polynomial in t with deg P3(t)=0,deg P4(t)=2,deg P5(t)=5,deg P6(t)=14,deg P7(t)=35,deg P8(t)=90,θ3=277/301,θ4=299/307,θ5=231/233,θ6=2460/2467,θ7=14881/14895,θ8=89765/89793,(?)3=157/181,(?)4=265/289,(?)5=397/421,(?)6=93/97.