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In this paper, for a square-free integer l>1, a even positive integer k and apositive integer N, we give a trace formula of the Hecke operator T(l) on thespace S_k^0(N) of all newforms of weight k and level \Gamma_0(N). Moreover, wegive a decomposition S_k^0(N)=\oplus_{i|N}S_k^0(N;i) and also give a traceformula of T(l) on S_k^0(N;i).

Trace formulas of the Hecke operator on the spaces of newforms