Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on special cases of this problem we use elementary methods for the computation of Petersson products and avoid using the Rankin $L$-function.