  Let $k,N \in \mathbb{N}$ with $N$ square-free and $k>1$. We prove an
orthogonal relation and use this to compute the Fourier coefficients of the
Eisenstein part of any $f(z) \in M_{2k}(\Gamma_0(N))$ in terms of sum of
divisors function. In particular, if $f(z) \in E_{2k}(\Gamma_0(N))$, then the
computation will to yield to an expression for the Fourier coefficients of
$f(z)$. Then we apply our main theorem to give formulas for convolution sums of
the divisor function to extend the result by Ramanujan, and to eta quotients
which yields to formulas for number of representations of integers by certain
families of quadratic forms. At last we give essential results to derive
similar results for modular forms in a more general setting.
