Let us consider the Petersson inner product $\left<-,-\right>:S_2(\Gamma_0(N))\times S_2(\Gamma_0(N))\rightarrow \mathbb{C}$, $$\left<f,g\right>:=\int_{\Gamma_0(N)\backslash \mathbb{H}}\bar{f}g\,dxdy.$$
$\textbf{My question is:}$ Is there an orthogonal basis for $S_2(\Gamma_0(N))$ with rational Fourier coefficients?
I have seen the proof that there exists a rational (and in fact integral) basis for $S_2(\Gamma_0(N))$ as a complex vector space. Via Gram-Schmidt we can then modify this basis into an orthogonal one. However, since the inner product takes values in $\mathbb{C}$, this process may kill the rationality of such basis.
I think we may need something other than Gram-Schmidt. Perhaps we should use one of the basis that we know to be orthogonal and do some clever manipulations? Or maybe do some tricks with Hecke operators?
Sorry if this is a trivial question, I may be missing something obvious.