$\DeclareMathOperator\GL{GL}$There are several books describing the classical Fourier/Whittaker expansion of automorphic forms on $\GL_3(\mathbb{R})$, e.g. Bump - Automorphic forms on $\GL_3(\mathbb{R})$, and Goldfeld - Automorphic forms and $L$-functions for the group $\GL_n(\mathbb{R})$. The expansions presented there are valid for forms invariant under $\GL_n(\mathbb{Z})$. Bump mentions on page 66 that there are analogous expansions for congruence subgroups as well, albeit more complicated ("[...] one must also sum over the cusps of the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$ [...]" ). Probably because of that, he does not consider the problem in the book.

Has the Fourier expansion been worked out for congruence subgroups somewhere since then? The usual proof goes through for the group $\Gamma_0(N)$ without any modification, but there are other subgroups one might consider, such as the analogue of $\Gamma(N)$ from $\GL_2$ (I am assuming that Bump was talking about this group).