Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character).

Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume $\chi$ is primitive mod $N$. You may also take $g$ to be an Eisenstein series $E(z,s,\chi)=\sum_{\gamma\in \Gamma_\infty\backslash \Gamma_0(N)}\overline{\chi}(\gamma) \Im(\gamma z)^s$.

If $g$ were level 1 (full modular group $\mathrm{SL}(2,\mathbb{Z})$), then we know that the Rankin-Selberg $L$-function of $F\times g$ is a nice integral \[\text{gamma factors}\cdot L(s,F\times g)=\int_{\mathrm{SL}(2,\mathbb{Z})\backslash \mathbb{H}} F\left(\begin{pmatrix}z&\\&1\end{pmatrix}\right)g(z)\det(z)^{s-1}\, \frac{dx \,dy}{y^2}.\]

The standard unfolding technique applies nicely to the Fourier-Whittaker expansion of $F$. See page 372 of Goldfeld, Dorian. Automorphic forms and L-functions for the group $\mathrm{GL}(n,\mathbb{R})$. Vol. 99. Cambridge University Press, 2006.

However, when $g$ has higher level with character, I don't know how to define the integral. That is my question.

Also, remind that if $f$ and $g$ are automorphic forms on $\mathrm{GL}(2)$ with characters $\chi_f$ and $\chi_g$ we can define their Rankin-Selberg $f\times \bar g$ by using some Eisenstein series $E(z,s, \chi_f\bar{\chi_g})$ to balance the characters of $f\bar g$.

And for my question, adelically, there is definite answer. But I am looking for something computable in classic settings.