Let $\mathbb{A}$ be the adele ring of $\mathbb{Q}$.
Let $f\in S_k(\Gamma_0(N),\chi)$ (possibly a newform) and $\phi_f:\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A})\to \mathbb{C}$ be its cuspidal automorphic representation. I will define $\phi_f$ as follows. For $g\in \text{GL}_2(\mathbb{A})$, we have the decomposition $g=\gamma hu$ where $\gamma\in\text{GL}_2(\mathbb{Q})$, $h\in\text{GL}_2(\mathbb{R})^+$, $u\in U_0(N)$ where $$ U_0(N) := \left\{ \begin{pmatrix}a & b \\ c & d\end{pmatrix}\in\text{GL}_2(\widehat{\mathbb{Z}}) \huge\mid\normalsize c\in N\widehat{\mathbb{Z}} \right\}.$$ Then, we may define $$ \phi_f(g) = \phi_f(\gamma hu):= (\det h)^{k/2}j(h,i)^{-k}f(hi)\omega_\chi(u) $$ where $\omega_\chi$ is the adelization of $\chi$ to $U_0(N)$. It can be checked this definition is well-defined.
The space of irreducible cuspidal automorphic representations can be decomposed into a restricted tensor product over the local places of $\mathbb{Q}$. I believe $\phi_f$ actually corresponds to a pure tensor. Is there a good way to see how to decompose $$ \phi_f = \bigotimes_v' \phi_{v} = \phi_\infty \otimes \bigotimes_{p}' \phi_p ?$$