I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\mathbb{R}^{n}$. The harmonic (or weighted) theta series attached to $(L,P)$ is defined as $$\theta_{L,P}(z)=\sum_{v\in L}P(v)e^{\pi i z q(v)},z=x+iy,y>0.$$ Of course with the help of Poisson summation formula and invariant theory, we can prove that $\theta_{L,P}$ is a classical modular form for $\mathrm{SL}_{2}(\mathbb{Z})$ of weight $n+d/2$. However I am confused with its connection with the global theta correspondence.
In the section 5 of Prasad's paper A brief survey on the theta correspondece, he defines a function $\theta_{\phi}$, which is an adelic analogue of the classical theta function, and use this as the kernel function to state the global theta lift for the dual pair $G_{1}\times G_{2}:=\mathrm{O}(V)\times \mathrm{SL}(2)$: for a vector $f$ in a cuspidal representation of $G_{1}$, define a function on $G_{1}(\mathbb{Q})\backslash G_{1}(\mathbb{A})$ $$\theta_{\phi}(f)(g_{2})=\int_{G_{1}(\mathbb{Q})\backslash G_{1}(\mathbb{A})}\theta_{\phi}(g_{1},g_{2})f(g_{1})dg_{1},$$ which gives us an automorphic form on $G_{2}=\mathrm{SL}(2)$. In the following example, Prasad claims that $\theta_{L,P}$ is such a lift from $\mathrm{O}(V)$ to $\mathrm{SL}(2)$.
So my question is: can we obtain the automorphic form generated by the modular form $\theta_{L,P}$ from the theta integral explicitly? I tried to do this but I failed to figure out which Schwartz function $\phi$ should I choose. Is there any reference for this, or some hint on it?
