On the Petersson norm $\langleθ_ψ,θ_ψ\rangle$

Let $K$ be an imaginary quadratic field of discriminant~$-d<-11$, and for $\ell\geq1$, let $\psi$ be a Hecke character of $K$ with trivial finite conductor and infinite type~$(2\ell,0)$. In this work we prove that for any prime $p>3$, the quotient $$\frac{\langle\theta_{\psi},\theta_{\psi}\rangle}{\Omega_{K}^{4\ell}},$$ is $\lambda$-integral for a prime ideal $\lambda$ over $p$, where $\langle\theta_{\psi},\theta_{\psi}\rangle$ is the Petersson norm of the theta function $\theta_{\psi}=\theta_{\psi}(\tau)$ attached to $\psi$, and $\Omega_{K}$ denotes the Chowla--Selberg period attached to~$K$, and the product $$\prod_{i=1}^{h_{K}}\frac{\langle\theta_{\psi_{i}},\theta_{\psi_{i}}\rangle}{\Omega_{K}^{4\ell}}$$ is rational, where the product is over all $h_{K}$ of the underlying Hecke characters.