it now for our purposes. Fix $\tau\in {\mathfrak}{b}S_+^{\triangledown}$ such that $c_f(\tau, 1)\neq 0$, let ${\mathfrak}{t}$ be an integral ideal satisfying $h^T(2\tau)^{-1}h\in 4{\mathfrak}{t}^{-1}$ for any $h\in\mathbb{Z}^n$, and let $\rho_{\tau}$ be the quadratic character associated to the extension $\mathbb{Q}(i^{[n/2]}\sqrt{|2\tau|})$. Choose $\mu\in\{0, 1\}$ such that $(\psi\chi)_{\infty}(x) = {\operatorname{sgn}}(x_{\infty})^{[k]+\mu}$.

The key ingredients of the integral are three modular forms: the eigenform $f$, a theta series $\theta_{\chi}$, and a normalised non-holomorphic Eisenstein series $\mathcal{E}(z, s)$. The definition of the theta series, taken from [@Shimuraexp (2.1)], is $$\begin{aligned}
    \theta_{\chi}^{\mu}(z; \tau) = \theta_{\chi}(z) := \sum_{x\in M_n(\mathbb{Z})}(\chi_{\infty}\chi^*)^{-1}(|x|)|x|^{\mu}e({\operatorname{tr}}(x^T\tau xz)), \label{theta}\end{aligned}$$ where we understand tht $(\chi_{\infty}\chi^*)(0) = 1$ if ${\mathfrak}{f} = \mathbb{Z}$. This has weight $\frac{n}{2}+\mu$, level $\Gamma[2, 2{\mathfrak}{tf}^2]$ determined by Proposition 2.1 of [@Shimuraexp], character $\rho_{\tau}\chi^{-1}$, and coefficients in $\mathbb{Q}(\chi)$.

The Eisenstein series of weight $\kappa\in\frac{1}{2}\mathbb{Z}$ is now defined more generally. Let $\Gamma' = \Gamma[{\mathfrak}{x}^{-1}, {\mathfrak}{xy}]$ be a congruence subgroup, contained in ${\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$, and let ${\operatorname{\varphi}}$ be a Hecke character such that ${\operatorname{\varphi}}_{\infty}(x) = {\operatorname{sgn}}(x_{\infty})^{[\kappa]}$ and ${\operatorname{\varphi}}_p(a) = 1$ if $a\in\mathbb{Z}_p^{\times}$ and $a\in 1+{\mathfrak}{y}_p$. This series is defined by $$\begin{aligned}
    E(z, s; \kappa, {\operatorname{\varphi}}, \Gamma') := \sum_{\alpha\in P\cap\Gamma'{\backslash}\Gamma'}{\operatorname{\varphi}}_{{\mathfrak}{y}}(|a_{\gamma}|)(\Delta^{s-\frac{\kappa}{2}}||_{\kappa}\alpha)(z), \label{eisenstein}\end{aligned}$$ for variables $z\in\mathbb{H}_n$ and $s\in\mathbb{C}$. This sum is convergent for ${\operatorname{\mathfrak{Re}}}(s)>\frac{n+1}{2}$ and can be continued meromorphically to all 