The Shimura correspondence central-value formula for level $p^2$

Let $p$ be the prime and let $f$ be the weight-$2$ eigenform introduced in the construction above. Fix $\sigma\in\{+,-\}$, and let $c_f(d)$ be the coefficient of $q^d$ in

$$\Theta_f^\sigma=\sum_{\left(\frac{d}{p}\right)=\sigma}c_f(d)q^d.$$

Let $\alpha_f$ be the rational constant defined according as $f$ is not a twist of a level-$p$ form, is the quadratic twist of a level-$p$ form, or is a level-$p$ form. The Shimura correspondence central-value formula. For every integer $d$ such that $-pd<0$ is a fundamental discriminant and $\left(\frac{d}{p}\right)=\sigma$,

$$L(f,-pd,1)L(f,1)=\alpha_f\,\frac{\langle f,f\rangle}{\sqrt{pd}}\frac{c_f(d)^2}{\langle\e{f}^{\sigma},\e{f}^{\sigma}\rangle}.$$

This formula relates central values of twisted $L$-functions to Fourier coefficients arising from the quaternionic Shimura correspondence, with the normalization depending on the choice of $\e{f}^{\sigma}$. The supplied text states the identity but gives no evidence resolving whether it is intended as a conjecture or as a theorem.