In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify linear combinations of Hecke forms with CM modular forms. For a weight $k$ newform $f=\sum a_nq^n$ with CM by an order $\mathcal O$ in a quadratic field, write for a prime $p$ $a_p=2p^{\frac{k-1}{2}}\cos(\theta_p)$. The equidistribution for the $\theta_p$ is [BFOR Theorem 15.4] \begin{equation}\lim_{n\to\infty}\frac{\#\{p\leq n:p \text{ splits in }\mathcal{O},\alpha\leq \cos(\theta_p)\leq \beta\}}{\#\{p\leq n:p\text{ splits in }\mathcal O\}}=\frac{1}{\pi}\int_\alpha^\beta \frac{d\theta}{\sqrt{1-\theta^2}}.\end{equation}
My question is: "What is an English-language reference for the rate of convergence for the Hecke equidistribution"? Since the result is classical I imagine it must have been done previously.
To get a precise sense for what I mean, consider the non-CM case which is described by the Sato-Tate conjecture, now a theorem. Let $ f $ be a newform without complex multiplication. Then we have the (effective) Sato-Tate distribution \begin{equation*} \frac{\#\left\{ p<x \colon \alpha \le \theta_p \le \beta \right\}}{\pi\left( x \right)} = \frac{2}{\pi} \int_{\alpha}^{\beta} \sin^2\theta d \theta + O\left( \frac{\log\left(kN \log x\right)}{\sqrt{\log x}} \right), \end{equation*}
from [Thorner]. What is a good reference for the effective Hecke equidistribution? A related MathOverflow question does not seem to provide effective results.
Thorner, Jesse, Effective forms of the Sato-Tate conjecture, Res. Math. Sci. 8, No. 1, Paper No. 4, 21 p. (2021). ZBL1465.11140.
Bringmann, Kathrin; Folsom, Amanda; Ono, Ken; Rolen, Larry, Harmonic Maass forms and mock modular forms: theory and applications, Colloquium Publications. American Mathematical Society 64. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1944-8/hbk; 978-1-4704-4313-9/ebook). xv, 381 p. (2017). ZBL1459.11118.
