# Effective Hecke Equidistribution

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] $$$$\lim_{n\to\infty}\frac{\#\{p\leq n:p \text{ splits in }\mathcal{O},\alpha\leq \cos(\theta_p)\leq \beta\}}{\#\{p\leq n:p\text{ splits in }\mathcal O\}}=\frac{1}{\pi}\int_\alpha^\beta \frac{d\theta}{\sqrt{1-\theta^2}}.$$$$

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

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.

• Please use a high-level tag like "nt.number-theory". I added this tag now. Regarding high-level tags, see meta.mathoverflow.net/q/1075 Commented Jun 3 at 15:00