I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.

**#1** Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then $\theta$ equidistributed as $p \to \infty$.

**#2** Let $\displaystyle T_n.z = \sum_{ad=n, 0 \leq b < d} \frac{az+b}{d}$ and extend to functions $\displaystyle [T_n ]f(y) = \sum_{y \in T_n.z} f(y)$. Then we have equidistribution of Hecke points over the modular surface:
$$ \lim_{n \to \infty} [T_n f](z) = \int_{X(1)} f(z) \, dz$$

Both of these results have Hecke's name attached. The first deals with Hecke characters and L-functions the second deals with Hecke operators and modular forms.

**Is result #1 considered an instance #2 of the equidistribution of Hecke points?**

Possibly I consider the theta function: $ \displaystyle \theta (z) = \sum_{(m, n) \in \mathbb{Z}^2} e^{2\pi i (m^2 + n^2) z} $

However I don't see how to act the Hecke operator in such a way as to prove the result.