TITLE: Coefficients of modular forms and the Sato-Tate distribution
QUESTION [5 upvotes]: Let $a(n)$ be the $n$th Fourier coefficient of a normalized Hecke eigenform $f(z)=\sum_{n=1}^{\infty}a(n)q^n$ of weight $k$ with respect to the full modular group, where $q=e^{i2\pi z}$.
A new paper [1] has the following result.
"We prove for $100\%$ of primes $p$ that
\begin{equation}
2p^{\frac{k-1}{2}}\frac{\log \log p}{
\sqrt{\log p}}
< |a_f (p)| < 2p^{\frac{k-1}{2}}.
\end{equation}
The authors claim that this is true on a subset of primes of density one, the complete details are given in Theorem 1.1.
Another recent paper [3] has the following result.
\begin{equation}
|a_f (n)| < 2p^{\frac{k-1}{2}}\left (\log n\right)^{-1/2+o(1)}
\end{equation}
on a subset of integers of density one, the complete details are given in Theorem 1.
And an old paper [2] has the following result.
"Corollary 2. For a positive density of primes $p$, we have
\begin{equation}
|a(p)|>
\left(\sqrt{2}-\epsilon\right)p^{\frac{k-1}{2}}.
\end{equation}
In [3], page 441, the author shows that
\begin{equation}
|a(n)|>
\left(\sqrt{2}-\epsilon\right)n^{\frac{k-1}{2}}e^{c\log n /\log \log n},
\end{equation}
for some constant $c>0$, which is true on a subset of positive density.
Question. Can you have a pair of disjointed subsets of primes of density one as in the Theorem 1.1, and positive density as in the Corollary 2?
In other words, does Corollary 2 contradict Theorem 1.1 in [1] or Theorem 1 in [3]?
Thanks for your valuable comments.
[1] Ayla Gafni, Jesse Thorner, Peng-Jie Wong; Almost all primes satisfy the Atkin-Serre conjecture and are not extremal;
arXiv:2003.09026; doi 10.1007/s40993-021-00258-w
[2] M. Ram Murty; Oscillations of Fourier Coefficients of Modular Forms; Math. Ann. 262, 431--446 (1983.
[3] Florian Luca, Maksym Radziwill, Igor E. Shparlinski; On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms;
arXiv:1308.6606.
Note: The question was edited per comments.

REPLY [10 votes]: Since $(\sqrt{2} - \varepsilon) p^{(k-1)/2} < 2 p^{(k-1)/2}$, there is no contradiction between the first and third statements.
The second statement as written certainly contradicts the third. On the other hand the second statement is not what is proven in the paper; that paper talks about the values for arguments at all integers $n$ not prime arguments. Since the set of primes has density zero there is no contradiction. Because of the multiplicative property of these coefficients, it is not surprising that that the behavior for general $n$ should not be exactly the same as for prime $n=p$. The entire point of the second paper is about how assuming a Sato-Tate conjecture for the prime coefficients and assuming a multiplicative condition implies various results for general coefficients.