Bounds on Hecke eigenvalues Let $\pi$ be an automorphic representation of a certain general linear group $GL(k)$. Write its L-function as
$$L(s,\pi) = \sum_{n>0} \frac{\lambda(n)}{n^s}$$
for $Re(s) \gg 1$. What is the best known bound (without assuming Ramanujan-Petersson) for $\lambda(n)$?
 A: Let $\pi$ be a cuspidal unitary automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\mathbb{A}_F$ denotes the ring of adèles of an algebraic number field $F$. Then
\[L(s,\pi) = \sum_{\substack{\mathfrak{a} \subset \mathcal{O}_F \\ \mathfrak{a} \neq \{0\}}} \frac{\lambda_{\pi}(\mathfrak{a})}{N_{F/\mathbb{Q}}(\mathfrak{a})^s}\]
converges absolutely for $\Re(s) > 1$. For $\mathfrak{a}$ a prime ideal $\mathfrak{p}$, we have that
\[\lambda_{\pi}(\mathfrak{p}) = \alpha_{\pi,1}(\mathfrak{p}) + \cdots + \alpha_{\pi,n}(\mathfrak{p})\]
for some complex numbers $\alpha_{\pi,1}(\mathfrak{p}), \ldots, \alpha_{\pi,n}(\mathfrak{p})$; these are known as the Satake parameters of $\pi$ at $\mathfrak{p}$. It is known that the product of the Satake parameters has absolute value $1$ (more precisely, the product is the central character) if $\pi$ is unramified at $\mathfrak{p}$, and otherwise it is smaller.
For $n = 1$, this means that the Ramanujan hypothesis holds. For $n = 2$, the best bound is $|\alpha_{\pi,1}(\mathfrak{p})|, |\alpha_{\pi,2}(\mathfrak{p})| \leq N_{F/\mathbb{Q}}(\mathfrak{p})^{7/64}$ (for $F = \mathbb{Q}$, this is due to Kim and Sarnak; for arbitrary $F$, this is due to Blomer and Brumley). Similarly, for $n = 3$, we have the bound $5/14$, and for $n = 4$, we have the bound $9/22$.
In general, it is not hard to show the inequality $|\alpha_{\pi,1}(\mathfrak{p})|, \ldots, |\alpha_{\pi,n}(\mathfrak{p})| < N_{F/\mathbb{Q}}(\mathfrak{p})^{1/2}$ (I think this goes back to Jacquet and Shalika). The best that is known now is the bound $N_{F/\mathbb{Q}}(\mathfrak{p})^{\frac{1}{2} - \frac{1}{n^2 + 1}}$, due to Luo, Rudnick, and Sarnak.
Some good references are these two papers.
