Suppose I have $f\in S\_2(\Gamma\_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as defined [here](https://www.lmfdb.org/knowledge/show/cmf.atkin-lehner).


I know the Atkin-Lehner operator commutes with the Hecke operators away from the level, but it seems like there should be some reason to care about it beyond just that it's something else one can do once one has an eigenform. I would guess that it relates in some way to the ramification at $p$ of the Galois representation associated to $f$, but I've had a look through some literature and found nothing beyond the definition of the operator. I'm sure it wouldn't have as much time dedicated to it if it weren't in some way important! Just what that importance is, though, is unclear to me.


I would also be interested in what the interpretation is for Bianchi and Hilbert modular forms, although I know this will probably be harder to specify due to the technical complications there are in these settings. Thanks!



Meaning of Atkin-Lehner eigenvalues
Write the Fourier expansion of the newform $f\in S\_2(\Gamma\_0(N))$ as


$\displaystyle\sum\_{n=1}^{\infty}\lambda\_f(n)n^{1/2}e^{2\pi inz}$


so that the Deligne bound is $|\lambda\_f(n)|\leq d(n)$, where $d(n)$ is the divisor function. If for a prime $p|N$ we let $\lambda\_p$ denote the eigenvalue of the Atkin-Lehner operator, then


$\displaystyle\lambda\_f(p) = \begin{cases}
-\lambda\_p p^{-1/2}&\mbox{if $p^2\nmid N$,}\\
0&\mbox{otherwise.}
\end{cases}$


If you let $f\in S\_k(\Gamma\_0(N))$ be a newform of even integral weight $k\geq 2$, then all of this carries through, except that we replace $n^{1/2}$ with $n^{(k-1)/2}$ in the Fourier expansion. This can be found in Chapter 2, Section 5 of Ken Ono's book "The web of modularity."


