we know that for $r \in \{1,2,3,4\},$ $\lambda\_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and Serre. My question is the following:
have automorphic forms Fourier expansions? 
Thanks in advance.



Fourier expansion of automorphic forms
I don't see the relation between the first sentence and the question, so perhaps I'm missing something, but the answer to the question is yes, regardless.


Fourier expansion is a very important tool in the study of automorphic forms, and works for any automorphic form over any reductive algebraic group.


As usual, the cuspidal and non-cuspidal cases are very different, and the $G=\mathrm{GL}\_n(\mathbb{A})$ case is particularly well understood.


For instance, for $\mathrm{GL}\_n(\mathbb{A})$ and $\varphi$ cuspidal, we have a beautiful theorem of Shalika and Piatetski-Shapiro:



> 
> $\varphi$ is globally generic, that is, has non-vanishing
>  Whittaker-Fourier coefficients, and the Fourier expansion is
>  absolutely convergent and uniformly converges on any compact set.
> 
> 
> 


For a nice exposition of the analytic theory on $\mathrm{GL}\_n$, see


* J. W. Cogdell, [Notes on L-functions for $\mathrm{GL}\_n$](http://users.ictp.it/~pub_off/lectures/lns021/Cogdell/Cogdell.pdf)


As a textbook reference, section 3 of Bump's classic book covers this aspect of automorphic forms:


* Daniel Bump, Automorphic Forms and Representations


A more concrete and advanced reference that I've found very useful:


* Baiying Liu, [Fourier Coefficients of Automorphic Forms and Arthur
Classification](http://www.math.utah.edu/~liu/Thesis-PhD.pdf)


The original references for the Shalika and Piatetski-Shapiro theorem:


* J. Shalika, [The multiplicity one theorem for $\mathrm{GL}\_n$](http://www.jstor.org/stable/1971071?seq=1#page_scan_tab_contents) (1974)
* I. Piatetski-Shapiro, Multiplicity one theorems (1979)


