Suppose that $f \in S\_k(\Gamma\_0(N)) $ be a Hecke eigenform whose Fourier expansion at $ i\infty $ is given by


$$
f(z) = \sum\_{n=1}^{\infty} \lambda(n) n^{\frac{k-1}{2}} \exp(2\pi i n z),
$$


normalized so that $\lambda(1)=1$. In this setting the Ramanujan-Petersson conjecture states that $ |\lambda(n)| \leq d(n) $ the number of divisors of $n$ (for all $ n $ coprime to the level $ N $).


Does the same bound hold if I consider the Fourier expansion of $f$ at some other cusp?



Ramanujan-Petersson conjecture at various cusps
The estimate $|\lambda(n)| \leq C\_N d(n)$ remains valid at all cusps, but $C\_N$ cannot in general be taken independent of $N$. See Remark 3.14 of [this paper](https://mathscinet.ams.org/mathscinet-getitem?mr=3110797) ([arxiv link](https://arxiv.org/abs/1205.5534)), where it is noted that for certain $(N,f,n)$, with $f$ a newform, one can have $\lambda(n) \gg n^{1/4}$ at some cusp. (One can take for $n$ a suitable power of a prime $p$ for which $p^2 | N$.)


