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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?

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    $\begingroup$ Note that for oldforms the first Fourier coefficient might vanish, in which case the normalization you indicate is impossible. At any rate, even for newforms, I am sure the answer to your question is "no". Check out the following paper by Goldfeld, Hundley, and Lee: Fourier expansions of GL(2) newforms at various cusps. $\endgroup$
    – GH from MO
    Feb 7, 2022 at 11:52
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    $\begingroup$ Thank you, I will take a look... $\endgroup$ Feb 7, 2022 at 12:17
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    $\begingroup$ @GH: I haven't looked at the paper you mention, but I do not understand your claim: the expansion at a cusp corresponds to the expansion of a form on the conjugate congruence group $\gamma^{-1}\Gamma_0(N)\gamma$, for which the R-P conjecture is true. $\endgroup$ Feb 7, 2022 at 12:44
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    $\begingroup$ Note that if the cusp is a so-called Atkin-Lehner cusp and the eigenform $f$ is a newform, then the Fourier expansion about that cusp is multiplicative and is essentially the same as the Fourier expansion at $\infty$ up to multiplication by an Atkin-Lehner eigenvalue, which necessarily has absolute value $1$. In particular, the Ramanujan conjecture at Atkin-Lehner cusps for newforms is equivalent to the Ramanujan conjecture at the cusp at $\infty$. $\endgroup$ Feb 7, 2022 at 17:04
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    $\begingroup$ Even if not multiplicative, one should still have $|\lambda(n)|\le C.d(n)$ for some constant $C$. $\endgroup$ Feb 7, 2022 at 17:12

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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 (arxiv link), 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 $p^2 | N$.)

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    $\begingroup$ Thank you for your answer and the reference. I have a question though. It seems that $C_N$ depends only on $N$, and therefore the estimate $|\lambda(n)| \gg n^{1/4}$, if at all true can hold for finitely many $n$ dependent on $N$. Am I correct? In particular in Remark 3.14 of the reference, you have a bound on $k$ for which the Ramanujan bound may not hold for $\lambda(p^k)$. For larger $k$ does the Ramanujan conjecture hold? $\endgroup$ Feb 7, 2022 at 19:13
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    $\begingroup$ Indeed, there are only finitely many values of $\gcd(n,N)$ (depending on $N$) for which such a lower bound can hold. $\endgroup$ Feb 7, 2022 at 21:13

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