Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture (proved in this case by Deligne) says roughly that $A(n) \ll n^{\epsilon}$ for any $\epsilon > 0$.

For more general cusp forms, there are lots of partial results of the form $A(n) \ll n^\theta$ towards the Ramanujan-Petersson conjecture. The best general result I'm aware of is due to Kim and Sarnak, showing that $\theta < \frac{7}{64} + \epsilon$.

I'm currently investigating some bounds involving Maass forms $\mu_j$ of level $N$ of the the form $$ \mu_j(z) = \sum_{n \neq 0} \rho_j(n)y^{\frac{1}{2}}K_{it_j}(2\pi \lvert n \rvert y)e^{2\pi i n x} $$ and I wonder: what is the best known Ramanujan-Petersson-style bound for the coefficients $\rho_j(n)$ for such a Maass form, and how does it depend on the level? Is it known that $\rho_j(n) \ll n^{\frac{7}{64} + \epsilon}$, with no dependence on the level $N$?

It seems more likely to me that we would currently have a bound of the form $\rho_j(n) \ll N^? n^{\frac{7}{64} + \epsilon}$, but I cannot seem to find what it may be.

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    $\begingroup$ (1) The Fourier expansion you write down will give a holomorphic form, in which case the Ramanujan-Petersson conjecture is known, due to Deligne. (2) The results for Maass forms (which is the relevant case) are proved by giving estimates for Satake parameters at (unramified) primes, hence have no dependency on $N$ if your form is a Hecke-eigenform. If it is not, then the problem is to express your form as a linear combination of Hecke eigenforms, and the dependency on $f$ cannot be controlled just in terms of $N$ (take $f=N^{100000000}g$, where $g$ is a Hecke eigenform). $\endgroup$ – Denis Chaperon de Lauzières Jun 7 '16 at 17:28
  • $\begingroup$ @DenisChaperondeLauzières Yes, that makes sense. I do indeed care about Hecke-eigenforms. Thank you for your fast reply. $\endgroup$ – davidlowryduda Jun 7 '16 at 17:34

There is no dependence on the level. If $\mu_j(z)$ is a Hecke-Maass newform of level $N$ normalized so that $\rho_j(1)=1$, then $\rho_j(n)$ is a multiplicative function satisfying at prime powers the bound $$ \left|\rho_j(p^k)\right| \leq \sum_{m=0}^k p^{\frac{7}{64}(k-2m)} \leq (k+1)p^{\frac{7}{64}k}.$$ In particular, $$ \left|\rho_j(n)\right|\leq d(n) n^{\frac{7}{64}},$$ where $d(n)$ is the divisor function. (These bounds are often stated for $p$ and $n$ coprime to $N$, but they hold in general. See Footnote 1 in Blomer-Brumley: On the Ramanujan conjecture over number fields. In fact at the ramified primes one has better bounds, not worse.)

Of course if you normalize $\mu_j(z)$ so that its $L^2$-norm is one, then it is a subtle (but pretty well-understood) question how large is $\left|\rho_j(1)\right|$, and all the bounds for $\left|\rho_j(n)\right|$ get multiplied by this constant. Also, clearly, the $L^2$-normalization depends on how the measure itself is normalized on the congruence hyperbolic surface $\Gamma_0(N)\backslash\mathcal{H}$.


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