Level dependence in the Ramanujan-Petersson Conjecture for GL(2) Maass forms

Question

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.

Answer 1

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 $\Gamma_0(N)\backslash\mathcal{H}$.