If $f(z)=\sum a_n q^n$ is a cusp form (of integer weight) normalized so that $a_1=1$, we have the inequality
$$\vert a_n \vert \leq d(n) n^{(k-1)/2},$$
known as the Deligne bound (in which $d(n)$ denotes the divisor function). One related bound that I am aware of is the following, due to Iwaniec:
$$\vert b_n \vert \ll_\epsilon n^{k/2-2/7+\epsilon},$$
in which $g(z)=\sum b_n q^n$ is a cusp form of weight $k$, a half-integer (again, normalized so that $b_1=1$). The term $2/7$ has subsequently been improved, but every bound that I have seen introduces implicit constants (here, dependent on $\epsilon$) that are rather ineffective.
I would be willing to introduce any amount of polynomial loss in $n$ to remove these implicit constants, getting instead a bound of the form
$$\vert b_n \vert\leq C n^A,$$
e.g., in which $A$ and $C$ can be effectively bounded (in terms of the weight $k$). I have attempted to make use of Waldspurger's extension of the Shimura correspondence to shift this problem to the integral weight case (where we have access to Deligne's bounds), but this technique requires us to effectively bound the Petersson norm of $g$ (here again, effective bounds would be more useful than theoretically sharp bounds), and this problem seems similarly untreated.