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The Conrey-Iwaniec bound is still the best known one. Originally it was restricted to $N=1$, but Petrow and Young extended the result to all square-free $N$'s (this is highly nontrivial). Also, in these results (including Iwaniec's bound) $n$ itself is also assumed to be square-free. Obtaining a bound for general $n$'s is somewhat tricky: see Corollary 2 in Blomer and Harcos and the references there. Finally, note that this last work features a weaker bound but in a more general situation (for example, there is no restriction on $N$ there).

The Conrey-Iwaniec bound is still the best known one. Originally it was restricted to $N=1$, but Petrow and Young extended the result to all square-free $N$'s (this is highly nontrivial). Also, in these results (including Iwaniec's) $n$ itself is assumed to be square-free. Obtaining a bound for all $n$'s is somewhat tricky: see Corollary 2 and the references in Blomer and Harcos. Note that this paper features a weaker bound but in a more general situation (for example, there is no restriction on $N$).

The Conrey-Iwaniec bound is still the best one. It has been generalized by Petrow and Young for all levels $4N$ with $N$ square-free (Conrey-Iwaniec is restricted to $N=1$). Note that these results are meant for $n$ square-free (including Iwaniec's result). Getting a bound for general $n$ is somewhat tricky, see Corollary 2 in Blomer and Harcos and the references there. Note also that the latter work contains a weaker bound, but it holds in a more general situation (e.g. there is no restriction on the level $4N$).

The Conrey-Iwaniec bound is still the best known one. Originally it was restricted to $N=1$, but Petrow and Young extended the result to all square-free $N$'s (this is highly nontrivial). Also, in these results (including Iwaniec's bound) $n$ itself is also assumed to be square-free. Obtaining a bound for general $n$'s is somewhat tricky: see Corollary 2 in Blomer and Harcos and the references there. Finally, note that this last work features a weaker bound but in a more general situation (for example, there is no restriction on $N$ there).

The Conrey-Iwaniec bound is still the best one. It has been generalized by Petrow and Young for all levels $4N$ with $N$ square-free (Conrey and Iwaniec restricted to $N=1$). Note that these results are meant for $n$ square-free (including Iwaniec's result). Getting a bound for general $n$ is somewhat tricky, see Corollary 2 in Blomer and Harcos and the references there. Note also that the latter work contains a weaker bound, but it holds in a more general situation (e.g. there is no restriction on the level $4N$).

The Conrey-Iwaniec bound is still the best one. It has been generalized by Petrow and Young for all levels $4N$ with $N$ square-free (Conrey-Iwaniec is restricted to $N=1$). Note that these results are meant for $n$ square-free (including Iwaniec's result). Getting a bound for general $n$ is somewhat tricky, see Corollary 2 in Blomer and Harcos and the references there. Note also that the latter work contains a weaker bound, but it holds in a more general situation (e.g. there is no restriction on the level $4N$).