We know the Poincare series are defined as the following:


The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is:
$$
P\_{m}^{k} (z) = \sum\_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}.
$$
The definition yields that, via the Petersson inner product, we obtain:
$$\langle f, P\_{m}^{k}\rangle = c\_{k,m}a\_{m},$$ 
where $c\_{k,m} = \frac{(k-2)!}{(4\pi m)^{k-1}}$, if $f = \sum\_{n \in \mathbb{Z}} a\_{n}e^{2 \pi inz}$ is a cusp form, and we can further show that the Poincare series generate $\mathcal{S}\_{k}$, the space of cusp forms of weight k.


I can't see that, we already know the $a\_{n}$'s, why can't we define the $m^{th}$ Poincare series of weight $k$ as:
$$
P\_{m}^{k} (z) = \frac{1}{c\_{k,m}}\sum\_{(c,d)=1}(cz+d)^{-k}e^{2 \pi in(\tau z)}\;?
$$
It seems to me that this definition also works.



Why are Poincare series defined as they are?
Ultimately this is bound to be a matter of taste -- you can't "prove" that one normalisation is better than another -- but let me try to justify why it is conventional to normalise this way.


The point of Poincare series isn't just to prove there exists a modular form satisfying $\langle f, P^k\_m \rangle = (const) \times a\_m(f)$ for all $f \in S\_k$. It's obvious that some such form exists (because the Petersson product is nondegenerate). The point is that the series having this property has a *nice explicit form*, and that allows you to prove various fun things you couldn't otherwise prove; renormalising as you suggest to make the constant be 1 makes the explicit form slightly less nice.


