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:

$<f, P_{m}^{k}> = c_{k,m}a_{m}$, where $c_{k,m} = \frac{(k-2)!}{(4\pi m)^{k-1}}$, if $f = \sum_{n \in \mathbb{Z}} a_{n}e^{2 \pi inz}$ 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.

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.