This is probably a confusion or a matter of notation. The congruence subgroup $\Gamma_0(N)$ is defined as $$\Gamma_0(N)= \left\lbrace \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \text{SL}_2(\mathbb{Z}) : c \equiv 0\text{ (mod } N\text{)} \right\rbrace.$$ In all the textbooks and other available materials, this is defined for $N$ positive. When $N$ is negative, the group is same right? i.e., if $N$ is any integer, $\Gamma_0(N) = \Gamma_0(-N)$?
I know $a\equiv b\text{ (mod } N\text{)}$ iff $a\equiv b\text{ (mod } -N\text{)}$; but I need to make sure if there are any caveats involved in the definition of $\Gamma_0(N)$.
$a\equiv b\pmod N$ means that $a-b\mid N$, or equivalently that $a=b+kN$ with $k\in\mathbb Z$. In the latter form it is particularly manifest that $(\operatorname{mod}N)$ and $(\operatorname{mod}-N)$ mean the same thing.