Remarkable subsets

Let $A \subset \mathbb Z$ be a non-trivial set of integers $($i.e. $A \not= \emptyset$ and $A \not= \mathbb Z).$ We call such a set $A$ a "remarkable set of type $N$" if it has the following properties:

• If $a$ is an element of $A,$ then $-a$ is also an element of $A$.

• If $a$ is an element of $A$, then $a+N$ is also an elements of $A$.

• If $a,b$ are elements of $A$ (not necessarily different), then $a + 2b$ is also an element of $A$.

How many (non-trivial) remarkable sets of type 18 are there?

Bonus: Generalize. If $N$ = $2^{e_2}\cdot 3^{e_3}\cdot 5^{e_5}\cdots$ is the prime factorization of $N$, how many non-trivial remarkable sets of type $N$ exist? What do they look like?