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?