Find $16$ complex numbers $x + yi, x,yin mathbb{Z}$ with norm $N(x + yi) =65$

I took some integer value for $x,y$, as $x=3, y=2$ with norm $N = \sqrt{13}$. But, no it is not. I hoped that by finding a relation between current value of $x,y, N$ to the desired one; would be able to get the desired $x',y'$. But could not find any such relation, as :
So, tried to square the values of $x,y$ to get $x'=x^2=9, y'=y^2=4$; as hoped that it will have norm $13$.

Definitely, the reason lies in adding $2x'\cdot y'$ also, whose square root need be split in $x',y'$.

To check this idea, $x'^2= x^4 = 81; y'^2 = y^4 = 16; 2x'y'= 72$ and $x'^2 + y'^2 + 2x'\cdot y' = 81+16+72 = 169$.

The square root of $72$ is not an integer, so cannot go further.

So, for getting a norm of $65$ cannot proceed from norm of $\sqrt{13}$.

Could not get any other way out. Please tell a logical approach to generate all such pairs.

---

Addendum Sorry, for flawed defn. of norm. Total question is created out of that wrong definition.

Answer

$$65=5\times 13=|2+i|^2|3+2i|^2=|(2+i)(3+2i)|^2 =|4+7i|^2$$ $$65=5\times 13=|2+i|^2|3-2i|^2=|(2+i)(3-2i)|^2 =|8-i|^2$$ etc.