Simplifying an Expression involving element-wise multiplication of matrix elements

Question

I started with the following, $$\zeta=\sum_{n=0}^{N-1}\sum_{p=0}^{N-1}\sum_{l=0}^{L-1}a_{nl}\left(a_{pl}\right)^*,$$ where $a_{ij}$ are complex numbers forming the matrix $\mathbf{A}$, $$\mathbf{A}=(a)_{ij}\in\mathbb{C}^{N\times L}.$$ I have been able to reduce $\zeta$ to give, $$\zeta=\sum_{l=0}^{L-1}\left\{\left[\sum_{n=0}^{N-1}\mathrm{Re}(a_{nl})\right]^2+\left[\sum_{n=0}^{N-1}\mathrm{Im}(a_{nl})\right]^2\right\}.$$ This looks "messier" mathematically, but I think it is going to run better computationally for larger $N$ and $L$. Looking at the equation, I feel there is a way to simplify this some more, but the answer is just not jumping out at me. Would welcome any suggestions on how to push this a bit further along!