Mean and concentration of $|\langle a, u\rangle|$ with Rademacher random vector $u$

Question

Suppose $u\sim\text{Unif}\{\pm1\}^n$ is a Rademacher random vector with iid components. Moreover, $a\in\mathbb{Z}^n$ is a fixed arbitrary integer vector. I know that we can easily analyze $\langle a, u\rangle^2$ But I would like to know the distribution of $|\langle a, u\rangle|$? Is the expectation a function of $\|a\|_1$?. What is the concentration around its mean?