Are there any examples of non-orientable closed 4-(or 3-)manifolds $M$ whose cohomology ring $H^*(M,\mathbb{Z}_n)$ and/or $H^*(M,\mathbb{Z})$ are known.

Here we assume that $M$ is not a product of lower dimensional manifolds, and is not a connected sum.

The following paper give a class of examples for 3-manifold based on torus bundle: https://arxiv.org/abs/1307.0518 . I wonder if there are examples for 4-manifolds and other examples for 3-manifolds.

This question is motivated from a physics problem. Manifolds with known cohomology ring are like probes that can detect topological orders in quantum matter. We like to have as much probes as possible in order to distinguish all topological orders.