Change Template For Category Listing

When I click on a post, my breadcrumb has a link to a page with that shows all the posts in that category.
I made a new Page that redirects to that link.
How do I make that page that lists all the posts in one category use a different template page?
I know how to do that with a Page with normal content, just create Page-Newname.php in my template directory.
Thanks!