Well in most cases logging into a bank site is done through HTTPS secure encrypted web pages and in this case avast doesn't scan that content, it can't as it is encrypted, so there is little point in trying to scan https pages.
Even if there were any left over registry settings there shouldn't be any underlying files to run and scan, etc. but you can use the uninstall utility as Alan suggests to be doubly sure.
You seem quick to jump to another AV, so I'm wondering if you had another before installing avast on this reloaded computer and if so what ?
Norton is probably one of the worst when it comes to remnants (depending on what version), based on what we see on the forums. So if you come to removing Norton that could be fun too.