bash: including both \n and double-quotes in a string variable

Question

I have a string that I want to enclose in double quotes and place a newline character before it. This is how I want the output to look:

 
"my string"

Here's what I thought would work:

#!/usr/bin/env bash

input='my string'
output=$"[\n\"$input\" ]"

But instead, I get this:

 \n"my string"

How can I persuade bash to convert \n to a newline character?