HMMT 2017 U24

At a recent math contest, Evan was asked to find $2^{2016}(\bmod p)$ for a given prime number $p$ with $100<p<500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it:

• Evan first tried taking 2016 modulo $p-1$, but got a value $e$ larger than 100 .
• However, Evan noted that $e-\frac{1}{2}(p-1)=21$, and then realized the answer was $-2^{21}(\bmod p)$.

What was the prime $p$ ?