jmc

We should begin by converting $527_{10}$ to base 4. The largest power of 4 that is less than $527$ is $4^4$, which equals $256$. Since, $(2\cdot 4^4)=512<527<(3\cdot 4^4)=768$, the digit in the $4^4$ place is $2$. We know that $527-512=15$, so there will be zeros in the $4^3$ place and $4^2$ place. The largest multiple of $4^1$ that goes into $15$ without going over is $3\cdot 4^1=12$, so $3$ will be the digit in the $4^1$ place. Since $15-12=3$, we can also see that $3$ will be the digit in the $4^0$ place. We now know that $527_{10}=20033_4$. The sum of the digits of $20033_4$ is $2+0+0+3+3=\boxed{8}$.