Working In Base Negative 31

From the hints in the problem and problem title, you may be able to guess that the answer is $404.$ Indeed, when converting to base $-31,$ $\frac{404}{-31} = -13, r=\boxed{1} \\ \frac{-13}{-31} = 1, r=\boxed{13} \\ \frac{1}{-31} = 0, r=\boxed{1} \\ 404 = 1:13:1_{-31}$ which is a palindrome. Repeating the process for $101,$ $202,$ and $303,$ it can be seen that these are not palindromes in base $-31.$ Therefore $\boxed{404}$ is the smallest 3-digit multiple of $101$ that is a palindrome in base $-31.$