1989 AIME Problem 14

Given a positive integer n^{}_{}, it can be shown that every complex number of the form r+si^{}_{}, where r^{}_{} and s^{}_{} are integers, can be uniquely expressed in the base -n+i^{}_{} using the integers 1,2^{}_{},\ldots,n^2 as digits. That is, the equation

r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0

is true for a unique choice of non-negative integer m^{}_{} and digits a_0,a_1^{},\ldots,a_m chosen from the set \{0^{}_{},1,2,\ldots,n^2\}, with a_m\ne 0^{}){}. We write

r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}

to denote the base -n+i^{}_{} expansion of r+si^{}_{}. There are only finitely many integers k+0i^{}_{} that have four-digit expansions

k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.

Find the sum of all such k^{}_{}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.