Sum of 5 squares

Brilli the ant is considering the complexities of a 5-dimensional universe. She wants to count the number of integer points that are distance $\sqrt{n}$ away from the origin. Let $T_n$ be the set of ordered 5-tuples of integers $(a_1,a_2,a_3,a_4,a_5)$ such that $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 = n.$ However, being an ant, she doesn't have enough toes to record numbers that are above 10. Let $D_n$ be the units digit of $\vert T_n \vert.$

Determine $\sum_{i = 1}^{5678} D_i.$

Details and assumptions

Note: You are not asked to find the last digit of the sum, but the sum of all the last digits.

For a set $S$, $| S |$ denotes the number of elements in $S$. You can read up on set notation.