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.

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