Compute the number of ordered quintuples of nonnegative integers \((a_1, a_2, a_3, a_4, a_5)\) such that \(0 \leq a_1, a_2, a_3, a_4, a_5 \leq 7\) and \(5\) divides \(2^{a_1} + 2^{a_2} + 2^{a_3} + 2^{a_4} + 2^{a_5}\).

Let's think step by step and output the final answer within \boxed{}.
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