Square divisors

Among the divisors of 400, exactly 6 are perfect squares:
- 1, 4, 16, 25, 100, 400

How many divisors of ${10}^9!$ are perfect squares?

Answer format: the 20 last digits of this huge number before the trailing 0s.

You are given: 54158086206534975488 for $1000!$


[My timing: 20 sec]
Beware of leading zeros