On the form of an odd perfect number

It has been known since the time of Euler that an odd perfect number N (if it exists) must have the form N = paQ2 where p is prime and p = a = 1 mod 4 (see, e.g., [1, pp. 3–33]). Further, it has been shown that N must equal 1 mod 12, or 9 mod 36 [3], [2]. However, we can do a little better than this.