Counting $K_4$'s in a Paley Graph

Is there a reason you need to do this, or are you just curious? This paper has a formula, but summarizing the argument would be way to long for me to put here: http://www.math.ucsd.edu/~revans/Pulham.pdf

Computational is nice too (computed in Magma):

p = 5 Number of K4 subgraphs: 0

p = 13 Number of K4 subgraphs: 0

p = 17 Number of K4 subgraphs: 0 *

p = 29 Number of K4 subgraphs: 203

p = 37 Number of K4 subgraphs: 555

p = 41 Number of K4 subgraphs: 1025

p = 53 Number of K4 subgraphs: 3445

p = 61 Number of K4 subgraphs: 6100

p = 73 Number of K4 subgraphs: 13140

p = 89 Number of K4 subgraphs: 31328

p = 97 Number of K4 subgraphs: 46560

[* It is worth noting that for $p=17$, the Paley graph is the largest graph $G$ for which neither $G$ nor $G^{C}$ contain a copy of $K4$.]