Many of the barrels are stamped what their twist is. It would also suggest from the length this might be an after market barrel. Green Mountain Barrel Co. makes a 32" barrel that will drop right into the stock of the Thompson Center Hawkins.

If it is a standard Thompson Center barrel it would have a 1:48 twist. They were meant to shoot roundball and conical. By your statement of it is a roundball barrel I am guess it is a 1:66 or 1:70 twist barrel after market barrel.

I know Thompson Center years back made some custom rifles in Hawkins I believe. They still had a 1:48 twist as I remember....