During Baam's fight against Data Jahad; Right after the ignition of Thorn, Data Jahad said : "It is just a faint feeling but It is like the power of the administrator" & "The power to control space,this is like the power of the administrator" ,not anytime before that.It was right after the Thorn ignition. Thus ,I think we have already been told the power of the Thorn is like that of the administrator.
For reference :Chapters 303&304
But before that, on his fight with the Hell Joe, it goes a little bit different.
The Red Thryssa says "This is the true power of the administrator, I`m going to suck all the shinsoo on the area".
Then we see Baam thinking about stopping HJ, thinking about more power, the power to control shinsoo.
We get a painel of the thorn and a painel of the souls.
HJ wonders if Baam's getting this much power from the thorn, then sees that's something else inside Baam... the colossal soul and the Blue Demon eye.
Then HJ freaks out and goes "Don't play dumb, That thing in your body!!! That's the administrator's power! How do you have that in your body?"
.
The Red Thryssa reaction is clearly about the blue demon/colossal soul than about the thorn fragment.
As we know that the thorn would have different specs on different people, I think that with Baam it boosts the Administrator like power that Baam already have.
season 2, episode 253, btw