The Bulls, beaten 2-0 at home by Northampton two weeks ago, currently lie 18th in League Two, 11 points above the relegation zone.
Chairman Graham Turner is back in charge of first-team affairs at Edgar Street.
"I was given the reasons behind the decision which I understand and accept," said Trewick in a club statement.
Elsewhere in the division, Colchester winger Simon Hackney has joined Morecambe on loan.
The Shrimps, who are four points behind the Cobblers at present but with a game in hand, travel to Sixfields on Saturday 20th March.