In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm of a Hecke Mass newform. As they indicate themselves, their method also works for classical holomorphic newforms of arbitrary level and weight.

Is this version explicitly written somewhere?

If $N$ is the level and $k$ is the weight, the bound is a function of $N$ and $k$, which is NOT exactly the same as in the Maass case (even if we replace $k$ by the corresponding eigenvalue). It would be useful to see the explicit form of the bound neatly written.

I know it follows from the Rankin-Selberg method, but the important point is to control a special value of an automorphic form on $GL_3$ and I am not familiar with such forms.