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In the prequel, a sharp bound in the level aspect on the fourth moment ofHecke--Maa{\ss} forms with an inexplicit (in fact exponential) dependency onthe eigenvalue was given. In this paper, we develop further the framework ofexplicit theta test functions in order to capture the eigenvalue moreprecisely. We use this to reduce a sharp hybrid fourth moment bound to anintricate counting problem. Unconditionally, we give a hybrid bound, which issharp in the level aspect and with a slightly larger than convex dependency onthe eigenvalue.

Theta functions, fourth moments of eigenforms, and the sup-norm problem III