  We develop the theory of modulated operators in general principal ideals of
compact operators. For Laplacian modulated operators we establish Connes' trace
formula in its local Euclidean model and a global version thereof. It expresses
Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate
the applicability of our main results in the context of log-classical
pseudo-differential operators, studied by Lesch, and a class of operators
naturally appearing in noncommutative geometry.
