  We establish effective versions of Oppenheim's conjecture for generic
inhomogeneous quadratic forms. We prove such results for fixed shift vectors
and generic quadratic forms. When the shift is rational we prove a counting
result which implies the optimal density for values of generic inhomogeneous
forms. We also obtain a similar density result for fixed irrational shifts
satisfying an explicit Diophantine condition. The main technical tool is a
formula for the second moment of Siegel transforms on certain congruence
quotients of $\operatorname{SL}_n(\mathbb{R})$ which we believe to be of
independent interest. In a sequel, we use different techniques to treat the
companion problem concerning generic shifts and fixed quadratic forms.
