Doubt about convergence of a sequence in $H^1(\mathbb{R}^3)$

Question

Let's consider a sequence $\{f_n\}_n$ of $C^\infty_0(\mathbb{R}^3)$ complex-valued functions and suppose thet $f_n\to f$ strongly in $H^1(\mathbb{R}^3)$. What can I say about the convergence of the sequence $\{\vert f_n\vert\}_n$? Is it true that $\nabla\vert f_n\vert\to\nabla\vert f\vert$ in $L^2(\mathbb{R}^3)$?