Making the integral over $\hat f$ small if $f$ is a test function

Question

Let $f \in C^\infty(\mathbb{R})$ be compactly supported with $\mathrm{supp}(f) \subseteq [-c,c]$ (i.e. $f$ is a test function). It follows that the Fourier transform is a Schwartz function. I'm wondering under which assumptions the integral over the absolute value of $\hat f$ outside an interval will be small, i.e. under which assumptions is $$\int_{[-a,a]^c} \left| \widehat f(\xi) \right| d\xi $$ small for $a>0$ a constant ?