  Let $U$ be the quantum group with divided powers in $p-$th root of unity for
prime $p$. For any two-sided cell $A$ in the corresponding affine Weyl group
one associates tensor ideal in the category of tilting modules over $U$. In
this note we show that for any cell $A$ there exists tilting module $T$ from
the corresponding tensor ideal such that biggest power of $p$ which divides
$dim T$ is $p^{a(A)}$ where $a(A)$ is Lusztig's $a-$function. In new version
some typos are corrected and exposition is improved following suggestions of
the referee.
