The dihedral group $D_{2n}$ has two dimensional irreducible representations for which the character values (on the cyclic group of order $n$) are $2\cos(2\pi k/n)$ for $1\le k\le n$. Clearly these values get arbitrarily close to zero for large $n$. See for example http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dihedral_groups .

The dihedral group $D_{2n}$ has two dimensional irreducible representations for which the character values (on the cyclic group of order $n$) are $2\cos(2\pi k/n)$ for $1\le k\le n$. Clearly these values get arbitrarily close to zero for large $n$ and $k$ close to $n/4$. See for example http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dihedral_groups .