Though it may seem strange (probably part of a polynomial trend analysis), you are considering the following contrast for a follow-up test for a 1-way ANOVA with 5 groups: $\displaystyle \psi_{{1}}:$  $\displaystyle {c}_{{{1}}}\cdot\mu_{{{1}}}-{2}\cdot\mu_{{{2}}}-\mu_{{{3}}}+{2}\cdot\mu_{{{4}}}+{2}\cdot\mu_{{{5}}}={0}$

Assuming a balanced design, what must the missing coefficient be for this to be a legitimate contrast?
$\displaystyle {c}_{{1}}=$