Complex no


putting values of $ω$ and $ω_{2}$ we get
$S=(a+2b+c )_{2}+43(b−c)_{2} $
we have to minimise this
now $(a+2b+c )_{2}$ cannot be zero as all $a,b,c$ cannot be zero at the same time
but $43(b−c)_{2} $ can be zero so let $b=c$
now we have to minimise $(a+2b+c )_{2}$
as two numbers can be equal, let $b=c=0$
and since $a$ cannot be zero and it is also an integer, $a=1$

@harshalpatil
How did you get S
Please can you show it's calculation