If I am given the inputs $u = [2,1,0,1,2,1,0,1]$ and asked to compute the real parts of the DFT outputs $U(m)$ I would need to apply the formula $$U(m) = \sum_{k = 0}^{N-1} u(k) e^{\frac{-j2 \pi mk}{N}} $$
When $m=0$ I get $U(m) =8$.
I tried calculating $m=1$ by hand, wrote out the entire summation and got $$U(1) = 2 + (\frac{1}{\sqrt2} - \frac{1}{\sqrt2}j) + (-\frac{1}{\sqrt2} - \frac{1}{\sqrt2}j) + (-2) + (-\frac{1}{\sqrt2} + \frac{1}{\sqrt2}j) + (\frac{1}{\sqrt2} + \frac{1}{\sqrt2}j) = 0$$
First of all, is this computation correct?
Second of all, is there another way to think about this problem without writing the summation out?