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Noam D. Elkies
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Trace is additive, and ${\rm Tr}(u)={\rm Tr}(u^2)$ for all $u$, so $ax^2+bx$ has the same trace as $(a+b^2)x^2$. Therefore the sum is $2^r$ if $a=b^2$ and zero otherwise.

Noam D. Elkies
  • 79.9k
  • 15
  • 281
  • 376