Question regarding Sum of squares in modular arithmetic

Question

Based on my previous question which turned out to be regarding modular arithmetic, I have another question: I have two functions $$f_1=\sqrt{\left(\frac{19(8m-1)+1}{3}\right)^2+\left(\frac{22(8m-1)+1}{3}\right)^2}$$ and $$f_2=\sqrt{\left(\frac{19(8m+1)+1}{3}\right)^2+\left(\frac{22(8m+1)+1}{3}\right)^2}$$ Using modular arithmetic, $f_1^2$ and $f_2$ can be reduced to $4m_1^2+4m_2^2$ and $4(m_1+1)^2+4(m_2+1)^2$ (for some $m_1$ and $m_2$) respectively and therefore $f_1$ and $f_2$ will always be even. Is my reasoning correct? If not, can someone please point out where I am wrong?