 Solution

Let $\zeta =e^{2\pi i/7}$ and let ${\lambda }_k(a)={\zeta }^k$. Conjugation by $b$ partitions the six nontrivial characters of $H=⟨a⟩$ into the two orbitsInducing one representative from each orbit gives two irreducible degree-three characters $\psi ,\overline{\psi }$. On the five classes represented by $1,a,a^3,b,b^2$ their values arewith the two cyclotomic sums interchanged for $\overline{\psi }$. Their squared degrees, together with the three linear characters, sum toso these are all irreducible characters.