= Solution
Put $\zeta=e^{2\pi i/7}$ and induce the <characters> $\chi_r(a)=\zeta^r$ of $H=\langle a\rangle$. Since $H$ is normal, the induced value outside $H$ is zero, while at $a^m$ it is $\zeta^{rm}+\zeta^{2rm}+\zeta^{4rm}$. The nonzero $r$ split into the two doubling orbits, so the two distinct induced <characters> have class values
$$
\boxed{(3,A,B,0,0),\qquad(3,B,A,0,0),}
$$
where $A=\zeta+\zeta^2+\zeta^4$ and $B=\zeta^3+\zeta^5+\zeta^6=\overline A$. Summing seventh roots gives $A+B=-1$, and multiplication gives $AB=2$, hence $|A|^2=|B|^2=2$. Their <character> <norms> are $(9+3\cdot2+3\cdot2)/21=1$, proving irreducibility. Together with the three degree-one <characters>, the sum of squared degrees is $3+2\cdot9=21$. Thus these are \b[all the remaining irreducible <characters>].
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