By reasoning on the multiplicity of the roots of $\chi_\sigma$ and $\chi_\tau$, show that if $P_\sigma$ and $P_\tau$ are similar, then, for all $q \in \llbracket 1, n \rrbracket$,
$$\sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\sigma) = \sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\tau)$$
(We sum over the values of $\ell$ that are multiples of $q$ and belong to $\llbracket 1, n \rrbracket$.)