Distinguishing Hecke operators (reviewed)

For a newspace $S^\mathrm{new}_k(N,\chi)$ we say that a set of Hecke operators $\mathcal T:=\{T_{p_1},\ldots,T_{p_r}\}$ distinguishes the newforms in the space if the sets $X_f(\mathcal T)$ of characteristic polynomials of the $T_p\in \mathcal T$ acting on the subspace $V_f$ spanned by the Galois orbit of $f$ in $S_k^\mathrm{new}(N,\chi)$ are distinct as $f$ ranges over (non-conjugate) newforms in $S_k^\mathrm{new}(N,\chi)$.

The set $\mathcal T$ can be identified by a list of primes $p$. For convenience we restrict to primes $p$ that do not divide the level $N$ and list the unique ordered sequence of primes $p_1,\ldots,p_n$ for which the sequence of integers $c_1,\ldots,c_n$ defined by \[ c_m := \#\bigl\{X_f(\{T_{p_i}:i < m\}): \mathrm{newforms}\ f\in S_k^\mathrm{new}(N,\chi)\bigr\} \] is strictly increasing. The length of the sequence $p_1,\ldots p_n$ is always less then the number of newforms in $S_k^\mathrm{new}(N,\chi)$ and we obtain the empty sequence when $S_k^\mathrm{new}(N,\chi)$ contains just one newform.