Hecke kernel (reviewed)

Each newform $f=\sum a_n^n$ necessarily lies in the kernel of the linear operator $g_n(T_n)$, where $g_n\in \Z[X]$ is the minimal polynomial of $a_n$ and $T_n$ is the $n$th Hecke operator acting on the newspace $S_k^{\rm new}(N,\chi)$ in which $f$ lies.

If $X$ is a set of distinguishing primes for $S^{\rm new}_k(N,\chi)$, then the Galois orbit of $f$ can be uniquely identified as the intersection of the kernels of the linear operators $\{g_p(T_p):p \in X\}$.

This characterization allows one to reconstruct the newform $f$ without computing a basis of newforms for $S_k^{\rm new}(N,\chi)$, which may be computationally useful when the dimension of $f$ is much smaller than that of $S_k^{\rm new}(N,\chi)$.