Galois conjugate newforms (reviewed)

Two newforms $f=\sum a_nq^n$ and $g=\sum b_nq^n$ are Galois conjugate if there is an automorphism $\sigma\in \Gal(\overline{\Q}/\Q)$ such that $b_n=\sigma(a_n)$ for all $n\ge 1$, in which case we write $g=\sigma(f)$.

The set $\{\sigma(f):\sigma\in\Gal(\overline{\Q}/\Q)\}$ of all Galois conjugates of $f$ is the Galois orbit of $f$; it has cardinality equal to the dimension of $f$, equivalently, the degree of its coefficient field