Maeda's conjecture for normalized Hecke eigenforms

Let $f\in S_K$ be a normalized Hecke eigenform. Let $K_f$ be the number field generated by its Fourier coefficients, and let $G(f)$ be the Galois group of the Galois closure of $K_f$ over $\mathbb{Q}$. For each $\sigma\in G(f)$, write

$$f^{\sigma}(z)=\sum_{n=1}^{\infty}a_f(n)^{\sigma}q^n.$$

Maeda's conjecture. The Galois conjugates satisfy

$$\operatorname{Span}\{f^{\sigma}:\sigma\in G(f)\}=S_K,$$

and $G_f$ is isomorphic to the symmetric group of degree $\dim S_K$. This conjecture is used in the paper to deduce non-vanishing of twisted central $L$-values, but no resolution status is supplied here.