Let $f\in S_k(\Gamma_1(N))$ be an eigenform, and $K_f$ be its number field, which is of finite degree over $\mathbb{Q}$. Consider the following statements.

1, $[K_f:\mathbb{Q}]=\#\{$Galois conjugates of $f\}$.

2, Any $n$-th coefficients of $f$ having degree $[K_f:\mathbb{Q}]$ will generate $K_f$, so could this $n$ always be $n=2$?

I guess statement 1 is true, but I have problem in proving it. I guess statement 2 is false, but I need a counterexample.