For the theory of classical modular forms, the space of new forms $S_k^{new}(\Gamma(N))$ has a basis of Hecke eigenforms $\{ f_i = \sum a_n q^n : a_1=1, a_n \in \bar{\mathbb{Q}}\}$

Given $k$ and $N$ (I'll take answers restricted to $\Gamma_0(N)$ or $\Gamma_1(N)$)

1) can we say anything about $\{dim_{\mathbf{Q}}[K_{f_i} : \mathbf{Q}] : f_i \}$, where $K_{f_i} = \mathbf{Q}[ a_n : n \geq 1 ]$? (Are they all equal? different? Are the fields themselves equal?)

2) Can we know an explicit $L = \cup K_{f_i}$ apriori? (My guess is that it's related to the defining equation of the projective curve $X(\Gamma)$).

We know that, for each $i$, the dimension of $K_{f_i}$ is finite, but what can we say about all of the $f_i$ collectively?

Simplified Question: I know that, via the modularity of rational elliptic curves, If $E/\mathbf{Q}$ is an elliptic curve of conductor N, then it's associated modular form $f \in S_2(\Gamma_0(N))$ has $dim_{\mathbf{Q}}[K_{f}:\mathbf{Q}] = 1$. Is it true that all of the $f_i$, new eigenforms, in $S_2(\Gamma_0(N))$ have $K_f = \mathbf{Q}$?