Dimension (reviewed)

The dimension of a space of modular forms is its dimension as a complex vector space; for spaces of newforms $S_k^{\rm new}(N,\chi)$ this is the same as the dimension of the $\Q$-vector space spanned by its eigenforms.

The dimension of a newform refers to the dimension of the $\Q$-vector space spanned by its Galois conjugates, equivalently, its orbit under the Hecke operators. This is equal to the degree of its coefficient field (as an extension of $\Q$).

The relative dimension of $S_k^{\rm new}(N,\chi)$ is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newforms $f\in S_k^{\rm new}(N,\chi)$.