Dimension (awaiting review)

The dimension of a space of Siegel modular forms is its dimension as a complex vector space; for spaces of newforms $S_{k,j}^{\rm new}(K(N),\chi)$ 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 its newform subspace, equivalently, the cardinality of its newform orbit. This is equal to the degree of its coefficient field (as an extension of $\Q$).

The relative dimension of $S_{k,j}^{\rm new}(K(N),\chi)$ is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newform subspaces.