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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 $f\in S_k^{\rm new}(N,\chi)$.

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• Last edited by Andrew Sutherland on 2019-01-30 16:01:38
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