Embedding of a modular form (reviewed)

The coefficients in the $q$-expansion $\sum a_nq^n$ of a newform $f$ are algebraic integers that generate the coefficient field $\Q(f)$ of $f$.

Each embedding $\iota\colon \Q(f)\to \C$ gives rise to a modular form $\iota(f)$ with $q$-expansion $\sum \iota(a_n)q^n$; the modular form $\iota(f)$ is an embedding of the newform $f$.

Distinct embeddings give rise to modular forms that lie in the same galois orbit but have distinct $L$-functions $L(s):=\sum \iota(a_n)n^{-s}$.

If $f$ is a newform of character $\chi$, each embedding $\Q(f)\to \C$ induces an embedding $\Q(\chi)\to \C$ of the value field of $\chi$. The embeddings of $f$ may be grouped into blocks with the same Dirichlet character; distinct blocks correspond to modular forms with distinct (but Galois conjugate) Dirichlet characters.