Label of a classical modular form (reviewed)

The label of a newform $f\in S_k^{\rm new}(N,\chi)$ has the format \( N.k.a.x \), where

• \( N\) is the level;

• \(k\) is the weight;

• \(N.a\) is the label of the Galois orbit of the Dirichlet character $\chi$;

• \(x\) is the label of the Galois orbit of the newform $f$.

For each embedding of the coefficient field of $f$ into the complex numbers, the corresponding modular form over $\C$ has a label of the form \(N.k.a.x.n.i\), where

• \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi\);

• \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character $\chi$.