The label complex embedded holomorphic cusp form \( f\) is \( N.k.a.x.c.j \) (sometimes shortened as \( a.j\) ), where

• \( N\) is the level,

• \(k\) is the weight,

• \(N.a\) is the label of the Galois orbit of the Dirichlet character,

• \(x\) is the Hecke Galois orbit label,

• \(N.c\) is the Conrey label for the character corresponding to the embedding, and

• \(j\) is the index for the embedding within those with the same Dirichlet character, these are ordered by the vector $\iota(a_n)$, where we order the complex numbers first by their real part and then by their imaginary part.