Spectral index of a Maass form (awaiting review)

The eigenvalues $\lambda = \frac{1}{4} + R^2$ (written in terms of the spectral parameter $R$) are discrete, real, and nonnegative. We order them, and the index of a particular eigenvalue $\lambda$ in this order is called the spectral index.

The index $0$ is reserved for when the eigenvalue $\lambda = \frac{1}{4}$, or equivalently for when the spectral parameter $R = 0$. For other eigenvalues, we notate the index differently based on whether the eigenvalue is exceptional or not.

• For standard eigenvalues, when $R$ is purely imaginary, we order the eigenvalues by the absolute value of the spectral parameter $R$. The first such eigenvalue with $R \neq 0$ is assigned spectral index $1$, and so on.

• For exceptional eigenvalues, when $R$ is purely real, we again order the eigenvalues by the absolute value of the spectral parameter $R$. But the index is encoded in base $26$ using the $26$ symbols $a, b, \ldots, z$. The first such eigenvalue with $R \neq 0$ is assigned the spectral index $b$.

Exceptional eigenvalues are conjectured to not exist. In practice, the latter indexing system is not necessary.