Reliability of rigorous Maass forms data (awaiting review)

Maass form data, in the LMFDB and elsewhere, is inherently approximate. The data are decimal approximations to numbers which (in general) are conjectured to be transcendental and not expressible in terms of well-known constants.

In the LMFDB, each rigorous Maass form is proven to be an accurate approximation to a true Maass form. Further, rigorous Maass forms are proven to be consecutive, i.e. there are no "missing" Maass forms with eigenvalues between known rigorous Maass forms.

Ranges of Maass forms

The database contains data for the Maass forms on a group with a range of eigenvalues. For example, the Maass forms on $\Gamma_0(5)$ and trivial character, for $0<R<10$.

There are two implicit claims in the above sentence: that each entry in the database corresponds to an actual Maass form, and that no eigenvalues are missing. Each eigenvalue of a rigorous Maass forms in the LMFDB is first approximated using an explicit form of Selberg's trace formula, and later improved using a refined form of Hejhal's algorithm. Numerical difficulties with the trace formula forms one major obstruction to extending ranges of consecutive eigenvalues rigorously.

An individual Maass form

A Maass form $f(z)$ on a group $G$ is specified by a spectral parameter $R$ and coefficients (Hecke eigenvalues) $a_1=1$, $a_2$, $a_3$, ..., appearing in its Fourier expansion. The parameter $R$ and the coefficients $a_n$ are found by using the transformation properties of $f(z)$ under the group $G$ to form a system of equations.

The decimal numbers listed for the spectral parameter $R$ and the coefficients $a_n$ are proven to be close to spectral parameter and coefficients of an actual Maass form. (Note that error intervals may be displayed on the website after truncation; be careful when considering less significant digits).

In practice, it is much easier to heuristically refine a given Maass form using a heuristic form of Hejhal's algorithm.