Certification of Maass cusp forms of arbitrary level and character

TL;DR

This paper introduces a method to certify the existence of Maass cusp forms of any level and character by bounding the difference in their eigenvalues, enabling the first certification of a non-CM level 5 form with quadratic character.

Contribution

It generalizes existing certification methods from level 1 to arbitrary levels and characters, allowing verification of Maass cusp forms beyond previously accessible cases.

Findings

First certified $ riangle$-eigenvalue for a non-CM level 5 form with quadratic character

Extended certification method to arbitrary level and character forms

Validated the existence of specific Maass cusp forms using the new method

Abstract

We present a method for certifying the existence of an arbitrary Maass cusp form for any level and character. This is accomplished by producing a bound on the difference between the $\Delta$-eigenvalue of an authentic Maass cusp form and a purported approximation of a $\Delta$-eigenvalue, arrived at by any means. We apply this method to a proposed non-CM level 5 form with quadratic character, to present the first certified $\Delta$-eigenvalue of such a form. This work generalises the method for certifying level 1 forms presented by Booker, Str\"ombergsson and Venkatesh, and is motivated by the production of purported Maass cusp forms of arbitrary level and character via methods developed by Hejhal and Str\"omberg.