Certification of Maass cusp forms of arbitrary level and character

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, Strömbergsson and Venkatesh, and is motivated by the production of purported Maass cusp forms of arbitrary level and character via methods developed by Hejhal and Strömberg.