Computation of Maass waveforms with nontrivial multiplier systems

Abstract

The aim of this paper is to describe efficient algorithms for computing Maass waveforms on subgroups of the modular group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper S upper L left-parenthesis 2 comma double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">PSL(2,\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with general multiplier systems and real weight. A selection of numerical results obtained with these algorithms is also presented. Certain operators acting on the spaces of interest are also discussed. The specific phenomena that were investigated include the Shimura correspondence for Maass waveforms and the behavior of the weight- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Laplace spectra for the modular surface as the weight approaches <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .