Quantum modularity of additive twists of Maass forms L values

Given the sequence $a(n)$ of Fourier coefficients of a Maass form for a Fuchsian group $\Gamma$ and a rational $x$, we consider the additively twisted central $L$-value

$$ L(x) = \sum_{n\geq 1} a(n) e^{2\pi i n x} / n^{1/2} $$

defined by means of analytic continuation. In joint work with Asbjørn Nordentoft, we study the relations between $L(\gamma x)$ and $L(x)$ for $\gamma\in\Gamma$, which generalizes various earlier works by Petridis-Risager, Nordentoft, Lee-Sun and Bettin-Drappeau. This has application to the normal distribution as $x$ varies among rationals of denominators at most $Q$ ($Q\to\infty$) when $\hbox{SL}(2, \mathbb Z)$; to the existence of reciprocity formulas for moments for Dirichlet twists of the $L$ function of $\phi$; and to "wide moments'' of the same Dirichlet twists.