Values ofL-series of modular forms at the center of the critical strip

On quadratic character twists of Hecke L-functions attached to cusp forms of varying weights at the central point

“…Except for again exploiting the results of [6][7][8], the proof we shall give for general D is different from that given in [13] for the case D = 1. In fact, we shall use Waldspurger's result relating the twisted central critical values to squares of Fourier coefficients of modular forms of half-integral weight in the more explicit version for level 1 given in [12], together with some simple estimates for Fourier coefficients of Poincaré series of half-integral weight.…”

Theta lifts and local Maass forms

“…Following their appearance in the theory of mock theta functions due to Zwegers [40], it has been shown that harmonic weak Maass forms have applications ranging from partition theory (for example [2,4,6,9,11]) and Zagier's duality [39] relating "modular objects" of different weights (for example [10]) to derivatives of L-functions (for example [14,15]). They also arise in mathematical physics, as recently evidenced in Eguchi et al [16] investigation of moonshine for the largest Mathieu group M 24 . The main difference between locally harmonic Maass forms and harmonic weak Maass forms is that there are certain geodesics along which locally harmonic weak Maass forms are not necessarily real analytic and may even exhibit discontinuities.…”

“…between S 2k and S + k+ 1 2 (Kohnen's plus space of weight k + 1 2 modular forms), which was defined by Kohnen and Zagier [24]. For g ∈ S + k+ 1 2 , the Petersson inner product g, Ω(−z, ·) equals (−1) k/2 2 2−3k times the Shimura lift of g. Kohnen and Zagier used Ω to explicitly compute the constant of proportionality in Waldspurger's result, in turn proving non-negativity of the central L-values of Hecke eigenforms.…”

Computation of Maass waveforms with nontrivial multiplier systems

“…e.g. [73,74], [35,34] [5] and Arakawa [2]. Of these, [30] and [37,38] are written in the more general setting of Hilbert modular forms for number fields.…”