A Kohnen-Zagier Formula in Level 1

Abstract

In 1985 Kohnen used the Shimura correspondence to prove a general formula relating the coefficients of half-integral weight modular forms of level $4N$ transforming with respect to the theta multiplier to central values of $L$-functions for twists of newforms of level $N$. This generalized previous work with Zagier. In this paper, we prove a Kohnen-Zagier type formula using Yang's Shimura correspondence for powers of the eta multiplier in level $1$, relating the coefficients of half-integral weight modular forms to central values of $L$-functions for twists of newforms of level $6$. To do so we develop some of the theory of modular forms transforming with respect to powers of the eta multiplier and introduce Shintani lifts of level $6$, with Fourier coefficients given in terms of cycle integrals.

arXiv comment28 pages