On newforms of half-integral weight and Jacobi forms

Abstract

Let N be an odd and square-free integer. Let α be a positive integer with α=2 or α=5. Let χ modulo N be a Dirichlet character and let χ0=(4χ(−1).)χ. Let(a)χ and χ2 are primitive characters mod N, if α=2;(b)χ is the principal character if α=5. In this paper, we set up the theory of newforms for the space of cusp forms of weight k+1/2 for Γ0(2αN) with character χ0.Moreover, we prove that the space of newforms of weight k+1/2 for Γ0(32N) is trivial. Also, we set up the theory of newforms for the space of Jacobi cusp forms and skew-holomorphic Jacobi cusp forms.