LAUSR

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 χ 0 = ( 4 χ ( − 1 ) . ) χ . Let (a) χ and χ 2 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 Γ 0 ( 2 α N ) with character χ 0 . Moreover, we prove that the space of newforms of weight k + 1 / 2 for Γ 0 ( 32 N ) is trivial. Also, we set up the theory of newforms for the space of Jacobi cusp forms and skew-holomorphic Jacobi cusp forms.