ON TWISTING OPERATORS AND NEWFORMS OF HALF-INTEGRAL WEIGHT II - COMPLETE THEORY OF NEWFORMS FOR KOHNEN SPACE

The author continues his previous work with the purpose of establishin g a theory of newforms in the case of half-integral weight. In this pa per, the author formulates and proves a complete theory of newforms fo r Kohnen space. Kohnen space is an important canonical subspace in the space of cusp forms of half-integral weight k + 1/2 (k > 0). Every He cke common eigenform in Kohnen space corresponds to a primitive form o f integral weight 2k and of odd level via Shimura Correspondence. Thes e newforms for Kohnen space satisfy the Strong multiplicity One theore m. Moreover, we explicitly determine the corresponding primitive form (of weight 2k) to each newform for Kohnen space. The space of oldforms is also explicitly described. In order to find all newforms for Kohne n space, the author needs a certain non-vanishing property of Fourier coefficients of cusp forms. Such property proves by using representati on theory of finite linear groups. The method of proof of newform theo ry is mainly based on trace formulae and trace relations.