We decompose the space of
newforms of half-integral weight into a direct sum of spaces of newforms of integral
weight which occur with multiplicity one or two. This not only demonstrates in a
precise way the failure of a multiplicity-one result to hold for half-integral weight
newforms, but moreover indicates which spaces occur with a given multiplicity. The
spaces occuring with multiplicity two are shown to be in one-to-one correspondence
with a collection of Kohnen subspaces. As a consequence, it is shown that under
the Shimura correspondence, the level a newform of half-integral weight
is not determined by the level of the integral weight newform to which it
corresponds.
Since a knowledge of the Hecke eigenvalues is insufficient to charectesize
half-integral weight newforms up to a scalar, we develop sufficient conditions on the
squarefree coefficients, augmenting the information on the eigenvalues, which allow
such a characterization. In the last section, these later results are carried over to the
Hilbert modular setting.
