On Shimura's decomposition

Let $k$ be an odd integer $\ge 3$ and $N$ a positive integer such that $4 \mid N$\mathrm{.}Let$\chi$beanevenDirichletcharactermodulo$N$. Shimura decomposes the space of half-integral weight cusp forms $S_{k\mathrm{/}2}(N,\chi )$ as a direct sum of $S_0(N,\chi )$ (the subspace spanned by 1-variable theta- series) and $S_{k\mathrm{/}2}(N,\chi ,\varphi )$ where $\varphi$ runs through a certain family of integral weight newforms. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating critical values of $L$-functions of quadratic twists of newforms of even weight to coefficients of modular forms of half-integral weight.