On Shimura's decomposition

Let $k$ be an odd integer $\ge 3$ and $N$ a positive integer such that $4 \mid N$. Let $χ$ be an even Dirichlet character modulo $N$. Shimura decomposes the space of half-integral weight cusp forms $S_{k/2}(N,χ)$ as a direct sum of $S_0(N,χ)$ (the subspace spanned by 1-variable theta- series) and $S_{k/2}(N,χ,ϕ)$ where $ϕ$ 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.

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