Imaginary quadratic twists

Let $\chi_d(n)$ with $d<0$ be character associated with the imaginary quadratic field $Q(\sqrt{d}).$ Let $f$ be a newform of even weight $k$ and level $N$ which has integral coefficients. Kohnen and Zagier showed, after work of Waldspurger, that

\begin{displaymath}L_f(1/2,\chi_d) = \kappa_f \frac{c_{\vert d\vert}^2}{d^{(k-1)/2}}\end{displaymath}

where $\kappa_f>0$ depends only on $f$ and where $c_{\vert d\vert}$ is an integer.

In fact,

\begin{displaymath}g(z)=\sum_{d<0} c_{\vert d\vert} e(\vert d\vert z)\end{displaymath}

is a cusp form of (half-integral) weight $(k+1)/2$ and level $4N$.



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