Simultaneous nonvanishing of quadratic twists via Rankin-Cohen brackets

Let $D$ be an odd fundamental discriminant, with $D=1$ permitted, and let $r\ge 1$ be fixed. We prove that, for every sufficiently large integer $\mathrm{\ell }$ satisfying $(-1{)}^{\mathrm{\ell }}D>0$, the first $r$ traced diagonal Rankin–Cohen brackets $${\mathrm{T}\mathrm{r}}_1^{\mathrm{\mid }D\mathrm{\mid }}[G_{\mathrm{\ell }-2e,D},G_{\mathrm{\ell }-2e,D}{]}_{2e},1\le e\le r,$$ are linearly independent in $S_{2\mathrm{\ell }}(S{L}_2(\mathbb{Z}))$. Here $G_{k,D}$ is the Eisenstein series of weight $k$, level $\mathrm{\mid }D\mathrm{\mid }$, and nebentypus ${\chi }_D$. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least $r$ normalized Hecke eigenforms $f\in S_{2\mathrm{\ell }}(S{L}_2(\mathbb{Z}))$ satisfy $L(f\otimes {\chi }_D,\mathrm{\ell })\mathrm{\ne }0$. For $D=1$, this gives, for every fixed $r$ and every sufficiently large $K\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, at least $r$ level-one eigenforms of weight $K$ with nonzero central value.