Uniform non-vanishing of prime quadratic twists of standard and symmetric square $L$-functions

Let $S_k$ be the space of cusp forms of weight $k$ for ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, with $\dim S_k\ge 1$, and let ${\chi }_p=\left(\frac{\cdot }{p}\right)$ for odd primes $p$. For fixed $s$ with $\mathrm{R}\mathrm{e}(s)>(k+1)\mathrm{/}2$, we prove that the proportion of primes $p\le X$ satisfying $L(h,{\chi }_p,s)=0$ tends to zero as $X\to \mathrm{\infty }$, uniformly over nonzero $h\in S_k$. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square $L$-values of the normalized Hecke eigenforms when $\mathrm{R}\mathrm{e}(s)>k$. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in $S_k$ for the kernels associated with twisted standard and symmetric square $L$-functions, with explicit basis criteria in specified right half-planes when $\dim S_k=2$ or $3$. We show that almost every tuple of $\dim S_k$ primes gives a basis of $S_k^{\ast }$ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of $S_k$ obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.