A case of simultaneous non-vanishing of automorphic $L$-functions

We show that for $k\geq 12$ even, there exists an effectively computable constant $q_k$ such that for any prime $q \gt q_k$, there exists a newform $f$ of weight $k$ and level $q$ such that $$ L(1/2,f)L(1/2,\mathop{\rm Sym}\nolimits^2 f)\neq 0. $$