  We calculate certain "wide moments" of central values of Rankin--Selberg
$L$-functions $L(\pi\otimes \Omega, 1/2)$ where $\pi$ is a cuspidal automorphic
representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $\Omega$ is a Hecke
character (of conductor $1$) of an imaginary quadratic field. This moment
calculation is applied to obtain "weak simultaneous" non-vanishing results,
which are non-vanishing results for different Rankin--Selberg $L$-functions
where the product of the twists is trivial.
  The proof relies on relating the wide moments to the usual moments of
automorphic forms evaluated at Heegner points using Waldspurger's formula. To
achieve this, a classical version of Waldspurger's formula for general weight
automorphic forms is derived, which might be of independent interest. A key
input is equidistribution of Heegner points (with explicit error-terms)
together with non-vanishing results for certain period integrals. In
particular, we develop a soft technique for obtaining non-vanishing of triple
convolution $L$-functions.
