  Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a
simple factor of $J_0(N)^{new}$ over $\mathbb{Q}$. Let $f$ be the newform of
level $N$ and weight 2 associated with $J_X$; assume $f$ has analytic rank 1.
We give analytic methods for determining the rational points of $X$ using
quadratic Chabauty by computing two $p$-adic Gross-Zagier formulas for $f$.
Quadratic Chabauty requires a supply of rational points on the curve or its
Jacobian; this new method eliminates this requirement. To achieve this, we give
an algorithm to compute the special value of the anticyclotomic $p$-adic
$L$-function of $f$ constructed by Bertolini, Darmon, and Prasanna, which lies
outside of the range of interpolation.
