Let
X
X
be a quotient of the modular curve
X
0
(
N
)
X_0(N)
whose Jacobian
J
X
J_X
is a simple factor of
J
0
(
N
)
n
e
w
J_0(N)^{new}
over
Q
\mathbf {Q}
. Let
f
f
be the newform of level
N
N
and weight
2
2
associated with
J
X
J_X
; assume
f
f
has analytic rank
1
1
. We give analytic methods for determining the rational points of
X
X
using quadratic Chabauty by computing two
p
p
-adic Gross–Zagier formulas for
f
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
p
-adic
L
L
-function of
f
f
constructed by Bertolini, Darmon, and Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148], which lies outside of the range of interpolation.