\contentsline {section}{\tocsection {}{1}{Introduction}}{2} \contentsline {subsection}{\tocsubsection {}{}{Plan of the paper}}{4} \contentsline {subsection}{\tocsubsection {}{}{Source code}}{4} \contentsline {subsection}{\tocsubsection {}{}{Acknowledgments}}{4} \contentsline {section}{\tocsection {}{2}{Preliminaries}}{5} \contentsline {subsection}{\tocsubsection {}{2.1}{De Rham cohomology}}{5} \contentsline {subsection}{\tocsubsection {}{2.2}{Iterated integrals}}{5} \contentsline {subsection}{\tocsubsection {}{2.3}{Chow-Heegner points}}{7} \contentsline {section}{\tocsection {}{3}{Chow-Heegner points on modular curves}}{8} \contentsline {subsection}{\tocsubsection {}{3.1}{Definitions}}{8} \contentsline {subsection}{\tocsubsection {}{3.2}{Algorithms}}{10} \contentsline {section}{\tocsection {}{4}{Details of the algorithm}}{12} \contentsline {subsection}{\tocsubsection {}{4.1}{Evaluating iterated integrals}}{13} \contentsline {subsection}{\tocsubsection {}{4.2}{Calculating a symplectic basis for $H^1_{\dR }(X/\mathbf Q)[g]$}}{15} \contentsline {subsection}{\tocsubsection {}{4.3}{Modular units and $\eta$-products}}{16} \contentsline {subsection}{\tocsubsection {}{4.4}{Computing the Poincar\'e dual $\gamma _f$ of $\omega _f$}}{17} \contentsline {subsection}{\tocsubsection {}{4.5}{Computing the adjustments $\DOTSI \intop \ilimits@ _{\gamma _f}\alpha$}}{18} \contentsline {subsection}{\tocsubsection {}{4.6}{Computing the coefficients $c_{ij}$}}{19} \contentsline {subsection}{\tocsubsection {}{4.7}{Computing the denominator $d_{g,n}$}}{19} \contentsline {subsection}{\tocsubsection {}{4.8}{Remarks on complexity}}{19} \contentsline {subsubsection}{\tocsubsubsection {}{4.8.1}{}}{20} \contentsline {subsubsection}{\tocsubsubsection {}{4.8.2}{}}{21} \contentsline {section}{\tocsection {}{5}{Numerical examples}}{22} \contentsline {subsection}{\tocsubsection {}{5.1}{Example: {\bf 37a1}}}{22} \contentsline {subsection}{\tocsubsection {}{5.2}{Example: {\bf 43a1}}}{24} \contentsline {subsection}{\tocsubsection {}{5.3}{Table}}{25} \contentsline {subsection}{\tocsubsection {}{5.4}{Further observations}}{25} \contentsline {subsubsection}{\tocsubsubsection {}{5.4.1}{}}{25} \contentsline {subsubsection}{\tocsubsubsection {}{5.4.2}{}}{26} \contentsline {subsubsection}{\tocsubsubsection {}{5.4.3}{}}{26} \contentsline {section}{\tocsection {}{6}{ Appendix: Chow-Heegner points associated to pairs of elliptic curves}}{27} \contentsline {subsection}{\tocsubsection {}{6.1}{Introduction}}{27} \contentsline {subsection}{\tocsubsection {}{6.2}{Example: $N=37$}}{28} \contentsline {subsection}{\tocsubsection {}{6.3}{The Formula of Yuan-Zhang-Zhang}}{28} \contentsline {subsection}{\tocsubsection {}{6.4}{Iterated Complex Path Integrals}}{29} \contentsline {subsection}{\tocsubsection {}{6.5}{A Numerical Approach to Computing $P_{E,F}$}}{29} \contentsline {subsubsection}{\tocsubsubsection {}{6.5.1}{Determining $\Gamma _0(N)$ equivalency}}{31} \contentsline {subsubsection}{\tocsubsubsection {}{6.5.2}{Data}}{31} \contentsline {subsubsection}{\tocsubsubsection {}{6.5.3}{Discussion}}{32} \contentsline {section}{\tocsection {}{}{References}}{33}