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Dec 7, 2022 at 16:38 comment added Maarten Derickx I was pointed also by Timo Keller via e-mail that William Stein has implemented the intersection pairing in magma a long time ago magma.maths.usyd.edu.au/magma/handbook/text/1675#18964 using the ideas from Loic Merel with a little bit of extra work. So this does answer my question for the purposes I was needing it.
Dec 4, 2022 at 16:57 comment added François Brunault Another approach (which I haven't looked at in detail) would be to use the fact that the Petersson scalar product on $S_2(\Gamma_0(N))$ is dual to the intersection pairing. Merel has proved an explicit formula for $\langle f_1, f_2 \rangle$ in terms of the modular symbols attached to $f_1$ and $f_2$, see Symboles de Manin et valeurs de fonctions L, Section 4.
Dec 4, 2022 at 16:57 comment added François Brunault There is a natural map $H_1(Y_0(N),\mathbb{Z}) \to H_1(X_0(N),\mathbb{Z})$ which is computed explicitly in Borisov-Gunnells, Toric modular forms and nonvanishing of L-functions, Section 2. Given two modular symbols $\gamma_1,\gamma_2$ in $H_1(X_0(N),\mathbb{Z})$, you can use linear algebra to compute $\widetilde{\gamma}_2$ in $H_1(Y_0(N),\mathbb{Z})$ mapping to $\gamma_2$, and then use Merel's formula to compute $\gamma_1 \bullet \widetilde{\gamma}_2$.
Nov 29, 2022 at 23:05 history edited Maarten Derickx CC BY-SA 4.0
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S Nov 29, 2022 at 16:55 history suggested little mouse CC BY-SA 4.0
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Nov 29, 2022 at 16:53 review Suggested edits
S Nov 29, 2022 at 16:55
Nov 29, 2022 at 16:45 history asked Maarten Derickx CC BY-SA 4.0