Title: Computing the "analytic order of $\Sha$" of abelian varieties of $\GL_2$-type over $\Q$ Abstract: In our talks on the verification of the strong BSD conjecture, we focused on the computation of the order of the Shafarevich-Tate group up to now. This talk is about the (easier) computation of the number (positive rational as a consequence of Gross--Zagier) it is expected to be equal to.