Abstract:
We shall discuss arithmetic properties of the modular abelian varieties $A_f$ when $f$ is a newform in $S_2(\Gamma_1(N))$ with complex multiplication.
More precisely, we shall focus on: (i) the minimum number field where $A_f$ has all its endomorphisms defined, (ii) the minimum number field where $A_f$ has an elliptic quotient, (iii) the cuspforms obtained as pullbacks of Weierstrass differentials, and (iv) the optimal CM-elliptic quotients of the Jacobian $J_1(N)$ which factor through $A_f$.
Some numerical examples will be presented.