It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ordinary integer. Of course, there are other induced splittings by the Fricke involution, however the finest one is by the Hecke eingenfunctions. However I´ve heard that each $E(f)$ has a real multiplication. Is it true? Any references?

Thanks in advance.