Abstract For each prime p, we show that there exist geometrically simple abelian varieties A over ${\mathbb Q}$ with . Specifically, for any prime $N\equiv 1 \ \pmod p$ , let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point P of order p, and let $B = A_f/\langle P \rangle $ . Then the number of positive integers $d \leq X$ with is $ \gg X/\log X$ , where $\widehat B_d$ is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of $\operatorname {\mathrm {GL}}_2$ -type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.