Small Heegaard genus and SU(2)

Let $Y$ be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if $H_1(Y;\mathbb{Z})$ has order $1$, $3$, or $5$, then there is a representation $\pi_1(Y) \to \mathrm{SU}(2)$ with non-abelian image. Similarly, if $H_1(Y;\mathbb{Z})$ has order $2$ then we find a non-abelian representation $\pi_1(Y) \to \mathrm{SO}(3)$. We also prove that a knot $K$ in $S^3$ is a trefoil if and only if there is a unique conjugacy class of irreducible representations $\pi_1(S^3\setminus K) \to \mathrm{SU}(2)$ sending a fixed meridian to $\left(\begin{smallmatrix}i&0\\0&-i\end{smallmatrix}\right)$.