Q-curves of degree 5 and jacobian surfaces of GL₂-type

We construct a parametric family {E^(±) (s, t, u)} of minimal Q-curves of degree 5 over the quadratic fields Q(√s² + st - t²), and the family {C(s, t, u)} of genus two curves over Q covering E⁽⁺⁾ (s, t, u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E⁽⁺⁾ (s, t, u) and its twist E^(-) (s. t, u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C (s, t, u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N = 29, 229, 349, 461, and 509.