  We give a rate of equidistribution of lifts of horocycles from the space
$\mathrm{SL}(2,\mathbb Z)\backslash \mathrm{SL}(2, \mathbb R)$ to the space
$\mathrm{ASL}(2,\mathbb Z)\backslash \mathrm{ASL}(2,\mathbb R)$, making
effective a theorem of Elkies and McMullen. This result constitutes an
effective version of Ratner's measure classification theorem for measures
supported on general horocycle lifts. The method used relies on Weil's
resolution of the Riemann hypothesis for function fields in one variable and
generalizes the approach of Str\"ombergsson to the case of linear lifts and
that of Browning and the author to rational quadratic lifts.
