Current results from the Lyman α forest assume that the primordial power spectrum of density perturbations follows a simple power-law form. We present the first analysis of Lyman α data to study the effect of relaxing this strong assumption on primordial and astrophysical constraints. We perform a large suite of numerical simulations, using them to calibrate a minimally parametric framework for describing the power spectrum. Combined with cross-validation, a statistical technique which prevents overfitting of the data, this framework allows us to reconstruct the power spectrum shape without strong prior assumptions. We find no evidence for deviation from scale-invariance; our analysis also shows that current Lyman α data do not have sufficient statistical power to robustly probe the shape of the power spectrum at these scales. In contrast, the ongoing Baryon Oscillation Sky Survey will be able to do so with high precision. Furthermore, this near-future data will be able to break degeneracies between the power spectrum shape and astrophysical parameters.