  The simplest model of primordial magnetogenesis can provide scale-invariant
magnetic fields that can explain the present abundances of it in the cosmic
scales. Two kinds of solutions of the coupling function can lead to such
phenomena and both of them suffer from the problems of either strong-coupling
or large backreaction. In this work, we consider the coupling function as a
linear combination of both kinds with a model parameter. We find that the
parameter needs to be as small as $\sim 10^{-20}$ in order to evade the
backreaction problem. On the other hand, requiring that the modes above Mpc
scales do not suffer strong coupling, we also obtain a weak constraint of the
model parameter to be greater than $10^{-60}$. For the allowed range of the
model parameter, we, then, analytically evaluate the cross-correlation
functions between the magnetic fields and the curvature perturbation. We find
that such a combination preserves the consistency relation. Also, the result
leads to enhanced non-Gaussianity in equilateral as well as flattened limits
with unique signatures that characterize the novelty of this model.
