  The data associated with the electromagnetic excitations of the nucleon
($\gamma^\ast N \to N^\ast$) are usually parametrized by helicity amplitudes at
the resonance $N^\ast$ rest frame. The properties of the $\gamma^\ast N \to
N^\ast$ transition current at low $Q^2$ can be, however, better understood when
expressed in terms of structure form factors, particularly near the
pseudothreshold, when the magnitude of the photon three-momentum vanishes
($|{\bf q}| =0$). At the pseudothreshold the invariant four-momentum square
became $q^2= (M_R-M_N)^2$, well in the timelike region $Q^2 =- q^2 < 0$ [$M_N$
and $M_R$ are the mass of the nucleon and of the resonance, respectively]. In
the helicity amplitude representation, the amplitudes have well-defined
dependences on $|{\bf q}|$, near the pseudothreshold, and there are
correlations between different amplitudes. Those constraints are often ignored
in the empirical parametrizations of the helicity amplitudes. In the present
work, we show that the structure of the transition current near the
pseudothreshold has an impact on the parametrizations of the data. We present a
method which modifies analytic parametrizations of the data at low $Q^2$, in
order to take into account the constraints of the transition amplitudes near
the pseudothreshold. The model dependence of the parametrizations on the
low-$Q^2$ data is studied in detail.
