  Let $f$ be a cuspidal eigenform of weight two and level $N$, let $p\nmid N$
be a prime at which $f$ is congruent to an Eisenstein series and let $V_f$
denote the $p$-adic Tate module of $f$. Beilinson constructed a class
$\kappa_f\in H^1(\mathbb Q,V_f(1))$ arising from the cup-product of two Siegel
units and proved a striking relationship with the first derivative $L'(f,0)$ at
the near central point $s=0$ of the $L$-series of $f$, which led him to
formulate his celebrated conjecture. In this note we prove two congruence
formulae relating the "motivic part" of $L'(f,0) \,(\mathrm{mod} \, p)$ and
$L''(f,0) \,(\mathrm{mod} \, p)$ with circular units. The proofs make use of
delicate Galois properties satisfied by various integral lattices within $V_f$
and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of
circular units and Beilinson--Kato elements and, most crucially, the work of
Sharifi, Fukaya--Kato and Ohta.
