  We construct a new anticyclotomic Euler system (in the sense of
Jetchev-Nekovar-Skinner) for the Galois representation $V_{f,\chi}$ attached to
a newform $f$ of weight $k\geq 2$ twisted by an anticyclotomic Hecke character
$\chi$. We then show some arithmetic applications of the constructed Euler
system, including new results on the Bloch-Kato conjecture in ranks zero and
one, and a divisibility towards the Iwasawa-Greenberg main conjecture for
$V_{f,\chi}$.
  In particular, in the case where the base-change of $f$ to our imaginary
quadratic field has root number $+1$ and $\chi$ has higher weight (which
implies that the complex $L$-function $L(V_{f,\chi},s)$ vanishes at the
center), our results show that the Bloch-Kato Selmer group of $V_{f,\chi}$ is
nonzero, and if a certain distinguished class $\kappa_{f,\chi}$ is nonzero,
then the Selmer group is one-dimensional. Such applications to the Bloch-Kato
conjecture were left wide open by the earlier approaches using Heegner cycles
and/or Beilinson-Flach classes. Our construction is based instead on a
generalisation of the Gross-Kudla-Schoen diagonal cycles.
