  The sine-Gordon model in 3+1 dimensions is known to admit two oscillons of
different energy and frequency but comparable lifetime. We show that the
oscillon spectrum includes more spherically symmetric ``states''. We identify
new high-amplitude oscillons by allowing the field profile to have a number of
nodes. For each number of nodes, we find 2 states with a comparable lifetime to
the nodeless ones. Oscillons with nodes are, however, unstable to non-spherical
perturbations and so their lifetime is significantly reduced. Interestingly,
these states are seen to fragment into a collection of nodeless oscillons. The
heavy nodeless oscillon is quite remarkable: despite its energy it is stable
against fragmentation. Moreover, it has considerably small oscillation
frequency, meaning that it can be interpreted as a rather relativistic bound
state.
