The symmetry of Cooper pairs is central to constructing a superconducting state. The demonstration of a d(x2-y2)-wave order parameter with nodes represented a breakthrough for high critical temperature superconductors (HTSs)(1,2). However, despite this fundamental discovery, the origin of superconductivity remains elusive, raising the question of whether something is missing from the global picture. Deviations from d(x2-y2)-wave symmetry(3,4), such as an imaginary admixture d(x2-y2) + is (or id(xy)), predict a ground state with unconventional properties exhibiting a full superconducting gap and time reversal symmetry breaking(5). The existence of such a state, until now highly controversial(6-10), can be proved by highly sensitive measurements of the excitation spectrum. Here, we present a spectroscopic technique based on an HTS nanoscale device that allows an unprecedented energy resolution thanks to Coulomb blockade effects, a regime practically inaccessible in these materials previously. We find that the energy required to add an extra electron depends on the parity (odd/even) of the excess electrons on the island and increases with magnetic field. This is inconsistent with a pure d(x2-y2)-wave symmetry and demonstrates a complex order parameter component that needs to be incorporated into any theoretical model of HTS.