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The signatures of topological superconductivity (TSC) in the superconducting materials with topological nontrivial states prompt intensive researches recently. Utilizing high-resolution angle-resolved photoemission spectroscopy and first-principles calculations, we demonstrate multiple Dirac fermions and surface states in superconductor BaSn$_3$ with a critical transition temperature of about 4.4 K. We predict and then unveil the existence of two pairs of type-uppercaseexpandafter{romannumeral1} topological Dirac fermions residing on the rotational axis. Type-uppercaseexpandafter{romannumeral2} Dirac fermions protected by screw axis are confirmed in the same compound. Further calculation for the spin helical texture of the observed surface states originating from the Dirac fermions give an opportunity for realization of TSC in one single material. Hosting multiple Dirac fermions and topological surface states, the intrinsic superconductor BaSn$_3$ is expected to be a new platform for further investigation of the topological quantum materials as well as TSC.
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