Deep imaging of the Sgr A complex at 6 cm wavelength with the B and C configurations of the Karl G. Jansky VLA† has revealed a new population of faint radio filaments. Like their brighter counterparts that have been observed throughout the Galactic center on larger scales, these filaments can extend up to ∼10 parsecs, and in most cases are strikingly uniform in brightness and curvature. Comparison with a survey of Paschen-α emission reveals that some of the filaments are emitting thermally, but most of these structures are nonthermal: local magnetic flux tubes illuminated by synchrotron emission. The new image reveals considerable filamentary substructure in previously known nonthermal filaments (NTFs). Unlike NTFs previously observed on larger scales, which tend to show a predominant orientation roughly perpendicular to the Galactic plane, the NTFs in the vicinity of the Sgr A complex are relatively randomly oriented. Two well-known radio sources to the south of Sgr A – sources E and F – consist of numerous quasi-parallel filaments that now appear to be particularly bright portions of a much larger, strongly curved, continuous, nonthermal radio structure that we refer to as the “Southern Curl”. It is therefore unlikely that sources E and F are Hii regions or pulsar wind nebulae. The Southern Curl has a smaller counterpart on the opposite side of the Galactic center – the Northern Curl – that, except for its smaller scale and smaller distance from the center, is roughly point-reflection symmetric with respect to the Southern Curl. The curl features indicate that some field lines are strongly distorted, presumably by mass flows. The point symmetry about the center then suggests that the flows originate near the center and are somewhat collimated.