  We present radio spectral analyses for a sample of 29 radio-quiet (RQ) and
three radio-loud (RL) narrow-line Seyfert 1 galaxies (NLS1s) detected with the
Australia Telescope Compact Array at both 5.5 GHz and 9.0 GHz. The sample is
characterized by L_bol/L_Edd > 0.15. The radio slopes in 25 of the 29 RQ NLS1s
are steep (alpha_5.5-9.0 < -0.5), as found in earlier studies of RQ high
L_bol/L_Edd AGN. This steep radio emission may be related to AGN-driven
outflows, which are likely more prevalent in high L_bol/L_Edd AGN. In two of
the three RL NLS1s, the radio slopes are flat or inverted (alpha_5.5-9.0 >
-0.5), indicating a compact optically-thick source, likely a relativistic jet.
Archival data at 3.0 GHz, 1.4 GHz, and 0.843 GHz are also compiled, yielding a
sample of 17 NLS1s detected in three bands or more. In nine objects, the radio
spectra flatten at lower frequencies, with median slopes of alpha_5.5-9.0 =
-1.21 +/- 0.17, flattening to alpha_3.0-5.5 = -0.97 +/- 0.27, and to
alpha_1.4-3.0 = -0.63 +/- 0.16. A parabolic fit suggests a median spectral
turnover of ~ 1 GHz, which implies synchrotron self-absorption in a source with
a size of only a fraction of 1 pc, possibly a compact wind or a weak jet. Two
objects show significant spectral steepening to alpha < -2 above 3 GHz or 5
GHz, which may suggest relic emission from past ejection of radio emitting
plasma, of the order of a few years to a few decades ago. Finally, two objects
present a single spectral slope consistent with star-forming activity.
