The trigger based on the Discrete Cosine Transform (DCT) allows recognition of ADC traces of a specific shape i.e., with a very short rise time and fast exponential attenuation related to a narrow, flat muon component of very inclined extensive air showers generated by hadrons and starting their development early in the atmosphere ("old" showers). Showers, crossing a surface detector composed of water Cherenkov tanks, such as those in the Pierre Auger Observatory generate Cherenkov light, which may hit photomultipliers (PMTs) directly or after several reflections. The geometry of the water Cherenkov tank imposes a type of the filter preferring the detection of very inclined showers (with very large zenith angle), where mainly two PMTs can be hit by direct light. The 3rd PMT is then hit by reflected light, but with some delay. For the fast sampling (80 MHz), this delay causes the signal to appear in the next time bin. Two-fold coincidences of DCT coefficients allow triggering signals that are ignored at that time due to either too high an amplitude threshold for three-fold coincidences in a single time bin (the standard Auger threshold trigger) or due to the de-synchronization of the signals in time caused by the tank geometry (delay on one or more time bins due to a longer light path corresponding to the reflection on the tank walls). Three DCT engines implemented into EP3C40F324I7 FPGA use all DSP blocks and generate the spectral trigger, when 8 DCT coefficients simultaneously are inside the acceptance lane in at least 2 channels. An additional veto signal (analyzing the amplitude) controls the trigger rate to avoid a saturation of a transmission channel. In a test performed in the Pierre Auger laboratory in Malargue (Argentina), the spectral trigger has been tuned for shapes corresponding to "old" showers. Both laboratory and long-term field measurements on a test tank confirmed a high efficiency of the recognition of expected patterns of ADC traces. The patterns of the spectral trigger for "young" showers are being optimized in simulations. We expect to detect "young" showers close to their cores.