
Samsung Galaxy Tab 4 10.1 LTE (AT&T)
The latest firmware for Samsung Galaxy Tab 4 10.1 LTE (AT&T) (SM-T537A) is T537AUCU1ANE1 with Android 4.4.2 (KitKat), released on April 30, 2014 for 1 region (CSC). Every official firmware build for SM-T537A, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/T537AUCU1ANE1
Box codes of SM-T537A for different colors and markets. Firmware is chosen by the model code SM-T537A and your region (CSC), not by the SKU.

SM-T537V

SM-T530X

SM-T532

SM-T638B

SM-T630
You can find the firmware on the model: SM-T537A page.
Yes, all firmware files for SM-T537A are 100% official, original, and obtained directly from Samsung servers.
To flash SM-T537A, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing SM-T537A.