How to received only primary labeled emails My Nexus 5 battery drains very fast when I am using the Gmail app. Even after I turned synced off. Is there any way to get emails that belongs to primary category?
Even though it's unlikely that Gmail is the cause of your battery drain, here are the steps to minimize the notification spam:


In Gmail app tap on the upper-left corner "hamburger" button to open the sidebar, then scroll down and tap on "Settings":
  
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Tap on your account and ensure the Notifications option is checked:

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Tap on Inbox sound &amp; vibrate menu, and un-check the Notify for every message option.
Go back to your e-mail list, and navigate to the "Primary" inbox.  Tap the 3-dot button on upper-right corner and tap on Label settings menu:

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Under the "Notifications" section enable Label notifications, and set up sound/vibration to your liking:

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Go back to your main inbox.
Open the sidebar (see step 1) and select one of the other labels.
Tap on menu button, go to Label settings, and un-check the "Label notifications" to disable them for this specific label only:

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Go back, and repeat steps 6 through 8 for every label (other than "Primary.")


You should now only receive notifications for your Primary inbox, and you will be only notified once until you dismiss the notification (as opposed to for every new e-mail.)