As per 'Varnashrama Viplava' by Sri Sitaramdas Omkarnath
Samo damas tapah saucham kshantir aarjavam eva cha/ Jnanam vijnanam astikyam brahmakarma-swabhabajam
meaning: sama, dama,tapasya,saucha,kshanti,aarjava, jnana and vijnana are brahmin's natural activities.
The special duties of brahmins are : adhyana(study), yajana(worshiping),daana(donation). Three pro3fessions are allowed : Adhyaapana(Teaching),Yaajana(Worshipping as representative of others) and pratigraha(accepting daana from others).
A brahmin must to Trisandhya Vandana regularly.Otherwise he will be fallen.
Reference: Omkarnath Rachanavali, Mahamilan Math.Vol 6, page 136
He also quotes from Devi-Bhagavatam 12/3
kuryad anyanna va kutyad anusthaanadikam tathaa/Gayatrimatranistham tu kritakrityo bhabed dwijah.
yad aksharaika-samsiddheh spardhate brahmanottamah/Hari-Sankara-Kanjottha-Surya-Chandra-Hutaashanair//
meaning : Even if a brahmin(Dvija) performs any other activity or not, he can get siddhi only by Gayatri. If he becomes siddha even in an akshara of Gayatri, He becomes equal to Chandra, Surya and Agni. (Ibid, page 275)
Manusamhita (2/87) also says
japenaiva tu samsiddhyed brahmano naatra samshayah/kuryaad annan na va kurtad maitro brahmano uchchyate
meaning : A brahmin can attain illumination by the [Gayatri] japa only, hether he does any any duty sacred duty or not.But who is friendly to everybody is a real brahmin.
Maahabharata (Maha-mokshaparva,chapter 297 says
naitadrisham brahmanasyaasti vittam yatha ekataa smataa styataa cha/ silam shitir dandanidhanam arjjavam tataschoparamah kriyabhah//
meaning: equality,Truthfulness, Viruous conduct, ahimsa, simplicity and abstaining from kriyas are the wealth of a brahmin.