On a congruence prime criterion for cusp forms on GL₂ over number fields

Let f be a normalized Hecke eigenform on GL 2 over a number field F and let 𝔓 be a prime ideal of a number field which contains the Galois closure of the number field which is generated by all Fourier coefficients of f over F . In this paper, we give a sufficient condition for 𝔓 to be a congruence prime for f . This criterion is a generalization of congruence prime criteria which were known for the case of elliptic cusp forms by Hida, for the case where F is an imaginary quadratic field by Urban and for the case of Hilbert cusp forms by Ghate and Dimitrov to arbitrary number fields.