Multiplicity of a place in a modulus ($m_v(\mathfrak{c})$)

For a modulus of a number field $\mathfrak{c}={\mathfrak{c}}_0{\mathfrak{c}}_{\infty }$, the multiplicity at a finite place is the exponent of its prime ideal in ${\mathfrak{c}}_0$. At a real Archimedean place it is one if that place occurs in ${\mathfrak{c}}_{\infty }$ and zero otherwise. Complex places have multiplicity zero. These effective modulus multiplicities encode congruences and positivity requirements in the ray class group.