A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map.
This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the unit circle in C ("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
The conductor of a Hecke character χ is the largest ideal m such that χ is a Hecke character mod m. Here we say that χ is a Hecke character mod m if χ (considered as a character on the idele group) is trivial on the group of finite ideles whose every v-adic component lies in 1 + mOv.
Read more about Hecke Character: Definition Using Ideals, Relationship Between The Definitions, Special Cases, Examples, Tate's Thesis, Algebraic Hecke Characters
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“Sometimes apparent resemblances of character will bring two men together and for a certain time unite them. But their mistake gradually becomes evident, and they are astonished to find themselves not only far apart, but even repelled, in some sort, at all their points of contact.”
—Sébastien-Roch Nicolas De Chamfort (1741–1794)