Coefficient ring (reviewed)

The coefficient ring ** of a modular form is the subring $\Z[a_1,a_2,a_3,\ldots]$ of $\C$ generated by the coefficients $a_n$ of its $q$-expansion $\sum a_nq^n$. In the case of a newform the coefficients $a_n$ are algebraic integers and the coefficient ring is a finite index subring of the ring of integers of the coefficient field of the newform. It is also known as the Hecke ring**, since the $a_n$ are eigenvalues of Hecke operators.