Hecke eigenform ($T_{n}f=a_n(f)f$)

A Hecke eigenform is a nonzero modular form that is an eigenvector for every Hecke operator. A cuspidal eigenform has nonzero first Fourier coefficient: $a_1(T_{n}f)=a_n(f)$ proves this. Normalize $a_1=1$, and then $T_{n}f=a_n(f)f$. The Hecke multiplication relations give multiplicativity and the prime-power recurrence of the coefficients.

For a normalized level-one cusp form Hecke eigenform of weight $k$, fix a prime number $p$ whose local quadratic has distinct roots. Hecke multiplication relations give $b_e=x{b}_{e-1}-b_{e-2}$ for $b_e=a_{p^e}/p^{e(k-1)/2}$ and real $x=a_p/p^{(k-1)/2}$. If $|x|<2$, write $x=2\cos \theta$; then $b_e=\sin ((e+1)\theta )/\sin \theta$. Periodicity when $\theta /(2\pi )$ is rational, and density of irrational rotations otherwise, give infinitely many $b_e\ge 1/2$. If $|x|>2$, the distinct real reciprocal roots give positive unbounded even-index terms. Reality follows from Hecke operators are self-adjoint for the Petersson inner product.

For a normalized weight-$k$ cuspidal Hecke eigenform, the local recurrence gives ${\sum }_{r\ge 0}{a}_{p^r}{X}^r=(1-a_{p}X+p^{k-1}{X}^2{)}^{-1}$. Multiplicativity therefore gives $L(f,s)={\prod }_p(1-a_p{p}^{-s}+p^{k-1-2s}{)}^{-1}$ in a right half-plane. Each degree-two local factor packages all the prime-power Fourier coefficients.

The rational space of level-one cusp forms has a full lattice of forms with integral Fourier coefficients. It has finite rank by the valence formula for the modular group, and is preserved by the Fourier coefficients of a composite-index Hecke operator formula. Thus every Hecke operator has a monic integral characteristic polynomial on this lattice, and its eigenvalues are algebraic integers. The common eigenvalue field is totally real by the Petersson inner product self-adjointness.