A family of operators mapping each space M_k of modular forms onto itself. For a fixed integer k and any positive integer n, the Hecke operator T_n is defined on the set M_k of entire modular forms of weight k by (T_n f)(τ) = n^(k - 1) sum_(d|n) d^(-k) sum_(b = 0)^(d - 1) f((n τ + b d)/d^2). For n a prime p, the operator collapses to (T_p f)(τ) = p^(k - 1) f(p τ) + 1/p sum_(b = 0)^(p - 1) f((τ + b)/p).
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