Quadratic polynomials, period polynomials, and Hecke operators Marie Jameson, Wissam Raji (2013) Acta Arithmetica For any non-square 1 < D ≡ 0,1 (mod 4), Zagier defined F k ( D ; x ) : = ∑ a , b , c ∈ ℤ , a < 0 b 2 - 4 a c = D m a x ( 0 , ( a x 2 + b x + c ) k - 1 ) . Here we use the theory of periods to give identities and congruences which relate various values of F k ( D ; x ) .