Let \(k\ge 2\) and \(N\ge 1\) be integers. Let D be a positive integer that is congruent to a square modulo 4N, and fix \(\rho \) with \(\rho ^2\equiv D\pmod {4N}\) . In this paper, we consider two weight 2k cusp forms \(f^{\pm }_{k,N,D,\rho }\) on \(\Gamma _0(N)\) defined by sums over binary quadratic forms and investigate the vector-valued period polynomial arising from these forms. Our first main result gives a closed formula for this vector-valued period polynomial. The identity component of this formula is particularly explicit: It separates as the sum of a finite algebraic part coming from some binary forms and a zeta part involving the values at \(s=k\) of certain zeta functions. Using this formula together with a symmetry of vector-valued period polynomials, we explicitly evaluate, for odd k, the difference between the zeta values corresponding to the two choices of square root of D modulo 4N, in terms of Bernoulli numbers and a finite quadratic form sum. Finally, under a vanishing condition on Fricke-invariant cusp forms at lower levels, we obtain a finite divisor sum formula for the Dedekind zeta values \(\zeta _{\mathbb {Q}(\sqrt{D})}(k)\) at even integers k.
