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초록

Let k >= 2$k\ge 2$ and N >= 1$N\ge 1$ be integers. Let D be a positive integer that is congruent to a square modulo 4N, and fix rho$ho$ with rho 2 equivalent to D(mod4N)$ho{<}^{>}2\equiv D(\mathrm{mod}4N)$. In this paper, we consider two weight 2k cusp forms fk,N,D,rho +/-$f{<}^{>}{\pm }_{k,N,D,ho}$ on Gamma 0(N)${\Gamma }_0(N)$ 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$s=k$ 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 Q(D)(k)${\zeta }_{\mathbb{Q}(\sqrt{D})}(k)$ at even integers k.

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