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The corrected PDF integrand isso the integral is a Petersson inner product. On compact subsets it is harmless, while at the cusp the exponential decay of $f$ dominates the polynomial growth of the Eisenstein series; therefore it converges absolutely.

Decompose each nonzero pair uniquely into a positive common divisor times a primitive pair. For even $k$ this writes $G_k$ as $2\zeta (k)$ times the Eisenstein sum over ${\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})$. Unfolding the fundamental domain gives a constant multiple ofThe inner integral is the constant Fourier coefficient of the cusp form and is zero. Thus the original integral is zero, expressing the orthogonality of cusp forms and holomorphic Eisenstein series.