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Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$ where $k$ is a positive integer.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.

Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$ where $k$ is a positive integer greater than 1.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.

Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.

Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$ where $k$ is a positive integer.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.

Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{m\tau+n}$$.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.

Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.