We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function \(\wp (z|\tau )\) as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups \(\Gamma _0(N)\) and \(\Gamma _1(N)\).

中文翻译：

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关于与Dirichlet字符相关的一类椭圆函数

我们构造了一类甚至与Dirichlet字符相关的伴随椭圆函数。使用经典的Weierstrass椭圆函数\（\ wp（z | \ tau）\）的众所周知的性质作为蓝图，我们将根据q级数和部分分数推导它们的表示形式。我们还探讨了它们的幂级数展开系数的重要性，并在算术组\（\ Gamma _0（N）\）和\（\ Gamma _1（N）\）的作用下建立了模数性质。