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As is well known, we can average the eigenfunction $y^s$ of the hyperbolicLaplacian on the hyperbolic plane by $\Gamma$ a lattice in$\mathbf{SL}(2,\mathbb{R})$ to obtain an automorphic form, the non-holomorphicEisenstein series $E_\mathfrak{a} (z,s)$.  In this note, we choose a particular eigenfunction $y^s dx$ of theHodge-Laplace operator for 1-forms on the hyperbolic plane. Then, we average by$\Gamma$ to define a 1-form $E_\mathfrak{a}^1 \big( (z,v), s \big)$.  We see that $E_\mathfrak{a}^1$ admits a Fourier expansion and calculates thecorresponding coefficients. Also, we evaluate the integral $\int_{\gamma}E_\mathfrak{a}^1$ for when $\gamma$ is a lifting of horocycles and closedgeodesics in the unit tangent bundle. Finally, we will obtain an analog to theRankin-Selberg method for $E_\mathfrak{a}^1$.

The Hodge Laplacian operator on 1-forms on $\mathbb{H}$ and 1-form $E_\mathfrak{a}^1$