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Abstract

In this paper, we analyze the theta series associated to the quadratic form $Q(\Vec{x}) \coloneqq x_1^2+x_2^2+x_3^2+x_4^2$ with congruence conditions on $x_i$ modulo $2,3,4$ and $6$. By employing special operators on modular, non-holomorphic Eisenstein series of weight 2, we construct a basis for Eisenstein space for levels $2^k, k\leq 7$, $3^{\ell}, \ell\leq 3$ and $p$, for odd prime $p$. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp form part of theta series corresponding to $Q$, we establish relation between the number of integer solutions to the equation $Q(\Vec{x}) = p$ and the number of $\F_p$-rational points on the associated elliptic curve under certain congruence conditions on $p$.