Weighted quadratic partitions modulo $P^m$- a new formula and a new demonstration

Let $Q({f{x}}) = Q(x_1 ,x_2 ,...,x_n )$ be a quadratic form over $mathbb{Z}$, $p$ be an odd prime. Let $V = V_Q = V_{p^m } $ denote the set of zeros of $Q({mathbf{x}})$ in $mathbb{Z}_{p^m }$ and $left| V ight|$ denotes the cardinality of $V$. Set $ phi (V_{p^m } ,{mathbf{y}}) = sum _{{mathbf{x}} in V} e_{p^m } ({mathbf{x}} cdot {mathbf{y}})$ for ${mathbf{y}} e {mathbf{0}}$ and $phi (V_{p^m } ,{mathbf{y}}) = left| {V_{p^m } } ight| - p^{m(n - 1)}$ for ${mathbf{y}} = {mathbf{0}}.$ In this paper we shall give a formula for the calculation of the function $phi (V,{mathbf{y}}).$