Constructing invariants of Weil representations associated with finite quadratic modules of square-free level

Constructing invariants of Weil representations associated with finite quadratic modules is basic and important in the topic of modular forms. In this article we construct a family of invariants of Weil representations associated with square-free level quadratic modules by computing the shadow of nonholomorphic Eisenstein series of vector-valued modular forms of weight two explicitly. Furthermore we prove that, for any finite quadratic module whose cardinality is a square of an odd prime p -power and the 𝔽 p -dimension is more than five, the space of invariants of Weil representation associated with this finite quadratic module must contain nonzero elements. Mathematics Subject Classification (2020) 11F27, 11F30.