A modularity criterion for Klein forms, with an application to modular forms of level 13

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ1(N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ1(13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).